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THE 



CIVIL ENGINEER'S 
POCKET-BOOK 



JOHN 0. TRAUTWINE 

CIVIL ENGINEER 

EKVISBD BY 
JOHN C. TKAUTWINE, Jb. 

AND 

JOHN C. TRAUTWINE, 3d. 

CIVIL ENGINEERS 



EIGHTEENTH EDITION, NINETIETH THOUSAND 



NEW YORK 

JOHN WILEY A SONS 
LovDoir: CHAPMAN & HALL, Limited 

1907 



\. 






Entered, according to Act of Congress, in the year 1882, hj 

JOHN C. TRAUTWINE, 
in the Office of the Librarian of Congress at Washingron. 



Copyright by John C. Trautwine, Jr., 1902. 



> 



WM. F. FELL COMPANY A. REED & CO. 

ELECTROTYPERS AND MIINTKRS BINDERS 

PHILADELPHIA PHILADELPHIA 



THE AUTHOR 



DEDICATES THIS BOOK 



TO THE MEMORY OF HIS FRIEND, 



THE LATS 



BENJAMIN H. LATROBE, Esq., 



CITIL ENQINEXK. 



No pains have been spared to maintain the position of this 
as the foremost Civil Engineer's Pocket-book, not only in the 
United States, but in the EngUsh language. 

JOHN 'WILEY & SONS, 

Scientific Publishers, 
IS East Nineteenth Street, New Yor^ City. 



PREFACE 

TO FIRST EDITION, 1872. 



QHOULD experts in engineering complain that they do not find 
^ anything of interest in this volume, the writer would merely 
remind them that it was not his intention that they should. The 
book has been prepared for young members of the profession ; and 
one of the leading objects has been to elucidate, in plain En^ish, a 
few important elementary principles which the savants have envel- 
oped in such a haae of mysteiy as to render pursuit hopeless to any 
but a confirmed mathematician. 

Comparatively few engineers are good mathematicians ; and in 
the writer's opinion. It is fortunate that such is the case ; for nature 
rarely combines high mathematical talent, with that practical tact, 
and observation of outward things, so essential to a successful 
engineer. 

There have been, it is true, brilliant exceptions ; but they are 
very rare. But few even of those who have been tolerable mathe- 
matidana when young, can, as they advance in years, and become 
engaged in business, spare the time necessary for retaining such 
accomplishments. 

Nearly all the scientific principles which constitute the founda- 
tion of civil engineering are susceptible of complete and satis- 
factory explanation to any person who reaUy possesses only so much 
elementary knowledge of arithmetic and natural philosophy as is 
Bupposed to be taught to boys of twelve or fourteen in our public 
schools.* 

* Let two little boys weigh each other on a platform scale. Then when thej 
iMdanoe each other on their board see-eaw, let them see (and measure for them- 
•elTbs) that the lighter one is farther from the fence-rail on which their boaid is 
placed, in the same proportion as the heavier boy outweighs the lighter one. 
Tfaey will then have learned the grand principle of the iever. Then let them 
measure and see that the light one see-saws farther than the heavy one, in the 
same proportion ; and they will have acquired the principle of virtual veloeiUa^L.^ 
Explain to them that eqwUUy qf moments means nothing more than that 

V 



VI PREFACE. 

^^^ • 

The little tbat is beyond this, might safely be intrusted to the 
savants. Let them work out the results, and give them to the engi- 
neer in intelligible language. We could afford to take their words 
for it, because such things are their specialty ; and because we 
know that they are the best qualified to investigate them. On the 
same principle we intrust our lives to our physician, or to the 
captain of the vessel at sea. Medicine and seamanship are their 
respective specialties. 

If there is any point in which the writer may hope to meet 
the approbation of proficients, it is in the accuracy of the tables. 
The pains taken in this respect have been very great. Most of the 
tables have been entirely recalculated expressly for this book ; and 
one of the results has been the detection of a great many errors in 
those in common use. He trusts that none will be found exceed- 
ing one, or sometimes two, in the last figure of any table in which 
great accuracy is required. There are many errors to that amount, 



they seat themselves at their measured distances on their see-saw, ikey balance 
each other. Let them see that the weight of the heavy hoy, when multiplied hy 
his distance in feet from the fence-rail amounts to just as inuch as the weight of 
the light one when multiplied by his distance. Explain to them that each of 
the amounts is in foot-pounds. Tell them that the lightest one, because he see- 
saws so much faster than the other, will bump against the ground Just as hard as 
the heavy one ; and that this means that their momentums are equal. The boys 
may then go in to dinner, and probably puzzle their big lout of a brother who 
has just passed through college with high honors. They will not forget what 
they have learned, for they learned it as play, without any ear-pulling, spanking, 
or keeping in. Let their bats and balls, their marbles, their swings, Ac, once 
become their philosophical apparatus, and children may be taught {really taught) 
many of the most important principles of engineering before they can read or 
write. It is the ignorance of these principles, so easily taught even to children, 
that constitutes what is popularly called " The Practical Enginkeb ; " which, 
in the great majority of cases, means simply an ignoramus, who blunders along 
without knowing any other reason for what he does, than that he has seen it done 
BO before. And it is this same ignorance that causes employers to prefer this 
practical man to one who is conversant with principles. They, themselves, were 
spanked, kept in, &jc, when boys, because they could not master leverage, equality 
of moments, and virtual velocities, enveloped in x's, p's, Greek letters, square- 
roots, cube-roots, &c, and they naturally set down any man as a fool who could. 
They turn up their noses at science, not dreaming that the word means simply, 
Juwwing why. And it must be confessed that they are not altogether without 
reason ; for the savants appear to prepare their books with the express object of 
preventing purchasers, (they have but few readers,) from learning why. 



PREFACE. Vll 

especially where the recalcalation was very tedious, and where, 
oousequently, interpolation was resorted to. They are too small to 
be of practical importance. He knows, however, the almost impos- 
sibility of avoiding larger errors entirely; and will be glad to be 
informed of any that may be detected, except the final ones alluded 
to, that they may be corrected in case another edition should be 
called for. Tables which are absolutely reliable, possess an in- 
trinsic value that is not to be measured by money alone. With this 
consideration the volume has been made a trifle larger than would 
otherwise have been necessary, in order to admit the stereotyped 
sines and tangents from his book on railroad curves. These have 
been so thoroughly compared with standards prepared independ- 
ently of each other, that the writer believes them to be absolutely 
correct. 

In order to reduce the volume to pocket-size, smaller type hat 
been used than would otherwise have been desirable. 

Many abbreviations of common words in frequent use have been 
introduced, such as abut, oen, diag, hor, vert, pres, &c, instead of 
abutment, center, diagonal, horizontal, vertical, pressure, &c. They 
can in no case lead to doubt ; while they appreciably reduce the 
thickness of the volume. 

Where prices have been added, they are placed in footnotes. They 
are intended merely to give an approximate or comparative idea of 
value ; for constant fluctuations prevent anything farther. 

The addresses of a few manufacturing establishments have also 
been inserted in notes, in the belief that they might at times be 
found convenient. They have been given without the knowledge 
of the proprietors. 

The writer is frequently asked to name good elementary books 
on civil engineering ; but regrets to say that there are very few 
such in our language. "Civil Engineering," by Prof. Mahan of 
West Point ; " Roads and Railroads," by the late Prof. Gillespie ; 
and the '* Handbook of Railroad Construction," by Mr. George L. 
Vose, Civ. Eng. of Boston, are the best. The writer has reason to 
know that a new edition of the last, now in press, will be far 



Viii PREFACE. 

superior to all predecessors ; and better adapted to the wants of 
the young engineer than any book that has appeared. 

Many of Weale's series are excellent. Some few of them are 
behind the times ; bat it is to be hoped that this may be rectified 
in iiitare editions. Among pocket-books, Haswell, Hamilton's 
Usefhl Information, Henck, Molesworth, Nystrom, W^^^®) ^f 
abound in valuable matter. 

The writer does not include Rankine, Moseley, and Weisbach, 
because, although their books are the productions of master-minds, 
and exhibit a profundity of knowledge beyond the reach of ordi- 
nary men, yet their language also is so profound that very few 
engineers can read them. The writer himself, having long since 
foigotten the little higher mathematics he once knew, cannot. To 
him they are but little more than striking instances of how com- 
pletely the most simple &cts may be buried out of sight under 
heaps of mathematical rubbish. 

Where the word *'ton '' is used in this volume, it always means 
2240 lbs. 

There is no table of errata, because no errors are known to exist 

except two or three of a single letter in spelling ; and which will 

probably escape notice. 

John C. Tbautwhi*. 

Philadelphia, November 13th, 1871. 



PREFACE TO NINTH EDITION. 

TWENTY-SECOND THOUSAND, 1885. 



CI INCE the appearance of its last edition (ihe twentieth thousand) 
'^ in 1883, the " Ppcket-Boo]c " has been thoroughly revised, and 
many important additions and other alterations have been made. 
These necessitated considerable change in the places of the former 
matter, and it veas deemed best to turn this necessity to advantage, 
and to make a thorough re-arrangement, putting all of the articles, 
as far as possible, in a rational order. 

The list of new matter and of revisions and extensions is condensed as 
foUows, 1902 : 

New articles on the steam-bammer pile driver, machine rock drills, air com- 
pressors, high explosives, cost of earthwork by drag and wheel scrapers and by 
steam excavators, iron trestles, track tanks, artesian well-boring and standard 
time, and new tables of railroad curves in metric measure, circumferences and 
areas of circles, thermometric scales, and fractions with their decimal equivalents. 

Articles revised and extended, on circular arcs, thermometers, flotation, flow 
in- pipes, waterworks appliances, velocities, d;c, of falling bodies, centrifugal 
force, strength of timber, strength of beams, riveting, riveted girders, trusses, 
Bospension bridges, rail joints, turnouts, turntables, locomotives, cars, railroad 
statistics and manufactured articles, including columns, beams, channels, angles 
and tees. 

Most of the new matter is in nonpareil, the larger of the two 
types heretofore used. Boldfoced type has been freely used ; 
but only for the purpose of guiding the reader rapidly to a desired 
division of a subject. For emphasis, italics have been employed. 

Illustrations which were lacking in clearness or neatness have 
been re-touched and re-lettered, or replaced with new and better 
cuts. The new matter is very freely illustrated. 

New rules have been put in the shape of formulae, and many of 
the old rules have been re-cast into the same form. 

ix 



X FB£:fAC£. 

The addition of new matter, and a number of blank spaces 
necessarily left in making the re-arrangement, have increased the 
number of pages about one-fifth. 

The new index is in stricter alphabetical order than that of 
former editions, and contains more than twice as many entries, 
although much repetition has been avoided by the free use of cross- 
references, without which this part of the work might have been 
indefinitely extended. 

The selection of articles of manufacture or merchandise for illus- 
tration, has been guided by no other consideration than their fitne^ 
for the purpose, and the courtesy of the parties representing them, 
in supplying information. 

The writer gratefully acknowledges the kindness of those who 
have assisted in furnishing and arranging data. 

Philadelphia, January, 1886. J. C. T., Jb. 



PREFACE TO EIGHTEENTH EDITION. 

(SEVENTIETH THOUSAND, 1902.) 



IN preparation for its eighteenth edition, The Civil Engineer's 
Pocket Book, the first edition of which appeared thirty years 
ago, has undergone a far more extensive revision than at any 
other time. More than 370 pages of new matter have been 
added ; and the new edition is larger, by about 100 pages, 
than its recent predecessors. 

Among the new matter in this edition will be found : 
Pages 

43- 46 Annuities, Depreciation, etc. 
70- 72 Logarithms. 

73- 77 Logarithmic Chart and Slide Kule. 
80- 91 New Table of Logarithms. 
228- 253 Conversion Table of Units of Meaaurement. 
300- 301 Isogonic Chart. 
532- 635 Venturi Meter. 

536 Ferris-Pitot Meter. 
546 Miner's Inch. * 
649 Water Consumption in Cities. 
658- 659 Cost of Water Pipe and Laying. 
745- 764 Digests of Specifications for Bridges and Buildings. 

816 Tie Plates. 
870- 873 Digest of Specification for Iron and Steel. 
905- 906 Gray Column. 

914 Trough Floor Sections. 
983- 995 Price List of Manufactured Articles. 
996-1007 Business Directory. 
1008-1023 BibUography. 

The following articles have been almost or entirely rewritten: 
Nkw Pages Old Pages 

35- 47 Arithmetic 33-37 

210-211 Specific Gravity 380-381 

265-266 Time 395 

282-283 Chains and Chaining 176 

284-290 Location of the Meridian 177-179 

322-325 Rain and Snow 220-221 

358-453 Statics 318 f-361, 370-375 

xi 



• • 



XU PREFACE. 

New Pages Old Pages 

466-494 Strength of Beams 478-520, 528-536 

499 Shearing Strength 476 

499-500 Torsional Strength 476-477 

501-503 Opening Remarks on Hydrostatics 222-224 

537-538 Effect of Curves and Bends on Flow in Pipes 255-256 

689-744 Trusses 647-614 

856-864 Locomotives 805-810 

865-866 Cars 811-813 

867-869 Railroad Statistics 814-818 

892-899 I Beams, Channels, Angles and T Shapes 521-527 

930-942 Cement 673-678 

943-947 Concrete 678-682 

954-956 Timber Preservation 425-425 a 

The articles on arithmetic are considerably extended, notably 
by the addition of new matter relating to interest, annuities, 
depreciation, etc., including several tables. 

The new and greatly enlarged table of five-place logarithms is 
arranged in a somewhat novel form. In constructing this table, 
the effort has been to obviate the difficulty, present in all tables 
where the difference between successive numbers is constant 
throughout, that the differences between successive logarithnas 
of the lower numbers are relatively very great. In the new table 
the differences between logarithms are much more nearly con- 
stant. For convenience in rough calculations, the old table of 
five-place logarithms, on two facing pages, is retained. 

The Conversion Tables contain the equivaleilts of both English 
and metric units, and of each of these in terms of the other; but, 
owing to the extreme ease with which one metric imit may be 
converted into others of the same system, it has been unnecessary 
to burden the table with many of the metric units. The tables 
have been separately calculated by at least two persons, and their 
results compared and corrected. One of these results has then 
been used by the compositor in setting the type, and the proofs 
have been compared with the other. 

The new article on the location of the meridian is much more 
complete than its predecessors, and a new table of azimuths of 
Polaris, corresponding to different hour-angles, has been added. 

Perhaps the most radical and extensive of all the changes in 
this edition are those in the articles on Statics, on Beams and on 
Trusses These have been almost entirely rewritten and com- 
pletely modernized. Under Trusses, modern methods of cal- 
culating the stresses in and the dimensions of the several 



FBEFAOS. xiii 



members, and modern methods of construction, are explained, 
and several modern roofs and bridges are described and illus- 
trated. One of the most notable features in the new article 
is the digest of prominent modem specifications for bridges 
for steam and electric railroads and for highways. The articles 
on the strength of beams are greatly simplified and brought into 
harmony with modern. methods of dealing with that subject. 

In preparing the digests of specifications for iron and steel, 
use has been made of the specifications recently adopted 
by the American Section of the International Association for 
Testing Materials; while those of the American Society of 
Civil Engineers and of the recent report of a Board of United 
States Army engineer officers have been similarly used in con- 
nection with cement. 

The price list of engineering materials and appliances has been 
prepared merely as a useful guide in roughly estimating the ap- 
proximate costs of work, and it is not to be supposed that it can, 
in any important case, take the place of personal inquiry and 
correspondence with manufacturers or their agents, nearly 700 
of whom are named in the accompanying list of names and 
addresses of manufacturers, etc. From its first appearance, the 
Pocket Book has undertaken to give prices of certain manufac- 
tured articles, and addresses of those from whom they may be 
obtained; but these, scattered as they were throughout the 
voliune, were necessarily desultory, and limited in their extent 
and usefulness. It is hoped that the present articles will be 
found at least an acceptable substitute for them. 

As in preceding editions, all new work and all revisions have 
been the subject of our personal attention, and " scissors-and- 
paste" methods have been scrupulously avoided. Even in using 
lists of manufactured articles, etc., although their statements 
have in general been left unchanged, the matter has in most or all 
cases been rearranged and classified, to suit the requirements of 
this work. 

For instance, the ''digests" of specifications for Cement, for 
Steel and Iron, for Railroad and Highway Bridges and for Steel 
Buildings, are by no means mere quotations from the originals; 
but, as their name implies, the result of careful digesting of the 
contents of the specifications selected for the purpose; their 
several provisions being carefully studied, in nearly all cases re- 
worded or reduced to figures, and tabulated in form convenient 



XIV PREFACE. 

for reference, the whole being arranged in such logical order as to 
facilitate reference. 

As in all cases heretofore, every rule or formula and every 
description of methods, etc., can be readily understood and ap- 
plied by any one, engineer or layman, understanding the use of 
common and decimal fractions, of roots and powers, of loga- 
rithms, and of sines, tangents, etc., of angles. On the other hand, 
one who is not possessed of this very meager stock of mathemati- 
cal knowledge will hardly approach engineering problems, even 
as an amateur; -and we have therefore followed the precedent, 
established seventeen years ago, of putting rules in the shape of 
formulas, which have " the great advantage of showing the whole 
operation at a glance, of making its principle more apparent, and 
of being much more convenient for reference" (From Preface to 
ninth edition, 1885). 

The new matter is very fully illustrated. As heretofore, all 
cuts have been engraved expressly for this work. 

As in preparing for the ninth edition (1885), all the matter 
of the book has been rearranged. This has necessitated a new 
paging; and, in making this, the lettering of pages, introduced 
from time to time as new editions have appeared in the past, 
has been eliminated. The rearrangement and the addition of 
so much new matter have of course necessitated the preparation 
of a new table of contents and a new index. 

In this, as in all previous editions since the eighth (1883), 
practically all new matter has been set in nonpareil, the larger of 
the two types hitherto used, and much of the old matter retained 
has been reset in the larger type. 

We take pleasure in acknowledging our indebtedness to many 
who have kindly assisted us in our work, notably to Messrs. Otis 
E. Hovey and Wm. M. White, of the American Bridge Co., for 
painstaking examination of the article on Trusses; to Mr. C. 
Robert Grimm and Professor E. J. McCaustland for similar as- 
sistance in connection with the article on Statics; to Misses Laura 
Agnes Whyte and Louise C. Hazen for suggestions respecting 
mathematics and astronomy ; and to the following gentlemen for 
valuable information respecting the subjects named : 

Isogonic Chart, Mr. O. H. Tittmann, Sup't, U. S. Coast and 
Geodetic Survey. 

Trusses, Messrs. Wm. A. Pratt, Engineer of Bridges, Pennsyl- 
vania Railroad; W. B. Riegner, Engineer of Bridges, Philadel- 



PREFACE. XV 

phia and Reading Railway; Paul L. Wolfel, Chief Engineer, 
American Bridge Co.; J. Sterling Deans, Chief Engineer, and 
Moritz G. Lippert, Assistant Engineer, Phoenix Bridge Co. ; Ralph 
Modjeski, Northern Pacific Railway; D. J. Whittemore, Chief 
Engineer, and C. F. Loweth, Engineer and Superintendent of 
Bridges and Buildings, Chicago, Milwaukee and St. Paul Railway. 

Specifications for Bridges and Buildings, Messrs. C. C. Schnei- 
der, Vice President, American Bridge Company; J. E. Greiner, 
Engineer of Bridges and Buildings, Baltimore and Ohio Railroad ; 
Theodore Cooper; W. K. McFarlin, Chief Engineer, Delaware, 
Lackawanna and Western Railway; Mason B. Strong, Bridge 
Engineer, Erie Railroad; F. C. Osborn, President, Osborn En- 
gineering Co. ; Wm. A. Pratt, Engineer of Bridges, Pennsylvania 
Railroad ; W. B. Riegner, Engineer of Bridges, Philadelphia and 
Reading Railway; W. J. Wilgus, Chief Engineer, New York 
Central Railroad. 

Locomotives, Baldwin Locomotive Works; Messrs. Wilson 
Miller, President, Pittsburgh Locomotive and Car Works ; Theo. 
N. Ely, Chief of Motive Power, Pennsylvania Railroad; A. 
E. Mitchell, C. W. Buchholz and A. Mordecai, of the Erie Rail- 
road; Edwin F. Smith, Wm. Hunter, A. T. Dice and Samuel F. 
Prince, Jr., of the Philadelphia and Reading Railway; and 
Thomas Tait, Manager, Canadian Pacific Railway; and Major 
E. T. D. Myers, of the Richmond, Fredericksburg and Potomac 
Railroad. 

Cars, Allison Manufacturing Co., Harlan & HoUingsworth Co., 
and Mr. Jos. W. Taylor, Secretary, Master Car Builders* Associa- 
tion. 

Railroad Statistics, Mr. Edward A. Moseley, Secretary, Inter- 
state Commerce Commission. 

Iron and Steel, Mr. Wm. R. Webster. 

Cement, Mr. Richard L. Humphrey. 

Concrete Beams, Mr. Howard A. Carson, Chief Engineer, Bos- 
ton Transit Commission. 

Preservation of Timber, Mr. O. Chanute. 

Building Material, Mr. John T. Willis. 

John C. Trautwine, Jr., 

John C. Trautwine, 3d. 
Philadelphia, October, 1902, 



Folios xvi to xxiv inclusive are 
left blank, to provide for future 
additions to prefaces. 



XTi 



CONTENTS. 



MATHEMATICS, paob 

Mathematical Ssnnbote 33 

Greek Alphabet 34 

Aritliinetie. 

Factors and Multiples 35 

Fractions 35 

Decimals 37 

Ratio and Proportion 38 

Progression 39 

Permutation, Combination, Al- 
ligation 40 

Percentage, Interest, Annuities 40 

Simple Interest 41 

Equation of Payments 42 

Compound Interest 42 

Annuity^ Sinking Fund, De- 
preciation, etc 43 

Equations and Tables. . .44r-46 

Duodenal Notation 47 

Reciprocals 48-52 

Roots and Powers. 
Square and cube. 

Tables ; 64 

Rules 66 

Fifth Roots and Powers .... 67 

LoKarithms 70 

Rules 70 

Logarithmic Chart and Slide 

Rule 73 

Two-page Table 78 

Twelve-page Table 80 



Geometry. Alensiiration, 
and Tnyonometrjr. 

liines. 

Definitions 02 



Angles- 
Definitions 92 

Construction 93 

Bisection 94 

Inscribed 94 

Complement and Supplement . 94 

In a Parallelogram 95 

Minutes and Seconds in Deci- 
mals of a Degree, Table of — 95 
Approximate Measurement of 

Angles 96 

Sine, Tangent, etc 97 

Definitions 97 

Table 98 

Ohonk. Table' d!-^ '.'.'.! !!'.'. 143 



PAOB 

Polygons. 

R^^ular — , Tables, etc.. of — 148 

Triangles. 

Dennitions. Properties 148 

Right-angled — 150 

Trigonometrical Problems . . 150 

Parallelogram '. 157 

Trapezoid. Trapezium 158 

Polygons 159 

Regular 159 

Reduction of Figures. . .159, 160 

Circle 161 

Radius. Diameter 161 

Area, Center, to Find — ... 161 

Problems 161. 162 

Tables of — . 

Diameter in Units, Eighths, 

etc 163 

Diameters in Units and 

Tenths 166 

Diameters in Units and 

Twelfths 172 

Arc. Circular. 

Chord, Length 179 

Radius, Rise, and Ordinates. 180 

Of Large Radius, to Draw — 181 

Tables of — 182-185 

Circular Sector, Ring, Zone, 

and Lune 186 

Circular Segment. 

Area of — ; to Find 186 

Area of — ; Table 187 

Ellipse. 

Properties of ^ 189 

Ordinates and Circumference 

of —; to Find — 189 

Elliptic Arc 189 

Tables of Lengths of — ... 190 

Area of; to Find — 190 

Construction. Tangents. . . 190 

Oval or False — 191 

C^ma Recta, Cyma Reversa, 

Ogee 191 

Parabola. 

Properties of — 192 

Parabolic Curve. Length of-^- 192 

Area 192 

Parabolic Zone or Frustum . 192 

Construction 193 

Cycloid 194 



Solids. 

Regular Bodies. Tetiahedron, 

Hexahedron, etc 194 

Guldinus Theorem 194 

Parallelopiped, Properties 105 



XXV 



XXVI 



CONTENTS. 



PAGE 

Priam .- 195 

Frustum 195 

Cylinder. 

Volume and Surface of — . . 196 

Volume. Table of — , in Cu. 

Ft. and U. S. Gala 197 

Wella; Contenta of — and 

Masonry in Walla of — ... 198 

Cylindrio Ungula 199 

Pyramid and Cone 200 

Frustums of 201 

Prismoid 202 

Wedge 203 

Sphere. 

Properties 204 

Volume, Surface, etc. 

Formulas for — 204 

Tables of — 205-207 

Segment and Zone of — . . . . 208 

Spherical Shell 208 

Spheroid or Ellipsoid 208 

Paraboloid 209 

Frustum of — 209 

Circular Spindle 209 

Circular Ring 209 



Specific OraTity. 

Principles 210 

Table 212-216 



Welgrbts and Measures. 

U. S., British and Metric — , 

Units of — 216 

Coins; Foreign and U.S. — 218 

Gold and Silver 219 

Weights; Troy, Apothecaries' 

and Avoirdupois — 220 

Long Measure 220 

Degrees of Longitude. Length. 221 
Inches Reduced to Decimals of 

a Foot. Table 221 

Square or Land Measure 222 

Cubic or Solid Measure 222 

Liquid Measures 223 

Diy Meaaure 223 

British Imperial Measures 224 

Volumes and Weights of Water 224 

Metric Units 226 

Systfeme Usuel, — Ancien 226 

Russian 227 

Spanish 227 

Conversion Tables 228 

Introduction and Explana- 
tion 228 

List of Tables 229 

Fundamental Equivalents . . 230 

Abbreviations 230 

Equivalents and Numbers in 

Common Use 231 

Metric Prefixes 231 

Tables 232 

Aorea per Mile and per 100 feet. 

Table 254 



PAGE 

Grades, Tables of — 255-257 

Heads and Pressures of 

Water; Tables of — 258-260 

Discharges in Gals, per Day 

and Cu. Ft. per Second; 

Tables 261-265 

Time. Definitions, etc 265 

Standard Railway — 267 

Dialing 268 

Board Measure. Table 269 



Survey infT. 

Testa of Accuracy, Distribution 

of Error, etc 274 

Chaining 282 

Location of Meridian 284 

By Circumpolar Stars 284 

Definitiona 284 

By Meana of Polaris 285 

By Means of Any Star at 

Equal Altitudes 287 

Times of Elonflnition and Cul- 
mination of Polaris 288 

Azimuths of Polaris, Table. . 289 
Polar Distances and Azi- 
muths of Polaris, Table. . 290 

Engineer's Transit 291 

Adjustment and Repairs. . . . 294 

Vernier 296 

Croas-hairs; to Replace 296 

Bubble Glasa; to Replace. . . 296 

Theodolite . . ; 296 

Pocket Sextant 297 

Compaaa. 

Adjustment 298 

Magnetic Declination and 
Variation. 

Isogenic Chart of U. S 300 

Declination 301 

Variation 301 

Demagnetization 302 

Leveling. 

Contour Lines 302 

Y Level 306 

Adjustment 307 

Forms for Notes 309 

Hand Level, Adjustment . . . 310 

Builder's Plumb Level 311 

Clinometer or Slope Inst .... 311 
Leveling by the Barometer 

or Boiling Point 312 

Table 316 



NATrRAI. PHENOMENA. 

Sound. 

Volocity of 316 

Heat. 

Expansion and Melting Points. 
Table 317 

Thermometer. 

Conversion of Scales 318 

Tables 318, 319 



CJ0NTENT8. 



XXVll 



Air. Atmospliere. page 

Properties 320 

Pressure in Diving Bells, etc. . . 321 

Dew Point 321 

Heat and Cold, Records of ... . 321 



Wind. 

Velocity and Pressure. Table. 321 



Bain »nd Snow. 

Precipitation. 

Average 322 

Effect of Climate on — 322 

and Stream-flow 323 

Maximum Rates of — 323 

Weight of Snow 323 

Rain Gau^ 324 

Precipitation, Details of — in 

U.S., Table 325 

Water. 

Composition, Properties 326 

Ice 326 

Effects of Water on Metals, etc. 327 

Tides 328 

KTaporatlon, ratration, 

lieakai^e 329 



MECHANICS, FOBCE IN 

RieiD BOBIES. 

Definitions 330 

Matter; Body 330 



Djmaiiiies. 

Motion, Velocity 331 

Force 332 

Action and Reactioti 333 

Acceleration 334 

Mass 336 

Impulse 337 

Density; Inertia 338 

Opposite Forces 339 

Work :. 341 

Power 842 

Kinetic Energy 343 

Momentum 345 

Potential Energy 346 

Impact 347 

Gravity, Falling Bodies 34$ 

Descent on Inouned Planes . . . 349 

Pendulums 350 

Center of Oscillation 351 

Center of Percussion 351 

Angular Velocity 351 

Moment of Inertia 351 

Radius of Gyration 352 

OnthfuffBd Force 354 



StatlctB. PAoa 

Forces .• 358 

Line of Action 359 

Stress 359 

Moments 360 

Classification of Forces 361 

Composition and Resolution 

of Forces 362 

Force Parallelogram 364 

Foi-ce Triangle 367 

Rectangular Components 369 

Inclined Plane 369 

Stress Components 371 

Applied and Imparted Forces . . 372 
Resolution, etc., by means 

of Co-ordinates 372 

Force Polygon 374 

Non-coneurrentCopUnarForoes 375 

Equilibrium of Moments 376 

Cord Polygon 377 

Concurrent Non - coplanar 

Forces 380 

Non-concurrent Non-coplanar 

Forces 381 

Parallel Forces 382 

Coplanar 382 

Non-coplanar 385 

Center of Gravitv 386 

Stable, Unstable, and Indif- 
ferent Equilibrium 387 

General Rules 387 

Special Rules 391 

Line of Pressure. Center of 

Force or of Pressure 399 

Position of Resultant 399 

Distribution of Pressure .... 400 

"Middle Third" 402 

Couples 404 

Friction 407 

Coefficient ' 408 

Morin's Laws 410 

Table of Coefficients 411 

Other Experiments 412 

Rolling Friction 414 

Lubricated Surfaces 415 

Friction Rollers 417 

Resistance of Trains 417 

Workof Overcoming Friction 418 

Natural Slope 419 

Friction of Revolving Shaft 419 

Levers 419 

StabUity 422 

Work of Overturning 422 

On Inclined Planes 424 

The Cord 425 

Funicular Machine 427 

Toggle Joint 427 

PuHey , . . . 428 

Loaded Cord or Chain 428 

Arches, Dams, etc. Thrust 

and Resistance Linec .... 430 

Arches 430 

Graphic Method 430 

Practical Considerations. . 432 

Masonrv Dam 433 

Graphic Method 435 

Practical Considerations. . 436 

The Rcrew 436 



zxviii 



OONTBKTB. 



PAOB 

Forces Acting upon Beams and 

Trusses 437 

Conditions of Equilibrium. . 437 

End Reactions 439 

Moments 440 

In Cantilevers 442 

In Beams 443 

Inclined Beams 445 

Curved Beams 446 

Shear 446 

Influence Diagrams 449 

For Moments 449 

For Shear 460 

Relation between Moment 
and Shear 452 



STREHGTS OF HATE- 
1IIAI.S. 

Ctoneral Principles. 454 

Stretch, Stress and Strain .... 455 

Modulus of Elasticity 456 

Limit of Elasticity 458 

Yield Point 459 

Resilience 460 

Suddenly Applied Loads 460 

Elastic Ratio 461 

Strengths of Sections 462 

Fatigue of Materials 465 

TransTerae Streng^tb 

Conditions of Equilibrium .... 466 

Neutral Axis 466 

Resisting Moment 467 

Modulus of Rupture 468 

Moment M Inertia. 468 

Table 469 

Section Modulus 473 

Loading. Strength 473 

Table 474 

Beam of Unit Dimensions .... 475 

Coefficients, Table 476 

Weight of Beam as Load 477 

Comparison of Similar Beams. 478 

Horizontal Shear 478 

Deflections 480 

Elastic Limit 482 

Elastic Curve •. . 482 

Deflection Coeffioi^it 483 

Eccentric Loads 484 

Uniform Loads • 486 

Inclined Beams ....'. 485 

Sirlindrical Beams 485 
aximum Permissible — . . . . 485 

Suddenly Applied Loads . . . 486 
Uniform Strength 486 

Cantilevers. Table 487 

Beams. Table 488 

Continuous Beams 489 

Table 490 

Cross-shaped Beam 492 

Plates 492 

Transverse and Longitudinal 

Stresses Combined 493 



PAoa 
Strengrtb of Piilam. 496 

Radius of Gyration 496 

Table 496 

Remarks 40S 



Slieariiiff Strentrtli . 499 



ToMtanal 8ir«iivtli. 490 



HTDBOSTATICfiL 

Principles 601 

Center of Pressure 601 

Air Pressure 602 

Horisontal and Vertical 

Components 603 

Pressure in Vessels 503 

Opposite Pressures 503 

Rules 604 

Transmission of Pressure 606 

Center of Pressure 609 

Walls to Resist Pressure 608 

Thickness at Base 609 

Stability 510 

Contents 510 

Liability to Crush 51Q 

Thickness for Cylinders 511 

Iron Pipes 512 

Lead Pipes 513 

Buovancy 513 

dotation. Metaeenter 614 

Draught of Vessels 515 



HTDRAUI«ICS. 

Flow Of W«ter tbrouffb 

Pipes 610 

Head of Water 616 

Velocity Head 616 

Entry Head 616 

Friction Head 616 

Pressure Head 618 

Piezometers 618 

Hydraulic Grade Line 519 

Siphon 620 

Velocity Formulae 622 

Kutter's Formukk 523 

Weight of Water in Pipes 526 

Areas and Contents of Pipes . . . 526 

Total Head Required 627 

Table of Velocity and Friction 

Heads and Discharge 628 

Compound Pipe 631 

Venturi Meter. 

Theory 632 

Tube 634 

Register 536 

Ferris-Pitot Meter 53ft 

Curves and Bends 637 



OONnsStB, 



PAOK 

Flow thronff li Ortflees 

Tbeoretical Velocities £39 

With Short Tubes 640 

Through Thin Partition 641 

Discharge from One Reservoir 

to Another 643 

Rectangular Openings 644 

Time of Emptying Pond. . . . 646 

Miner's Inch 646 



Flow OTor Wolrs 

End Contractions 647 

Measiu«ment of Head 648 

Formulae , 649 

Francis 660 

Table of Discharges 561 

Basin 662 

Values of m 663 

Submerged Weirs 664 

Velocity of Approach 666 

Iztelined Weirs 668 

Broad-crested Orerf all 669 

Triangular Notch 669 

Trapezoidal Notch 669 

Flow In Open Channels 

ligations of Velocities 660 

Steam Gauging 660 

Pitot Tube, etc 661 

Wheel Meter 662 

Abrasion of Channel 663 

Theory of Flow 663 

Kutter's Formula 664 

Coefficient of Roughness 664 

Coeffs of Roughness. Table 666 

Coefficient, e. Table 666 

To Draw Kutter Diagram. 670 

Flow in Sewers 674 

Flow to Sewers 676 

Flow in Drain-pipes 676 

Constriction of Channel 676 

Scour 677 

Obstruction's in Streams 677 

Power of Falling Water 678 

Water Wheels. 678 

Hydraulic Ram 678 

Power of Running Stream .... 678 



COVSTBVCnONS, ETC. 

"Dredging* 

Cost of Dredging 680 

Horse Dredges 681 

Weight of Material 681 

Foundations. 

Foundations 682 

Borings in Common Soils 682 

Unreliable Soils 683 

Resistanoe of Soils. . , 688 



PAOB 

Rip-rap 583 

Protection from Scour 683 

Timber Cribs 684 

Caissons 685 

Coffer-dams 686 

Earth Banks 686 

Crib Coffer-dams 687 

Mooring Caissons or Cribs 689 

Sinking through Soft Soil 689 

PUes 689 

Sheet Piles 690 

Grillage 690 

Pile Drivers 690 

Resistance of Piles 592 

Penetrability of Soils 693 

Driving 693 

Screw Piles 694 

Drivin/s by Water Jet 695 

Hollow Iron Cylinders 696 

Pneumatic Process 696 

Timber Caisson 598 

Masonry Cylinders 699 

Fascines 699 

Sand-Piles 699 

Stonework. 

Cost, etc 600 

Retaining Walls. 

General Remarks 603 

Theory 606 

Surcharged Walls 609 

Wharf Wails 611 

Transformation of profile 611 

Sliding, etc 612 

Stone Bridg^es. 

Definitions 613 

Depth of Keystone 613 

Pressures on Arch-stones 614 

Table of Arches 615 

Abutments 617 

Abutment Pi^s 619 

Inclination of Courses 620 

Culverts 622 

Wing Walls s, 624 

Foundations 627 

Drains 627 

Drainage of Roadway 62S 

Contents of Piers 62$ 

Brick Arches 62P 

Centers 631 

Timber Bams. 

Primary Requisites 642 

Examples 642 

Abutments. Sluices, Ground 

Plan, Cost 645 

Measuring Weirs 64i 

Trembling 648 

Thickness of Planking Re- 
quired 648 



CONTENTS. 



WATER SUPPI<T. PAGE 

Consumption, Use and Waste. 649 
Waste Restriction ; Water 

Meters 649 

Water for Fire Protection . . . 650 

Reservoirs 650 

Leakage through — , Mud 

in— 651 

Storage Reservoirs 652 

Valve Towers, etc 652 

Comj^ensation 653 

Distributing Reservoirs .... 653 

Water Pipes 653 

Concretions in — , preven- 
tion of — 655 

Weights of Cast Iron Pipes . . 666 

Wrought Iron Pipes 656 

Wooden and Other Pipes . . . 657 
Costs of Pipes and Laying . . 658 

Pipe Joints 660 

Pipe Jointer 660 

Flexible Joints.. 661 

Special Castings 661 

Repairs and Connections. . . 662 

Air Valves 662 

Air Vessels, Stand-pipes 663 

Service Pipes 664 

Tapping Machines 664 

Anti-bursting Device 665 

Valves, Gates 666 

Fire Hydrants 668 

TEST AND WEI^Ii BORING. 

Test Boring Tools 670 

Artesian Well Drilling 671 

ROCK DRII4I1S. 

Diamond . Drills 675 

Percussion DrjUs 676 

Hand Drills 681 

Channeling 681 

Air Compressors 681 

TRACTION, ANIMAIi 
POWER. 

On Roads, Canals, etc 683 

TRUSSES. 

Introdnetion. 

General Principles 689 

Loading, Counterbraoing 690 

Cross bracing 691 

Types of Trusses 691 

Camber 696 

Cantilevers 696 

Movable Bridges 696 

Skew Bridges 697 

Koof Trusses 698 

Stresses in Trnss Mem- 
bers 

€(eneral Principles 698 

Method by Sections 700 

Chord Stresses, Moments, 

Chord Increments 701 



FACIB 

Shear 702 

Influence Diagram 702 

Dead Load Stresses 703 

Live Load Stresses 705 

Typical Wheel Loads 706 

Cooper's 706 

Live Load Web Stresses 706 

Live Load Chord Stresses. . . 709 

Wind Loads 710 

Impact, etc 711 

Maximum and Minimum 

Stresses 712 

Effect of Curves 712 

Counterbracing 713 

Stresses in Roof Trusses 713 

Weights and Loads 713 

Wind Pressures 714 

Graphic Method 715 

Timber Roof Trusses 716 

Deflections 718 

Redundant Members 720 

Brtdg^e I>etalls and Con- 
struction 

General Principles 720 

Floor System and Bearings. . 720 

Design 721 

Flexible and Rigid Tension 

Members 721 

Compression Members 721 

Pin and Riveted Connec- 
tions 721 

Floor Beam Connections 721 

Tension Members, Detail . . . 722 

Compression Members, De- 
tail 722 

End Post and Portal Bracing 723 

Joints 724 

Pin Plates 724 

Pins 725 

Expansion Bearings 725 

Loads, Clearance, etc., for 

Highway Bridges 726 

Camber 726 

Examples 726 

Weights of Steel Railroad 

Bridges 731 

List of Large Bridges 732 

Timber Trusses 732 

Joints 733 

Howe Truss Bridges 736 

Examples 738 

Metal Roof Trusses 740 

Broad Street Station, Phila. . 740 

List of Large Arched Roofs. 742 

Timber Roof Trusses 742 

Transportation and Erection . . 743 

Digests of Speelfleations for 
Brldgres and Buildings. 

For Steel Railroad and 
Highway Bridges. 

General Design 745 

Material 751 

i Loads 755 



C0NTEKT8. 



PAOB 

Btreeses and Dimensioos 759 

Protection 763 

Erection 763 



For Combination Railroad 
Bridyes. 

General Design 763 

Material 763 

Loads 764 

Stresses and Dimensions 764 

Protection 764 



For Roofli, Bulldlngns* etc. 

General Design, Material, etc.. 764 

Sl^SPENSIOM BRIDOIS. 

Data Required 765 

- Formulas 766 

Anchorages 770 

RITETS AND RITETINe. 

Rules and Tables 772 

RAIIiROADS. 

Carves. 

Definitions 780 

Tables, etc 784 

EartliworlK. 

Table of Level Cuttings 790 

Shrinka^ of Embankment .... 799 
Cost of Earthwork 800 

Tunnels. 

Coostruction 812 

Trestles. 

Construction 813 

Track. 

Ballast 815 

Ties 816 

Tie Plates 816 

Rails 817 

Spikes 818 

Rail Joints 819 

Turnouts 824 

Eqnlpment. 

Turntables I 845 

Water Stations 851 

Track Tanks 853 

Track Scales, Fences, etc 854 

Cost of Mile of Track 855 



Rolling Stoe 



J? 



XXXI 



PASS 



Locomotives. 

Dimensions. Weights, etc. . . 856 

Performance 860 

Tonnage Rating 862 

Fast Runs 863 

Running Expenses 864 

Cars 865 



Statistics. 

Earnings, Expenses, etc. 



867 



MATERIAUS). 

Metals. 

Iron and Steel. 

'Requirements. International 

Ass'n for Testing Materials. 870 

Cast Iron 874 

Weight 875 

Weight of Cast Iron Pipes. . 876 
Weight of Wrought Iron and 

Steel 877 

Roofing Iron 880 

Corrugated Iron 881 

Wrought Iron Pipes and Fit- 
tings 882 

Screw Threads, Bolts, Nuts 

and Washers 883 

Lock-nut Washers 885 

Buckle Plates 885 

Bolts. Weight and Strength, 

Table 886 

Wire Gauges 887 

Circular Measure 889 

Wire, Table , 891 

Structural Shapes. 

I Beams 892 

Channels 894 

Angles and T Shapes 896 

Separators for I Beams 900 

Z-Bar Columns 901 

Phcenix Segment Columns . . 904 

Gray Column 906 

Strengths of Iron Pillars, 

Tables 907 

Floor Sections 914 

Chains 915 



0kber Metals. 

Tin and Zinc 916 

Copper,' Lead, etc 918 

Tensile Strengths, Table 920 

Compressive Strengths, Table. 921 



Stone, etc. 

Tensile Strengths. Table 922 

Compressive Strengths, Table 923 

Transverse Strength, ^^able. . . 924 



XXXll 



CONTENTS. 



lIortov,Briclu»efe. page 

Lime Mortar 926 

Bricks 927 

Cement 930 

Cement Mortar 931 

Sand 935 

Effects on Metab 936 

Efflorescence 936 

Silica Cement 937 

Recommendations, Am. Soc. 

C. E 937 

Tests 938 

Report of Board of U. S A. 

Engineer Officers 940 

' Tests 941 

Requirements 942 

Concrete 943 

Properties 943 

Handling 946 



Explosives. 

Nitro^ycerine and Dynamite. 948 

Blasting Powders 951 

Firing 962 

Gunpowder 963 



Timber. 

Decay and Preservation 954 

Tensile Strength 957 

Compressive Strength 958 

Transverse Strength 959 

Strength as Pillars 963 



B«lldlii|r Materials and 

Op^V^^OXS. PAOS 

Plastering 966 

Slating 969 

Shingtes 971 

Painting 971 

Glass and Glasing 973 

Sundry Materials. 

Rope 976 

Wire Ropes 976 

Paper 978 

Blue Prints, etc 979 



Price lilst and Business Bt- 
rectoiry. 

Prieelist 984 

Business Directory 996 



Biblioffrapiiy. 

List of Engineering Books 1008 



GLOSSARY 1026 

INDEX 10» 



KATHEMATIGS. 

MATHEMATICAI. STMBOIA. 

•f Pins, positive, add. 1.414+ means 1.414 -f other decimala. 

— Minas, nejg^ative, subtract. 

± Plus or minus, positive or negative. Thus, y^a* — ±a. 

7 Minus or plus. 

X Multiplied by, times. Thus, x'Xy = x:.y=x7;3X4 = 12, 

: vDivided by. Thus, a -4- b = a : b = a/b = -r-- 

y) ^ 

: : : Proportion. Thus, a : b : : c : <2, as a is to 6, so is to <<. 
-= Equals, is equal to. 
> Is ffreater than. Thus, 6 > 5. 
< Is less than. Thus, 5 < 6. 

'^ Is not equal to. 

:^ Is greater or less than. 

j^ Is not greater than. 

^ Is not less than. 

;^ Is equal to or greater than. 

^ Is equal to or less than. 

oc la proportional to, varies with. 

00 Innnity. 

J. Is perpendicular to. 

^ \ Angla 

'v Is similar ta 

I la parallel to. 

V l^~Root of. Thus, "i/oor r/o^ square root of o, i/ o =* 8d or cube root of a, 

** J a s— nth root of a. 
Parenthesis. 



11 



Braclcets. I Quantities enclosed or covered by the symbol are to be 
I taken tpgether. 

-Vinculum. J 

*.* Since, because. 

.*. Hence, therefore. 

o Degrees. 

' Minutes of arc,* feet. 

" Seconds of arc,* inches. * 

/ ff /// gtc^ Prime, second, third, etc Distinguishing accents. Thus, a', 

a prime ; of', a second, etc. 

Circumference „-..,„„««.. r • • 1 <«»«« 

n — y- 7 = 8. 14159265 +, arc of semicircle, or 180°. 

Diameter ' 

E, Modulus of elasticity. 

e c, Base of Napierian, natural or hyperbolic logarithms = 2.718281828. 

g, Acceleration of gravity = approximately 32.2 feet per second per second » 

approximately 9.81 meters per second per second. 

* Minutes and seconds of time, formerly also denoted by ' and '', are now de- 
noted by m aud «, or by min and sec, respectively. 

3 33 



34 



OBEEK ALPHABET. 



THE eREEK AI.PHABET. 

This alphabet is inserted for the benefit of those who have occasion to consult 
scientific works in which Greek letters are used, and who find it inconvenient 
to memorize the letters. 



Greek letters. 












Name. 


Approximate 
equivalent. 


Commonly used to designate 






Capital. 


Small. 




* 




A 


a 


Alpha 


a 


Angles, Coefficients. 


B 


^ 


Beta 


b 


it u 


r 


y 


Gamma 


g 


" " Specific gravity. 


A 


i 


Delta 


d 


« " Density, Variation. 
/Base ot hyperbolic logarithms » 


s 


« 


Epsilon 


e (short) 


-j 2.7182818. 

V Eccentricity in conic sections. 


z 


< 


Zeta 


* 


Co-o'rdinates, Coefficients. 


H 


n 


Eta 


e (long) 


ii (I 


e 


9& 


Theta 


th 


Angles. 


I 


I 


loU 


i 




K 


iC 


Kappa 


k 




A 


A 


Lambda 


1 


Angles, Coefficients, Latitude. 


M 


Jtt 


Mu 


m 


tt t< 


N 


V 


Nn 


B 


t( 


B 


f 


Xi 


X 


Co-ordinates. 


O 


o 


Omicron 


(short) 




n 


w 


Pi 


P 


Circumference -i- radios.* 


p 


p 


Bho 


r 


Badius, Batio. 


2 


o-« 


Sigma 


• 


Distance (space).t 


T 


T 


Tau 


t 


Temperature, Time. 


Y 


V 


Upsilon 


u or y 




« 


* 


Phi 


ph 


Angles, Coefficients. 


X 


X 


Chi 


ch 




♦ 


^ 


Psi 


P8 


Angles. 


o 


w 


Omega 


o (lon«) 


Angular velocities. 



* The small letter fr (pt) is universally employed to designate the number of 
times (= 3.14159265 . . .) the diameter of a circle is oootained in the circum- 
ference, or the radius in the semi-circumference. In the circular measure of 
angles, an angle is designated by the number of times the radius of any circle is 
<k>ntained iu an arc of the same circle subtending that angle. ir then stands for 
an angle of 180° (= two right anglesX because, in any circle, ir X radius = the 
semi-clrcumferenoe. 

The capital letter n (;>i) is used by some mathematical writers to indicate the 
product obtained by multiplying together the numbers 1, 2, 3, 4, 5 . . . etc., up to 
any given point. Thus, n 4 = 1 X2 X 3X4 = 24. 

t The capital letter 2 (sigma) is used to designate a mm. Thus, in a system 
of pandlel forces, if we calf each of the forces (irrespective of their amounts) F, 
then their resultant, which is equal to the (algebraic) sum of the forces, may he 
written B = 2 F. 



AssTButata. ' 35 



ABITHMETIO. 

FACTORS AND MVI4TIPI1ES. 

(1) Factors of any number, n, are numbers whose product is = n. Thus, 
17 and 4 are factors of 68 ; so also are 34 and 2 ; also 17, 2, and 2. 

<3) A prime number, or prime, is a number which has no factors, 
except itself and 1 ; as 2, 3, 5, 19, 2S&. 

(8) A common HicAor, common diwiflor or common meaanre, 
of two or more numbers, is a number which exactly divides each of them. Thus, 
8 is a common dirisor of 6, 12, and 18. 

(4) Tlie hiipiieBt common fiictor or nreatest common diwiaor, 
of two or more numbers, is called their H. C. F. or their O. C I>. Thus, 6 is 
the H. C. F. of 6, 12, and 18. 

(5) To find (lie H. ۥ F. of two or more numbers ; find the prime factors 
of each, and multiply together those factors which are common to all, taking 
••di factor only once. Thus, required the H. C. F. of 78, 126, and 284 

78 = 2 X 8 X 13 
126 = 2X3X3X7 
284 = 2X3X3X13 
and H. G. F. * 2 X 8 — 6. 

(6) To find tlie H. C. F* of two large numbers ; divide the greater by the 
less ; then the less by the remainder, A : A by the second remainder, B ; B by 
the third remainder, G ; and so on until there is no remainder. The last divisor 
Is the H. G. F. Thua, required the H. a F. of 575 and 782. 

675)782(1 
575 

A 207)575(2 
414 

B 161)207(1 
161 

G 46)161(8 
188 

D 28)46(2 H. G. F. =- D » 2& 

46 



(7) A comnMMi maltiple of two or more numbers is a number which is 
exactly divisible by eaoh of tn^m. 

(8) Tbe least common maltiple of two or more numbers is called 
iheir li. €. M. 

(9) To find the !<• C. M. of two or more numbers ; find the prime factors 
of each. Multiply the factors together, taking each as many times as it is con- 
tained in that number in which it is oftenest repeated. Thus, required the 
L. G. M. of 7, 80, and 48. 

7 = 7 
30 = 2X3X5 
48 = 2 X2X 2X2X3 
L. C. M. = 7X2X2X2X2X8X5 = 1680. 

(10) To find the !<• C M. of two large numbers; find the H. C. F., as 
above ; and, by means of it, find the other factors. Then find the product of the 
fKtors, as before. Thus, required the L. G. M. of 575 and 782. As above, 

H. G. F. =23; ^ = 25; and^ = 34. Hence, 

575 = 23 X 25 
782 = 23 X 34 
and L. G. M. = 28 X 25 X 84 = 19,660. 

FRACnOBTS. 

CI) A conuBfMi denominator of two or more fhictions is a common 
moltiple of their denominators. 

(2) The least common denominator, or !<• €• D«, of two or more 
firactions is the L. G. M. of their denominators. 



36 ARITHMETia 

(8) To rednce to a oommoii denominfttor. Let 

N °. the new numerator of any fraction 

n = its old numerator 

d a its old denominator 

C » the common denominator 

Then _, C 

Thus, ^t -f-> j-* C *" L* C* ^' o^ denominators «« 24. 

S ^ ^^ 4 8X6 18. 5 5X4 20. 7 ^ 7X8 ^ 21 

4~^,,24"4X6''24* 6"'6X4''a4' 8 8X8~24* 
4X-4 

If none of the denominators have a common factor, then C^the product of all th« 
denominators, -: = the product, P, of all the otAer denominators, and N » P n. 

Thu8,|,l^,f c = 84 

2 _ 2X4X7 se. 1 _ 1X3X7 _ 21. 5 _ gX3X4 eo 
t ~ 84 T¥' T ~ 84 ^^' 7" 84 TT* 

(4) Addition and Subtraction. If necessary, reduce the fractions to 
a common denominator, the lower the better. Add or subtract the numerators. 
Thus, 

1 4.1 _2_i.8 4.1_4_i.8 4.5_27 .20_47_i 11. 

3_l7_64.7_13_,6 

f_l— 2_1.8_5_2_7_20_jr.7_3_7 6_1 

(5) Multiplication. Multiply together the numerators, also the denomi- 
nators, cancelling where possible. Thus, 

lvl_l. 8vl S_« 3 V 5 v^ 2 _ 6 . 

84 X i| = ^ X I = Jjft^ = 5|f ; I X f = |; 
|of|of|of^ = f X^Xf X| = |. 

(6) IMvision. Invert the divisor and multiply. Thus, 

l^l=,lv2_2_,. 3^1_8v4_8„-. 

i;^7_Bv8_40_e5 
o-7--g- — oXir — 7V- — 5-S-. 

(7) A fraction is said to be in its lowest terms, or to be simplified* 
when its numerator and denominator have no common factor. Thus, 

1^ simplified = |-. 

(8) To reduce to low<$st terms. Divide numerator and denominatox 

34 
by their H. C. F. Thus, required the lowest terms of ^. 

H. C. F. Of 34 and 85=^17; and ?^ « ?i:ti? = ?. 

85 85 + 17 S 



ARITHMETIO. 



87 



(9) Mnltlplleatlon. The prodnct has as many decimal places as th« 
factors combined. Thus, 

. Factors: 100X3X3.5X0.004X465.21 = 1953.882000 

Number of decimal places: + 0+1+ 8+ 2= 6 

(10) DiTisloii. The number of decimal places in the quotient = those in 
the dividend minus those in the divisor. Thus, 

5.125 ,„_ 5 5.00 i„^.3 3.00 ■^. 0.42 _ 0.4200 _ 

^^ = 1.25; -= — =1.25; 4 = "X = ^'^^ 00021 "" 0:0021 ^ ^' 
When the divisor is a fraction or a mixed number, we may multiply both 
divisor and dividend by the least power of 10 which will make the divisor a 
whole number. Thus, 

2.679454 26,794.54 .^ ,_ 

0.0062 62 

(11) To rednee a common fraction to decimal form ; dividt 

the numerator by the denominator. Thus, ^ = 0.8 ; 1-|- = -|. =» 1.6. 



Table 1. Decimal eqniTalents 


Of common 


fractions. 


8thB 


16tha 


SMi 


64t]u 


, 


8ths 


lethg 


82dB 


64tlis 










1 


:015625 








S3 


.515625 






1 


2 
3 


.03125 
.046875 






17 


34 . 
35 


.53125 
.546875 




1 


2 


*4 
5 


.0625 
.078125 




9 


18 


36 
,37 


.5625 
.578125 






8 


6 

7 


.09375 
.109375 






19 


38 
39 


.59375 
.609875 


1 


2 


4 


8 
9 


.126 
.140625 


5 


10 


20 


40 
41 


.625 
.640625 






5 


10 
11 


.15625 
.171876 






21 


42 
48 


.65625 
.671875 




' 8 


6 


12 
13 


.1875 
.203125 




11 


22 


44 
45 


.6875 
.708125 






7 


14 
15 


.21875 
.234375 






23 


46 
47 


.71875 
.734375 


2 


4 


8 


16 
17 


.25 
.265625 


6 


12 


24 


48 
49 


.75 
.765625 






9 


18 
19 


.28125 
.296875 






25 


50 
51 


.78126 
.796875 




5 


10. 


20 
21 


.3125 
.328126 


• 


13 


26 


52 
63 


.8126- 
.828125 






11 


22 
23 


.34375 
.359375 






27 


54 
55 


.84376 
.859375 


8 


6 


12 


24 
25 


.375 
.390625 


7 


14 


28 


56 
57 


.875 
.890625 






13 


26 

27 


.40625 
.421875 






29 


58 
59 


.90625 
.921875 




7 


14 


28 
29 


.4375 
.453125 




15 


30 


60 
61 


.9375 
.958125 






15 


80 
31 


.46875 
.484375 






31 


62 
63 


.96875 
.984375 


4 


8 


16 


82 


.5 


8 


16 


82 


64 


1. 



(12) To reduce a decimal fraction to common form. Supply 
the denominator (1), and reduce the resulting fraction to its lowest terms. Thus : 



0.25 



0.25 
1.00 



25 
100 



1 
4' 



= . ; 0.75 = 



To 
100 



3 

4' 



^ : 0.800626 = 



890626 
1000000 



57 
64* 



38 ABITHMETIO, 

(IS) Becnriinff, etrealattny, or repeattny decimals are those in 
which certain digits, or series of digits, recur indefinitely. Thus, ^ =» 0.8338...., 

and so on ; ^^ ^ 1.428571428571 and so on. Becurring decimals may be in« 

dicated thus : 0.3, 1.428571 ; or thus : 0.*3, l.*428571. 

RATIO AND PlU^PORTIOir* 

(1) Batio. The ratio of two quantities, as A and B, is expressed by their 
qaotient, ^ or •-. Thus, the ratio of 10 to 5 is =» - =a 2 : the ratio of 5 to 10 

A* 

(2) Dapllcate ratio is the ratio of the tquares of numbers. Thus, ^-s 

is the duplicate ratio of A and B. 

(S) Proportion is equality of ratios. Thus, ^ = -^. = ^A*? = 2. 
I9 the figure, which represents s^ments, A, B, C, and D, between parulel lines ; 

A : B : : C : D, or 5 = ^. 

(4) The first and fourth terms, A and D, are called the extremes, and the 
second and third, B and C, are called the means. The first term, A or C, 
of each ratio, is called the antecedent, and the second term, B or D, is called 
the consequent. D is called the fonrtli proportional of A, B, and C. 

(5) In a proportion, A : B = C : D, we have : 

Product of extremes = product of means. A D >= 

..... A C A B 
Alternation. 3 = 5; c " D* 

_ , B D B A D C 

Inversion. ^ = ^; ^ - ^; 5 = ;^. 




^ ... A + B C +D . A-f B 

Composition. — - — = — ^ — ; — g— 

-., ,, A-B C-D A — B 
Diyision. — ^ = -^ ; -^^ 

A 4- B 
'Composition and division. =, = _ ... 

We have, also : 
mA ^ A ^ C ^ n^ ^ nC, mA^mC^ ^^^, */a ^ ^y/g 
mB B D nB nl)' nB ~nD' b*~D"' **|/B "" *i/D 

(6) If, in the proportion, A : B = C : D, we have B = C = m, then A : m « 

TO : D, or — = - or m * ■" A D, or m = 1/ A D. 
ml) 

(7) In such cases, m is called the mean proportional between A and D, 
mnd D is called the tbird proportional of A and m. 

A «M»ntinned proportion is a series of equal ratios, as 

A:B = C:D = E:F, etc. = R; or ^ = ~ = y, etc «- E 

In continued proportion, 

A + C -f E + etc. _AC_E _ 
B 4- D + F + etc. "^ B ~ D ~ F ^^^' ~ '^ 

„ A _ C A' C' A" _ C^' A A^ A» _ C C^ €<> 

B "■ D' b' ~ D'*' i3'' ~' iy' B B' B» - DDT)"®^ 

(8) Let A, B, and C be any three numbers. Then 

A_AB AAC 

C ' B • C' *°** B " C • B" 

■"^ ♦ 0.*8, l.*428571, etc., sUnding for 0.3333...., 1.428571428671...., etc. 



ABITHMETIC. 39 

(0) Reciprocal or inverae proportion. Two quantities are said 

to be redproeally or inversely proportional, when the ratio ^ of two values, A 

B' 
and B, of the one, is => the reciprocal, -j-,^ of the ratio of the two corresponding 

values of the other. Thus, let A = a velocity of 2 miles per hour, and B == 3 

miles per hour. Then the hours required per mile are respectively. A' = — = i» 

andB' = | = -J-. HereA: B = B' : A', or | = ?^„ or | = | = i = l-s-^'. 

(10) If two variable numbers, A and B, are reciprocally proportional, so that 
A' : B' = B" : A", the product, A' A", of any two values of one of the numbers 
is equal to the product, B' B'' pf the two corresponding values of the other. 

(11) The application of proportion to practical problems is sometimes called 
the rale 01 three. Thus : sing^le rule of tbree : If 3 men lay 10,000 
bricks in a certain time, how many could 6 men lay in the same time? 

As 3 men are to 6 men, so are 10,000 bricks to 20,000 bricks; or, 10,000 

bricks X -g- = 20,000 bricks. 

If 3 men require 10 hours to lay a certain number of bricks, how many hours 
would 6 men require to lay the same number? 

As 6 men are to 3 men, so are 10 hours to 5 hours ; or, 10 hours X -|- = 5 hours. 

(12) Double rule of tbree. 

If 3 men can lay 4,000 bricks in 2 days, how many men can lay 12,000 bricks 
in 3 days? Here 4,000 bricks require 3 men 2 days, or 6 man-days, and 12,000 

12 000 
bricks will require 6 X XaSa = 6 X 3 = 18 man-days ; and, as the work is to be 

done in 3 days, -^ = 6 men will be required. 

PROGRESSION. 

(1) Aritbmetteal Prog^ression. A series of numbers is said to be in 
arithmetical progression when each number differs from the preceding one by 
the same amount. Thus, —2. —1, 0, 1, 2, 8, 4, etc., where diff'erence = 1 ; or 4, 3, 
2, 1, 0, —1, —2, etc.. where diflTerence == —1 : or —4, —2, 0, 2, 4, 6, 8, 10, where 
dlffiapence = 2 ; or % 1%, 1, %, %, %, 0, -% —3^, etc., where diffference = —^ 

(2) In any such series the numbers are called terms. Let a be the first term, 
I the last term, d the common differdnce, n the number of terms, and s the sum 
of the terms. Then 



i = a + (n — 1) d 



Required 
I 


Given 
a d n 


I 


ads 


s 


a d n 



? = — l.rf±|/2d* + (a — ^cf)S 
, = 1. n [2 a + (n — 1) d] 

dls o=»-|-(f± l/(/-|-^d)8 — 2d* 



d — 2 a ± ^(2 a — d)8 -I- 8 d * 
n ads n =a 



2d 



n dls 



21 + d ± >/(2/ + d)2— 8dj 
2d 



(S) ISeometrieal Progression. A series of numbers is said to be in 
geometrical progression when each number stands to the preceding one in the 

same ratio. Thus: •^, -J-, 1, 8, 9, 27, 81, etc., where ratio => 8; or 48, 24, 12, 6, 

J, 1^, 4, f, etc., where ratio =- -J-; or ^, 1-J-^, 3|, 6|, 13^, 27, etc., where 

iatio = 2. 



40 AKITHMETIO. 

(4) Let a be the flnt term, I the last term, r the constant ratio, n the numbet 
of terms, and 4 the sum of the terms. Then : 



Bequired 
I 


Given 
a r n 


I 


art 


1 


r H * 



^^ g + (r- 1)* 
r 

^ r" — 1 

a n Z «=> 






r n I * = 



r«-.r*~* 



«»-i 



PiaKMVTATIOH, Ete. 

(1) Permatation shows in how many positions any namber of things oatt 
be arranged in a row. To do this, multiply together all the numbers used in. 
counting the things. Thus, in how many positions in a row can 9 things be 
placed? Here, 

1X2X3X4X6X6X7X8X9 = 362880 positions. Ans. 

(2) Combinatton shows how many combinations of a few things can be 
made out of a greater number of things. To do this, first set down that number 
which indicates the greater number of things; and after it a series of numbers, 
diminishing by 1, until there are in all as many as the number of the few thinga 
that are to form each combination. Then beginoing under the last one, set down 
said number of few things \ and going backward, set down another series, also^ 
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 ones to 
form another. Divide the upper product by the lower one. 

Ex. How many combinations oi 4 figures each, can be made from the 9 figure* 
1, 2, 3, 4, 5, 6, 7, 8, 9, or from 9 any things? 

9X8X7X6 3024 ,„^ ., ^, . 

r x 2 X 8 X 4 ^'2r^ combinations. Ans. 

(3) AlUg^tion shows the value of a mixture of different ingredients, When 
the quantity and value of each of these last is known. 

Ex. What is the value of a pound of a mixture of 20 fi>s of sugar worth 15 ots 
per lb ; with 80 lbs worth 25 cte per fi>? 

fts. cts. cts. 

20 X 15 = 800 _, - 1050 „, , 

80 X 25 = 750 Therefore, -^ = 21 cts. Ans. 

60 lbs. 1050 cts. 

PEBCENTAOE, INTEREST, ANNUITIES. 

Percentagre* 

(1) Batio is often expressed by means of the word " per." Thus, we speak of 
a grade of 105.6 feet per mile, i. e., per 5280 feet. When the two numbers in the 
ratio refer to quantities of the same kind and denomination, the ratio is often 
expressed as a percentage (perAundredage). Thus, a grade of 105.6 feet per mile,. 

* Equations involving powers and roots are conveniently solved by means of 
logarithms. 



AMTtBUmiC. 41 

or per 6280 feet, is equivalent to a grade of 0.02 foot per foot,* or 2 feet per 100 
feet, or simply (since botli dimensions are in feet) 2 per 100, <» 2 per " cent.'* 

(2) One-fiftietli, or 1 per 50, is plainly equal to two hundredths, or 2 per Atm- 
dred, or 2 per cetU. Similarly, 3^ = 25 per cent, % =,3 X 26 per cent. = 75 per 
cent., etc Heace, to reduce a ratio to the form of percentage, divide 100 times* 
the first term by the second. Thus, in a concrete of 1 part cement to 2 of sand 
and 5 of broken stone, there are 8 parts in all, and we have, by weight— f 

Cement = X » 0.126 = 12.6 per cent, of the whole. 
Sand =2. = 0.260= 26.0 " " 

Stone =|. = 0.626= 62.6 " " 

Concrete = f = 1000 = 100.0 " " 

(3) Percentage is of very wide application in money matters, payment for 
service in such matters being often based upon the amount of money involved. 
Thus, a purcliasing or selling agent may be paid a brokerage or commission 
which forms a certain percentage of the money value of the goods bought or 
sold ; the premium paid for insurance is a percentage upon the value of the goods 
insured; etc. 

Interest. 

(4) Interest is hire or rental paid for the loan of money. The sum loaned is 
caDea the -prlneiiMftl, and the number of cents paid annually for the loan of 
each dollar, or of dollars per hundred dollars, is called the rate of interest* 
The rate is always stated as a percentage. 

(5^ If the interest is paid to the lender as it accrues, the money is said to be 
at siniple interest ; but if the interest is periodically added to the princi- 
pal, so that it also earns interest* the money is said to be at eomponncl 
Interest, and the interest is said to be compounded. 

Simple Interest. 

(6) At the end of a year, the interest on the principal, P, at the rate, r, is » 
P r, and the Amoant, A, or sum of principal and interest, is 

A =- P + P r = P (1 + r). 

(7) At the end of a number, n, of years, the interest is » P rn (see right- 
hand side of Fig. 1), and 

A = P + P rn =» P (1 + rn). 

Thus, let P = $866.32, r = 3 per cent., or 0.03, n=l year, 3 mouths and 10 

days =» 1 year and 100 days = 1-J^ Y^axB =» 1.274 years. Then A — P (1 + rn) 

— S866.82 X (1 + 0.03 X 1.274) » $866.32 X 1.08822 => 8898.39. 

(8) For the present worth, principal, or eapltallEatlon, P, of 

the amount, A, we have 



p 



1 + rn 



Thns, for the sum, P, which, in 1 year, 8 months, 10 days, at 8 per cent. 

898 39 
simple interest, will amount to S898.39, we have P «- , ^ no v^ i otA = ^^866.32. 

(9) In commercial business, interest is commonly ealenlatecl approxl* 
nuktely by taking the year as consisting of 12 months of 30 days each. Then, 
at 6 per cent., the interest for 2 months, or 60 days, = 1 per cent; 1 month, or 30 
days, = Hp^ cent.; 6 days = 0.1 per cent. Thus, required the interest on 
$1264.35 for 6 months, 28 days, at 6 per cent. 

*A.Jraetianj as ^^ •^, etc., or its decimal equivalent, as 0.125, 0.3126, etc., 

is compared with unUy or one; but in percentage the first terra of the ratio is 
compared with one hwndred units of tue second term. Mistakes often occur 
through n^lect of this distinction. Thus, 0.06 (six per cent, or six per hundred) 
is sometimes mis-read six one-hundredths of one per cent, or six oue-hun- 
dredths per cent, 
t For proportions by volume, see pp 936 and 943. 



42 



ARITHMETIC. 



Principal .tl264.85 

Interest, 2 mos, 1 per cent 12.64 

2mo8, 1 " 12.64 

" Imo, h " 6.82 

" 20 days, I " 4.21 

" 6 days, 0.1 " 1.26 

" 2 days, ^ " 0.42 

Interest at 6 per cent $37.^ 

Deduct one-sixth 6.25 

Interest at 5 per cent $31.24 

Equation of Paymente. 

(10) A owes B $1200 ; of which $400 are to be paid in 3 months ; $500 in 4 
months; and $300 in 6 months; all bearing interest until paid; but it has been 
Agreed to pay all at onc& Now, at what time must this payment be made so that 
neither party shall lose any Interest? 

$ months. 

400 X 3 = 1200 . _.. 6000 ^.. ., . 
500 X 4 = 2000 Average time = T^ = ^ months. Ans. 

300 X 6 = 1800 

1200 5000 

Compound Interest. 

(11) Interest is usually compounded annually, semi-annually, or quarterly. 
If it is compounded annually, then (see left side of Fig. 1) 

at the end of 1 year A = P (1 + r) 

2 years A = P (1 + r) (1 + r) = P (1 + r)« 

n years A = P (1 + r)**; and 

^=(T:n^n=A(i + r)- 

p = (l+r)« 

(12) If the int^est is compounded g times per year, we have 

(la) The principal, P, is sometimes called the |»i*esent worth or present 

Talue of the amount, A. Thus, iu the following table, $1.00 is the present 
worth of $2,191 ^ue iu 20 years at 4 per cent, compound interest, etc, etc 



<i 



M 



(( 



«( 



i 


k / 


z. 








21 


y 


x*- 


rTv 


<^ 


F(l + r)n ^ 










5 




^ 


r 






J 


?r 


i 


». 




^ 


^ 














« 




^ 


w^^ 
















<M 




_r^^ 


















at 




^^ 




















>; 


\ 


r^^ 


*— 
















\ 


' 


•8^' 


^ 


















1 


I 


























^L 





























t 


J 


3 






















o 












• 










J 


> 


> 


' > 


' 


















> 


r < ' 




i 


> J 


\ A 


; ii 


r 4 


\ i 


F < 


:; 


1 t 


r 1 


9 » 


I 



Years 



Figr. 1. 



ABITHHETIC. 



43 



Ttible S« CompouiMl Interest. 

Amount of 81 at Compoand Interest. 





8 


»H 


4 


^ 


6 


6H 


6 


«H 


Yean. 


per 


per 


per 


per 


per 


per 


per 


per 




cent. 


oent. 


cent. 


cent 


cent 


oent 


cent. 


cent 


1 


1.030 


1.035 


1.040 


1.045 


1.060 


1.066 


1.060 


1.065 


2 


1.061 


1.071 


1.082 


1.002 


1.103 


1.118 


1.124 


1.134 


8 


1.098 


1.109 


1.126 


1.141 


1.168 


1.174 


1.191 


l.i08 


4 


1.126 


1.148 


1.170 


1.193 


1.216 


1.239 


1.262 


l.f86 


5 


1.159 


1.188 


1.217 


1.246 


1.276 


L807 


1.338 


1.870 


6 


1.194 


1.229 


1.265 


1.302 


1.340 


1.379 


1.419 


1.459 


7 


1.230 


1.272 


1.316 


1.361 


1.407 


1.455 


1.504 


1.654 


8 


1.267 


1.817 


1.869 


1.422 


1.477 


1.635 


1.594 


1.655 


9 


1.805 


1.863 


1.423 


1.486 


1.651 


1.619 


1.689 


1.763 


10 


1.844 


1.411 


1.480 


1.553 


1.629 


1.708 


1.791 


1.877 


11 


1.384 


1.460 


1.539 


1.623 


1.710 


1.802 


1.898 


1.999 


18 


1.426 


1.511 


1.601 


1.696 


1.796 


1.901 


2.012 


2.129 


18 


1.469 


1.564 


1.665 


1.772 


1.886 


2.006 


2.133 


2.267 


14 


1.518 


1.619 


1.732 


1.852 


1.980 


2.116 


2.261 


2.415 


15 


1.558 


1.675 


1.801 


1.935 


2.079 


2.282 


2.397 


2.672 


16 


1.606 


1.734 


1.878 


2.022 


2.188 


2.355 


2.540 


2.739 


17 


1.653 


1.795 


1.948 


2.113 


2.292 


2.486 


2.693 


2.917 


18 


1.702 


1.868 


2.026 


2.208 


2.407 


2.621 


2.854 


3.107 


19 


1.754 


1.923 


2.107 


2.308 


2.527 


2.766 


3.026 


3.309 


98 


1.806 


1.990 


2.191 


2.412 


2.653 


2.918 


3.207 


3.524 


91 


1.860 


2.069 


•2279 


2.520 


2.786 


8.078 


8.400 


3.753 


92 


1.916 


2.132 


2.370 


2.634 


2.925 


3.248 


3.604 


3.997 


98 


1.974 


2.206 


2.465 


2.752 


3.072 


3.426 


8.820 


4.256 


94 


2.033 


2.283 


2.563 


2.876 


3.225 


3.615 


4.049 


4.533 


95 


2.004 


2.863 


2.666 


3.005 


3.386 


a8i8 


4.292 


4.828 


98 


2.157 


2.446 


2.772 


3.141 


3.556 


4.023 


4.549 


5.141 


97 


2.221 


2.532 


2.883 


3.282 


3.733 


4.244 


4.822 


5.476 


98 


2.288 


2.620 


2.999 


3.430 


3.920 


4.478 


6.112 


5.832 


98 


2.857 


2.712 


3.119 


3.584 


4.116 


4.724 


5.418 


6.211 


80 


2.427 


2.807 


3.243 


3.745 


4.822 


4.984 


&743 


6.614 


81 


2.500 


2.905 


3.373 


3.914 


4.538 


6.268 


6.088 


7.044 


89 


2.575 


3.007 


3.508 


4.090 


4.765 


6.547 


6.453 


7.502 


88 


2.652 


8.112 


3.648 


4.274 


5.008 


5.852 


6.841 


7.990 


84 


2.732 


8.221 


3.794 


4.466 


5.253 


6.174 


7.251 


8.509 


85 


2.814 


8.834 


3.946 


4.667 


5.516 


6.514 


7.686 


9.062 


88 


2.898 


3.450 


4.104 


4.877 


6.792 


6.872 


8.147 


9.651 


87 


2.985 


3.671 


4.268 


5.097 


6.081 


7.250 


8.636 


10.279 


88 


8.075 


3.696 


4.439 


5.826 


6.385 


7.649 


9.154 


10.947 


89 


3.167 


3.825 


4.616 


6.566 


6.706 


8.069 


9.704 


11.658 


40 


8.262 


3.959 


4.801 


6.816 


7.040 


8.613 


10.286 


12.416 



Compoand interest on M dollars, at any rate r for n years =» M X compoand 
interest on $1 at same rate, r, and for n years. 

AnBnity, Sinkinir Fand, Amortisatloii, ]>epreeiaftloii. 

(14) Under "Interest" we deal with cases where a certain sum or "prin- 
cipal,** P, paid once for all, is allowed to accumulate either simple or compound 
interest ; but in many cases equal periodical payments or appropriations, called 
•mnaltiee, are allowed to accumulate, each earning its own interest, usually 
compoand. 



44 



ARITHMEnO. 



(15) Thua, a sum of money is set aside annually to accumulate oompoand 
interest and thus form a stiikliiil^ ftind, in order to extinguish a debt. In 
this way, the cost of engineering works is frequently paid virtually in instal- 
ments. This process is called amortlBatlon. 

(16) In estimating the operating expenses of engineering works, an allowance 
is made for depreelatlon. In calculating this allowance, we estimate or 
assume the life-time, n, of the plant, and find that annuity, p, which, at an 
assumed rate, r, of compound interest, will, in the time n, amount to the cost of 
the plant, and thus provide a fund by means of which the plant may be replaced 
when worn out or superseded. 

(17) The present wortb, present walae, or capltaliBation, W. 

Fig. 2, of an annuity, p, for a given number, n, of years, is that sum whidi. if 
now placed at compound interest at the assumed rate, r. will, at the end of that 
time, reach the same amount, A, as will be reached by tnat annuity. 



i 


> 1 


1 


z 


• 


1 


I 


(*+'>'■ ^ 


^ 

^ 


^ 


^ 


1 1 i 




.^ 


^ 






J 


V 


t 


1 


1 
L^ 

a 


,^ 


f 














> 


r 

> 








f 














J 


y 


f \ 


f 


















1 


r \ r 




< 


> J 


[ J 


\ a 


( 4 


\ I 


S i 


i : 


r « 


r I 


i » 


% 




Years 
Flff.l. 



O X 2 3 4 s a 
Year* 

FlV. 2. 



7 S 9 n 



(18) Equations for Compoand Interest and Annnltles. (See 
Figs, land 2.) 

P = principal ; r => rate of interest ; n = number of years ; 
A =■ amount ; p = annuity ; W = present worth. 

The interest is supposed to be compounded, and the annuities to be set aside, 
at the end of each year. 

Compound Interest. 

(1) The amount. A, of $1, at the end of n years, see (11), is A => (1 + r)". 

(2) Since the present worth of (1 + r)\ due in n years, is $1, see (1), it 
Uows, by proportion, that tlie present worth, W, of $1, due in n yean, 



fol 



isW = 



(1 + r)' 



= (1 + r) 



Annuities. 



(3) In n years, an annuity of $r will amount to (1 + r)** — 1.* Hence, the 
amount. A, of an annuity of $1, at the end of n years, is 



*In the case of compound interest on $1, the rate, r, may be regarded as an 
annuity, earning its interest; and, at the end of n years, the amount of the 
several annuities (each = the annual interest, r, on the $1 principal) with the 

interests earned by them, is = the amount, (1 + r)", of $1 in n years at rate, r, 

minus tiie $1 principal itself; or, amount of annuity = (l -f r)** •— 1. 



ARITUMETIG. 



45 



(4) For the present wortli, W, of an nnnnity of $1 for n years, 
we oave, trom Eqaations (1) and (3) : 

1 i— 

(l + r)*:l = ^^^^:^^^: i-iW. Hence. W = )-f-f^i jr-^^ 

r (1 + r) r 

See Table 3. 

(6) Tlie annuity for n years, which $1 will purchase, is * 

1* r 
P='W^ i — 



1 — 



(6) Tlie annnl^ which, in n years, will amount to $1, is 



jf = p -T 



W 









ft 



1 — 



(l + r)*-l 



See Table 4. (1 + r) * 

Table 8. Present Talne of Annuity of $1000. See Equation (4). 

Bate of Interest (Compound). 





2^ 


8 


8H 


4 


4^ 


6 


6Ji 


6 


Tears. 


per 


per 


per 


per 


per 


per 


per 


per 


• 


cent. 


cent. 


cent 

• 


cent. 


cent. 


oent. 


cent. 


cent. 


6 


4,646 


4,580 


4,515 


4,452 


4,390 


4,829 


4,268 


4,212 


10 


8,752 


8,580 


8,816 


8,111 


7,913 


7,722 


7,688 


7,360 


16 


12,381 


11,938 


11,517 


11,118 


10,740 


10,380 


10,037 


9,712 


ao 


15,589 


14,877 


14,212 


13,590 


18,008 


12,462 


11,950 


11,470 


26 


18,424 


17,413 


16,482 


15,622 


14,828 


14,094 


13,414 


12,783 


80 


20,930 


19,600 


18,392 


17,292 


16,289 


15,372 


14,534 


13,765 


S6 


23,145 


21,487 


20,000 


18,664 


17,461 


16,374 


15,391 


14,498 


40 


25,103 


23,115 


21,865 


19,793 


18,401 


17,159 


16,045 


16,046 


46 


26,833 


24,519 


22,495 


20,720 


19,156 


17,774 


16,648 


15,456 


60 


28,362 


25,730 


23,456 


21,482 


19,762 


18,256 


16,982 


15,762 


100 


36,614 


31,599 


27,655 


24,505 


21,950 


19,848 


18,096 


16,618 



(19) In comparing the merits of proposed systems of improvement, it is 
usual to add, to the operating expenses and to the cost of ordinary repairs and 
nuUntenance, (1) the interest on the cost, (2) an allowance for depreciation, and 
sometimes (3) an annuity to form a sinking fund for the extinction of the debt 
incurred by construction. The cilpitalization of the total annual expense, thus 
obtained, is then regarded as the true first cost of the construction. Ail the 
elements of eost are thus reduced to a common basis, and the several propositions 
become properly comparable. 

(20) Thus, in estimating, in 1899,^ the cost of improving the water supply of 
Fliiladelphia, the rate, r, of interest was assumed at 3 per cent, and depreciation 
was assumed as below. Under "Life" is given the assumed life-time of each 
class of structure or apparatus, and under *' Annuity " the sum which must be 
set aside annually in order to replace, at the expiration of that life, $1,000 of the 
corresponding value. 



Present worth Annuity 
* Because, W $1.00 

Equation (4) 



Present worth Annuity 

Sl.OO : p. Hence, j9 

Equation (5) 



1 



Annuity .Amount Annuity Amount ^ 

tBecause, r : (1 + r) " — 1 : : p' : $1.00. Hence, p' = .^ , \ n — 7. 
Equation (8) Equation (6) (1 + r) " — 1 

X Report by Rudolph Hering, Samuel M. Gray, and Joseph M. Wilson. 



46 



▲BITHMEnO. 



BTBUCVDBm, Apparatus, etc. Lvb, Ahkoitt 

in years f 

Masonry conduits, filter beds, reservoirs ^..Indefinite 0.00 

Permanent buildings 100 1.65 

Cast iron pipe, railroad side-tracks 80 8.11 

Steel pipe, valves, blow-o£b, and gates 85 16.M 

Engines and pumps 30 21.02 

Boilers, electric light plants, tramways and equipment, 

iron 'fences 20 87.22 

Telephone lines, sand-washer, and regulating apparatus.... 10 87.24 

(21) Calculated upon this basis, two projects, each designed to fiimish 450 
million gallons per day, compared as follows : 

BiVER Watkb, takkn within City 
Ldcixb and Filtbbkd. 



Unfiltbbed Watbb, by Aqubduct. 

First Out. 

8toraffe leservoirs. 930,900,000 

Aqueducts 47,730,000 

Distribution 8,655,000 

Distributing reservoir 1,000,000 

Total $88,185,000 

Annual. 
Interest on |68,185,00a 82,485,550 



I%rstCbH. 

Filter plants 828,174,680 

Mains ^ » 10,980,000 



Depreciation 

Operation and Maintenance. 

Analyses and inspec- 
tion 841,620 

Ordinary repairs ^,150 

Pumping and wages 140,770 



198,640 



281,540 



Total $84,154,68^ 

AninuaL 
Interest on 884,154,680 $1,024,840 



Depreciation 



206,540 



Operaiion and MaMenanee. 

Pumping 81,216,021 

Filtration 525,600 



82,925,780 






1,741,621 
82,971,801 



It will be noticed that, although the first cost of the filtration project was much 
less than half that of the aqueduct project, its large proportion of perishable 
parts made its <diarge for depreciation somewhat greater, while its cost for oper- 
ation and maintenance was more than seven times as great, and its total annual 
charge a little greater. 

Table 4. Anniilty required to redeem $1000. See Equation (6). 

Bate of Interest (Compound). 





1 


2 


2^ 


t 


«K 


4 


5 


6 


Years. 


per 


per 


per 


per 


per 


per 


per 


per 




cent. 


cenL 


cent. 


cent 


cent. 


cent. 


cent. 


cent. 


5 


196.04 


192.16 


190.24 


188.36 


186.49 


184.63 


180.98 


177.80 


10 


95.58 


91.33 


89.25 


87.23 


85.24 


83.29 


79.60 


75.87 


15 


62.12 


57.83 


66.77 


53.77 


61.82 


49.94 


46.34 


42.90 


20 


45.42 


41.16 


89.14 


37.22 


85.36 


33.58 


30.24 


27.18 


85 


85.41 


31.22 


29.27 


27.43 


25.67 


24.01 


20.96 


18.28 


SO 


28.75 


24.65 


22.78 


21.02 


19.37 


17.83 


15.05 


12.65 


S5 


24.00 


20.00 


18.20 


16.54 


15.00 


13.68 


11.07 


8.97 


40 


20.46 


16.55 


14.84 


13.26 


11.88 


10.62 


8.28 


6.46 


45 


17.71 


13.91 


12.27 


10.79 


9.45 


8.26 


6.26 


4.70 


50 


15.51 


11.82 


10.26 


8.87 


7.63 


6.55 


4.78 


8.44 


60 


12.24 


8.77 


7.35 


6.18 


6.09 


420 


2.83 


1.88 


70 


9.93 


6.67 


5.40 


434 


3.46 


2."74 


1.70 


1.08 


80 


8.22 


5.16 


4.03 


8.11 


2:88 


1.81 


1.08 


0.578 


90 


6.91 


405 


8.04 


2.26 


1.66 


1.21 


0.627 


0.318 


100 


5.87 


3.20 


2.31 


1.65 


1.16 


0.808 


0.383 


0.177 



ARITHMETio. 47 

I>rODENAI« OB BUOBBNART NOTATION.* 

(1) In the Arabic system of notation 10 is taken as the base, but in dnodenal 
notation 12, or " a dozen," is the base. While 10 is divisible only by 0, and (once 
only) by 2, 12^s divisible twice by 2, and ouce by 8, by 4, and by $. This accounts 
for tne popularity of the dozen as a basis of enumeration ; of weights, as in the 
Troy pound of 12 ounces ; of measures, as in the foot of 12 inches ; thoTear of 12 
months, and the half day of 12 hours ; and of coinage, as in the British shilling 
of 12 pence. 

(S) The dnodenal notation uses the dozen (12), the gross (12^ = 144), and the 
great gross (12^ == 12 gross =» 1728), as the decimal system uses the ten (10), the 
hundred (10^ = 100), and the thousand (10^ =» 10 hundred => 1000). Two arbitrary 
single characters, such as T and E, represent ten and eleven respectively ; the 
symbol 10 represents a dozen ; 11 represents thirteen, and so on. Thus, the num- 
erals of the two systems compare as follows : 

Decimal 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 20 21 22 28 24 25 36 48 60 
Duodenal 1 2 3 4 5 6 7 8 9 T E 10 11 12 ... 18 19 1T1E20 21 30 40 50 

Decimal 72 84 96 99 100 108 109 110 111 112 113 117 118 119 120 121 122 
Duodenal 60 70 80 83 84 90 91 92 93 94 95 99 9T 9E TO Tl T2 

Decimal 129 130 131 182 133 138 140 141 142 143 144 288 1728 20736 etc. 
Dnodenal T9 TT T£ EO El E6 E8 E9 ET EE 100 200 1000 10000 etc. 

(8) IHiodeclmaUL Areas of rectangular figures, the sides of which are 
eadbressed in feet and inches, are still sometimes found by a method called 
*' Duodecimals," in which the products are in square feet, in twelfths of a square 
toot (each equal to 12 square inches) and in square inches ; but, by means of our 
table of *' Inches, reduced to decimals of a foot." page 221, the sides may be taken 
in feet and decimals of a foot, and the multiplication thus more conveniently 
performed, after which the decimal fraction of a foot in the product may, if 
oesired, be converted into square inches by multiplying by 144. 

•See Elements of Mechanics, by the late John W. Nystroa. 



48 



RECIPBOCALS OP NUMBERS. 



Table of Reetprocate of STuinbers. 8m p. 9S. 



No. 


Reciprocal. 


No. 


Reciprocal. 


No. 


fleciprocal. 


No. 


ReciprocaL 


1 


1.000000000 


56 


.017857148 


Ill 


.009009009 


166 


.006024096 


2 


0.500000000 


57 


.017543860 


112 


.008928571 


167 


.005988024 


3 


.333333333 


58 


.017241379 


113 


.008849558 


168 


.005952381 


4 


.250000000 


59 


.016949153 


114 


.008771930 


169 


.005917160 


5 


.200000000 


60 


.016666667 


115 


.006695652 


170 


.005882353 


6 


.166666667 


61 


.016393443 


116 


.008620690 


171 


.005847953 


7 


.142857143 


62 


.016129032 


117 


.008547009 


172 


.005813953 


8 


.125000000 


63 


.015873016 


118 


.008474576 


173 


.005780347 


9 


.111111111 


64 


.015625000 


119 


.006403361 


174 


.005747126 


10 


.100000000 


65 


.015384615 


120 


..008333333 


175 


.005714286 


11 


.090909091 


66 


.015151515 


121 


.008264463 


176 


.005681818 


12 


.083333333 


67 


.014925373 


122 


-.008196721 


177 


.006649718 


18 


.076923077 


68 


.014705882 


123 


.008130081 


178 


.005617978 


14 


.071428571 


69 


.014492754 


124 


.008064516 


179 


.005586592 


15 


.066666667 


70 


.014285714 


125 


.008000000 


180 


.005565656 


16 


.062500000 


71 


.014084507 


126 


.007936608 


181 


.005524862 


17 


.058828529 


72 


.013888889 


127 


.007874016 


182 


.005494505 


18 


.055555556 


73 


.013698630 


128 


.007812500 


183 


.005464481 


19 


.052631579 


74 


.013513514 


129 


.007751988 


184 


X)05434788 


20 


.050000000 


75 


.013333333 


130 


.007692308 


185 


.005405405 


a 


.047619048 


76 


.013157895 


181 


.007633588 


186 


.005876844 


•22 


.045454545 


77 


.012987013 


132 


.007575758 


187 


.005347594 


28 


.043478261 


78 


.012820513 


138 


.007518797 


188 


. .005319149 


24 


.041666667 


79 


.012658228 


134 


.007462687 


189 


.005291005 


25 


.040000000 


80 


.012500000 


135 


.007407407 


190 


.005263158 


26 


.038461538 


81 


.012345679 


136 


.007352941 


191 


.005235602 


: 27 


.037037037 


82 


.012195122 


137 


.007299270 


192 


.005208333 


28 


.035714286 


83 


.012048193 


138 


.007246377 


198 


.005181847 


29 


.034482759 


84 


.011904762 


139 


.007194245 


194 


.005154639 


30 


.033333333 


85 


.011764706 


140 


.007142857 


195 


.005128205 


31 


.032258065 


86 


.011627907 


141 


.007092199 


196 


.005102041 


32 


.031250000 


87 


.011494253 


142 


.007042254 


197 


.005076142 


33 


.030303030 


88 


.011363636 


143 


.006993007 


198 


.005050505 


34 


.029411765 


89 


.011235955 


144 


•UUOU's^.'z 4fB 


199 


.005025126 


35 


.028571429 


90 


.011111111 


145 


.006896552 


200 


.005000000 


36 


.027777778 


91 


.010989011 


146 


.006849815 


201 


.004975124 


37 


.027027027 


92 


.010869565 


147 


.006802721 


202 


.004950495 


38 


.026315789 


93 


.010752688 


148 


.006756757 


203 


.004926108 


39 


.025641026 


94 


.010638298 


149 


.006711409 


204 


.004901961 


40 


.025000000 


95 


.010526316 


150 


.006666667 


205 


.004878049 


41 


.024390244 


96 


.010416667 


151 


.006622517 


206 


.004854369 


42 


.023809524 


97 


.010309278 


152 


.006578947. 


207 


.004830918 


43 


.023255814 


98 


.010204082 


153 


.006535948 


208 


004807692 


44 


.022727273 


99 


.010101010 


154 


.006493506 


209 


.004784689 


45 


.022222222 


100 


.010000000 


155 


.006451613 


210 


.004761905 


46 


.021739130 


101 


.009900990 


156 


.006410256 


211 


.004739336 


47 


.021276600 


102 


.009803922 


157 


.006369427 


212 


.004716981 


48 


.020833333 


103 


.009708738 


158 


.006329114 


213 


.004694836 


49 


.020408163 


104 


.009615385 


159 


.006289308 


214 


.004672897 


60 


.020000000 


105 


.009523810 


160 


.006250000 


215 


.004651168 


bl 


.019607843 


106 


.009433962 


161 


.006211180 


216 


.004629680 


52 


.019230769 


107 


.009345794 


162 


.006172840 


217 


.004608295 


£3 


.018867925 


108 


.009259259 


163 


.006134969 


218 


.004587156 


£4 


.01851&'>19 


109 


.009174312 


164 


.006097561 


219- 


.004566210 


Sb 


.018181818 


110 


.009090909 


165 


.006060606 


220 


.004545455 



BECIPROCALS OF NUMBEBS. 



49 



Table of BeeiproMOa of Hnmbom.— {Cbn/imied.) See p. 62. 



Ka 


BedprooaL 


No. 


Reciprocal. 


Na 


Beciprooal. 


No. 


BeciprocaL 


221 


.004524887 


276 


.003623188 


831 


.008021148 


886 


.002590674 


222 


.004504505 


277 


.008610108 


832 


.008012048 


887 


.002588979 


228 


.0044848a'> 


278 


.003597122 


888 


.003008003 


888 


.002577320 


224 


.004464286 
.004444444 


279 


.008584229 


834 


.002994012 


889 


.002570694 


225 


280 


.008571429 


83l> 


.002965075 


890 


.002564103 


226 


.004424779 


281 


.003558719 


836 


.002976190 


901 


.002557545 


227 


.004405286 


282 


.008546099 


887 


.002967859 


892 


.002551020 


228 


.0048a5965 


283 


.003533569 


838 


.002958580 


893 


.002544529 


229 


.004366812 


284 


.003521127 


339 


.002949853 


394 


.002538071 


280 


.004347826 


285 


.008508772 


340 


.002941176 


895 


.002531646 


231 


.004329004 


286 


.003496503 


341 


.002982551 


396 


.002525258 


232 


.004310345 


287 


.003484321 


842 


.002923977 


897 


.002518892 


238 


.004291845 


288 


.003472222 


343 


.002915452 


896 


.002512568 


234 


.004278504 


289 


.008460208 


344 


.002906977 


399 


.002506266 


235 


.004255819 


290 


.008448276 


845 


.002898551 


400 


.002600000 


236 


.004237288 


291 


.003436426 


846 


.002890173 


401 


.002493766 


237 


.004219409 


292 


.003424658 


347 


.002881844 


402 


.002487562 


238 


.004201681 


293 


.008412969 


818 


.002873563 


408 


.002481890 


289 


.004184100 


294 


.003401861 


349 


.002865330 


404 


.002475248 


240 


.004166667 


295 


.003888831 


350 


.002857143 


405 


.002469186 


241 


.004149878 


296 


.003378378 


351 


.002849008 


406 


.002463054 


242 


.004132231 


297 


.008367003 


352 


.002840909 


407 


.002457002 


243 


.004115226' 


298 


.003855705 


858 


.002832861 


408 


.002450960 


244 


.004098861 


299 


.008344482 


354 


.002824859 


409 


.002444988 


245 


.0040K1638 


800 


.008338833 


355 


.002816901 


410 


.002439024 


246 


.004065041 


301 


.003322259 


856 


.002808989 


411 


.002438090 


247 


.004048583 


802 


.008811258 


857 


.002801120 


412 


.002427184 


a<8 


.004082258 


308 


.003300830 


358 


.002798296 


418 


.002421808 


249 


.004016064 


804 


.008289474 


3591 .002785515 


414 


.002415459 


250 


.004000000 


805 


.008278689 


360 


.002777778 


415 


.00240968t 


251 


.008984064 


306 


.003267974 


361 


.002770088 


416 


.002408846 


252 


.003968254 


307 


.003257829 


362 


.002762431 


417 


.002398062 


258 


.003952569 


308 


.003246753 


363 


.002754821 


418 


.002392344 


254 


.003987008 


809 


.003236246 


364 


.002747253 


419 


.002386685 


255 


.003921569 


810 


.003225806 


365 


.002739726 


420 


.002380962 


256 


.003906250 


811 


.008215434 


866 


.002782240 


421 


.002375297 


267 


.003891051 


812 


.003205128 


867 


.002724796 


422 


.002369668 


258 


.003875969 


813 


.003194888 


868 


.002717391 


428 


.002864066 


259 


.003861004 


314 


.008184718 


869 


.002710027 


424 


.002358491 


260 


X)08846154 


81d 


.003174603 


370 


.002702703 


425 


.002352941 


261 


.008881418 


816 


.008164557 


371 


.002695418 


426 


.002347418 


2G2 


.003816794 


817 


.003154574 


872 


.002688172 


427 


.002341920 


268 


.008802281 


818 


.003144654 


873 


.002680965 


428 


.002336449 


264 


.003787879 


319 


.003134796 


374 


.00267^97 


429 


.002381002 


265 


.003778585 


320 


.003125000 


375 


.002666667 


430 


.002825561 


266 


.008759398 


321 


.003115266 


376 


.002659574 


431 


.002320186 


267 


.003745318 


322 


.008105590 


377 


.002652520 


432 


.002314^5 


268 


.003731348 


323 


.008095975 


378 


.002645503 


433 


.002309469 


269 


.003717472 


324 


.008086420 


379 


.002638522 


484 


.002304147 


270 


.003703704 


325 


.008076923 


380 


.002681579 


485 


.002298851 


271 


.003690037 


826 


.008067485 


381 


.002624672 


436 


.0022Sfe578 


272 


.003676471 


327 


.008058104 


882 


.002617801 


437 


.002288380 


273 


.003668004 


328 


.003048780 


888 


.002610966 


438 


.002283105 


274 


.000649685 


329 


.008039514 


384 


.002604167 


489 


.002277904 


275 


.003636864 


830 


.008080808 


885 


.002597408 


440 


.002272727 



50 



BEOIPROCALS OF mTHBXltS. 



TftM« of meetpra9M9 «ff Bfanibei«b--KOMiiMiin£> 996'^9i. 



Kc 


Recipi^ooal. 


N<y. 


Beeipvocal. 


No. 


Reoiprocul. 


No. 


Recipf^dal 


441 
44JJ 
443 
444 
445 


.002267574 
.002262443 
.002237836 
.0022622B2 
.002247191 


496 

m 

496 
499 
500 


.002016129 
.002012072 
.002000032 
.002004008 
.002600000 


651 
5S2 
668 
564 

565 


.001814882 
.001811594 
.001806818 
.001806054 
.001801802 


606 
697 

668 
609 
610 


.0016S(n6i 
.001643tt6 
.00164087 
.0016«aD86 
.001639344 


446 
447 
446 
441> 
450 


.002242152 
.002287136 
.002282143 
.002227171 

.002222222 


601 
602 
503 
501 
505 


.001996008 
.001992032 
.001988072 
.001984127 
.001980198 


556 
657 
558 
559 
660 


.001798561 
.001795332 
.001792115 
.001768909 
.0Q198S714 


611 
612 
618 
614 
615 


.001686661 
.001633887 
.00169Uei 
.001628664 
.001626016 


m 

492 
458 
464 
469 


.002217295 
.002212889 
.002207506 
.002202643 
.002197802 


506 
507 
506 
509 
510 


.001976285 
.001972387 
.001968504 
.001964637 
.001960784 


561 
562 
568 
564 
665 


.001782581 
.001770859 
.001776199 
.001773050 
.001769912 


616 
617 
618 
619 
620 


.001628877 
.001620746 
.001618128 
.001615909 
.00161f990S 


46^ 
467 
4$^ 
«9 

4m 


.002192982 ' 

.0021881S4 

.002188406 

.002178649 

.002178918 


511 
512 
618 
514 
515 


.001966047 
.001958125 
.001949818 
.001945629 
.001941748 


566 

5<fr 

568 
569 
570 


.001766784 

.001769661 

.0O176056J 

.001767469 

.00175488^ 


621 
622 

m 
ess 


.001610806 
.001607717 
.00160006 
.00160SN4 
.001600000 


m 
4m 


.002169W7 
.002164502 
.002159827 
.002155172 
' .002150638 


516 
517 
518 
519 
620 


.001937984- 
.001934236 

.0019805021 
.001926782 
.001928077 


571 
572 
573 
574 
676 


.001761813 
.001748252 
.001745201 
.001742160 
.001789130 


626 
637 
628 
629 
680 


.001597444 
.001594606 
.001692067 
.001588(325 
.001587602 


46$ 
467 
468 
469, 
470 


.002145923 
.002141828 
.002136752 
.002182i96 
.002127660 


521 
522 
528 
524 
525 


.001919886 
.0019157091 
.001912046 
.001908897 
.001904762 


676 
577 
578 
579 
680 


.0017961111 
.001788102 
.001799104 
.001727116 
.001724138 


681 
682 
688 
694 
685 


.001j5847d6 
.00158^8 
.001679779 
.001577B67 
.00157^08 


471 
472 

474 
476 


.002128142 
.002118644 
.002114165 
.002109705 
.002106263 


526 
527 
528 
529 
530 


.001901141 
.001897533 
.001898939 
.001890859 
.0018867921 


681 
582 
588 

584 
885 


.001721170 
.001718213 
.001716266 
.001712329 
.001709403 


636 
637' 
638 
639 
640 


.001572827 
.001569869 
.0015^7$98 
.0015a«945 
.001562500 


476 
477 
478 
479 
480 


.002100840 
.002096436 
.002092050 
.002087683 
.002088833 


531 
532 
598 
534 
585 


.001»^39l 
.001879699 
.001876173! 
.001872659 
.001869159 


586 
587 
588 
589 
590 


.00170648.1 
.001703678 
.001700680 
.001697793 
.001694913 


641 
642 
643 
644 
645 


.001566062 
.001557602 
.001558^0 
.001592796 
.001556808 


461 
482 
408 
464 
48& 


.002079002 
.002074689 
.002070393 
.002066116 
.002061856 


536 
537 
538 
589 
540 


.001865672 
.001862197 
.001858736 
,001855288 
.001851852 


591 
592 
598 
694 
595 


.001692047 
.001689189 
.001686841 
.001683502 
.001680672 


646 
647 
648 
649 
650 


.001547908 
.001546695 
.001548^0 
.001546682 
.001538^2 


486 
487 
468 
489 
490 


.002057613 
.002053888 
.002049180 
.002044990 
.002040816 


541 
542 
•543 
544 
545 


.001848429 
.001846018 
.001841621 
.001838235 
.001834862 


596 
597 
598 
599 
600 


.001677852 
.001676042 
.001672241 
.001669449 
.001666667 


651 
652 
653 
654 
665 


.0015360^ 
.001538742 
.001531894 
.001529062 
.001526718 


491 
492 
493 
494 
495 


.002036660 
.002032520 
.002028398 
.002024291 
.002020202 


546 
547 
548 
549 
550 


.001831502 
.001828154 
.001824818 
.001821494 
.001818182 


601 
602 
608 
604 
605 


.001663894 
,001661130 
.001658875 
.001655629 
.001662893 


666 
667 
668 
669 
660 


.001524890 
.001622070 
.001619787 
.001517461 
.001515162 



RECIPROCALS OP NUMBERS. 



51 



Tfil»l« of kl««tpii#caf8 ^VKataiierik-^aMiMraMl.) Seep. S2. 



WtJL 



Beciprotat 



668 
66S 
664 

000- 



667 
668 
668 
«70 

671 
672 
67B 
674 
«7fi? 

€fS 

ersf 



«S2 
088 
«M 

<I85 

685 
697 
688 

689 



691 



6M 

695 

696. 

697 

698 

699 

700 

701 
702 
7€8 i 

706 

25§ 

707 
708 
709 
710 

711 
712 
713 
714 
715 



.001512869 
.601510874 
;*0015e8996 
.001506024 
i0015aS759 

;OOi5erso2. 

.001499250 
i001497Q06 
.00149<?68' 
.00149e§37 

.001490813 

.001488095 

.001488884: 

.001498660; 

.00148M81, 

.001479990 
.001477105 • 
.001474926 
.001472754. 
.001470088; 

.00146M29. 

.001466976. 

.00146«129 

.001461988 

.001459654 

.00145V726 

.0014S6604, 

.001488488; 

.00146*979' 

.001449075 

.001447178 
.001446087 
.6014^19001 
.0014409!» 

.001438849 

.091436382 
)1484720 




.001426534 
.0OT424501 
.001422475 
J001420455 
.001418440 

:06l4i643i 

.001414427 
.001412429 
.001410437 
.001408451 

.001406470 
.001404494 
.001402525 
.001400560 
.001398601 



M«. 



716 
717 
718 
719 
720 

721 

72» 
72» 
724 
726 

726 
727 
728 
729 
730 

73t 
732 
738 
734 
786 

796 
7S7 
788 
789 
740 

741 

742 
748 
744 
745 

746 
747 
74B 
749 
750 

761 
j2 

754 
766 

756 
757 
758 
759 
760 

7^ 
762 
763 
764 
765 

766 
767 
7^8 
769 
770 



Beciporocal. 



.001386648 
.001394700 
.001392758 
.00139082]; 
.001388889 

.001386963 
.001385042 
J0013a8126 
.001381213 
X)013?9810 

.001377410 
X)013755l6 
X)01373626 
X)0137174!| 
•.0Q1869863 

.00186798* 
-.001366120 
.001364250 
.001862398 
.001860644 

i001358696 
.001866852 
.001865014 
i001$58186 
• .001351351 

.00134952 

.0019477C 

.00184689( 

.001844086 

.00184228^ 

•iO0l84O488: 
.001388688 
:001886e98 
.001835113 
.001333333, 

.001331558 
.001329787 
.001328021 
.001326260 
.001324508 

.001322751 
.001321004 
.001819261 
.001317523 
.001315789 

.0013l4d60 
.001812386 
.001310616 
.001308901 
.001307190 

.001305483 
.001303781 
.001302083 
.001300390 
.001298701 



No. 



771 
772 
778 
774 
775 

776 

777 
778 
779 
780 

781 
782 
788 

784 
785 

786 
787 
788 
789 
790 

791 
792 
793 
794 
795 

796 

797 
798 
799 
800 

^ 

802 

•«03 

804 

805 

806 
807 
808 
809 
810 

811 
812 
813 
814 
815 

816 
817 
818 
819 
820 

821 
822 
823 
824 
825 



Becipracal. 



.001297617 
.001296337 
.001296661 

.001291990 
J0O129O323 

J00128866Q 
J0O12870O1 
J001283847 
.001283697 
.001282051 

.001280410 
.001278772 
.00127713^ 
.001276610 
.001278686 

.001272265 
.001270648 
.001269036 
.001267427 
.001265828 

.00126422$ 
.001269626 
.00126)031 
.001269446 
.00126786^ 

.001266281 
.001254705 
.00126813$ 
.001261564 
.001260000 

.001248439 
.001246883 
.001146880' 
\ .001943781' 
J001242236 

.001240695 
.001239157 
.001237624 
.001286094 
:00l2$4d68 

.001233046 
.001231527 
.001230012 
.001228501 
.001226994 

.001228990 
.(X)1222494 
.001221001 
.001219512 

.001218027 
".001216545 
.001215067 
.001213592 
.001212121 



No. 



826 

827 



829 
880 

891 

832 
838 
834 
885 

886 
887 
838 
839 
640 

841 
842 
843 
844 
846 

846 
847 
848 
849 
850 

861 
862 
863 
854 
855 

856 
857 
889 
869 
860 

861 
862 
863 
864 
865 

866 
867 
868 
869 
870 

871 
872 
873 
874 

875 



B^ciproOftL 



.001210664 
.001200190 
.001207729 
.0012062^ 
.001204819 

.001208869 
.001201928 
.001200480 
.0011990a 
.001197605 

.001196172 
.001194743 
.001193817 
.001191885 
.001190476 

.001189061 
.001187648 
.001186240 
.001184884 
,001183432 

.001182083 
.001180688 
.001179245 
.001177866 
X)01176471 

.001176088 
.001178709 
.001172883 
.001170960 
.00116S691 

.001168224 
.001166861 
'.O0I168GO1 
.001164144 
.001162791 

.001161440 
.001160093 
.001158749 
.001157407 
.001156060 

.001154734 
.0011584031 
,001152074 
.001100748 
.001149426 

.00114fflOe 
.001146789 
.001145475 
.001144165 
.001142857 



876 .001141558 

877 .001140251 

878 .001138952 

879 .001137656 

880 t .001136864 



62 



BECIPROCALS OF NUMBEB8. 



Table of Reelproeals of If ambers.— {ObnMniMtf.) See below. 



No. 


Reciprocal. 


No. 


Reciprocal. 


No. 

941 


Reciprocal 


No. 


Redproesl 


881 


.001138074 


911 


.001097695 


.001062699 


971 


.001029666 


882 


.001133787 


912 


.001096491 


942 


.001061571 


972 


.001028807 


888 


.001132nO3 


913 


.001005290 


943 


.001060445 


973 


.001027749 


884 


.001131222 


914 


.001094092 


944 


.001059322 


974 


.001026694 


886 


.001129944 


915 


.001092896 


945 


.001058201 


975 


.001025641 


886 


.001128668 


916 


.001091703 


946 


.001057062 


976 


.001024590 


887 


.001127396 


917 


.001090513 


947 


.001065966 


977 


.001023541 


888 


.001126126 


918 


.001089325 


948 


.001054852 


978 


.001022495 


889 


.001124859 


919 


.001088139 


949 


.001063741 


979 


.001021450 


880 


.001123596 


920 


.001086957 


950 


.001052632 


960 


.001020408 


891 


.001122334 


921 


.001065776 


951 


.001051526 


961 


.001019368 


892 


.001121076 


922 


.001084599 


952 


.001060420 


982 


.001018380 


893 


.001119821 


923 


.001068424 


953 


.001049318 


983 


.001017294 


894 


.001118568 


924 


.001062251 


954 


.001048218 


964 


.001016200 


895 


.001117318 


925 


.001061081 


955 


.001047120 


966 


.001015228 


896 


.001116071 


926 


.001079914 


956 


.001046026 


986 


.001014199 


897 


.001114827 


927 


.001078749 


957 


.001044932 


987 


.001018171 


898 


.001113586 


928 


.001077586 


958 


.001043841 


988 


.001012146 


899 


.001112347 


929 


.001076426 


959 


.001042753 


989 


.001011122 


900 


.001111111 


930 


.001075269 


960 


.001041667 


990 


.001010101 


901 


.001109878 


931 


.001074114 


961 


.001040588 


991 


.001009062 


902 


.001108647 


932 


.001072961 


962 


.001039501 


992 


.001008065 


903 


.001107420 


933 


.001071811 


963 


.001038422 


998 


.001007049 


904 


.001106195 


934 


.001070664 


964 


.001087344 


994 


.001006086 


905 


.001104972 


935 


.001069519 


965 


.001036269 


995 


.001005026 


906 


.001103753 


936 


.001068376 


966 


.001035197 


996 


.001004016 


W7 


.001102536 


937 


.001067236 


967 


.001034126 


997 


.001003009 


908 


.001101322 


988 


.001066098 


968 


.001083058 


998 


.001002004 


909 


.001100110 


939 


.001064963 


969 


.001031992 


999 


.001001001 


«10 


.001098901 


940 


.001063830 


970 


.001030928 


1000 


.001000000 



BECIPBOCAIiS. 



(a) Tbe reeiproeal of a number Is the quantity obtained by divid- 
ing unity or 1 by that number. In other words, if n be any number, then 

Recip n = — . Thus, Redp 40 = — =s= 0.025 ; Recip 0.4 = — = 2.5, etc., etc 

Hence, Recip — = — , because Recip -- = l-»- — =«1X — =* — • 

Thus, since 1 yard = 36 inches, 1 inch = ^ yard = .027777778 yard, for Recip 
36 = .027777778. Again, 1 foot head of water gives a pressure of .4336 lbs. per 
square inch. Hence a pressure of 1 lb. per square inch corresponds to a head 

of -^^ feet = 2.306805 feet, for Recip .4335 = 2.306805. (See b, below.) 

(b) It follows that if any number in the column headed *' No." be taken as 
the denominator of a common fraction whose numerator is 1, the corresponding 
reciprocal is the value of that fraction expressed in decimals.* Thus, ^ » .03126. 
Hence, to reduce a eommon fraction to decimal form, multiply 
the reciprocal of the denominator by the numerator. Thus, ^ sa .63125, because 
Recip 82 = .03125, and .03125 X 17 = .53125. 

(e; Conversely, if the reciprocal of a number n be taken as a number, then the 

number n itself becomes the reciprocal. In other words, Recip — =» n. Thus, 
Recip 0.025 = Recip -^ — 40 ; Recip 2.5 = Recip ~ = 0.4, etc., etc. 

* The numbers 2 and 5, and their powers and products, are the only ones whose 
reciprocalB can be exactly expressed in decimaUi 



BEGIPROOALS OF NI7MBEBS. 53 

(d) The prodnet of any nmnber by its own redpioeal is equal to unity or 1 ; 

•r, n X — =r — = 1. 
* n n 

(e) Any number, a X Becip of a number, n = o x — = — . 

Hence, to aToid the labor of dlTiding, we may multiply by the redp- 
roctU of the divisor. Thus, 
200 -+■ 48750 = 200 X Becip 48750 » 200 X.00002051282 (see ll, below)=.004102564 

(f ) Any number, a -5- Becip of a number, n — a-i- — = an. 
fieiice, a -f- Becip a = a -*- — =»aX'7~ = a*. 

Thus. Eedp2 = 0.6.and-g^ = ^ = 4-2.. 

(:g) The numbers in the foregoing table extend from 1 to 1000 ; bat the recip- 
rocals of maltlples of these nmnbers by 10 may be taken from the 
table by adding one cipher to the left of the reciprocal (after the decimal point) 
for each cipher added to the number. Thus, 

Becip 390 = .002564103 ; 

Becip 3900 = .0002564103; 

Becip 39000 » .00002564108 ; 
and the reciprociJs of nambers eontaining' decimals may be taken 
firom the table by shifting the decimal point in the tabular reciprocal one place 
to the right for each decimal place in the number. Thus: 

Becip 227 = .004405286; 

Becip 22.7 = .04405286; 

Becip 2.27 » .4405286; 

Becip .227 = 4.405286; 

Becip .0227 = 44.05286. 

(k) The reciprocal of a number of more than three fkgnrea may be 

taken firom the table approxinvately by interpolation. Thus, to find Becip 236.4: 

Becip 236 =.004237288 
Becip 237 =■ .004219409 

Differences: 1, .000017879, 286.4—^ = 0.4. 
Then, 0.4 X .000017879 = .000007152, 

and Becip 236 =.004237288 

minus .00 0007152 

= Becip 236.4 =.004230136 by interpolation. 
The correct reciprocal is .004280118. 

(1) The reciprocals of numbers not in the table may be conveniently found 

bjr means of logarithms. Thus, to find the Becip 236.4 = : 

Log 1 =0.000000 
Subtract Log 236.4 = 2.373647 

'7.626353 = liOg 0.00428012 

Becip 286.4 — 0.00423012. 

-, ^ J « , M24 286.4 
To iUid Becip .^^^ =-3524", 

Log 236.4 = 2.378647 
Subtract Log 8424 = 8.925518 

"5.448129 = Log 0.0280627. 

fU24 
Re«ip-^^ = 0.0280627. 

' (J) Position of the decimal point. For the Nos. 10, 100, 1000, etc., 
the number of the decimal place occupied by the first significant figure in the 
teciprocal is equal to the number of ciphers in the No. ; but for all other Nos. it 
is equal to the number of the digrUs in the integral portion of the No. Thus : 
Becip 148.7 = .0069.., etc. Here the number of digits in the integral portion 
(143) of the No. is 8, and the first significant figure (6) of the reciprocal occupies 
tin Hdrd decimal place. 



BQU^KE AKD CUBE SOOTS. 
a Itoau awl Coke M*ot* vf BWnber* f) 



N 

1 



i 
It 



8qui.be and cube boots. 

< and «!■»• Bwata t Ktemben fiKMH .1 WgjL 



1 
i 


Ml 

■i" 


a 

S 

1 

1 


1 

i 
i 


i 


"1 

i 

1 

i 
s 


i 

i 

i 

i 


J 


is 

ii 

Is 
IS 

i| 

Is 

IS 


i 

ii 



9QjUAB3S&l>.CUBS9, AND BO0;r8. 



5d 



TAMBUE of Sqinares. Cnbes, Square Roots, and Cube Boots. 

of Vumbers f^om 1 to lOOO. 

BuMAfF OH «Hi( I90LL0VIX* Tabuc. WbtToy^f \he «#eoi of a fifth 4(Bclmal in (he roots voQld lie M 
tad 1 td the fottrth aad flnel decimal tn the taole, (he addition has been made. 1ft errors. 





Bqpamf. 


Cpbe. 


6<l. Rt. 


C. Rt. 


No. 


Sqnave. 


Cube. 


6q. Kt. 


CBt. 




1 


1 


1.0000 


1.0000 


61 


.9721 


220981 


7.8102 


3.9965 




4 


8 


1.4i«3 


1.3999 


68 


9844 


238328 


7.8740 


3.9679 




• 


2T 


J.79»l 


l.(«29 


J98 


9999 


250047 


7.9873 
8.0000 


3.9791 




U 


64 


9.0990 

isooi 


1.5874 


«4 


4096 


262144 


4. 




15 


1» 


I.TI90 


65 


4226 


274625 


8.0623 


4.0207 




96 


316 


^4495 


1.8171 


66 


4356 


287496 


8.1240 


4.0411 


1 


S 


SM 


9.M58 


1.9129 


67 


4480 ■ 


300789 


. 8-1854 


4.0615 


# 


51S 


2.8384 


2.9890 


68 


4624 


31449S 


8.2462 


4.9617 




(a 


7fl» 


8.0909 


2.0691 


69 


4761 


328909 


8.3016 


4.1016 


10 


IM 


1600 


9.1939 


9.1544 


70 


4899 


843090 


8.3996 


4.1218 


B 


m 


1331 


9.S166 


2.2240 


71 


5041 


357911 


8.4261 


4.1406 


144 * 


1738 


8.4941 


2.2804 


72 


5184 


373248 
389017 


8.4866 


4.1602 


19 




3I6T 


8.6966 


2.3513 


78 


5476 


8.5440 


4.1798 


n 


^^ 


3944 


8.741t 


2.4101 


74 


495224 


8.6028 


4.1989 


ii 


0n 


8.8^90 


2.4692 


75 


5625 


421975 


8.6696 


4.2179 


•» 


^ 


4096 


40000 


2.5198 


76 


5776 


498076 
466593 


8.7178 


4.2858 


^9 


Svn 


*^& 1 


2.5713 


77 


5999 


8.7760 


4.25a 


19 


32 


56^ 




2.9907 


78 


6094 


474652 


8.8318 : 


4.9717 


» 


Ml 


fMiO 


2.9684 


79 


6341 


498080 


8.8883 


4.2996 


'» 


«D 


MM 


iiSsi 

i 


2.7144 


80 


6400 


512009 


8.9448 


4.8099 


n 


m 


1^107 




2.7589 


81 


6561 


591441 


9. 


4.3267 


.» 


2^ 


H.mNM ' 


fi.9920 


«2 


6794 


551868 


9.0554 


4.8445 


9 


^S 


4.7^58 


2.9489 


88 


6889 


571787 


9.1104 ; 


4.3631 


M 


. i 




5!^ 


9.9945 


84 


7066 


592T04 


9.1952 


4.8796 


» 


2.9940 


85 


7395 


614196 


9.2195 


4.8968 






J7S7fi 


siSS- 


B.9625 
8.9900 


86 
87 


7396 
7569 


686056 
658608 


9.2736 
9.3274 


4.4140 
4.4810 


.19 


M| 


3H9I 


5.'^ ] 


8.0866 
8.6783 


88 


7744 


681472 


9.3808 


4.4480 


» 


«g[ 


B4Mfr 




80 


7931 


704999 


9.4840 


4.4647 


» 


^ 


MWJ 


^!4772 


8.1072 


90 ' 


8100 1 


729000 


9.4868 


4.4814 


n 


w 


bSos 


5.5678 


8.1414 ' 


91 


8381 


758571 


9.5394 


4.4879 


s> 


'iSi 


5.6599 • 


8.1748 


99 


8464 


778688 


9.5017 


4.6144 


ft 


iohII 


piBST 


6.7446 < 


8.ao75 


99 


8649 


804857 


9.6487 , 


4.5307 


n 


ySH 


IN804 


5.8610 


B.S896' 
B.8711 


94 


8886 


880584 


9.6954 


4.5480 


•? 


in» 


«af5 


5.9161 


95 


9025 


857875 


9.7466 


44690 


M 


1U0 


44556 


6.0000 


8.3019 


96 


9316 


884796 


9.7980 


4.5780 


n 


Xjg^ 


fi^^ 


6.08eB 


8J822' 


VJ 


0499 


912678 


0.8488 


4.6047 


» 


144ft 


oSSui 


6.1644 


3.8620 


«8 


9604 


941192 


9.8895 


4.0104 


SI 


isn 


6961^ 


6.3469 


S.8S12 


90 


0891 


970299 


. 9.9499 


4jKt61 


40 


1600 


64000 


6.3946 


8.i«a00 


100 


10000 


1000900 


10. 


4.6416 


41 


1Q61 


tSS 


B.4081 


8.4482 


101 


10201 


1080301 


10.0499 


4.6570 


tt 


17M 




8.4160 


192 


10404 


1061206 


10.0095 


4.6728 


48 


1M» 


'TttitfT 


Jgg 


3.5034 


169 


10609 


1002727 


10.1480 


4.0875 


44 


^ 


. m 


3.5903 


104 


10816 


1124864 


10.1980 


4.7037' 


a 


6!T98a 


BJ>5«9 


105 


U025 


1157925 


10.2470 


4.7li77 


« 


2U6 


iSSS 


6.7828 


3.5830 


106 


11236 


1191016 


10.2956 


4]7336 


4r 


a)()^ 


6.885T 


S.9068 


107 


114l9f 


1225048 


10.3441 


4.7475 


48 




11«SM 


6.9989 


8.0842 


106 


11604 


1359712 


10.3933 


4.7023 


II 


^ 


1350W 


7.0000 


3.6663 


109 


]!l881 


1295029 


10.4408 


4.7760 


* 


7.0^11 


8.8840 


110 


12109 


lasiooo 


10.4881 


4.7914 


51 


3Q01 


1183661 


7.1414 


8.7084 


111 


12321 


1867631 


10.5357 


4.8080 


b 


2T04 


140986 


7.aHii 


8.7S25 


112 


12644 


1404928 . 


10..'i880 


4.8988 


tt 


aaoir 


149B77 


7.aB01 


3.7B63 


118 


12789 


1442897 


10.9801 


'4.8B46 


& 


mn 


167464 


7.3485 


3.n98 


114 


12908 


1481544 


10.6771 


4.84B8 


3035 


166875 


7.4162 


3.8030 


115 


13225 


1520875 


10.7238 


4.8639 


M 


3136 


175616 


7.4883 


3.8259 


116 


13456 


1560896 


10.7703 


4.8770 





SM9 


185198 


7.5496 


3.8485 


117 


18689 


1601613 


10.8187 


4.8910 


tt 


8«64 


19511S 


7.6158 


3.8709 


118 


13924 


1643082 


10.8628 


4.9040 


g 


S4S1 
S600 


305979 


7.6811 


S.88S0 


110 


14161 


1686169 


10.9087 


4.9187 


o 


%k9m 


7.7460 


8.0149 


120 


14400 


1728000 


10.9545 


4.9896 



56 



SQUARES, CUBES, AND ROOTS. 



TABUE of Squares, Cabes, Square Boota, aud Cube 
oi^umbers firom 1 to lOOO— (Continued.) 



JTo. 



m 
m 

123 
134 

i» 

IM 
137 
128 
139 
180 

ISl 
1S3 
1S8 
1S4 
IM 

1»6 
18T 

in 

IM 

140 

141 
148 
148 
144 
146 

14« 
14T 
148 
149 
IM 

ISl 
US 
168 
164 
166 

16« 
16T 
168 
16» 
100 

181 
108 
188 
164 
166 

166 
167 
168 
160 
170 

m 

173 
178 
174 
176 

176 
177 
178 
170 
180 

181 
183 
188 
184 
186 



Sqnmre. 



14641 
14864 
15139 
16876 
16636 

16876 
16138 
16884 

16641 
16800 

17161 
17434 
17680 
17866 
18336 

18486 
18710 
19044 
19831 
19600 

19681 
80164 
80440 

30780 
81086 

81816 
31600 
31904 
33301 
38600 

33801 
23104 
23400 
23716 
34026 

84886 
84640 
34864 
86381 
25600 

35931 
36344 
36660 



37226 

37656 
37880 
28324 
28561 
S8900 

892^ 
29684 
39939 
30276 
80626 

80976 
81839 
81664 
82041 
83400 

32761 
33124 
33488 
33856 
84226 



Cube. 



1771561 
1815848 
1860867 
1906624 
1868136 

8000876 

3048388 
3007153 
3146680 
3197000 

3348091 
3399968 

3853687 
3406104 
3460376 

3616466 
3671868 
3638073 
3886619 
8744000 

3808331 



3034907 

3886964 
8048636 

8113186 
8176638 
3341793 
8307949 
8875000 

8442961 
3511806 
8581577 
3663364 

8733875 

8796416 
8869898 
3944813 
4019679 
4096000 

4173381 
4351528 
4830747 
4410944 
4483125 

4674396 
4657463 
4741633 
4836800 
4913000 

6000811 
5068448 
617ni7 
5268034 
5359375 

5461776 
5645233 
6639752 
5736339 
6832000 

5029741 
6028668 
6128487 
6229604 
6831625 



8q. Bt. 



1. 

1.0464 

1.0906 

1.1366 

1.1808 

1.2260 
1.2694 
1.8187 
1.8578 
1.4018 

1.4466 
1.4891 
1.5326 
1.6758 
1.6190 

1.6619 
1.7047 
1.7478 
1.7806 
1.8833 

1.8748 

1.9164 

1.9588 

3. 

3.0416 

3.0680 
3.1344 
3.1666 
3.3066 

3.3474 

3.2883 

3.8388 
3.8688 

3.4007 
3.4499 

3.4900 
3.6300 
2.5696 
2.6006 
3.6491 

2.6866 
2.7279 
3.7671 
2.8062 
3.8452 

2.8841 

2.9238 

2.9616 

8. 

3.0884 

3.0767 
8.1148 
3.1529 
3.1909 
8.2288 

3.2666 
8.3041 
3.3417 
8.3791 
3.4164 

3.4586 
3.4907 
3.5277 
3.5647 
3.6015 



O.Bt. 


KO. 


4.9461 


186 


4.9687 


187 


4.9783 


188 


4.9866 


189 


6. 


190 


6.0188 


191 


6.0365 


193 


6.0897 


198 


6.0638 


194 


6.0658 


196 


6.0788 


196 


6.0016 


197 


6.1045 


196 


6.1172 


199 


6.1399 


300 


6.1436 


301 


6.1551 


303 


6.1676 


808 


6.1801 


304 


6.1936 


306 


6.3048 


306 


6.3171 


307 


6.3398 


306 


6.3416 


309 


6.2636 


310 


6.3666 


311 


6.3776 


313 


6.3886 


318 


6.3015 


814 


6.3183 


316 


6.8351 


316 


6.8868 


317 


6.8486 


318 


6.8601 


319 


601717 


220 


6.8882 


331 


6.8947 


333 


6.4061 


338 


6.4176 


324 


6.4288 


336 


6.4401 


236 


6.4614 


837 


6.4626 


338 


6.4787 


338 


6.4848 


330 


6.4860 


381 


6.5068 


233 


6.5178 


338 


6.52R8 


334 


6.5397 


286 


5.5605 


386 


6.5613 


287 


6.5721 


238 


5.5828 


280 


6.5934 


240 


6.6041 


341 


6.6147 


242 


6.6252 


248 


5.6357 


244 


6.6462 


246 


5.6567 


346 


5.6671 


247 


5.6774 


248 


5.6877 


249 


5.6880 


250 



Sqiure. 



84596 
84869 
86344 
35721 
86100 

86481 
86864 
87249 
87686 

88026 

88416 
88809 

89304 
88601 
40000 

40401 
40804 
41300 
41616 
43036 

43486 

43848 
43364 
43681 
44100 

44621 

45869 

45796 

46226 

46666 

47069 
47524 
47961 
48400 

48841 
49384 
49729 
60176 
60625 

61076 
61629 
61984 
63441 
63800 

68861 
68824 
64289 
64756 
65225 

65696 
56169 
56644 
57121 
67600 

bdOei 
68564 
69049 
69686 
60025 

60516 
61009 
61604 
62001 
62500 



Cube. 



6434856 
6539308 
6644672 
6751269 
6659000 

6867871 
7077888 
7189067 
7301884 
7414875 

7629586 
7646873 
7762892 
7880599 
8000000 

8120601 

8842408 
8366427 
8489664 
8615136 

8741816 
8869743 
8998913 
9129829 
9961000 

9893931 
9628138 
9663597 
9800844 
9938376 

0077696 
0318318 
0360333 
0503459 
0648000 

0793861 
0941048 
1089567 
1339434 
1890625 

1543176 
1697083 
1852852 
3008969 
2167000 

2826391 
2487168 
2649887 
2813904 
3877875 

8144266 
3313053 
3481272 
16651919 
8824000 

3997521 
4172488 
4348907 
4526784 
4706125 

4886936 
5069223 
5252992 
5438248 
5625000 



Bq. Ht. 



13.6383 
13.6748 
13.7113 
13.7477 
18.7840 

18.8308 
18.8564 
18.8934 
13.9284 
13.9643 

14. 

14.0067 

14.0713 

14.1067 

14.1431 

14.1774 
14.3137 
14.3478 
14.3898 
14.8178 

14.8637 
14.3876 
14.4333 
14.4568 
14.4914 

14.6258 
14.5603 
14.5046 
14.6387 
14.6628 

14.6069 
14.7800 
14.7648 
14.7866 
14.8834 

U.8661 
14.8887 
14.9832 
14.9666 
16. 

15.0333 
16.0665 
16.0907 
16.1327 
16.1668 

16.1987 
15.2316 
16.2643 
16.2971 
16.8297 

16.3638 
16.3948 
16.4272 
16.4686 
16.4819 

16.5243 
15.5663 
15.5885 
16.6306 
15.6626 

15.6844 
15.7162 
15.7480 
16.7797 
15.8114 



as*^ 



6.7088 
6.7186 
6.7287 
6.738a 
6.7480 

6.760(> 
6.7680 
6.7790 
6.789a 
6.7989 

6.8e8» 
6.8189 
6.838S 
6.8808 

6.8486 

6.8678 
6.8676 

6.8771 

6.8ei» 

6.8864 

6.9089 
6.9156 
6.9360 
6.9846 
6.9489 

6.9689 
6.S6ST 
6.9731 
6.9614 
6.990T 

6. 

6.0008 

6.0186 

6.037T 

•.0869 

6.04i» 
6.066« 
6.0641 
6.0782 
6.0633 

6.0912 
6.1002 
6.1091 
6.1180 
6.1369 

6.1368 
6.1440 
6.1634 
6.1633 
6.1710 

6.1797 
6.1885 
6.1972 
6.3068 
6.2145 

6.2281 
6.2817 
6.3408 
6.3488 
6.367» 

6.3668 
8.2743 
6.868» 
6.3912 
6.2996 



SQUABES, CUBES, AND BOOTS. 



67 



TABliE of Sqimres, Onbes, Square Boots, and €al»e Roots* 
of A ambers troBa. 1 to 1000->(OoNTiNin£D.) 



No. 


BqxiAra. 


Cabe. 


Sq. Bt. 


C. Bt. 


No. 


Square. 


Cube. 


Sq. Bt. 


CBt. 


S61 


68001 


15818251 


15.8480 


6.3060 


316 


99856 


31554496 


17.7764 


6.8113 


3tU 


68904 


16003006 


15.8745 


6.3164 


817 


100489 


31R55018 


17.8045 


6.8185 


253 


«4UW 


16194277 


15.9060 


6.3847 


818 


101124 


32157432 


17.8886 


6.8256 


S64 


64516 


16387064 


15.9374 


&3880 


819 


101761 


32461759 


17.8606 


6.8328 


355 


65035 


16581875 


15.9687 


6.3413 


830 


102400 


32768000 


17.8886 


6.8899 


S66 


665S6 


16777216 


16. 


6.3486 


821 


103041 


83076161 


17.9165 


6.8470 


957 


06048 


16974598 


16.0313 


6.3579 


822 


103664 


33386248 


17.9444 


6.8541 


SS8 


66564 


17178512 


16.0624 


6.8661 


828 


104829 


33698267 


17.9722 


6.8612 


»e 


67061 


17378879 


16.0885 


6.3748 


824 


. 104976 


34013824 


18. 


6.8683 


160 


67600 


17576000 


16.1245 


6.8835 


825 


105625 


34328185 


18.0278 


6.8763 


381 


68121 


17779561 


16.1555 


6.3907 


826 


106276 


84645976 


18.0555 


6.8824 


363 


6R644 


17984728 


16.1864 


6.3968 


827 


106929 


34965783 


18.0831 


6.8884 


968 


68169 


18191447 


16.2178 


6.4070 


828 


107584 


35287562 


18.1108 


6.8964 


964 


OTUlfD 


18390744 


16.2461 


6.4151 


829 


106241 


35611380 


18.1884 


6.9684 


966 


70325 


18608625 


16.2788 


6.4282 


880 


108900 


35987000 


18.1659 


6.91M 


966 


70766 


18821096 


16.3085 


6.4312 


881 


109561 


36264691 


18.1934 


6.9174 


967 


712H0 


19084163 


16.3401 


6.4898 


832 


110224 


36594868 


18.2209 


6.9944 


968 


71824 


19248882 


16.3707 


6.4478 


888 


110889 


36926087 


18.2483 


6.9813 


960 


72361 


19465109 


16.4012 


6.4558 


884 


111556 


37259704 


18.2757 


6.9382 


970 


72900 


19688000 


16.4317 


6.4688 


885 


112225 


37695375 


18.3080 


6.9461 


871 


73441 


19902511 


16.4621 


6.4718 


886 


112886 


37938056 


18.8308 


6.9531 


973 


73964 


20129648 


16.4824 


6.4792 


887 


113669 


88272758 


18.3576 


6.958» 


973 


74528 


20346417 


16.5227 


6.4872 


888 


114244 


38614472 


18.3848 


6»9668 


974 


76076 


20570824 


16.5529 


6.4851 


888 


114921 


38958219 


18.4120 


6.9727 


375 


75625 


20796875 


16.5881 


6.5080 


840 


115600 


89304000 


18.4391 


6.9796 


976 


76176 


21024576 


16.6132 


6.5106 


841 


116881 


39651821 


18.4662 


6.9664 


977 


76729 


21258883 


16.6488 


6.5187 


842 


116864 


40001686 


18.4832 


6.9088 


378 


77384 


21484852 


16.6788 


6.5265 


843 


. 117649 


40858607 


18.5203 


7. 


379 


77841 


21717688 


16.7063 


6.5843 


844 


118886 


40707584 


18.5472 


7.0068 


360 


78400 


21952000 


16.7832 


6.5421 


345 


119035 


41068635 


18.5742 


7.0136 


981 


78861 


22188041 


16.7681 


6.5499 


846 


119716 


41421786 


18.6011 


7.0206 


383 


79524 


22435768 


16.7829 


6.5577 


847 


120409 


41781928 


18.6279 


7.0271 


3RS 


80689 


22665187 


16.8226 


6.5654 


848 


121104 


4214492 


18.6648 


7.0838 


984 


80656 


22906804 


16.8528 


6.5731 


849 


121801 


42508549 


18.6815 


7.0406 


885 


81285 


28146135 


16.8819 


6.5806 


850 


122500 


42875000 


18.7088 


7.0478 


3R6 


81796 


23398666 


16.9116 


6.9865 


851 


123801 


43248651 


18.7350 


7.0540 


987 


82369 


23688908 


16.9411 


6.5962 


862 


123904 


43614206 


18.7617 


7.0607 


«8 


82944 


23887872 


16.9706 


6.6089 


858 


124609 


48966977 


18.7888 


7.0674 


968 


83521 


24187568 


17. 


6.6115 


854 


125316 


44361864 


18.8149 


7.0740 


»0 


saoo 


24S8800O 


17.0394 


6.0191 


855 


126025 


44788R75 


18.8414 


7.0807 


sn 


84681 


24643171 


17.0587 


6.6267 


856 


126736 


45118016 


18.8680 


7.0878 


983 


85264 


24897068 


17.0680 


6.6348 


857 


127449 


45499298 


18.8944 


7.0940 


386 


85848 


25158757 


17.1172 


6.6419 


858 


128164 


45882712 


18.9209 


7.1006 


984 


86488 


25412184 


17.1464 


6.6494 


859 


128881 


46268379 


18.9473 


7.1072 


985 


87025 


25673875 


17.1756 


6.6569 


860 


129600 


46656000 


18.9737 


7.1138 


986 


87616 


25884886 


17.2047 


6.6644 


861 


130821 


4704S881 


19. 


7.1804 


987 


88209 


36198078 


17.2887 


6.6719 


862 


131044 


47437828 


19.0263 


7.1260 


986 


88804 


36468582 


17.2627 


6.6794 


863 


131760 


47882147 


19.0526 


7.1386 


988 


88401 


26780699 


17.2916 


6.6869 


864 


132496 


48228544 


19.0788 


7.1400 


800 


90000 


27000800 


17.3305 


6.6948 


865 


133225 


48627125 


19.1050 


7.1466 


801 


90601 


27276801 


17.8484 


6.7018 


866 


. 183956 


48027896 


19.1811 


7.1531 


803 


91304 


27548806 


17.3781 


6.7092 


867 


134689 


49439863 


19.1572 


7.1586 


806 


91809 


27818137 


17.4068 


6.7166 


868 


135424 


49836082 


19.1888 


7.1661 


804 


93416 


28084M4 


17.4856 


6.7240 


369 


136161 


50248409 


19.2094 


7.1726 


806 


83025 


28373635 


17.4643 


6.7818 


870 


136900 


50658000 


19.2354 


7.1791 


806 


9S686 


38659816 


17.4929 


6.7887 


871 


137641 


51064811 


19.2614 


7.1866 


«rr 


94348 


28884448 


17.5214 


6.7460 


872 


188884 


51478848 


19.2873 


7.1920 


806 


94864 


29316112 


17.5499 


6.7588 


878 


188129 


51895117 


19.3132 


7.1984 


808 


85481 


apfiowM 


17.5784 


6.7606 


874 


189676 


53818624 


19.3391 


7.2048 


AO 


86100 


29791000 


17.6068 


6.7678 


875 


140635 


52784875 


19.3649 


7.2112 


811 


86731 


80080881 


17.6868 


6.ni2 


876 


141876 


58157876 


19.8907 


7.2177 


813 


87844 


808n828 


17.6685 


6.7834 


877 


142129 


53583688 


19.4165 


7.2240 


818 


87888 


80684987 


17.6818 


6.7887 


878 


142884 


54010152 


19.4422 


7.2804 


814 


86686 


80860144 


17.7300 


6.7968 


879 


143641 


54439969 


19.4679 


7.2368 


816 


96996 


81366676 


17.T48S 


6.8041 


880 


144400 


5487360(r 


19.4986 


7J4S2 



58 



SQUARES, •CUBES^ Ain> BOOTS. 



TABIiE off SqiiaveSy Cubes, flqvave Boots, oad Cube 
of srambem Drom 1 to 10O<^^*<OMrenriTXD.) 



STo. 



S81 
M3 
188 
884 
886 



887 



•qiuura. 



itfin 

140M 

147456 
148»5 



S»7 
806 



401 
408 
408 
404 
405 



40T 
408 
400 
•410 

411 
418 

418 
414 
4U 

43M 
41T 
419 
4It 



411 
41t 

4» 
484 

435 

496 
487 
488 
498 
480 

■ai 

4S2 

488 
434 
436 

486 
487 
488 
480 
440 

441 
448 
'448 
444 
446 



140768 
150644 
161S21 
158100 

153881 
15S664 

154440 
155386 
156035 

156816 
157600 
158404 
150301 
160000 

160601 
161604 
163400 
168316 
164036 

164886 

166640 
166464 
167881 
168100 

168031 
160744 
170660 
171886 
172335 

173066 
178889 
174734 
175661 
176400 

in341 
178064 
178038 
179776 
180686 

181476 
183880 
18S184 
184041 
184900 

185761 
186624 
187480 
188S66 

189386 

190096 
190060 
191844 
192721 
193600 

194481 
195804 
190349 
1971S0 
196026 



Gab«. 



8q. Bt. 



65306841 
65748068 
66181887 
66623104 
67066616 

67513«< 
67960608 
68411073 
68868860 
69318000 

69770471 
60236388 
60696467 
61169804 
61639075 

63099186 
62570778 
63044793 
63531199 
64000000 

64481901 
64864800 
65460087 
65880864 
66480196 

06038410 
674101tt 1 
67917818 
68417929 
68891000 

09436681 
68034638 
70444007 
7096V0a 
71478875 

71991386 
72511718 
73034603 
73660660 
74088000 

74610101 
75161440 
75686007 
76338094 
76766096 

7780en6 
77854488 
78403763 
78058680 
79507000 

80003001 
80631566 

81183T87 
81746604 
83319876 

82881866 
83458458 

84027673 
84604619 
85184000 

85766131 
86350888 
869S8807 
87528884 
\ 881211<t5 



19.5192 
19.5448 
19.5704 
19.6950 
19.6314 

19.6468 
19.6738 
19.6977 
19.7381 
19.7484 

19.7787 
19.7990 
19.8343 
19.8494 
19.8746 

19.8887 
18.9349 
10.9499 
19.8760 
80. 

30.0860 
30.0480 
30.0740 
90.0908 
a0.18«6 

30.1494 
30.1743 
30.1990 
20.2387 
90.3485 

90.2731 
90.9078 
90.3834 
90.8470 
90.8716 

20.8881 
30.4806 
20.4460 
20.4096 
90.4088 

90.5183 
30.5496 
20.5670 
90.5918 
90.6166 

20.6396 
20.6640 
20.6882 
20.7138 
20.7864 

20.7606 
20.7846 
20.8087 
20.8387 
20.8607 

20.8806 
20.9046 
20.9884 
20.9638 
20.9708 

21. 

21.0838 

21.0476 

21.0713 

21.0050 



cut. 



7.2495 
7.2558 
7.2028 
7.2086 
7.2748 

7.2811 

7.2874 
7.J 
7.3 
7.8061 

7.3134 
7.8186 
7.8946 
7JS10 
TJ87B 

7.3484 
7.3490 
7.8660 

7.3610 
7.8681 

7.8743 

7.1 

7.J 

7. 

7. 

7.4047 
7.4100 
7.4100 
7.4299 
7.4990 

7.4860 

7.4410 

7.4470 

1.4 

1.4 



K«. 



7.4660 
7.4710 
7.47T0 
7.4899 
1.4 



T.4048 
7.5007 
7.5067 
7.5190 
7.6186 

7.5944 
7.5800 
7.58a 
7.5410 

7.5478 

7.5587 
7.5585 
7.5664 
7.6711 
7.5770 

7.5898 
7.5886 
7.5044 
7.6001 
7.6060 

7.6117 
7.6174 
7.6Sa2 
7.6289 
7.6846 



446 
447 
448 



460 

451 
469 
468 

454 

466 

466 

457 
468 
460 
460 

401 
463 
468 
484 
486 



SqtuuM. 



467 
468 
460 

470 

471 
473 
473 
474 
475 

476 
477 
478 
479 
480 

481 
483 
488 

484 
485 

486 
487 
488 
480 
480 

491 
493 
483 
494 
485 

496 
tf7 
498 
499 
fiOO 

601 
50S 
SOS 
804 
606 

6DB 

607 
SOB 
609 
610 






196916 
190800 
300704 
301601 

tM500 

S0B4O1 
304904 
3a30O 

306110 

307085 

307866 
308840 

300764 
310681 
311600 

313531 
318444 
314368 
315296 
316396 

117166 
318080 
310034 
310061 
330900 

221841 
222784 
223720 
224676 
235636 

nssfm 

237590 
928484 
2204^ 
280400 

231361 
232304 
233280 
234256 
235235 

236106 
237109 
238144 
239121 
240100 

241081 
242064 
249040 
244036 
245035 

346016 
347000 
348004 
249001 
260000 

261001 
252004 
253600 
254016 
255026 

256086 
257040 
258064 
259081 
860100 



Gubo. 



Bq.Bt. 



88716586 
89614688 
88015803 
90516649 
911360004 

91788861 
93845406 
93968077 
9S57W64 
94186675 

94816616 
95448088 

9a(moi8 

96703570 



21.1107 
81.1404 
21.1080 
81.1880 
11.2183 



97071181 
98611118 
99361847 
9988T844 
100644096 

101194880 

103608883 
103161760 
10380000 

104487111 
105164048 
105818817 
106486434 
107in676 

107850176 
106581888 

108315861 
108001180 
llOSOMOO 

111384041 
1I1980168 
113078567 
113370804 
114OOC10 

114701360 
115601808 
116214372 
116880160 
117040800 

iioswrri 

119006488 

iioeiffcr 

120651104 
121381tf» 

122098806 
123708478 
123506082 
124211408 
125000000 

123761601 
126506089 
127288617 
128014804 
128787086 

128564116 
13QS13M8 
131006618 

isisfrsaso 

133651000 



CL&U 



11. 

11. 

31.! 

91.3078 

81.8987 

31.3543 
21.3776 
21.4000 
21.4248 
31.4476 

11.4709 
11.4043 
31.5174 
31.5407 
31.5680 

11.5890 
21.6103 
31.6888 
21.6664 
21.6705 

31.7025 
31.7366 
31.7486 
31.7n6 
31.7045 

81.8174 
31.8408 
31.8681 
21.8061 
11.0000 

11.9317 

11.9545 

31.9778 

32. 

33.0337 

32.0454 
32.0661 
83.0907 
22.1188 
22.1860 

32.1585 
32.1811 
22.3080 
32.2381 
12.3486 

22.2711 
23.3085 
33.3160 
32.8883 
32.3607 

23.3880 
22.4054 
23.4877 
22.4480 
28.4732 



23.4844 4 
22.5181 < 
22.5680 ' 
23.5610 
28.6881^ 



7.6400 
7.0480 
7.6517 

7.6574 
7j 



7. 

7.6744 

7.6801 

7.6867 

7.6014 

7.6870 

7.74 

7.71 

7.718B 

7.7U6 

7.7860 
7.7800 
7.7801 
7.7410 
7.747t 

7.7610 
7.7604 
7.7030 
7.708t 
7.7750 

7.780i 
7.7800 
7.7015 
7.7070 
7.8005 

7.80IO 
7.8184 
7.8180 

7.8140 
7. 



7. 
7.8100 
7.8400 
7.8U4 

7.J 



7.f 

7.80M 

7.8110 

7.8784 

7.8 



7.8801 

l.t 

1.1 

7.0051 

7.M06 

7.0108 
7.0811 

7.0184 
7.90lt 

7.9870 



i: 

7.9tTi 
l.i 
1.1 
7.S 



l.i 

7.9700 

7.9701 

7.1 

1.1 



BqUAK£8» CUBES, AKD BOOX8. 



5ft 






HI 
•18 



•18 



«n 

618 
St8 



Mi 



8ifr 



584 



•8T 



Bfl 



i4« 

Mi 



««r 






C«8 



6tT 

» 

WO 

m 

IT8 



Sqoas*. 


Oab*. 


Sq. Bk 


CBt. 


K«. 


961111 
983144 
388168 
364188 


1SS4S38S1 
184317198 
185006687 
M6788744 
186B86B26 


83.6068 

82.6374 
33.6485 
83.6710 
33.6660 


7.9048 

8. 

&0062 

&01O4 

8.0166 


670 
677 
678 
670 
600 


387388 
36BS84 

3m4B8 


1S78B6B88 
138168fi8 

isaoHoao 

13910686* 
140686888 


83.7160 
32.7870 
32.7908 
n.7810 
33.6006 


8.0060 
8.QBtl 
8.0008 
8.0416 


6»3 

6tt4 
666 


sn4a 

373184 
318S88 
374898 
375895 


1414MI61 
143g664B 

1438TSaM 
144761185 


33.8354 
32.8418 
32.8003 
33.8810 
22.9130 


8.04(86 
8.0617 
8.0609 
8.00aD 
8.0671 


688 

667 
680 
600 
600 


3166n 

37770 
378784 

378641 


1456SIBK 
146860188 

147181668 
148085668 
148811088 


22.9015 
23.0780 
33. 
33.0817 


8.0723 
8.0774 
8.6036 
8.0876 
8.0037 


001 

608 
608 
604 
605 


iiil! 


148mS81 

15050aM8 
16141M87 
163818684 
168188875 


SS.0404 
23.0651 
23.00n 
23.1004 
3S.I801 


8.6878 
8.1038 
8.1019 
8.1180 
8.U80 


686 
607 
100 
660 
000 


387n8 
388868 
388444 
380131 
381680 


ISItWHB 
164854158 
155780018 
166688018 

167461680 


ss.iosr 

23.1738 
23.1048 
23.2M4 
2S.281t 


8.1331 
8.1381 
8.U32 
8.M82 
8.1683 


001 
003 
008 
004 
085 


383681 
98784 
384648 
386886 
387895- 


168840481 

15|66066B 
lflM06688 
166668184 
161M60a5 


23.3804 
28.3010 
28.S0U 
28.3388 
28.8458 


8.1083 
8.1533 
8.1568 
8.U8S 
8.1)603 


000 
007 
006 
000 
•10 


386118 
388988 
800684 
801401 
809500 


103191006 

10808038 
164806068 
1666681188 
166891000 


28.8666 
28.3688 
23.4004 
28.4807 
2S.4ft8t 


8.1783 
8.1783 
8.1883 
8.1883 
6.1983 


Oil 
013 
018 
014 
015 


808661 

904704 
806608 
800818 
806686 


10f80tl51 

168080868 
16BUlBn 
170681464 
170666835 


28.48U 
28.4061 
28.6100 
23.6888 
23.5604 


8.8862 
8.3881 
8.8861 
8.8180 
8.0180 


OM 

017 
010 

a.* 


808186 
810848 
811864 
813481 
818680 


171878818 

1787I1UI 
1740M8IO 


28.6709 
28.0808 
23.0080 
28.8888 
28.0840 


8.3839 
8.8378 
8.3837 
8.3377 
8.3436 


031 

688 
038 
034 
OK 


814181 

816844 
816888 
818086 
818935 


178506681 
171004838 
178U0649 
17MI0lt4 
180069186 


2S.6854 
28.7066 
28.7376 
28.7487 
28.7607 


6.9475 
6.3684 
6.3673 
6.8681 
8.«70 


686 

687 
638 
680 
680 


890658 

831488 
823634 
838701 
894800 


181831406 
183984868 
188880483 
1841890000 
1861800QO 


2B.780O 
28.8110 
2S.8080 
23.8687 
2B.87«r 


8.8719 

8.W66 
84il6 
8.0106 
84^3 


mt 

088 

884 
036 


ilii! 


186100411 

187140948 

laouasu 

188110894 
ll8M60875 


23.8066 
33.0166 
33.8074 
33.9688 
88.8798 


8.89B2 

6.aoao 

6JW59 
8.8107 
8.U66 


080 
087 
688 

089 
040 



3S1770 
333900 
884004 

886341 



888714 



34238( 

843300 
S446a 

346744 
3488101 
S4810O 

840981 
860404 
85104O 
862880 
864035 

866310 
860400 

S6700i 
868001 




870681 
8T3M» 

873331 
874644 
876700 
876600 
8783)16 

879450 
880680 
381984 

883101 
884400 




8918X0 
8081^ 
884884 
385041 
886000 

S86101 
309444 
400660 
4OUIS1O 
403836 

404480 
406700 
407044 
400881 
409600 




19U0887O 
1931000SS 
193100663 
10a04680 
196118000 




311708786 
8137U178 
218849|M 
2140U7W 
210000080 







237U066O 

338188000 



389488001 
240|B1S8 
241004107 
2430^0^4 
244Um 



247Mp 
260O430QO 




267360466 
266474868 
2&80940;3 
200017110 

sraSior 



24. 

34.0306 
34.0410 
34.0034 
34.0688 

84.1060 
34.1847 
34.104 
34.1081 
34.1868 




94.SM5 
34.88U 
34.S5U 
34.Sm 

34.4181 
34.4836 
24.4640 
24.47I& 
24.4040 




34.0171 

34!8n 
34.0t70 
34.0068 




C. 



34.8tt6 
S4.8m 
34.8000 



35.2190 
25.2380 
35.3687 
35.3384 
36.3M8 



8.3208 

8.sasi 

8.3900 

8.SS48 
0.S386 

0.8448 
8.3401 
8.3^ 
8.3607 
8.8084 



8.4 

8.: 



m 

8.4104 

tt& 

8.i 
8. 
8.^ 
8. 
8.4 

8.4 
8.4 
8.4 
8.4 
0.^ 

8.4688 

1:1 

0.400 
ImIO 

8.1 

ei 
8.r 
8.1 
8-H?o 

6.6816 
8.BV3 
8J 
8-i 

oj 

8.C 
8J 
8^ 
8.1 
8.1 

84|773 
eSts 

6.6807 
6.6048 
8.00B8 



60 



SQUARES, CUBES, AND BOOTS. 



TABI4E ofBonarea, Cabes, Sqnar« Root*, and Cube 
of iVambers from 1 to lOOO — (CoimiruBD.) 



No. 


SQnmre. 


Cube. 


Bq. Bt. 


O.Bt. 


No. 


Square. 


Cube. 


Bq.&t. 


CBU 


Ml 


410881 


363374721 


25.3180 


8.6233 


706 


498436 


861886816 


86.5707 


8.90tt 


643 


412164 


264609388 


35.3377 


6.6267 


707 


498848 


353996343 


86.6896 


8.9686 


643 


413449 


265847707 


25.3574 


8.6818 


708 


601364 


854894818 


96.6068 


8.9187 


644 


414TS6 


267089984 


25.3773 


8.6867 


709 


602681 


856400838 


66.6371 


8.9168 


646 


416026 


368336135 


25.3969 


8.6401 


710 


604100 


867911000 


96.6468 


8.99U 


646 


417316 


269586136 


35.4165 


8.6446 


711 


606681 


859436481 


96.6646 


8.996S 


647 


418608 


270840033 


35.4363 


8.6480 


713 


606844 


860844138 


86.6889 


8.9386 


648 


419904 


372097793 


85.4558 


8.66S6 


m 


608969 


362467097 


86.7081 


8.9897 


649 


431201 


273359449 


86.4766 


8.6679 


714 


6097W 


863884844 


86i7a06 


8.9878 


6&0 


432500 


274625000 


85.4961 


8.6684 


716 


611336 


366686876 


96.7996 


8.8498 


661 


483801 


375894451 


85.5147 


8.6668 


716 


61366« 


967061686 


96.7689 


8.8489 


652 


48S104 


377167808 


35.5343 


8.6718 


717 


614080 


368601813 


36.7769 


8.9609 


65S 


426409 


378445077 


25.5539 


8.6767 


718 


615584 


370146888 


36.7966 


8.9646 


«4 


427716 


279736364 


35.5734 


8.6801 


719 


616961 


371684869 


36.8148 


8.9687 


•66 


439025 


381011375 


25.5990 . 


8.6846 


730 


618400 


973348000 


36.8838 


8.9899 


666 


430336 


383900416 


35.6135 


8.6890 


721 


519641 


374806961 


36.8514 


8.9nt 


<6T 


431649 


283593893 


35.6330 


8.6834 


733 


631884 


376967048 


36.8701 


8.8711 


668 


432964 


284890313 


35.6515 


8.6878 


733 


633789 


377999067 


36.8887 


8.876? 


66» 


434281 


286191179 


36.6710 


8.7033 


724 


634176 


379609434 


36.9073 


8.8T84 


660 


436600 


387496000 


86.6906 


8.7066 


736 


635686 


981078136 


36.9868 


8.8696 


661 


436931 


388804781 


35.7099 


8.7110 


786 


637076 


883667176 


^m*w%Aw 


8.987« 


662 


438244 


390117538 


35.7394 


8.7164 


737 


638639 


884840689 


26.9689 


8.981S 


«3 


439569 


391494347 


85.7488 


8.7196 


738 


689984 


386888963 


36.9816 


8.986» 


664 


440896 


393754944 


25.7683 


8.7341 


739 


631441 


387480489 


37. 


8. 


665 


443326 


394079636 


85.7876 


8.7886 


730 


633800 


389017000 


37.0186 


8.00a 


666 


4436S6 


395406396 


85.8070 


8.7389 


731 


6S496I 


890617891 


37.0970 


8.6089 


667 


444889 


396740968 


85.8363 


8.7373 


733 


636884 


893888168 


37.0666 


8.0199 


668 


446324 


298077633 


25.8457 


8.7416 


733 


637888 


393833897 


37.0740 


8.01«4 


668 


447661 


399418309 


25.8650 


8.7460 


784 


638766 


395448804 


37.0884 


8.8906 


mo 


448900 


600763000 


36.8844 


8.7606 


736 


640886 


397066976 


87.1108 


8.6a4S 


671 


460241 


303111711 


36.9037 


8.7647 


736 


641686 


99e688SS« 
400816669 


87.1896 


8.09W 


673 


461684 


303464448 


26.9830 


8.7690 


787 


6491«8 


87.1477 


8.689B 


67S 


463939 


304831317 


35.9433 


8.7634 


738 


644644 


401947878 


87.1668 


8.Q868 


674 


464376 


306183034 


35.9616 


8.7677 


739 


646181 


409689418 


87.1846 


9.0410 


676 


465626 


30764687S 


86.9806 


8.7791 


740 


647600 


406884000 


37.9088 


9.04SO 


676 


466876 


308915776 


26. 


8.7764 


741 


649081 


406868081 


87.8819 


9.04n 


677 


468339 


310288733 


26.0198 


8.7807 


743 


650664 


4086IS488 


87.3997 


9.0699 


678 


469684 


311665752 


36.0384 


8.7860 


743 


668048 


410179407 


87.8680 


8.06T9 


679 


461041 


313046839 


36.0676 


8.789S 


744 


663696 


411890784 


87.8764 


9.90M 


680 


463400 


314433000 


36.0768 


8.7987 


746 


666086 


418486696 


37.8847 


8.06M 


661 


463761 


815831341 


36.0960 


8.7960 


746 


666616 


416160886 


37.9190 


8.0604 


663 


465134 


317314568 


26.1151 


8.8083 


747 


668008 


416689789 


37.9913 


8.0796 


683 


466489 


318611967 


36.1343 


8.8066 


748 


669604 


418608893 


37.3486 


8.0n6 


684 


467856 


330013504 


36.1534 


8.8109 


749 


(^1001 
563600 


420189748 


37.8679 


8.06U 


686 


469335 


331419136 


26.1735 


8.8163 


750 


421876000 


37.9861 


8.QM6 


686 


470596 


333818866 


36.1916 


8.8194 


751 


564001 


488664761 


37.4044 


8.068S 


687 


471969 


S3434370S 


36.3107 


8.8887 


763 


665604 


ITSXMK 


87.4886 


8.0987 


688 


473344 


336660673 


36.3896 


8.8380 


753 


667008 


430967777 


37.4406 


8.0877 


689 


474721 


337083769 


26.3488 


8.8S2i 


764 


668616 


428661664 


37.4691 


8.1017 


600 


476100 


338609000 


86.3679 


8.8866 


766 


670036 


430968876 


37.4773 


8.1067 


691 


477481 


339939S71 


36.3869 


8.8406 


756 


671686 


432061816 


37.4966 


8.1086 


692 


478864 


SS1373888 


36.3069 


8.8461 


757 


573048 


433796098 


37.5136 


8.118ft 


698 


480249 


333813657 


86.3349 


8.8483 


758 


574664 


436619619 


37.6318 


8.117ft 


694 


481636 


334365384 


26.3439 


8.8686 


758 


676081 


43784647* 


37.6600 


8.191ft 


•96 


483035 


S36703S75 


26.3639 


8.8678 


760 


577600 


438976000 


37.6681 


8.196ft 


696 


484416 


SS7158586 


26.3818 


8.8621 


761 


579181 


440711061 


37.5868 


8.138ft 


•97 


4A5809 


338608873 


26.4008 


8.8663 


763 


580644 


449460788 


37.6043 


8.199ft 


•98 


4B7204 


340068392 


88.4197 


8.8706 


768 


583168 


444194847 


37.6326 


8.197S 


•99 


488601 


841533099 


26.4886 


8.8748 


764 


688686 


446948744 


27.6405 


8.141ft 


700 


490000 


848000000 


26.4575 


8.8790 


766 


685885 


447697136 


37.6686 


8.146ft 


701 


491401 


344472101 


26.4764 


8.8883 


766 


686756 


448466006 


37.6767 


9.1488 


702 


492804 


346948408 


26.4963 


8.8875 


767 


688989 


461817668 


37.6848 


8.16ST 


70S 


494309 


347438837 


26.5141 


8.8817 


768 


689634 


468864888 


37.7188 


8.1677 


T04 


495616 


348913664 


36.5380 


8.8868 


769 


681361 


4547&6B08 


37.7806 


8.1ttT 


t06 


497026 


850403635 


36.6618 


8.8001 


770 


683800 


466539000 


37.7489 


8.1861 



SQUARES, CUBES, AKD BOOTS. 



61 



TABUB of Sqiiares, Cubes, S4|nare Roots, and Cabe Roots, 
of Nnmbers from 1 to lOOO— (Continued./ 



No. 



807 



810 

811 
812 
818 
814 
815 

810 
817 
818 
810 
830 

821 



814 
836 

888 
827 
838 



880 

881 
888 



884 

fl86 



Square. 



50M41 
S9S884 
&075W 
599076 
600625 

602176 
603720 
606284 
606841 
608400 

609861 
611534 
613060 
614666 
616225 

617796 
619369 
620944 
622521 
624100 

625681 
627264 
628848 
630436 
632025 

688616 
635209 
636804 
638401 
640000 

641601 
643204 
644809 
646416 
648035 

648636 
651248 
662864 
664481 
6S6100 

657731 
668344 



662586 
664235 

665856 

667480 
669134 
670761 
672400 

674041 
676684 
677339 
678976 
680625 

682276 



685584 
687241 
688800 

600561 
602234 



MUUUM 
OMKKIO 

007225 



Onbe. 



458314011 
460090648 
461889917 
463684824 
465484375 

467388576 
469007433 
470010962 
473730139 
474653000 

476870541 
478311768 
480048687 
481800304 
483736635 

485687666 

487443403 
488803873 
491168060 
493039000 

494913671 
496798088 
4086n257 
600606184 
602459675 

604858336 
506361573 
608160603 
510063399 
512000000 

613822401 
515849608 
617781637 
619718464 
521660135 

523606616 
535667948 
637514113 
529476130 
681441000 

533411731 
635387838 
637367797 
538353144 
541343375 

643338486 
545338513 
647343433 
649358259 
661368000 

653387661 
666412248 
657441767 
550476224 
561616625 

563559076 
565600388 
567663552 
568723780 
571787000 

573866191 
675930868 
678009537 
580093704 
683183875 



8q. Bt. 



27.7660 
27.7848 
27.8029 
27.8300 
27.8388 

27.8568 
27.8747 
27.8927 
27.9106 
27.9285 

27.9464 

27.9643 

27.9831 

28. 

38.0179 

28.0367 
28.0535 
28.0713 
28.0691 
28.1069 

28.1847 
28.1425 
28.1608 
28.1780 
28.1957 

38.2185 
28.2312 
38.2488 
28.2666 
28.2848 

28.8019 
28.3106 
38.3378 
38.3540 
28.3735 

38.3901 
28.4077 
28.4353 
38.4439 
38.4606 

38.4781 
28.4956 
38.5182 
38.5307 
38.5482 

88.5667 
38.5832 
28.6007 
38.6183 
28.6366 

38.6581 
38.6705 
88.6880 

38.7054 
38.7338 

38.7402 

28.7576 
28.7760 
28.7924 
28.8097 

28.8271 
28.8444 
28.8617 
38.8791 
38.8864 



C. Bt. 


No. 


9.1696 


886 


9.1736 


837 


9.1775 


838 


9.1815 


839 


9.1855 


840 


9.1894 


841 


9.1933 


843 


9.1973 


843 


9.2013 


844 


9.3063 


845 


8.2091 


846 


9.3130 


847 


9.3170 


848 


9.3300 


849 


0.3348 


850 


9.3287 


851 


9.2326 


863 


9.2365 


853 


9.2404 


854 


9.2443 


855 


9.2482 


856 


8.2521 


867 


9.2560 


858 


9.3599 


850 


9.3638 


860 


9.2677 


861 


9.27ie 


862 


9.2754 


863 


9.2793 


864 


9.3832 


865 


9.3870 


866 


9.3900 


867 


9.3948 


868 


9.2986 


868 


9.3025 


870 


0.3063 


871 


9.3102 


872 


9.3140 


873 


9.3179 


874 


9.8217 


875 


9.3355 


876 


9.3394 


877 


9.8332 


878 


9.3370 


879 


9.3406 


880 


9.3447 


881 


9.8486 


882 


9.3533 


883 


9.8561 


884 


9.8599 


885 


9.8637 


886 


9.3675 


887 


9.3713 


888 


9.3751 


889 


9.8789 


890 


9.3827 


891 


9.3865 


893 


9.S902 


893 


9.3940 


894 


9.3978 


895 


9.4016 


896 


9.4053 


897 


9.4091 


898 


8.4129 


899 


9.4166 


900 



Square. 



700569 
702344 
703921 
705600 

707381 
706964 
710649 
.712336 
714025 

716716 
717409 
719104 
730801 
732500 

724201 
735904 
737609 
739316 
731025 

782736 
734449 
736164 
737881 
738600 

741321 
743044 
744760 
746486 
748225 

748966 
751689 
758434 
755161 
756900 

758641 
760384 
762129 
763876 
766625 

767376 
769139 
770884 
772641 
774400 

776161 
777924 
779689 
781456 
783225 

784996 
786760 
788544 
790S2I 
792100 

793881 
796664 
797449 
799236 
801025 

802816 
804609 
806404 
808201 
810000 



Cube. 



8q. Bt. 



584277056 
586376258 
588480(72 
590688719 
592704000 

594828321 
596947688 
599077107 
601311584 
603851135 

606496786 
607646423 
600800193 
611900049 
614135000 

616395061 
618470308 
6206504n 
622835864 
626026875 

637233016 
629432793 
631628713 
633839779 
636056000 

638377381 
640608938 
642735647 
644872644 
647214626 

648461896 
651714368 
658972032 
656284809 
658608000 

660n6311 
668064848 
665838617 

667627624 
660921875 

678321376 
674636183 
676836152 
679151439 
681473000 

683797841 
686128868 
688465387 
690807104 
698154125 

695506456 
697864106 

700227072 
702595369 
704969000 

707847971 
709732288 
712121957 
714516984 
716817375 

719323136 
721734273 
724150792 
726673699 
739000000 



28.9137 
28.9310 
28.9483 
28.9655 
28.9828 

29. 

29.0172 

29.0345 

29.0517 

29.0689 

19.0861 
29.1083 
29.1204 
29.1376 
28.1648 

29.in9 
28.1890 
29.2062 
29.2283 
29.2404 

29.2675 
29.2746 
39.2916 
29.3087 
29.3258 

89.8488 
29.3598 
29.8769 
88.8939 
29.4109 

89.4279 
29.4448 
29.4618 
29.4788 
29.4958 

29.5137 
29.5296 
29.5466 
29.5635 
29.5804 

29.5973 
29.6142 
29.6311 
29.6479 
29.6648 

29.6816 
29.6985 
29.7153 
29.7321 
29.7489 

29.7668 
29.7825 
29.7993 
29.8161 
29.8329 

39JB496 
29.8664 
29.8881 
29.8998 
29.9166 

29.9383 
29.9500 
29.9666 
29.9833 
SO. 



O.BK 



8.4204 
8.4241 
9.4279 
9.4316 
8.4854 

8.4391 
9.4429 
9.4466 
8.4503 
9.4541 

8.4578 
9.4615 
8.4652 
9.4690 
9.4727 

8.4764 
0.4801 
9.4888 

9.4875 
9.4918 

9.4948 
9.4968 
9.5028 
9.5000 
9.508T 

9.5184 
9.5171 
9.5207 
9.5244 
8.5281 

9.5817 
9.6864 
8.5S9I 
8.5427 
9.6464 

8.6601 
9.6537 
9.5574 
9.5610 
9.5647 

9.5688 
9.5719 
9.5756 
9.5792 
9.5828 

9.5865 
9.5901 
9Ji937 
9.5973 
9.6010 

9.6046 
9.6062 
9.6118 
8.6154 
8.6190 

9.6226 
9.6262 
9.6298 
9.6S34 
9.6370 

9.6406 
9.6442 
9.6477 
9.6513 
9.6549 



62 



8QUABEB, OUBB8, ANI> ROOXfiL 



VAMMmE of Stt«Mr«i» €«1>es« tenape Boots, mmA CqIm 
of N ambers from 1 to 14l0O--(Oo)(TunjEi>.) 



ITa 



Sqiuun. 



901 

m 

903 
904 
905 

906 
*W , 
908 * 
900 
910 

911 
912 
9IS 
9U 
916 

tie 

917 
•18 

m 

M0 
Ml 



&7 
931^ 
999 

Mo 

961 
913 
913 
9M 

m 

966, 

9«r' 

9S8 

941 
94S 
94S 
944 
943 

946 
W 
948 
948 
950 



811801 
813604 
815409 
817316 
816036 

830836 
833648 
834464 
838381 
838100 

839931 
831744 
833569 
835386 
837335 

839Q6ft 
840889 
84273 



8464d0 

84B3a 
8500M 

851939 
85B776 
856625 

857476 
859339 
861184 
868041 
664900 

866761 
868624 
870489 
87235i 
874335 

876086 
877968 
8798U 
881731 
383600 

885481 
887364 
889249 
891136 
893025 

894916 
896800 
898704 
900601 
903500 



Cul>«. , 8q. &t. 



731433701 
733870808 
736314337 
738763364 
741217635 

743677416 
746143643 
7486LS312 
751089439 
753571000 

75606808] 
758550638 
761048497 
763661944 
766060875 

768676386 
7710063X3 
773^32 
77616I56» 
778688000 

781338861 
78S777448 
78633(Mff7 
788888034 

791^25 

79402976 
796597983 
799178762 
801765089 
8O436710OO 

806964481 
8O95&7608 
8131607 
814780604 
817400876 

8200; 

82 

825283612 

8279S60I9 

830584000 

833337621 

8S5886tt8 
838561 W7 
Ml 233384 
843908625 

846590536 
849278123 
851971392 
854670349 
857375000 




30.0167 
30.0333 
30.0500 
80.0666 
30.0832 

30.0998 
80.1164 
30.1330 
30.1486 
30.1663 

30.1838 

so.iwi 

30.2159 
30.2334 
30.3490 

30.2666 
30.2830 
30.2986 
30.3160 
30.3316 

30.3480 
30.3«I5 
3a3809 
30.3974 
30.4138 

3O.4S0A 
30.4467 
30.4631 
30.4796 
30.4959 

30.5133 
30.5287 
30.5460 
30.5614 
30.5778 

30.5941 
30.610$ 
30.62i8 
30.6431 
30.6594 




30.70< 

30.7246 

30.7409 

S0.75T1 
80,7734 
30.7896 
30.8058 
30.8221 



cut. 



Ko. 



fkiaave. 



9.6586 
9.6630 
9.6666 
9.6693 
9.6737 

9.6763 
9.6799 
9.6834 
9.6970 
9.6906 

9.^1 

9.7013 
9.7047 
9.7083 

9.7118 
9.7153 
9.7188 
9.7334 
9.7359 

9.738i 
9.7338 
9.73W 
9.7400 
9.7436 

9.7470 
9.7505 
9.7540 
9.7575 
9.7610 

9.7645 
9.7680 
9.7716 
9.7750 
9.7785 

9.7819 
9.7864 
9.7889 
9.7934 
9.7959 

9.7983 
9.8038 
9.8063 
9.8087 
9.8132 

9.8167 
9.8201 
9.8236 
9.8270 
9.8305 



951 
962 
963 
954 
956 

956 
967 
968 
969 
960 

% 

9B3 
964 

966 

% 

966 

909 

970 

971 
973 
973 
974 
976 

976 

977 
rf78 
9T9 
980 

981 

d83 
984 
965 

986 
987 

9e9 

990 

991 
993 
993 
991 

996 

•fvQ 

998 

999 

1000 



904401 
906304 
908309 
9L0U6 
913036 

918986 
91689 
917784 

933LeW 




938166 
936088 
83T034 
838861 
840800 

943841 
944784 
946739 
94867« 
956636 

953576 
954539 
956484 
958441 
968400 

963361 
964334 
96628» 
968256 
9702^ 

972196 
974169 
976144 
978121 
980100 

9830^ 
984094 
986049 
988036 
990036 

993016 
994009 
99000% 
998001 
1000000 



CulM. 



Sq.su. 




860085351 
862801408 
865633177 
868360664 
870083876 

873733816 
876467^ 
87921^913 
881874579 
884736000 

887603681 
89037t(28 
dKOBHsm 

89684IS64 
898683135 

901438686 
9O4S3t06S 
907089333 
909863309 
913679000 

91 

91 

931 

t2401«i434 

9368S98T6 

939714176 
9336Y4883 
9S5«US63j 
9383; 
941 ll 

944076141 
946966168 
949863087 
952763904 
966671635 

958686356 

961604803< 

964430373 

9678616m 

970299000 

973248371 

976191488' 

979146667 

982107784 

985074875 

968047936 
991030*73 
994011992 
997008999 
1000000000 




30.8383 
30.8545 
80.8707 
30.8869 
30.9031 

30.9192 

sasaN 

8O.8$0 
80.8877 
80.8838 

31. 

81.0161 

81.0333 

81.04B 

31.0644 

31.0806 
31.0893 
81.113T 
31.1288 
81.1448 

31.1608 
31.176» 
31.183» 
31.2090 
31.3360 

31.3410 
31.36T0 
31.3730 
81.3890 
31.3060 

31.3208 
31.3369 
81.3538 
31.3688 
31.8847 

31.4006 
31.41« 
31.4335 
31.4484 
31.4643 

31.4803 
31.4966 
31.6lf9 
S1.627« 
31.5438 

31.5595 
31.5753 
31 .5911 
81.6070 
31.6228 



CS^. 



9.8339 
9.8374 
9.8408 
9.8443 
9.84!t 

9.86U 
9.8M« 
9.8660 
9.8614 
9.8848 



•••^ 




9.9631 
9.96M 

ISZ 

9.9698 
9.9738 
9.97W 

9.98» 
9«V^Bo 



.9666 

.9m 



9. 
9.1 

9.9988 
9.99Wr 
10. 



To find tbe sonaro or eabo of any whole nnmber endlMP 
wltb cipbers. First, omit all the final ciphers. Take from the table w 

sqiMire or oub« (as the oaae maj be) of the rest of tbe number. To tbU tquare add twice M mt.nf 
ciphers as there were final ciphers in the original number. To the cube add three times as many at 
m the orlgioal number. Thus, for 905003; 9053 = 819025. Add twice 3 cipher*, obtaiuiog 8190250000. 
For iH)5803, go&3 = 741217625. Add 3 times 2 ciphers, obtaining 741217625000000. 



SQUABi: AND GITBB BOOTS. 



63 



No CTTora. 



Num. 


Sq. Rt. 


Ca. Rt. 


Num. 


Sq. Rt. 


Ca. Rt. 
11.20 


Nam. 


Sq. Rt. 


Cu. Rt. 


Nam. 


Sq. Rt. 


Cu.Rt. 


ido& 


81.70 


10.02 


1405 


87.48 


1805 


42.49 


12.18 


2205 


46.96 


1102 


XOlO 


31.78 


10.03 


1410 


87.56 


11.21 


1810 


42.54 


12.19 


2210 


47.01 


1?« 


1015. 


91.86 


10.05 


1416 


87.62 


U.23 


1815 


42.60 


12.20 


2216 


47.00 


19.04 


low 


31 .04 


10.07 


1420 


87.68 


11.24 


1820 


42.66 


12.21 


2220 


47.12 


1«.05 


10» 


82.0S 


' 10.06 


1426 


87.76 


11.26 


1826 


42.72 


12.22 


2226 


47.17 


I9.0ft 


U»0 


82.oe 


10.10 


1430 


87.82 


11.27 


1830 


42.78 


12.23 


2230 


47,22 
47.28 


i$.oe 


1036. 


32.17 


10.12 


1436 


87.88 


11.28 


1836 


42.84 


12.24 


2236 


19.07 
19.08 


1040 


82.25 


10.13 


1440 


87.96 


11.29 


1840 


42.90 


12.25 


2240 


47.99 


lOtf 
106O 


38.88 


10.15 


1446 


88.01 


lUl 


1845 


42.96 


12.20 


2246 


47.98 


19.00 


82.40 


10.16 


1450 


88.08 


11.32 


1850 


43.01 


12.28 


2250 


47.43 


13.10 


iioo 


32.48 


10.18 


1456 


38.14 


U.33 


1856 


43.07 


12.29 


2256 


47.^ 


19.11 


82.56 


10.20 


1460 


88.21 
88.21 


11.34 


1860 


43.13 


12.30 
12.81 


8260 


47.64 


14.12 


106& 


$2.68 


10.21 


1466 


11.36 


1866 


43.19 


2266 


47.89 


1^13 


I074» 


82.71 


10,23 


1470 


38.34 
88.41 


14.37 


1870 

1876 


4S.2i 


12.32 


2270 


47.64^ 


lil4 


^ 


82.70 


10.24 


. 1476 


11.38 


43.30 


12.33 


2876 


47.70^ 


l£lS 


$2.86 


10.26 


1480 


38.47 


U.40 


1860 


43.36 


12.34 


2280 


47.75 


19.10 


1066 


82.04 
83.08 


10.28 


' 1486 


98.60 
88.6t 


U.41 
11.42 


1886 


43.42 


12J5 


2286 


47,80 
47.86 


i9.n 


109V 


10.29 


1490 


1890 


.43.47 


18.36 


2290 


19!S 


1 06 


83.00 


10.31 


1496 


11.43 


1896 


43.53 


12.37 


2296 


47.91 


l<N>^ 


83.17 


10.82 


1500 


38.73 


U.46 


1900 


43.50 


12.30 


2300 


47.0^ 


19.20 


101^ 


89.34 


10.84 


1506 


38.79 


U.46 


1906 


43.3 


12.40 


2906 


48.01 


19.21 


Ul« 


33.8S 


10,36 


1510 


88.86 
98.99 


11.47 


1910 


43.7)1 


12.41 


3310 


48.00 


19.22 


uw 


88.30 

is.47 


10.87 


1516 


U.49 


1916 


43.71 


12.42 


' 2315 


49.11 


li29 


UM 


10.38 
10.40 
10.42 


1520 


89.12 


11.50 


19« 


43.8! 


12.43 


zS20 


49.17 


isjit 




88.54 
88.68 


1526 
■ 1530 


11.51 
li.63 


1926 
1930 


49.8! 
43.9: 


12.44 
12.46 


2330 


49.22 
49.92 


19:25 

19.26 


1 sfr 


3^.60 


10.43 


1536 


98.18 


U.54 


1936 


43.9) 


12.40 


. 2336 


19.27 


1 40 


83.76 


UL46 


1540 


38.24 


U.56 


1940 


44.06 


lt47 


2940 


48-97 


19.28 




83.84 


10.46 


1646 


S.'S 


11.66 


1946 


44.10 


12.48 


2945 


48.43 


19.29 


liso 


83.01 


10.48 


1550 


11.57 


1950 


44.16 


12.19 


2950 


48.48 


19.90 


1^6 


83.00 


10.40 


. 1656 


89.49 


U.59 


1956 


44.23 


12.60 


2856 


48.63 


19.90 


ifiS 


84.06 


10.51 


1560 


S9.g 


11.60 


1960 


44.27 


12.51 


2360 


48.58 


19*91 


fj/i^ 


84.18 


10.63 


1566 


99!62 


11.61 


1966 


US 

44.U 


12.63 


2366 


48.69 


19.92 


^ 


84.21 
84.26 


10.64 
1(^65 


1570 
1575 


11.62 
11.69 


1970 
1976 


ll54 
12.66 


2970 

2376 


48.68 

48.79 


19.98 

19.94 


UJBO 


84.36 


10.57 


1680 
16^ 


S9.7& 


ll.((5 


1980 


44.50 
44.56 


12,80 


2380 


48.70 


li.3S 


n^K 


84.43 


10.58 


ov.u 


11.66 


1986 


12-§T 


3986 


48.84 


13.98 


iSo 


«4.5<) 


10.60 


1690 


^.87 


11.67 


1990 


^•^ 


lite 

liM 
12.00 


2S9D 


48.89 


iljst 


nj6 


84.57 


10.61 
10.63 


1696 


g.9i 


11.66 


1996 


44.fl» 


2995 


48.94 


iSJiS 
13.89 


Qoo 


U.U 


1600 
1606 


4o!m 


11.70 


2000 


44.72 


MOO 


48.99 


U06 


84.71 
34.70 


10.04 


11.71 
11.72 


2006 


44.78 


12.61 


2106 


49.04 


13.40 


uso 


10.60 


1610 


40.12 


2010 


44.83 


12.62 


»10 


49L<[» 


18.41 


♦jll^ 


94.80 


10.67 


1616 


40.li 
40.25 


11.19 


2016 


44.n 


12.09 


i&5 


4a.u 


19.42 


y<£3i 


84.08 


10,69 


1620 


11.74 


2020 


44.94 


12.64 


2480 


tt.24 


19.48 




196 


35.00 


10.70 


1626 


40.31 
40.St 


11.76 


2025 


45.0D 


12.66 


2485 


19.« 




y^ 


36.21 


10.71 


1630 


11.77 


2030 


45.0B 


12.60 


2430 


40.ao 


18.44 




S£ 


10.73 
10.74 


1636 
1640 


40.44 
40.60 


11.78 

i;.7d 


2036 
2040 


45.11 
45.17 


12.67 
12.68 
12.« 


2436 
2440 


4».& 


1I45 
ll4ft 




15.20 


10.76 


1646 


40.59 


11.80 


2046 


45.22 


2445 


4^*45 


19.47 


;Ei6d 


85.30 


10.77 


1650 


40.62 


11.82 


2050 


45.28 


12.70 


2460 


4S.8O 


19.48 


466 


95.43 


10.79 


1656 


40.68 


11.83 


2055 


45.33 


12.71 


2460 


«^.60 


19J2 




85.50 


10.80 


1660 


40.7i 


11.84 


2060 


45.39 


12.72 


2470 


49.70 


!M6 


35.67 


10.82 


1066 


10.80 


11.83 


2066 


45.44 


12.73 


2480 


48.80 


19.64 


S9» 


86.64 


10.83 


1670 


40.87 


11.86 


2070 


45.50 
45.55 


12.74 


2490 


49.90 


19.66 


U76 


86.71 


10.84 


1675 


40.99 


11.88 


2075 


12.75 


2500 


60.00 


19.67 




85.78 


10.86 


1680 


40.99 


11.89 


208O 


45.61 


12.77 


2610 


90.10 


19.59 


3B6 


35.86 


10.87 


1686 


41.06 


11.90 
11.91 


2086 


45.66 


12.78 


2520 


60.20 


13.61 


aoo 


85.92 


10.89 


1690 


41.11 


2090 


46.72 


12.79 


26SO 
2540 


80.30 


19.63 


85.90 


10.90 


1695 


41.17 


11.92 


2095 


45.77 


12.M 


50.40 


19.64 


s 


36.06 


10.91 


1700 


41.23 


11.93 


2100 


45.89 


12.8T 


2650 


60.30 


1166 




80.13 


10.99 


1705 


41.29 


11.93 


2105 


43.88 


12.82 


2560 


60.60 


19.68 


^DO 


86.10 


10.94 


1710 


41.36 


11.96 


2110 


45.93 


12.83 


2570 


50.70 


1170 


lljiy 


ioiS 


10.96 


1715 


41.41 


11.97 


2116 


45.99 


12.84 


2580 


50.79 


1172 


ICW 


58S 


1720 


41.47 


11.98 


2120 


46.04 


12.83 


2590 


50.89 


19.79 


S5 


96.40 


1726 


41.63 


11.99 


2125 


46.10 


12.86 


2600 


60.99 


1175 


S5o 


36.47 


11.00 


1730 


42.59 


12.00 


2130 


46.15 


12.87 


2610 


61.09 


19.7T 


x56 


96.54 


U-Ol 


1736 


41.65 


12.02 


2135 


46.21 


.12.88 


2620 


51.19 


1179 


1|M# 


90.61 


11.02 


1740 


41.71 


12.03 


2140 


46.26 


12.89 


2630 


51.28 


19.80 


iMft 


36.67 


11.04 


1746 


41.77 


12.04 


2145 


46.31 


12.90 


2640 


51.38 


1182 


itso 


96.74 


11.06 


1750 


41.83 


12.05 


2150 


48.37 


12.91 


2650 


61.48 


1184 


S{ 


96.81 


11.07 


1755 


41.89 


12.06 


2155 


46.42 


12.92 


2660 


61.58 


1I86 


SS 


90.88 


11.08 


1760 


41.96 


12.07 


2160 


46.48 


12.93 


2670 


51.67 


1187 


Mt 


90.96 


11.09 


1765 


42.01 


12.09 


2165 


46.53 


12.94 


2680 


61.77 


ll89 


SM 


97.01 


11.11 


1770 


42.07 


12.10 


2170 


46.58 


12.95 


2690 


51.87 


ll91 


Bo 


97.08 


11.12 


1776 


42.13 


12.11 


2175 


46.64 


12.96 


2700 


61.96 


18.92 


97.U 


11.13 


1780 


42.19 


12.12 


2180 


46.69 


12.97 


2710 


52.06 


18.94 


IW 


97.82 


1U6 

lOo 


1786 


42.23 


12.13 


2185 


46.74 


12.98 


2720 


52.15 


18.90 


m 


97.28 


1790 


4i.U 


12.14 


2190 


46.80 


12.99 


2730 


52.25 


19.98 


m 


97.86 


11.17 
11.10 


1795 


42.37 


12.15 


2195 


46.85 


13.00 


2740 


52..35 


19.99 




um 


87.42 


1800 


42.43 


12.10 


2200 


46.90 


13.01 


2730 


62.44 


14.01 



8QUAKE A.ND CUBE £ 



SQUABB AND CUBB BOOTS. 



66 



SQUARE AND CUBE ROOTS. 



Square Boots and Cube Roots oflf nmbem fWmi 1000 to lOOM 

— (GONTIirUXD.) 



Hun. 


Sq.Bt. 


Co. Bt. 


Nora. 


Sq.Bt. 


Od. Bt. 


Nam. 


8q. Bt. 


Ca.Bt. 


Num. 


Bq.Bi. 


01I.B4 


tow 


W.29 


».M 


0990 


M.64 


21.04 


9660 


97.79 


21.22 


97M 


96.M 


I1J» 


MM 


96.S4 


ao.87 


OSM 


M.6e 


91.06 


96W 


97.78 


21.22 


97M 


.96.94 


S1.8t 


91M 


96.89 


ao.ae 


9S40 


M.04 


91.M 


9670 


97.88 


21.28 


9eM 


W.M 


31.M 


9110 


96.46 


30.89 


9060 


M.70 


91.07 


9680 


97.88 


21.24 


9810 


M.06 


si.a 


9iao 


95.60 


ao.H9 


99M 


M.76 


91.07 


96M 


97.M 


21.26 


9820 


M.10 


si.a 


91M 


96.66 


M.M 


n7o 


M.M 


U.W 


96M 


97.98 


21.26 


98W 


M.16 


tl.4t 


9140 


96.M 


90.91 


OSM 


M.86 


Sl.M 


WIO 


W.M 


21.26 


9840 


M.20 


tLU 


9160 


96.M 


90.09 


99M 


M.M 


91.10 


WJO 


96.08 


21.27 


9660 


M.26 


81.44 


91M 


96.71 


90.99 


9400 


M.M 


91.10 


96M 


W.1S 


21.28 


OSM 


M.M 


31.44 


9170 


96.7C 


90.M 


9410 


97.01 


91.11 


9840 


06.18 


21.28 


W70 


M.85 


S1.4ft 


•IM 


96.81 


90.94 


94M 


97.M 


91.12 


9850 


96.38 


21.29 


96M 


M.40 


21.46 


91M 


96.W 


90.96 


94M 


97.11 


91.1S 


98M 


W.39 


21.M 


9eM 


M.45 


S1.4T 


9»0 


96.92 


90.W 


9440 


97.18 


91.18 


9870 


W.84 


21.80 


99m 


M.60 


21.4T 


9910 


96.97 


90.M 


9460 


97.91 


91.14 


9880 


98.89 


21.81 


MIO 


M.66 


21.48 


9no 


M.03 


90.97 


94M 


97.96 


91.15 


98W 


86.44 


21.82 


M20 


M.M 


21.49 


9B0 


90.07 


90.98 


9470 


97.81 


91.16 


9700 


96.48 


21.88 


99M 


M.86 


21.4S 


9140 


W.13 


90.M 


94M 


97.8T 


91.16 


9710 


96.64 


21.88 


9940 


M.70 


UM 


91M 


M.18 


90.M 


94M 


97.49 


91.17 


9720 


W.69 


21.84 


9960 


M.76 


tLM 


tMO 


W.23 


31.M 


96M 


97.47 


21.18 


97M 


96.84 


31.36 


90M 


M.M 


tun 


9970 


W.» 


91.01 


9610 


97.69 


91.19 


9740 


96.W 


21.88 


9970 


M.86 


S1.6S 


9180 


W.SS 


91.01 


9690 


97.57 


91.19 


9750 


98.74 


21 J6 


99M 


M.M 


21.6t 


99M 


M.38 


91.09 


9680 


97.83 


91.90 


97M 


98.79 


21.87 


99M 


M.M 


S1.64 


9iW 


M.U 


91 .OS 


9640 


97.87 


31.31 


9770 


98.84 


21 JK 


lOOM 


1M.00 


1144 


HIO 


M.49 


91.04 





















To find Square or Cube Roots of larire numbers not eoa- 
tained in tlie column off numliers of tlie table. 

Booh roots mmj MmetimM be taken at onoe from the table, b7 merelr regarding the oolnmns of 
powen as being oolamne of namber* ; and thoie of nambera aa being those of roota. Thna, if tte 
•q ft of 9BI81 ia reqd, ilrat iiiid that nnmber in the column of tquaru ; and opposite to it, In th« 
eolumn of oamben, ii its sq rt 160. For the evhe rt of 857876. find that namber in the eolumn of 
eu5M ; and opposite to it, in the eol of numbers, is its onbe rt 95. When the ezaot nnmber is not con- 
tained in the oolnmn of sqnares, or onbes, as the ease may be, we maj nse instead the nnmber nearest 
to it, if no great aoouraey is reqd. But when a oonsiderablo degree of aoonraoj is necessary, tk* 
following Tery oorreet methods may be need. 

For the squfufe root. 

This rale applies both to whole nnmbers. and to those which are parlor (not wholly) decimal. Flntt 
la the foregoing manner, take out the tabular number, which is nearest to the giren one ; and also tM 
tabular sq rt. Mult this tabular nnmber by 8 ; to the prod add the given number. Call the sum M» 
Then mult the given naml)«r by 8 ; to the prod add the tabular number. Call the sum B. Then 

A : B : : Tabular root : Beqd root. 

Sx. Let the given nnmber be 946.58. Here we find the nearest tebnlar number to bo 947 : aaA Mi 
Ubvlar sq rt M.7784. Henee, 



947 = ub nam 
8 



3841 
940.68 = gl 

8787.68 = ▲. 



and 



948.58 = given num. 
8 



2889.58 
947 = tab nam. 



.8786.59 ^^ B. 



A. 

S787.5I 



B. Tab root. Beqd root. 

Then S787.5I : 8786.89 : : M.7784 : m!7657 +. 

The root as found by aetual mathematical process is also M.7667 -(-. 

For the cube root. 

This rale applies both to whole nnmbers, and to thoee which are par«v decimal. Flrat take ovt tM 
Ubnlar number whioh is nearest to the given one; and also its tabular onbe rt. If nit this tabular 
number by 3 ; and to the prod add the given number. Gall the snm A. Then mull the given anmber 
by 1 ; and to the prod add the tabular number. Gall the sum B. Then 

A : B : : Tabular root : Reqd root. 

Bz. Let the given nnmber be 7368. Here we fiuu cne nearest tabalar number (ia tike Mlaan •( 
ettftes) to be 6860; and iu tabalar cube rt 19. Hence, 



= tab nam. 



18718 y and 

7868 = given nam. 

310Mr:A. 

B. Tab Boot. BeqdBt. 
21696 — 



7868 = given num. 

2 



14788 
8859 = Ub nam. 

. 21696 =:B. 



Then, as 210M 21696 19 19.4585 

Tke root as fbond by oorreet mathematioal prooess is 19.4Mi. The engineer rarely raqoiree 



BQCABE AND CUBE BOOTS. 67 

UtilllirMof HHiTatfyi ll>r Ub pwroHi, IktHfoH, tUi pfWM ll tvMttr pnUBnbU tfp I^ DrAury 

To and ttte aqaBrs r»o( of n number wbleb !■ wIioIIt 
declaaal. 

hwl fln OiarH, foitntifkg from Ikejtrti ji'ummrai.Hi^ h^viudtna it, wld au or mors cIpbHra to nuJa 
luj rnlDlcf Ihlf UbulBt rmllo LbBHn, Jkl^ at UBDJ I>lUM la lUB riBBU7 Doa^ad [bctmaf nDDbCT 

■If h( ^am J ano-IHir of wblDh la' I ; tlHnf&K, mora tha dmlmftl niat or ibH nni iij. ^qr pluu H 
the ton; biUbi tt .OUT. tbla la U« Tsqd vq rt or .0(a> Dornci tg iha third bamvm] TJDp]Ddad- 
T• Bad UlC «nb« rootof «D«ml>erwhlcliIawboll7deeliUal. 

Tsrj ibiipla, ud SDmn u Ua OltA mmanl loolHlia. 

ir iW nDBbar data not aonlUii •! Mut Bn Oiuna, aamiUDi rrom Iba Biat nuiaiil, and 1iialudlB| 



Fin 


b roo 










tr 


,.„, 


Sir 


,™ 


! 




j 


1 


1 


i 


1 


1 


i 


3 


1 
1 


ill 
11 





68 



ROOTS AND POWEBB. 



Fiftli roots and flftb powero— (Continued). 



Power. 


No. Ot 
Boot. 


Power. 


Rio^j f o'«r- 


No. or p„_^ 
Boot. ^«'«'* 


No. Of 
Boot. 


Power. 


No. or 
Boot. 


Power. 


No. Of 
Root. 


88.2735 


2.45 


2824.75 


4.90 


86873 


9.70 


2609193 


19.2 


20511149 


^.0 


459165034 


54. 


V1.ao6-i 


2.5U 


2y71.84 


4.95 


9U392 


9.80 


2747949 


19.4 


21228258 


29.2 


508284376 


56. 


107.b20 


2.55 


3125.00 


3.00 


95099 


9.90 


2892547 


19.6 


21965275 


'29.4 


550731776 


66. 


118 bl4 


2.60 


3450.25 


5.10 


100000 


10.0 


3043168 


19.8 


22722628 


29.6 


601693067 


57. 


130.(>d« 


2.65 


3802.04 


5.20 


110408 


10.2 


3200000 


20.0 


23500728 


29.8 


656356768 


68. 


lU.MIt 


2.70 


4181.95 


5-30 


121665 


10.4 


3363232 


20.2 


24300000 


30.0 


7149-24299 


69. 


167.276 


2.73 


4591.65 


5.40 


133823 


10.6 


3533059 


20.4 


26393634 


30.5 


777600000 


60. 


172.104 


2.80 


5032.84 


5.50 


146933 


10.8 


3709677 


20.6 


28629151 


81.0 


844696301 


61. 


188.(Md 


2.85 


5507.32 


5.60 


161051 


11.0 


3893289 


20.8 


31013642 


31.5 


916132832 


62. 


203.111 


2.90 


6016.92 


5.70 


176234 


11.2 


4084101 


21.0 


33554432 


32.0 


992436543 


63. 


U9.4U 


2.95 


6563.57 


5.80 


192541 


11.4 


4282322 


21.2 


36259082 


32.5 


1073741824 


64. 


243.000 


3.00 


7149.24 


5.90 


210034 


11.6 


4488166 


21.4 


39135393 


33.0 


1160290625 


66. 


263.936 


3.0a 


7776.00 


6-00 


228776 


11.8 


4701850 


21.6 


42191410 


33.5 


1252332576 


66. 


286.292 


3.10 


8445.96 


6.10 


248832 


12.0 


49-23597 


21.8 


45435424 


84.0 


1850125107 


67. 


810.136 


3.15 


9161.33 


6.20 


270271 


12.2 


515.3632 


22.0 


48875980 


34.5 


1463933568 


68. 


835.54i 


3.20 


9924.37 


6.30 


298163 


12.4 


5392186 


22.2 


52521875 


35.0 


1564031349 


69. 


962.391 


3.25 


10737 


6.40 


317580 


12.6 


5639493 


22.4 


56382167 


35.5 


1680700000 


70. 


891.334 


3.30 


11603 


650 


343597 


12.8 


5895793 


22.6 


60466176 


360 


1804229361 


7L 


421.419 


3.35 


12523 


6.60 


371293 


13.0 


6161327 


22.8 


647&3487 


365 


19S49176B2 


7*. 


454.354 


3.40 


13501 


6.70 


400746 


13.2 


6436343 


23.0 


69343957 


37.0 


2073071593 


7i 


488.760 


3.45 


145.39 


6-80 


432040 


13.4 


6721093 


23.2 


74167715 


37.5 


2219006624 


74. 


525.219 


3.50 


15640 


6.90 


465259 


13.6 


7015834 


23.4 


79235168 


38.0 


2373046876 


76- 


563.822 


8.55 


16807 


7.00 


500490 


13.8 


7320825 


23.6 


84587005 


36.5 


7535525376 


76. 


604.662 


3.60 


18042 


7.10 


537824 


14.0 


7636332 


23.8 


90224199 


39.0 


2706784157 


77. 


647.835 


3.65 


19319 


7.20 


577353 


14.2 


7962624 


24.0 


96158012 


39.5 


-2887174368 


781 


693.440 


3.70 


20731 


7.30 


619174 


14.4 


8299976 


24.2 


102400000 


40.0 


3077056399 


79. 


T41,577 


3.75 


22190 


7.40 


663383 


14.6 


8648666 


24.4 


108962013 


40.5 


3276800000 


80l 


792.352 


3.80 


23730 


7.60 


710082 


14.8 


9008978 


24.6 


115856201 


41.0 


3486784401 


81. 


845.870 


3.85 


25355 


7.60 


759375 


15.0 


9381200 


24.8 


1-23096020 


41.5 


3707398432 


83. 


902.242 


3.90 


27068 


7.70 


811368 


15.2 


9765625 


25.0 


130691232 


42.0 


3939040643 


83. 


961.380 


3.95 


28872 


7.80 


866171 


15.4 


10162550 


25.2 


138657910 


42.5 


4182119424 


84. 


1024.00 


4.00 


30771 


7.90 


923896 


15.6 


10572278 


25.4 


147008443 


43.0 


4437053125 


86. 


1089.62 


4.05 


32768 


8.00 


984658 


15.8 


10995116 


25.6 


155756538 


48.5 


4704270176 


86. 


1158.56 


4.10 


34868 


8.10 


1048576 


16.0 


11431377 


25.8 


164916224 


44 


4984209207 


87. 


1230.95 


4.15 


37074 


8.20 


1115771 


16.2 


11881376 


26.0 


174501858 


44.5 


5277319168 


88. 


1306.91 


4.20 


39.390 


.8.30 


1186367 


16.4 


12345437 


26.2 


1845281-25 


45.0 


5584059449 


89. 


1386.58 


4.25 


41821 


8.40 


1260493 


16.6 


12823886 


26.4 


195010045 


45.5 


5904900000 


90. 


1470.08 


4.30 


44371 


8.50 


1.338278 


16.8 


13317055 


26.6 


205962976 


46.0 


6240321451 


91. 


1557.57 


4.35 


47043 


8.60 


1419857 


17.0 


1.3825281 


26.8 


217402615 


46.5 


6590815232 


92. 


1649.16 


4.40 


49842 


8.70 


1505366 


17.2 


14348907 


27.0 


229345007 


47 


6956883693 


93. 


1745.02 


4.45 


52773 


8.80 


1594947 


17.4 


14888280 


27.2 


241806543 


47.5 


7.339040224 


94. 


1845.28 


4.50 


55841 


8.90 


1688742 


17.6 


15443752 


27.4 


254803968 


48.0 


7737809375 


96. 


1950.10 


4.55 


59049 


9.00 


1786899 


17.8 


16015681 


27.6 


J68.354383 


48.5 


8153726976 


96. 


2059.63 


4.60 


62403 


9.10 


1889568 


18.0 


16604430 


27.8 


^>8'2475249 


49.0 


8587340257 


97. 


2174.03 


4.65 


65908 


9.20 


1996903 


18.2 


17210368 


2M.0 


•297184.391 


49.5 


9039207968 


• 98. 


2293.45 


4.70 


69569 


9.30 


2109061 


1H.+ 


17833868 


28.2 


U2500000 


50.0 


9509900499 


99. 


2418.07 


4.75 


73390 


9.40 


•2'2?«203 


18 6 


1 8475:^09 


28.4 


345025251 


51. 






2548.04 


4.80 


77378 


9.aO 


234«493 


18.8 


19135075 


28.6 


380-204032 


62. 






9683.54 


4.85 


81537 


9.60 


2476099 


19..0 


19813557 


28.8 


418195493 


63. 







Square roots of fifth powers of numbers, j/n^, 

or % powers of numbers, n^^. 

See table, page 69. 

The column headed " 12 n " facilitates the use of the table in oases where, 
for instance, the quantity is giveti in inoheSf and where it is desired to obtain 
the % power of the same quantity in feet. Thus, suppose we have a % inch 
pipe, and we require the % power of the diameter in feet. Find ^ (the 
diameter, in, inches) in thecolumn headed/' 12 n," opposite which, in the column 

headed *'n," is 0.041666 (the diameter. In feet), and, in column headed "n%,'» 
0.00035 (the % power of the diamet«r, 0.041666, in feet). 

Values of n, ending in or in 5, are exact values. All others end in repeat- 
ing decimals. Thus: n = 0.052083 signifies n«» 0.052083333 



BOOTB AITD POVEBB. 
>qnar« roata of BfUi powers of nnmbCTM 



(1) Tables itT lOE^rltbioi gteatl}' facilitate multipIIcatloD anil dlTlsionuid 
the findlDC of powera and roots of iiumben* 

(2) Thelabl^pp. 78 to 81 ccinlalutlie eommOB.dMlnalor Brl«ca 
■ 'fl»lin|i»ornui)ibe™. The coinmim logartitim ofatmia'-— '- •'- 



paDentorladeiorthalnmnberuapowerofKI. Bee (IB). ThuB:lD0O = 
and log lOOO (logarilbm of lOOO) = S.CWOOO. Similarly, 28.7 = 10 Lii ;bI, i 
lo«.28.f =1.«7S. 
(S) In geneiil, let A and B b« an; two uumben, and jt any Bzponi 



(1) log \B = log A + log B ; (a) log g = log A — log B ; 
(3) log A» = t (log A) ; W log y-l = ^-^ 



or loEB of tecton. 

.,„jt dividend -log of 

log of rractloa = log of numerator — log of deaominatoT. 



!) Log of quotient = logot dividend — log of divisor 



(1) L^ of povper =■ log of number, multiplied by ei . 
(4) Log of root — log of number, divided by exponent. 
(4) From wbat baa been aald, It followc tbat 
Log 100 = loglO" = 2.00 too I Log 0.1 = log »-• - l.MOOOt 
Log 10 ^ log 101 = i.oaooo Log 0.01 - log Iff^ - 2.00 000 
Log 1 =^ log 10= =- O.OOOOOt I Log 0.001 = log lO"" = S.OOOOO 
1 number, conBlstlng of an inUffral 
ii Index (prrarliTip tbe declmml 
BtmaiiBaw^i following the decimal 
ISO of eacU lag. the cbaracteiisLia 
mantMa is Klwaya positibe- The 
miad number, is poaiiive, and la 
lole number, minus l; while the 
r Is TKijotiue, and is Qumerically 
imedlalel)' followiog the decimal 

log !870 = 3.45 788 log 0.287 - 1.45- 7S8: 

" 287 = 2.45 788 " 0.0287 - 2.15 788 

•' 2S.7 - 1.45 788 " 0.00287 = 3.45 788 

2.87 = 0.46 788 " 0.0002B7 = 4.45 788 

It win be noticed that the mantissa remains constant thr any given com- 
hiaatlon of signtfloaut figurea lu a number, wherever the decimal point In 
the number he placed ; while the cbsraeteristic depends solely upon the 
podtlou of the decimal pnlut in the number. 
(6) Let the number be resolved into two factors, one of which is m 
itegei power of ID, while the other is greater than 1 and less than 10. Then 
le indei of the power of 10 is the oharaclerlatic of the logarithm, and the 
logarithm of the other factor Is the mantissa. Tbns, 2370 = IDOO x 2.ST -^ 
l(^ X 2.87, and the Iwarlthm of 2870 (3.46 78*1 is the sum of the exponent 3 
' 3.00 000) and the log (0.45 79S) of 2.87,t 



* LuEBTlthms not being exact quantities, operations performed *lth them 

tra subject to soma ins/ionracy, especially where a logRrfthm la multiplied 
y a large number, the existing error being thus magnified. Logarlthmaof 
only five places in the mantissa usually BulDce for calculations with nuU- 
ben of four or five places. Greater accuracy is obtained by the \ii» of 
tables of logarithms carried out to seven places. 

t Log 1 = log 18 - log 10— log 10 = 1— 1 = ; ot 1 - 10». 
Log 0. 1 = log A = tog 1 — 'og 10 = — 1 = 1.0 : or ai - 10- 1. 
1 0287 = 2.S7 -^ 10. Hence, log 0.287 = log 2.87 - log 10 =■ 0,45 783 - 1, 
which, for convenience. Is written 1^45 788. See (16). Slmilarty, log O.OIST 
■ log 2.87 — log 100 - 0,45 788 — 2 = 8.45 788, 



LOGARITHMS. 



71 



(7) To find tbe lovaritbiu of a number. The short table on pages 
78^ 79 gives logs of numbers up to 1000. The longer table, pages 80 to 91, 
giyes 

(1) The mantissa for each number from 1000 to 1750 

(2) The mantissa for each even number fh>m 1750 to 3750 

(3) The mantissa for each ^th number from 3750 to 10000 

(8) Logs of numbers Intermediate of those given in the tables are 
found by simple proportion. The procedure necessary in these cases is 
explained in the examples given in connection with the tables, but it will 
often be found sufficiently accurate to use the log of the nearest number 
given in the table, neglecting interpolation. 

Tbe antilog^ariinm or nnm log^ {numerus logarithmt) is the num- 
ber correspondinfT to a given logarithm. Thus, log. 2 = 0.80 108, and 
antilog 0.30 l(fe = 2. 

(9) Mnltiplicatlon. To multiply together two or more numbers, add 
together their logs and find the antilog of their sum. See t'roportion 
(11) below. 

(10) AiTision. Subtract the l<^ of the divisor from that of the dividend, 
and find the antilog of the remainder. See Proportion (11) below. 

The reciprocal of any number, n, = . See page 62. Thus, recip 2 => 

w 

- = 0.5. Hence, log recip n = log - = log 1 — log n = — log n. 
Similarly, log recip — = log — — — = o — log . 

Since n«-i = ni = - , n^-i = n« = " = 1, n^-^ =n-i = - , and no-a = n-« 

= -j it follows that log w-i = log = log recip n ; log n-* = log zj = "^og 
recip 7*2, etc. • 

(11) Proportion. Example. 6.3023 : 290.19 = 1260.7 : ? 



xr w 1 xr y ^e 290.19 =2.46 269 

Multiply Nos, J i* 1260.7 = 3.10 062 

Add Logs. I j^^ 290 jg ^ J260.7 = 5.56 331 



{ 



Divide Nos. f Log 6.3023 = 0.79 95 

Subtract Log. \ Log 58051 =4.76 381 

The true value is 58049.05 + 

(19) Instead of subtracting the log of the divisor, we may add its coloipa- 
ritlim or arithmetical complement, which is log of reciprocal 
of divisor, = — log divisor = 10 — log divisor — 10. Thus :. 

1523 _ 

3.382 X 8.655 

Log 1523 = 3.18 270 

Colog 8.382 = 10 — log 3.332 — 10 = 10 — 0.52 270 — 10 = 9.47 730 — 10 
Colog 8.655 = 10 — log 8.655 — 10 = 10 — 0.93 727 — 10 = 9.06 273 — 10 

Sum of logs and cologs = 21.72 273 — 20 

= Log 52.813 = 1.72 273 
The true value is 52.8114 + 

(13) Involution, or findinf^ powers of numbers. Multiplv log of 
given number by the exponent of the required power, and find the anti- 
log of the product. Thus : 36^ = ? 

Log 36 = 1.55 630. 1.55 630 X 3 = 4.66 890. Antilog 4.66 890 = 46656. 

(14) Evolution, or finding roots of numbers. Divide log of given 
number by exponent of required root, and find antilog of quotient. Thus : 

s 

V46656 = ? Log 46656 = 4.66 890. 4.66 890-5-3 = 1.55 680. Antilog 1.55 630 = 36. 
(tJi) In finding roots of numbers, if the given number is a whole or mixed 



72 



LOGARITHMS. 



number, the division of the log is performed in the usual way, as in the 
preceding example, even where, as in that example, the characteristic ia 
not exactly divisible by the exponent of the required root. But if tl&e 
namber is a fraction, and the characteristic of ita log therefore nega- 
tive, and if the characteristic is not exactly divisible by the exponent, 
division in the usual wav would give erroneous results. In such cases we 
may add a suitable number to the mantissa and deduct the same number 

from the characteristic, thusj to find Vo.00048. Log 0.00048 = 4.68 124 = 
0.68 124 — 4 = 2.68 124 — 6 = 6 + 2.68 124, which, divided by 8, = 2 + 0.89 375 
= 2.89 375 = log 0.0783. Or, see (16) and (17). 

(16) To avoid inconvenience from the use of negatiTe character- 
istics, it is customary to modify them by adding 10 to them, afterward 
deducting each such 10 from the sum, etc., of the logarithms. Thus : in 
multiplying or dividing 7425 by 0.25, we have 

Multiplying. Dividing, 

either log 7425 = 3.87 070 = 8.87 070 

log 0.25 = 1.39 794 = 1.39 794 

3.26 864 4.47 276 



or log 7425 = 3.87 070 = 3.87 070 

modified log 0.25 = 9.39 794 — 10 = 9.89 794 — 10 

13.26 864 — 10 6.47 276 + 10 
= 3.26 864 = 4.47 276 

In most cases the actual process of deducting the added tens may be 
neglected, the nature of the work usually being such that an error so great 
as that arising from such neglect could hardly pass unnoticed. 

(17) To dlTide a modified loiparithm, add to it such a multiple of 
10 as will make the sum exceed the true log by 10 times the divisor. Thus : 

to divide log 0.00048 by 3. Log 0.00048 = 4.68 124, which, divided by 3, = 

2.89 375. See (15). • 

Log 0.00048= 4.68 12 4 

Modified log 0.00048 = 6.68 124 — 10 
Add 2 X 10 20 — 20 

Dividing by 3) 26.68 124 — 30 

we obtain 8.89 375 — 10, which is 2.89 375 modified. 

(18) Except 1, any number can (like 10) be made the base of a system of 
logarithms. The base of the byperbolic, Napierian, or natural 
lograritiims, much used in steam engineering, is 

1 + 1 + 1-^2 + lX-^3 + 1X2X3X4 + ' " ' " = ^'^ «^ + 
and is called « (epsilon) or e. 

M = logi oC (common log e) = 0.43 429 ; ^ =log « 10 (hyperbolic log 10) =2.30 250. 
For any number, n, 
loge n = — 1^ = 2.30259 logio n ; logjo n = M loge n = 0.43429 loge n 

(19) Whatever may be the base chosen for a system of logs, the man> 
tissas of the logs of any given numbers bear a constant ratio to each 
other. Thus, in any system of logs, log 4 is always = 2 X log 2, and 
=• K X log 8, etc., etc. 

(20) liOffarithmic sines, tansrents, etc. of angles are the logs of 
the sines, tangents, etc. of those angles. Thus, sin 80° = 0.5000000, and log 

sin 30° = log. 0.5 = 1.69 897, usually written 9.69 897 — 10, or simply 9.69 897. 
(ai) Since no power of a positive number can be negative, negative num- 
bers properly have no logs ; but operations with neyatl-ve nnm- 
bern ran nevertheless be performed by means of logs, by treating all the 
numbers as positive and taking care to use the proper sign ,+ or — , in the 
result. 



LOGARITHMIC CITART AND SLIDE RULE. 



73 




1,1- 

JLog». 
l.O- 

OJO- 

OJS 

0.7- 

0.0- 

OJi 

oa 

OJO- 

IJDr- 

J 1 r 1 1 \ 1 1 1 1 1 1 r 

JLog9,lJ> 0/» 0,1 OJi OJ3 0,4 OJg OM 0.7 O^ 0.9 1.0 la 



I 



I 



0,9 0,4 

t L_ 



o.e 0,8 

—J L_ 



1,0 

L_ 



1.9 

I 



1.4 

I 



2.0 

— I 



1.9 



9.0 

1__ 



9.9 



jro«. 



Mo9* 



E 



2 3 4 S 97801 

-l^ I I . f I ,1 I I I 



2 



r 
Bl 

C 



T 

9 



1 — I 1 M I 
4 5 7891 



3 4 5 07891J\ 

I I I I .1 I r I I 



+ 



Dl 



i ri^-^ 



T 1 1 ■ I ' I I I 1 I 

2 3 4 S G78»l 
S e 7 8 9 



lA. 



9 



3 



-T 

5 



e 



-T 
7 



T — 1 I ■ f 
8 9 lU] 



J»L 



i3 



-« ' 1 1 1 1 1 r— 

ij} 0.0 0.1 0.9 0.3 0.4 ojs o,e 

Log»» 



0.7 



— I r- 

0:8 0.9 



1.0 



— I 
1.1 



Tb« ttOgnrfthmic Chart and th« S11d« Ral«. 

(1) By means of a logarithmic chart or diagram (often miscalled lo«i- 
rtthmic cross-section paper) logarithmic operations are performed graphi- 
cally, and by means of the slide rule mechanically, without reference 
to the logarithms themselves *. But see t. P 76. Their use greatly facili* 
tales many hydraulic and other engineering computations. 



(•) The ratio between the mantissas of the logs of any given numbers 
being constant for all systems of logs, the ratio between the distances laid 
off on the chart or slide rule is the same for all systems, and the use of the 
chart or rule is independent of the system of logs used. 



74 



LOGARITHMIC CHART AND SLIDE RULE. 



(2) The lofrarlttamle eliart consists primarily of a square,* on the 
sides oi which the distances marked 1-2, 1~3, etc., are laid off by scale 
according to the logs (0.30 103, 0.47 712, etc.) of 2, 3. etc. Ordinary 
"squared" or cross seetlon |mper may of course be used for loga- 
ritmnio i>lotting, by plotting on it the loo9 instead of their Not. Lines 
representing Nos. may be drawn in their proper places as dedired. 

(3) As ordinarUv constructed.^ the slide rule consists essentially of 
four scales. A, B, G, and D, see (17), scales A and D being placed on the 
** rule," while B and C are placed upon the sliding piece, or " sUde." As 
in the logarithmic chart, see (2), the scales are divided loearithmically 
(see figure), but marked with the numberB corresponding to the logs. Scales 
A and B are equal, as are also scales C and D, but a given length on A or B 
represents a logarithm, twice as great as on C or D. See (4). Hence, each 
number marked on A is the aquare of the coinciding number marked on £>. 

(4) A single logarithmic scale is usually numbered from 1 to 10, or from 
10 to 100; but it may be taken as representing any series embracing the 
niunbers from 10* to 10**+ ^; as from 0.1 to 1.0 (n = —1); or from 1.0 to 
10.0 (n "» 0); or from 10.0 to 100.0 (n = 1); or — etc., etc. Here n and 
n + 1 are the cliairaeteristlcs of the corresponding logarithms. 

A single scale would therefore serve for all values, from to infinity ; 
but for convenience several contiguous scales are sometimes added, as in 
the log chart*. 

When a line reaches the limit of a square, the next square may be 
entered* or the same square mav be re-entered at a point directly opposite. 
Thus, in the case of line xH (= iTS'y. 



TiiTiP Trifi.i*1rAi^ 


between 

• 


correspondi to values of 


xH 


xttom 


xH from 


(1) 
(2) 
(3) 
(4) 


1 and S 
8} and S, 
S, and S. 
Ss and H 


Ito 10 
10 to 31.62 
81.02 to 100 
100 to 1000 


1 to 4.64 
4.64 to 10 
10 to 21.54 
21.54 to 100 



Note that the numbers, marked on any given scale, must be taken as 10 
times the corresponding numbers marked In the next scale preceding, and 
the characteristics therefore as being greater by 1, and vice verm. Thus, in 
our figure, log 1.5 + log 2 = 1-1.5 + 1-2 = log 8 = distance 1-M. But 
log 15 + log 20 = (1-1.5 + 1-10) + (1-2 + 1-10), so that the characteristic 
ofthe resulting log is greater by 2, and the 3 representing the product of 15 
and 20 is really in the second square to the right of that shown. In finding 
powers and roots, remember that multiplying or dividing the number by 
0.1, 10, 100, etc. a. e., changing the charactensttc of its log), changes also the 

mantissa of the log of its power or root. Thus, 1^277 = 1.39 . . , (log = 0.14 379) ; 

but T>'27'== 3, aog = 0.47 712) and 1^270 = 6.46 . . , (log = 0.81 023). The 
chart or rule gives aU such possible roots, and care must be taken to select 
the proper one. Most operations exceed the limits of one scale, and fi&cility 
in using either instrument depends largely upon the ability to pass readily 
and correctly from one scale to another. This ability is best gained by prac- 
tice, aided by a thorough grasp of the principles involved. Where several 
successive operations are to be performed, a sliding runner or marker 
(furnished with each slide rule) is used, in order to avoid error in shifting 
the slide. Detailed instructions are usually famished with the slide rule. 

(*) A common form of chart has four or more similar squares Joined 
together. See (4). Our figure represents one complete square, with por- 
tions of adjoining squares. For actual use, both charts and slide rules 
are, of course, much more finely subdivided than in our figures, which are 
given merely to illustrate the principles. Carefully engraved charts are 
published by Mr. John R. Freeman, Providence. R. I. 

(X) Other forms embodying the same principle are : The " Reaction Scale 
and Gteneral Slide Rule," bv W. H. Breithaupt, M. Am. Soc. C. E. ; Sexton's 
Omnimeter or Circular Slide Rule, bv Thaddens Norris : The Goodchild 
Computing Chart ; The Thacher Calculating Machine or Cylindrical Slide 
Rule : The Cox Computers, designed for special formulas ; and the Pocket 
Calculator, issued by " The Mechanical Engineer," London. 



LOGABrrHMIC CHABT AND SUBB BI 



<5) Mvltliiltcattoii aad dlvlsiofli. For example, 
1-X* in the chart, or on C or D, in the alide rule, the diatf 
sents by scale the logarithm (0.17 600) of 1.5, and 1-1 
losaiithxn (0.30 103) of 2. If now we add these two dis 
by laylnflT off 1-2 ttom 1.5 on 1-X of the chart, or by placl 
In the figure, we obtain the distance 1-3 = .47 712 = the m 
or of log (2 X 1.5).* Conversely, to divide 3 by 2, we graphica 
cally subtract 1-2 fh>m 1-3. 




(•) In tbe l4»9Arftliinlc chart, the scales of both axes, 
1-Y, being equal, a line 1-H, marked x, bisecting the square ai 
ing an angle of 45<' with each axis (tan 45° = l),t will bisect also tl 
sections ox all equcU co-ordinates. Thus, points In the line x, imm 
over 2, 3, 4, etc.. in 1-X, are also opposite 2, 3, 4, etc., respect!' 
1-Y. 8ee (4). 

g*) If lines 2-A\ S-K, etc. (marked 2x, 8a;, etc.), parallel to m 
, be drawn through 2, 8, etc., on 1-Y, then points in such li 
mediately over any number, x, in 1-X, will be respectively oppo 



(*) In the slide rule, with the slide as shown, ea/:k number on 
1.5 X the coinciding number on C. 

(t) In disenssing tangents of angles on log chart, we refer to th< 
measured distanoes, as shown on the equally divided scales of tog 
flgnres, and not tb the numbers, which, for mere convenience, are 

C B 10 li 

on lb« cljart. TJius, in )ine 1-B, tan C 1 B = ,~^ = ;;-^-, not — 



I C 0.38 : 



2. 



76 LOGARITHMIC CHART AND SLIDE RULE. 

numbers giving the products 2x, 2x, etc., on 1«Y; while similar lines, 
drawn below 1-H and through 2, S, etc., on 1-X, give. values of ^^ ?, etc., 

respectively. If these lines ^^ «• etc., be produced downward, they will 

cut 1-Y (produced) at 0.5 (= }4), 0^ . . (= V^, etc!, respectively * See (4). 

(8) Powers and roots. If a line z^ be drawn through 1, at an angle 

s — s 
So 1-X, whose tangent, f-^ is 2, it will give values of z*. Thus, the ver- 

tical through 3, on 1-X, cuts the line x* opposite 9 (= 3*) on 1-Y. Simi- 
larly, line x^ (tangent = 3) gives values of «' ; and line ^x (tangent = *^ 

gives values of a;' <*' T/'ir See (4). 

(9) Any equation of the form y = C.x" in which log y = log C + n log «, 
(such as : area of circle = ir radius*), is represented, on a logarithmic chart, 
by a straight line so drawn that the tangent T of its angle with 1-X is = n, 
and intersecting 1-Y at that point which represents the value C. Thus, 
the line marked v x^, (tangent = 2) is a line of squares, and, being drawn 
through IT (= 3.14. .) on 1-Y, it gives values of w x*. Thus, for a circle of 



radius 2, we find, in the line n x^ over 2, a point L opposite E, or 12.57. . . . the 
area of such circle.t Conversely, having area = 12.57. . . , we obtain, from 
the diagram, radius = 2. 

(10) If a chart is to be used for solving many equations of a single 
kind, such asy = C a:", where C is a variable coefficient, and n a constant 
exponent, parallel lines, forming the proper angle with 1-X, should be perma- 
nently ruled across the sheet at short intervals. 

(11) For any log, as 1-8 (= log 3), we may substitute its equal. M-N 
or 3-N, extending to the central diagonal line 1-H, marked x; and then, 
since, for instance, 1-1.2 = N-Q, 1-3 = N-K, etc., we may add any log 
(as 1-3) by moving upward from line x (as from N to K) or to the right, 
and siw^act any log (as 1-1.2) by moving downward (as from N to Q) or to 
the l^. This facilitates the performance of a series of operations. 

Thus: 

To multiply 1.5 by 2 (= 3). by 3 (= 9), and divide by 2 (= 4.5). 

F-G = 1-F = log 1.5. Add G-J = 1-2 = log 2 ; sum = F-J = log 3 = 1-3 = 
M-N. Add N-K = 1-3 = log 3 ; sum = M-K = log 9 = 1-9 = 9-R. Subtract 
R_T = 1-2 = log 2 ; remainder = 9-T = log 4.5. 

For an example of the application of this principle to engineering prob- 
lems see " Diagrams for proportioning wooden beams and posts," by Carl 
S. Fogh, " Engineering News^', Sept. 27, 1894. 

(la) If eipatiTe exponents. If a: is in the dm«or, the line will slope 
in the opposite direction, or downward from left to right. Thus, line 4-2 
leaving 1-Y, at 4, and forming, with 1-X, the angle X, 2. 4, with tangent 

= ^^ ' ■ • ^ = — 2, represents the equation : j/ = - , = 4 x-*. 

(IS) If the lines of products, powers, and roots, C «, a?», and y^ etc., 
be drawn at angles whose tangents are less by 1 than those of the angles 
formed by the corresponding lines in our figure, the resmts may be read 
directly from oblique lines drawn parallel to 2-2. Lines (C x) giving multi- 
ples and sub-multiples of the first power of x then become horwmial lines 

(14)" Powers and roots by tbe slide rale. Scales C and D being 
twice as large as scales A and B, these scales, with their ends coinciding, 
form a table of squares and of square roots. See (3). By moving the slide 
we solve equations of the forms jy = (C x)^ and y = C x^. Thus, with the 



(*) In each of these lines, the product of the two numbers at its ends is 
= 10. Thus, in line 2-A. 2 X 5 = 10 ; in 3-K, 8 X 3.38 ... = 10, etc. The 
chart thus furnishes a table of reciprocals. . , 

(t) Even with full-size charts and slide rules for actual use, accuracy is 
not to be expected beyond the third or fourth significant flgure. 

(t) A chart of this kind, prepared by Major Wm. H. Bixby, U. S. A., 
atter the method of L6on Lalanne. Corps de Fonts et Chaussees, France, 
is published by Messrs. John Wiley & Sons, New York. Price, 25,centi. 



LOOARITHHIO CBABT AND SLIDE RULE. 



77 



slide M shown, each nmnber oa A is «= the sqaftre of (1.6 X the coinciding 
number on G) ; while, with 1 on B opposite 1.5 on A, each number on A is = 
1.5 X the square of the coinciding number on C. 

(15) Since x» = *" X x, we find cubes or third powers by placing the 
slide with 1 on B opposite x^ on A ({. e., opposite x on D), see (3), and read- 
ing «■ f^om A opposite x on B. Thus, 1.5* = ?. Place 1 on B opposite 1.5 on 
D ; t, «., opposite 1.5* (= 2.25) on A. Then, on A, opposite 1.5 on B, find 
8.875 = 1.5*. Or, turn the slide end for end. Place 1.5 on B opposite 1.5 
on D, t. e., opposite 1.5* = 2.25 on A. Then, adding log 1.5 (on B) to log 2.25 
on A, we find 3.375 (= 1.5') on A opposite 1 on B. 

(16) Conversely, to find v'iT we shift the slide (in its normal position) 
until we find, on B, opposite x on A, the same number as we have on I) op- 
posite 1 on 0, and this number will be =° f/3c7 . Or, turn the slide end 
for end,* place 1 on C opposite x on A, and find, on B, a number wl^ich 

coincides with its equal on D. This number is = i^zT See also (17), (18). 

(17) On the back of the slide is usually placed a scale of logs (see scale 
shown below the rule in figure) and two scales of angles, marked " S " and 
" T " respectively, for finding sines of angles greater than 0*^ 34' . . . ", and 
taxigents of angles between 5° 42' . . . " and 45°. 

(18) Placing 1 on C opposite any number a; on D (with slide in its normal 
pofiitiou), log X IS read from the scale of logs by means of an index on the 
Sack of the rule. The logs may be used in fitidlng powers and roots. 

ZtogB. 

t^ 0.0 OJf 0,4, 0,e 0.8 1,0 1,9 1^ X.e 1,8 s,o 9J» 

J I I I I I 1 t I I ' I « 

J«0«. Cfi 5 8 4 H €7891 3 3 

-U » I . I I .1 I It .... 1 ■ I 



1 » L ' L L 1 11 * — ^ — ^ — ' I ' ' I • 1 'I I I L r 

7. ^ 



4 J 078»ljA 

' . ' ■ i '' i ' r' i 



JBl » 8 dS87891 2 8 4S87891M 

r^ U — ,"^ f , ?, f , ^ f J.Mfg ) 

^00. tPJ IJf 9 3 4 5 7 8 llA 

-I > 1 1 1 1 1 1 1 r r 1 r 

ij> 0.0 0,1 0,2 0,3 0,4 o^ o.e 0.7 oa 0,9 ijo .1,1 

(19) To find the sine or tang^ent of an angle a ; bring a, on scale S or 
T, as the case may be, opx>osite the index on back, and read the natural 

inot logarithmic) sine or tangent opposite 10 at the end of A or D : sines on 
S, and tangents on C. Or, invert the slide, placine S under A, and T over 
D. with the ends of the scales coinciding. Then the numbers on A and ]> 
are the sines and tangents, respectively, of the angles on S and T. 
Caution. Sines of angles less than 5° 45' ... " are less than 0.1. 

Tangents " " betw. 5° 42' . . . " and 45° are betw. 0.1 and 1.0. 

(90) On the back of the rule is usually printed a table of ratios of num- 
bers in common use, for convenience in operating with the slide rule. Thus : 
diameter 118 U. S. gallons 3 .. . ...... 

circumference = »5 = "i^nl^ ' 25 <"" * «''«° ""*""*>' of water). 

(31) Soaping the edges of the slide and the groove in which it runs, will 
often cure sticking, wnich is apt to be very annoying. If the slide is too 
loose, the groove may be deepened, and small springs, cut from narrow 
steel tape, inserted between it and the edge of the slide. 



(*) With the slide thus reversed, and with the ends of the scales coin- 
ciding, the numbers on A and Bare reciprocals (page 62), as are also 
those on C and D. 



TABLE or LOOABITHHB. 



TABLE OF LOOARITHMS. 



79 



Commoii or Brlgrs* I«oir»i4<l>iM>* 



1«. 



No. 





M 


81954 


«7 


82607 


68 


83250 


60 


83884 


70 


84609 


71 


86135 


72 


86783 


73 


86S32 


74 


86023 


76 


87606 


76 


88081 


77 


88649 


78 


89209 


79 


89762 


80 


90800 


81 


90848 


82 


91381 


83 


91907 


84 


92427 


86 


92041 


86 


98449 


87 


93961 


88 


94448 


80 


94939 


00 


96424 


01 


96904 


02 


96378 


93 


96848 


94 


97312 


95 


97772 


96 


08227 


97 


98677 


98 


99122 


99 


99668 



82020 
82672 
83314 
83947 
84671 

86187 
86703 
86891 
86981 
87664 
88138 
88705 
89266 
89817 
90663 

90902 
01434 
91960 
92479 
92993 
03600 
94001 
94497 
94987 
96472 

06951 
96426 
96806 
97369 
97818 
98272 
98721 
99166 
99607 



82085 
82736 
83378 
84010 
84633 

86248 
86853 
86461 
67040 
87621 
88195 
88761 
89320 
89872 
90417 

90966 
91487 
92012 
92531 
98044 
93560 
94051 
94546 
95036 
96620 

05999 
96473 
96041 
97405 
97863 
98317 
98766 
99211 
99651 



S 



82161 
82801 
83442 
84073 
84696 

85309 
86913 
86610 
87098 
87679 
88262 
88818 
89376 
89927 
90471 

91009 
91640 
92064 
92682 
93095 
93601 
94101 
94596 
96085 
05568 

96047 
96620 

97461 
97909 
98362 
98811 
99266 



82216 
82866 
83505 
84136 
84767 

86369 
85978 
86569 
87157 
87737 
88309 
88874 
89431 
89982 
90626 

91062 
91592 
92116 
92634 
93146 
93651 
94161 
94646 
95133 
95616 

96094 
96667 
97034 
97497 
97964 
98407 
98866 
99299 
99738 



82282 
82930 
83669 
84198 
84818 

86430 
86033 
86628 
87216 
87794 
88366 
88930 
89487 
90036 
90679 

91115 
91646 
92168 
92685 
93196 
93701 
04200 
94694 
96182 
96664 

96142 
96614 
97081 
97543 
98000 
98452 
98900 
99348 
99782 



6 



82347 
82994 
83632 
84260 
84880 

86491 
86093 
86687 
87273 
87852 
88422 
88986 
89542 
90091 
90683 

91169 
91608 
92220 
92737 
93247 
93751 
94260 
94748 
96230 
96712 

96189 
96661 
97127 
97689 
98046 
98497 
98946 
99387 
99826 



82412 
83068 
83696 
84323 
84941 

86661 
86153 
86746 
87332 
87909 
88479 
89042 
80697 
90146 
00687 

91222 
91750 
92272 
92788 
93298 
93802 
94300 
04792 
96279 
96760 

96236 
96708 
97174 
97636 
98091 
98642 
98989 
99431 
99869 



s 





82477 


82542 


83123 


83187 


83758 


83821 


84385 


84447 


86003 


85064 


86612 


85672 


86213 


86272 


86806 


86864 


87890 


87448 


87966 


88024 


88636 


88692 


89098 


89163 


89662 


89707 


90200 


90264 


90741 


90794 


01276 


91328 


01808 


91866 


92324 


92376 


92839 


92890 


93848 


93399 


93862 


93902 


94840 


94398 


94841 


94890 


95327 


06376 


95808 


96866 


06284 


96331 


96754 


96801 


97220 


97266 


97680 


97726 


08136 


98181 


98587 


98632 


99033 


99078 


09475 


99619 


99913 


99966 



Prop* 

66 
66 
64 
63 
62 

61 
60 
60 
68 
67 
66 
66 
•66 
64 
64 

63 
68 
62 
61 
61 
60 
49 
4f 
48 
48* 

48 
47 
47 
46 
46 
46 
46 
44 
44 



For extended table of lofpaiittoms see pages 80-91. The table 
above, being given on two opposite pages, avoids the necessity of turning leaves. 
It contains no error as great as 1 in the final figure. The proportional parts, in 
the last column, eive merely the average difi'erence for each line. Heuce, when 
dealing with small numbers, and using 5-place logs, it is better to find difTer- 
enoes by subtraction : but where a two-page table » used, interpolation is often 
auneoeasary. Indeed, the first four, or even the first three, places of the man- 
tissas here f^ven will often be found sufficient. If rhe first number dropped is 
S or more, increase by 1 the last figure retained. Thus, for log 660, mantissa 
» 81954, or 8195, or 820. 

Miiltlplleatioii. Log a 6 = log a + log b. 
Dlvtoton. Ix>g ^ s log a — log b. 
Involatlon (Powers). Log of* — n. log a. 
BTOlntion (Roots). Log^^s^ * ^^^ 



Log 2870 


-8.45788 


u 


287 


= 2.46788 


«l 


28.7 


» 1.45788 


u 


2.87 


»= 0.45788 



n 
sristtes. 








Log 0.287 


= 0.45788 - 


1 


= 1.46788 


" 0.0287 


= 0.46788 - 


2 


= 2.45788 


" 0.00287 


= 0.45788 - 


8 


= 8.45788 


" 0.000287 


= 0.46788 - 


■4 


= 4.4578^ 



80 



LOQARITHMS. 



O^mniMi or Brim* I^OffaritliimB, Brnio » lO. 



90. Log. 



,1000 

01 
02 
03 
04 
09 
06 
07 
08 
09 

1010 

11 

12 
13 
14 
15 
16 
17 
18 
19 

1020 

21 
22 
23 
•• 24 
25 
26 
27 
28 
29 

1030 

31 
32 
33 
34 
36 
36 
37 
88 
89 

1040 

41 
42 
43 
44 
45 
46 
47 
48 
49 



00000 

043 

— Q87 

130 

173 

—217 

—260 

—303 

346 

389 



432 

475 

518 
—561 
—604 
—647 

689 

732 
—775 

817 

860 
—903 

945 
—988 
01030 

072 

1571^2 
199:^2 

— 242,t^ 

42 



43 
44 
43 
43 
44 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
42 
43 
43 
42 
43 

43 



42 
43 
42 

42 
43 



—284 

—326 

—368 

410 

452 

494 

—536 

—578 

—620 

-«62 

703 

745 

—787 

828 

870 

—912 

953 

—995 

02036 

—078 



42 
42 
42 
42 
42 
42 
42 
42 
42 
41 

42 

42 
41 
42 
42 
41 
42 
41 
42 
41 



No. 



Log. 



1090 02119 7: 

160 J* 

—202 ;f 

53 — 243j} 



51 
52 



54 
55 
56 
57 
58 
59 

1060 

61 
62 
63 
64 
65 
66 
67 
68 
, 69 

1070 

71 
72 
73 
74 
75 
76 
77 
78 
79 

1080 

81 
82 
83 
84 
85 
86 
87 
88 
89 

1090 

91 
92 
93 
94 
95 
96 
97 
98 
99 



284 
325 
366 
407 

■■^'1 45/ 

—490 

-^31 

-572 

612 

653 

694 

-735 

—776 

816 

857 

—898 

938 
—979 
03019 
—060 

100 
—141 

181 
—222 
—262 

302 

342 
—383 
—423 
—463 
—503 
—543 
—583 
—623 
-663 
—703 

—743 
782 
822 
862 

-902 
941 
981 

04021 
060 

—100 



41 
41 
41 
42 
41 
41 

41 
40 
41 
41 
41 
41 
40 
41 
41 
41 

41 
40 
41 
40 
41 
40 
41 
40 
40 
40 

41 

40 
40 
40 
40 
40 
40 
40 
40 
40 

39 
40 
40 
40 
39 
40 
40 

39 
40 
39 



No. 



1100 

01 
02 
03 



Log. 



^ 



04139 

—179 

218 

—258 



04 —297 



05 


336 


06 


—376 


07 


—415 


08 


—454 


09 


493 


1110 


532 


11 


571 


12 


610 


13 


—650 


14 


—689 


15 


727 


16 


766 


17 


805 


18 


844 


19 


883 


1120 


—922 


21 


—961 


22 


999 


23 


05038 


24 


-077 


25 


116 


26 


—154 


27 


192 


28 


—231 


29 


269 


1130 


—308 


31 


346 


32 


—385 


33 


—423 


34 


461 


35 


—500 


86 


—538 


37 


576 


38 


614 


39 


652 


1140 


690 


41 


—729 


42 


—767 


43 


—805 


44 


—843 


45 


—881 


46 


918 


47 


956 


48 


994 


49 


06032 



40 
39 
40 
39 
39 
40 
39 
39 
89 
39 

39 
39 
40 
39 
38 
39 
39 
39 
39 
39 

39 
38 
39 

39 
38 
39 
38 
39 
38 
39 

38 
39 
38 
38 
39 
38 
38 
38 
38 
38 

89 
38 
38 
38 
38 
37 
38 
38 
38 
38 



No. 



Log. 



IISO 06070 

51 —108 

52 145 

53 1—183 

54 —221 



56 
56 
57 
58 
59 

1160 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1170 

. 71 
72 
73 
74 
75 
76 
77 
78 
79 

1180 

81 
82 
83 
84 
85 
86 
87 
88 
89 

1190 

91 
92 
93 
94 
95 
96 
97 
98 
99 



258 
—296 

333 
—371 

408 

—446 

483 

—521 

—558 

595 

—633 

—670 

707 

744 

781 

—819 
—856 
—893 
—930 
—967 
07004 
^^41 
—078 
—115 
151 

188 

—225 

—262 

298 

335 

—372 

408 

445 

—482 

518 

—660 
591 

—628 
664 
700 

—737 
773 
809 

—846 

—882 



5 

38 
37 
38 
38 
37 
38 
37 
88 
87 
88 

87 
38 
37 
37 
38 
37 
87 
37 
87 
38 

37 
37 
37 
37 
37 

37 
37 
37 
36 
37 

37 
37 
36 
37 
37 
36 
87 
87 
36 
37 

86 
87 
36 
36 
37 
86 
36 
37 
36 
36 



No. 



1200 

01 
02 
08 
04 
05 
06 
07 
* 08 
09 

1210 

11 

It 

13 

14 

15 

16 

17 

18 

19^ 

1220 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1230 

31 
32 
33 
34 
35 
36 
37 
.38 
39 

1240 

41 
42 
43 
44 
45 
46 



liOg. s 



07918 36 

954 36 
990 37 

36 



08027;;^ 



-099!^ 
— 135 on 
—171^ 



48 
49 



—207 
—243 

—279 
314 
350 
386 

—422 

—468 
493 
629 

—565 
600 

—636 
—672 

707 
—743 

778 
—814 

849 

884 
—920 

965 

—991 

09026 

061 

096 

—132 

—167 

—202 

—237 

272 

307 

342 
377 
412 
447 
482 
—517 



.47^-687 



621 
656 



36 
36 
36 

85 
36 
36 
86 
86 
36 
86 
86 
86 
86 

86 
86 
36 
35 
86 
36 
36 
36 
35 
86 

35 

85 
85 
36 
85 
85 
85 
35 
35 
35 

35 
85 
35 
35 
86 
36 
35 
84 
35 
85 



Example: 

To find Log. 11826 : 
Log. 11830 = 07298 
Dif. = 10 36 

Log. 11820 = 07262 

11826 — 11820 e= 6 
Dif. for 6 under 36 

= 22 
Log. 11826 = 

07262 + 22 = 07284 



1 
2 
3 

4 
5 
6 

7 
8 
9 



44 

4 
9 
13 
18 
22 
26 
31 
Z5 
40 



43 

4 
9 
13 
17 
22 
26 
30 
34 
39 



42 

4 
8 
13 
17 
21 
25 
29 
84 
88 



41 

4 
8 
12 
16 
21 
25 
29 
33 
87 



40 

4 
8 
12 
16 
20 
24 
28 
32 
36 



39 

4 
8 
12 
16 
20 
23 
27 
31 
35 



38 

4 
8 
11 
15 
19 
23 
27 
30 
84 



37 

4 
7 
11 
15 
19 
22 
26 
80 
33 



36 

•4 
7 
11 
14 
18 
22 
25 
29 
32 



35 

4 
7 
11 
14 
18 
.21 
25 
28 
32 



84 

3 

7 

10 
14 
17 
20 
24 
27 
81 



1 
2 
3 

4 
5 
6 

7 

8 

9 



LOGABITHM8. 
r BrlCK* Irf»s*'"l>»»- Base = 



LOQAKITHU8 
CMnnB*n •r Brigita LoynrlMiiii 



liOOABITHHS. 



83 



Oommoii or Brlns Ij<»s»rltliiiis. Base » 10. 



9o. 



1790 

62 
64 
66 
68 

1760 

62 
64 
66 
68 

1770 

72 
74 
76 
78 

1780 

82 

84 
86 
88 

1790 

92 
94 
96 
98 

1800 

02 
04 
06 
08 

1810 

12 
14 
16 
18 

1820 

22 
24 

26 
28 

18S0 

32 
34 
86 
88 

1840 

42 
. 44 

46 

48 



Log. 



24304 
853 

k-403, 
462' 

—602 

551 

—601 

—650 

699 

748 

797 
846 
895 
944 
993 

26042 

—091 

139 

188 

—237 

286 
—334 

382 
—431 
—479 

627 
675 

—624 
-672 

—720 

—768 
—816 
—864 
—912 
969 

26007 
—055 
102 
150 
—198 

245 
—293 
—340 

387 
—436 

—482 

—629 

676 

623 

670 



S3 

49 
50 
49 
50 
49 

60 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 

48 
49 
49 
48 

49 
48 
49 

48 
48 

48 
49 
48 
48 
48 

48 
48 
48 
47 
48 

48 
47 
48 
48 
47 

48 
47 
47 
48 
47 

47 
47 
47 
47 
47 



Ko. 



1850 

52 
64 
56 
58 

1800 

62 
64 
66 
68 

1870 

72 
74 
76 
78 

1880 

82 
84 
86 
88 

1800 

92 
94 
96 
98 

1900 

02 
04 
06 
08 

1910 

12 
14 
16 
18 

1920 

22 
24 
26 
28 

1930 

32 
34 
36 
38 

1940 

42 
44 
46 

48 



Log. 



26717 
764 
—811 
—868 
—905 

951 
—998 
27045 

091 
—138 

184 
—231 
—277 

323 
—370 

-416 

—462 

508 

564 

600 

646 

692 

—738 

—784 

—830 

875 

921 

—967 

28012 

—068 

103 
—149 

194 
—240 
—285 

330 

375 

—421 

—466 

—511 

-656 
—601 
—646 
—691 
735 

780 
—825 
—870 

914 
—959 



S3 
O 

47 

47 
47 
47 
46 

47 
47 

46 

47 
46 

47 
46 
46 
47 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
45 

46 
45 
46 
45 
45 

45 
46 
45 
45 
45 

45 
45 
45 
44 

45 

45 
45 
44 
46 
44 



ToflDdLog. 18117: 

Log. 18120 ==25816 
Bif 20 48 
Log. 18100 = 25768 
18117 — 18100 = 17 
Under 48 
Dif. for 10 — 24 
7 = 17 



u 



" " 17 = 41 
Lttj. 18117 = 
^68 + 41 =- 26809. 



No. 



1 
2 
8 

4 
6 
6 
7 
8 
9 
10 



00 

3 
5 
8 

10 

13 

16 

18 

20 

23 

26 



49 

2 
6 
7 
10 
12 
15 
17 
20 
22 
26 



1900 

52 
54 
66 
58 

1960 

62 
64 
66 
68 

1970 

72 
74 
76 
78 

1980 

82 
84 
86 
88 

1990 

92 
94 
96 
98 

2000 

02 
04 
06 
06 

2010 

12 
14 
16 
18 

2020 

22 
24 
26 

28 

2030 

32 



Log. 



29008 
—048 

092 
—187 

181 

—226 

—270 

314 

358 

—403 

—447 
—491 
—636 
—679 
—628 

—667 
710 
754 

—798 
—842 

886 
—929 
—973 
30016 
—060 

—108 
146 

—190 
233 
276 

—820 

—363 

—406 

449 

492 

635 

678 

621 

—664 

—707 

—750 
792 



34 


8a5 


36 


—878 


38 


920 


2040 


963 


42 


31006 


44 


048 


46 


—091 


48 


—183 



45 
44 
46 
44 

46 

44 
44 
44 
46 
44 

44 
44 
44 
44 
44 

43 
44 
44 

44 

43 

44 
44 

48 
44 

48 

48 
44 

43 
43 
44 

48 
48 
43 
43 
43 

43 
43 
43 
43 
43 

42 
43 
43 
42 
43 

43 

55 
43 
42 
42 



No. 



2000 

62 
54 
66 
68 

2060 

62 
64 
66 
68 

2070 

72 
74 
76 
78 

2080 

82 
84 
86 
88, 

2090 

92 
94 
96 
98 

2100 

02 
04 
06 
08 

2110 

12 
14 
16 
18 

2120 

22 
24 
26 
28 

2130 

32 
34 
36 
38 

2140 

42 
44 
46 

48 



Log. 



31176 

—218 

260 

802 

—846 

—887 

—429 

-471 

618 

665 

697 
—689 
—681 
—723 
—765 

806 
848 
—890 
981 
973 



32016 41 



I 

43 
42 
42 
48 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
41 

42 
42 
41 
42 
42 



056 
—098 

189 
—181 

—222 
263 
—306 
—846 
887 

428 
469 
610 
—652 
—593 

—684 

—675 

715 

756 
797 

-888 

—879 

919 

960 

33001 

041 
—082 

122 
—163 

203 



42 
41 
42 
41 

41 
42 
41 
41 
41 

41 
41 
42 
41 
41 

41 
40 
41 
41 

41 

41 
40 
41 
41 

40 

41 
40 
41 
40 
41 



No. 


Log. 

88244 


2100 


62 


284 


64 


—825 


66 


—866 


68 


406 


2160 


445 


62 


—486 


64 


—626 


66 


—666 


68 


—606 


2170 


--646 


72 


—686 


•74 


—726 


76 


—766 


78 


-806 


2180 


-«46 


82 


886 


84 


926 


86 


966 


88 


84005 


2190 


044 


92 


084 


94 


—124 


96 


168 


98 


—203 


2200 


242 


02 


—282 


04 


821 


06 


—861 


08 


—400 


2210 


489 


12 


—479 


14 


—618 


16 


—667 


18 


696 


2220 


686 


22 


674 


24 


718 


26 


—768 


28 


—792 


2230 


880 


32 


869 


34 


908 


86 


947 


38 


986 


2240 


85026 


42 


—064 


44 


102 


46 


—141 


48 


—180 



15 

41 
40 
40 
40 

41 
40 
40 
40 
40 

40 
40 
40 
40 
40 

89 
40 
40 
40 

89 

40 
40 
89 
40 
89 

40 
89 
40 
89 
89 

40 
89 
89 
39 
39 

39 
89 
40 
8f 
38 

39 
89 
89 
39 
39 

39 
38 
39 
39 
38 



48 

2 

5 

7 
10 
12 
14 
17 
19 
22 
24 



47 

2 
5 

7 
9 

12 

14 

16 

19 

21 

24 



46 

2 

6 

7 

9 
12 
14 
16 
18 
21 
23 



40 

2 
5 
7 
9 

11 

14 

16 

18 

20 

23 



44 

2 

4 

7 

9 
11 
13 
16 
18 
20 
22 



43 

2 

4 
6 
9 

11 

13 

15 

17 

19 

22 



42 

2 
4 
6 
8 

11 

18 

16 

17 

19 

21 



41 

2 

4 

6 

8 

10 

12 

14 

16 

18 

21 



40 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 



39 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 



88 

2 

4 

6 

8 

10 

11 

18 

16 

17 

19 



1 
2 
S 

4 
5 
6 
7 
8 
9 
10 



84 



LOOABITHMB. 



CommoB or Brlns I«oirftiltli; 



10. 



Ho. 


Log. 


3200 


85218 


02 


—267 


64 


295 


56 


—834 


68 


372 


2360 


—411 


62 


449 


64 


—488 


66 


—526 


68 


564 


3370 


-603 


72 


—641 


74 


679 


76 


717 


78 


765 


33S0 


793 


82 


—832 


84 


—870 


86 


—908 


88 


—946 


3390 


—984 


d2 


36021 


M 


059 


m 


097 


98 


185 


3300 


—173 


02 


—211 


04 


248 


Ort 


—286 


08 


—324 


3310 


361 


12 


—399 


14 


436 


16 


—474 


18 


511 



3320, 

22 1 
24 
26 ; 
28 

3330 

32 

34 
36 
88 

2340 

42 
44 
46 

48 



—549 

686 
—624 
—661 

698 

—786 

—773 

810 

847 

884 

—822 
—959 
—996 
37033 
—070 



89 
38 
39 
38 
39 

38 
39 
38 
38 
39 

38 
38 
38 
38 
38 

39 
38 
38 
38 
38 

37 
38 
38 
38 
38 

38 
37 
38 
38 
37 

38 
37 
38 
37 
38 

37 
38 
37 
37 
38 

37 
37 
37 
37 
38 

37 
37 
87 

S7 
87 



No. 



3850 

62 
64 
66 
68 

3360 

62 
64 
66 

68 

3370 

72 

74 
76 

78 

3380 

82 
84 
86 
88 

3390 

92 
94 
96 
98 

3400 

02 
04 
06 
08 

3410 

12 
14 
16 
18 

3430 

22 
24 
26 

28 

3480 

32 
34 
36 
88 

3440 

42 
44 

46 

48 



Lof. 



87107 
—144 
—181 
—218 
264 

291 

—828 

—366 

401 

488 

—476 
611 
648 

—585 
621 

—658 
694 

—731 
767 
803 

—840 

876 

912 

—949 

—986 

38021 

057 

093 

—130 

—166 

-202 
—238 
—274 
—810 
—346 

—382 
417 
453 
489 

—625 

—661 
596 
632 

—668 
703 

—739 

—775 
810 

—846 
881 



87 
37 
37 
36 
37 

87 
37 
36 
37 
37 

36 
37 
37 
36 
37 

36 
37 
36 
36 
37 

36 
37 
36 
36 

86 
36 
37 
36 
36 

36 
36 
36 
36 
36 

35 
36 
36 
36 
36 

a5 

36 
36 
35 
36 

36 
35 
36 
35 
36 



No. 



3450 

62 
64 
66 

68 

3460 

62 
64 
66 
68 

3470 

72 
74 
76 
78 

3480 

82 
84 
86 

88 

3490 

92 
94 
96 
98 

3500 

02 
04 
06 
08 

3510 

12 
14 
16 
18 

3530 

22 
24 

26 
28 

3530 

32 
34 
36 
38 

3540 

42 
44 
46 

48 



L09. 



38917 
962 
987 

39023 
068 

—094 

—129 

164 

199 

—236 

—270 

—805 

—840 

875 

410 

446 
480 
615 

660 
685 

—620 

—666 

—690 

724 

769 

794 

—829 
863 
898 

—933 

967 
40002 
—037 

071 
—106 

140 
—176 
—209 

243 
—278 

312 
346 

—381 

—415 

449 

483 
—518 
—562 
—586 
—620 



86 
86 
86 

35 
36 

86 
35 
35 
36 
35 

35 
35 
35 
»5 
36 

35 
35 
35 
35 
35 

35 
35 
34 
36 
35 

86 
34 
85 
35 
34 

35 

35 
34 
35 
34 

35 
34 
34 
35 
34 

34 
35 
34 
34 
34 

35 
34 
34 
34 
34 



No. 



3550 

62 
64 
56 
68 

3560 

62 
64 
66 
68 

3570 

72 
74 
76 

78 

3580 

82 
84 

36 
88 

3590 

92 
94 
96 
98 

3600 

02 
04 
06 
08 

3610 

12 
14 
16 
18 

3630 

22 
24 
26 
28 

3630 

32 
34 
36 
38 

3640 

42 
44 
46 

48 



Lof. 



40654 
688 
722 
766 
790 

—824 
—868 
—892 
—926 
—960 

993 
41027 
—061 
—096 

128 

—162 
—196 

229 
—263 

296 

-4J30 
363 

—897 
430 

—464 

497 
—631 
664 
697 
—631 

664 
697 
—731 
—764 
—797 

830 
863 
896 
929 
—963 

—996 
42029 
—062 
—095 
127 

160 

193 

226 

—269 

—292 



84 
84 
84 
34 
34 

34 
34 
34 
34 
33 

34 

34 
34 
38 
34 

34 
83 
34 
33 
34 

33 
34 
33 
84 
33 

34 
33 
33 
34 
33 

38 
34 
33 
33 
33 

33 
33 
33 
34 
33 

33 

33 
32 
33 

33 
33 
33 
33 
33 



No. 



3650 

62 
64 

66 
68 

3660 

62 
64 
66 
68 

3670 

72 
74 
76 
78 

3680 

82 
84 
86 
88 

3690 

92 
94 
96 
98 

3700 

02 
04' 
06 
08 

3710 

12 
14 
16 
18 

3730 

22 
24 
26 
28 

3730 

32 
34 
36 
38 

3740 

42 
44 
46 
48 



Log. 



42826 
867 
890 

—423 
456 

488 
—621 
663 
686 
-619 

661 
—684 

716 
—749 

781 

813 
—846 

878 
-911 

-943 

976 
43008 
—040 
—072 

104 

136 
—169 
—201 
-233 
—265 

—297 
—829 
—861 
—393 
—426 

—467 
-489 
—521 
—653 
684 

616 

648 

—680 

—712 

743 

776 

—807 

888 

870 

—902 



32 
33 
33 
32 
33 

33 
32 
33 
33 
32 

83 
82 
33 
32 
32 

38 
32 
38 

32 
32 

^ 
32 
82 
32 
32 

33 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
31 
32 

32 
32 
32 
31 
32 

32 
31 
82 
82 
81 



To find Log. 23335 : 

LoK. 23340 = 36810 
Dif. 20 37 
Log. 23320 = 36773 
23385 — 23820 = 15 
Under 37 
Dif. for 10 = 19 
" " 5 =__9 

" " 15 = 28 
' '^^. 23335 = 

-78 + 28 = 36801. 





39 


38 


37 


36 


85 


34 


38 


83 


31 


1 


2 


2 


2 


2 


2 


2 


2 


2 


2 


3 


4 


4 


4 


4 


4 


8 


8 


3 


8 


3 


6 


6 


6 


6 


6 


6 


6 


6 


6 


4 


8 


8 


7 


7 


7 


7 


7 


6 


6 


5 


10 


10 


9 


9 


9 


9 


8 


8 


8 


6 


12 


11 


11 


11 


11 


10 


10 


10 


9 


7 


14 


18 


13 


13 


12 


12 


12 


11 


11 


8 


16 


15 


15 


14 


14 


14 


13 


18 


IS 


9l 18 


17 


17 


16 


16 


16 


16 


14 


14 


10 


\70 


19 


19 


18- 


18 


17 


17 


16 


16 



1 

2 

8 

4 
5 
6 
7 
8 
9 
19 



LOOABTTHMS. 



85 



Common or Brigrips I«ograrittams. Base « 10. 



No. 



»750 

52 
54 
56 
58 

62 
64 
66 
68 

«770 

72 
74 
76 

78 

»780 

82 
84 
86 
88 

9790 

92 
94 

96 
98 

98O0 

02 
04 
06 
08 

12 
14 
16 

16 

98»0 

22 
24 
26 

28 

»S30 

32 
84 
36 
38 

42 
44 
46 
48 



Log. 



43933 
—965 

996 
44028 

059 

-091 
122 

—154 
185 

—217 

—248 

279 

—311 

—342 

373 

404 
—436 
—467 
498 
529 

560 
—692 
—623 
—664 
—686 

—716 

—747 
—778 
—809 
—840 

—871 

—902 

982 

963 

994 

45025 

—056 

086 

117 

—148 

—179 

209 

—240 

—271 

301 

—332 
362 

—393 
423 

—454 



S3 



32 
31 
d2 

31 
32 

31 
32 
31 
32 
31 

31 

32 

31- 

31 

31 

32 
31 
31 
31 
31 

32 
31 
31 
31 
31 

31 
31 
31 
31 
81 

31 
30 
31 
31 
31 

31 
30 
31 
31 
31 

30 
31 
31 
30 
81 

30 
31 
30 
31 
30 



No. 



$8850 

52 
54 
56 

58 

2860 

62 
64 
66 
68 

2870 

72 

74 
76 

78 

2880 

82 
84 
86 
88 

2800 

92 
94 
96 
98 

2900 

02 
04 
06 
08 

2910 

12 
14 
16 
18 

2920 

22 
24 
26 

28 

2930 

32 
34 
36 
38 

2940 

42 
44 
46 

48 



Log. 



45484 
—515 

545 
—576 

606 

—637 
—667 
697 
—728 
—758 

788 
818 
—849 
—879 
909 

939 
969 
46000 
—030 
—060 

—090 
—120 
—150 
—180 
—210 

—240 
—270 
—300 
—330 
359 

389 
419 

—449 
-479 

—509 

538 
568 
^598 
627 
657 

—687 
716 
746 

—776 
805 

—835 
864 

—894 
923 

—953 



31 
30 
31 
80 
31 

30 
30 
31 
30 
30 

30 
31 
30 
30 
30 

30 
31 

30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
29 
30 

30 
30 
30 
30 
29 

30 
30 
29 
30 
80 

29 
30 
30 
29 
30 

29 
30 
29 
30 
29 



No. 



2950 

52 
54 
56 
58 

2060 

62 
64 
66 
68 

2970 

72 

74 
76 
78 

2980 

82 
84 
86 
88 

2990 

92 
94 
96 

98 

3000 

02 
04 
06 
08 

3010 

12 
14 
16 
18 

802O 

22 
24 
26 

28 

3030 

32 
34 
86 
38 

3040 

42 
44 
46 

48 



Log. 



46982 

47012 

041 

070 

—100 

129 

-159 

—188 

217 

246 

—276 
—305 
334 
363 
392 

—422 

—451 

-480 

—509 

538 

567 
596 
625 
654 
683 

712 

741 

—770 

—799 

—828 

—857 
885 
914 
943 

—972 

48001 

029 

058 

-087 

—116 

144 
—173 
—202 

230 
—259 

287 
-316 

344 
—373 

401 



Cm 



30 

29 
29 
30 
29 

30 
29 
29 
29 
30 

29 
29 
29 
29 
30 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

28 
29 
29 
29 
29 

28 
29 
29 
29 

28 

29 
29 
28 
29 
28 

29 
28 
29 
28 
29 



No. 



3050 

52 
54 
56 
58 

3060 

62 
64 
66 

68 

3070 

72 
74 
76 
78 

3080 

82 
84 
86 
88 

3090 

92 
94 
96 
98 

3100 

02 
04 
06 
08 

3110 

12 
14 
16 
18 

3120 

22 
24 
26 
28 

3130 

32 
34 
36 

38 

3140 

42 
44 
46 

48 



Log. 



48430 
458 

—487 
515 

,-^44 

572 
—601 
—629 

657 
—686 

—714 
742 
770 
—799 
—827 

855 
883 
911 
—940 
—968 

—996 
49024 
052 
080 
108 

136 
164 
192 
220 
248 

276 
—304 
—332 
—360 
—388 

415 

443 

471 

-499 

—527 

554 
582 
—610 
—638 
665 

—€93 
—721 

748 
—776 

803 



(M 



28 
29 
28 
29 
28 

29 
28 
28 
29 
28 

28 
28 
29 
28 
28 

28 
28 
29 
28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
27 

28 
28 
28 
28 
27 

28 
28 
28 
27 
28 

28 
27 
28 
27 
28 



No. 



3150 

52 
54 
56 
58 

3160 

62 
64 
66 
68 

3170 

72 
74 
76 
78 

3180 

82 
84 
86 
88 

3190 

92 
94 
96 

98 

3200 

02 
04 
06 
08 

3210 

12 
14 
16 

18 



Log. 



49831 28 



—859 
886 

—914 
941 

—969 
996 

50024 
051 

—079 

—106 
133 

—161 
188 
215 

—243 
270 
297 
—325 
—352 

379 
406 
433 
—461 

—488 

—515 
542 
669 
596 
623 

—651 
—678 
—705 
—732 
—759 



3220 —786 



22 


—813 


24 


—840 


26 


866 


28 


893 


3230 


920 


32 


947 


34 


974 


36 


51001 


38 


—028 


3240 


—055 


42 


081 


44 


108 


46 


—135 


48 


—162 



27 
28 
27 
28 

27 
28 

27 
28 
27 

27 
28 
27 
27 
28 

27 
27 

28 
27 
27 

27 
27 
28 
27 
27 

27 
27 
27 
27 
28 

27 
27 
27 
27 
27 

27 
27 
26 
27 
27 

27 
27 
27 

27 
27 

26 
27 
27 
27 
26 



To find Log. 29019: 

Log. 29020 == 46270 
Dil 20 30 

Log. 29000 = 46240 
»019 — 29000 = 19 
Under 80 
Dif. tor 10 = 16 
»' " 9 = J4 

" " 19 = 29 
Log. 29019 = 
4G240 + 29 = 46269. 



|3S 
1 2 



2 

3 

4 
5 
6 

7 

^ 
9 

10 



32 

2 

8 

5 

6 

8 

10 

11 

18 

14 

16 



31 30 29 28 



2 

3 

6 

6 

8 

9 

11 

12 

14 

16 



2 

3 

5 

6 

8 

9 

11 

12 

14 

16 



1 
3 
4 
6 

7 

9 

10 

12 

13 



1 

3 

4 

6. 

7 

8 
10 
11 
13 



15 14 



27 

1 

3 

4 

5 

7 

8 

9 

11 

12 

14 



26 

1 

3 

4 

5 

7 

8 

9 

10 

12 

13 



1 
2 
3 
4 
5 
6 
7 
8 
9 
10 



A dasli before 
or after a log. de- 
notes that its true 
value is less thAu 
the tabular Value 
by less than half a 
unit in the last 
place. Thus : 
Log. 3128=4952667 

*^ 3130=4956448 



86 



IX)GARITHMS. 



Common or Brlffss I«oir*>'itlimB. Base = 10. 



No. 



39150 

62 
M 
56 
58 



00 290 

8a60--322 
62 348 



Log. 

51188! „- 

215, 27 

-2421 S 

268; 26 

295 27 

26 
27 



348 
375 

428 26 



d/o 
—402 

j 428 

—455 
481 
-508 
534 



62 
64 
66 

68 

8»70 

72 -^M. 
74 —508 
76 534 
78 —661 

8980 587 

82 —614 
84 640 
86 —667 
88 693 

8990 —720 „. 
92 —746 S 
94 772 *? 
96 —799 g 
98 825 *^ 

8800 

02 
04 
06 



24 
26 
28 

8880 

32 
84 



I 26 
27 
26 
27 
26 

27 
26 
27 
2C 
27 



*51 27 
■»'« 2fi 

930 ^ 
08 -957 27 

8810 —983 
12 52009 „- 
14 035 il 
16 061 ;6 
18 -088; 2^ 

8880 —1141 ^ 
22 —140 -6 
166 26 

^26 

270 26 
-297 -' 



36 —323 26 
38 —349 ^^ 



38 —349 ^ 

8840 —375 „- 
42 —401 26 

46 -453! 26 

48 -H179| 5° 



No. 



3300 

52 
54 
56 
68 

3360 

62 
64 
66 
68 

3370 

72 
74 
76 
78 

3380 

82 
84 
86 
88 

8300 

92 
94 
96 
98 

8400 

02 
04 
06 
08 

8410 

12 
14 
16 
18 

8490 

22 
24 
26 
28 

3430 

32 
84 
86 
38 

3440 

42 
44 
46 

48 



Log 



52504 
530 
556 
582 
608 

—634 

—660 

—686 

711 

737 

—763 

— 7«i9 

—815 

840 

—866 

—892 
917 
943 

-969 
994 

58020 
—046 

071 
—097 

122 

—148 
173 

—199 
224 

—260 

275 
—801 

326 
—852 

877 

—403 
428 
453 

—479 
504 

529 

—655 

586 

605 

-631 

—666 

681 

706 

—782 

-757 



26 
26 
26 
26 
26 

26 
26 
25 
26 
26 

26 
26 
25 
26 
26 

25 
26 
26 
25 
26 

26 
25 
26 
25 
26 

25 
26 
25 
26 
25 

26 

25 
26 
25 
26 

25 
2o 
26 
25 
26 

26 
26 
25 
26 
25 

25 
25 
26 
25 
25 




3740 

42 
44 

46 

48 



Log. 



56229 
263 
—277 
—301 
824 

348 
—372 
—396 

419 
-443 



—467 
490 
—614 
—538 
661 

—585 
608 
—632 
—656 
679 

—703 
726 

—750 
773 

—797 

820 
—844 

867 
—891 
—914 

937 
—961 

984 
67008 
-031 

054 
—078 
—101 

124 
—148 

—171 
194 
217 

—241 

—264 

287 
810 
—834 
—857 
—880 



To find Log. 36114: 
Log. 36120 = 65775 
Log. 86100 = 55751 

Dif. 20 94 
3B114 — 36100=»14 

Under 24 
Dif. for 10 = 12 



(> (I 



4= 6 



'• " 14 = 17 
Log. 36114 = 
66751 + 17 = 55768. 





27 


1 


1 


2 


3 


3 


4 


4 


5 


5 


T 


6 


8 


7 


10 


8 


11 


9 


12 


10 


14 



A dash before 
or after a log. de* 
notes that ito true 
value is leu than 
the tabular yalue 
bj lees than half a 
unit in the last 

£lace. Thus : 
>g. 3490 = 6428264 
3492 = 5480742 



Comin*!! or Brines IiOK«rltkma. Base — U 



LOGARITHHB. 



LOOARlTHMa. 



8S 

90 

10 
20 

n 

SB 

40 
15 

flO 
SB 

70 
79 

ao 
le 

' 90 

OB 

MO* 

OB 
» 

30 

■X 

«o 

4S 

«0 
6B 

70 

811 
SB 

w 

OS 


;| 

s u 

- i 

- > 

- n 


i 

35 
85 

31 

3! 
I 

35 

S4 
39 
34 
» 
34 
36 
SI 
34 

S. 
M 

Si 


Nb. 
•BOO 

IS 
80 

40 

45 

SKIM 

«0 

70 
7S 
80 

w 

6«00 

10 

so 

40 
46 

«a 

«0 

M 

S700 

15 

. 85 

40 


Log. 

4M 

-sst 

723 

— 76T 

-m 

~MI 

—086 

—119 

161 

249 
2N 

-«18 

880 
441 

«H 
— S4( 

—70; 
-73; 

-8oi 
-ss. 

891 


1 

38 
S8 
S3 
S 

S 
33 

s; 


Ne.' 
•7BO 

76 
10 

ss 

45 
•BSO 

SB 

95 
10 

ao 

SB 

30 

M 
4B 

«»se 

80 

1 


Lw. 

-9« 

- K 

i 

n 

w 

85^ 

-11 ■ 


5 

33 
K 

3; 

32 
32 
32 
32 
31 

32 
3 

32 
32 


Ho. 

40 
4fi 

™. 

60 

80 

90 

9fi 

7100 

06 
10 

20 
2fi 
30 
8B 

*^ 
71S« 

60 

90 

Taoo 

16 

80 
35 

45 


Log 

-54 
—57 

—63 

7S 
SB 

-94 
SSOO 

09 

~11 
-24 

33 

40 

-46 

-82 
-66 

61 

IS 
-82 

-«a 

-91 
—94. 
-91 


S 

81 
30 

31 

30 

30 


No. 

raao 

«o 

7300 

10 

so 

80 

40 

46 

7SS0 

60 
S6 

80 
86 
90 
95 
7400 

EO 
3S 
SO 

40 

45 

74<tO 


Log. 

86wi 
—064 
— OM 

-37! 
361 

4S1 

~6»! 
SB) 

7i: 
—74: 

-801 
— 95J 

on 

-216 
-24! 


1 

30 
M 

90 
80 
30 
9> 
30 
30 

SO 

so 

29 

20 
30 

20 

39 
30 

39 
30 
39 
30 

29 
20 
39 

so 

39 

30 

39 

29 

29 
39 
29 
29 
20 




MOD 


s 
Its 


MS 

-as 

J»BSO. 


3 

S 

i 

7 
8 
9 
■0 


39 


': 


s: 


3a 

0.6 


31 

0.( 


8 


) 

i 
3 


6.8 


3 


i 


•testbi 
ue Is 
e Ubu 

iHBth 


ta be 
Bnh: 


1" 

ioa 

but 

i61 



LOOABTTHUS. 



LOOARITHMa. 
Common or Brlna IiOS»'tt)»i>a> 



92 



eSOXBTBT. 



QEOMETBI. 



I^lnes, Fifiriire*, Solldii, defined. Strictly speaking a geometrical 11b« 

ii limply length, or disUnoe. The Unes we draw on paper have not only length, bat breadth and 
thiokneas ; still they are the most oonTeoient Bymbol we can employ for denoting a geometrioAl line. 

Stralirlit lines are also called rl|pb t lines. A vertical line is one that points 
toward the center of the earth ; and a horisontnl one is at right angles to a 
vert one. A. plane finrnre is merely any flat surface or area entirely enclosed 

by lines either straight or ourred ; which are ealled its oatline, boandary, oiroomf, or pcnphery. We 
often oonfoond the ootline with the tig itself a* when we speak of drawing eirolee, sqnans, «e ; for 
we aotaally draw only their outlines. Oeometrieally speaking, a Og has length and braadth only ; n* 

thickness. A solid is any body ; it has length, oreadth, and thickness. 

Geometrically nlmllar figs or solias, are not necessarily of the same 
slse; but only of precisely the same sbape. Thus, any two squares are, scien- 
tifically speaking, similar to each other ; so also any two circles, eobes, 4ko, no matter how diflbrenft 
ther may be in aiie. When they are not only of the same shape, bat of the same siie, they are said 

to Ibe similar, and eqaal. 
The qaantltles or lines are to each other simply as their leng^ttas; but 

the quantities, or areas, or surfaces of similar flipnreSy are as, or in proportion 
to, the squares of any one of the corresponding lines or aides which enclose the 
figures, or which may he drawn upon them : and the quantities, or solidities of 
similar solids, are as the enbes of any of the corresponding lines which form 

their edges, or the figures by which th^ are enclosed. 

Bem«~Simple as the following operations appear, it is only by care, and good instmrnenta, that 
they are made to give accurate results. Several of them can be much better performed by means of a 
metallic triangle haying one perfectly accurate right angle. In the field, the (ape-llne, ehain, or a 
■Masuring-rod will take the place of the dividera and ruler used indoors. 



Te divide a si wen line, a b, into two equal pmrUu 

From Its ends a and h as centers, and with any rad greater than one-half of • ft, 
describe the area e and d, and Join e/. If the line a & is very long, first lay on 
eqaal dists a o and i g, each way from tba ends, so as to approach conveniently 
near to each other ; and then proceed as if o y were the line lo be divided. Ov 
ineaiare a b by a seale, and thns aaoertain its eenter. 



To divide a siwen line, «» a, into anj' 
ffiven number of equal parts. 

From m and n draw any .two parallel lines m o and n c, 
te an' indefinite dist ; and on them, tmrn m and n step off th« 
reqd number of eqaal parts of any convenient length : final- , 
ly. Join the eorresponding points thus stepped on. Or only 
one line, as mo, may be drawn and stepped oif, as to «; 
then Join «n; and draw the other short lines parallel to It. 

To divide a ^iren line, fa n, into two parts wbieb sball liawo 
a yiven proportion t^ eacb otber. 

This is done on the same principle as the last ; thns, let the proportion be as 1 to 8; First draw 
any line m o ; and with any convenient opening of the dividers, make m s equal to one step ; and •• 
equal to three steps. Join « n ; and parallel to it draw z c. Then m e is to c n as I is to 3. 

AJlGIaES. 

Aniples. When two straight, or right lines meet each other at any lncUn»- 
tion, the inclination is called an anicle; and is measured by the d^n^ees con- 
tained in the arc of a circle described from the point of meeting as a center. Since all circles, whether 
large or small, are supposed to be divided into SCO degrees, it follows that any number of degrees of a 
small circle will measure the same degree of inclination as will the same number of a large one. 

When two straight lines, as o n and a h, meet in such a manner that the inclination o n a is eqaal 
to the inclination o n 6, then the two lines are said to be 
perpendienlar to each other; and the angles on a and 
onh, are called rlgbt angles ; and are each measd by, or 

are equal to, W>, or one-fourth part of the circumf of a circle. Any angle, 
tMced, smaller than a right angle, is called acute or sharp ; 
and one c «/, laraer than a right angle, is called obtuse, or 

blant. When one line meets another, as in the first Fig on opposite page, the two angles on tha 
same side of either line are called contiguous, or a^iyacent. Thus, vus and 
* u w are adjacent ; also tut and tuw ; tut audit uv ; vout and wuv. The sum of two a<!yaoaat 
angles is always equal to two right augled ; or to 1H0°. Therefore, if we know the number of de* 
frees contained in one of them, and subtract it from 180°, we obtain the other. 





laanon o n 



Z 



QEOHETBY. 



93 




When two straight lines crow each other, forming four 
angles, either pair of those angles which point in exactly 
opposite directions are called opposite, or irertlcal 
angles ; thus, the pair a « < and vuw are .opposite an- 
gles ; also the pair suv and t u C9. The opposite anglet 
of any pair are always equal to each other. 

When a straight line a b crosses two parallel lines e <2, 
«/, the alternate angles which form a kind of Z are 
equal to each other. Thus, the angles don and on/ are 
equal : as are also con and one. Also the sum of the 
two internal angles on the same side of a 6, is equal to two 
right angles, or 180°; thus, co n + on/ =» 180°; also 
don -\- one = 180°. 

An interior angle* 

• 

In any fig, Is any angle formed intid* of that fig, by the meet- 
ing of two of its sides, as the angles c a b, a b c, b e a, of this 
triangle. All the interior angles of any straight-lined figure of 
any number of sides whaterer, are together eqaal to twice al 
many right angles minus four, as the figure has sides. Thus, a 
triangle has 3 sides ; twice that number is 6 ; and 6 right angles, 
or 6 X 9(P=b4(P; ffom which take 4 right angles, or 360° ; and 
there remain 18(P, which is the number of degrees in eraty 
plane, or straight-lined triangle. This principle furnishes ao- 
easy means of testing our measurements of the angles of any 
fig; for if the sum of all our measurements does not agree with 
ihc torn, given bj th« mie, It is a proof that we have committed some error. 

An exterior angle 

Of any straight-lined figure, is any angle, as a & d, formed by the meeting of 
any side, as a b, with the prolongation of an adjacent side, as c b; so likewise 
the angles c a a and b c to. All the exterior angles of any slraight-lined fig, 
no matter how many sides it may have, amount to 860° ; but, In (he case of 
a re-entering angle, as gij, the interior angle, g ij, exceeds 180°, and the 
"exterior" angle, g i x, being = 180° — interior angle, is negative. Thus 
ab d + 6cto-fca« = 360° ; and yhj+xji — gix + igie = 380°. 
Angles, as a, b, c, g, h, and^, which point outward, are called •alientl. 





From any given point, p, on a line « t, 
to draw a perp, p a. 

From p, with any oonvenient opening of the dividers, step off the 
•qvals po,p§. From o and g as centers, with any opening greater 
Ahan half o g, describe the two short arcs b and c ; and Join a p. 
Or still better, describe four arcs, and join a y. 

Or from p with any conyenient scale describe two 
•hori area g and e either one of them with a radius 3, and the other 
with a rad 4. Then from g with rad 6 describe the arc b. Join p a. 



tS tbe point p is at one end of the line, 
or very near it, 

■ztfend the line, if possible, and proceed as above. But if this 
•aanot be done, then ftom any convenient point, w, open the divid- 
er* to p, and describe the semicircle, « p o ; through o to draw o «o 
«;JeiBf»«. 

Or use the last foregoing process with 

rada 8, 4, and 5. 



Front a given point, o, to let fall a 
perp o «» to a given line, m n. 

From o, measure to the line m n, any two equal dists, o e, 
• « ; and troxa e and « as centers, with any opening greater 
than half of e e, describe the two arcs a and b ; join o t. Or 
from any point, as d on the line, op<m the dividers to o, and 
the arc o g ; make i x equal to < o ; and Join o x. 



b>ft^c 



P 



^^ftK 



V^e 




94 



eXOMETBT. 



If thm line, a b, !■ on tbe rronnd, 

Up«- Un«, or chaio. m»n; then Ughtea oat the striiiff, ko. u ■hown 
^ m . n ; • belDg lu oeatar. Tben will • e be therMd peroT Or if 

SS^J.'inH'u'"**.'^^'*.** '*L"* '««'•• thenholdlnftheendof °UJif 
£!f . i f :5"** **■ °* ?•• '*•' "i*"^ •* »'• »"»'* *»«e four f<^t mark at «, ko»i 

r Inl iS'u^TJ* *?!k ***•" *' V»«»»t-*«»«l«d triangle. JwiuSd of S, 4, and 
», la, 16, *o : aJ«o instead of feet, we niaj use jarde, chaina, Ao. 



Throairb a fflTen point, a, to draw m 
line, a c, parallel U 




6 n 




10 



y 8 



rsTi— W 



«/. 



to anotber line. 



With t)>« P*rp diet, a «, from any point, n. In •/, dew^rlbe 
■a arc, I ; draw a e Jut toaoblng the arc. 



At any point, a, In a line a b, 
to make an angrle «a fr^eqnal 
to a irlven anyle, mno. 

From n and a, with any oonvenlentrad, deeoribe 
??/"f ««.<*«; measure s t, and make • d equal 
to 11; through a d draw a e. 



7^^^ 



e 



n 





To biseet, or divide any ani^le, wxy, Into 
two equal parts. 

From X aet off any two ei^a&l dists, xr,x*. From r and « with any ra4 
describe two aroe interseeting, as at o ; and Join o x. If the two sides of 
the angle do not meet, fis e / and g h, either first extend them until th«« 
do meet; or else draw lines x to, and xy, parallel to them, and at equal 
disu from them, so as to meet; tben proceed as before. 



All angles, han am,n o m, at ttaeciroamf of a semicircle, and stand' 
ing on its diam n m, are right angles ; or, as it is usually expressed, 

all angrles in a semicirele are rig^bt ang^les. 

An angle n « z at the center of a circle, is twice as great as an angle 
n n» z at the circumf, when both stand upon the same arc n x. 



All angles, as y dp. y e p, y ^ p, at the oiroumf of a circle, and aUndlng 
upon the same are. as y p, are equal to eaeh other ; or, as usually expressed. 

all ang^les In tbe same segment of a cfreleare 
equal. 




But ordinarily we may neglect the signs -4- and — . before eomplementa iiii 
supplements, and call tbe complement of an angle its dilT from W>' matt 
the supplement lU dvtf^ from 180°. 



AITGLES. 



95 




Aayles fln a ParaUeloffimm. 

A pamllelogTam is any four-aided Btraight-UBed flg< 
ure whose opposite sides are equal, as a b c d ; or a 
square, &c. Any line drawn across a parallelogram 
between 2 opposite angles, is called a diagoneU^ as a & 
orb d. A diag divides a parallelogram into two equu 
parts ; as does also any line m n drawn through the 
center of either diag ; and moreover, the line m «• 
itself is div into two equal parts by the diag. Two 
diags bisect each other ; they also divide the parallel- 
ogram into four triangles of equal areas. The sum 
if the two angles at the ends of any one side is = 180^ ; thus, dab + abc^abo-i- 
hed==- ISfP; and the sum of the four angles, dab,abc^bed^cdaf= 360^. 

The sum of the squares of the four sides, is equal to the sum of the squares of the 
two diags. 

T« reduce Minutes and Seconds to Beyrees and decimals 

of a Degree, etc. 

In any given angle — 

Hnmber of degrees ^ Number of minutes -!- 60. 

SB Kumber of seconds -^ 3600. 

» 

Hnmber of mlnntes = Number of degrees x 60. 

= Number of seconds -^ 60. 



H'nniber of seconds 



Number of degrees X 3600. 
Number of minutes X 60. 



Table of Hinntes and B€»conds in Decimals of a Degree, 
and of Seconds in Decimals of a Minute. 

(The columns of Mins and Degs answer equally for Sees and Mins.) 



Mlns. Deg. Hins. Deg. Mins'. Deg. 



Sees. Deg. 



Sees. Deg. Sees. Deg, 



In each equivalent, the last digit repeats indeflnitely. See * below 



1 


0.016 


21 


0.350 


41 


0.683 


1 


0.00027 


21 


0.00583 


41 


0.01138 


2 


0.033 


22 


0.866 


42 


0.700 


2 


0.00055 


22 


0.00611 


42 


0.01166 


8 


0.060 


23 


0.383' 


43 


0.716 


3 


0.00083 


23 


0.00638 


43 


0.01194 


4 


0.066 


24 


0.400 


44 


0.733 


4 


0.00111 


24 


0.00666 ; 44 


0.01222 


5 


0.083 


25 


0.416 


45 


0.750 


5 


0.00138 


25 


0.00694 45 


0.01250 


6 


0.100 


26 


0.433 


4e 


0.766 


6 


0.00166 


26 


0.00722 46 


0.01277 


7 


0.116 


27 


0.450 


47 


0.783 


7 


0.00194 « 


27 


0.00750 47 


0.01305 


8 


0.133 


28 


0.466 


48 


0.800 


8 


0.00222 


28 


0.00777 48 


0.01333 


9 


0.150 


29 


0.483 


49 


0.816 


9 


0.00260 


29 


0.00805 49 


0.01361 


10 


0.166 


30 


0.500 


50 


0.833 


10 


0.00277 


30 


0.00833 , 60 


0.01388 


11 


0.183 


31 


0.516 


51 


0.850 


11 


0.00305 


31 


0.00861 ! 51 


0.01416 


12 


0.200 


32 


0.533 


52 


0.866 


12 


0.00333 


32 


0.00888 I 52 


0.01444 


13 


0.216 


33 


0.550 


53 


0.883 


13 


0.00361 


33 


0.00916 53 


0.01472 


14 


0.233 


34 


0.566 


54 


0.900. 


14 


0.00388 


34 


0.00944 


54 


0.01600 


15 


0.250 


85 


0.583 


55 


0.916 


15 


0.00416 


35 


0.00972 


55 


0.01527 


16 


0.266 


36 


0.600 


56 


0.933 


16 


0.00444 


36 


0.01000 


66 


0.01555 


17 


0.283 


87 


0.616 


57 


0.950 


17 


0.00472 


37 


0.01027 


67 


0.01583 


18 


0.300 


88 


0.633 


58 


0.966 


18 


0.00500 


38 


0.01055 


58 


0.01611 


19 


0.816 


39 


0.650 


59 


0.983 


19 


0.00527 


39 


0.01083 59 


0.01638 


20 


0.383 


40 


0.66G 


60 


1.000 


20 


0.00555 


40 


0.01111 


60 


0.01666 


- 
Sees. Mio. 


Sees 


. Min. 


Sees, 


Min. 


Sees 


. Deg. 


Sees. Deg. 


Sees. Deg. 



* Each equivalent is a repeating decimal, thus : 

2 minates = 0.0333333 .... degree 
7 " = 0.1166666 .... " 
12 " =0.2000000 .... " 



12 seconds = 0.2000000 

1 second = 0.0002777 

50 seconds = 0.0138888 



minute 
degree 



96 



ANGLES. 



Approzimate Measurement of Angrles. 

(1) The foar flnarerfl of the hand, held at right angles to the arm and 

at arm's length from the eye, cover about 7 degr<^ea. And an angle of 7° corre- 
sponds to about 12.2 feet in 100 feet ; or to 36.6 feet in 100 yards ; or to 645 feet in a 
mile. 

(S) By means of a two-foot rnle, either on a drawing or between dis- 
tant objects in the field. If the inner edges of a common two-foot rule be opened 
to the extent shown in the column of inches, they will be Inclined to each other 
at the angles shown in the column of augles. iSince an opening of ^ inch (up 
to 19 inches or about 105°) corresponds to from about U° to 1° no great accuracy 
is to be expected, and beyond 105° still less ; for the liability to error then in- 
creases very rapidly as the opening becomes greater. Thus, the last ^ inch cor- 
responds to about 129. 

Angles for openings intermediate of those given may be calculated to the 
nearest minute or two, by simple proportion, up to 28 inches of opening, or 
about 147«. 

Table of Angles correspondlntr to openinipi of a 2-foot rule. 

(Original). 

Correet. 



Ini. 


Deg. mio.| 


lD>. 


Deg. mln.| 


Ins. 


Deg. min.] 


Ids. 


Dsg.min.] 


Ins. 


Deg.mln.] 


Ins. 


Dag. min. 


H 


1 


12 


<y* 


20 


24 


8M 


40 


IS 


l2Ji 


61 


23 


16K 


85 


14 


20 Ji 


115 6 




1 


48 




21 






40 


61 




62 


5 




86 


S 




116 » 


H 


2 


24 


H 


21 


37 


H 


41 


29 


H 


62 


47 


H 


86 


52 


H 


117 » 




8 


00 




22 


13 




42 


7 




«3 


28 




87 


41 




118 30 


H 


8 


86 


H 


22 


60 


H 


42 


46 


H 


64 


11 


H 


88 


81 


H 


119 40 




4 


11 




23 


27 




43 


24 




04 


58 




89 


21 




120 52 


1 


4 


47 


5 


24 


3 


9 


44 


t 


13 


66 


35 


17 


90 


12 


21 


122 • 




6 


33 




24 


39 




44 


42 




66 


18 




91 


8 




123 20 


H 


6 


58 


H 


25 


16 


H 


45 


21 


y* 


67 


1 


H 


91 


64 


H 


124 ZS 




« 


34 




25 


53 




45 


59 




67 


44 




92 


46 




125 64 


H 


7 


10 


H 


26 


90 


H 


46 


88 


H 


68 


28 


H 


96 


88* H i 


127 14 




7 


46 




27 


7 




47 


17 




69 


12 




94 


81 




128 36 


H 


8 


22 


H 


27 


44 


H 


47 


66 


H 


69 


55 


H 


95 


24 


H 


129 59 




8 


58 




28 


21 




48 


35 




70 


38 




96 


17 




131 2ft 


s 


9 


34 


6 


28 


58 


10 . 


49 


15 


14 


71 


22 


18 


97 


11 


22 


132 ftS 




10 


10 




29 


35 




49 


54 




72 


6 




96 


6 




184 M 


H 


10 


46 


H 


30 


11 


H 


60 


34 


H 


72 


61 


H 


99 


00 


H 


135 6S 




11 


22 




30 


49 




51 


13 




78 


86 




99 


65 




187 36 


H 


11 


58 


Vi 


31 


26 


H 


61 


63 


H 


74 


21 


H 


100 


61 


H 


189 1% 




12 


34 




32 


8 




62 


83 




75 


6 




101 


48 




141 1 


H 


18 


10 


H 


32 


40 


H 


53 


13 


H 


75 


51 


H 


102 


45 


H 


142 51 




IS 


46 




83 


17 




63 


63 




76 


86 




103 


48 




lU 4f 


1 


14 


22 


7 


33 


54 


11 


64 


34 


15 


77 


22 


19 


104 


41 


28 


146 46 




14 


68 




34 


83 




55 


14 




78 


8 




106 


40 




148 6B 


34 


16 


34 


H 


35 


10 


Vi 


65 


65 


}i 


78 


54 


H' 


106 


89 


H 


151 ir 




16 


10 




85 


47 




56 


35 




79 


40 




107 


40 




153 41 


H 


16 


46 


H 


36 


25 


H 


57 


16 


H 


80 


27 


H 


106 


41 


H 


156 Si 




17 


22 




37 


8 




67 


57 




81 


14 




109 


48 




159 41 


H 


17 


59 


H 


37 


41 


H 


58 


38 


H 


82 


2 


H 


110 


46 


H 


168 27 




18 


35 




38 


19 




59 


19 




82 


49 




111 


49 




168 18 


4 


19 


12 


8 


38 


67* 


12 


60 


00 


16 


83 


37 


20 


112 


53 


24 


180 00 




19 


46 




39 


86 




tiU 


41 




84 


26 




118 


58 







(3) With the same table^ using: feet instead of inches. From 
the given point measure 12 feet toward * each object, and place marks. Measure 
the distauce in feet between these marks. Suppose the first column in the table to 
be feet instead of inches. Then opposite the distauce in feet will be the angle. 

^ foot = 1.5 inches. 



1 in. « .083 ft. 


4 ins. = .333 ft. 


7 ins. -= .583 ft. 


10 ins. « .833 ft. 


2 ins. — .167 ft. 


5 ins. = .416 ft. 


8 ins. = .667 ft. 


Hins. =» .917 ft. 


3 ins. = .25 ft. 


6 ins. >« .5 ft. 


9 ins. — .76 ft. 


12 ins. = l.O ft. 



(4) Or, measure toward * each object 100 or any other number of 
feet, and place marks. Measure the distance in feet between the marks. Then 

Sine of half _ half the distance between the marks 

the angle ~* the distance measured toward one of the objecta* 

Find this sine in the table pp. 98, etc. ; take out the corresponding angle and 
multiply it by 2 
(0) See last paragraph of foot-note, pp 152 and 153. 



_ * If it Is inconvenient to measure toward tbe objects, measare directly /Vom them. 



SnfTBS, TAKQENTS, B70. 



97 




Sines, Tans^nta, Ac. 

Sine* a », of any angle, a e 5, or vUeh is th* same thing, the sine of any oiroolar aro, • », 
vhieh subtends or measures the angle, ix.a straight line drawn from one end, as a, of the aro, at right 
•ftgles to, and terminating at, the rad c 6, drawn to the other end b of the are. It is, therefore, eqoal 
lo half the chord a n, of the aro a 5 n, which is equal to twice the aro a b ; or, the sine of an angle ia 
•lw»n equal to half the obord of twice that angle; and Tioe vena, the ohord of an angle is alwajt 

a Ml to twioe the sine of half the angle, 
e sine < c of an angle ( c b, or of an are 
fa ft, of iW, is equal to the rad of the aro 
or of the oirele ; and this sine of 90° is 
y ter than that of any other angle. 

Cosine e < of an angle acb^ 

Is that part of the rad which lies between 
the sine and the oenter of the oirole. It 
is always equal to the sine y a of the 
complement tcaotaeb; or of what a 
e b wants of being 90°. The prefix co be- 
fore sines, Ao, means oompiemeni ; thus, 
cosine means sine of the complement. 
Tersed sine «b of any angle 

• e 6, is that part of the diam whieh lies 
between the sine, and the outer end 6. 
It is T«ry common, but erroneous, when 
■peaking of bridges, Ao, to call the rise 
or height « fr of a caronlar areb a 6 n, its 
Tersed sine; while it is actually the versed 
■ineofonly half the arch. This absurdity 
•hoald.oease ; for the word rise or height 
is not only more ezpressiTe,but is correct. 

Tanicen tbworad, of any angle 

« « fr. is a line drawn from, and at right 
angles to, the end 6 or a of either rad c 6, 
or c a, which forms one of the legs of the 
sn^ ; and terminating as at to, or d, in 
the prolongation of the rad which forms 
die other leg. This last rad thns pro- 
lonfBd, that is, c w, or e d, as the case may 

W, is the secant of the angle 

• e i. The angle (eft being loppeaed 
to-be equal to 90°, the angle tea becomes the complement of the angle a o ft, or what a e ft wanta 
of being 90° ; and the sine y a of this complement ; its versed sine t y ; its tangent < o; and its seoaat 
e o, are respeotirely the eo-sine, co-rersed sine ; co-tangent; and oo-«ecant, of the angle a e ft. Or, 
viee versa, the sine, 4o, of aeb, are the cosine, Ac, of tea; because the an^le a e ft is the oomple* 
ment of the angle tea. When the rad e ft, e a, or c t, is assumed to be equal to unity, or 1, the cor> 
responding sines, tangents, Ac. are called natural ones ; and their several lengths for diff angles, 
for said rad of unity, have been calculated ; constituting the well-known tables of nat sines, fto. In 
any eirele whose rad is either larger or smaller than 1, the sines, Ac, of the angles will be in the 
amme proportion larger or smaller than those in the tables, and are consequently found . by mult tlM 
■iae. M, of the table, by said larger or smaller rad. 

The followinir table of natural sines, Ac. does not contain nat 
Tened sines, co-versed sines, secants, nor cosecants, but these may be found thus ; 
Cnr any angle not exceeding 90 degrees. 

Vened 9bu. From I take the nat cosine. 
Oo-verted Sine. From 1 take the nat sine. 
Seeant. Divide 1 by the nat cosine. 
OoaeeaiAt. Divide I by the nat sine. 

Wmr «Bftfe« ezeee4bur M^ t to find the sine, eosine, tangent, ootang, secant, or coseo, (but not 
the versed sine or co-versedsine), take the angle trota 180° : if between 180° and 370° take 180° fk-om 
the angle : if bet 270° and 360°, Uke the angle from 860°. Then in each ease take trom the tebie the 
sine, ooeine, tang, or ootang of the remainder. Find Its leoant or coseo as directed above. Far the 
^ ttnm ; if between 90(^and 270°, add cosine to 1 ; if bet 270° and 360°, take eosine from 1. (The 
ddem needs sines, Ae, ezoeoding 180°. 

To find tbo nat sine* cosine, tans, secant* Tersed sine, ^fcc, 
of an anvle containing seconds. First find that due to the given deg 

sad min ; tbea the next greater one. Take their diff. Then as 60 see are to this diff, so are the see 

only of the given angle to a dec quantity to be added to the one first taken out 
if it ia a sine, tang, secant, dec ; or to be subtracted from it if it is a cosine, 
cotang, cosecant, &c. 

The tjanfpents in the table are strict triiponometrical ones ; that is, 
tsBcents to given anglts ; and which must extend to meet the secants of the angles 
towbich they belong. Ordinary, or ipeometrical tangents, as those on 
p 162, may extend as far as we please. In the field practice of railroad 
earvea* two trigonometrical tangents terminate where they meet each other. 
Iseb oftnese tangs is the tang of half the curve. It is usually, but improperly, 
called '' the tang of the eurM. ' ** Apex dist of the curve," as suggested by Mr 
Shank, woald be better. 



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3 



TABUS OF CHOBDS. 



143 




below, fkinkisbes the meaoBoflaying down angles on 
paper more accurately than by an ordinary protractor. To do this, after having drawn 
and measured the first side (say ac) of the figure that is 
to be plotted ; from its end c as a center, describe an arc 
ny of a circle of sufficient extent to subtend the angle at 
that point. The rad en with which the arc is described 
should be as gjeat as conyenience will permit ; and it is to 
be assumed as unity or 1 ; and must be decimally divided, 
and subdivided, to be used as a scale for laying down the 
chords taken fh>m the table, in which their lengths are 
given in parts of said rad 1. Having described the arc, find 
in the table the length of the chord n t corresponding to 
the angle act. Let us suppose this angle to be 46^; then 
we find that the tabular chord is .7654 of our rad 1. There- 
fore fiom n we lay oif the chord nt, equal to .7654 of our radius-scale ; and the lint 
et drawn through the point t will form the reqd angle act of 46^. And so at each 
angle. The degree of accuracy attained will evidently depend on the length of the 
rad, and the neatness of the drafting. The method becomes preferable to the com- 
mon protractor in proportion as the lengths of the sides of the angles exceed the rad 
of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not 
much exceeding the same limits, the protractor will usually be preferable. The di- 
viders in boxes of instruments are rarely fit for accurate arcs of more than about 6 
ins diam. In practice it is not necessary to actually describe the whole arc, but 
merely the portion near t, as well as can be Judged by eye. We thus avoid much use 
of the India-rubber, and dulling of the pencil-point. For larger radii we may dis- 
pense with the dividers, and use a straight strip of paper with the length of the rad 
marked on one edge ; and by laying it from c toward «, and at the same time placing 
another Jtrip (witii one edge divided to a radius-scale) from n toward t, we can 
by trial find their exact point of intersection at the required point t. In such mat* 
ters, practice and some Ingenuity are very essentlial to satisfactory results. We can' 
not devote more space to the subject. 



m' » 



CHORDS TO A RADIUS 1. 



M. 


OO 


1° 


HP 


SO 


4° 


5° 


e° 


70 


80 


90 


10° 


M. 


0' 


.0000 


.0175 


.0849 


.0584 


.0098 


.0872" 


.1047 


.1221 


.1395 


.1669 


.1748 


0' 


2 


.0000 


.0180 


.0855 


.0589 


.0704 


.0878 


.1063 


.1227 


.1401 


.1675 


.1749 


8 


4 


.0012 


.0186 


.0061 


.0585 


.0710 


.0884 


.1068 


.1288 


.1407 


.1581 


.1756 


4 


6 


.0017 


.0192 


.0M6 


.0541 


.0715 


.0890 


.1064 


.1288 


.1418 


.1587 


.1761 


6 


8 


.0023 


.0198 


.0872 


.0547 


.0721 


.0896 


.1070 


.1244 


.1418 


.1692 


.1766 


6 


10 


.0029 


.0204 


.0878 


.0668 


.0727 


.0901 


.1076 


.1860 


.1484 


.1596 


.1772 


10 


IS 


.0035 


.0200 


.0884 


.0558 


.0738 


.0907 


.1082 


.1256 


.1430 


.1604 


.1778 


18 


14 


.0041 


.0215 


.0890 


.0564 


.0739 


.0913 


.1087 


.1262 


.1436 


.1610 


.1784 


14 


16 


.0047 


.0221 


.0896 


.0570 


.0745 


.0919 


.1093 


.1267 


.1442 


.1616 


.1789 


16 


18 


.0052 


.0227 


.0401 


.0576 


.0750 


.0025 


.1009 


.1278 


.1447 


.1621 


.1706 


18 


SO 

2S 


] .0058 


.0238 


.0407 


.0682 


.0766 


.0981 


.1105 


.1279 


.1468 


.1627 


.1801 


20 


-.0004 


.0239 


.0413 


.0588 


.0762 


.0936 


.1111 


.1285 


.1459 


.1683 


.1807 


22 


24 


.0070 


.0244 


.0419 


.0598 


.0768 


.0942 


.1116 


.1291 


.1465 


.1639 


.1813 


24 


ss 


.0076 


.0250 


.0426 


.0599 


.0774 


.0948 


.1182 


.1296 


.1471 


.1645 


.1818 


26 


28 


.0081 


.0256 


.0430 


.0605 


.0779 


.0954 


.1128 


.1302 


.1476 


.1660 


.1824 


28 


80 


.0087 


.0262 


.0486 


.0611 


.0785 


.0960 


.1134 


.1808 


.1482 


.1666 


.1830 


30 


S2 


.0008 


.0268 


.0442 


.0617 


.0791 


.0965 


.1140 


.1314 


.1488 


.1662 


.1836 


32 


94 


.0000 


.0273 


.0448 


.0622 


.0797 


.0671 


.1145 


.1320 


.1494 


.1668 


.1842 


34 


SS 


.0105 


.0279 


.0464 


.0628 


.0808 


.0977 


.1151 


.1325 


.1500 


.1874 


.1847 


86 


88 


.0111 


.0285 


.0460 


.0684 


.0808 


.0983 


.1157 


.1831 


.1505 


.1679 


.1858 


36 


40 


.0116 


.0291 


.0465 


.0640 


.0614 


.0989 


.1168 


.1887 


.1511 


.1685 


.1859 


40 


42 


.0122 


.0297 


.0471 


.0646 


.0620 


.0994 


.1169 


.1343 


.1517 


.1691 


.1865 


42 


44 .0128 


.0303 


.04n 


.0651 


.0826 


.1000 


.1175 


.1349 


.1523 


.1697 


.1871 


44 


48 


.0134 


.0308 


.0483 


.0857 


.0832 


.1006 


.1180 


.1366 


.1529 


.1703 


.1876 


46 


48 


.0140 


.0314 


•fPMHp 


.0463 


.0838 


.1012 


.1186 


.1360 


.1534 


.1708 


.1882 


48 


M) 


.0146 


.0320 


.0404 


.0660 


.0843 


.1018 


.1192 


.1366 


.1540 


.1714 


.1888 


50 


(2 


.0151 


.0386 


.0500 


,0675 


.0849 


.1023 


.1198 


.1372 


.1546 


.1720 


.1894 


5S 


64 


.0157 


.0382 


.0606 


.0681 


.0856 


.1029 


.1204 


.1.378 


.1552 


.1726 


.1900 


54 


M 


.0163 


.0387 


.0512 


.0686 


.0861 


.1035 


-.1209 


.1384 


.1658 


.nn 


.1905 


56 


68 


.0160 


.0848 


.6618 


.0092 


.0867 


.1041 


.1215 


.1389 


.1561 


.1737 


.1911 


58 


•D 


join 


U»49 


.0524 


.0096 


.0872 


.1047 


.1221 


.1396 


.1569 


.1743 


.1917 


60 



144 



TABLE OF CHORDS. 



Table of Cbords, In 


parte 


of aradl; 


for protractlng^-Gontinued. 


M. 


11° 


12° 


13° 


14° 


15° 


1«° 


17° 


18° 


1©° 


20° 


M. 


0* 


.1917 


.2091 


.2264 


.2437 


.2611 


.278S 


.2966 


.3129 


.3301 


.3478 


0' 


2 


.1»2S 


.2096 


.2270 


.2443 


.2616 


.2789 


.2961 


.8134 


.8807 


.8479 


2 


4 


.1928 


.2102 


.2276 


.2449 


.2622 


.2796 


.2968 


.8140 


.8812 


•••Cm 


4 


6 


.19S4 


.2108 


.2281 


.2455 


.2628 


.2801 


.2973 


.3146 


.3318 


.8480 


6 


8 


.1940 


.2114 


.2287 


.2460 


.2634 


.2807 


.2979 


.8152 


.3824 


Jt496 


8 


10 


.1946 


.2119 


.2293 


.2466 


.2639 


.2812 


.2986 


.3167 


.8330 


.8502 


20 


n 


.1962 


.2125 


.2299 


.2472 


.2645 


.2818 


.2901 


.S16S 


.8386 


.8607 


12 


u 


.1957 


.2131 


.2305 


.2478 


.2651 


.2834 


mMK^9 


.S169 


.8341 


.8618 


14 


16 


.1963 


.2137 


.2310 


.2484 


.2657 


.2830 


.8002 


.8176 


JU47 


.S6I» 


16 


18 


.1960 


.2143 


.2316 


.2489 


.2662 


.2836 


.3008 


.8180 


.3353 


.8526 


18 


20 


.1975 


.2148 


.2322 


.2495 


.2668 


.2841 


.SOU 


.3186 


.8366 


.8630 


20 


22 


.1981 


.2154 


.2328 


.2501 


.2674 


.284T 


.9019 


.8192 


.8364 


.3536 


22 


M 


.1986 


.2160 


.2333 


.2507 


.2680 


.2853 


.3026 


.8198 


.8370 


.3542 


34 


26 


.1992 


.2166 


.2339 


.2512 


.2685 


.2858 


.3081 


.8208 


.3376 


.3547 


36 


28 


.1998 


.2172 


.2345 


.2518 


.2691 


.2864 


.3087 


.8200 


.8381 


.3553 


38 


SO 


.2004 


.2177 


.2351 


.2524 
.2530 


.2697 


.2870 


.8042 


.8215 


.3387 


.3659 


80 


32 


.2010 


.2183 


.2357 


.2703 


.2876 


.3048 


.3221 


.3398 


.3565 


S3' 


Si 


.2015 


.2189 


.2362 


.2536 


.2709 


.2881 


.3054 


.3226 


.3398 


.3570 


84 


36 


.2021 


.2195 


.2368 


.2541 


.2714 


.2887 


.3060 


.3233 


.3404 


.3576 


86 


38 


.2027 


.2200 


.2374 


.2547 


.2720 


.289S 


.3065 


.8288 


.3410 


.3688 


88 


40 


.2033 


.2206 


.2380 


.2553 


.2726 


.2890 


.3071 


.8244 


.3416 


.3587 


40 


42 


.2038 


.2212 


.2385 


.2559 


.2732 


.2904 


.3077 


.8249 


.3421 


.3693 


43 


44 


.20U 


.2218 


.2391 


.2564 


.27.'57 


.2910 


.3088 


.8255 


.8427 


•oOW 


44 


46 


.2050 


.2224 


.2397 


.2570 


.2743 


.2916 


.3088 


.8261 


.3433 


.8606 


4C 


48 


.2056 


.2229 


.2403 


.2576 


.2749 


.2922 


.3094 


.3267 


.8439 


.8610 


48 


60 


.2062 


.2235 


.2409 


.2582 


.2755 


.2927 


.3100 


.3272 


.3444 
.3450 


.3616 


60 


52 


.2067 


.2241 


.2414 


.2587 


.2760 


.2933 


.3106 


.3278 


.3622 


63 


54 


.2073 


.2247 


.2420 


.2593 


.2766 


.2989 


.3111 


.8284 


.3456 


.8626 


66 


56 


.2079 


.2253 


.2426 


.2599 


.2772 


.2945 


.3117 


.3289 


.8462 


.3633 


5ft 


58 


.2085 


.2258 


.2432 


.2605 


.2778 


.2950 


.3123 


.3295 


.3467 


..H639 


58 


60 


.2091 


.2264 


.2487 


.2611 


.2783 


.2956 


.3129 


.8801 


.3473 


.3645 


60 



M. 


21° 


22° 


28° 


24° 


25° 


26° 


27° 


28° 


2»° 


so° 


"-. 


0' 


.3645 


.3816 


.3967 


.4158 


.4329 


.4489 


.4609 


.4838 


.5008 


.5176 


0' 


3 


.3650 


.3822 


.3898 


.4164 


.43^4 


.4606 


.4675 


.4844 


.5013 


.5182 


2 


4 


.3656 


.3828 


.3999 


.4170 


.4340 


.4510 


.4680 


.4850 


.5019 


.5188 


i 


6 


.3662 


.3833 


.4004 


.4175 


.4346 


.4616 


.4686 


.4855 


.6034 


.5193 


• 


8 


.3668 


.3839 


.4010 


.4181 


.4352 


.4523 


.4608 


.4861 


.5030 


.5199 


8 


10 


.8673 


.3845 


.4016 


.4187 


.4357 


.4527 


.4697 


.4867 


.6036 


.5204 


10 


12 


.3679 


.3850 


.4022 


.4192 


.4363 


.4538 


.4703 


.4872 


.5041 


.5210 


12 


14 


.3686 


.3856 


.4027 


.4198 


.4369 


.4539 


.4708 


.4878 


.6047 


.5816 


14 


16 


.3690 


.3862 


.4033 


.4204 


.4374 


.4544 


.4714 


.4884 


.5063 


.5221 


16 


18 


.3696 


.3868 


.4039 


.4209 


.4.180 


.4550 


.4720 


.4888 


.5058 


.6227 


18 


ao 


.3702 


.3873 


.4044 


.4215 


.4386 


.4556 


.4725 


.4885 


.6064 


.5233 


30 


22 


.3708 


.8879 


.4050 


1 
.4221 


.4391 


.4661 


.4731 


.4901 


.5070 


.5238 


22 


24 


.3713 


.3885 


.4056 


.4226 


.4397 


.4567 


.4787 


.4906 


.5075 


.5244 


34 


26 


.8719 


.3890 


.4061 


.4232 


.4403 


.4573 


.4742 


.4812 


.5081 


.5249 


36 


28 


.3725 


.3886 


.4067 


.4238 


.4408 


.4578 


.4748 


.4917 


.5086 


.5255 


38 


SO 


.3730 


.8902 


.4073 


.4244 


.4414 


.4584 


.4754 


.4923 


.6092 


.5261 


SO 


32 


.3736 


.3908 


.4070 


.4249 


.U20 


.4590 


.4759 


.4929 


.5098 


.5266 


S3 


34 


.3742 


.3913 


.4084 


.4255 


.4425 


.4595 


.4765 


.4934 


.5108 


.6272 


34 


36 


.3748 


.3919 


.4090 


.4261 


.4431 


.4601 


.4771 


.4940 


.5109 


.5277 


36 


88 


.3753 


.3936 


.4096 


.4266 


.4487 


.4607 


.4776 


.4946 


.5115 


.52b3 


80 


40 


.3759 


.3980 


.4101 


.4272 


.4442 


.4612 


.4782 . 


.4061 


.5120 


.5269 


40 


42 


.3765 


.3936 


.4107 


.4278 


.4448 


.4618 


.4788 


.4957 


.5126 


.5294 


48 


44 


.8770 


.3942 


.4113 


.4283 


.4454 


.4624 


.4793 


.4963 


.6131 


.5300 


44 


46 


.3776 


.3947 


.4118 


.4289 


.4459 


.4629 


.4799 


.4<M8 


.5137 


.5306 


40 


48 


.3782 


.8953 


.4124 


.4295 


.4465 


.4635 


.4805 


.4974 


.5143 


.5311 


40 


.1. 


.3788 


.3959 


.4130 


.4800 


.4471 


.4641 


.4810 


.4979 


.5148 


.5317 


60 


52 


.3798 


.3065 


.4135 


.4.HG6 


.4476 


.4646 


.4816 


.4985 


..M54 


..^322 


fit 


54 


.3799 


.3970 


.4141 


.4312 


.4482 


.4652 


.4822 


.4991 


.5100 


.58?8 


M 


56 


.9806 


.8976 


.4147 


.4317 


.4488 


.46.')8 


.4827 


.4996 


.6166 


.5834 


60 


58 


.3810 


.3982 


.4153 


.4323 


.4493 


.4663 


.4888 


.6003 


.6171 


.5839 


60 


00 


.3816 


.3987 


.4158 


.4329 


.4499 


.4669 


.4888 


.5008 


.6176 


.5846 


00 



TABLE OF CHOBDB. 



145 



Tftble of ehovdOflii parte off a rad 1^ for protractlnv— ContliraeC 



M. 


81° 


as*" 


Sso 


Z4P 


99° 


86° 


87° 


88° 


89° 


40° 


M. 


•• 


.5846 


.5613 


.5680 


.5847 


.6014 


.6180 


.6346 


.6511 


.6676 


.6840 


0' 


3 


.5850 


.5618 


.5686 


.5868 


.6030 


.6186 


.6363 


.6517 


.MH'X 


•OBVQ 


2 


A 


.5856 


.5534 


.5601 


.6868 


.6035 


.6191 


.6357 


.6633 


.6687 


.6851 


4 


« 


.5868 


.5630 


.5697 


.5864 


.6081 


.6197 


.6363 


.6538 


.6693 


.6867 


6 


8 


.5867 


.6685 


.5708 


.5870 


.6036 


.6303 


.6368 


.6633 


.6606 


.6863 


8 


M 


.5878 


.5541 


.5706 


.5676 


.6042 


.6306 


.6874 


.6630 


.6704 


•0888 


10 


13 


.5878 


.5646 


'.5714 


.5881 


.6047 


.6314 


.6379 


.6544 


.6709 


.6873 


12 


14 


.5884 


.5562 


.5719 


.6886 


.6063 


.6310 


.6385 


.6560 


.6715 


.6879 


14 


U 


.5880 


.5667 


.5786 


.5893 


.6058 


.6335 


.6390 




.6730 




16 


18 


.5395 


.5568 


.5780 


.5897 


.6064 


.6280 


.6396 


.6661 


.6725 


.6890 


18 


90 


.5401 


.5569 


.6796 


.5803 


.0070 


6236 


.6401 


.6666 


.6731 


.6895 


20 


S 


.5406 


.S6T4 


.5743 


.5600 


.6075 


.6241 


.6407 


.6673 


.6736 


.6901 


22 


M 


.5413 


.5580 


.5747 


.5814 


.6081 


.6247 


.6412 


.6677 


.6743 


.6906 


24 


» 


.5418 


.5686 


.6758 


.5830 


.0086 


.6353 


.6418 


.6683 


.6747 


.6911 


26 


» 


.M2S 


.5501 


.6768 


.5936 


.6002 


.6258 


.6438 


.6588 


.6763 


.6917 


28 


JO 


.54*29 


.5507 


.6764 


.6981 


.6097 


.6263 


.6439 


.6694 


.6758 


.6923 


80 


n 


.5484 


.5608 


.6769 


5986 


.6103 


.6260 


.6484 


.6589 


.6764 


.6838 


82 


a 


.5440 


.6606 


.6775 


.5843 


.6108 


.6374 


.6440 


.6605 


.6769 


.6933 


81 


» 


.5446 


.5613 


.6781 


.5047 


.6114 


.6280 


.6445 


6610 


.6775 


.6039 


M 


18 


.5451 


.5619 


.6786 


.6963 


.6119 


.6386 


.6451 


.6616 


.6780 


.J944 


38 


40 


.5457 


.5625 


.6793 


mngg\ 
•OWOV 


.6135 


.6391 


.6456 


.6631 


.6786 


.6950 


40 


43 


.5463 


.5630 


.6797 


.5964 


.6130 


.6396 


.6463 


.6637 


.6791 


.6955 


42 


44 


•9voO 


.6686 


.6806 


.6870 


.6136 


.6303 


.6467 


.6632 


.6797 


.6061 


44 


46 


.5474 


.5641 


.6808 


.5075 


.6143 


.6307 


.6473 


.6638 


.6803 


.q8od 


46 


48 


.5479 


.5647 


.5814 


.5061 


.6147 


.6313 


.8476 


.6643 


.6806 


.6971 


48 


fiO 


.5485 


.5653 


.5820 


.5866 


.6153 


.6318 


.6484 


.6649 
.6654 


.6613 


.6977 


50 


51 


.5490 


.5668 


.6826 


.5983 


.6158 


.6334 


.6489 


.6619 


.6983 


52 


64 


.5486 


.5664 


.6861 


.5087 


.6164 


.6330 


.6495 


.6660 


.6824 


.6988 


54 


M 


.5502 


.5660 


.5886 


.6006 


.6169 


.6336 


.6600 


.6665 


.6829 


.6993 


56 


W 


.5507 


.5675 


.6648 


.6000 


.6175 


.6841 


.6606 


.6671 


.6835 


.6999 


66 


40 


.5513 


.5680 


.6847 


.6014 


.6160 


.6846 


.6611 


.6676 


.6840 


.7064 


60 



0' 
3 
4 
6 
8 
10 

i7 

14 
16 
U 



21 
24 

28 
28 
10 

HT 

J4 

16 
18 
40 



46 
46 
50 

IS' 

54 

M 



41° 



48° 



.7004 
.7010 
.7015 
.7020 
.7026 
.7081 



, .7167 
I .7171 
I .7176 

.7184 
I .7188 

.7186 




.7200 
.7206 
.7211 
.7216 
.7222 



.7227 
.7232 
.7238 
.7343 
.7249 



.7081 


.7254 


.7097 


.7280 


.7102 


.7265 


.7106 


.7270 


.7113 


.7276 



.7118 
.7124 
.7129 
.7135 
.7140 



.7281 
.7387 
.7282 
.7388 
.7803 



.7146 
.7151 
.7156 
.7162 
.7187 



.7806 
.7314 
.7819 
.7126 
.TIM 



.7380 
.7335 
.7341 
.7346 
.7362 
.7357 



44' 



.7482 
.7486 
.7606 
.7608 
.7614 
.7518 



.7362 
.7368 
.7878 
.7379 
.7384 



.7390 
.7385 
.7400 
.7406 
.7411 



.7417 
.7432 
.7427 
.7433 

.7488 



.7524 
.7580 
.7536 
.7541 
.7546 



.7551 
.7557 
.7562 
.7568 
.7573 



.7578 
.7584 
.7588 
.7596 
.7600 



.7443 
.7448 
.7464 
.7460 

.7466 



.7471 
.7476 
.7481 
.7487 
.7493 



.7605 
.7611 
.7616 
.7631 
.7637 



.7683 
.7638 
.7648 
.7648 
.7664 



45° 


48° 


.7664 


.7816 


.7659 


.7820 


.7664 


.7826 


.7670 


.7831 


.7675 


.7836 


.7681 


.7841 


.7686 


.7847 


.7691 


.7852 


.7687 


.7857 


.7703 


.7868 


.7707 


.7868 


.7713 


.7873 


.7718 


.7879 


.7733 


.7884 


.7739 


.7890 


.7784 


.7895 


.7740 


.7900 


.7746 


.7906 


.7750 


.7911 


.7756 


.7916 


.7761 


.7933 


.n66 


.7987 


.7773 


.7933 


.7777 


.7938 


.7783 


.7948 


.7788 


.7948 


.7793 


.7954 


.7799 


.7959 


.7804 


.7964 


.7809 


.7970 


.7815 


.7975 



47° 48'= 



.7975 
.7960 
.7966 
.7991 
.7996 
.8003 

.8007 
.8013 

.8018 
.8033 
.8028 

.8084 

.8030 
.8044 
.8050 
.8065 

.8060 

.8071 
.8076 
.8083 

.8067 
.8093 
.8098 
.8103 
.8108 

.8118 
.8119 
.8134 
.8139 
.8136 



.8135 
.8140 
.8145 
.8151 
.8156 
.8161 

.8167 
.8173 
.8177 
.8183 
.8188 

.8193 
8198 
.8204 
.8209 
.8314 

.8320 
.8235 
.8230 
.6236 
.8341 

.8246 
.8351 
.8257 
.8263 
.8367 

.8273 
.8278 
.8383 
.8389 
.8394 



49° 


59° 


.8394 


.8453 


.8299 


.8458 


.8304 


.8463 


.8310 


•o40d 


.8315 


.8473 


.8320 


.8479 


.8336 


fUAL 


.8331 


.8489 


.8336 


.8495 


.8341 


.8500 


.8347 


.8505 


.8353 


.8510 


.8357 


.8516 


.8363 


.8521 


.8368 


.8526 


.8373 


.8531 


.8378 


.8537 


.8.^84 


.8543 


.8389 


.8547 


.8394 


.8552 


.8400 


.8558 


.8405 


.8563 


.8410 


■8668 


.8415 


.8573 


.6431 


.8579 


.8436 


.8584 


.8431 


.8589 


.8437 


.8694 


.8443 


.8600 


.8447 


.8605 


.8453 


.8610 



V 
3 
4 
6 

8 
10 

13 
14 
16 
IB 
20 

33 
34 
36 
38 
30 

82 
34 
36 

98 
40 

42 
44 

48 

48 
50 

~M 
54 
68 

58 



10 



146 



TABLE OF CHORDS. 



VsMe of ebordSy in parts of a rad 1 ; for ^rotrmmUmg >- Contiiiiisd 



M. 


n° 


6SO 


MP 


54'' 


Ofto 


56° 


57« 


Sfio 


59° 


•o° 


0' 


MIO 


.8767 


.8934 


.9060 


.9286 


.9889 


.9648 


.9696 




1.0000 


3 


.8615 


.8778 


.8939 


.9066 


.9340 


.9396 


.9648 


.9701 


.9864 


1.0006 


4 


.8621 


.8778 


.8984 


.9090 


.9345 


.9400 


.9568 


.9706 


.9860 


1.0010 


« 


.8636 


.8783 


.8940 


.9096 


.9260 


.9405 


.9569 


.9711 


UCMBJ 


1JW16 


8 


.8681 


.8788 


.8946 


.9101 


.9256 


.9410 


.9564 


.9717 


•vonP 


1.0030 


10 


.8686 


.8794 


.8960 


.9106 


.9281 


.9416 


.9669 


.9733 


.9674 


1.0036 


13 


.8642 


8790 


.8966 


.9111 


.9266 


.9430 


.9674 


.9737 


.9879 


1.0060 


14 


.8647 


.8804 


.8960 


.9116 


.9271 


.9436 


.9679 


.9782 


.9884 


1.0066 


16 


.8662 


.8809 


•8D0D 


.9131 


.9276 


.9480 


.9684 


.9737 


•VSoV 


1.0040 


18 


.8667 


.8814 


.8971 


.9136 


.9281 


.9486 


.9689 


.9742 


■INNM 


1.0046 


30 


.8668 


.8830 


.6976 


.9183 


.9287 


.9441 


.9694 


.9747 


.9899 


1.0060 


38 


■8Od0 


.8836 


.8961 


.9187 


.9292 




OMf 


.9763 


.9904 


1.0065 


34 


.867S 


.8880 


•cWBo 


.9143 


.9297 


.9461 


•9604 


.9767 


.9909 


1.0060 


as 


.8678 


.8885 


.8993 


.9147 


.9302 


.9466 


.9610 


.9763 


.9914 


1.0065 


38 


.8684 


.8841 


.8897 


.9163 


.9807 


.9461 


.9616 


.9767 


.9919 


1.0070 


M 


.8688 


.8846 


.9003 


.9167 


.9312 


.9466 


.9630 


.9773 


.9934 


1.0076 


83 


•OWv 


.8851 


.9007 


.9168 


.9817 


.9473 


.9626 


.9778 


.9939 


1.0060 


M 


.8690 


.8866 


.9013 


.9168 


.9823 


.9477 


.9680 


.9788 


■VvV* 


1.0066 


86 


.8706 


.8861 


.9018 


.9178 


.9828 


.9483 


.9685 


.9788 


.9989 


1.0061 


88 


.8710 


.8867 


.9038 


.9178 


.9833 


.9487 


.9640 


.9798 


.9946 


1.0096 


40 


.8716 


.8872 


.9038 


.9183 


.9888 


.9493 


.9646 


.9798 


.9960 


1.0101 


43 


.8720 


.8877 


.9088 


.9188 


.9843 


.9497 


.9660 


.9808 


.9955 


1.0106 


44 


.8736 


.8882 


.9088 


.9194 


.9348 


.9503 


.9666 


.9608 


.9060 


1.0111 


46 


.8781 


.8887 


.9044 


.9199 


.9853 


.9607 


.9661 


.9618 


.9965 


1.0116 


48 


.8786 


.8888 


.9049 


.9304 


.9869 


.9512 


•VOBo 


.9818 


.9970 


1.0131 


60 


.8741 


ftflOfi 


.9064 


.9309 


.9364 


.9518 


.9671 


.96X8 


.9976 


1.0136 


63 


.8747 


.8908 


.9069 


.9314 


.9869 


.9623 


.9676 


J638 


.9980 


1.0181 


64 


.8762 


.8908 


.9064 


.9319 


.9874 


.9638 


.9681 


.9668 


.9986 


1.0186 


66 


.8757 


.8914 


.9069 


.9335 


.9379 


.9638 


.9686 


.9888 


.9990 


1.0141 


68 


.8762 


.8019 


.9076 


.9330 


.9884 


.96a6 


.9691 


.9648 


.9996 


1.0146 


60 


.8767 


.8924 


.9080 


.9336 


.9880 


.9548 


•VQVD 


.9648 


1.0000 


1.0161 



9 

3 
4 
6 
8 
10 

13 

14 
1« 

18 



IS 

94 
16 
SB 
10 

ss 

84 



44 



6S 
64 



M. 


en.o 


62° 


•8° 


64° 


65° 


e^° 


•7° 


•SO 


er> 


700 


M. 


0' 


1.0151 


1.0801 


1.0450 


1.0698 


1.0746 


1.0693 


1.1089 


1.1184 


1.1838 


1.1473 


0- 


3 


1.0156 


1.0306 


1.0455 


1.0608 


1.0761 


1.0898 


1.1044 


1.1189 


1.1888 


1.1476 


s 


4 


1.0161 


1.0811 


1.0460 


1.0608 


1.0756 


1.0903 


1.1048 


1.1194 


1.1888 


1.1481 


4 


6 


1.0166 


1.0316 


1.0466 


1.0613 


1.0761 


1.0907 


1.1063 


1.1198 


1.1S43 


1.1486 


e 


8 


1.0171 


1.0321 


1.0470 


1.0618 


1.0766 


1.0912 


1.1068 


1.1203 


1.1S47 


1.1491 


s 


10 


1.0176 


1.0826 


1.0475 


1.0623 


1.0771 


1.0917 


1.1063 


1.1208 


1.U63 


1.1496 


M 


13 


. 0181 


1.0331 


1.0480 


1.0628 


1.0775 


1.0923 


1.1068 


1.1213 


1.IS67 


1.1500 


IS 


14 


1.0186 


1.0336 


1.0485 


1.0683 


1.0780 


1.0927 


1.1073 


1.1218 


1.1963 


1.1606 


U 


16 


1.0191 


1.0841 


1.0490 


1.0688 


1.0785 


1.0982 


1.1078 


1.1222 


1.1866 


1.1610 


16 


18 


1.0196 


1.0346 


1.0495 


1.0643 


1.0790 


1.0937 


1.1082 


1.1227 


1.1371 


1.1614 


U 


20 


1.0301 


1.0361 


1.0500 


1.0648 


1.0795 


1.0942 


1.1067 


1.1232 


1.1876 


1.1619 




33 


1.0206 


1.0356 


1.0504 


1.0653 


1.0800 


1.0946 


1.1093 


1.1237 


1.1381 


1.1634 


S8 


34 


1.0211 


1.0361 


1.0609 


1.0658 


1.0605 


1.0951 


1.1097 


1.1242 


1.1386 


1.1529 


S4 


26 


1.0216 


1.0866 


1.0614 


1.0662 


1.0810 


1.0956 


1.1102 


1.1246 


1.1390 


1.1683 




38 


1.0221 


1.0870 


1.0619 


1.0667 


1.0615 


1.0961 


1.1107 


1.1351 


1.1395 


1.1538 


36 


80 


1.0236 


1.0876 


1.0534 


1.0672 


1.0620 


1.0966 


1.1111 


1.1366 


1.1400 


1.1643 


80 


83 


1.0231 


1.0380 


1.0529 


1.0677 


1.0824 


1.0971 


1.1116 


1.1261 


1.1406 


1.1548 


83 


84 


1.02S6 


1.0385 


1.0534 


1.0682 


1.0829 


1.0976 


1.1121 


1.1266 


1.1409 


1.1562 


84 


86 


1.0241 


1.0390 


1.0539 


1.0687 


1.0834 


1.0980 


1.1126 


1.1271 


1.1414 


1.1667 


86 


88 


1.0246 


1.0896 


1.0644 


1.0692 


1.0839 


1.0985 


1.1131 


1.1275 


1.1419 


1.1662 


Si 


40 


1.0251 


1.0400 


1.0648 


1.0697 


1.0644 


1.0990 


1.1136 


1.1280 


1.1434 


1.1567 


46 


43 


1.0256 


1.0406 


1.0554 


1.0702 


1.0649 


1.0995 


1.1140 


1.1285 


1.1439 


1.1571 


4S 


44 


1.0361 


1.0410 


1.0659 


1.0707 


1.0654 


1.1000 


1.1145 


1.1290 


1.1433 


1.1576 


44 


46 


1.0266 


1.0416 


10664 


1.0712 


1.0859 


1.1006 


1.1150 


1.1295 


1.1438 


1.1681 


4ft 


48 


1.0271 


1.0420 


1.0568 


1.0717 


1.0863 


1.1010 


1.1165 


1.1299 


1.1443 


1.1586 


4ft 


60 


1.0376 


1.0425 


1.0574 


1.0721 


1.0868 


1.1014 


1.1160 


1.1304 


1.1448 


1.1690 


60 


63 


1.0281 


1.0430 


1.0579 


1.0726 


1.0673 


1.1019 


1.1165 


1.1309 


1.1453 


1.1506 


63 


64 


1.0286 


1.0435 


1.0584 


1.0781 


1.0678 


1.1024 


1.1169 


1.1314 


1.1467 


1.1600 


64 


66 


1.0391 


1.0440 


1.0589 


1.0736 


1.0683 


1.1029 


1.1174 


1.1319 


1.1462 


1.1606 


M 


16 


1.0396 


1.0445 


1.0598 


L0741 


1.0888 


1.1034 


1.1179 


1.1833 


1.1467 


1.1600 


6B 


•0 


1.0801 


1.0460 


1.0666 


1.0746 

•- 


1.0693 


1.1039 


1.1184 


1.1828 


1.1473 


l.ljSU 


•ft 



TABLE OF CHORDS. 



147 



Table of Cbovda, in parte of a rad 1 } i 


for protractlnfT— 


-Continued 


M. 


71° 


TSB® 


7SO 


740 


750 


7«o 


770 


78° 


7V> 


80° 


ML 


0' 


1.1614 


1.1756 


1.1896 


1.2036 


1.2175 


1.2313 


1.2450 


1.9586 


1.2722 


1.2856 


» 


•i 


1.1619 


1.1700 


1.1901 


1.2041 


1.2180 


1.2318 


1.2455 


1.2691 


1.27a 


1.2860 


3 


i 


1.1624 


1.17« 


1.1906 


1.2046 


1.2184 


1.2322 


1.2459 


1.2505 


1.2731 


1.2865 


4 


• 


1.1628 


1.1770 


1.1910 


1.2050 


1.2188 


1.2327 


1.2464 


1.2600 


1.2735 


1.2869 


« 


• 


1.163S 


1.1775 


1.1916 


1.2056 


1.2194 


1.2882 


1.2468 


1.2604 


1.2740 


1.2874 


8 


10 


1.1638 
1.1642 


L1770 
1.1704 


1.1920 


1.9060 


1.2198 


1.2886 


1.2473 


1.2609 


1.2744 


1.2878 


10 


u 


1.1934 


1.9004 


1.3303 


1.2841 


1.3478 


1.M14 


1.2748 


1.2882 


IS 


14 


1.1647 


1.1T80 


1.1939 


1.2060 


1.2208 


1.2346 


1.24«i 


1.2618 


1.2763 


1.2887 


14 


U 


1.1663 


1.170S 


1.19S4 


1.3073 


1.2212 


1.2360 


1.24H7 


1.2623 


1.2757 


1.2891 


10 


18 


i.nsT 


1.1706 


1.1IS8 


1.3078 


1.2217 


1.3364 


1.2491 


1.9627 


1.2763 


1.2896 


18 


30 


1.1661 


i.isa 


1.1948 


1.3086 


1.9991 


1.3869 


1.9496 


1.9I83 


1.27M 


1.2900 


n 


n 


1.1666 


1.1807 


1.1949 


1.3067 


1.3236 


1.2364 


1.2500 


1.2636 


1.2771 


1.2905 


39 


34 


1.1671 


1.1813 


l.MM 


1.2003 


1.9381 


1.3368 


1.2506 


1.9641 


1.2776 


1.2909 


84 


3S 


1.1676 


1.1817 


1.1W7 


1.9007 


1.3236 


1.2873 


1.2600 


1.3646 


1.3780 


l.»I4 


a 


a 


1.1680 


lun 


11063 


1.2101 


1.8340 


1.2377 


1.2514 


1.2660 


1.2784 


l.»18 


a 


» 


1.U86 


1.18M 


1.1866 


1.3106 


1.32a 


1.2389 


1.K18 


1.9664 


1.1789 


1.2933 


so 


n 


l.ltM 


1.1BS1 


1.1971 


1.3111 


1.93a 


1.2886 


1.2523 


1.2659 


1.3798 


1.2937 


88 


M. 


1.1604 


LUM 


1.1976 


1.3116 


1.2254 


1.2891 


1.2528 


1.2663 


1.2798 


1.2931 


84 


M 


LU99 


1.1840 


1.1980 


1.2120 


1.2268 


1.2896 


1.2539 


1.2668 


1.2802 


1.2936 


M 


M L1T04 


1.1846 


1.1986 


1.2124 


1.22tt 


1.2400 


1.2687 


l.a72 


1.2807 


1.29a 


M 


40 i.no0 


1.1860 


1.1990 


1.2129 


1.3967 


1.2406 


1.2641 


i.an 


1.2811 


1.2945 


a 


43 


LHU 


1.1864 


1.1994 


1.2134 


1.2272 


1.2409 


1.2546 


1.9B8I 


i.aie 


1.2949 


48 


44 


1.1718 


1.1659 


1.1900 


1.2138 


1.2277 


1.2414 


1.2550 


1.2686 


1.2820 


1.2954 


44 


a 


Ln23 


1.1864 


1.9004 


1.2143 


1.2281 


1.2418 


1.2555 


1.2690 


1.2825 


1.2958 


a 


a 


1.1727 


1.1868 


1.3006 


1.2148 


1.2286 


1.2428 


1.2559 


1.2695 


I.28» 


1.2962 


a 


w 


1.17S2 


1.187S 


1.201S 


1.2152 


1.2290 


1.2428 


1.2564 


1.2690 


1.3838 


1.2967 


60 


it 


1.1TS7 


1.1878 


I.90I8 


1.2157 


1.2296 


1.2432 


1.3668 


1.2704 


1.2838 


1.2971 


68 


u 


1.174S 


1.188t 


1.9022 


1.2161 


1.2299 


1.2437 


1.2573 


1.2706 


1.3842 


1.2976 


64 


M 


1.1746 


1.1887 


1.9037 


1.2166 


1.3304 


1.2441 


1.2577 


1.2713 


1.2847 


1.2980 


66 


M 


1.1761 


1.1803 


l.aOS2 


1.2171 


1.2309 


1.2446 


1.2582 


1.2717 


1.2861 


1.2985 


68 


m 


L1756 


1.1896 


1.3066 


1.2176 


1.3311 


1.2450 


1.2586 


1.2722 


1.2866 


1.2989 


M 



0' 

9 
4 
• 
8 
10 

18 
14 
16 
18 
80 

"m" 

84 

a 
a 

M 

38 
84 



48 
44 

a 
a 

w 

IT 

H 

16 



«1« 



.3903 



1. 

1. 

1. 

1.9008 

1.M07 

1.9011 



1.8015 
1.3030 
1.S024 

i.soa 
i.soa 



i.9oa 

1.3048 

i.soa 

1.8061 
1.3056 



1.3060 
1.3064 
1.3068 
1.8073 
1.S0T7 



1.3068 

1.8086 

1.3000 

1.! 

1. 



1.3104 
1.3106 
1.8118 
1.1117 
1.S181 



1.8181 
1.3ta 
1.3ia 
1.3134 
1.3ia 

i.8ia 



1.3147 
1.3158 
1.3156 
1.3161 

i.3ia 



i.3ia 

1.9174 
1.3178 
IJia 

1J187 



1.3191 
1.S1M 
1.3800 
1.3904 
1. 



1.8213 
1.3318 
1.8828 
1.33a 
1.8881 



1.3336 
1.3239 
1.9844 

1.83a 
i.3a8 



i.8a9 
i.3a7 
i.3ai 

1.32tt 
1.8270 
1.3274 



1.3379 
1.3388 
1.3287 
1.3293 

Lsao 



1.3800 
1.8306 
1.38a 
l.ai8 
l.ttl8 



l.a28 

i.aa 

1.3W1 
1J886 
1. 



1.044 
1.83a 
1.3868 
1.3367 
1.3M1 



1.3865 

1.3370 
l.a74 
1.3878 
1.8383 



840 



1.1 

1.3387 

i.8ai 

18896 
1.3400 
1.3404 



1.3409 
1.3413 
1.3417 
1.3481 
1.84a 



1.3430 
1.8484 
l.S4a 

i.Ma 

1.3U7 



1.3468 
1.84S6 

1.8460 
1.8466 
1. 



1.8473 
1.3477 
1.3a8 
1.34a 
1.3490 



1.3486 
1.3499 
1.8608 
1.3508 
l.ai2 



85^ 



1.3612 
1.3516 
1.85W 
1.3525 
l.a29 
1.3533 



1.3538 
1.3542 
1.3546 
1.3560 
1.8665 



1.85a 

1.3663 
1.3667 
l.a72 
l.tt76 



1.8580 
1.8586 
1.85a 
1.86a 
1.3697 



1.8a2 
1.3606 
1.K10 
1.3614 
1.3619 



1.8623 

1.3627 
1.3631 
1.3636 
1.3640 



8«° 



1.86W 
1.3644 
l.S6a 
1.3668 
1.3657 
1.3M1 



1.3665 
1.M70 
l.a74 
1.M78 
1.3682 



1.3687 

Lsai 

1.36M 
IJMW 
1.3704 



1.37a 
1.3712 
1.3716 
1.8721 
1.87a 



1.37a 
1..H73S 
1..17a 
1.3742 
1.37a 



1.8750 

1.3754 
1.37a 
1.3783 
1.3767 



870 


880 


880 


1.8767 


1.88a 


i.ai8 


1.3771 


1.3897 


1.4028 


1.3776 


1.3902 


i.4oa 


1.37a 


1.39M 


1.4031 


1.3784 


i.aio 


1.4035 


1.87a 


i.ai4 


i.4oa 


1.8792 


i.ai8 


1.4043 


1.3797 


1.3922 


1.4047 


1.3801 


i.a27 


1.4051 


1.3806 


i.3ai 


1.4055 


1.8800 


1.3966 


i.4oa 


i.ais 


i.aso 


1.4064 


i.ai8 


1.3943 


1.4068 


i.Mn 


1.3947 


1.4072 


i.a26 


1.3952 


1.4076 


1.8830 


1.3966 


1.4080 


1.3»4 


l.S9a 


1.4084 


1.38a 


1.3964 


1.4089 


1.38a 


1.39a 


1.4O03 


1.8847 


l.a72 


1.4097 


1.3861 


l.a77 


1.4101 


1.3855 


i.3ai 


1.4105 


1.3860 


1.3985 


1.4109 


1.8864 


1.38a 


1.4113 


1.3868 


1.3993 


1.4U7 


1.3872 


i.sa7 


1.4122 


l.a76 


1.4002 


i.4ia 


1.3881 


1.4006 


1.41M 


1.3885 


1.4010 


1.4134 


1.3889 


1.4014 


i.4ia 


1.8808 


1.4018 


1.4148 



0' 

8 
4 
6 
8 
10 

18 
14 
16 
16 
M 

22 
24 

a 
a 
a 

88 
34 
M 

a 

40 

48 
44 

a 

a 
a 

la 
54 
a 
a 

a 



F0LYG0N8. 



m. HfiuaH. BipUoam. Dctiun. 

nsBlar. Of coarvf tfin aambn af poljfOQK U IbBoLH. ' 

T»I>I« orBeroluP Polygons, 



X 




■.itTk- 






*^ar 


M 


tiiugla. 

Deongon, 
UndKBgon. 


J .«.„ 


Ji77»M 


eo° 


ISO" 




Ii;i96152 


:»so6si 


108° 

isn° 
H7° is.sese' 

180° 


90" 
60° 

.,.».„ 

40° 
32°43.«3M' 


^^rr^ij'/x'K.rf^ES'JKS^ ''°*'' "' "■ ■«•■ • 'X p^ ' *"" 


nx.«,^«,l. 


S« Bf lawriar astf «, ■ b <!. m, ar mar poljB«. respUr « In. 


■■Ur = iaa°x 






TBIASTOIiES. 




*A »/K /K* h\i> IV^ 


E 


7 


f\ /^ 


\/r \i^ ^ 


^ r\ 


\ 



i<B^; <c IbDH biTlii) itoml^t 






TBIANaLES. 



149 



^•o find area, baTlnflr one aide and tbe A angles at its ends. 

Add the t anglM together; take the sam from lW>f the rem will be the angle opp the given ilde. 
Find the nat BUte of tfals angle ; also find the nat ainea of the other angles, and mult them together. 
Then ai the nat alne of the alngle angle, ia to the prod of the nat sinei of the other 2 anglea, ao ia the 
tfumre of the given side to tUnM* tbe reqd area. 

To find area, bavlngr two sldes^ and tbe Inelnded ang^le* 

Ifnlt together tbe two eidee, and the nat sine of the tnoloded angle ; dlr by 2. 

Ez.~8ides 650 ft and 980 ft; included angle W* 20'. By the table we find the nat tine .9856 1 

therefore* ^j s= 397988.6 aqnare ft area. 

To find area^ baTlnc tbe tbree ang^les and tbe 
o perp belybt, a b. 

Find tbe nat sines of the three angles ; mult together the sines of the anglae 
d and : dlT the sine of the angle h by tbe prod ; mult the qnot by the squari 
of the perp height a & ; dlr by 2. 

To find any side, as tf o> baTing^ tbe tbree 
angles, d, h and Of and tbe area. 

(Sine of d X rine of o) | sine of b 1 1 twlee the area t aware of d o. 




The perp height «fmm eqvilatenU irlansle is eqaal to one aide X .860025. Hence one of 
its Bidea is equal to the perp height div by .8660-25 or to perp height X 1.1M7. Or, to find £ at4«i 
BHdt the sq rt of its area by 1.61967. The side of an equilateral triangle, mult by .658037 = side of* 
I of ue same area } or mult by .742517 it gives the diam of a eircle of the same area. 



n 



C a B 





The following apply to any plane triangle, whether oblique or right-angled 
S. The three angles amount to 180°, or two right angles. 
9l Any Mcterior angle, as A C n, is equal to the two interior and opposite 
aoes, A and B. 
C The greater side is opposite the greater angle. 

4i Tha sides are as the sines of tbe,opposite angles. Thus, the side a is to 
the Mm 6 as the sine of A is to the sine of B. 

ik If any angle as s be biseeted by a line • o, tbe two parts me, o n of 
thfi eppaeite side m n will be to eaeh other as the other two aides »m, an; 
•r, »•:« n::s m:s n. 

4L If ttnes Iw drawn tnm eaoh angle r • < to the 
~ eenter of tbe onposite side, they will eross eaoh 

other at one punt, a, and the abort part of each 
of the lines will be tbe third part of the whole line. 
Alao, « is the eea of sntT of the triangle. 

T. If lihoa be drawn bisecting the three angles, they will meet at a point 
perpendionlarly equidistant from eaeh aide, and consequentlj the centev 
ai^ V — a^ f of tke sreateet etr<de that ean be drawn in the triangle. 

•^ ^^* 8. If a line « n be drawn parallel to any side e a, 

«iie two trianglM ran^re€i, will be similar. 

•. To divide any triangle aer into two equal parts by a line s n parallel to 

any en* of its sides c a. On either one of the other aides, as a r, as « diam, 

dsMrIb* a samiairele a o r/ and find its middle e. From r (opposite e a), with 

radiusre, deaerilM theareon. From n draw n s. par- 

Q allel to e a. 

y\ 10. To And the grcatast parallelogram that ean be 

y^ \ drawn in any jriven triangle onh. Bisect the tbree sidea at a e s, and join 

<V^ jf o e> « «i a 0* Then either aehe, aeeo, or a ean, eaoh equal to half the 

^\ y^\ triangle, will be tbe reqd parallelogram. Any of these parallelograms can 

^ \^ \ plainly t>e converted into a rectangle of equal area, and the greatest that ean be 

% t 1% drawn in the triangle. * 

lOX. If a line a e bisects any two sides o i, o n, of a triangle, it will be par* 
allel to the third aide n b, and half as long as it. 

11. To find the greatest square that ean \m drawn in any triangle a ae r. From 
an angle as a draw a perp a n to the opposite side «r, and find its length. Then 

9 n, or a side v I of the square will = . 

BeBU~*If the triangle la such that two or three suoh perps ean be drawn, thM 
two or three equal squares may be found. 



an r 




;\5(\«5-«;''t.^- 



150 



FLANE TBIGOKOMETBT. 



Bifflit-aiiirle^ Tri»iiirlefl« 

4.U the foregoing appw also to right-angled triangles : hat what foUew 




the right angle A, and the othen B and C ; and eali 
oppoelte to them a, i, and e. Then Is 

ft = a X Sine B = aXOoeC = eXCotOs«X Tana S, 
cs«XSineO = aXGoaB-=»XTangO. 



them only. 
>e sidM nwMtlfelf 



e h 

Also Sine of = -; OoeO = ~/ 
a 



Tang I 
TangOi 






h § 5 

And Sine ttrBs-zOoeBs-/ Tang B = j. 

- ,. -w ^ _.\»* **■;••' 4. <>''**° = ': CoiA=0. Tang A rrlndnHy. SeeAstalBl^. 
1* If from the right angle o a line o w be drawn perp to the hypothenuie or long side * «, then the 
two small triangles owh.owg, and the large one oka. will be similar. 
Or Mr : 10 : : IP o : w A; and gwXwhszwoi. 

t. A line drawn from the right angle to the oeater of the long side will 
be hair as long as sa>d side. 

8. If on the three sides oh, og, gh me draw three sqnarae (, u, m, or 
three oireles, or triangles, or any other three figs that mm siadlar, thtp the 
area of the largest one is eq^oal to the sum of the areas of the (wo othfsn. 

4* In a triangle whose sides are as S, 4, and 6 Cas are thoee of the Irt* 
angle ABC), the angles are rery approximately MP; 5tor4S.nw; nad 
36° 52' 11.62'/. Their Sines, 1. ; .8} and .6. Their Tangs, inOnitj ; l.SaM : 
and .73. 

ft. One whose sides are as 7, 7, and 9.9, has rery appror one angle of 90» 
and two «r W* eaoh, near enoogh for all prsctical purposes. 



' «\ 


h 


^^ 


•■ 


;\ 


^ u 


/. 


9 



►-•- 



PLANE TEIGONOMETEY. 




P&Aira trigonometry teaohee how to find certain unknown parts of plane, or straight • aldnd M> 
•ni^, by means of other parts which are known ; and thus enables us to measure inaooessiUe dla> 
tanoes, Ao. A triangle oondsu of six parts, namely, three sides, and three ancles ; and If we know 
any three of theee. (except the three angles, and in the ambiguous case under "Case S,") we can flad 
the other three. The following four oases include the whole sulyeot ; the student shon^i oommlt then 
le memory. ^ 

■ ' C pH<» va 

Case 1. HaTlna: any two angles, and one side^ ^ **' 

to find the oilier sides and an^le. 

Add the two angles together ; and subtract their sum from 180^; the rem 
•vill be the third angle. And for tbe sides, as 

Sine of the angle . Sine of the angle . , ^„ .^ . .^^ ,,j- 
opp the given side • opp the reqd side • • «»»•«» "<» • '^l*^ •»<»* 

Use the tide thus found, as the given one ; and in the same manner And 
Ihe third side. 

Case 2. HaTlngr two sides, ba,ae, Vi^ X, and the ani^le a be, 
opposite tooneof tiiem, to find the other side and angles. 

Side a c opp The other Sine of the Sine of angle hdaor 

the given an* I given side I * given angle I icaopposite the other 
^tr gle a b c ba ab e given side b a. 

Having fonnd the sine, take out the oorreeponding angle from the labia af 
nat sines, but, in doing so, if the side • e opp the given aagto Is 

shorter than the other given side b a, bear in mind that an angle and Its snp« 
plement have the same sine. Thus, in Fig X, the sine, ai found above, is 

opp the angle & e a in the table. But a e, if sJtortsr than b a, can evidently be 
laid off in the opp direction, a d, in which case I «I • is the sappltment of ( c s. 
If a c is as long as, or longer than, b a, there can be no doubt ; for In that i 
It oannot be drawn toward b, but only toward n, and the angle A « « will 
ftMind ec onoe in th« table, opp the sine as fonnd abovib 




PLJLKE TRIOONOMETBT. 



161 



When th« two angtei, ahe,heo, have been (band, find th* remalalnK side hj Cue 1* 
IW the remaining angle, hae, add together the angle abc flrtt given, and the one, i e s. 
M abOTO. Oedoet their aam from 180<*. 

Case 3. KaTlniT ^wo sides, and the an^le included 

between tbem. 

Take the angle trem 180''; the rem will be the sum of (he two uDknown angles. Dlr thU sum bf 
t; and find the nat tang of the qaou Then as 

The »m of the . mw«|_ ^nr . • Tang of half the earn of . Tang of half 
two giTon sides • ^""■i^«"' . . the two unknown angles • their dlff. 

Take flrem the table of nat tang, the angle opposite this last tang. Add this angle to the half sum 
•f the two unknown angles, and it will give the angle opp the longest given side ; and subtraot it 
firem the same half sum, for the angle opp the shortest given side. Having thus found the angles, 
lad the third side by Case 1. 

As a praetieal example of the use of Case S, we oan asoertain the dist n m across a deep pond, by 
measuring two lines n o and mo; and the angle n e m. From these data we may calculate nm ; or 
by drawing the two sides, and the angle on paper, by a soale, we can afterward measure » m ea 
•he drawing. 





€ase 4. Kaviuir ^b® tbree sides* 

lb And tte three aaglM; upon one side • ( as a base, draw (or suppose to be drawn) a perp eg tnm 
the oppoaita angle c Find the diff between the other two sides, a c and c b ; also theLr sum. Then, as 

Sum of the , . Diff of other . Diff of the two 

other two sides • • two sides • parts ag and bg, of the base. 



The base 



Add half this diff of the parU, to JuU/ the base a &; the sum will be the longest part ag; which 
taken tnm the whole base, gives the shortest part g 6. By this means w« get in each of the small tri- 
angles a eg and egb, two sides, (namely, a c and a gi and c b and gb;) and an angle (namely, the 
right angle cga,megb) opposite to one of the given sides. Therefore, use Case 2 for flnding the 
a and e. When that is done, take their sum fMm WV>, tor the angle • c *. 

Or* Sd ■§•<« t call kalf the sum of the three sides, si and call the 
two sides which form either angle, mt and m. Then the nat sine of 




hiOf that angle wUl be equal to \ /C — *»)XJs 



-«> 



Fiir.i. 




Tig.fi. 



Ex. 1. To find tbe dlst from a to an Inae* 
eesslble objeet e. 

Measure a line ab; and from its ends measure the angles eab and 
eba. Thus having found one side and two angles of the triangle a > c, 
ealenlate a c by means of Case 1. Or if extreme aqonracy is not read, 
draw the line a I on paper to any convenient scale ; then by means of a 
protraeter lay off the angles c ab,eba; and draw a e and eb; thaa 
measure • e bj the same scale. 

Ex. 3. To find the helgrli^ of a veffioal 
objeet, n a. 

Place the instmmeni for measnrlng eagles, at any oenve. 
nlent spot o ; also meas the distea ; orif oa cannot be actually 
measd in consequence of some obstacle, calculate it by the 
same process as a e in Fig 1. Thm, first directing the instra< 
ment horizon tally,* as o s, measure the angle of depreesioa, 
to a, say liP ; also the angles o n, say 80°. These two anises 
added together, give the angle a on, 42°. Kow. in the small 
triangle o « a we have the angle o « a equal to 90O, because a n 
is vert, and o a hor ; and ninoe the three angles of any triangle 
are equal to 180p, if we subtract the angles ota <90O), and s e « 
(12°) from 180°. the rem (78°) will be the angle o a « or o a «. 
Therefore, in the triangle one, we have one side o a; and twe 
angles a on, and o a «i, to calculate tbe side a n by Case 1. 



i dlsts on sloping ^ronnd must be measured hor- 

Ison tally. The graduated hpr 
clrole of the instrument evideafly meaa> 

fr-rj *-'-*-'*TtP \ ures the angle between two ob}eets horl 

1 :^- /\r \ tonully, no matter bow much hlirher one 

— ^i^/. \ of them may be than the othf>r ; one pes* 

haps requiring the telescope of the iastra* 
ment to be directed upward toward it; 
and the other downward. If. thereforek 
the sides of trianglen lying upon sloping 
C \ ground, are not also meiuid hor, there can 

be no accordance between the two. Tba« 




PLANE TBIOONOMETKY. 




PLANE TRIGONOMBTBY. 



153 




its sngle iftt of incUuftUoa with the horison foand u before i 
in whioh cue the dut a n is caloolated. Or if the vert height c n 
is sought, the point o may first be found bj sighting upward 
along a plumb-line held abore the head. 

Ex. 3. To iind tlie approximate belifht^ 
9 00; of a moantain. 

Of whioh, perhaps, only the very summit, x, is visible abova 
interposing forests, or other obstacles ; but the dist. mi, of whioh 
is known. In this case, first direct the instrument hor, as m k; 

and then meainre the anglb i m x. 
Then in the triangle i m z we have 
one ^de mi: the measd angle <ms, 
and the angle mix (90°), to find ir 
by Case 1. But to this » z we must 
add 1 0, equal to the height y m of the 
-instrument above the ground; and 
also o «. Now, o s is apparently due 
entirelv to the curvature of the earth, 
whioh is equal to very nearly 8 ins, or 
.667 ft in one mile : and iaoreases aa 
the squares of the dists; being 4 
times 8 ins in 2 miles ; 9 times 8 ins 
is S mflM, ito. Bat thts It MBMVhat dinlnlshed bv the refraotion of the atmosphere ; whioh variee 
with temperature, moisture, &o ; but alwaya teaos to make the obieet x appear higher than it 

■otoallj is. At an average, this deoeptive elevation amovmts to aboat-=-th part of the enrvatuie of 

the earth ; and like the latter, it varies with the ■qnarea of the dists. Consequently if we subtract -=- 

part from 8 ins, or .667 ft, we have at onoe the combined effect of curvature and reft-action for one 
mile, eqaal to 6.867 Ins, or .5714 ft; and for other dists, as shown in the following table, by the UM 
of which we avoid the neoessity of making »q}arate allowances for curvature and refraction. 

Table of allowances to be added for carvature of tbe eartb ; 

and for refraction ; combined. 




Fig.7. 



Dist. 


Allow. 


Dist. 


Allow. 


Dist. 


AUow. 


Dist. 


Allow. 


inyarda. 


feet. 


in miles. 


feet. 


in miles. 


feet. 


in milee. 


feet. 


100 


.002 




.036 


6 


20.6 


20 


229 


150 


.004 


xt 


.143 


7 


28.0 


22 


277 


200 


.007 


y^ 


.321 


8 


86.6 


25 


357 


800 


.017 


1 


.572 


9 


46.3 


30 


614 


400 


.080 


11^ 


.803 


10 


57.2 


35 


700 


500 


.046 


\Xc 


1.29 


11 


69.2 


40 


916 


600 


.066 ' 


1% 


1.75 


12 


82.3 


45 


1168 


700 


.090 


2 


2.29 


13 


96.6 


60 


1429 


800 


.118 


2H 


3.67 


14 


112 


55 


1729 


goo 


.140 


3 


5.14 


15 


129 


60 


2058 


1000 


.185 


3K 


7.00 


16 


140 


70 


2801 


1200 


.266 


4 


9.15 


17 


165 


80 


3659 


1500 


.415 


4^ 


11.6 


18 


185 


90 


4631 


2000 


.738 


6 


14.3 


19 


206 


100 


5717 



, If a person whose eye is 5.1i ft, or 112 ft above the sea. sees an object just at the sea'b 

korixoB, that object will be about 3 miles, or 14 mites distant from him. 

A borlBOntal line is not a leirel one, for a straight line cannot be a 

level one. The carve of the earth, as exemplified in an expanse of quiet water. Is level. In Fig T, 
If we suppoee tiie enrved line tp»gio represent the sarfaoe of the sea, then tbe points ty » and g aae 
on a level with each other. They need not be equidistant ft-om the center of the earth, for the sea at 
the poles is about IS miles nearer it than at the equator ; yet its surface is everywhere on a level. 

Up. and down, refer to sea level. IjCTcI means parallel to the curvature 
of the sea ; and boriaontal means tangential to a level. 

Ex. 4. If tbe inaccessible irert beiffbt e d, Flip 8, 

A $o lUuated thai v>* cannot reach it at aU, then place the instrument for measuring angles, at any 
oonveoient spot n ; and in range between n and d, plant two staffs, whose tops o and i shall range 
praeiaely with n, though they need not be on the same level or hor line with it. Measure n o : also 
from n meaaore the angles on d and one. Then move the instrument to the precise spot previously 

• — I — ' ■ — ' ~i 

which he had no idea. For allowance for curvature and refraction see above Table. 
A triangri® wbose sides are as 3, 4, and 5, is right angled ; and one 

'hose sides are as 7 : 7 ; and 9. 9 ; eontains 1 right angle ; and 2 angles of iffi each. At it is fre* 
<|eently' necessary to lay down angles of 45° and 9QP on the ground, these proportions may be used for 
the purpose, by shaping a portion of a tape-line or chain into suoo a triangle, and driving a stake at 
eaehani^ 



154 



PLANE TBIQOKOMETBY. 



ipted by tbe top o of the lUff; and trvm o mearan th« aaftat <• 4 kdA40c 

tract tbe angle < o e ftom 
180° ; tbe rem will be tbe 
angle e • n. Cenaeqaent- 
ly in tbe triangle nee, we 
bare one side n o, and two 
angles, «no and e o n, to 
find by Case 1 tbe aide o e. 
Again, take tbe angle iod 
from 180° ; tbe remainder 
will be tbe angle n o d, ao 
that in tbe triangle dno 
we bare one side n o, and 
tbe two angle* dno and 
» d, to find br Case 1 
tbe tide od. Finally, in 
tbe triangle cod, we hare 
two aides CO and od, and 
tbcir included angle cod, 
to find d, tbe reqd rerfe 
bfligbt. 





Figr.a. 



Figr.9. 




Jttd were in a valley, or on a bill, and tbe obserrationi reqd to be made tnm either hlgta«r 
•r lower groond, tbe operation would be precisely the same. 

£x. 5. See Sx 10. 

To find (be dlst ao. Tig 9, betwe«M two oiitirely inaceemiMe 

oliJecUi, 

Meaiwre asldenm; at n measure the angles a nm and onm: also at mnMasore the angles o mm, and 
• M fk This being done, we have in tbe triangle anm, one side n m, Fig 9, and tbe anglee •«»••, and 
nma; benoe, br Case 1, we can calculate the side an. _ 
▲gain, in tbe triangle o m n we have one side n m, and P 
the two angles omn, and mno; hence, by Case 1, we can 
•alenlate the side n e. This being done, we have in the 
triangle ano, two sides an, and n o ; and their included 
angle a n o ; hence, br Case 8, we can oalcnlate tbe side 
ao, which is the reqd dist. It Is plain that in this manner 
we may obtain also the position or direction of tbe inacces- 
sible line a o ; for we ean calculate tbe angle nao; and can 
therefrom deduce that of ao; and thus be enabled to ran 

a line parallel to it, if required. By drawing n m on pa- T!itr If) 

per bT a scale, and laying down the four measd angles, 'iK- -lu* 

Che dist a • may be measd upon tbe drawing bj tbe same scale. 

If the position of the inaccessible dist c n. Fig 10, be such that 
we can place a stake p in line with it, we may proceed thus : Place 
the instrument at any suitable point «, and take tbe angles ptc 
and cnn. Also find the angle eps, and measure tbe distps. Then 
In the triangle p t c find « e by Case 1 ; again, the exterior angle 
n e «, being equal to tbe two interior and opposite angles cp «, 
and j> « c, we have in the triangle eon^ one side and two angle* 
to find e n by Case 1. 

Ex. 6. To flnd a dlst ah, Flgr II9 of whieh 
the ends only Mre accessible. 

From a and 6, measure any two lines a e, & c meeting at e ; also 
measure the angle a eh. Then in the triangle aft c we have two 
sides, and tbe included angle, to find the third side a 6 by Case S. 

Ex. 7. To And tbe vert beigbt o nt^ of a FfflT- U. 

bill, above a i^iven point i. 

Flaoe the instrument at i ; measure a m. Directing 
the instrument hor, as an, take tbe angle nam. Then, 
since a n m is 9P Fig 12, we bare one side a m, and 
two angles, nam and a n m, to find n m by Case 1. 
Add n o, equal to a <, the height of the instrument. 
Also, if tbe bill Is a long one, add for cnrrature of the 
earth, and for reh-action, as explained in Example 3, 
Fig 7. Or tbe instrument may be plaoed at the top of 
the bill ; and an angle of depression measured ; instead 
of tbe angle of elevation nam. 

Bxu. 1. It is plain, that if tbe height o m be previously 
known, and we wish to ascertain tbe dist from its Bum- TiMir 72 

mit m to any point i, the same measurement as before, * ' * 

of the ancle nam, will enable us to calculate a m by 

Case 1. So in Ex. 2, if the height na be known, the angles measd in that example, wfU enable «k 
to compute the dist a ; so also In Figs S, 4, 6, and 7 ; La all of which tbe process is so plain as to 
raqnire no further explanation. 

Bbm. 2. Tbe height of a vert object by UieanS Of its SbadOW. Plant one end of 
a straight stick vert in the ground ; and measure ts shadow ; also measure tbe length of tbe shadow 
of the object. Then, as the length of the shadow of the stick is to tbe length of the stick abovt 





PLANE TRIGONOMETRY. 



156 



gnvaA, lo to tlM toagtli of IIm ahadov of tht ol^oot, to its helgbt 
moBk bo eqvftUy iaolinod. 



If the ob|}«et It inoHiMd, the itiek 



xu 1 my Rem. 8. Or tb« beiffbtof a irert object mn^ 

'^£r* Ji^H , Fig l^^whoee distance r m is known, may be found by 
ZJ^ Iti rellection in a vessel of water, or in a piece of 

.'"y^ looking f iUB plaoed perteotW borixontal at r ; fttr •■ r als to tlM balglUI 
[^ a < of the eye above the refliMtor r, w to r m to^^ ^ 13*1 <» "i^Xd. 




the height m n of the ol^eot above r. 

Rem. 4. Or 
n pl»nied pole, or a rod held yert 

staod at a proper dlit baok tnm It, and keeping the ^ee eteadj, let marks 
made at o and e, where the lines of sight i n aad iae strifea tht rod. Then 
ieistoeo, soisimtomn. 



»r let c. Fig 12K 

by an assistant. T 



"•"fir.. Pifir.l2> 

-ksbe I 6L-->* 

sn •m^.Mex::^ — ' 1, 




flff.lS. 



The following examples may be regarded as tabetitntei for strict trigonome- 
try : and will at times be nsefhl. in ease a table of sines, fto, to not at hand for 
making trigenometrieal ealoulations. 

Ex. 8. To And tbe dlst a h^ of wbicb one end only 

Is accesftlble. 

Drive a stake at any eonvenient point a ; ft!>om a lay off any angle i a e. In 
the line « e, at any coDvenient poini c, drive a stake ; and fh>m c lay off an angle 
acd, eqaal to the angle b ac. In the line e d. at any oonrenient point, as dt 
drive' a stake. Then, standing at d, and looking at h, plaoe a stake o in raoft 
with d h ; and at tbe name time in the line a c. Measure ao,oc, and cd\ 
from the principle of similar triangles, as 

o e \ e d I X a o X Ah. 




Fiff.lfi. 



Or tbnss 

VIg 14, » A being tbe dtot, plaoe a stake at n ; and lay off tbe angle b n m VP. 
At any convenient dlst n tn, place a stake m. Make the angle it m y =90° ; and 
plaoe a stake at y, in range with h n. Measure n y and n m ; then, fh>m tht 
principle of similar trianglea, as 

n]f:tt»»t:nn»:nA. 

Or tbns. Fig 14. Lay off the angle hnm=^ 90°, placing a stake 

m, at any ooaventent dtot n m. Measure n m. Also measure the angle n m A. 
Find nat tang of » m A by Table Mult thto nat tang by n «. The prod 

will ben A. 

Or tbns. Lay oflT angle A n m » 90^. From m measure the 
angle n m A, and lay off angle n m y equal to tt, plaolag a ttnkt at y la raagt 
with A n. Then to n y = n A. 

Or tbns, without measurlnir 
any ang^le ; 

t « being the dlst. Make it v of any convenient 
length, in range with ( u. Measure any v o ; and 
o % equal to It, in range. Measure u o ; and «« 
equal to it in range. Plaoe a stake s in range with 
both X y, and ( o. Then will y jt be both equal to 
t u, and parallel to it. 





Or tbna, witbont meiisarlnir ^ny anffle. 

Drive two stakes I and «, in range with the object s. From ( lay off any 
eonvenient diet t x, in any direction. From « lay off w w parallel to < s, 
placing 10 in range with z <. Make « v equal to ( «. Measure w •, v s, and 
X t. Then, as 

vpifxvaBx xett xt0. 

Or tbiifl. At a lay off angle oac » S^ 48^ Lay 

v « off 00 at right angles to ao. Measure oe. Then 

_, • _ 00 » lOoe, too long only 1 part in 935.6, or 5.643 feet 
Ylg. 16, in a mile, or .1069 foot (full U Inches) in 100 feet. 




PLANE TRIGONOMETRY. 






Ex. lO. See Bx. 4. To And U* •ntlreir 
iDMcewlble dlBt — ------ 

lu dlr«ei 










FABALLELOGBAHB. 



167 



Square. 



PARAI«IiEI.OOBAMB. 

Rectangle. Bhombus. 





Rhomboid. 



]^""*--, 



8 




A PAKALLELOORAX is any figure of four straight sides, the opposite ones of wbtch 
are parallel. There are bat four, as in the above figs. l%e rhombus, lilce the rhom- 
bf^odron. Fig 3, p 106, is sometimes called ** rhomb." In the square and rhombus 
all the foar sides are equal ; In the rectangle and rhomboid only the opposite ones 
are equal. In any parallelogram the four angles amount to four right angles, or 
360^ ; and any two diagonaUy opposite angles are equal to each other ; hence, having 
one angle given, the other three can readily be found. In a square, or a rhombus, a 
diag divides each of two angles into two equal parts ; bat in the two other parallel- 
ograms it does not. 

To flnd tbe area of any parallelosram. 

Mnltlply any ilde, m 8, bv the perp height, or dUt p to ihs opposite aide. Ovk multiply tocathar 
two sMm and nmt alne of their inoladad aagla. 

The 4Smm a b of any s^aare is equal to one side molt by 1.41421 ; and a side is eqaal to 
diacooal 
^^^31 ; er, to diag mult by .707107. 

'31ie side ef a B««are eqval tn area to a aUrem elrele» is equal to dSam X .89Stn. 

Tke dide of file sreateet aoaare, tMat can h*in»erib«d in 
•^MM HreU, is eqnal todlaoi X .707107. 

Tha side of a sanara molt by 1.51967 gives the aide of an equi- 
lateral trtanue of the same area. All paraUelosraau as a. 
aad C, whiek littve eq^al baaea» a c, and eqnal psrp heights n 
e, haTe also equal areas ; and the area of ea«h Is twice tbat of a tri> 
angle baring the same base, and perp height. The area of a 
■raare laserlbed In a elrele i« equal to twioe tbe square of the 

In every parallelosranM the 4 squares drawn oh its sides have a united area «qu^ to that of 
tha tvo squares drawn on iu 2 diags. If a Inrcer aqnare be drawn on tha diag a 6 of a a mailer 
square, ite area will be twioe tbat of said smaller square. Either dlas of any parallelosram 
tfridea IMato two eqnal triangles, and the S diags div it inte 4 triangles of eonal areas. The two 
ly MiraUelo|trani divide each other Into two equal parte. Any Une drawn throach 
iter of a 41aC divides the parallelogram into two equal parte. 

1.— The urea of any fiff whatever as B that la eneloeed bylbnr atralcht 

. __j may be found thus : Mult together the two diags mm,nb: and the nat sine of tbe least angle 

«oi;ori»e«H fbnnad by their interseotion. Div the prodnet by 3. This Is useful Id land surveying, 
whan ohataelaa, as is often tha aaaa^ make it dilBauU to measara tha sides of the flg or flald ; while Ik 
may be easy to measure the diags ; and after finding their point of interseotion o, to measure the re* 
qnbed angle. Bnt If the flgr 1* to be drawn, the porta o «, o 6, o n, o m of the diags must also 
be measd. 

Boh. 9.— The sidee of a parallelogram, trlani^e, and many other !«■ may he 
Ibnnd, when only the area an4 aanlea are ftven, thus : Assume some partloular one of ite 
•ides to be of tbe length 1 ; and oaleulaw what ite area would be if that were the ease. Then as the 
sq rt of the area thus found is to this side 1, so Is the sq rt of the aotual given area, to the oorre* 
•pondtaig aotoal aide of the fig. 



On a iriTen line tcr0e,to ^vww a M|aare^ 

From w and x, with red ts x, describe the aros xrp and to r e. 
From their intersection r, and with rad equal to H of w«. deaeribe 
M»». From ts and s draw tvn and 0m tangential to «s«, 
ending at the other aros j Join n «i. 



the 




158 



TRAPEZOIDS AND TBAPBZIUM8. 



TBAPEZOmS. 

fi t m n 





a « e at 

A trupmM menm,l» Miy flfwe with tour ttrmighl ildM, only two of ▼bioh, m me mad » *, art 
paraUd. 

To And tbe area of any trapoaold. 

Add toffBthar the two panlM tidoaf a « aad m n; malt ika aaai by tha parp diat • i 
tliam ; div Um prod bj S. Saa tha faUowiog mloa far trapaaiaKB, whlah ara all aqnally 
totoapasolda; alM laa BaoMrlu aftar Parallalofraau. 

TRAPCZIUMS. 




A trapaaiam a & e o, ia any flg with foor atralght ildaa, of which no two ara parallal. 

To find the area of any trapoBlnm, taaTlnir griven tbe diac 
5o, or a e, between eliber pair of opposite an^lee; and alia 
the two perpe, n, ft, fW>ni the other two anirlee. 



Add togathar thaae two parpo ; molt the som by the diag; dlT the prod by i. 

SlaTiniT the fonr sides i and either pair of opposite anirlcs* 

mm a be, a o eg or bao, and beo, 

Conaider the trapeiiam aa diridad into two trlanglaa, in aaeh of whieh ara givaa two lidae and tte 
Inoladed ancle. Find tbe area of eaoh of theae triangfea as direoted under the preoading head " Trt* 
aaglea," and add them together. 

HaTlnfp the fonr angples, and either pair of opposite sides. 

Begin with one of the aidea, and the two anglea at its enda. If the aam of these two aaglea exeeeds 
180O, aabtraet aaeh of tbem from 180°. and make use of the rema Inataad of the angles tbemaalTaa. 
Than oonslder this side and its two adjaoant anglea (or the two reau, as tha oaae aMT be) aa tbn— 
af atriangia; and And ila area aa diraeted far thai aaaa under tha praead lag head "friangla." D* 
a* aama with the alhar glvao aida, and ita twa adjaoent angles, (ar their reau, aa tha oaae may ha.) 
Subtraot the least of the areas thus, found, from the greatest; the rem will be the raqd area. 

Havinff three sides ; and the two included anfrles. 

Mult together the middle side, and one of the adjaoent sides ; mult tbe prad by the uat sine of their 
ineloded angle ; call the result a. Do the same with the middle aide and its other a^aaaut aida, 
and the nat sine of the other included angle; call the result b. Add the two anglea together ; fln4 
the diir between their sum and 180(>, whether greater or less ; find the nat sine of this diff; malt 
together the two given sides whieh ara appostta one another ; molt the prod by the nat aine just found ; 
eall the result e. Add together the results a and ft ; then, if the sum of the two given angles is lass 
than 180°, subtract e from the anm of a and 6 ; Aof/the rem will be the area of tha trapeiTum. Bat 
if the aum of the two given anglea be greater than 180°, add together the three reanlta a, ft, and a; 
half their aum will be the area. 

Havlnff the two diayonalSy and either ann^le formed by their 

intersection. 

Sea Bamarka affear Parmllalegrams. 
In railroad measurements 

Of ezearation and embankment, the trapeslum 
imno frequently ooours ; as well as the two 6-sided 
figures { a» « o < and { m n o a ; in all of which m n 
represents the roadway ; rt.rc, and r ( the center- 
depths or heights ; I u and o v the lide-deptha er 
heigbta, aa given by the level ; Im and no the aide- 
alopea. 

The aame general rule for area appliea to all three 
of theae flga ; namely, mult the extreme hor width 
« « by ko^ the center depth r «, r e, or r t. an the 
oaae may be. Also molt one fawih of the width of 
roadway m n, by the mm of tbe two aide-depths I u 
and «. Add the two proda together ; the sum is the 
reqd area. Thia rule appliea whether tbe two side- 

slapas at I and n o have the same angle of inelination or DOC IB ndlvMtd work* 0t«H tka nIC* 
way hor width, eeatar depth, and aida depths of a prismOld ara respectively tm tIm half nm» «| 
ttia aorreaponding end ones, and thus ean be found without actual meaaurament. 




1 



POLYGON& 



169 



To draw a hezason, eacb nide of whteh shall 
be eqaal to a ffiven line, a b. 

From a and h, with rad a h, dosoribe the two arcs; from their Jntersectien, 
i, with Um oaBe rad, deaoribe aolreloi aroand the oireumf of which, step off 
the same rad. 

Side or a bexagon ts^nnX ^7795. 



T» draw 



side 



an oetaflpon, with each 
equal to a grlven line, e e. 

Prom c and e draw two perps, cp, ep, Aiso prolong c« toward 
/ and g; and ftrom c and e, with rad equal e «, draw the two 
onadraats : and find their centers h h : join e A, and e h ; draw 
« • and h t parallel to e j> ; and make each of them equal to c 0; 
aaka c Qt and « o, each equal to h h ; Join oo^o*, and o <. 

tSlde of an oetaffon ^nnX .41421354. 



To draw an oetaffon in a irlTen oqnare. 

Vrom each comer of the square, and with a rad equal to half its diag, 
deicribe the few arcs; and Join the points at which they out the sides of the 
•qaare. 

To draw anjr reirnlar |M>1yson, with each side 

e^inal to «n n« 

IHr MQ degrees by the anmber of sides ; take the qoot fh>ro IBffi ; div the 
Km br t. Thil will give the angle c m n, or e n m. Mm and n la; down these 
ancles hr » protractor: the side* of these angles will meet ata point, c. f^m 
which desoribe the circle m m y ; and aronad it* drcumf step off disu equal to 
mn. 

In any circle* m m y, to draw any reffular 

polycfon. 

JHfWlP'tj the number of sides ; the qoot will be the aa^^le m c m, aithe cen ler. 
Ltf eff this angle bj a protraeiw ; and its chord m n will be one side ; which 
atep dff arooad tbrcironmf. 

To reduce any polyiron, asa50^e/a^toa triani^Ie of the 

same area. 




W 





Fig. 2. 



If- *• ai^oco the side /a toward w; and draw b g parallel to a c, and join g c. we get equal trl* 
inclas a e'fr and a eg, both on the same base a c ; and both of the same perp height, inssmuch aa 
Iherare between the two parallels a c and g 6. But the part a e i forms a portion of both these irt* 
aa^ or in other iravde. Is eommpn to botk. Tber«rore, if it be tak«i away from both triangles, 
IheremalnlBC parts, < e 6 of one of them, and < y a of the other, are also equal. Therefore, if the 
•srt7e b be left off from the p^ygon, and the part igabe Mken into it, the polygon g/edcigviM 
■Me the « »i T«*> area as a/« d e 6 a; but it will have but five sides, while the other has six. Again, 
tt«s Indrawn parallel to 4/, and d* joined, we have upon the same base es, aud between the same 
mut^MM e a aadd/. the two equal triangles e • d. and e •/. with the part eot common to both ; and 
iMmMBay the rewaintaig part e o d or one. and o «/ of the other, are equal. Therefore, if o «/ be 
AaffftMn the polygon, and so d be taken into it, the new polygon gad eg, Fig 2. will have the same 
Mas a/ e d eo ; but It has but fbor sides, while the other has five. Finally, if g t, Fig 2, be 
ttmZJl u>wa«d)t: aad d » drawn parallel to c s : and c n joined, we have on the same base c «, and 
tSMsa lAe aaMt paraMtlt e s and d n, the two equal triangles etn, and ttd, with the part c s I 
MHaM le hoth. Tberefore, If we leave out c d (, and take ltt.s f n, we have tbe triangle gne equal 
•theaolfBOBjadcy.Pigi; orto o/«dc6a, FIgl. , , „ 
TM/ffT^P'* method it applicable to polygons of any number of aides. 



Wtel 



160 



POLYGOKS. 



IU^hede fg, to a ■mailer 



To reduee a larire 

nlmllar one. 

From Any interior point o, which had better be near the center, draw line* 
to all the angles a, h, c, ko. Join these lines by others parallel to the sides 
•f the fig. If it should be reqd to enlarge a small fig, draw, from any point 
• within it, lines extending beyond its angles ; and Join these lines by others 
fsnllsl to the sides of the small fig. 



To redaee a map to one on a smaller seale. 

The best meth9d is by dividing the large map into squares by faint lines, with a rery soft leadi 
penoil; and then drawing the rednoed map upon a sheet of 
smaller squares. A pair of proportional dividers will assist 
mueh in nzing points intermediate uf the sides of the squares. 
If the large map would be injured by drawing and rubbing 
•n# the squares, threads may be stretched across it to form the 
aqnares. 




In a reetanfpnlar tk§;^ ghsd, 

Bepresenting an open panel, to find the points • o o o In Ua 
•ides ; and at equal dists firom the angles g. and « ; Cor inserting 
a diag piece o o o o, of a given width 1 1, measured at right 
angles to its length. From g and « as centers, describe several 
ooncentrio arcs, as in the Fig. Draw upon transparent paper, 
two parallel lines a a, c e, at a distance apart equal to II; and 
placing these lines on top of the panel, move them about until it 
18 shown by the ares that the four dists g o, go, t o, s o, are 
equal. Instead of the transparent paper, a strip of common 
paper, of the width { I may be used. 

Rbm. Many problems which would otherwise be very diflBcult, 
■Bay be thus solved with an aoouraoy suffloient for praotieal 

purposes, by means of transparent paper. 





To find tbe area of any irreffnlar poly* 
§fon, anb e m. 

Div it into triangles, as anhfame, and a b e; in oaoh of 
wliloh find the perp dlst o, between its base a &, a e, or 6 e; and 
tbe opposite angle n, m, or a ; mult eaoh base by its perp dist; 
add all tbe prods together ; div by 2» 

*" To find approx tbe area of a lon^r tr^ 
reg^nlar fiK, as a 6 e d. Between it* ends «&,« 4, 




mc:r 



apace off equal dists, (the shorter they are the more accurate will be the result,) through whioh 
draw the intermediate parallel lines 1. 2, S, &o, across the breadth of the fig. Measure the lengths 
of these intermediate lines : add them together : to the sum add ht^/ the sum of the two end breadths 
• 6 and c d. Mult. the entire sum by one of the equal. spaces between the parallel lines. The prod 
will be the area This rule answers as well if either one or both the ends terminate in points, as at m 
and n. In the )ast of these cases, both a b and c d will be included In tne kntormodiate linos ; «nd 
kalf the two end breadths will be 0, or nothing. 

To find tbe area of any irre^nlar fiynre. 



Draw around it lines whioh shall enclose within them (as nearly as 
ean be judged by the eye) as much spaoe not belonging to the flgnro as 
they exclude space belonging to it. The area of the simpUflod flgnro 
thus formed, being in this manner rendered equal to that of the eom- 
plicated one, may be calculated by dividing it into triangles, Ao. By 
using a piece of fine thread, the proper position for the now bovndary 
lines may be found, before drawing them in. 
Areas of irregular figures may be found from a drawing, by Inyinc 
noon it a piece of transparent paper garefnUy ruled into small squares, eaoh of agivon area, say u 
M, or 100 sq. ft. eaoh ; apd by first oounting the whole squares, and then adding the fHkoUona of 
squares. 



cn 



dBCLESb 



161 



CIBCIiES. 

A •iNto Is Um area Ineladed within s onrred Him or aueh a eharMtw fhst evwy pofnt In it ts 
«|a«Uy ditunt from » c«rt«iD {lOiDt within It, cilUbA ita oontor. Tb« oorred line ItMlf la eaUed tlio 
airouBferoaoe, or peripherj of the circle ; or verj common! j It la called tbe oirole. 

T* And tbe circnmrerenee. 

Malt dlam bj S.1416, which givea too maoh by only .148 of an Inoh In a mlla. Ov, aa 113 la to SM 
- to is diam to elreaaif ; too graat 1 Inch in 186 niUea. Or* molt dlam h7 9^i too grpat bj about 1 
part in UBS. Or* mnlt area by IS.MW, and take aq root of prod. 

To find tbe diam. 

DiT the •Irounf by S.14I6 ; or. aa SS5 la to US, ao la cireumr to diam ; or, molt the elrenmf. by 7: 
aaddlT »k» prod by tt, whish (Ivao thediaih toe anali by only abont om part ia S48&; or, mnlt the 
area by l.STSl; aad take th* aq rt of tiie prod. 

The dlam la to the olroamf more exactly aa 1 to S. 14159366. 

To find tbe area of a cflrele. 

Square the dlam; malt tbia aqoare by .7864; or more accarately by .786S9816; ^r aqnare the dr- 
eanf; mnlt thla aquare by .071)68 : or more accurately by .07957747 ; or mult half the diam by half the 
eirenmf ; or refer to the following table of areaa of olrdea. Alao area = an of rad X S.I416. 

The area of a drele la to the area of anr etreumaorlbed atraight-alded flg, aa the circumf of the 
drsle la to the elrenmf or periphery of the ig. Tbe area of a aquare Inaeribed in a circle, ia equal to 
twice the aqnare of the rad. Of a circle in a square, =r square X .7864. 

It Is eonvenient to remembatv In rmmdlnt off a aquara ooroer a h «, by a quarter of j 
a drele, that the shaded area • b c la equal to about 1 pan (correctly .3146) of the " 
wholA aqnare ahed. o 




To find tbe dlam of a circle eqoal In area to a ylTon sqaare. 

Mnit one aide of the aqnare by 1. 128S8. 

To find tbe rad of a circle to drcamscrlbe a i^lTcn eqaare. 

Mult one aide by .7071 ; or take H tbe diag. 

To find tbe side of a square equal In area to a fflYcn circle. 

Malt the diam by .8863S. 

To find tbe side of tbe (rre^^^st square in a siven circle. 

Malt dlam by .7071. The area of the greatest aquare that can be inscribed in a drele la equal to 
toiae the equare of tbe rad. The diam X by 1.3468 glvea tbe aide of an eqallatoral trianglf of equal area. 




To find tbe center e, of a nrf Ten dr^sle. 

Draw any chord a b ; and from the middle of it o, draw at r^ght angles t* 
it, a dlam d g ; find tbe center e of thla diam. 




11 



To describe a circle tbrongb any tbree 
points, abe, not in a straiipbt line. 

Join the pointo by the linea a6, ie; from the centers of these linea draw 
the dotted perpa meeting, as at o, which will be the center of the circle. 
Or from b, with any convenient rad. draw the arc m n; and from, a and c, 
with the aame rad. draw arcs y and jr; then two linea drawn through the 
iatoraeotiona of these area, will meet at the center o. 

To describe a circle to toucb tbe tbree 
ancles of a triangle is plainly the same as this. 

To inscribe a circle In a trianirle draw two lines 
blaeeting any two of tbe anglea. Where theae linea meet ia the eentor of 
the drele. 



162 



OEMXJLBBm 




T9 4i»W a tonyent* i€i,fm circle, firom any 
i^lven point, e, in its circnnMi. 

Through the center n, and the glren point «. dr»w n e ; "»*^ » •9"*J J* 
e n ; from n and o, with any rad creatar than half of o n, dewnrihe tha twa 
oairs of arc <<: Join their IntarMoUona iU 

Here, and in the following three flgt. the («n««nt« are ordinary vrjuo- 
mtrical one*; and may end where we pleaae. But the mgonometrum 
tangent of a given angU, must end in a Meant. 

Or ftom c lay off two equal distances c c, e < ; ana draw i i 
parallel to c t. 

To draw a tangr, « « ft, to a circle, ftnom a point. 
a, wblcii la onUiide of tlie circle. 

Draw a e, and on it deacrihe a •emiolrcle ; through the intaneetieB, «, drma 
a • 6. Here e is the oenter of the oirole. 



To draw a tangr* gh,ttonk a circnlar arc,sr«0» 

Of which n a is the rise. With rad g a, describe an are, • • o. lUH f « 
•qual ta • a. Through t draw g h. 



To draw a tani; t6 two circles. 

First draw the line m «, just touching the two 
•irales; this gives the direction of the Ung. Then 
from the centers of the circles draw the rsdil. o •^V^rP 
to n» n. The potato ( t are the Ung points. If the 
tang is in the position of the dotted line, • y, the ope- 
ration is the same. 





If any two chords, as a b, o c, cross eacli otkier, 

then as on : n 6 :: o n : n c. Hence, n ft X a n = onX ne. That 
f is the product of the two parts of one of the lines, is «- tlkS pro- 
h 4uct ofthe two parts of the other line. 






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1? 






H 












Ml 


















>K 


1 






« 








M.SM 




















St.OM 
















!»: 




».IH 








^ 










3M. 












"'sa* 




s 


Iffl.BM 


3(1. 




w™ 
















ISO.JIO 






go.fu 










OoiBl 




K 


130.™ 




1 












m.im 




N 






«;™ 








^ 




Bie.a 










tl.TK 








oi:?™ 












K.1BS 


1 






1 


02.102 


ulii 


S 


taiTM 


toa; 


i 


KJIVV 


iS'^ 


SS 


mieli 
















193.™ 


»~ 




0J.5M 








118. 






^ 




W 








1 




«!.■ 


H 




imIiw 


■a 










US. 


^ 


ItMl 




1H.J70 


•0 












«!. 


afA 




19S.17I 
















g 




1 


1«.S«1 












11 




Sl.Tl* 


H6.0S7 




g 


«■§! 


9«:Bt 


S 


™- 


M." 






IM.W) 


B 








M 










ISfilJS 










K 






1 


n.ao 




i»7.ta 










ft 










1»7.™ 










S 










IM.SI9 
















1 




1 


























































U 










W.3S 






















J 




















sra'.w 














^ 


1 






















!«'» 






sm:. 


















11 TW 


LOO. 












1 


1 ilil 


imi' 


^ 


Hi 


IS: 


11* 


i!™ 


■ s 


!S:™ 








1 




836. 






■ « 
















3 


J.Hi 


H 






1 


:m 


1^1 


\yxf, 


Si! 


fS! 




IHltH 






iint.1 




11 SIS 






I.7SI 


g 








lomi 




11 MM 


S?* 




Mil 






g 








u m 






S.IM 












1 








!.*» 


1 










idKa 








won 












g 






















11 B«0 




1 










! 












tiBOO 






i 


ili 


IfSli 


"i< 


iiS 


!«■ 


J 


"iS 


1 


nun 




s 


si 


iil 


1 


■is 


7ds: 


S 


KJS 





StSi 



I ^ 
' H mi:!: 

M 



r: 



166 



CtBCLEB. 



TABIDS 3 OF cmCIiES. 
IMameters in anita and tenths* 



DUu 


Ctreamf. 


Area. 


mm. 


Cireanf. 


Area. 


Dia. 


Ciroinf. 


Area. 


•.1 


.814159 


.007854 


6.3 


19.79208 


81.17245 


12.5 


89.26991 


122.7185 


.2 


.628319 


.031416 


.4 


20.10619 


82.16991 


.6 


39.58407 


124.6898 


^ 


.942478 


.070686 


.6 


20.42085 


83.18307 


.7 


39.89823 


126.6769 


.4 


1.256637 


.125664 


.6 


20.73451 


34.21194 


.8 


40.21239 


128.6796 


Jb 


1.570796 


.196360 


.7 


21.04867 


35.25652 


.9 


40.52655 


130.6981 


A 


1.884956 


.282743 


.8 


21.36288 


36.31681 


18.0 


40.84070 


132.7323 


.7 


2.199115 


.384845 


.9 


21.67699 


37.89281 


.1 


41.15486 


134.7822 


.8 


2.513274 


.502655 


7.0 


21.99115 


38.48451 


o 


41.46902 


136.8478 


.9 


2.827433 


.636173 


.1 


22.30531 


30.59192 


.8 


41.78318 


138.9291 


1.0 


3.141593 


.785398 


.2 


22.61947 


40.71504 


.4 


42.09734 


141.0261 


.1 


3.455752 


.950332 


.8 


22.93363 


41.85387 


.5 


42.41150 


143.1388 


^ 


3.769911 


1.13097 


.4 


23.24779 


43.00840 


.6 


42.72566 


145.2672 


^ 


4.084070 


1.32732 


.5 


28.56194 


44.17865 


.7 


43.03982 


147.4114 


.4 


4.398230 


1.53938 


.6 


23.87610 


45.36460 


.8 


43.35398 


149.5712 


.5 


4.712389 


1.76715 


.7 


24.19026 


46.56626 


.9 


43.66814 


151.7468 


,6 


5.026548 


2.01062 


.8 


24.50442 


47.78362 


14.0 


43.98230 


163.9880 


.7 


5.34070» 


2.26980 


.9 


24.81858 


49.01670 


.1 


44.29646 


156.1460 


.8 


5.654867 


2.54469 


8.0 


25.13274 


50.26548 


.2 


44.61062 


158.3677 


.9 


5.969026 


2.83529 


.1 


25.44690 


51.52997 


^ 


44.92477 


160.6061 


2.0 


6.283185 


8.14159 


.2 


25.76106 


62.81017 


.4 


45.23893 


162.8602 


.1 


6.597345 


3.46361 


.8 


26.07522 


54'.10608 


.5 


45.55309 


165.1300 


;2 


6.911504 


3.80133 


.4 


26.38938 


55.41769 


.6 


46.86725 


167.4165 


.8 


7.225663 


4.15476 


.5 


26.70354 


66.74502 


.7 


46.18141 


169.7167 


A 


7.539822 


4.52389 


.6 


27.01770 


58.08805 


.8 


46.49657 


172.0336 


Jb 


7.858982 


4.90874 


.7 


27.33186 


59.44679 


.9 


46.80973 


174.3662 


A 


8.168141 


5.30929 


.8 


27.64602 


60.82123 


15.0 


47.12389 


176.7146 


.7 


8.482300 


5.72555 


.9 


27.96017 


62.21139 


.1 


47.4.3805 


179.0786 


^ 


8.796459 


6.15752 


9.0 


28.27433 


63.61725 


.2 


47.76221 


181.4584 


.9 


9.110619 


6.60520 


.1 


28.58849 


&'>.03882 


.8 


48.06637 


183.8539 


3.0 


9.424778 


7.06858 


.2 


28.90265 


66.47610 


.4 


48.38053 


186.2660 


J 


9.738937 


7.54768 


.8 


29.21681 


67.92909 


.5 


48.69469 


188.6919 


^ 


10.05310 


8.04248 


.4 


29.53097 


69.39778 


.6 


49.00885 


191.1345 


^ 


10.36726 


8.55299 


.5 


29.84513 


70.88218 


.7 


49.32300 


193.5928 


.4 


10.68142 


9.07920 


.6 


30.15929 


72.38229 


.8 


49.63716 


196.0668 


.5 


10.99557 


9.62113 


.7 


30.47345 


73.89811. 


.9 


49.95132 


198.5565 


A 


11.30973 


10.17876 


.8 


30.78761 


75.42964 


16.0 


60.26648 


201.0619 


J 


11.62389 


10.75210 


.9 


31.10177 


76.97687 


.1 


60.57964 


203.5831 


.8 


11.93805 


11.84115 


10.0 


81.41593 


78.53982 


.2 


60.89380 


206.1199 


.9 


12.25221 


11.94591 


.1 


31.73009 


80.11847 


.8 


61.20796 


208.6724 


4.0 


12.56637 


12.56637 


.2 


32.04425 


81.71282 


.4 


61.52212 


211.2407 


.1 


12.88053 


13.20254 


.8 


32.35840 


83.32289 


.5 


61.83628 


213.8246 


.2 


13.19469 


13.85442 


.4 


32.67256 


84.94867 


JR 


62.15044 


216.4248 


.3 


13.50885 


14.52201 


.5 


32.98672 


86.59015 


.7 


62.46460 


219.0307 


.4 


13.82301 


15.20531 


.6 


33.30088 


88.24734 


S 


52.77876 


221.6708 


.5 


14.13717 


15.90481 


.7 


33.61504 


89.92024 


.9 


63.09292 


224.3176 


.6 


14.45133 


16.61903 


.8 


33.92920 


91.60884 


17.0 


63.40708 


226.9801 


.7 


14.76549 


17.34945 


.9 


34.24336 


93.31316 


.1 


63.72123 


229.6583 


^ 


15.07964 


18.09557 


11.0 


34.55752 


95.08318 


.2 


64.08539 


232.3522 


.9 


15.39380 


18.85741 


.1 


34.87168 


96.76891 


S 


64.34955 


235.0618 


6.0 


15.70796 


19.63495 


.2 


35.18584 


98.52035 


A 


64.66371 


237.7871 


.1 


16.02212 


20.42821 


.8 


35.50000 


100.2875 


£ 


64.97787 


240.5282 


^ 


16.33628 


21.23717 


.4 


35.81416 


102.0703 


.6 


65.292as 


243.2849 


.8 


16.65044 


22.06183 


.5 


36.12832 


103.8689 


.7 


55.60619 


246.0574 


-.4 


16.96460 


22.90221 


.6 


36.44247 


105.6832 


.8 


65.92035 


248.8456 


^ 


17.27876 


23.75829 


.7 


36.75663 


107.5182 


.9 


56.23451 


251.6494 


j6 


17.59292 


24.63009 


.8 


87.07079 


109.3588 


18.0 


56.54867 


264.4690 


.7 


17.90708 


25.51759 


.9 


37.38495 


111.2202 


.1 


56.86283 


267.3048 


.8 


18.22124 


26.42079 


ISO 


37.69911 


113.0978 


J2 


57.17699 


260.1558 


.9 


18.53540 


27.33971 


.1 


38.01327 


114.9901 


A 


67.49116 


268.0220 


«.o 


18.84956 


28.27433 


.2 


38.32743 


116.8967 


A 


67.80580 


265.9044 


.1 


19.16372 


29.22467 


.8 


38.64159 


118.8229 


Jb 


68.11946 


268JN)25 


.2 


19.47787 


80.19071 


.4 


88.96575 


120.7628 


^ 


6&48862 


271.71168 



CIBGI«EB. 



167 



TABIiS 8 OF €IB€I«BiM00BtiBiw4). 
Dittinetem in unite and tenths. 



Ma. 


droinf. 


Atmu 


DIft. 


Ctreamf. 


Area. 


Mft. 


Ctreanf. 


kntu 


18.7 


68.74778 


274.6459 


24.9 


78.22566 


486.9647 


81.1 


97.70B53 


759.6460 


.8 


59.06194 


277.59U 


86.0 


78.53982 


490.8789 


.2 


98.01769 


764.6880 


.9 


59.37610 


280.5621 


.1 


78.85388 


494.8087 


.8 


98.38185 


769.4467 


19.0 


59.69026 


283.5287 


.2 


79.16818 


498.7592 


.4 


98.64601 


774.8712 


.1 


60.00442 


286.5211 


.8 


79.48229 


502.7255 


.5 


98.96017 


779.3118 


.2 


60.31858 


289.5292 


.4 


79.79645 


506.7075 


.6 


99.27438 


784.2672 


^ 


60.63274 


292.5530 


.5 


80.11061 


510.7052 


.7 


99.58849 


789.2388 


. A 


60.94690 


205.5925 


.6 


80.42477 


514.7185 


A 


99.90266 


794.2260 


J5 


61.26106 


298.6477 


.7 


80.73803 


518.7476 


.9 


100.2168 


799.2290 


.6 


61.57582 


301.7186 


.8 


81.05309 


522.7924 


88.0 


100.5310 


804.2477 


.7 


61.88986 


304.8052 


.9 


81.36725 


526.8529 


.1 


100.8451 


809.2821 


JR 


62.20363 


307.9075 


86.0 


81.68141 


580.9292 


.2 


101.1503 


814.3322 


S 


62.51769 


311.0255 


.1 


81.99557 


535.0211 


.8 


101.4734 


819.3980 


80.0 


62.83185 


314.1598 


.2 


82.30973 


539.1287 


.4 


101.7876 


824.4796 


.1 


68.14601 


317.3067 


.3 


82.62389 


5482521 


.5 


102.1018 


829.6768 


J2 


68.46017 


320.4730 


.4 


82.93805 


547.8911 


.6 


102.4159 


834.6898 


Jl 


68.77438 


323.6547 


.5 


83.25221 


55L5459 


.7 


102.7301 


839.8184 


.4 


6108848 


326.8513 


.6 


83.56686 


565.7163 


.8 


106.0442 


844.9628 


A 


64.40266 


380.0636 


.7 


83.88052 


569.9025 


.9 


103.8584 


850.1228 


JS 


64.71681 


383.2916 


.8 


84.19468 


564.1044 


88.0 


103.6726 


855.2986 


.7 


66.03097 


336.5353 


.9 


84.50884 


568.3220 


.1 


103.9867 


860.4901 


.8 


65.34518 


339.7947 


87.0 


84.82300 


572.6653 


.2 


104.3009 


865.6973 


.9 


65.65929 


343.0698 


.1 


85.13716 


576.8043 


.8 


104.6150 


870.9202 


tl.O 


65.97S45 


346.3606 


.2 


85.45132 


581.0690 


.4 


104.9292 


876.1588 


.1 


66.28760 


849.6671 


.3 


85.76548 


585.3494 


.5 


105.2434 


881.4131 


.2 


66.60176 


852.9894 


.4 


86.07964 


589.6455 


.6 


105.6575 


886.6831 


^ 


66.91592 


356.3278 


.6 


86.39880 


593.9574 


.7 


105.8717 


891.9688 


.4 


67.23008 


359.6809 


.6 


86.70796 


598.2849 


.8 


106.1858 


897.2708 


^ 


67.54tt4 


363.0608 


.7 


87.02212 


602.6282 


.9 


106.5000 


902.5874 


^ 


67.85840 


366.4354 


.8 


87.33628 


606.9871 


84.0 


106.8142 


907.9208 


.7 


68.17256 


369.8861 


.9 


87.65044 


611.8618 


.1 


107.1288 


918.2688 


.8 


68.48672 


873.2526 


88.0 


87.06459 


615.7522 


.2 


107.4426 


918.6331 


.9 


68.80088 


376.6848 


.1 


88.27875 


620.1582 


.3 


107.7666 


924.0181 


M.0 


69.U504 


380.1327 


.2 


88.59291 


624.5800 


.4 


108.0708 


929.4088 


.1 


69.42920 


388.5963 


.8 


88.90707 


629.0175 


.5 


108.8849 


934.8202 


.2 


69.748SS 


887.0756 


.4 


89.22123 


638.4707 


.6 


108.6991 


940.2478 


^ 


70.06788 


300.5707 


A 


89.58539 


637.9397 


.7 


109.0138 


945.6901 


A 


70.37168 


394.0814 


.6 


89.84955 


642.4243 


.8 


109.3274 


951.1486 


Jb 


70.68688 


397.6078 


.7 


90.16371 


646.9246 


.9 


109.6416 


956.6228 


j6 


70.99999 


401.1600 


.8 


90.47787 


651.4407 


86.0 


109.9557 


962.1128 


.7 


71.81415 


404.7078 


.9 


90.79203 


655.9724 


.1 


110.2699 


967.6184 


^ 


71.62881 


408.2814 


88.0 


91.10619 


660.5199 


J2 


110.5841 


973.1397 


.9 


71.94247 


411.8707 


.1 


91.42035 


665.0830 


.8 


110.8982 


978.6768 


tt.O 


72.26668 


415.4756 


.2 


91.73451 


669.6619 


.4 


111.2124 


984.2296 


.1 


72.57079 


419.0068 


A 


92.04866 


674.2565 


.6 


111.5265 


989.7980 


.2 


72.88496 


422.7827 


A 


92.86282 


678.8668 


.6 


111.8407 


995.3822 


^ 


78.19911 


426.3848 


A 


92.67698 


683.4928 


.7 


112.1649 


1000.9821 


A 


78.51827 


480.0526 


A 


92.99114 


688.1345 


.8 


112.4690 


1006.5977 


& 


78.82M8 


488.7861 


.7 


93.30530 


692.7919 


.9 


112.7832 


1012.2290 


A 


74.14169 


487.4854 


.8 


98.61946 


697.4650 


86.0 


113.0973 


1017.8760 


.7 


74.45695 


441.1608 


.9 


98.93362 


702.1538 


.1 


113.4115 


1023.5387 


^ 


74.76001 


444.8809 


80.0 


94.24778 


706.8583 


.2 


113.7257 


1029.2172 


.9 


75.06406 


448.6278 


.1 


94.56194 


711.5786 


.3 


114.0898 


1034.9118 


M.0 


75.30822 


452.8808 


.2 


94.87610 


716.3145 


.4 


114.3540 


1040.6212 


.1 


75.71238 


466.1671 


.8 


05.19026 


721.0662 


.5 


114.6681 


1046.3467 


a. 


76X>2I64 


459.9606 


.4 


95.50442 


725.8886 


.6 


114.9828 


1052.0880 


z 


76.84090 


468.7698 


.5 


95.81858 


730.6166 


.7 


115.2965 


1057.8449 


A 


76.66418 


467.5947 


.6 


96.13274 


735.4154 


.8 


115.6106 


1063.6176 


J» 


76.90182 


471.4862 


.7 


96.44689 


740.2299 


.9 


115.9248 


1069.4060 


A 


77.S8n8 


475.2916 


.8 


96.76105 


745.0601 


87.0 


116.2889 


1075.2101 


a 


77J0li4 


479J686 


.9 


97.07521 


749.9060 


.1 


116.5531 


1081.0299 


M 


97.01160 


4K.DG18 


81.0 


97.38937 


754.7676 


.2 


116.8672 


1086.8664 



168 



CIBCLES. 



TABIiE 3 OF cmCIiKIMOontiaiMd). 
Diameters in iiniUi and tenths. 



Dis. 


Ciroumf. 


Are*. 


DU. 


Cirenaf* 


Area. 


DU. 
49.7 


arcamf. 


▲res. 


87.3 


117.1814 


1092.7168 


48.5 


136.6593 


1486.1697 


186.1372 


1940.0041 


.4 


117.4956 


1098.5835 


.6 


136.9734 


1493.0105 


.8 


166.4513 


1947.8189 


A 


117.8097 


1104.4662 


.7 


137.2876 


1499.8670 


.9 


166.7655 


1965.6493 


.6 


118.1239 


1110.3645 


.8 


137.6018 


1606.7393 


60.0 


167.0796 


1963.4964 


.7 


118.4380 


1116.2786 


.9 


187.9159 


1513.6272 


.1 


167.3938 


1971.3572 


.8 


118.7622 


1122.2083 


44.0 


138.2301 


1520.5308 


J2 


157.7080 


1979.2348 


.9 


119.0664 


1128.1538 


.1 


138.5442 


1527.4502 


.5 


158.0221 


1987.1280 


88.0 


119.3805 


1134.1149 


.2 


138.8584 


1534.3853 


.4 


158.3363 


1995.0370 


.1 


119.6947 


1140.0918 


.8 


139.1726 


1541.3360 


.5 


168.6504 


2002.9617 


.2 


120.0088 


1146.0844 


.4 


139.4867 


1548.3025 


.6 


168.9646 


2010.9020 


.8 


120.3230 


1152.0927 


.5 


139.8009 


1555.2847 


.7 


169.2787 


2018.8581 


.4 


120.6372 


1158.1167 


.6 


140.1150 


1562.2826 


.8 


169.6929 


2026.8299 


.5 


120.9513 


1164.1564 


.7 


140.4292 


1569.2962 


.9 


159.9071 


2034.8174 


.6 


121.2655 


1170.2118 


.8 


140.7434 


1576.3255 


61.0 


160.2212 


2042.8206 


.7 


121.5796 


1176.2a30 


.9 


141.0575 


1583.3706 


.1 


160.5364 


2050.8395 


.8 


121.8938 


1182.3698 


46.0 


141.3717 


1590.4313 


.2 


160.8495 


2058.8742 


.9 


122.2080 


1188.4724 


.1 


141.6858 


1597.5077 


.3 


161.1637 


2066.9245 


89.0 


122.5221 


1194.5906 


.2 


142.0000 


1604.5999 


.4 


161.4779 


2074.9906 


.1 


122.8363 


1200.7246 


.8 


142.3141 


1611.7077 


.5 


161.7920 


2083.0728 


J2 


123.1504 


1206.8742 


.4 


142.6283 


1618.8313 


.6 


162.1062 


2091.1697 


Ji 


123.4646 


1213.0396 


.6 


142.9425 


1625.9705 


.7 


162.4203 


2099.2829 


A 


123.7788 


1219.2207 


.6 


143.2566 


1633.1255 


.8 


162.7345 


2107.4118 


.5 


124.0929 


1225.4175 


.7 


143.5708 


1640.2962 


.9 


163.0487 


2115.5663 


.« 


124.4071 


1231.6300 


.8 


143.8849 


1647.4826 


62.0 


163.3628 


2123.7166 


.7 


124.7212 


1237.8582 


.9 


144.1991 


1654.6847 


.1 


163.6770 


2131.8926 


J& 


125.0354 


1244.1021 


46.0 


144.5133 


1661.9025 


.2 


163.9911 


2140.0848 


.9 


125.3495 


1250.3617 


.1 


144.8274 


1669.1360 


.3 


164.3063 


2148.2917 


40.0 


125.6637 


1256.6371 


.2 


145.1416 


1676.3853 


.4 


164.6196 


2166.5149 


.1 


125.9779 


1262.9281 


JS 


145.4557 


1683.6502 


.6 


164.9386 


2164.7587 


.2 


126.2920 


1269.2348 


.4 


145.7699 


1690.9308 


.6 


165.2478 


2173.0082 


.8 


126.6062 


1275.5573 


.6 


146.0841 


1698.2272 


.7 


166.6619 


2181.2785 


.4 


126.9203 


1281.8955 


.6 


146.3982 


1705.5392 


.8 


166.8761 


2189.5644 


.6 


127.2345 


1288.2493 


.7 


146.7124 


1712.8670 


.9 


166.1908 


2197.8661 


.6 


127.5487 


1294.6189 


.8 


147.0265 


1720.2105 


68.0 


166.5044 


2206.1884 


.7 


127.8628 


1301.0042 


.9 


147.3407 


1727.5697 


.1 


166.8186 


2214.5165 


.8 


128.1770 


1307.4052 


47.0 


147.6549 


1734.9445 


.2 


167.1327 


2222.8658 


.9 


128.4911 


1313.8219 


.1 


147.9690 


1742.3351 


.8 


167.4469 


2231.2296 


41.0 


128.8053 


1320.2543 


.2 


148.2832 


1749.7414 


.4 


167.7610 


2239.6100 


.1 


129.1195 


1326.7024 


.8 


148.5973 


1757.1635 


.5 


168.0752 


2248.0059 


Jl 


129.4336 


1333.1663 


.4 


148.9115 


1764.6012 


.6 


168.3894 


2256.4175 


.8 


129.7478 


1339.6458 


.5 


149.2257 


1772.0546 


.7 


168.7035 


2264.8448 


.4 


130.0619 


1346.1410 


.6 


149.5398 


1779.5287 


.8 


169.0177 


2273.2879 


.5 


130.3761 


1352.6520 


.7 


149.8640 


1787.0086 


.9 


169.3318 


2281.7466 


.6 


130.6903 


1359.1786 


.8 


150.1681 


1794.5091 


64.0 


169.6460 


2290.2210 


.7 


131.0044 


1365.7210 


.9 


160.4823 


1802.0254 


.1 


169.9602 


2298.7112 


.8 


131.3186 


1372.2791 


48.0 


160.7964 


1809.5574 


.2 


170.2743 


2307.2171 


.9 


131.6327 


1378.8529 


.1 


151.1106 


1817.1050 


.8 


170.5885 


2315.7386 


4S.0 


131.9469 


13)85.4424 


J2 


161.4248 


1824.6684 


.4 


170.9026 


2824.2769 


.1 


182.2611 


1392.0476 


.8 


151.7389 


1832.2476 


.5 


171.2168 


2882.8289 


.2 


132.5752 


1398.6685 


.4 


152.0531 


1839.8423 


.6 


171.5810 


2341.8976 


Ji 


132.8894 


1405.3051 


.6 


162.3672 


1847.4528 


.7 


171.8451 


2849.9820 


.4 


183.2035 


1411.9574 


.6 


162.6814 


1855.0790 


.8 


172.1593 


2358.5821 


Jb 


133.5177 


1418.6254 


.7 


152.9956 


1862.7210 


.9 


172.4784 


2967.1979 


A 


183.8318 


1425.8092 


.8 


153.3097 


1870.8786 


66.0 


172.7876 


2375.8294 


.7 


184.1460 


1432.0086 


.9 


153.6239 


1878.0519 


.1 


173.1018 


2884.4767 


.8 


184.4602 


1438.7288 


48.0 


153.9380 


1885.7410 


.2 


173.4159 


2893.1396 


.9 


134.7743 


1445.4546 


.1 


154.2622 


1893.4457 


.8 


173.7801 


2401.8188 


48.0 


185.0886 


1452.2012 


.2 


154.5664 


1901.1662 


.4 


174.0442 


2410.6136 


.1 


1S5.4026 


1458.9685 


.8 


154.8805 


1908.9024 


.6 


174.8584 


24192227 


JZ 


186.7168 


1465.7416 


.4 


155.1947 


1916.6543 


.6 


174.6726 


2427.9485 


J 


186.0810 


1472.6352 


.6 


155.5088 


1924.4218 


.7 


174.9867 


2486.6899 


4 


186.3451 


1479.8448 


.6 


155.8230 


1932.2061 


.8 


175.3009 


2445.4471 



GIBCLES. 

TABIA 2 OF €lB€I<iaiMOoi»tliiii«dX 
I^lamet^vs in nnlts waA tentha. 



169 



ma. 



56.9 


175.6160 


56.0 


175.9292 


.1 


176.2433 


.2 


176.6576 


.3 


176.8717 


.4 


177.1858 


.5 


177.5000 


.6 


177.8141 


.7 


178.1283 


r ^ 


178.4425 


.9 


178.7566 


67.0 


179.0708 


.1 


179.3849 


.2 


179.6991 


.8 


T80.0133 


.4 


180.3274 


.5 


180.6416 


.6 


180.9557 


.7 


181.2699 


.8 


181.5841 


.9 


181.8982 


68.0 


182.2124 


.1 


182.5265 


.2 


182.8407 


.3 


183.1549 


.4 


188.4690 


.5 


188.7832 


.6 


184.0973 


.7 


184.4115 


.8 


184.7256 


.9 


185.0398 


69.0 


185.3540 


.1 


185.6681 


.2 


185.9823 


^ 


186.2964 


.4 


186.6106 


.6 


166.9248 


.6 


187.2389 


.7 


187.5531 


.8 


187.8672 


.9 


188.1814 


io.o 


188.4956 


.1 


188.8097 


.2 


189.1289 


.3 


189.4880 


.4 


189.7522 


.5 


190.0664 


.6 


190.3805 


.7 


190.6947 


.8 


191.0068 


.9 


191.8280 


§1.0 


191.6672 


.1 


191.9518 


.2 


192.2666 


^ 


192.6796 


.4 


192.8868 


^ 


- l'(PS.20'99 


.6 


VfSi.S^ki 



.7 
.8 



Cireaiif. 




Areft# 



2454.2200 
2463.0086 
2471.8130 
2480.6330 
2489.4687 
2498.3201 
2507.1873 
2516.0701 
2524.9687 
2533.8830 
2542.8129 
2551.7586 
2560.7200 
2569.6971 
2578.6899 
2587.6985 
2596.7227 
2605.7626 
2614.8183 
2623.8896 
2632.9767 
2642.0794 
2651.1979 
2660.8321 
2669.4820 
2678.6476 
2687.8289 
2697.0259 
2706.2386 
2715.4670 
2724.7112 
2733.9710 
2743.2466 
2752.5378 
2761.8448 
2771.1675 
2780.5058 
2789.8599 
2799.2297 
2808.6152 
2818.0165 
2827.4384 
2836.8660 
2846.8144 
2855.7784 
2865.2582 
2874.7536 
2884.2648 
2898.7917 
2903.8343 
2912.8926 
2922.4666 
2982.0568 
2941.6617 
2951.2828 
2960.9197 
2970.6722 
2980.2406 
2989.9244 
21^.6241 
6009.8896 
6019.0706 



Dift. 



62.1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

68X) 
.1 
.2 
.8 
.4 
.5 
.6 
.7 
.8 
.9 

64.0 
.1 
.2 
.8 
.4 
.6 
.6 
.7 
.8 
.9 

66.0 
.1 
.2 
.8 
.4 
.5 
.6 
.7 
.8 
.9 

66.0 
.1 
.2 
.3 
.4 
.6 
.6 
.7 
.8 
.9 

67.0 
.1 
.2 
.8 
.4 
.6 
.6 
.7 
.8 
.9 

68.0 
.1 
,2 



Cireumf. 



195.0929 
195.4071 
196.7212 
196.0364 
196.3495 
196.6637 
196.9779 
197.2920 
197.6062 
197.9203 
198.2346 
198.5487 
198.8628 
199.1770 
199.4911 
199.8053 
200.1195 
200.4336 
200.7478 
201.0619 
201.3761 
201.6902 
202.0044 
202.3186 
202.6327 
202.9469 
203.2610 
203.5752 
203.8894 
204.20a'> 
204.5177 
204.8318 
205.1460 
205.4602 
205.7743 
206.0885 
206.4026 
206.7168 
207.0310 
207.3451 
207.6593 
207.9734 
208.2876 
208.6018 
208.9159 
209.2301 
209.5442 
209.8584 
210.1725 
210.4867 
210.8009 
211.1160 
211.4292 
211.7483 
212.0575 
212.3717 
212.6858 
213.0000 
213.3141 
213.6283 
213.9425 
214.2566 



Area. 


DU. 


8028.8178 


68.8 


3038.5798 


.4 


3048.3580 


.6 


3058.1520 


.6 


8067.9616 


.7 


3077.7869 


.8 


3087.6279 


.9 


S097.4847 


69.0 


3107.3571 


.1 


3117.2453 


J2 


8127.1492 


.8 


3137.0688 


.4 


3147.0040 


.5 


3156.9560 


.6 


3166.9217 


.7 


3176.9042 


.8 


3186.9023 


.9 


3196.9161 


70.0 


3206.9456 


.1 


3216.9909 


.2 


3227.0518 


.3 


3237.1285 


.4 


3247.2209 


.6 


3257.3289 


.6 


3267.4527 


.7 


3277.5922 


.8 


3287.7474 


.9 


3297.9183 


llJO 


3308.1049 


.1 


3318.3072 


.2 


3328.5253 


.8 


3338.7590 


.4 


3349.0086 


.5 


3859.2786 


.6 


3869.5545 


.7 


3379.8510 


.8 


3390.1683 


.9 


3400.4913 


72.0 


3410.8350 


.1 


3421.1944 


.2 


8431.5695 


.3 


3441.9603 


.4 


3452.8669 


.6 


3462.7891 


.6 


8473.2270 


.7 


3483.6807 


.8 


3494.1500 


.9 


3504.6351 


78.0 


8515.1359 


.1 


3625.6524 


.2 


3536.1845 


.8 


8546.7324 


.4 


3557.2960 


.6 


3567.8764 


.6 


3578.4704 


.7 


8589.0811 


.8 


3899.7075 


.9 


3610.8497 


74.0 


3621.0075 


.1 


3631.6811 


.2 


3642.3704 


.3 


3658.0754 


.4 



Circomf. 

214.5708 
214.8849 
215.1991 
216.5133 
215.8274 
216.1416 
216.4557 
216.7699 
217.0841 
217.3982 
217.7124 
218.0265 
218.3407 
218.6548 
218.9690 
219.2882 
219.5973 
219.9115 
220.2266 
220.5398 
220.8540 
221.1681 
221.4823 
221.7964 
222.1106 
222.4248 
222.7389 
223.0531 
223.3672 
223.6814 
223.9956 
224.3097 
224.6239 
224.9880 
225.2522 
225.5664 
225.8805 
226.1947 
226.5088 
226.8230 
227.1871 
227.4518 
227.7655 
228.0796 
228.3938 
228.7079 
229.0221 
229.3363 
229.6504 
229.9646 
280.2787 
230.5929 
230.9071 
231.2212 
231.5354 
231.8495 
232.1687 
232.4779 
232.7920 
233.1062 
233.4203 
233.7345 



Area. 

3663.7960 
3674.5324 
3685.2845 
3696.0623 
3706.8369 
3717.6361 
3728.4500 
3739.2807 
3750.1270 
3760.9891 
3771.8668 
3782.7603 
3793.6695 
3804.5944 
3815.5360 
3826.4913 
3837.4633 
3848.4510 
3859.4544 
3870.4736 
3881.5084 
3892.5690 
3903.6252 
3914.7072 
3925.8049 
3986.9182 
3948.0473 
3959,1921 
3970.3526 
3981.5289 
3992.7208 
4003.9284 
4015.1518 
4026.3908 
4037.6456 
404S.9160 
4060.2022 
4071.6041 
4082.8217 
4094.1550 
4105.5040 
4116.8687 
4128.2491 
4139.6452 
4151.0571 
4162.4846 
4173.9279 
4185.3868 
4196.8615 
4208.3519 
4219.8579 
4231.8797 
4242.9172 
4254.4704 
4266.0394 
4277.6240 
4289.2243 
4300.8403 
4312.4721 
4324.1195 
4335.7827 
4347.4616 



170 



TABUB S OF €lII€IdB»-(OcmtlBiMdX 
Dtentetem In unite and tenths. 



M«. 


Clrennf. 


Area. 


DU. 

80.7 


CirewBi; 


Area. 


DU. 


Cirenni: 


Arab 


74.5 


284.0487 


4359.1562 


288.5265 


6114.8977 


86.9 


278.0044 


5931.0206 


.6 


284.3628 


4370.8664 


A 


258.8407 


5127.5819 


87.0 


278.8186 


5944.6787 


.7 


234.6770 


4382.5924 


.9 


254.1548 


5140.2818 


.1 


273.6327 


6968.8525 


.8 


234.9911 


4384.8841 


81.0 


254.4690 


5152.9974 


.2 


273.9469 


6972.0420 


.9 


235.3053 


4406.0016 


.1 


254.7832 


6165.7287 


.8 


274.2610 


6985.7472 


75.0 


235.6194 


4417.8647 


.2 


255.0973 


5178.4767 


.4 


274.5762 


5999.4681 


.1 


235.9336 


4429.6535 


.8 


255.4115 


5191.2884 


.5 


274.8894 


6013.2047 


a, 


286.2478 


4441.4580 


.4 


255.7256 


6204.0168 


.6 


276.2035 


6026.9570 


^ 


236.5619 


4453.2788 


.5 


256.0398 


5216.8110 


.7 


275.6177 


6040.7250 


A 


236.8761 


4465.1142 


.6 


256.3540 


5229.6208 


.8 


276.8818 


6054.5088 


Jb 


287.1902 


4476.9659 


.7 


256.6681 


5242.4463 


.9 


276.1460 


6068.3082 


.6 


237.5044 


4488.8832 


.8 


256.9823 


5255.2876 


88.0 


276.4602 


6082.1284 


.7 


237.8186 


4500.7168 


.9 


257.2964 


6268.1446 


.1 


276.7743 


6096.9542 


.8 


238.1327 


4512.6151 


89.0 


257.6106 


6281.0178 


.2 


277.0886 


6109.8008 


.9 


238.4469 


4524.5296 


.1 


257.9248 


6293.9066 


.8 


277.4026 


6123.6631 


fl.0 


238.7610 


4536.4598 


.2 


258.2389 


6306.8097 


.4 


277.7168 


6137.5411 


J 


239.0752 


4548.4067 


.8 


258.5531 


6819.7295 


.6 


278.0309 


6151.4348 


2. 


239.3894 


4660.3678 


.4 


258.8672 


6332.6650 


,6 


278.3451 


6165.3442 


^ 


239.7035 


4572.3446 


.6 


259.1814 


6345.6162 


.7 


278.6593 


6179.2698 


A 


240.0177 


4584.3377 


.6 


259.4956 


5358.5882 


.8 


278.9734 


6193.2101 


Ja 


240.3318 


4596.3464 


.7 


259.8097 


6371.5658 


.9 


279.2876 


6207.1666 


.6 


240.6460 


4608.3708 


.8 


260.1239 


6384.6641 


89.0 


279.6017 


6221.1380 


.7 


240.9602 


4620.4110 


.9 


260.4380 


6897.6782 


.1 


279.9159 


6235.1268 


A 


241.2748 


4632.4669 


88.0 


260.7522 


6410.6079 


.2 


280.2301 


6249.1804 


.9 


241.5885 


4644.5384 


.1 


261.0663 


6423.6534 


.8 


280.6442 


6263.1498 


77.0 


241.9026 


4656.6257 


.2 


261.3805 


6436.7146 


.4 


280.8584 


6277.1849 


.1 


242.2168 


4668.7287 


.3 


261.6947 


6449.7915 


.6 


281.1725 


6291.2356 


.2 


242.531C 


4680.8474 


.4 


262.0088 


6462.8840 


.6 


281.4867 


6305.3021 


.8 


242.8461 


4692.9818 


.5 


262.3230 


6475.9923 


.7 


281.8009 


6319.3843 


.4 


243.1593 


4705.1319 


.6 


262.6371 


6489.1163 


.8 


282.1160 


6333.4822 


Jb 


243.4734 


4717.2977 


.7 


262.9513 


6502.2561 


.9 


282.4292 


6347.6958 


.6 


243.7876 


4729.4792 


.8 


263.2655 


5516.4115 


90.0 


282.7483 


6361.7251 


.7 


244.1017 


4741.6765 


.9 


263.5796 


6528.6826 


.1 


283.0575 


6375.8701 


.8 


244.4159 


475S.8894 


84.0 


263.8938 


6641.7694 


.2 


283.3717 


6390.0909 


.9 


244.7301 


4766.1181 


.1 


264.2079 


6554.9720 


.8 


283.6868 


6404.2073 


18.0 


245.0442 


4778.3624 


.2 


264.5221 


5568.1902 


.4 


284.0000 


6418.8995 


.1 


245.3584 


4790.6225 


.8 


264.8363 


5581.4242 


.5 


284.3141 


6432.6073 


a 


245.6725 


4802.8988 


.4 


265.1504 


6594.6789 


.6 


284.6283 


6446.8309 


A 


245.9867 


4815.1897 


.5 


265.4646 


5607.9392 


.7 


284.9425 


6461.0701 


A 


246.3009 


4827.4969 


A 


265.7787 


6621.2208 


.8 


285.2566 


6475.3251 
6489.6968 


.6 


246.6150 


4839.8198 


.7 


266.0929 


5634.6171 


.9 


285.6708 


.6 


246.9292 


4852.1584 


.8 


266.4071 


5647.8296 


91.0 


286.8849 


6503.8822 


.7 


247.2488 


4864.5128 


.9 


266.7212 


6661.1578 


.1 


286.1991 


6518.1848 


.8 


247.5575 


4876.8828 


85.0 


267 0354 


6674.5017 


.2 


286.5188 


6532.6021 


S 


247.8717 


4889.2685 


.1 


267.8495 


6687.8614 


.3 


286.8274 


6546.8856 


99.0 


248.1858 


4901.6699 


.2 


267 6637 


6701.2367 


.4 


287.1416 


6561.1848 


a 


248.5000 


4914.0871 


.8 


267.9779 


6714.6277 


.6 


287.4657 


6575.6498 


.2 


248.8141 


4926.5199 


.4 


268.2920 


6728.0346 


.6 


287.7699 


6589.9804 


.3 


249.1283 


4938.9685 


.5 


.268.6062 


6741.4569 


.7 


288.0840 


6604.8268 


.4 


249.4425 


4951.4328 


.6 


268.9203 


6754.8951 


.8 


288.3982 


6618.7388 


.6 


249.7566 


4963.9127 


.7 


269J2345 


6768.8490 


.9 


28a7124 


6633.1668 


.6 


250.0708 


4976.4064 


.8 


269.5486 


5781.8185 


92.0 


289.0265 


6647.6101 


.7 


250.3849 


4988.9198 


.9 


269.8628 


5795.8038 


.1 


289.8407 


6662.0602 


.8 


250.6991 


5001.4469 


8A.0 


270.1770 


6808.8048 


.2 


289.6548 


6676.6441 


.9 


251.0133 


5013.9897 


.1 


270.4911 


5822.8215 


.8 


289.9690 


6691.0347 


io.o 


251.3274 


5026.5482 


.2 


270.8053 


6835.8539 


.4 


290.2882 


6705.5410 


.1 


251.6416 


5039.1225 


.8 


271.1194 


6849.4020 


.5 


290.5978 


6720.0680 


.2 


261.9557 


5051.7124 


.4 


271.4336 


6862.9659 


.6 


290.9116 


6734.6008 


.8 


252.2699 


5064.8180 


.5 


271.7478 


5876.6454 


.7 


291.2256 


6749.1542 


.4 


252.5840 


5076.9394 


.6 


272.0619 


6890.1407 


.8 


291.5898 


6768.7288 


^ 


252.8982 


5089.5764 


.7 


272.3761 


5908.7516 


.9 


291.8540 


6778iKW2 


A 


253.2124 


5102.2292 


.8 


272.6902 


6917.8788 


98.0 


292.1681 


6792.9087 



CIBGLE8. 



171 



TABIDS 9 OF ClBCIiES-<Ooiittniiad). 
Blameters in nnlts and tenths. 



Ma. 


Clrcnnf. 


Area. 


ms. 


Gtrennf. 


ArMU 


Dia. 


Cirenmf. 


Area. 


iM.1 


292.4823 


6807.5250 


05.5 


800.0221 


7163.0276 


97.8 


307.2478 


7512.2078 


a, 


292.7964 


6822.1569 


.6 


300.8363 


7178.0366 


.9 


307.5619 


7527.5780 


.3 


293.1106 


6836.8046 


.7 


300.6504 


7193.0612 


98.0 


307.8761 


7542.9640 


.4 


298.4248 


6851.4680 


.8 


900.9646 


7208.1016 


.1 


308.1902 


7558.3656 


.6 


293.7389 


6866.1471 


.9 


301.2787 


7223.1577 


.2 


308.5044 


7573.7830 


.6 


294.0531 


6880.809 


96.0 


801.5929 


7238.2295 


.3 


308.8186 


7689.2161 


.7 


294.3672 


6895.5524 


.1 


801.9071 


7253.3170 


.4 


309.132'3: 


7604.6648 


.8 


294.6814 


6910.2786 


.2 


302.2212 


7268.4202 


.0 


309.4469 


7620.1293 


.9 


294.9956 


6925.0205 


.3 


302.5354 


7283.5391 


.6 


309.7610 


7635.6095 


M.0 


295.3097 


6939.7782 


.4 


802.8495 


7298.6737 


.7 


310.0752 


7651.1054 


.1 


295.6239 


6954.5515 


.5 


803.1637 


7313.8240 


.8 


310.8894 


7666.6170 


.2 


295.9880 1 6969.3406 1 


.6 


803.4779 


7328.9901 


.9 


310.7035 


7682.1444 


.3 


296.2522 


6984.1453 


.7 


803.7920 


7844.1718 


99.0 


311.0177 


7697.6874 


.4 


296.5663 


6998.9658 


.8 


304.1062 


7859.3693 


.1 


311.3318 


7713.2461 


.5 


296.8805 


7013.8019 


.9 


304.4203 


7374.5824 


.2 


311.6460 


7728.8206 


.6 


297.1947 


7028.6538 


97.0 


304.7345 


7889.8113 


.8 


811.9602 


7744.4107 


.7 


297.5088 


7043.5214 


.1 


305.0486 


7405.0559 


.4 


312.2743 


7760.0166 


.8 


297.8230 


7058.4047 


.2 


805.8628 


7420.3162 


.5 


812.5885 


7775.6382 


.9 


298.1371 


7073.3037 


.3 


305.6770 


7435.5922 


.6 


812.9026 


7791.2754 


•5.0 


298.4513 


7088.2184 


.4 


305.9911 


7450.8839 


.7 


813.2168 


7806.9284 


.1 


298.7655 


7103.1488 


.5 


306.3053 


7466.1913 


,8 


813.5309 


7822.5971 


.2 


299.0796 


7118.0950 


.6 


306.6194 


7481.5144 


.9 


313.8451 


7838.2815 


.3 


299.3938 


7133.0568 


.7 


306.9336 


7496.8532 


100.0 


314.1593 


7853.9816 


.4 


299.7079 


7148.0343 















Cirenmferenees when the diameter has more than one 

place of decimals. 



Dian. 


1 
Giro. 


Dlun. 


Circ. 


Diam. 


Clro. 


1 
Diam. 


Giro. 


Diam. 


Giro. 


.1 


.314169 


.01 


.031416 


.001 


.003142 


.0001 


.000314 


.00001 


.000031 


.2 


.628319 


.02 


.062832 


.002 


.006283 


.0002 


.000628 


.00002 


.000063 


.8 


.942478 


.03 


.094248 


.003 


.009425 


.0003 


.000942 


.00003 


.000094 


.4 


1.256637 


.04 


.126664 


.004 


.012566 


.0004 


.001257 


.00004 


.00012$ 


Ji 


1.570796 


.05 


.157080 


.005 


.015708 


.0005 


.001571 


.00005 


.000157 


.6 


1.884956 


.06 


.188496 


.006 


.018850 


.0006 


.001886 


.00006 


.000188 


.7 


2.199115 


.07 


.219911 


.007 


.021991 


.0007 


.002199 


.00007 


.000220 


^ 


2.513274 


.08 


.251827 


.008 


.025133 


.0008 


.002513 


.00008 


.000251 


3 


2.827433 


.09 


.282743 


.009 


.028274 


.0009 


.002827 


.00009 


.000283 



Examples. 



Diameter = 3.12699 

Circumference «■ 

Cire for dia of 3.1 

.02 
.006 
iK)09 
.00009 



M 



Snm of 

9.788937 
.062832 
.018850 
.002827 
.000283 

9.823729 



Clrcnmfte — 
Diameter — 

Dia for circ of 



9.823729 



9.738937 

.084792 
.062832 



.021060 
.018860 

.003110 
.002827 

.000283 
.000883 



Sum of 

3.1 

.02 

.006 

.0009 

.09009 
3.19699 



172 



CIRCLES. 



TABUB a OF CIBCIiKS. 



Diams in unite and twelfths) as in feet and inehea. 


Dia. 


Circumf. 


Area. 


Dia. Cirenmf. 


Area. 


Dia. 


Clrcamf. 


Area* 


irt.in. 


Feet. 


Sq. ft. 


Ft.In.l Feet 


Sq.ft. 


Ft.In. 


Feet. 


Sq. ft. 








5 ' 15.70796 


19.63495 


10 


31.41593 


78.53982 


1 


.261799 


.005454 


1 15.96976 


20.29491 


1 


31.67773 


79.85427 


2 


.523599 


- .021817 


2 16.23156 


20.96577 


2 


81.93953 


81.17968 


8 


.785398 


.049087 


3 ' 16.49336 


21.64754 


3 


32.20132 


82.51589 


4 


1.047198 


.087266 


4 


16.75516 


22.34021 


4 


32.46312 


88.86307 


5 


1.308997 


.136354 


5 


17.01696 


28.04380 


5 


32.72492 


85.22115 


6 


1.570796 


.196350 


6 


17.27876 


23.75829 


6 


32.98672 


86.59015 


7 


1.832596 


.267254 


7 


17.54056 


24.48370 


7 


33.24852 


87.97005 


8 


2.094395 


.349066 


8 


17.80236 


25.22001 


8 


33.51032 


89.8606S 


9 


2.356195 


•441786 


9 


18.06416 


25.96723 


9 


33.77212 


90.76258 


10 


2.617994 


.545415 


10 


18.32596 


26.72535 


10 


84.03392 


92.17520 


11 


2.879793 


.659953 


11 


18.58776 


27.49439 


11 


84.29572 


98.59874 


1 


3.14159 


.785398 


6 


18.84956 


28.27433 


11 


34.55752 


95.08818 


1 


3.40339 


.921752 


1 


19.11136 


29.06519 


1 


34.81982 


96.47858 


2 


3.66519 


1.06901 


2 


19.37315 


29.86695 


2 


36.08112 


97.98479 


8 


3.92699 


1.22718 


3 


19.63495 


30.67962 


8 


85.34292 


99.40196 


4 


4.18879 


1.39626 


4 


19.89675 


31.50319 


4 


35.60472 


100.8800 


5 


4.45059 


1.57625 


5 


20.15855 


32.33768 


5 


35.86652 


102.8690 


6 


4.71239 


1.76715 


6 


20.42035 


^33.18307 


6 


36.12832 


103.8689 


7 


4.97419 


1.96895 


7 


20.68215 


34.03937 


7 


36.39011 


105.3797 


8 


5.23599 


2.18166 


8 


20.94395 


34.90659 


8 


36.65191 


106.9014 


9 


6.49779 


2.40528 


9 


21.20575 


35.78470 


9 


36.91371 


108.4840 


10 


6.76959 


2.63981 


10 


21.46755 


36.67373 


10 


87.17551 


109.9776 


11 


6.02139 


2.88525 


11 


21.72935 


37.57367 


11 


37.43731 


111.5320 


S 


6.28319 


3.14159 


7 


21.99115 


38.48451 


12 


37.69911 


113.0973 


1 


6.54498 


3.40885 


1 


22.25295 


39.40626 


1 


37.96091 


114.6736 


2 


6.80678 


3.68701 


2 


22.51475 


40.33892 


2 


38.22?71 


116.2607 


3 


7.06858 


3.97608 


8 


22.77655 


41.28249 


8 


38.48451 


117.8588 


4 


7.33038 


4.27606 


4 


23.03835 


42.23697 


4 


38.74631 


119.4678 


5 


7.59218 


4.58694 


5 


23.30015 


43.20235 


5 


39.00811 


121.0877 


6 


7.85398 


4.90874 


6 


23.56194 


44.17865 


6 


39.26991 


122.7185 


7 


8.11578 


5.24144 


7 


23.82374 


45.16585 


7 


39.53171 


124.3602 


8 


8.37758 


5.58505 


8 


24.08554 


46.16396 


8 


39.79351 


126.0128 


9 


8.63938 


5.93957 


9 


24.34734 


47.17298 


9 


40.05631 


127.6763 


10 


8.90118 


6.30500 


10 


24.60914 


48.19290 


10 


40.31711 


129.3507 


11 


9.16298 


6.68134 


11 


24.87094 


49.22374 


11 


40.57891 


131.0360 


S 


9.42478 


7.06858 


8 


25.13274 


50.26548 


18 


40.84070 


132.7328 


1 


9.68658 


7.46674 


1 


25.39454 


51.31813 


1 


41.10250 


134.4894 


2 


9.94838 


7.87580 


2 


25.65634 


52.38169 


2 


41.36430 


136.1575 


8 


10.21018 


8.29577 


3 


25.91814 


53.45616 


8 


41.62610 


137.8865 


4 


10.47198 


8.72665 


4 


26.17994 


54.54154 


4 


41.88790 


189.6263 


5 


10.73377 


9.16843 


5 


26.44174 


55.63782 


5 


42.14970 


141.8771 


6 


10.99557 


9.62113 


6 


26.70354 


56.74502 


6 


42.41160 


143.1888 


7 


11.25737 


10.08473 


7 


26.96534 


57.86312 


7 


42.67:^30 


144.9114 


8 


11.51917 


10.55924 


8 


27.22714 


58.99213 


8 


42.93510 


146.6949 


9 


11.78097 


11.04466 


9 


27.48894 


60.13205 


9 


43.1^90 


148.4893 


10 


12.04277 


11.54099 


10 


27.75074 


61.28287 


10 


43.45870 


150.2947 


11 


12.30457 


12.04823 


11 


28.01253 


62.44461 


11 


43.72050 


152.1109 


4 


12.56637 


12.56637 


• 


28.27433 


68.61725 


14 


48.98230 


158.9388 


1 


12.82817 


13.09542 


1 


28.53613 


64.80080 


1 


44.24410 


155.7761 


2 


13.08997 


13.63538 


2 


28.79793 


65.99526 


2 


44.50590 


157.6250 


8 


13.35177 


14.18625 


3 


29.05978 


67.20063 


8 


44.76770 


159.4849 


4 


13.61357 


14.74803 


4 


29.32153 


68.41691 


4 


45.02949 


1 61.8557 


5 


13.87537 


15.32072 


5 


29.58333 


69.64409 


6 


45.29129 


168.2374 


6 


14.13717 


15.90431 


6 


29.84513 


70.88218 


6 


45.55809 


165.1801 


7 


14.39897 


16.49882 


7 


30.10693 


72.13119 


7 


45.81489 


167.0831 


8 


14.66077 


17.10423 


8 


30.36873 


73.39110 


8 


46.07669 


168.9479 


9 


14.92267 


17.72055 


9 


30.63053 


74.66191 


9 


46.88849 


170.8738 


10 


15.18486 


18.84777 


10 


30.89233 


75.94364 


10 


46.60029 


172.8094 


U 


15.44616 


18.98591 


11 


31.15413 


77.23627 


11 


46.86209 


174.7665 



OIBOLEB. 



173 



Mmmam In nnlt* and tw«lftiift| as tn ft«i and ineliea. 



Miu 


Cirenaf. 


Arcs. 


ma. 


Cireoinf. 


Ar«ft. 


Dte. 


Olreimf. 


IrMU 


FUn, 


Feet. 


Sq.ft. 


Ftln. 


Feet. 


Sq. ft. 


Ft.In. 


Feet. 


Sq.ft. 


16 


47.12389 


170.7146 


20 


62.88185 


314.1598 


25 


78.53982 


490.8739 


1 


47.38589 


17&6835 


1 


63.09865 


816.7827 


1 


78.80162 


494.1518 


2 


47.64749 


180.6634 


2 


63.35545 


819.4171 


2 


79.06342 


497.4407 


3 


47.90929 


182.6542 


3 


63.61725 


322.0623 


8 


79.32521 


500.7404 


4 


48.17109 


184.6558 


4 


63.87905 


324.7185 


4 


79.58701 


504.0511 


5 


48.43289 


186.6)S84 


5 


64.14085 


827.8856 


6 


79.84881 


607.8727 


6 


48.60469 


188.6919 


6 


64.40265 


830.0636 


6 


80.11061 


510.7052 


7 


48.95649 


190.7263 


7 


64.66445 


832.7525 


7 


60.37241 


514.0486 


8 


49.21828 


192.7716 


8 


64.92625 


335.4523 


8 


80.68421 


517.4029 


9 


49.48008 


194.8278 


9 


65.18805 


838.1630 


9 


80.89601 


520.7681 


10 


49.74188 


196.8950 


10 


65.44985 


340.8816 


10 


81.15781 


524.1442 


11 


50.00868 


198.9730 


11 


65.71165 


843.6172 


11 


81.41961 


527.5312 


le 


50.26548 


201.0619 


21 


66.97345 


346.3606 


28 


81.68141 


530.9292 


1 


50.52728 


203.1618 


1 


66,23525 


349.1149 


1 


81.94321 


534.3380 


2 


60.'^3908 


206.2725 


2 


66.49704 


351.8802 


2 


82.20501 


537.7578 


8 


51.06068 


207.3942 


3 


66.75884 


354.6564 


3 


82.46681 


541.1884 


4 


51.31268 


209.5268 


4 


67.02064 


357.4434 


4 


82.72861 


544.6300 


5 


51.67448 


211.6703 


5 


67.28244 


360.2414 


5 


82.99041 


548.0825 


,6 


51.83628 


213.8246 


6 


67.54424 


363.0503 


6 


83.25221 


551.5459 


7 


52.09808 


215.9899 


7 


67.80604 


365.8701 


7 


83.51400 


555.0202 


8 


52.85988 


218.1662 


8 


68.06784 


368.7008 


8 


83.77580 


558.5054 


9 


52.62168 


2W.3533 


9 


68.32964 


371.5424 


9 


84.03760 


562.0015 


10 


52.88348 


2X>..5513 


10 


68.59144 


374.3949 


10 


84.29940 


565.5085 


11 


58.14528 


224.7602 


11 


68.85324 


377.2584 


11 


84.56120 


569.0264 


17 


58.40708 


226.9801 


22 


69.11504 ' 380.1327 1 


27 


84.82300 


572.5558 


1 


53.66887 


229.2108 


1 


69.37684 


383.0180 


1 


85.08480 


576.0960 


2 


58.93067 


231.4525 


2 


69.68864 


385.9141 


2 


85.34660 


579.6457 


8 


54.19247 


233.7050 


3 


69.90044 


388.8212 


8 


85.60840 


583.2072 


4 


54.45427 


235.9685 


4 


70.16224 


391.7392 


4 


85.87020 


586.7797 


5 


54.71607 


238.2429 


5 


70.42404 


394.6680 


5 


86.13200 


590.3631 


e 


54.97787 


240.5282 


6 


70.68583 


397.6078 


6 


86.^9380 


593.9574 


7 


55.23967 


242.8244 


7 


70.94763 


400.5585 


7 


86.65560 


597.5626 


8 


55.50147 


246.1315 


8 


71.20943 


403.5201 


8 


86.91740 


601.1787 


9 


55.76327 


247.4495 


9 


71.47123 


406.4926 


9 


87.17920 


604.8057 


10 


56.02507 


249.7784 


10 


71.73308 


409.4761 


10 


87.44100 


608.4436 


, 11 


56.28687 


252.1183 


11 


71.99483 


412.4704 


11 


87.70279 


612.0924 


18 


56.54867 


254.4690 


28 


72.25663 


415.4756 


28 


87.96459 


615.752? 


1 


56.81047 


256.8307 


1 


72.51843 


418.4918 


1 


88.22639 


619.4228 


2 


57.07227 


259.2032 


2 


72.78023 i 421.6188 


2 


88.48819 


623.1044 


8 


57.38407 


261.6867 


8 


73.04203 


424.5568 


3 


88.74999 


626.7968 


4 


57.59587 


263.9810 


4 


73.30383 


427.6057 


4 


89.01179 


630.5002 


6 


57.85766 


266.8863 


5 


73.56563 1 430.6654 


5 


89.27359 


634.2145 


6 


58.11946 


268.8025 


6 


73.82743 


433.7361 


6 


89.53639 


637.9397 


7 


58.88126 


271.2296 


7 


74.08923 


436.8177 


7 


89.79719 


641.6758 


8 


58.64806 


273.6676 


8 


74.35103 


439.9102 


8 


90.05899 


645.4228 


9 


68.90486 


276.1165 


9 


74.61283 


443.0137 


9 


90.32079 


649.1807 


10 


59.16666 


278.5764 


10 


74.87462 


446.1280 


10 


90.58259 


652.9495 


11 


59.42846 


281.0471 


11 


75.13642 


449.2532 


11 


90.84439 


656.7292 


t» 


50.69026 


288.5287 


24 


75.39822 


452.3893 


29 


91.10619 


660.5199 


1 


59.96206 


286.0213 


1 


76.66002 


455.5364 


1 


91.36799 


664.3214 


2 


60.21886 


2885247 


2 


76.92182 


458.6943 


2 


91.62979 


668.1339 


8 


60.47566 


291.0891 


3 


76.18362 


461.8632 


8 


91.89159 


671.9572 


4 


6a7S?46 


293.5644 


4 


76.44542 


465.0430 


4 


92.15338 


676.7915 


8 


60.99926 


296.1006 


5 


76.70722 


468.2337 


5 


92.41518 


679.6867 


8 


61.2fa06 


298.6477 


6 


76.96902 


471.4352 


8 


92.67698 


683.4928 


7 


81.52286 


801.2056 


7 


77.23082 


•474.6477 


7 


•92.93878 


687.8597 


8 


61.78486 


808.7746 


8 


77.49262 


477.8711 


8 


98.20058 


691.2377 


9 


8Z0IM6 


806.3544 


9 


77.75442 


481.1055 


9 


93.46238 


695.1266 


10 


62.80895 


808.9451 


10 


78.01622 


484.3607 


10 


93.72418 


699.0262 


11 


82.G99D5 


811.54ff7 


11 


78.27802 


487.6068 


11 


98.98598 


702,9868 



174 



CDtBOUBIL 



TABUB S 0F CMMCIMM (OontlmMdr). 
DlaoM in mats wad tweUftb*; m in Wft and immU 



Ma. 


Clreuif. 


Am. 


Ua. 


Cireunf. 


ArtA. 


Utu 


Ctreamf. 


Aim. 


Vt.In. 


Fe«t. 


Sq.ft. 


FUn. 


FMt. 


Sq.ft. 


Vt.Tn. 


Feet. 


8q.ft 


•0 


94.24778 


706.a'j88 


t6 


109.9657 


962.1128 


400 


125.6687 


1266.6871 


1 


94.50958 


710.7908 


1 


110.2175 


966.6997 


1 


126.U266 


1261.8785 


2 


94.77188 


714.7841 


2 


110.4793 


971.2975 


2 


126.1878 


1267.1809 


8 


95.08318 


718.6881 


8 


110.7411 


975.9063 


3 


126.4491 


1272.3941 


4 


95.29498 


722.6536 


4 


111.0029 


980.6260 


4 


126.7109 


1277.6688 


5 


95.55678 


726.6297 


6 


111.2647 


985.1566 


• 5 


126.9727 


1282.9684 


6 


95.81858 


780.6166 


6 


111.6265 


989.7980 


6 


127.2345 


1288.2498 


7 


96.08038 


734.6145 


7 


111.7883 


994.4504 


7 


327.4963 


129&6662 


8 


96.34217 


788.6233 


8 


112.0601 


999.1187 


8 


127.7681 


1298.8740 


9 


96.60397 


742.6431 


9 


112.8119 


1003.7879 


9 


128.0199 


*1804.2027 


10 


96.86577 


746.6787 


10 


112.6737 


1008.4731 


10 


128.2817 


1809.5424 


11 


97.12757 


750.7152 


11 


112.fi3r>5 


1013.1691 


11 


128.6435 


1314.8929 


SI 


97.38937 


764.7676 


M 


118.0973 


1017.8760 


41 


128.8063 


1820.25tt 


1 


97.65117 


758.8810 


1 


113.3591 


1022.6939 


1 


129.0671 


1825.6267 


2 


97.91297 


762.9052 


2 


113.6209 


1027.3226 


2 


129.8289 


1831.0099 


8 


98.17477 


766.9904 


8 


113.8827 


1032.0623 


3 


129.5907 


1886.4041 


4 


98.43657 


771.0865 


4 


114.1445 


1036.8128 


4 


129.8626 


1841.8091 


5 


98.69887 


775.1984 


5 


114.4063 


104L6748 


5 


130.1143 


1847.2251 


6 


98.96017 


779.8118 


6 


114.6681 


1046.8467 


6 


130.8761 


1862.6625 


7 


99.22197 


783.4401 


7 


114.9299 


1051.1800 


7 


130.6379 


1868.0808 


8 


99.48877 


787.6798 


8 


115.1917 


1055.9242 


8 


130.8997 


1363.6885 


9 


99.74557 


791.7304 


9 


115.4635 


1060.7293 


9 


131.1616 


1868.9981 


10 


100.0074 


795.8920 


10 


115.7153 


1065.5458 


10 


131.4238 


1874.4686 


11 


100.2692 


800.0644 


11 


115.9771 


1070.3728 


11 


131.6851 


1879.9600 


tt 


100.5310 


804.2477 


87 


116.2389 


1075.2101 


42 


131.9469 


1886.4424 


1 


100.7928 


808.4420 


1 


116.5007 


1080.0588 


1 


182,2087 


1890.9458 


2 


101.0546 


812.6471 


2 


116.7625 


1084.9185 


2 


132.4705 


1896.4698 


8 


101.8164 


816.8632 


8 


117.0243 


1089.7890 


3 


182.732S 


1401.9848 


4 


101.6782 


821.0901 


4 


117.2861 


1094.6705 


4 


132.9941 


1407.5208 


5 


101.8400 


825.8280 


6 


117.5479 


1099.5629 


5 


133.2569 


1418.0676 


6 


102.1018 


829.6768 


6 


117.8097 


1104.4662 


6 


133.5177 


1418.6254 


7 


102.3636 


833.8365 


7 


118.0715 


1109.3804 


7 


133.7796 


1424.19a 


8 


102.6254 


838.1071 


8 


118.3338 


1114.8055 


8 


134.0413 


1429.7787 


9 


102.8872 


842.8886 


9 


118.6951 


ni9StAib 


9 


134.8031 


1436.8642 


10 


103.1490 


846.6810 


10 


318.8569 


1124.1884 


10 


184.6649 


1440.9656 


11 


103.4108 


850.9844 


11 


119.1187 


1129.1462 


11 


134.8267 


1446.5780 


S8 


103.6726 


865.2986 


88 


119.3805 


1134.1149 


48 


185.0885 


1452.2012 


1 


103.9344 


859.6237 


1 


119.6423 


1139X)946 


1 


135.3603 


1457.8858 


2 


104.1962 


863.9598 


2 


119.9041 


1144.0851 


2 


185.6121 


1463.4804 


8 


104.4580 


868.3068 


3 


120.1659 


1149.0866 


3 


185.8739 


1469.1364 


4 


104.7198 


872.6646 


4 


120.4277 


1154.0990 


4 


136.1357 


1474.8082 


5 


104.9816 


877.0334 


5 


120.6895 


1159.1 2*?? 


5 


1S6.3975 


1480.4810 


6 


105.2434 


881.4131 


6 


120.9513 


1164.1564 


6 


186.6593 


1486.1697 


7 


105.5052 


885.8037 


7 


121.2131 


1169.2015 


7 


136.9211 


1491.8698 


8 


105.7670 


890.2052 


8 


121.4749 


1174.2575 


8 


137.1829 


1497.5798 


9 


106.0288 


894.6176 


9 


121.7367 


1179.3244 


9 


137.4447 


1508.8012 


10 


106.2906 


899.0409 


10 


121.9985 


1184.4022 


10 


137.7065 


1509.0835 


11 


106.5524 


903.4751 


11 


122.2603 


1189.4910 


11 


137.9688 


1614.7767 


S4 


106.8142 


907.9203 


89 


122;5221 


1194.5906 


44 


138.2301 


1520.6308 


1 


107.0759 


912.3763 


1 


122.7839 


1199.7011 


1 


138.4919 


1526.2969 


2 


107.3377 


916.8433 


2 


123.0457 


1204.8926 


2 


138.7687 


1532.0718 


8 


107.5995 


921.8211 


8 


123.3075 


1209.9550 


8 


189.0166 


1687.8587 


4 


107.8613 


925.8099 


4 


123.5693 


1215.0982 


4 


189.2778 


1548.6666 


5 


108.1231 


930.8096 


6 


123.8811 


1220.2524 


6 


189.6891 


1549.4651 


6 


108.3849 


934.8202 


6 


124.0929 


1226.4175 


6 


189.8009 


1556.2847 


7 


108.6467 


989.3417 


7 


124.3547 


1230.5935 


7 


140.0627 


1561.1152 


8 


108.9085 


943.8741 


8 


124.6165 


1285.7804 


8 


140.3245 


1666.9566 


9 


109.1703 


948.4174 


9 


124.8783 


1240.9782 


9 


140.6863 


1572.8069 


10 


109.4321 


952.9716 


10 


125.1401 


1246.1869 


10 


140.8481 


1578.6721 


U 

1 


109.6989 


957.6867 


U 


125.4019 


1251.4065 


11 


141.1099 


1584.5462 



GIBCIiBS. 



175 



TAIUUB 8 OF €IB€I<BI^(Coiitila«0d). 
Wiaumm in unite mnd twelftlm; m in feet mnd incli 



DIa. 


Cireunf. 


Area. 


Dia. 


drenmf. 


Area. 


Dia. 


Circnnf. 


Area. 


Ftln. 


Feet. 


Sq.ft. 


Ft.Iii. 


Feet 


Sq.ft. 


Ft.Tn. 


Feet. 


Sq.ft. 


46 


141.8717 


1590.4313 


50 


157.0796 


1968.4964 


56 


172.7876 


2375.8294 


1 


141.6885 


1596.3272 


1 


157.3414 


1970.0458 


1 


173.0494 


2383.0344 


2 


141.8953 


1602.2841 


2 


157.6032 


1976.6072 


2 


173.3112 


2390.2502 


8 


142.1571 


1606.1518 


8 


157.8650 


1983.1794 


8 


173.5730 


2397.4770 


4 


142.4189 


16140805 


4 


158.1268 


1989.7626 


4 


173.8348 


2404.7146 


5 


142.6807 


1620.0201 


5 


168.3886 


1996.8567 


5 


1740966 


2411.9632 


6 


142.9426 


1625.9705 


6 


158.6504 


2002.9617 


6 


1743584 


2419.2227 


7 


143.2048 


1681.9319 


7 


158.9122 


2009.5776 


7 


1746202 


2426.4931 


8 


143.4661 


1687.9042 


8 


159.1740 


2016.2044 


8 


174.8820 


2433.7744 


9 


148J279 


1648.8874 


9 


159.4358 


2022.8421 


9 


175.1438 


2441.0666 


10 


143.9897 


1649.8816 


10 


159.6976 


2029.4907 


10 


175.4066 


2448.8607 


11 


144.2515 


1655.8866 


11 


159.9594 


2036.1602 


11 


175.6674 


2456.6887 


46 


144.5133 


1661.9025 


51 


160.2212 


2042.8206 


66 


175.9292 


2463.0086 


1 


144.7751 


1667.9294 


1 


160.4830 


2049.5020 


1 


176.1910 


2470.3446 


2 


145.0369 


1678.9671 


2 


160.7448 


2056.1942 


2 


176.4528 


2477.6912 


8 


145.2987 


1680.0158 


8 


161.0066 


2062.8974 


8 


176.7146 


2485.0489 


4 


145.5605 


1686.0753 


4 


161.2684 


2069.6114 


4 


176.9764 


2492.4174 


5 


145.8223 


1692.1458 


5 


161.5302 


2076.8364 


5 


177.2382 


2499.7969 


6 


146.0841 


1698.2272 


6 


161.7920 


2083.0723 


6 


177.5000 


2507.1878 


7 


146.8459 


1704.8195 


7 


162.0538 


2089.8191 


7 


177.7618 


25145886 


8 


146.6077 


1710.4227 


8 


162.3156 


2096.5768 


8 


178.0236 


2522.0008 


9 


146.8696 


1716.5368 


9 


162.5774 


2103.8454 


9 


178.2854 


2529.4239 


10 


147.1818 


1722.6618 


10 


162.8392 


2110.1249 


10 


1785472 


2536.8579 


11 


147.8931 


1728.7977 


11 


163.1010 


2116.9153 


11 


178.8090 


25443028 


47 


147.6649 


17849445 


63 


163.3628 


2123.7166 


67 


179.0708 


2551.7586 


1 


147.9167 


1741.1023 


1 


163.6246 


2130.5289 


1 


179.3326 


2569.2254 


2 


148.1785 


1747.2709 


2 


163.8864 


2137.8520 


2 


179.5944 


2566.7030 


8 


148.4403 


1753.4505 


3 


164.1482 


2144.1861 


8 


179.8562 


25741916 


4 


148.7021 


1759.6410 


4 


1644100 


2161.0310 


4 


180.1180 


2581.6910 


5 


148.9689 


1765.8423 


5 


1646718 


2157.8869 


5 


180.3798 


2589.2014 


6 


149.2257 


1772.0546 


6 


164.9336 


2164.7537 


6 


180.6416 


2596.7227 


7 


149.4875 


1778.2778 


7 


166.1954 


2171.6314 


7 


180.9034 


2604.2549 


8 


149.7492 


17845119 


8 


165.4572 


2178.5200 


8 


181.1662 


2611.7980 


9 


150.0110 


1790.7569 


9 


165.7190 


2185.4195 


9 


181.4270 


2619.3520 


10 


150.2728 


1797.0128 


10 


165.9808 


2192.3299 


10 


181.6888 


2626.9169 


11 


150.5346 


1803.2796 


11 


166.2426 


2199.2512 


11 


181.9506 


2634.4927 


48 


150.7964 


1809.5574 


68 


166.5044 


2206.1834 


58 


182.2124 


2642.0794 


1 


151.0582 


1816.8460 


1 


166.7662 


2213.1266 


1 


182.4742 


2649.6771 


2 


151.3200 


1822.1456 


2 


167.0280 


2220.0806 


2 


182.7360 


2657.2856 


8 


151.6818 


1828.4560 


3 


167.2898 


2227.0456 


3 


182.9978 


26649051 


4 


151.8436 


18347774 


4 


167.5516 


2234.0214 


4 


183.2596 


2672.5354 





152.1064 


1841.1096 


5 


167.8134 


2241.0082 


5 


183.5214 


2680.1767 


6 


152.3672 


1847.4528 


6 


168.0752 


2248.0059 


6 


183.7832 


2687.8289 


7 


152,6290 


1853.8069 


7 


168.3370 


2255.0145 


7 


184.0450 


2695.4920 


8 


152.8908 


1860.1719 


8 


168.5988 


2262.0340 


8 


184.3068 


2703.1669 


9 


153.1626 


1866.5478 


9 


168.8606 


2269.0644 


9 


1845686 


2710.8508 


10 


163.4144 


1872.9346 


10 


169.1224 


2276.1057 


10 


184.-8304 


2718.5467 


11 


153.6762 


1879.3324 


11 


169.3842 


2283.1679 


11 


185.0922 


2726.2534 


48 


153.9380 


1885.7410 


64 


169.6460 


2290.2210 


69 


185.3540 


2733.9710 


1 


1541998 


1892.1605 


1 


169.9078 


2297.2951 


1 


185.6158 


2741.6996 


2 


164.4616 


1898.5910 


2 


170.1696 


2304.3800 


2 


185.8776 


2749.4390 


8 


1647234 


1905.0323 


3 


170.4314 


2311.4759 


3 


186.1394 


2757.1893 


4 


1549852 


1911.4846 


4 


170.6932 


2318.5826 


4 


186.4012 


2764.9506 


5 


165.2470 


1917.9478 


5 


170.9550 


2325.7003 


5 


186.6630 


2772.7228 


6 


156.6068 


19244218 


6 


171.2168 


2332.8289 


6 


186.9248 


2780.5058 


7 


156.7706 


1980.9068 


7 


171.4786 


2339.9684 


7 


187.1866 


2788.2998 


8 


166.0824 


1987.4027 


8 


171.7404 


2347.1188 


8 


187.4484 


2796.1047 


9 


166.2942 


1948.9095 


9 


172.0022 


2354.2801 


9 


187.7102 


2803.9205 


10 


156.6660 


1960.4273 


10 


172.2640 


2361.4523 


10 


187.9720 


2811,7472 


11 


156.8178 


1966.9569 


11 


172.5258 


2368.6854 


11 


188.2338 


2819.5849 



176 



CIBCLB8. 



TABI<E S OF €IRCIiE8->(Gonttiiii«d). 
Dlamsi In units and twelfths; a4s In feet and inches. 



DIa. 


Circumf. 


Area. 


Dia. 


Circumf. 


Area. 


Dia. 


Circomf. 


Ares. 


Ft.Iu. 


Feet. 


Sq. ft. 


Ft. 111. 


Feet. 


Sq.ft. 


Ft.Iij. 


Feet. 


Sq.ft. 


60 


188.4956 


2827.4334 


65 


204.2085 


8318.3072 


70 


219.9116 


8848.4510 


1 


188.7574 


2885.2928 


1 


204.4658 


3826.8212 


1 


220.1733 


8857.6194 


2 


189.0192 1 2848.1632 


2 


204.7271 


3335.3460 


2 


220.4861 


3866.7988 


3 


189.2810 2851.0444 


8 


204.9889 


8848.8818 


8 


220.6969 


8876.9890 


4 


189.5428 2858.9:^ 


4 


205.2507 


8362.4284 


4 


220.9587 


8886.1902 


5 


189.«04() ' '.^866.8397 


5 


205.5126 


3360.9860 


5 


221.2206 


3894.4022 


6 


190.0664 ' 2874.7536 


6 


205.7748 


3369.6546 


6 


221.4823 


8903.6262 


7 


190.3282 2882.6786 


7 


206.0861 


8378.1889 


7 


221.7441 


8912.8591 


8 


190.5900 ; 2890.6143 


8 


206.2979 


3386.7241 


8 


222.0069 


3922.1089 


9 


190.8518 ! 2898.5610 


9 


206.5597 


3895.8263 


9 


222.2677 


3981.8506 


10 


191.1136 2906.5186 


10 


206.8215 


8403.9876 


10 


222.6296 


8940.6262 


11 


191.3754 ; 2914.4871 


11 


207.0833 


3412.5605 


11 


222.7918 


8949.9087 


•1 


191.6372 2922.4666 


66 


207.3451 


3421.1944 


71 


223.0681 


3969.1921 


1 


191.8990 2930.4569 


1 


207.6069 


8429.8392 


1 


228.8149 


8968.4915 


2 


192.1608 2938.4r)81 


2 


207.8687 


3438.4950 


2 


223.6767 


8977.8017 


3 


192.4226 , 2946.4703 


3 


208.1806 


8447.1616 


3 


228.8885 


3987.1229 


4 


192.6843 2954.4934 


4 


208.3928 


8455.8392 


4 


224.1008 


8996.4549 


n 


192.9461 , 2962.5273 


6 


208.6641 


8464.5277 


5 


224.8621 


4006.7970 


C 


193.2079 i 2970.5722 


6 


208.9159 


8473.2270 


6 


224.6239 


4016.1618 


7 193.4697 2978.6280 


7 


209.1777 


3481.9873 


7 


224.8867 


4024 5165 


8 19:^.7815 2986.6947 


8 


209.4895 


3490.6686 


8 


225.1475 


4088.8022 


9 


193.9933 2994.7723 


9 


209.7018 


8499.8906 


9 1 225.4093 


4048.2788 


10 


194.2551 3002.8608 


10 


209.9631 


8508.1386 


10 1 225.6711 


4052.6768 


11 


194.5169 3010.9602 


11 


210.2249 


*351 6.8876 


11 


225.9329 


4062.084S 


62 


1 94.7787 i 3019.0705 


67 


210.4867 


8525.6524 


72 


226.1947 


4071.5041 


] 


195.0405 


3027.1918 


1 


210.7485 


8534.4281 


1 


226.4566 


4080.9848 


2 


195.3023 


3035.3289 


2 


211.0108 


3643.2147 


2 


226.7188 


4090.3766 


8 


195.5641 


3048.4670 


8 


211.2721 ; 8552.0128 1 


8 


226.9801 


4099.8275 


4 


195.8259 


3051.6209 


4 


211.5339 


3560.8207 


4 


227.2419 


4109.2906 


5 


196.0877 


3059.7858 


5 


211.7957 


a569.6401 


5 1 227.5037 


4118.7648 


6 


196.3495 


3067.9616 


6 


212.0575 


8578.4704 


6 227.7656 


4128.2491 


7 


196.6113 


8076.1483 


7 


212.3198 


3587.8116 


7 228.0273 


4187.7448 


8 


196.8731 


:S084.8459 


8 


212.5811 


8596.1687 


8 228.2891 


4147.2514 


9 


197.1349 8092.55441 


9 


212.8429 


8606.0267 


9 


228.6509 


4156.7689 


10 1 197.3967 3100.7738 


10 


213.1047 


3618.9006 


10 


228.8127 


4166.2978 


n 1 197.6585 3109.0041 


11 


213.8665 


8622.7864 


11 


229.0746 


4175.8866 


68 


197.9203 3117.2453 


68 


213.6283 


3631.6811 


78 


229.8868 


4185.8868 


1 


198.1821 3125.4974 


1 


213.8901 


3640.6877 


1 


229.5981 


4194.9479 


2 


198.4439 


3183.7605 


2 


214.1519 


8649.6068 


2 


229.8699 


4204.5200 


3 


198.7057 


3142.0344 


3 


214.4187 


8(h')8.4887 


3 


280.1217 


4214.1029 


4 


198 9675 


3150.3193 


4 


214.6755 


3667.3781 


4 


230.8886 


4228.6968 


6 


199.2293 


3158.6151 


5 


214.9373 


8676.8284 


5 


280.6458 


4283.8016 


6 


199.4911 


3166.9217 


6 


215.1991 


8685.2845 


6 230.9071 


4242.9172 


7 


199.7529 


3175.2393 


7 


215.4609 


3694.2566 


7 


231.1689 


4252.5488 


8 


200.0147 


8183.5678 


8 


216.7227 


8708.2396 


8 


231.4307 


4262.1818 


9 


200.2765 


3191.9072 


9 


215.9845 ^ 3712.2385 


9 


281.6925 


4271.8297 


10 


200.5383 


3200.2575 


10 


216.2463 ; 3721.2388 


10 231.9643 


4281.4890 


11 


200.8001 


3208.6188 


11 


216..'>081 1 3730.2540 


11 


282.2161 


4291.1592 


64 


201.0619 


3216.9909 


60 


216.7699 3739.2807 


74 


282.4779 


4800.8408 


1 


201.3237 


8225.3739 


1 


217.0317 3748.8182 


1 


282.7397 


4310.6824 


2 


201.5855 


3283.7679 


2 


217.2935 3757.86(>6 


2 


238.0015 


4320.2858 


S 


201.8473 


3242.1727 


8 


217.5558 


3766.4260 


3 


288.2683 


4829.9492 


4 


202.1091 


3250.5886 


4 


217.8171 


3776.4962 


4 


283.6261 


4839 6789 


5 


202.3709 


3259.0151 


5 


218.0789 


3784.5774 


5 


288.7869 


4849.4096 


6 


202.6327 


3267.4527 


6 


218.3407 


3798.6696 


6 


234.0487 


4859.1562 


7 


202.8945 


3275.9012 


7 


218.6025 


3802.7726 


7 


234.8105 


4368.9186 


8 


203.1563 


3284.3606 


8 


218.8643 


3811.8864 


8 


234.6728 


4878.6820 


9 


203 4181 


3292.8809 


9 


219.1261 


8821.0112 


9 


2^.8341 


4388.4618 


10 


203.6799 


8801.8121 


10 


219.1^79 


38.30.1469 


10 


235.0959 


4896.2S15 


11 


203 9417 


8309.8042 


11 


219.6497 3839.2936 


11 


235.3576 


4408.0626 



CIRCLES. 



177 



TABI.1: S OF €IB€I.Efll-(ContIniied). 
Dlams In imtta and twelftbat w In feet and Inelies. 



PU. 


Cirvnnif. 


JLrfMU 


Dia. 


Cireunf. 


Area. 


Dia. 


Clrenmf. 


ArMU 


run. 


teeU 


Sq.ft. 


FUn. 


Veet. 


8q.ft. 


Ft.In. 


Feet. 


Sq.ft. 


96 


235.6194 


4417.8647 


80 


251.8274 


5026J>482 


86 


267.0354 


6674.5017 


1 


285.8812 


4427.6876 


1 


251.5892 


5037.0257 


1 


267.2972 


5685.6337 


2 


236.1430 


4437.5214 


2 


251.8510 


6047.5140 


2 


267.5590 


5696.7765 


8 


236.4048 


4447.8662 


8 


252.1128 


5068.0188 


8 


267.8208 


5707.9302 


4 


286.6666 


4457.2218 


4 


252.3746 


5068.5284 


4 


268.0826 


5719.0949 


5 


236.9284 


4467.0884 


6 


2524»64 


5079.0445 


6 


268.3444 


5780.2706 


6 


2S7.1902 


4476.9659 


6 


252.8982 


5089.5764 


6 


268.6062 


5741.4569 


7 


287.4520 


4486.8548 


7 


253.1600 


5100.1193 


7 


268.8680 


5752.6543 


8 


2S7.7138 


4496.7536 


8 


253.4218 


5110.6731 


8 


269.1298 


5763.8626 


9 


287.9756 


4506.6637 


9 


253.6886 


5121.2378 


9 


269.8916 


5775.0818 


10 


238.2374 


4516.5849 


10 


253.9454 


5131.8184 


10 


269.6534 


5786.3119 


11 


288.4992 


4526.5169 


11 


254.2072 


5142.3999 


11 


269.9152 


5797.5529 


n 


238.7610 


4586.4598 


81 


254.4690 


5152.9974 


86 


270.1770 


5808.8048 


1 


289.0228 


4546.4136 


1 


254.7808 


5163.6057 


1 


270.4388 


5820.0676 


2 


289.2846 


4556.3784 


2 


254.9926 


5174.2249 


2 


270.7006 


5831.3414 


8 


289.5464 


4566.3540 


8 


255.2544 


5184.8551 


8 


270.9624 


5842.6260 


4 


289.8082 


4576.3406 


4 


255.5162 


5195.4961 


4 


271.2242 


5853.9216 


6 


240.0700 


4586.3380 


5 


255.7780 


5206.1481 


5 


271.4860 


5865.2280 


6 


240.8318 


4596.3464 


6 


256.0398 


5216.8110 


6 


271.7478 


5876.5454 


7 


240.5936 


4606.3657 


7 


256.8016 


5227.4847 


7 


272.0096 


5887.8787 


8 


240.8554 


4616.3959 


8 


256.5634 


5238.1694 


8 


272.2714 


5899.2129 


9 


241.U72 


4626.4370 


9 


256.8252 


5248.8650 


9 


272.5332 


5910.5680 


10 


2a^790 


4636.4890 


10 


257.0870 


5259.5715 


10 


272.7950 


5921.9240 


11 


241.6408 


4646.5519 


11 


257.8488 


5270.2889 


11 


273.0568 


5983.2959 


17 


241.9026 


4656.6257 


81 


257.6106 


5281.0178 


87 


278.8186 


5944.6787 


1 


242a644 


4666.7104 


1 


257.8724 


5291.7565 


1 


278.5804 


5956.0724 


2 


242.4262 


4676.8061 


2 


258.1342 


5302.5066 


2 


278.8422 


5967,4771 


8 


242.6880 


4686.9126 


8 


258.8960 


5313.2677 


8 


274.1040 


5978.8921 


4 


242.9498 


4697.0801 


4 


258.6578 


5324.0396 


4 


274.8658 


5990.3191 


6 


248.2116 


4707.1584 


6 


258.9196 


5334.8225 


5 


274.6276 


6001.7564 


• 


248.4784 


4717.2977 


6 


259.1814 


5345.6162 


6 


274.8894 


6018.2047 


7 


248.7862 


4727.4479 


7 


259.4432 


5356.4209 


7 


275.1512 


6024.6689 


8 


248.9970 


4787.6090 


8 


259.7050 


5367.2365 


8 


275.4130 


6086.1340 


9 


244.2588 


4747.7810 


9 


259.9668 


5378.0630 


9 


275.6748 


6047.6149 


10 


244.5206 


4757.9639 


10 


260.2286 


5388.9004 


10 


275.9366 


6059.1068 


11 


244.7824 


47681577 


11 


260.4904 


5399.7487 


11 


276.1984 


6070.6087 


38 


246.0442 


477&3624 


88 


260.7522 


5410.6079 


88 O: 276.4602 


6082.1284 


1 


245.8060 


47885781 


1 


261.0140 


5421.4781 


1 


276.7220 


6093.6480 


2 


246.6678 


4798.8046 


2 


261.2758 


5432.8691 


2 


276.9838 


6105.1885 


8 


245.8296 


4809.0420 


8 


261.5376 


5443.25U 


8 


277.2456 


6116.7800 


4 


246.0914 


4819.2904 


4 


261.7994 


5454.1589 


4 


277.5074 


6128.2878 


5 


2463582 


4829.5497 


6 


262.0612 


6465.0677 


5 


277.7692 


6189.8556 


6 


246.6150 


4839.819B 


6 


262.3230 


5475.9923 


6 


278.0309 


6151.4348 


7 


246.8768 


4850.1009 


7 


262.5848 


5486.9279 


7 


278.2927 


6163.0248 


8 


247.1386 


4860.3929 


8 


262.8466 


5497.8744 


8 


278.5545 


6174.6258 


9 


247.40Q4 


4870.6058 


9 


2631084 


55088318 


9 


278.8163 


6186.2877 


10 


247.6623 


4881.0096 


10 


263.3702 


5519.8001 


10 


279.0781 


6197.8605 


U 


247.9240 


4881.8348 


11 


263.6320 


5580.7793 


11 


279.8899 


6209.4942 


n 


24&1868 


4901.6699 


84 


263.8938 


5541.7694 


89 


279.6017 


6221.1889 


1 


24&4476 


4912.0165 


1 


264.1556 


5552.7706 


1 


279.8635 


6232.7944 


2 


248.7094 


4922.8739 


2 


264.4174 


5563.7824 


2 


280.1253 


6244.4608 


8 


248.9712 


4982.7423 


8 


264.6792 


5574.8058 


8 


280.3871 


6256.1882 


4 


249.2K0 


4943J215 


4 


264.9410 


5585.8390 


4 


280.6489 


6267.8264 


5 


249.4948 


4958.5117 


5 


266.2028 


5596.8887 


5 


280.9107 


6279.5266 


6 


249.7566 


4968.9127 


6 


265.4646 


5607.9892 


6 


281.1725 


6291.2856 


7 


250.0184 


4974.8247 


• 7 


265.7264 


5619.0057 


7 


281.4343 


6302.9566 


8 


250.2802 


4984.7476 


8 


265.9882 


5630.0881 


8 


281.6961 


6314.6885 


9 


250.^420 


4996.1814 


9 


266.2500 


5641.1714 


9 


281.9579 


6326.4813 


10 


250^088 


5005.6261 


10 


266.5118- 


6652.2706 


10 


282.2197 


6888.1860 


11 


25L0668 


6016.0817 


11 


266.7736 


5663.3807 


11 


282.4815 


6849.9496 



1? 



I/O 



CIBCLES. 



TABUS 8 OF €IRCI<B»-<CoBtinQed% 
Dlams in anlts and twelfUisi m in Wtet nnd lnck( 



DIa. 


Cirenmf. 


Area. 


Dia. 


Cireunf. 


Area. 


Dla. 


dreumf. 


Area. 


Ft.In. 


Feet. 


Sq. ft. 


Pt.In. 


Feet. 


Sq. ft. 


Ft.In. 


Feet. 


Sq. ft. 


•0 


282.7433 


6361.7251 


98 5 


29a4771 


6858.9134 


96 9 


908.9491 


7851.7686 


1 


283.0051 


6378.5116 


6 


293.7889 


6866.1471 


10 


804.2109 


7864.4881 


2 


283.2669 


6885.8089 


7 


294.0007 


6878.8917 


11 


804.4727 


73770196 


3 


283.5287 


6397.1171 


8 


294.2625 


6890.6472 


97 


804.7345 


7889.811S 


4 


283.7905 


6408.9863 


9 


294.5243 


6902.9135 


1 


804.9963 


7402.5140 


5 


284.0623 


6420.7663 


10 


294.7861 


6915.1908 


2 


8a').2581 


7416.2277 


6 


284.3141 


6432.6078 


11 


295.0479 


6927.4791 


8 


905.5199 


7427.9522 


7 


284.5759 


6444.4592 


04 


295.8097 


6989.7782 


4 


805.7817 


7440.6877 


8 


284.8377 


6456.3220 


1 


295.5715 


6952.0682 


5 


806.0485 


7458.4840 


9 


285.0995 


6468.1957 


2 


295.8333 


6964.4091 


6 


.306.8053 


7466.1913 


10 


285.3613 


6480.0803 


8 


296.0951 


6976.7410 


7 


306.5671 


7478.9595 


11 


285.6231 


6491.9758 


4 


296.3569 


6989.0887 


8 


806.8289 


7491.7386 


tl 


285.8849 


6503.8822 


5 


296.6187 


7001.4874 


9 


807.0907 


7504.6286 


1 


286.1467 


6515.7995 


6 


296.8805 


7013.8019 
7026.1774 


10 


807.3525 


7517.8294 


2 


286.4085 


6527.7278 


7 


297.1423 


11 


807.6143 


7530.1412 


8 


286.6703 


6539.6669 


8 


297.4041 


7038.5638 


98 


807.8761 


7542.9640 


4 


286.9321 


6551.6169 


9 


297.6659 


7050.9611 


1 


908.1879 


7555.7976 


5 


287.1989 


6563.5779 


10 


297.9277 


7063.8693 


2 


808.3997 


7568.6421 


6 


287.4657 


6575.5498 


11 


296.1895 


7075.7884 


8 


808.6615 


7581.4976 


7 


287.7175 


6587.5325 


Wi 


298.4513 


7088.2184 


4 


808.9238 


7594.8689 


8 


287.9793 


6599.5262 


1 


298.7131 


7100.6593 


5 


809.1851 


7607.2412 


9 


288.2411 


6611.5808 


2 


298.9749 


7118.1112 


6 


809.4469 


7620.129S 


10 


288.5029 


6623.5468 


8 


299.2367 


7125.5739 


7 


309.7087 


7688.0284 


11 


288.7647 


6685.5727 


4 


299.4985 


7138.0476 


8 


809.9705 


7645.9884 


fS 


289.0265 


6647.6101 


5 


299.7603 


7150.6321 


9 


810.2323 


7658.8598 


1 


289.2883 


6659.6588 


6 


800.0221 


7163.0276 


10 


810.4941 


7671.79n 


2 


289.5501 


6671.7174 


7 


.300.2839 


7175.5340 


11 


810.7559 


7684.7888 


•« 


289.8119 


6683.7875 


8 


300.5457 


7188.0518 


99 


311.0177 


7697.6874 


4 


290.0737 


6695.8684 


9 


300.8075 


7200.6794 


1 


311.2795 


7710.6519 


5 


290.3355 


6707.9603 


10 


301.0693 


7213.1185 


2 


311.5418 


7723.6274 


6 


290.5973 


6720.0630 


11 


801.8811 


7225.6686 


8 


811.8031 


7736.6187 


7 


290.8591 


6732.1767 


96 


301.5929 


7238.2295 


4 


812.0649 


7749.6109 


8 


291.1209 


6744.8013 


1 


301.8547 


7250.8018 


5 


312.3267 


7762.6191 


9 


291.3827 


6756.4368 


2 


302.1165 


7263.8840 


6 


312.5885 


7775.68R2 


10 


291.6445 


6768.5882 


8 


302.3783 


7275.9777 


7 


812.8503 


7788.6681 


11 


291.9063 


6780.7405 


4 


302.6401 


7288.5822 


8 


813.1121 


7801.7090 


M 


292.1681 


6792.9087 


5 


302.9019 


7301.1977 


9 


313.3739 


7814.7606 


1 


292.4299 


6805.0878 


6 


303.1637 


7313.8240 


16 


313.6857 


7827.8286 


2 


292.6917 


6817.2779 


7 


303.4255 


7326.4613 


11 


318.8975 


7840.8971 


S 


292.9535 


6829.4788 


8 


303.6873 


7339.1095 


100 


314.1593 


7858.9816 


4 


293.2153 


6841.6907 













Diam. 


Ciroamf, 


Diam, 


Ciroamf, 


Diam, 


iMk. 


Ibot. 1 


Ineh. 


■ Ibot. 


Ineh. 


1-64 


.004091 


7-32 


.057269 


27-64 


1-32 


.008181 


15-64 


.061359 


7-16 


8-64 


.012272 


ili 


.065450 


29-64 


1-16 


.016362 


.069640 


16-32 


«-64 


.020463 


0-82 


.073631 


81-64 


8^ 


.024644 


10-64 


.077722 


8^ 


7^ 


.028634 


6-16 


.081812 


Hu 


.032726 


21-64 


.086908 


17-32 


U)36816 


11-32 


.089994 


86-64 


6-32 


.040908 


23-64 


.094084 


9-16 


11-64 


.044997 


1^ 


.098176 


87-64 


8-16 


.049087 


.102266 


19-32 


IM4 


.068178 


13-32 


.106366 


39-64 



Giroamr, 
_lbat._ 

.110447 
.114637 
.118628 
.122718 
026809 
030900 
034990 
039081 
048172 
047262 
061868 
056448 
059534 



Diam. 

6-8 
41-64 
21-32 
43-64 
11-16 
46-64 
28-82 
47-64 

Jii 

26-32 
61-64 
13-16 



Ciroamf, 



.163626 

067715 

.171806 

.176896 

079987 

084078 

.188168 

092269 

.196360 

.200440 

.204531 

.208621 

.212712 



Diam, 
Inelu 

63-64 
27-32 
65-64 
7-8 
67-64 
29-32 
69-64 
15-16 
61-64 
81-32 
68-64 
1 



Cireomr* 



.216808 
.220808 
.224064 
.229074 
.238161 
.237266 
.241346 
.246487 
.249688 
.263618 
.267700 
.261799 



dBCULAB ARCS. 
CIBCVI.AB ARCS. 



179 




S^itf.l 




BnlM for Fig. 1 apply to all arei •qnal to, or l€w than, a Bemi-circle. 
** " Fig. i «« *• «• or greater than, a ■emi-cirelt^ 



Cltordy a b, ot -vrlfcole aircy mdb, 

2 X \/raditi«s — (radiua — rise)^. Fig. 1. 

2 X \/iadia«> — (rise — radiiis)^. Fig. 2. 

2 X \/rise X (2 X radius — rise). Figs. 1 and 2. 
2 X radius X >ine cf}4acb. Figs. 1 and 2. 
rise 



— 2 X 



Figs. 1 and 2. 



tangent of a b d.* 
2 X dbl X cosine of a&d.* Figs. 1 and 2. 

2 X >/db9 — rise*. Figs. 1 and 2.§ 

approximately 8 X db^ — 3 X Length of arc adb^. Fig. 1. 



— 2 « radius X 



JjmiMjgOkf adb, 

arc a d 5 in degrees 



360 



. Figs. 1 and 2; 



•^ .01746 X radius X arc a d b in degrees. Figs. 1 and 2. 



drenmference of circle — length of mnaU arc subtending angle aeb. Fig. 2. 

. 8 X d&§ — ohordaft.** ^ , 
approximately 5 Fig. 1. 



•abdis — ^ofttie angle a b, subtended by the arc. In Fig. 2 the latter angle 
exceeds 180°. 

2<I6 — chord of dib^ or of half ad&— \/rlBe« + (i^ab)*. Figs. 1 and 2, 



flf rise — 
^ chord, 
.4 « 

..833 « 
.8 « 

••If rise — 
.6 chord 
.4 « 
.833 ** 
.8 •* 



multiply the rsaolt l^ 
1.036 
1.0196 
1.0114 
l.t083 

multiply the rasnli by 

1.012 
1.0066 
1.00B8 
1.0t28 



If rise — 
.26 chord, 
.2 « 
.126 « 
.1 « 

If lisa — 
.26 chord 
.2 « 
.126 « 
.1 « 



multiply the result by 
1.0044 
1.0021 
1.00036 
1.00016 

multiply the result Hr 
1.0015 
1.0007 
1.00012 
1.00006 



180 



OIBGULAB ABGB. 





Ooattnwd from p. 179. 

Bolts for Fig. 1 appij to all arcs equal to or less than a semi-circle. 
M u pig^ 2 ** ^ ** or greater than a 8emi<clrclo. 

R adimiy eOfC^pi or cbp 

. (H «<>)« + ri»e« ^ ij-jga. 1 and 2. 
2 X rise 

. ^^§_ , Pigs. 1 and 2. 
2 X rise 



%ab 



, Figs. 1 and 2. 

sine of ^ a e 6 

1 — cosine of ^ a e 6 



- ^<^^? , ngs. 1 audi, 
sineof >^6e<i| 

risedc 



1 4- cosine of ^ a o d f 



, FIg.x 



Rifle* or middle ordliisite» d9p 

radius — \/radius« — Q^ab]^, Fig. 1. 

radius + \/ndiwfl — Q^aS^, Fig. 2. 
radius X (1 — cosine of 6 e d ||), Fig. 1. 
radius X (1 + cosine of b e d ||),t Fi^. 2, 

^^^ , Figs. 1 and 2. 
2 X radius 

liab X tangent cf abd,* Figs. 1 and S. 

approximately ^^^^ ' '*«• 1- 
2 X radius 

When radius — chord a b, the resftit is 6.7 parts In lUO too shwrt. 
** *^ — 3X chord a b, the result is 0.7 parts in 100 too ahoft; 



Side ordimatey as n <» 

= >/radiu8> —en* + rise — radlni, Figfc 1 and S. 
= proximately /^ ^^ . Fig. l.t 



* a b d is s 3>^ of the angle acb^ subtended by the arc. 

t Strictly, this should read 1 mimu cosine; but the ooslBes of angles between 90* 
and 270^ must then be regarded as mimu or negative. Our rule, therefore, amonnta 
to the same thing. 

^db '^ chord of dib, or of half adb, — \/rUe» + (^a^)'- Xig>- 1 and 2. 

I be d — half the angle eob subtended by the are, la Fig. 2, the latter angle 
exceeds 180°. 
\ When radius = chord a b, this makes de 6.7 parts in 100 too short 

'< «< = 3 X chord a b, this makes d e 0.7 parts in 100 too short 
The proportionate error is greater with the side ordinates. 



CflBCDLAB ABGB. 



181 



Angley acb, sabtended lay Arc* adb. 

An angle and its supplement (as 5 e « and bed, Fig. 2) have the same «ine, the 
same cosine and the same tangmU. 

CAUtlon. The following sines, etc., are those of only half aob. 



fflneof J^oc6 — H?^ . Figs.land2. 



radius 
radius — rise 



rise — radius 
radius 



, ng.2. 



Cosineof Jiac6 J^aST" *^«-^* 

Tangent of >^ a c6 ^,^"^^ , Tig.l; - ^ ^**^^. , Fig. i 

^* radius ~ rise ® * rise — radius ' 



Versed sine of ^ a« 6 ■— 



rise 
radius 



, Figs. 1 and 2. 



Vo dMwrilM ttie mve sf m elrde too Isury* ftnr Um dl-rtders. 

Let a c 1m the choordy and o b the height, of the required arc, as 




laid down om the drawing. On a separate sMp af paper, «• m n, drawa c. o h. and aft. 
•Ibo b e, parallel to the chord a c. It Is well to make b«,and b e, each a little longer 
than a b. Then cut off the paper earefhUy along the lines 8 h and 6 «, so as to leare 
renaaining only the strip tabemn. Now, if the straight sides s b and 6 e be applied 
to tlie drawing, so that any narts of them shall touch at the same time the points a 
and 6, or b and e, the point h on the strip will be in ttie circumference of the arc, 
and may be prldced off. Thus, any number of points in the arc may be found, and 
afterward united to form the corre. 

31d Hi ottiodt Draw tteOMn a b; the rise re; and a 0^6 a From c with radios 




e r describe a drele. Make each of the arcs o I and i I equal to ro or r i; and draw 
c C cL DiTide eC, eZ, er, each into half as many equal parts as the curre is to be divided 
into. Draw the lines 61, 52, 2>3; and a4, a5, a6, extended to meet the first ones at 
e, «, A. Then e, «, A, are points in one half the curve. Then for the other half, draw 
simUar lines flrom a to 7» 8, 9; and others from b to meet them, as before. Trace 
tte ennro by hand. 



182 



CIRCULAR ARCS. 



^It DMj firaquentlj b* of um to 



'afhattaiABjMedoi^nol ' 




azMeding 29<*, or in o<:her wordi, whou cluyrd be it of Uad tiadUm Umm iUriM, th* 
nUddle oratnate a o, will be one-half of a c, quite near enovgh fbr manj pap* 
poses; b c and < e boinir tangenta to the arc.f And Tica Tena, if in tnch an arc we 
make o c equal a o, then will o be, rwj nearly, the point at which tangents fh>m th« 
ends of the arc will meet. Also the muUlle oxdlnate n, ot thm ikmlt uno ob,or 
ott will be approximately 3^ of a ft, the middle ordinate of the whole arc. Indeed, 
this last obserTadon will apply near enough for many approximate uses even if the 
arc be as great as 46°; for if in that case we take ^ of o a fbr the ordinate n, n wlU 
then be but 1 part in 1U3 too small; and therefore the principle may often be used 
in drawings, for finding points in a curve of too great radius to be drawn by the 
diTiders ; for in the same manner, V^ of n will be the middle ordinate for the arc n h 
or n o; and so on to any extent. Below will be f>uud a table bjr nrldelk tbe 
rlae or middle ordliuite ot a ludf mrc can be obtained with greater 
accuracy when required for more exact drawings. 

CIRCUIjAR arcs in FBSMiUKlIT ITSIB. 

The fifth column is of use for finding points for drawing arcs too \argB fbr tiM 
beam-compass, on the principle giren above. In even the largest cfllce drawings it 
will not be necessary to use more than the first three decimals of the fifth column ; 
and after the arc is subdirided into parts smaller than about 86° each, the first two 
decimals .25 will generally su£Bce. OriginaL 



BlM 






For 


ForriM 


BiM 






For 


Fer 


in 


De(r«ei 


For nA 


length of 


of half 


In 


Dogreei 


For rad 


length of 


rlMoff 


paru 


in whole 


mult rise 


aro malt 


aro 


paru 


in whole 


multrlM 


aro nalt 


halfara 


of 


•ro. 


by 


oborA 


mnltriM 


of 


are. 


iv 


ehord 


bibIS 


dioid. 


• 




by 


by 


sherd. 






bj 


ti—hf 


1-60 


o / 
9 9.76 


313. 


1.00107 


.2601 


u 


o / 
66 8.70 


6^ 


1.04116 


• 

.2688 


1-46 


10 10.76 


263.626 


1.00132 


.2501 


63 46.90 


6.626 


1.06366 


.2649 


1-40 


11 26.98 


200.6 


1.00167 


.2602 


.165 


68 63.63 


6.70291 


1.06288 


.2667 


1-36 


13 4.92 


163.625 


1.00219 


.2502 


1-6 


73 44.89 


6. 


1.07260 


.26t6 


1-30 


15 16.38 


113. 


1.00296 


.2503 


.18 


79 11.73 


4.36803 


1.08428 


.2676 


1-26 


18 17.74 


78.626 


1.00426 


.2504 


1-6 


87 12.34 


3.626 


1.10847 


.2693 


1-20 


22 60.54 


60.6 


1.00666 


.2506 


.207107 


90 


3.41422 


1.11072 


.2699 


1-19 


24 2.16 


46.026 


1.00737 


.2607 


.226 


96 64.67 


2.96913 


1.12997 


.2616 


1-18 


26 21.65 


41. 


1.00821 


.2508 


.2^6 


106 16.61 


2.6 


1.16912 


.2639 


1-17 


26 60.36 


36.626 


1.00920 


.2609 


116 14.69 


2.15289 


1. J 9083 


.2666 


1-16 


28 30.00 


82.6 


1.01088 


.2510 


.3 


123 6130 


1.88889 


1.22496 


.2692 


1-16 


30 22.71 


28.626 


1.01181 


.2611 


^ 


134 46.62 


1.626 


1.27401 


.2729 


1-14 


32 31.22 


26. 


1.01366 


.2613 


144 30.08 


1.43827 


1.32413 


.2766 


1-13 


34 69.08 


21.626 


1.01671 


.2516 


.4 


154 38.35 


1.28125 


1.^322 


.2808 


1-12 


37 60.»6 


18.6 


1.01842 


.2517 


.426 


161 27.52 


1.10204 


1.42764 


.2838 


1-11 


41 13.16 


16.626 


1.02189 


.2620 


.45 


167 66.93 


1.11728 


1.47377 


.2868 


1-10 


46 14.38 


•18. 


1.02646 


.2625 


.476 


174 7.49 


1.06402 


1.62162 


.2899 


1-0 


60 6.9II 


10.625 


1.03260 


.2630 


.6 


180 


1. 


1.67080 


.2929 



V At 29° o • thus fbond will be bat about 8 parti too tiiort in 100. 



MENaUKATION, 183 

bniStbB af elpenlH »f«s. If itrc«zce«da aaeialelrel«,H*p IS4 

riMolii lu obon) ud bdibb dlrtd> Iha fal«tt bj lb« Uud. Ttaa In Uu MoBn dT balibli Iki 

MiUpIj llu Uit EiBbir bj ili> Itatlh of U> Jru lEonL * « omn <> Dt>U 

TABLE OF CIKOVLAB ABCS. H«nn». 



Uvi^i. P'lbu. I'Oiiftb*. H'ibli. l^nctbB. B'lliUr L«DBI^ 



184 MESSUBATIOH. 

TABI.B «F CIKCIJI.AB ARCH— 



n arc of 1° if tbe eartb's Krent circle Is but 4.3354 feet loBcrr tbni lt> 
1. lu lsiijiUiiiO.lt lindi>riiuniumnn. ■ulli'i«|»virli>lnil:^>HI.b10Siiill«. Polir 3*«><fT. 



MENBUKATIOI'. 185 

T« Bad tbe Ie>|rUi of > circular src br tbe followliic teUe- 



I'EireTBS or circdi.ak abcs to bad i 



mi 



186 



MENSURATION, 



CmCVttAR BBCTORSy BINGMS, SBOmSRVS, SSTCX 

^ * Area of a eironiar ■eetor, adbe^ Fig. A, 

arc adh 




X radlua o a. 
— area of entire drole X 



Fig. B. 




aro g d 6 In degrees; 
S60 



Area of a clrc«lar ving. Fig. B, 
.—1 area of larger circle, d, — area of smaller one, a b. 
1^ — .7854 X (sam of diams. cd + ah)X (cUfil of diams. e d^a 6.) 
— 1.5708 X thickness e « X *<i°^ <^ diameters « d and a h. 



To And. the rmdi«a of a clrele -vrhleli aliall have the aanie 
as a giyrevk elrciilar rln|^ c» dab. Fig. B, 

Draw any radius n r of the outer circle ; and from where said radins cuts tht 
bner circle at t, drew < « at right angles to it. Then will t « be the required ladins. 

Bresultl&y ea^mbd, of a circular rl»|ft Flf. ^ 

iM. V^ difference of diameters e d and a &. 

« ^ (diameter ed—w 1.2732 area of circle a 6.) 

Area of a eirenlar xone abed^ 

0m area of circle m n — areas of segments am 5 and end, 
(for areas of segments, see below.) 

A circular Inne is a crescent-shaped 
figure, comprised between two arcs abe 
. and a o e of circles of different radii, a d 
and AM. 



of a drcvlar lume uheo 

^ area of segment ahe — area of segment a oc^ 
(fix arcM of segments sea bcloir.) 





Pig.D. 




V»flndflio 



«f »olreiilMP 



it^mbodf Figi.O^Di. 



Area of Segment adbn, Fig. A (at top of page) 
■■ Area of Sector a d 5 e — Am of Triangle a 5 0. 
•^^iiAroadb X tadinaa* — en X cbordafty. 

Vmwinff the area of a aeKment required to bo ent mtt 
gkvewk clrelcy €0 flnd tta chord suad rise. 



^ IHTide the area hj the square of the diameter of the clrele : look for the qnotleot 
In th9 column of areas in the table of areas, opposite; taice out from the table 
Che corresponding number In the column of risei. Mnltipljr this nninbar bgr the 
diameter. The product will be the required rise, Thea 

ahord — 2 X V^ (dUmeter — rte) X 



MENSURATION. 



187 



TABUB OP AREAS OF CIB€UI«AR SEOlIEjnni, Fiffi C, Dl 
' If the seyment exeeeda a semieirelef it* are« i* = %nm <a eireie— i 

of • aegmant whose riie Is = (dUm of eirelt — rise of giren segment). Dlaai of eird* * (eqiian 
ef hair ohord t> rise) 4* rise, whether the segment exeeeds a eemieirole or not. 



Rise 


Area= 


Rise 


.Areas 


Rise 


Areas 


Rise 


Area» 


Rise 


Areap* 


dlrhf 


(sqnare 


diYby 


(sqaare 
of diam) 


dlTby 


(Bonare 
of diam) 


diT by^ 


(square 
of diam) 


dirby 


(sqnare 


diamef 


ef diam) 


diam of 


diam of 


liaaof 


diam of 


of diam 


•irele. 


malt by 


oirole. 


moltbj 


eiioto. 


moltby 


einia. 


mult by 


oirole. 
.25^ 


BMritby 


.001 


..000042 


.064 


.021168 


.127 


.057991 


.190 


.103900 


.166149 


.002 


.000119 


.065 


.021660 


.128 


.058658 


.191 


.104686 


.254 


.157019 


.003 


.000219 


.066 


.02'2;55 


.129 


.059328 


.192 


• .106472 


.255 


467891 


.004 


.000337 


.067 


.022663 


.130 


.059999 


.193 


.106261 


.256 


.168768 


.005 


.000471 


.068 


.023156 


.131 


.060673 


.194 


.107051 


.257 


469686 


.006 


.000619 


.060 


.023660 


.132 


.061349 


.196 


.107843 


.258 


460511 


.007 


.000779 


.070 


.024168 


.133 


.062027 


.196 


.108636 


.269 


461386 


.008 


.000952 


.071 


.024680 


.134 


.062707 


.197 


.109431 


.260 


462268 


.009 


.OOllSft 


.072 


.025196 


.136 


.063389 


.198 


.110227 


iS61 


.168141 


.010 


.001329 


.073 


.025714 


.136 


.064074 


.199 


.111025 


.262 


464020 


.011 


.001633 


.074 


.026236 


.137 


.064761 


.200 


.111824 


.263 


464900 


.012 


.001746 


.076 


.026761 


.138 


.065449 


.201 


.112626 


i264 


.166781 


JQIS 


.001969 


.076 


.027290 


.139 


.066140 


.202 


.113427 


.266 


.166688 


mt 


.002199 


.077 


.027821 


J40 


.066833 


.203 


.114231 


.266 


487646 


XH6 


.002438 


.078 


.028356 


.141 


.067528 


.204 


.115036 


.267 


.188481 


Me 


.002685 


.079 


.028894 


.142 


.068225 


.205 


.115842 


.268 


.109816 


.017 


.002940 


.080 


.029435 


.143 


.068924 


.206 


.116651 


.260 


.170202 


.018 


.008202 


.081 


.029979 


.144 


.069626 


.207 


.117460 


.270 


471090 


.019 


.003472 


.082 


.030526 


.146 


.070329 


.208 


.118271 


.271 


.171978 


.020 


.003749 


.083 


.031077 


.146 


.071034 


JHOd 


419084 


.272 


.172868 


joai 


.004032 


.084 


.031630 


.147 


.071741 


.210 


419898 


.273 


.173768 


JOZ 


.004322 


.086 


.032186 


.148 


.072450 


.211 


420718 


.274 


474660 


JOSS 


.004619 


.086 


.032746 


.149 


J073162 


.212 


.121530 


.276 


.176542 


J024 


.004922 


.087 


.033308 


.160 


.073876 


.213 


422348 


.276 


476486 


J0fi6 


.005231 


.088 


.033873 


.181 


.074590 


.214 


423167 


.277 


477830 


JM 


.005546 


.089 


.034441 


.152 


.076307 


.216 


.123988 


.278 


478226 


Ml 


.005807 


.090 


.035012 


.163 


.076026 


.216 


424811 


.279 


479122 


xas 


.006194 


.091 


.035586 


.164 


Wfl747 


.217 


.126634 


.280 


480020 


M9 


.006627 


.092 


.036162 


.165 


.077470 


.218 


426469 


.281 


.180918 


J06O 


.006866 


.003 


.036742 


.166 


.078194 


.210 


.127286 


.282 


481818 


JOSL 


.007209 


.094 


.037824 


.157 


.078921 


.220 


428114 


.283 


482718 


M2 


.007660 


.096 


.037909 


.168 


.079660 


.221 


428948 


.284 


488619 


JOBS 


.007913 


.006 


.038497 


.169 


.080380 


.222". 


.129778 


.286 


484622 


J084 


.008273 


.097 


.039087 


.160 


.081112 


.223 


430606 


.286 


.186426 


j066 


.008638 


.098 


.039681 


.161 


.081847 


.224 


431488 


.287 


486329 


JOM 


.009006 


.099 


.040277 


.162 


.062682 


.225 


.132278 


.288 


487236 


.037 


.009388 


.100 


.040875 


.163 


.088320 


.226 


483109 


.289 


488141 


JOSS 


.009764 


.101 


.041477 


.164 


.084060 


.227 


.133946 


.290 


489048 


.080 


.010148 


.102 


.042081 


.165 


.084801 


.228 


434784 


.291 


.189956 


J040 


.010638 


.103 


.042687 


.166 


.085545 


.229 


.136624 


.292 


.190866 


041 


.010932 


.104 


.043296 


.167 


.086290 


.230 


.136466 


.293 


.191774 


J042 


.011831 


.106 


.043908 


.168 


.087037 


.231 


.137307 


.294 


492685 


.048 


^11734 


.106 


.044623 


.169 


.087785 


.232 


.138151 


.296 


493597 


J044 


.012142 


.107 


.045140 


.170 


.088536 


.233 


438996 


.296 


494509 


.046 


.012555 


.108 


.045759 


.171 


.089288 


.234 


439842 


.287 


.196428 


.046 


.012971 


.109 


.046381 


.172 


.090042 


.235 


140689 


.298 


496337 


j047 


.013303 


.110 


.047006 


.173 


.090797 


.236 


.141538 


.299 


497262 


.048 


.013818 


.111 


.047633 


.174 


.091556 


.237 


.142388 


.300 


.198168 


J04» 


.014248 


.112 


.048262 


.175 


.092314 


.238 


443239 


.301 


.199086 


.060 


.014681 


.118 


.048894 


.176 


4»8074 


.239 


.144091 


«302 


.200008 


JO&I 


.016110 


.114 


.049529 


.177 


.093837 


.240 


.144945 


.308 


.200922 


M>2 


.016661 


.115 


.060165 


.178 


.094601 


.241 


.145800 


.804 


.201841 


J06» 


.016008 


ai6 


.060805 


.179 


.095367 


.242 


446656 


.366 


.202762 


J064 


.016468 


J17 


.061446 


.180 


.096135 


.243 


.147513 


.306 


.203688 


j06§ 


.016013 


.118 


.062090 


.181 


•090804 


.244 


448371 


.307 


.204606 


iNM 


M79n 


419 


.062737 


.182 


.097675 


.246 


.149231 


.308 


.206628 


jm 


.017881 


.120 


.063886 


.183 


.098447 


.246 


.160091 


.309 


.206462 


MB 


.018907 


.121 


.0640:7 


.184 


.099221 


.247 


460953 


.310 


.207376 


JOM 


.018766 


.122 


.064690 


.186 


.099997 


.248 


461816 


.311 


.208302 


JIMO 


.019188 


428 


.066846 


.186 


.100774 


.249 


452681 


.312 


.209228 


jOd 


.oime 


J24 


.066004 


.187 


.101553 


.250 


463546 


.313 


.210166 


jm 


/mm 


096 


.066664 


.188 


.102.334 


.261 


.154413 


.314 


.211083 


Ml 


iMMBI 


Jfl6 


.087827 


.189 


J03116 


1 .262 


.166281 


.816 


.212011 



188 MENSURATION. 

TABLK OF AKEAS OF CIRCDE.AK SBONEVTS-tCoHTHiDH: 



Urn 






1« 












A««_ 


dl.BT 
















irdi™ 


«lui< 












orai>~i 






^nli. 


"^imi 


.363 


i!73« 


.380 


.383603 


'427 


mint. 


Biujt 








56e730 












.284H9 




.320940 


406 


5677 2S 


£li 


.314S0'J 






.302 












JiU 


















.369723 




isieeea 
















.3607a 




il7«» 




.!BaB3.; 


.SOS 








4«9 


.3fll7M 


xa 


^86W 


















XSi 


^wee 


isao 




.ae; 










!363IU 


Mt 


33SMM 


.Ml 
















su 






SMHli 


-390 


592390 










M» 


iaZMTS 


iVi 














!36fl711 




.tzg^S 


.8M 


.168385 








.330S6fe 




.367710 


MS 


.2M1M 


.366 


JJB9SM 


•402 






.331861 






jm 


,236094 


















too 


.ismi 












!38383« 




.?I070S 


^1 






.2«S249 


■406 






.834829 






JS2 


iersi 






406 












.33S 










.300238 






!4S1 








571 






.301221 




.837810 




sitm 


.S3B 








'409 












xe 




















JSl 






!»l^04e 




504171 


^143 






5T7T01 


jaa 














M17SB 






xss 




















MO 
Ml 


liwai 


is78 


.i710Bl 


A\t 


1307126 
JOSllD 


462 


I4477S 


4m1 


asijoo 


MS 


.affi3«9 








.309096 




M673S 




.382700 


Ma 


.238319 


.380 








464 








M* 


jawss 


.381 


.ii*sa 


-418 


aiiow 


466 




!492 


:384eM 




.M0Z19 












.848766 




,3S68» 






!3S3 














.38Se»0 


Ml 




.3S4 


^77748 


J21 




468 






.381390 




1213074 


.38* 






.316017 




■.3fil7«. 




5<W«90 


MS 




.383 










562142 




.389300 


Xba 


.24U80 




jHoero 














J»i 


3MaS6 




.28HM3 


426 


.317981 


!4«2 


.364736 


!499 


.S913W 


Mi 


.MflSM 


.380 






.118970 


4«3 


566733 


JiOO 


xMm 



ELLIPSE (page 139). 



Focal dlBMiice^/0 = 



HENSURATIOir. 



189 



THK BIiI.IPSfi» 




An «B!tM« Is m enrra, • «««, Fig I. formed by an obllqae Mctioa of eltlMr • oone or s eylinder, paaa* 
Ins throngh Ita ourred Mrfaee, withoat cattiog the base, lu nature la luoh that if t«o linei, aa 
n/ and n g. Fig. 8, be drawn from any point n in Ita periphery or etraamf, to two oertain points/ 
nnd g, in iu long diam o w, (and called the foei of the eiUpie,) their ram will be eqnal to that of any 
other two lines, as i/, and b g, drawn from any other point. a« 6, in the clreumf, to the fooi/aad ^j 
slao the snm of any two snch lines will be equal to the long diam « w. The line e w diriding the ellioso 
Into two eqnal parte lengthwise, is oalled its transverse, or major axis, or long diam ; and • i, whieh 
dirtdee it equally at right-angles to e io, is called the oonjogate, or minor axis, or diort 41ain. To 
find the position of the tool of an ellipse, from either end, as 6, of the short diam, memsnre olf the 
diets ft /and 6 g. Fig S, each equal to o c, or one-haif the long diam. 

The parameter of an ellipse is a oertain length obtained thus ; as the long diam i short diam : : 
short diam : parameter. Any line r v, or • d, Fig S, drawn from the eireamf* to, and at right angloa 
to, eliher diam, is ealled an ortUnau; and the parts e v and 9W,b» and • «, of that diam* between 
the ord and the eiroumf, are oalled al^teUam, or a&seiseei^ 

To flnil tlie leufftli of any ordinate, rvovsd, drawn to eitbetf 

dianif e W or h a* Knowing (h« ahecisa, « • or « a, and tiM two diams, e w, ft •{ 



ew*:fta<::cvXvwiFA 



ftd^i««!*::fr« X « a:g<i>. 



To lind the elreumf of an elHpse. 

Mathe— HelnM have fhmisked praodeal men with no simple working rale Ibr this pvrpoae. The 
•e-ealled appvMdmate mlea do not deserre the name. They are as foUowa, D being the long diam ; 
4 the aiiorteino. 



RvLB 1. Circamf =8.141« R±A. • Rvlb S. S.M16 / f^^^-\ • Buu t. «.2ai6y' DS^hP: 

thte if tiie nme aa Bnle 2, bnt In a dllT shape. Sou4.2X|/ DS+ 1.1874 A Now, in an elUpse 

vhoae long and short dlams are 10 and S, the oirenmf Is MtnaUy 11, very approximately; bnt rule 1 
(ires it = 18.85 ; rale 2, or 3, == 22.65 ; and rule 4. =: 30.68. Again, if the diams-be 10 and 6, the dr. 
•omf aotnallT = 25.50; but rule 4 gives 24.72. These examples show that none of the rales nsnaUy 
SiT0n are reliable. The following one by the writer, is snfflclently exact for ordinary pnrpoaes; Ml 
Mag iasrrer probably more than 1 part in 1000. When D la not more than 6 ttaass as long as 4, 






If D ezeeeda 5 times if, then in- fr 

stead of dividing (D — d^ by 8.8, div i^ by Si m 

the number in ibis table. o 

The following rule originated with Mr. M. 
Arnold Pears, of New South Wales, Australia, 



s;«S«««SSm68SSS!:fl« 

stetSkeisteCaieiSeisieiee^ee 



and was by him kindly communicated to the author. Although not more accu* 
rate than our own, it is much neater. 

3.1416 d + 2(D — d) — d(D — d) 



Circumf 



V<(D -f d) X (D + 2d) 



The following table of senii»elllptle arcs was prepwvd by oar niik 

To nse this table, div the height or rise of the are, by its span or ehord. The qnet 
will be the height of an are whose span is 1. Find this quot in the oolnmn of 
heights ; and Uke out the oorresponding number ft*om the ool. of lengths. Halt this 
number by the actoal span. The prod will be thereqd lenRth. 

When the height becomes .500 of the chord fas at the end of the table) the ellipse 
beeomee a eirole. When the height exceeds .500 of the chord, as in a b e, then take 
a o, or half the ehord, as the rise ; and dir this rise by the long diam 6 d, for the 
qnot to be looked ror in the ool of heights ; and to be mult by long diam. We tfens 
get the aro had, which is evidently equal to a 6 c 




190 



MENSUIUTIO>. 



TABI.E OF I^ENOTHB OF 8EMI.EI«I«IPTI€ ABCB. 


ftnrlglnal4 


Height 


Lengtl^a 


Hdght 


Lengths 


Height 


Length v 


Height 


Lengths 


•I'SlAn. 


spanxby 


. •A'lpftn. 


■pan X by 


•fr span. 


■pan X by 


4- ■pan. 


■pan X by 


JOOb 


1.000 


.130 


1.079 


.266 


1.219 


.880 


1.390 


M 


1.001 


.136 


1.084 


.260 


1.226 


.385 


1.897 


.015 


1.002 


.140 


1.089 . 


.266 


1.233 


.890 


1.404 


.02 


1.003 


.145 


1.094 


^0 


1.239 


.396 


1.412 


026 


1.004 


.160 


1.099 


.276 


1.245 


.400 


1.419 


.03 


1.006 


.166 


1.104 


.280 


1.262 


.406 


1.425 


.036 


1.008 


.160 


1.109 


.286 


1.259 


AIO 


1.434 


X)4 


1.011 


.166 


1.116 


.290 


1.265 


.416 


1.441 


X)46 


1.014 


.170 


1.120 


.295 


1.272 


.420 


1.44P 


.06 


1.017 


.176 


1.126 


.300 


1.279 


.425 


1.456 


.066 


1.020 


.180 


1.131 


.306 


1.286 


.430 


1.464 


.06 


1.023 


.186 


1.137 


.310 


1.292 


.436 


1.471 


.066 


1.026 


.190 


1.142 


.316 


1.298 


.440 


1.47» 


..07 


1.029 


.196 


1147 


.320 


1.306 


.446 


1.486 


.076 


1.032 


.200 


1.153 


.326 


1.312 


.460 


1.494 


.08 


1.036 


.206 


1.169 


.330 


1.319 


.455 


1.50i 


.086 


1.039 


.210 


1.166 


JXif> 


1.325 


.460 


1.509 


.09 


1.043 


.216 


1.171 


.340 


1.332 


.465 


1.517 


.096 


1.046 


.220 


1.177 


•346 


1.339 


.470 


1.624 


.100 


1.061 


.226 


1.183 


.350 


1.346 


476 


1.582 


.105 


1.066 


.230 


1.189 


.365 


1.368 


.480 


1.540 


aio 


1.069. 


.236 


1.196 


.360 


1.361 


.486 


1.547 


J16 


1.064 


.240 


1.202 


.365 


1.368 


.490 


1.556 


.120 


1.069 


.1?45 


1.207 


.370 


1.376 


.495 


1.568 


.126 


1.074 


.260 


1.213 


.375 


1.382 


.500 


1.571 



Area of an ellipse = prod of dlam^ X .78M. Bz. D = lO ; d = «. Then 10 X 6 X .T§6« 
c 47.124 area. The area of an elUpiie la a mean proportional between the areae of two cirelae, d«* 
■eribed on its two dlama ; therefore it may be found by mult together the areaii of.thote two -eirolaa ^ 
and taking the aq rt of the prod. The area of ah ellipse ii therefore always greater than that of th« 
eircolar seotion of the cylinder f^om which it may be supposed to be derived. 

Dlam of circ of same area as a given ellipse = i^Long diam x ahort diaml 
To find tbe area of an elliptic segment wbose iNwe is paral. 

lei to eitlier dlam. DIt the height of the segment, bT that diam of which wid height 
!■ a part. From the table of circular segments take out the tabular area opposite the qnot. If nil 
together this area, the long diam, and the short diam. 

To drair an ellipse. Having its long and short dtaas a b and e d, Pig. 4. 

BoLB 1. From either end of the short 
diam., as c, lay off the dists. ef, ef, each 
equal to « a, or to one-half of the loug diam. 
The points/, /' are the foci of the ellipse. 
• Prepare a string, fn/.orfgf. with a loop 
at each end ; the total length of string from 
end to end of loop, being equal to the long 
diam. Place pins at /and/'; and placing 
the lloops over them, trace the curve by a 
pencil, which in every position, as at n, org, 
keeps the string/' n /, or /' gf stretched all 
the time. 

Note. Owing to the diflDoulty of keeping 
the string equally stretched, this method is 
not as satisfactory as the following. 

Bulb 2. On the edge of a strip of paper 
«0 «, mark w I equal to half the short diam. ; 
and IS a equal half the long diam. Then in 
whatever position this strip be placed, keep- 
ing I on the long diam., and s on the short 
diam., te will mark a point in the eircumf. of the ellipse. We may thna obtain at many each polnu 
as we please ; and then draw the curve through them by hand. 

Bdlb 8. From the two foci / and /', Fig. 4, with a rad. equal to any part whatever of the long 
diam. describe 4 short arcs, o o o o; also with a rad. equal to the remaining part of the lon^ diam., 
describe 4 other arcs, iiii. The intersections of these four pairs of ares, will give four points in tha 
eircumf. In this manner any number of such pointt may be found, and the curve be drawn by hand. 

To draw a tanarent 1 1, at any point n of an ellipse. Draw n / 

and n /', to the foci ; bisect the angle / n /' by the line xp ; draw < n ( at right angles to xp. 

To draw a Joint n p^ of an elliptic arcli, f^om any point a, im 

tbe arcb. Proceed as ic the foregoing rule for a tangent, only omitting (I; np will be 
required joiac 




I?ig-4. 



IfBNSUBATHnr. 



191 




To draw an OTal, or felse ellipse. 

When only tbo long diam a b It given, tbe fbllowing 
will give agreeable caires, of wbicb tbe span a h wiU 
not exceed abont tbree times tbe riie e o. On a & d»> 
■eribe two Intersecting circle* of any rad; through 
their Interseetiona t, 9, draw ay; make • g and r • 
each eqnal to tbe dtam of one of the eirelea. Tbrongb 
the center* of tbe circles, draw «f,*h,gd,gU FroB 
edeioribeA<y; and from y dMoribe d o I. 



"Wiieii the span, «nn^ and tlio 
rise* s t, are boUi yliren. 

Make any f w and mr, eqnal to each otbei;^ 
but each less than t ». Draw r w; and throngn 
its center o draw tbe perp toy. Draw y r «• 
Make n « equal mr, and draw tfxb. From sand 
r describe n e and m m; and fh>m y describ* 
ate. By making « d eaaal to « y, we obtain 
the center Ibr tbe other side of the oral. 

Tbe beaaty of tbe canre will depend npon 
what portion of I « is taken for m r and t m. 
When OB oval le verf flat, more than three cen- 
ters are reqnired for drawing a gracefbl enrre ; 
bat the flnflng of these centers Is qaite aa tron* 
bleseme as to draw tbe oorrect ellipse. 



€tai the §:!▼«>■ line, a 9, to draw a 
cyma reeta^ aes. 

Find the eenter e, of a ». From «, e, and $, with one-half 
ef • • aa rad, draw the fonr small arcs ato. o. The inter* 
o, «, are the oenters Ibr drawing the oyma, with 
I ra4. By rerersing the position of the ares, w« 
oreyee, 4 </. 





192 



MENSURATION. 



THB PAIIABOI.A* 





The eommoii or eonle iiarabola, 

o b e. Fif 1, is a onrre formed by oatting • oone in a dlreetlon b a, parallel to ita lida. 
•arred line obe itself is called theptrimt«r of the parabola ; the line o e is called ita bcwe ; ft • iti 
height or axta ; b its apex or vertex i any line e s, or o a. Fig S, drawn from theonrve, to, and at right 
angles to, the axis, is an ordinate ; and the part s 6, or a i, of the axis, between the ordinate and the 
apax b, is an abscissa. The /ooms of a parabola is that point in the axis, where the abaoisaa 6 «, is 
oqual to one-half of the ord e ». The dist from apex to focus, called the focal diet, is found thus: 
square auy oid, as o a; div this sauare br the abscissa i a of that ord; diy the quot by 4. The 

Cature of the parabola is such that its absoiBsas, as 6 s, 6 a, fto, are to each other as, or in proportion 
», the sanares of their respective ords s s, o a, Ac; that is, as i s : ba : : ss* :o<i>;orbs:ss>::b«: 
• a* . If the square of any ord be divided by iu abscissa, the qnot will ho a constant qnantltj ; that 
Is, it wHl bo equal to the sqoaro of any other ord dlTlded by Its abscissa. This qnot or oonstantqaan* 
tfty Is also equal to a eertsln quantity oallod the pmrameter of the parabola. Thersfbra tho p^'^nwtsr 
may be found by squaring s s, or e a, (one>ha^ of the base,) and dividing said square bv tho height 
i s. or b a, as the case may be. If the square of any ord be divided by tho panoMtar, tbt qnot wff 
he the abscissa of that ord. 

To And (lio lenyth of a parabolle enrre. 

The approximate rule given by various pocket-books, is as IbUows t 

Length — 2 X V(H '^>^e)a + \% Umes the (Height^ 

(g Where the height does not exceed 1-lOth of the base, thls'mle may, for praetlMi 

purposes, be called exact. With ht = )^ base, it gives about H par oeat tos 
Bueh; ht s M base, about 3^ percent; htsbase, about 8K per coot; ht = 
%«tee the base, about 11% percent; ht= 10 X base, or more, about 15)t( per oeat 

The flillewlas \ij the writer U eo r reel 
within perhaps 1 part in aOO, in all eases ; and will 
therefore answer for many purposes. 

Let a d b. Fig S, orik a d. Fig 4, be the parabola. 
In whioh are given the base abvtndt and tte 
height c li or c a. Imagine the eonpleteflg ad bs, 
or » a 4< b, to be drawn ; and in sttAsr ease, aaanms 
Us loMi^ dlam a b to be the chord or base; and one- 
half the short diam, or e <i, to be the heightt of a 
circular arc. Find the length of this circular are, 
by means of the rule and table given for that pur* 
pose. Then div the chord or Immo a b, or n d of 
the parabola, by its height c d or e a. Look for 
the qnot in the column of bases in the following 
table, and take from the table the correspondiag 
multiplier. Mult the length of the eireolar aro by 
this ; the prod will be the length of are a d b, or 
n a cl, as the case may be. For bases of parabolas 
less than .05 of the hdght, or greater than lOtimea 
the height, the multiplier is 1, and is very approx> 
imate; or in other words, the parabola will be 
of almost exactly the same length as the eiroular 
are. 

To find the area of a |»arabola ta a n l^. 

Mult iU base m n, Fig 5, by its height a h ; and Uke %^^M of the prod. 
The area of any segment, as « b v, whose base tt v is parallel to as n, is 
found in the same way, using u « and s b, instead of iw i» and a b. 

To find the area of a parabolic aone, or fl^as- 

tam, as t>» n t« V. 

RuLx 1. First find by the preceding rule the area of the whole pambola 
m b n ; then that of the segment « b « ; and subtract the last mm the 
flmt. 

RuLK 1. From the cube of m n, take the eubo of « v; eall the difP %, 
From the square of m n, take the square of m « ; eall the dlff «. Div e bf 
«. Mult the quot by ^ds of the height • s. 





MENSURATION, 



193 



1 



Table lor I^enytlis off Parabolic Curves. See opp page. (Original.) 



Baa«. 


Mole 


BM6. 


Molt. 


Bue. 


Molt. 


, Base. 


Molt. 


.05 


1.000 


1.10 


.999 


2.15 


.949 


8.20 


.983 


.10 


1.001 


1.16 


.997 


2.20 


.951 


3.30 


.984 


J6 


1.002 


1.20 


.995 


2.25 


.954 


3.40 


.986 


.20 


1.004 


1.25 


.993 


2.30 


.956 


3.50 


.986 


.25 


1.006 


1.80 


.990 


2.S5 


.958 


3.60 


.987 


JSO 


1.007 


1.35 


.987 


2.40 


JMM) 


8.70 


.988 


JB6 


1.007 


1.40 


.984 


2.45 


.002 


3.80 


.989 


AO 


1.008 


1.45 


.980 


2.50 


.963 


3.90 


.990 


.45 


1.009 


1.50 


.977 


2.55 


.965 


4.00 


.991 


.60 


1.010 


1.55 


.974 


2.60 


.967 


4.25 


.992 


.65 


IMO 


1.60 


J>70 


2.65 


.969 


4.50 


.993 


jOO 


1.010 


1.65 


.966 


2.70 


.970 


4.75 


.994 


.66 


1.011 


1.70 


.963 


2.75 


.972 


6.00 


.996 


.70 


1.011 


1.75 


.960 


2.80 


.973 


5.25 


.996 


.76 


1.010 


1.80 


.957 


2.85 


.975 


6.50 


.997 


.80 


1.009 


1.85 


.953 


2.90 


.976 


6.76 


.908 


.85 


1.008 


1.90 


.950 


2.95 


.978 


6.00 


.998 


.00 


1.006 


1.95 


.946 


3.00 


.979 


7.00 


.999 


.96 


1.004 


2.00 


.942 


306 


.980 


8.00 


1.000 


1.00 


1.002 


2.05 


.944 


3.10 


.981 


10.00 


1.000 


1.05 


1.001 


2.10 


.946 


3.15 


.982 







To draw a parabola) having base o t and height « o. 

••«, Flc6. Make e I eqoal to the height «e. DraweCand 
• I; and dlride each ofthem into aoT number of equal parte; 
BmnberlDg them as in the Fig. Join 1,1; 2, 2 ; 3, 3, Ao ; 
then draw the oorve by hand. It will be obeenred that Um 
itttereeetions of the lines 1,1; 1, 3, &o, do not give pointi in 
the eurre ; but a portion of each of those lines forms a tan. 
gent to the eurre. By increasing the number of diri^iona 
on e < and « t, an almost perfect oorre is formed, scaroelj 
teqnlring to be tooohed up by hand. In practice it is best 
first to draw onlr the center portions of the two lines whioh 
•rasa eaeh other Just aboTO o ; and trom them to work down* 
ward; aetnally drawing oalj that small portion of eaeh 
low« Une, whioh is neoessary to indioate th« 



bo drawn 




Fifir.tt. 



Or the i»araboIa ma 

tbasx 

Let ft «, Fig T, be the base ; and a d the height. Draw th» 
leetangie hnine; dir each half of the base into an j nom. 
ber of equal parts, and number them ftom the center each 
vmT. DIt n h, and m e into the same number of equal parts ; 
■ad number them from the top, downward. From the points 
on b e draw rert lines ; and trom those at the sides draw lines 
to d. Then the interseetions of lines 1,1; 2. 3, ke, 
will form points in the parabola. As in the pre- 
esding ease. It is not necessary to draw the entire 
lines ; but merely portions of them, as shown be. 
teeeu d and c. 

Or a parabola may be drawn by first dlT the 
height a h. Fig 5, into any number of parts, either 
equal or unequal; and then ealoulating the ordi. 
aatea u»,Ao; thus, as the height a h : square of 
half base am : : any absciss b s : square of iu 
erd « «. Take the sq rt for ««. 

I. —When the height of a parabola is not 
ir than 1.10th part iu base, the eurre eoin- 
■o very eloeely with jlhat of a drcntar are, 
that in the preparation of drawings for suspen> 
rieo bridges. Ac., the eironlar are may be em. 
ployed ; or if no groat aoenraoy is veqd, the olrole 
■ay be need eren when the hMghfe la aa great •• 
«e^«igfath of the base. 

To dra^w a tangr^nt w v, TIk- 5, to a parabola, from any point v. 

Draw V » perp to axis a h ; prolong a h until b w equals s b. Join v> v. 

13 




194 



MENSURATION. 



a 




Tlie Cycloid, 

^^h i-the curve deacribed by a point a in the circumference of a circle, 
.^'d'ix^fonr^^^S^^o.uLn'S the clro.e.^roU^^^ 

d h cycloid. 

Tlie vertex of the cycloid is at e. 
Base, a 6, =s circumference of generat- 
ing circle a u 
=s diameter, cd, of generat- 
ing circleXir = 3.1416«i. 

Axis, or taeli^lit, cd=^an. 
lieuiTtli, oc6, = 4cd. 

I, a c 6 d = 3 X area of generating circle, o n 
= 3?^ = ca8 X 3ir = cci« X 2.3562. 
Center of sravity of surface at g. cg = t\ c d. Center of gravity oi 
cydoid (curved line a c 6) in axis c d at a point (as ») distant J c d ttom c. 

To draw a tangent, «o, from any point e in a cjrcloid; draw « » at right 
anTlM to the axScd; one d describe the generatingcircle dc<; join /c; from 
J draw CO parallS to / c. The cycloid is the curve of a uickest descent ; 

So thit a ESdy would fall from"^ h to c along the curvelm c, in less time than 
along the inclined plane 6 ic, or any other line. 

TKE REGVIiAB BOBIES. 

A revnlar body, or reffular polyhedron, is one which has all its 
dies, and its solid angles, resnectively similar and equal to each other. There 
'e but five such bodies, as follows : 



■ides 
are 



Name. 



Tetrahedron ......... 

Hexahedron or cube 

Octahedron 

Dodecahedron 

Icosahedron • 



Bounded by 



4 equilateral triangles. 
6 squares. 

8 equilateral triangles, 
12 " pentagons, 
|20 " triangles. 



Surface 

(—sum of surfaces 
of all the faces). 

Multiply the square 
of the length of 
one edge by 



1.7320 
6. 

3.4641 

20.6458 

8.6602 



Tolnme. 

Multiply the 

cube of the 

length of one 

edge by 



.1178 
1. 

.4714 
7.6681 
2.1817 



Ouldinus' Tbeorem. 

Fig. A. Fig. B. 

I 



To find the volume of any body <as the 

irregular mass a 6 c w. Fig A, or the rinft 
abom^ Fig B), generated by a complete 
or partial reyofution of any figure (as 
_ ahca) around one of its sides (as/ie, 

Fig A), or around any other axis (as 
a;v,FigB). 

volume =3 surface ahcaY. length 
of arc described by its center of grar^ 
ity G. 

If the revolution is complete, the arc 
described is = circumference = radius 
G* X 2ir = radius o G* X 6.283186 ; and 

Tolume =surface a6ea X radius 
oG*X 6.283186. 

If the revolution is incomplete, 

complete . incomplete . . circumference . mo 
revolution ' revolution * ' found as above * described 




* Measured perpendicularly to the axis of revolution. 



HEMBirRA.TION. 
PABA1.1.EI.OPIPEDS 



&r^^f^ 



nlt^Fig 1,Dhl£h)u 



iglM right iDgleB, each pair of 



;1*>> right 



1 nil 1(1 ildco eqoil rhombn , , 

loalled-'itaomb"; iba EJumbia prism. Fig 4; Ita lluiei, rbomJ 
loibolds. well pilr ot oppoilte bon aqosl, but not *11 ila Kwes eqi 



(rrm. Fig 3, 
UB, p 15?. Is 



^ ^WTVrxJvuJar dl 
'^ Cs tlig oppodi 



A piiBm ig aoy solid irhaM 

>Dd equal ; and whose iida 
art pwaUeiogTami, »a Flga G 
to 10. Cansequ«ntlT the for^ 
n faint pBrBllelopipeds are 
prlnns. A HgU prism is i>d« 
wh«e Bldu are perpeodic- 



bnn the cuds are equal, aod the anglea included bati 

eqnjd, the prJam la aaid to ' -"■ "- 

T«Inni« cf mMT prii 



ngnlu or Irr^ular. right or oblique) 

,^., lataDOe,p.totb<otheTend. 

— area of cfOM NCtioii perpeDdlculat ta tbe ddea x utnal length, aft, Figi 

H 8 X TfduiFi* of prnmld vhoae biae aod height are ^ those ol the prism. 



idlcnlar to Iti nlH*. 
ly pirallelog 



J Dumber nf sldi 



" lanale ; any piraUelognii 
1u^> 01 a reffiUar paljgo 



reffiUar paljgan of 



goflenKthaofporolWedgea, "S*""*- 

i~f + Ti + S~i + T^ "fe* of <!«™ section 

nDmberotauchedgea ^ ^SH^rf^ 



196 



MEKSURATION. 



fl 

■ 

# I 

dL g 



Fig. 10J4 




This rule may be used for aacertainins beforehand, the Quantity of earth to 
be removed from a "borrow pit." The irregular surface of the ground is first 
staked out in squares; (the tape-line being stretched horizontally ^ when meas* 

uring o£f their sides). These squares should be of such 
a suse that without material error each of them may be 
considered to be a plane surface, either horizontal or in- 
clined. The depth of the horizontal bottom of the pit 
being determined on, and the levels being taken at every 
^b corner of the squares, we Hre thereby furnished with the 
lengths of the four parallel vertical edges of each of the 
resulting Arnstums of earth. In Figs 10^ y may be sup- 
posed to represent one of these Arustums. 
If the frustdm is that of an irregular 4-sided, or polyg- 
onal prism, first divide its cross section perpendicular to \ts sides, into tri- 
angles, by lines drawn frpm any one of its angles, as a, Fisr 10^. Calculate the 
area of each of these triangles separately ; then consider the entire frustum to 
be made up of so many triangular ones; calculate the volume 
(•;\ of each of these by the preceding rule for triangular frustnms; 

and add them together, for the volume of the entire frustum. 

Tolnme of any frnstam of any prism. 

Or of a cylinder. Consider either end to be the base ; and find its 
area. Also fipd the center of gravity c of the other end, and the 
perpendictUar distance n c, from the base to said center of g^ravity. 

Then Volame of frnstam = area of base X»«, Fig 10^. 

The slant end, c, is an ellipse. Its area is greater than that of the circular end. 
Snrfaee of any prism. Figs 5 to 10, whether right or oblique, regular 
or irregular 

/ circumference measured s^ «-*„-i iA«»ti, >. A i »tt™ of the areas 
" Vperpendicular to the sides ^ *®^"" lengin, a <> j + of the two enda. 

CTIilHTBERS. 

. If A cylinder is any solid whose ends are 

^h^-^_^ jC ^ parallel, similar, and equal curved fignires ; 

and whose sections parallel to the ends 
are everywhere the same as the ends. 
Hence there are circular cylinders, ellip- 
P tic cylinders (or cylindroids) and many 
others ; but when not otherwise expressea, 
the circular one is understood. A right 
cylinder is one whose ends are perpen- 
dicular to its sides, as Fig. 11 ; when otner- 
Fig. 11. Fig. 12. wise, it is oblique, as Fig 12. If the ends 

of a right circular cylinder be cut so as to 
make it oblique, it becomes an elliptic one ; oecause then both its ends, and aJl 
sections parallel to them, are ellipses. An oblique circular cylinder seldom 
occurs ; it may be conceived of by imagining the two ends of Fig 12 to be circlet^ 
united by straight lines forming its curved sides. 
A cylinder is a prism having an infinite number of sides. 

Volume of any cylinder (whether circular or elliptic, Ac, right or obliqa^ 
= area of one end X perpendicular distance, j9, to the other end, 

-{rJZ^^^Zi^ X actual length, « 6. Figs U and 12. 

^ 3 X volume of a cone whose base and height are » those of the cylinder. 
Snrface of any cylinder (whether circular or elliptic, &c, right or oblioue) 

(circumference ^ g^m ^f ^^jje areas 

measured perpendicularly X actual length, o 6 1 + ^f the two ends 
to the sides, as at c o. Fig 12, f 

RIfirlit circular cylinder whose lieiirb^ " diameter. 

Volume = H X volume of inscribed sphere. 

Curved surface = surface of inscribed spltere. 

Area of one end == \ surface of inscribed sphere =«= \ curved surface. 

Entire surface = U X surface of inscribed sphere =« IJ X curved surfkee. 




CJONTENTB OP CTUNDBRS, OB PIPEa. 



197 



ContentB for one fi»ot tn lenstti, in Cub Ft, and in U. 8. Gallons of 

Ml oab ins, or 7.4806 Galls to a Cub Ft. A e«1» Rof water wei«lu aboat 62M lbs ; and a gallon 
altoat 6H IlM. IHaaw »• 8» or 10 Hmm m svMt* «iTe i, 9. or 100 times tbe (Mutant. 







For in. in 






For I ft in 






For 1 ft. im 






length. 






lengtH. 






length. 


Dlam. 


Dlam. 
in deoi- 




Diam. 
in 


Dlam. 
in deci- 




Diam. 


Dlam. 
in deci- 






in 


-•3 


• 

^ ■ 


-5 


*s 2 


^ v^^ 


^ 9 


Ins. 


malsof 


• H * 


o a 


Ins. 


mals of 


^a^ 


o a 


in 


mal* of 


8?*i 


a 




afoot. 


h 




afoot. 




ii 


Ins. 


afoot. 


^s^ 


5 . 






•§sS' 


5" 






■pg- 


=50 






•Ss^ 


^0 


t 




"3 


^n 


i 




«3 


5s 






«3 


*a 


^Yt 


.0206 


.0003* 


.0025 


% 


.5625 


.2485 


1.869 


19. 


1.683 


1.969 


14.73 


.0260 


.0005 


.0040 


7. 


.6833 


.2673 


1.999 


H 


1.626 


2.074 


16.61 


,-!i 


.0313 


.0008 


.0057 


' ^ 


.6042 


.2867 


2.146 


20. 


1.667 


2.182 


16.32 


.0366 


.0010 


.0078 


.6260 


.3068 


2.296 


34 


1.708 


2.292 


17.15 


».^ 


0417 


.0014 


.0102 


yi 


.6466 


.3276 


2.460 


21. 


1.750 


2.406 


17.99 


.0409 


.0017 


.0129 


8. 


.6667 


.3491 


2.611 


H 


1.792 


2.621 


16.86 


nM 


.0521 


.0021 


.0180 


§ 


.6876 


.3712 


2.777 


22. 


1.833 


2.640 


19.76 


.0673 


.0026 


.0193 


.7083 


.3941 


2.948 


H 


1.875 


2.761 


20.66 


4 


.0625 


.0031 


.0230 


% 


.7292 


.4176 


3.125 


23. 


1.917 


2.885 


21.68 


.0677 


.0036 


.0209 


9. 


.7500 


.4418 


3.306 


}4 


1.968 


3.012 


22.63 


is-fi 


.0729 


.0042 


.0312 


H 


.7708 


.4667 


8.491 


24. 


2.000 


3.142 


23.60 


.0781 


.0048 


.0359 


.7917 


.4922 


3.682 


25. 


2.083 


3.400 


25.60 


1. 


.0633 


.0065 


.0408 


74 


.8126 


.5185 


3.879 


26. 


2.167 


3.687 


27.66 


8 


.1042 


.0085 


.0638 


10. 


.8333 


.5464 


4.060 


27. 


2.260 


3.976 


29.74 


.1260 


.0123 


.0918 


i 


.8542 


.5730 


4.266 


26. 


2.333 


4.276 


31.90 


H 


.1458 


.0167 


.1240 


.8760 


.6018 


4.498 


29. 


2.417 


4.687 


34.31 


2. ^* 


.1667 


.0218 


.1632 


Z4 


.8968 


.6303 


4.716 


30. 


2.600 


4.009 


36.72 


/4 


.1876 


.0276 


.2066 


11. 


.9167 


.6600 


4.937 


31. 


2.683 


6.241 


39.21 




.2063 


.0841 


.2650 


H 


.9375 


.0903 


5.164 


32. 


2.667 


6.585 


41.78 


5i 


.2-292 


.0412 


.3085 


.9683 


.7213 


6.S96 


33. 


2.760 


6.940 


44.43 


a. ^* 


.2500 


.0491 


.3612 


7* 


.9792 


.7680 


5.638 


34. 


2.833 


6.306 


47.15 




.2708 


.0670 


.4300 


12. 


1 Foot. 


.7854 


5.876 


36. 


2.917 


6.681 


49.98 


.2917 


.0668 


.4906 


H 


1.042 


.6522 


6.376 


36. 


3.000 


7.060 


62.68 


5k 


.3125 


.0767 


.5738 


18.^ 


1.083 


.9216 


6.896 


37. 


3.068 


7.46T 


66.86 


i. * 


.3333 


.0873 


.6628 


u^ 


1.126 


.9940 


7.436 


36. 


3.167 


7.876 


68.92 




.8542 


.0986 


.7360 


1.167 


1.069 


7.997 


39. 


3.260 


8.206 


62.06 


.3750 


.1104 


.8263 


H 


1.208 


1.147 


8.678 


40. 


3AS3 


8.727 


65.28 


5i 


.9958 


.1231 


.9206 


15. 


1.250 


1.227 


9.180 


41. 


3.417 


9.168 


68.68 


5. ^* 


^167 


.1864 


1.020 


H 


1.292 


1.310 


9.801 


42. 


3.600 


9.621 


71.97 




.4375 


.1508 


U26 


16.^ 


1.383 


1.396 


10.44 


43. 


3.683 


10.085 


76.44 


.4583 


.1650 


1.234 


H 


1.375 


1.485 


11.11 


44. 


3.667 


10.659 


76.99 


X* 


.4792 


.1808 


1.340 


17. 


1.417 


1.576 


11.79 


46. 


8.760 


11.046 


82.62 


«. 


.5000 


.1903 


1.469 


u 


t.458 


1.670 


12.49 


46. 


3.833 


11.641 


86.33 




.5208 


.2131 


1.594 18.' " 


1.600 


1.767 


13.22 


47. 


3.917 


12.046 


90.13 


.5417 


.2804 


1.724 }4 


1.642 


1.867 


13.96 


48. 


4.000 


12.666 


94.00 



TaMo oontlniied, bat wtth tbe dlanui In feet. 





Gab. 


U.S. 


Dlam. 


Onb. 


U.S. 


DU. 


Gab. 


U.S. 


Dia. 


«ab. 


U.S. 


Feet. 


Feet. 


Gallfl. 


Feet. 


Feet. 


Oallfc 


Feet. 


Feet. 


Galls. 


Feet. 


Feet. 


Galla. 


4 


12^ 


04.0 


7 


S8.48 


287.9 


12 


113.1 


846.0 


24 


452.4 


8884 


1^ 


14.19 


106.1 




41.28 


808.8 


18 


132.7 


992.9 


25 


490.9 


8672 


xt 


10.90 


119U) 


23 


44.18 


330.5 


14 


153.9 


1152. 


26 


530.9 


3972 


/i 


17.72 


182.6 


& 


47.17 


852.9 


16 


176.7 


1822. 


27 


672.6 


4288 


% 


19.0S 


146.0 


8 


60.27 


876.0 


16 


201.1 


1604. 


28 


616.8 


4606 


W 


21.66 


161.9 


M 


66.75 


424.5 


17 


227.0 


1698. 


29 


660i» 


4941 


I4 


28.76 


177.7 


0^ 


63.62 


475.9 


18 


254.6 


1904. 


30 


706.9 


5288 


/• 


25.97 


1912 


K 


70.88 


580.2 


19 


283U$ 


2121. 


31 


764.8 


6646 


f 


2&27 


211A. 


10 


78.54 


687.5 


20 


814.2 


2850. 


32 


8012 


6018 


l^ 


80.68 


22BA 


K 


86.59 


647.7 


21 


346.4 


2591. 


33 


865.3 


6398 


xc 


88.18 


248^ 


11 


06.08 


710.9 


22 


380.1 


2844. 


84 


907.9 


6792 


% 


8&78 


287.7 


% 


108.87 


777.0 


28 


415.5 


8108. 


85 


962.1 


7197 



198 CONTENTS AND LININ08 OF WELI*. 

COSTENT8 AKD LIJriHeB OF VELIA. 

For lIuH WlBe u irul u IkaH In Ih. U-Ut. Ibr »>• n» JiU iC Unliil. Uli ml tbm onuM 
OM ka|f dT lh> inuH dim ; u< khU IMM In 4, Tkm, iH- Un gDl ill [u (loli tvA of d>pi£ id • 
vallfll r««tlB«w, llmUbautrniiaillauW*lbo«a«»a<U«lli*dtioar]A4fcfaK; nu»J/, A.Hi. 
Tbn t.Mi X t ~ n.aU anb Jill ngd tor UnHlft^Um. BHItattlK uoni llnlDf •rnUlu HliU 
ar pUiHiiH, BiU tbg laJmUr gguilu •hkhIH half U» ininr OiH. br 1. TllL thi HnMi tf 
IKH nlUMf Ht aKik HM gf «iplk ors will of II 6 diuD. wlU la LOn X I = I.IM. Ir tht nil li 





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ItV«r«bea u« named In ■ ' 



CYLINDRIC CNOULAB, ETC. 199 

CIBCUI^B CVLIITDBIC UNQITI^B. 

■ the enttlng plaoe dvea mot eat tbe baae. Flp l^ 14 

1 -m 



]ft J I perp u ildw, u z, '^ jr'n,al,nieai<Jili>iieChB>id«. 



Add arena *t ends If required. 

r«r area* of SAetlaiis perpendioulir to tbe ildM, see GIrelH. 

r«r areaa of aecUoaa oblfqu* to tbs ildea, •»• Tbu ElltpN. 

II. Wb«n the enttlns plane to>«taea tbe baae. Flgi A l« IX 






.^--* 





Talome FlgA-(^at 



X*ra*a<l«t otbau)- 



(«ta.ih« 



-«»WX«ii. 



FigD-H>">(>f (sIrcIayM X"" 
~ ^ TolDins of cfUiider c y m n. 

Kg A - foi X "in - o< X length of are imh ) |^. 
(^,, FisB-»,X-. 

nBgoU FlgC — (lift Xn>n + 00 X length of «jc dm* )— —. 

oflj) "- 



200 



PYRAMIDS AND COVES, 



PTH^ttlDft AND COITEB. 




4 5 

A pjrainid, Fin. 1, 2, 8. Is any solid which has, for Its base, a plane figure 
of any number of sides, ana, for its sides, plane triangles all terminating at one 
point d, called its apex, or top. When the base is a regular figure, the pyramid 
Is regular ; otherwise irr^uCar. 

A cone, Figs. 4 and 5, is a solid, of which th6 base is a curved figure; and 
which may be considered as made or generated by a line, of which one end is 
stationary at a certain point d, called the apex or top, while the line is being 
carried around the circumference of the base, which may be a circle, ellipse, 
or other curve. A cone may also be regarded as a pyramid with an infinite 
numoer of sides. 

The axis of a pyramid or cone. Is a straight line eZ o In Figs. 1, 2, 4 ; and diiA 
Fi^s. 8 and 5, from the apex e2, to the center of gravity of the base. When the 
aj^s is perpendicular to the base, as In Figs. 1, 2, 4, the solid is said to be a right 
one ; when otherwise, as Figs. 3. 5, an oblique one. When the word cone is used 
alone, the right circular cone. Fig. 4, is understood. If such a cone be cut, as at 
1 1, obliquely to its base, the new base 1 1 will be an ellipse; and the cone dtt 
becomes an oblique elliptic one. Fig. 6 will represent either an obUtiue <^ular 
eoiie, or an oblique elliptic one, according as its base Is a circle or an ellipse. 

V oliune or pyramtd or co£e, regular or Irregnlai^ right or obliqu«. 

Volume mm ^ «rea of base X perpendicular height d o. Figs. 1 to i. 

-» ^ volume of prism or cylinder having same area of base and 
same perpendicular height. 

— K volume of hemlsphJBre of same base and same height 

Or, a oone. hemisphere and cylinder, of the same base and same height, havt 
volumes as 1, 2 and 3. 

Area of anrlkec of sides of right regular pyramid or right dicular ooiM. 

Area — J^ circumference of base X slant height.*^ 

In the cone, this becomes I Add area of bass 







Area of sarfoce of oblique elliptic eone, dtt, 

Fig. 6i, cut from a rieht circular cone, dss. From the point 
c where the axis d o of the right circular cone cuts the elliptio 
base t L measure a perpendicular, r, in any direction, to the 
curved surface of the cone. Let v = the volume of oblique 
elliptio cone, dti; let a — the area of its elliptic base t (.and 
let A = the height d u measured perpendicularly to said nase. 
Then 

Carved snrlkiee = = . 

r r 

Add area of base if required 

No measurement, has been devised for the surface of an 
oblique circular cone. 



*In the pyramid, this slant height must be measured along the middle of one 
of the sides, and not along one of the edges. 



PYRAMIDS AND CONES. 201 



To And thm surfiwe of mat IrvcffiKlar p jramld. 

Whether right or oblique, each side must be calculated as a separate triangle (i 
p. 148); and we several areas added together. Add the area of base if required. 



FRUSTUMS OF PYRAMIDS AND CONES. 



Flff.0. Fig. 7. 

Frastam at pjnunld (Fig. 6) or of oono (Fig. 7) with haw aad Uff 
pnaUeL 

Tolmne (regular or irregular, right or oblique) 

my ^, perpendicular v- / area i area i / area v/ area \ 

— >* P^ height oo ^ ^of top •" ©f base t" V of top -^ of base/ 

^ w vr perpendicular w / area' i "«» i * ^ areaof aBection \ 

— X X *helght oo X V of top + of base + l^^^ to, and midway I 

>» ' between, base and top / 

»^ (for ffmstam of right or oblique circular cone only; Hee Fig. 7) 
« X "^SSS^ X M4M X (•<• 4 •»* + •« . o.) 



of frustum of righi fgiAjur pjmunid or ooue, with top And base paiallelt 
9|0k 6 and 7. 

J. /diemnferenoe _i oirouinlbrenoeX v^ dant • 
>^\ oftop T ofbaM y X iMigiitfC 

Aid MiM «f top and Inuo If nq«li«4. 



Im tlM finuitoaA of a vl|^t etreolar oono^ tUibMoaat 

"^ Vof top T^ of basej X hdght f f 
(ir * 8.1416) . Add areas of top and base !f reqafawd. 



of IwegiUsur or o1»liq«« pjnroiBld or ooim. Sorlhee •• 
■an of smrfiwes of sldsi, each of which must be treated as a trapeasoid. 

•In the frustum of the jpframld (fig 8), this slant height must be measured along 
of Ite MM (M at <s), Mid net along one of tha edgsib 



202 



PBI6HOID& 



PBIBHOIDB. 




Flff.L 




VtK.2. 



A prUnnoUl is sometimM d<iHwtl M AfBlid bttdng Ibr Hi ends two paralWI 
plane figures, connected by other plane flfiuns on which* and through every point 
of which, a straight line may be drawn nom one of tho two parallel ends to ^s 
other. These connecting planes msj bo parallelograms or not. and parallel to each 
other or not. 

Tbla doflnltlon iroiild Imolndo the cube and all other parallelopipeds; 
the prism : the cylinder (considered as a prism baring an infinite namber of sides); 
the pyramid and cone (in whieb one of the two parallel endl^ i« theonelbiminiftiio 
apex, is considered to be infinitely small), and their frnstams with top and boso 
parallel ; and the wedge. 

But the use of the term prlanaold is frequently restxietod to siz-eided aolidd, 
in which the two parallel ends are unequal quadrangles; and the connecting plane^ 
trapezoids; as in Figs. 1 and 2; and, by soma writers, to cases where the patalkl 
quadrangular ends are rtetatiffies. 

The following •'prlsmoldal fbrmnla** i^Uas to all tbo ftregolng •olidi^ 
and to others, as noted below. 

Let A — the area of one of the two parallal ends. 

a — <* ** the other of the two panUlel ends. 
M — « *< a cross section midway between, and panllil to^ Hm tm 

parallel ends. 
L — the peipendicnlar distance between tfao two psnlU < 

Then 



Tolmiae — L X 







^ L X mean area of enm section. 



The following six flgnrss repvstent a few of the irregular solids which ftlltBderlht 
aboye broad definition of '< prismoid,*' and to which the prismoidal formnla appUiC 
They may be regarded as one-chain lengths of raihroad cutttnga; a o being^the loogUv 
sr perpendicular (horiaontal) distance between the two parallel (Tertloal) ' 





WEDGES. 



203 



The prismoldal ft»rmii]» applies also to the qihere) hemiiphere, and 
ether qpE«rlcel segmeiite; also to any aeotlf joe each aeafroi^aiid onidbct ai the 





In which the ddee ad^ ae, or od, <<i^ are itraiffhii tM ttuj are onty when the 
•atttng plane ade paaaes Umugh ike apes or top a. Also to ih» cylliiidev 

when a plane paraUd to the tides passes through both ends; but not if the plane 
«s is obHquet as in the fig., though never erring more than 1 in 142. In tl&la last 
case we must imagine the plane to be extended until it cuts the side of the cylinder 
likewise extended ; and then by page 199 find the solidity of the uegnlathus formed. 
Then find the solidity of the snuUl nngnla above to, also thus formed, and subtract 
it fh>m the large one. 

This very extended applicability of the prismoidal formula was first discorered, 
and made known* hy KUwood Morris, a B., of Philadelphia, in 1840. 



WEDGES* 




m n m 



SI m m 

Fiff.]a 



m 



Fiff.n. 



b neaally defined to be a solid. Figs. 8 and OjjKenerated by a plane triangle, anei, 
moving; parallel to itself; In a straight line. This definition requires that the twe 
triamgnlar ends of the wedge should be parallel; but a wedge may be shaped as in 
ng. 10 or 11. We wouid therefore propose the following definition, which embraces 
sll the figs.; besides vuious modifications of them. A solid of five plane faces ; one 
sf which is a parallelogram abed, two opposite sides of which, as a e and h d, are 
onlted by means of two triangular foces aen, and frdm, to an edge or line « m, 
parallel to the other opposite sides ab and ed. The parallelogram abed maj be 
eitlier rectangular, or not ; the two triangular Ikces may be similar, or npt ; and the 
with r^ard to the other two fhces. The following rale appUss equally to all : 



SunoTleDgths 
— K X oftheSedges 



peiphtj^from 
edgetobaok 



width of 

back {abed^ 

massed neip to « it 



204 



lOENBURATIOV. 



SPHERES OR GLOBES. 

A Sphere 

Is a solid generated by the revolation of a semicircle around its diameter. E^ery 
point in the surface of a sphere is equidistant (h)m a certain point called the center. 
Any line passing entirely throns;h a sphere, and through its center, is called its axis, 
or diameter. Any circle described on tlie surface of a sphere, fh>m the center ol 
the sphere as the center of the circle, is called a great eirde of that sphere i in other 
words any entire circumference of a sphere is a great circl«f. A «phere has a greatei 
content or solidity 'than any other solid with the same amount of surface ^so that i| 
the riiape of a sphere be any way changed, its content will be reduced. The inter- 
section of a sphere with any plane is a circle. 



Tohune of sphere 

— J TT radios* 

— )^ TT diameter* 

, ^ circumference * 
■" •« zr5 



— 4.1888 

— 0.5236 



radius' 
diameter* 



-» 0.01689 circomferenoe* 



— 3^ diameter X area of surface 
"" ^ diameter X area of great circle 
«- % Tdlume of circumscribing cylinder 
^ 0.6236 Tolnme of circumscribing cube. 



ot avtrfiace of sphere 

— 4 TT radius* 

— w diameter* 
circumference* 



— 12.6664 radius* 

— 8.1416 diameter* 

•^ 0.8183 circumference* 



— diameter X circumference 
■- 4 X area of great circle 

^ area of circle whose diameter is equal to twloe diameter of 

— curved surfkce of circumscribing cylinder 
6 X volume 

diameter. 



Badlw of sphere 
s t 

= * f 



volume 



= O.e2036 'v^volna* 



= / 



Area of surface 

47r 



= ^.07968 X anaof Boxflwe 



Gireiinalbrenee of sphere 
=s \/6 TT* volume 

a« ^TT Area of surfisoe 

_^ area of snrikoe 
~^ diameter. 



=r '^/59.2176 VolWM 

=s ^8.1416 are* of ioifiMe 



MEKBUBATION. 



J_ 


j_ 


1 


1 


ill 




IMl 


,1» 


r.Bie 


S "!:?; 




T-« 




7.Ma 




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H iUJS 




n.f 


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auH 


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ill. 16 1 










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s 


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UENBUBATION. 











8PB 


EBEI 


1 


i 


1 


•1 


1 


f 


J 


ii 

ii 

1 


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i 

IS 

§ 


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-(OoRninmi,) 

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208 



■BOXENTS, STG., OF SFHSRIB. 



To find the solidity of a splierieal seviiieiit. 

RiTM 1. Bqaar* the radon, of its baie; multthla tqnarebjS; to 
the prod add the iquare of ita hole ht o « ; mult tke bud by the helghfe 
o « : and mult this last prod by .5286. 

Bulb S. Malt the diam ah ofth* 4)ker«byS; flrom the prod 
take twiee the height o « of the Mcmeat; mult the rem by the Mioare 
ef the height o « ; and malt thle prod br .&SS6. 

The ■oUdtty of a sphere being HAa that of Its draamwnibiiic ^Un- 
der, If we add to any solidity In the Ubie. Ita half, we obtain that 
of a cylinder of the same dlam as the sphere, and whose height 
equals ita dlam. 




To And the enrved sarftM^e off a ■ptaerleal seirneiit. 

RvLi 1. Mult the diam a b of the sphere fk«m whleh the segment is out, by S.141C; 
MBit the prod bT the height e « of the seg. Add area of base If reqd. Ban. Having the diam n f 
•f the seg, and ita height o «, the diam a 6 of the sphere may be found thus: Div the square of half 
the dlam n r, by Ita height o • ; to the qoot add the height o : Bulb 3. The eurvmi surf of either 
B segment, last Fig, or of a lone, (nest Fig,) bears the same proportion to the surf of the whole 
■phere, that the height of the aeg or tone bean to the diam of the sphere. Therefore, first find the 
snrf of the whole sphere, either by rule or from the preoeding table ; mult it by the height of the aeff 
or Bone ; dir the prod fatr diam of s|riiere. Bin^ S. Molt the oiroumf of the splierB by the height e • 

of the sag. 

To find tbe solidity of * spberieal cone. 

Add together the square of the rad • d, the square of rad o &, 
and H<1 of ^^^ square of the perp height «o; mult the earn by 
1.&706; and mult this prod by the height «•. • 

To find the carved snrflsee off a spiier- 

ical sone. 

BvLB 1. Mult together the diam m n of the sphere ; the height 
e of tbe sone, and the number S.U16. Or nee preoeding Rule t 
tor surf of segmenta. Bale S. Mult the etroamf of the sphere, by 
the bf^lghtof the zone. • 

To find the solidity off a hollow spher- 
ical shell. 

Take f^m the fbregolng table the loltditlee of two aphorae haTlBf 
the diams a &, and e <L Snbtraot the least fhmi the grMtMi. B«i« 
a c or » 4 U the Ihiokneie «r tha ahMtt. 




THE ElililPSOID, OR SPHEROID, 

Is a solid generated by the rerolution of an ellipse around either Ita long or ita short dlam. When 
around the long (or transverse) diam, as at a. Fig 1, it is an oblons" or pr< 
late spheroid; when around the short (or co^Jaffate) one, as at m, in Fig % 
it is oblate. 




Fiir.i. 






Flg.2. 



For the solidity in either case, mult the fixed diam or ^tU br the •quare 

of the revolving one ; and mult the prod by .5336. ^ 

—— ■ — — . . » ___^_^ 

*This rule applies, whether the zone includes the equator (as in our figure) or 
not, as in the earth's temperate zoues. 



PAAABOliOWa, 



THE PAHABOLOID, OH PARABOI.IC COKOID, 

r lU ■olldltj' mult the ires of Ita bue, bj batr lu belaht, re. Oi 

Pop tbe •olldlt^ af a (TiutaB, 

I. (bl mil ot wlloll ««r.r]J Hi IMUjiril IiMHIIMIwlM 



To and the anraMm ofK pBraboloM, 




To And IM sarlkee. 

1b« dirt a« rroBi tbe«at4r ortha oIhU w tb« cvrtUr ar Eb« iplndle. CkQ 

To Nnd tbe ■oIMIly of » mldfUe cone ofn elrnilsp Bplndlo, 



((—'?)«.•)-(••«—'■■■))"- 





Tvlnme- of ,),iei, rfng ia mads '< dlamBlflm, ooand 6t XI-MIWB. 
-__-.__ _ drcumfflivrMfl of bar ^ 1 Bum of Inner HOd out^r w a tjifau 



210 SPECIFIC GRATITT. 



SPEOmO GEAVm. 



1. The specific gravity, or relative density, D*. of a sabstancei 

is the ratio between the weight, W, of any given volume of that substance and 
the weight, A, of an equal volume of some substance adopted as a standard of 

w 

comparison. Or: D = -^. 

2» For ffaseous substances, the standard substance is air, at a temper- 
ature of 0° Cent. =s ^29 Fahr., with barometer at 760 millimeters = 29.922 incnes. 

3. For solids and liquids, the standard substance is distilled water, at its 
temperature (4^ Cent = 39.2° Fahr.) of maximum density. 

4. For all ordinary purposes of civil engineering, any clear fresh 
water, at any ordinary temperature, may be used. Even with water at 
SOP Cent., = 86^ Fahr., the result is only 4 parts in 1000 too great. 

5. When a body is immersed in water, the upward forc& or '* buoyancy*** 
exerted upon it by the water, or the **loss of weight " of the body, due to its 
immersion, is equal to the weight of the water displaced by the immersion of 
the body f ; or, if 

W = the weiffht of the body in air, 

u; = its weight in water, 

D ■= its relative density or specific gravity, 

A = the weight of water displaced ; 

then A =- W — ic ; and D = -r- — tT? • 

' A W — w 

6. Since the volume, V, of a body, of given weight, W, is inverselv aa iti 
density, or specific gravity, D ; the specific gravity is equal also to the ratio 
between the volume V, of an equal weight of the standard substance, to the 

volume, V, of the body in question ; or D = ^'. 

7. The specific gravities of substances heavier than water are ordi- 
narily determined by weighing a mass of the substance, first in air (obtain- 
ing its weight, W), and then when the mass is completely submerged in water 

W 

(obtaining its diminished weight, w). Then D = -^ , as in If 5. 

8. If the body Is lighter than water » it must be entirely immersed, 
and held down against its tendency to rise. Its weight, «>, in water, or ita 
upward tendency, is then a negative quantity, and means must beprovided for 
measuring it, as by making it act upward against the scale pan. We then have^ 
A = W — (— w) = W + M? ; or 

Loss due to immersion = weight of body in air, pltis its buoyancy. 

9. Or, first allow the body to float upon the water, and note the resulting di»- 
placeraent, t>, of water, as by the rise of its surface level in a prismatic vessel. 
Then immerse the body completely, and again note the displacement, V. Now 
V, the volume displaced by the body when floating, and V, tne volume displaced 
by the body when completely immersed, are proportional respectively to the 
weight, W, of the body, and to the weight, W — tr, of a mass of water of equal 

volume with the body. Hence D == = ^^. 

W — w V 

10. Or, attach to the light body, b, a heavier body, or sinker, S, of such den* 
sity and mass that both bodies together will sink in water. Let W be the 
weight of the light body, 6, in air ; Q the weight of both bodies in air, and q 
their combined weight in water. Then Q — ^ = the weight of a mass of water 
of equal volume with the two bodies, and Q — W =». the weight, S, of the sinker 
in air. By immersing the sinker alone, find the weight, fc, of water equal in 
volume to the sinker alone, — loss of weight in sinker, due to immersion. 
Then, for the weight, A, of water of equal volume with the light body, fr, or tor 

* Strictly speaking, " specific gravity " refers to weight, and " relative density »» 
to mcus (see Mechanics, Art. 14 a); but, as specific gravity and density 
numerically equal, they are often treated as identicaL 

t See Hydrostatics, Art. 18. 



1 



SFifiOIPIC GRAVITY. 211 

the low of weight of b, due to immersion, we have A = Q --0 — k ; and, for 
the specific gravity, D, of the light body, 6, we have D =- a_ ^ . =" ^_^ 
where to — the (unknown) buoyancy of b. 

11. A granular body, as a mass of saw-dust, gravel, sand, cement, etc., 
or a porous body, as a maas of wood, cinder, concrete, sandstone, etc., is a com- 
posite body, consisting partly of solid matter and partly of air. Thus, a cubic 
foot of quartz sand weighs about 100 fi>s.; while a cubic foot of quartz weighs 
about 165 lbs. 

12. The specific sraTltj of porous substances is usually taken 
as that of the composite mass of solid and air. Thus, a wood, weighing (with 
its contained air) 62.5 ttw. per cubic foot, or the same as water, is said to have a 
specific gravity of 1. The absorption or water, when such bodies are immersed 
forthe purpose of determining their specific gravities, may be prevented by a 
thin coat of varnish. 

13. The specific grravity of granular substances is sometimes taken 
as that of the solid part alone. Thus, Portland cements ordinarily weigh (in 
air) from 75 to 90 fts. per cubic foot, which would correspond to specific gravities 
of from 1.20 to 1.44 ; out the specific gravity of the solid portion ranges from 
8.00 to 3.25 ; and the latter figures are usually taken as representing the speciflo 
gravities. 

14. In determining the specific gravities of substances (such as cement) 
which are soluble in water or otherwise affected by it, the substances are 
weighed in some liquid (such as benzine, turpentine or alcohol) which will not 
affect them, instead of in water. The result, so obtained, must then be multi' 
plied by the ratio between the density of the liquid and that of water. 

15. The specific ifpavity of a liquid is most directly determined by 
weighing equal volumes of the liquid and of water. 

16. Or weigh, in the liquid, some body, whose weight, W, in air, and whose 
specific gravity, d, are known. Let u/ = its weight m the liquid. Then, for 
the specific gravity, D, of the liquid, we have 

d(W— «/) 
W:W — «/ = d:D; or D = -^— ^^^ — •'. 

17. Or, let the body, in f 16 (weighing W in air), weigh %o in water, and (as 
before) to' in the liquid in question. Then, since specific gravity of water =s 1, 
we have 

W — m;:W — u/^lrD; orD = 3~"^ » 

w — to 

18. The specific gravities of liquids are commonly obtained by observing the 
depth to which some standard instrument (called a hydrometer) sinks when 
allowed to float upon the surface of the liquid. The greater the depth, the less 
the specific gravity of the liquid. In Beaum4('s hydrometer tne depth 
of immersion is shown by a scale upon the instrument. The graduations of the 
scale are arbitrary. For liquids heavier than water, 0^ corresponds to a specific 
gravity of 1, and 76^ to a specific gravity of 2. For liquids lighter than water, 
10° correspond to a specific gravity of 1, and 60° to a specific gravity of 0.745. 

19. In Twaddell's hydrometer, used for liquids heavier than water, 

.- ,^ 6 X No. of degrees + 1,000 
specific gravity = -^ J — 

Thus, if the reading be 90°, 

,- 4* 5 X 90 -f 1,000 1,450 , ^„ 
specific gravity j^^^^-i- ^ ^^ 1.46. 

20. In Nicholson's hydrometer, largely used also for solids, the specific 
gravity is deduced from the weights required to produce a standard depth of 
immersion. It consists of a hollow metal float, trom which rises a thin but stiff 
▼ire carrying a shallow dish, which always remains above water. From the 
float is suspended a loaded dish, which, like the float, is always submerged. On 
tile wire supporting the upper dish is a standard mark, which, in observations, 
is alwavB brought to the surface of the water. The specific gravity is then deter- 
miiied by means of the weights carried in the two dishes respectively. 

21. The determination of the specific graYltles of ffaseous sub- 
■tanees requires the skill of expert chemists. 



212 



8PE0IFIC GRAVITY. 



Table of speelfle ^mvitiefl, and w«lirlita* 

In this table, the sp gr of air, and gases also, are oompared with that of watec 
instead of that of air ; which last is usual. 



Th« specific gravity of any substance Is « its weicht 
in fframs per enbie «eiitlmetre. 



»••••••• 



<« 
« 



Air, atmoapbario ; MfiO° Fkh, and ander tbe pnMve ef oat atmMph«r> or 

14.7 Afl per aq iaoh, weigh* j\j part as mooh aa water at 00° 

Aleobol, pure 

" of oommeroe 

" proof spirit ^ * ', 

▲ab, perfcotly dry. V.V.V.V.V.V.'.'aTe'iife. * 

1000 ft board meaaore weighs 1.748 tons. 

Aab, American white, dry " 

1000 ft board meaaare weigha 1.414 tona. 
Alabaater, fklaely ao ealled; bat reaUy MarUea 

" real; a eonpaot white plaster of Paria aTerage.. 

Alamlnlom 

Antimony, caat,'6.86 to 6.74 averace .. 

" natlTe •• .. 

Anibraoite. See Coal, below. 

Aaphaltom, 1 to 1.8 

Baaalt. See Limeatonea, qnarrled 

Bath Btone, Oolite. ..................................... 

lUamoth, oast. Alaonatlre 

gltamen, aolid. See Aaphaltom. 
rasa, (Copper and Zinc,) oast, 7.8 to 8.4 " 

" rolled «« 

Brooie. Copper 8 parte; Tin 1. (Gun metal.) 8.4 to 8.6 '• 

Brick, beat pressed 

" common hard 

** ■eft, inferior 

Brickwork. See Masonry. 

Boxwood, dry 

Oaloite, transparent.. , 

Carbonic Acid Oas. is IM times as beary aa air " .. 

Cement. (See T IS.) 

•• Portland, 8.00 to J.tft.-. » 

•• Natural, 2.75 to 8.00 

Chalk, S.'i te 3.8. Bee Limestones, quarried m 

Charcoal, of pines andoak«.~ 

Cherty, perfectly dry 

Chestnut, perfectly dry ......^ 

Goal. See also page S15. 

Anthracite, 1.8 to 1.7 

" piled loose 

Biinmlnons, 1.8 to 1.4 m««*.««m..... 

" piled loose ....M...M 

€oke 

** piled loose 

In ooUag, coals swell from 86 to 60 per sent. 

Copper, oast, 8.6 to 8.6 , 

" roUed, S.8to».0 

Crystal, pure Qnarti. See Quartz. 



.1 

4» 



.*••••••*...«•.•..• 



Cork. 

Diamond, 8.44 to 8.66 ; asaaUy8.61 to 3.66 

■arth ; common loam, perfectly dry, loose 

" •' " shaken 

" ** " moderately rammed.... 

*' slightly moist, looae. 

" more moiat, " • 

" •« ahaken 

" *' moderately packed • • 

" aa a aofl flewinr. mad 

" aaaaoftmud, well preaaed into a box........ 



il 


4< 




4< 




M 




«4 




M 




M 




M 




U 



Ether 

Blm, peribctly 4rr. 

1000 ft board measnre weicbs 1.803 teas. 

Bbeny, dry 

Emerald, 3.68 to 2.76 

Fat. 



.average. 



flint •• 

Feldspar, i.5tot.8 •* 

Qarnet,8.5to4.8; Preoions, 4.1 to 4.8 ** 

Qtaas, 8.6 to 8.46 « 

" oommon window .' *' 

" Mill viUe, Kew Jersey. Thiek flooring glass " 

Oranite, 8.66 to 8.88. See Limestooe. 160 to 180 " 



ATerage 
BpOr. 



.00188 

.798 

.884 

.916 
.768 

.61 

8.7 

8.81 

2.6 

6.70 

6.67 

1.4 
8.9 

8.1 

9.74 

8.1 
8.4 

8.6 



J6 
8.788 

.00187 

8.19 

2.87 
840 

0.67 
0.66 

1.60 

1.80 

1.00 



S.T 
&9 

.96 
8.68 



•••• •»••! 



.716 
.66 

1.28 
8.7 
.08 
9.6 

9.66 

9.W 
9.69 
946 
9.79 



ATerage 

Wtof a 

Cab Ft. 

Lbs. 



.6766 
4i>.48 
63.1 
67.8 
4T. 

88. 

US. 

14i. 
163. 
418. 
41«. 

87.8 
181. 
181. 
607. 

804. 
694. 
629. 
160. 
196. 
100. 

60 
169.9 



TSteSO 
60 to 66 

1B6» 
16 to 90 
48. 
4L 

nteVM 
4TtoM 

Tswaa 

«4toU 
CiJ6 



79 to 80 

89to 99 

90 to 100 

70 to 76 

66 to 68 

76 to 90 

00 to 100 

104 to 119 

UOtoUO 

44.6 

86. 

W.1 
0B. 

tot. 

10k 

106. 

167. 

160. 

t ITOw 



8PEOIFIO GBAVITY. 



213 



T»¥le of speelflc frnkvttlea, mnA welffbtfi— (Ooutiiiiifld.) 



The specific gravity of any anbatance is » ttm welfllt 
in grains per cubic eentiHietre. 



(I 
«< 



ftneiMt oommoa* t.68 to 2.76 

** In looM piles 

" Hornbtondlo 

'* " quarried, in loose piles. 

Oyponia, Plaster of Paris, 2.24 to 2.80 

** in irregular lamps " 

'* gronnd, loose, per straok Iraahel, 70 " 

M «• well shaken '* *' 80.... •• " 

'• " Oaloined, loose, per stniokbaVhVtt to ral *.'.*. II " 1! 

GtMnstone. trarr *>8 to 3.2 «' ,. 

'* " fnarried, in loose piles ** 

Oravel, abont the same as sand, which see. 

Gold, oast, ptixv, or 34 earau '* 

" native, pure, 19.3 to 10.34 '* .. 

" *' freqaentiy oontaining silrer, 15.6 to 19.8 *' 

" pure, hammered, 19.4 to 19.6. > " 

OnttaPeroha ** .. 

HomUende, blaok, 8.1 to 8.4 '* 

Hydrofm G«s, is 14)^ times lighter than air ; and 16 times lighter than 

o^gea average.. 

Hendoek, perfeotljdrr. " 

1000 reet board measure weighs .930 ton. 

Hlekorj, perfeotly dry. " 

1000 feet board measure weighs 1.971 tons. 
Inn, and steel. 
•• Pig and oast iron and cast steel 

•* Wvoaght iron and steel, and wire, 7.6 to 7.9 •..••.. 

Ivory ' 

lee, .911 to .922 

fiidiarobber '* 

Lignum vita, dry *< 

Lard " .. 

Lead, of eoBaMree,U.80ta 11. 4T; either rolled or east '• 

UmMtanee and Marbles, 3.4 to 2.M,U0 to 17&8 

" " •* ordinarily about 

** ** ** quarried in irregular fragments. 1 oub yard solid, 

makes abont 1.9 cab yds perfeotly loose : or about 
1^ yds piled. In this last oase. 571 of the pile 
is solid; aod the Nmaining .429 part of it is 

voids piled.. 

UmBt qafBk, ground, loose, per straok bushel 62 to 70 lbs 

•• •• " well shaken. •• »• ....80 " 

♦• " " thoroughly shaken, '* ...MH " 

ICahogaay, Spanish, dry*..... ....« «•.•... ...average. • 

** Honduras, dry " 

Ibpte, di7« ♦' .. 

MarMei, sea Limestones. 

Maaoiuy, of granite or limestones, well dressed throngheal. 

** *' " weU>scabbled mortar rubble. About 4 of the mass 

will be mortar 

- f ** wen-seabbled drr nibble 

M •< M roughly soabbled morur rabble. About H to Mi P^^ri 

will be mortar 

M M M rsntfily soabbled drv rabble 

▲t 156 lbs per eub n, a cub y trd weighs 1.868 tons ; and 14.46 oub ft, 
1 ton. 
Masouy of sandstone ; about H part less than the fbregolnf . 

*' briokwork, press e d briok, fine Joints average. . 

medlam quality •« 



(1 M <t 

•• " •* eoarse; infbrlor soft bricks " 

At 135 fl>s per eub ft, a oub yard weighs 1.607 tons; and 17.98 eub 



fl. 1 ton. 

IbraaiT.atSSOFah 

» 60° " 

•< tijo •« 

llka.2.75toS.l.... 

Mortar, hardened, 1.4 to 1.9k.. 
Mad, dry, close 

moderately pressed. 

fluid 



Average 
BpGr. 



3.69 

'2.8*" 
*2.'27' 



8. 



19.268 
19.32 

19.6 
.96 
3.35 



A 

.86 

7.2 
7.76 
1.82 
.92 
.98 
1.38 
.96 
11.88 
3.6 
2.7 



.86 

.66 
.79 



19.62 

13.58 

13.88 

2.93 

1.66 



Average 

Wt of a 

Cub Ft. 

Lbs. 



168. 

96. 
176. 
100. 
141.6 

8Z 

66. 

64. 

52 to 66 
187. 
107. 

1204. 

1206. 

1217. 
61.1 
203. 

.00531 
25. 

53. 



450. 
4T6ta4 

114. 

57.4 

58. 

83. 

59.3 
709.6 
164.4 
168. 



96. 
61. 

64. 
76. 
63. 
86. 
40. 

166. 

154. 
138. 

150. 
135. 



140. 
135. 
100. 



849. 
.846. 

8S6. 

183. 

103. 

80 to 110 
110 to 130 
104 to 120 



• Green timbers asually weigh from one-fifth to nearly one-half more than 

4fT;and ordinary building timbers when tolerably seasoned about one-sixth morethao perfectly dry 



214 



SPEOIFIO GRAVITY. 



Table of speelflc ffravitleB, and wetybUi— (Oontinaed.) 



The specific gravity of any sQbfltance is » its weiifllt 
in i^rams per cnbie centimetre. 



ATenge 
. Sp Or. 



Naphtlia 

Viirog«D Gas is about -^ part lighter than air 

Oak. live, perfeotly dry, .88 to 1.02* averafQ.. 

" r«d. blacli, 4o« " .. 

Oils, irhale; olive •• 

" oftarpentine " 

Oolites, or Boestones, 1.9 to 2.6 " 

Ozygeu Oas, a little more than JL part heavier thau air 

Petroleum 

Peat, dry, unpressed 

Pine, white, perfectly dry, .86 to .46* 

1000 ft board measure weighs .080 ton.* 

" yellow, Northern, .48 to .62 

1000 ft board measure weighs 1.276 tons.* 

• " Southern, .64 to .80 

1000 ft board mean u re weighs 1.674 tons.* 
Pine, heart of long-leafed Southern yellow, luueai. ... 

1000 ft board measure weighs 2.418 tens. 

Pitch 

Plaster of Paris ; see Gypsum. 

Powder, slightly shaken 

Porphyry, 2.66 to 2.8 

Platinum 21 to 22 

" native, in grains 16 to 19 

Qnarti, common, pure 2.64 to 2.67 

*' " finely pulveriied, loose 

** *' " " well shaken 

" " " " well packed 

" quarried, loose. One measure solid, makes full IK broken and 

piled 

Baby and Sapphire, 8.8 to 4.0b^ 

Bosin 

8alt...... 

Sand, pure quarts, perfectly dry, loose 

•* <• ** •* •* slightly shaken 

•« «« rammed, dry.... 

Natural sand consists of grains of differeat sixes, and weighs more, per 
unit of volume, than a sand sifted from it and having grains of 
uniform site. Sharp sand with very large and rery ■mall grains 

may weigh as much as < 

Sand is very retentive of moisture, and, when in large bulk, its natural 
moisture may diminish its weight from 6 to 10 per eent. 

** perfectly wet, voids full of water ->»- 

Sandstones, fit for building, drv, 2.1 to l.YS 131 to 171. 

'* quarried, and piled, 1 measure solid, makes about IH piled... 

Serpentines, good 2.5 to 2.66 

Bnow, fresh fallen 

** moistened, and compacted by rain... 

Sycamore, perfectly dry. 

1000 ft board measure weighs 1.S76 tons. 

Shales, red or black 2.4 to 2.8 average.. 

** quarried, in piles " .. 

Slate t.Tto2.9 • ** .. 

Silver " .. 

Soapstone, or Stea|ite 2.66 to 2.8 *' .. 

Steel, 7. T to 7.9. The heaviest oon tains least earbon " .. 

Steel is not heavier than the iron from which it is made; onless the 
iron had impurities which were expelled daring its oonversion into 
steel. 

Svlphur ...., •.••.....•...•«••■■.•.....•..••..••..... average.. 

Spruce, perftiotly drr. • " .« 

1000 ft board raeasore weighs .990 ton. 

Spelter, or Zinc 6.8 to 7.2. 

Sapphire; and Ruby, 3.8 tQ i...« 

Tallow " 

Tar " 

Trap, compact, 2.8 to 3.2 ** 

" quarried; in piles " 

Topaz. 8.46 to 8.66 " 






.95 

.77 



.92 
.87 
2.2 

.00186 
.678 
•"•'-'•■ 

.65 

.72 
1.04 
1.16 

1. 

2.78 
21.6 
17.5 

8.66 



S.9 
1.1 



M 
U 



t.41 

• • • • 

S.6 



•••••• 



.60 
S.6 



2.8 
10.6 
S.TS 
7.66 



1. 
.4 

7.00 
8.9 
.94 
1. 
8. 

*S!65* 



Average 

Wt of a 

Cob Ft. 

Lbs. 



6X.9 

.0741 
50.3 
48. 
82 to4B 
57.3 
54.8 
137. 
.0648 

54.8 
20 to SO 
25. 

34.8 

45. 

66. 

Tl.T 

82.8 
170L 
1343. 

185. 

90. 

105. 

112. 

94. 

88.t 

60 to 70 

90 to 108 

92 to 110 

100 to 180 



117. 



U8 to ISO 
111. 

88. 
182. 

6 to IS 
16 to 50 

87. 

161. 
92. 
17i. 
656. 
17QL 
480. 



1S5. 



487.6 

68.8 
02.4 

187. 
107. 



* Green timbers usually weigh from one-fifth to nearly ODe-half more than 

dry ; and ordinary building timbers when tulerably seasoned about cue-sixth more than perfectly dry. 



WEIGHT OF COAL. 



215 



Table of apeclfle gravities, and weiffbta— (Continued.) 



The specific gravity of any substance is » its welgrli^ 
In yrams per eubie eentimetre. 



Tin, oast, 7.2 to 7.5 arerage. 

Turf, or Peal, dry, unpreaaed 

Water. Sm pagA 3*i6. 

Wax. bees average. 

Wine*. .993 to 1.04 •» , 

WalDOt, blaok, perfectly dry. " . 

1000 ft board measnre weighs 1.414 tons. 

Zlno, or Spelter, 6.8 to 7.2.... < «« . 

Zirooo, 4.0 to 4.9 ** . 



Average 

8pOr. 



7.35 



OQR 
•wo 

.61 

7.00 
4.45 



Averace 

Wt of a 

Cub Ft. 

Lbs. 



459. 

90 to 80 
68.417 

eo.5 

63.8 

38. 

4S7.6 



S|Miee oeenpied by eoal. In cubic feet per ton of 2240 pounds. 

PennsylTanla Anthracite. 



Hard white ash* 

Free-burning white ash *. 
Shamokin * , 



Schuylkill white ash *. 
" red " *. 



Lykens Valley * 

Wyoming free-bumingf * 

Lehigh t 

Lehigh ; Reading C. & I. Co. *... 
Lehigh : f Lump, 40.5 ; cupola, 40 



Bro- 
ken. 


Egg. 


Stove. 


Nut. 


Pea. 


Buck- 
wheat. 


f 


38.6 


39.2 


39.8 


40.5 


41.1 


' 39.4 


39.6 


39.6 


39.6 


89.8 


39.8 




39.0 


39.6 


40.2 


40.8 


41.6 


' 39.6 


39.6 


39.6 


41.2 


41.9 


42.4 




39.3 


39.9 


40.5 


41.2 


41.9 


39.0 


39.9 


42.6 


45.7 


46.5 


47.7 




39.6 


40.3 


40.9 


41.6 


42.3 




40.0 


40.5 


41.1 


41.7 


42.3 


{44.2 


44.8 


45.2 


45.7 


46.2 


46.7 


443 


44.3 


45.0 


46.1 


46.5 




40.0 


39.8 


39.4 






39.4 


38.8 


38.5 


38.4 


42.1 


41.4 


38.5 


38.8 


40.1 


40.3 


40.3 


40.5 


0.3; du 


Lst, 39.] 


.• 









Aver- 
age. 

39.8 
39.6 
40.2 
40.7 
40.6 
43.6 
40.9 
41.1 
45.7 
45.1 
39.7 
40.0 
39.7 



3itaininoas« 



From Coxe Bros. & Co. f 



Pittsburg 48.2 

Erie 46.6 

Hocking Valley 45.4 

Ohio Cannel 45.5 

Indiana Block 51.1 

Dlinols 47.4 



From Jour. XJ. S. Ass'n Charcoal Iron Workers. 
Vol. Ill, 1882.g 



Pittsburg 47.1 

Cumberland, max 42.3 

min 41.2 

Blossburg, Pa 42.2 



Clover Hill, Va 49.0 

Richmond, Va. 

(Midlothian) 41.0 

Caunelton, Ind ,....47.0 

Pictou,N. S 45.0 

Sydney, Cape Bretou.47.0 



Logarithm. 
1 cubic foot per ton of 2240 pounds = 

0.89286 cubic foot per ton of 2000 pounds 1.950 7820 

2240 (exact) pounds per cubic foot 3.350 2480 

1 cubic foot per ton of 2000 pounds = 

1.12 (exact) cubic reet per ton of 2240 pounds 0.049 2180 

aOOO (exact) pounds per cubic foot .3.801 0800 

1 pound per cubic foot = 

2240 (exact) cubic feet per ton of 2240 pounds 8.850 2480 

2000 " " " 2000 " 3.301 0300 



•From Edwin F. Smith, Sup't A Eng'r, Canal Div., Phila. and Reading R. R. 

fFrom very oarefiil weighings in the Chicago yards of Coxe Bros. & Co. 
Kote the irregular variation with size of anthracite In Coxe Bros.' figures. 

^Quoted from ITie Mining Record. On the authority of *• many years' experi- 
ence" of "a prominent retail dealer in Philadelphia," the Journal gives also 
figures requiring from 4 to 13 per cent, less volume per ton than those here 
quoted from the Journal and from other authorities. 



216 WEIGHTS AND KEASITKE8. 

WEIGHTS AND MEASURES. 

United States and Brttisb measures of lengrtli and weiirbt» 

of the same denomination, may, /or all ordinarp pttrposeSf be ooncidered as equal ; 
but the liquid and dry measures of the same denomination differ widely 
in the two countries. Ttaie standard measure of leng^tb of both coun- 
tries is theoretically that of a pendulum vibratiiig seconds at the level of the 
sea, in the latitude of Loudon, in a vacuum, with Fahrenheit's thermometer at 
629. The length of such a pendulum is supposed to be divided into S9.1393 
equal parts, called inches ; and 36 of these inches were adopted as the standard 
yard of both countries. But the Parliamentary standard having been destroyed 
by fire, in 1834, it was found to be impossible to restore it by measarement of a 
pendulum. The present British Imperial yard, as determined, at a temperature 
of 629 Fahrenheit, by the standard preserved in the Houses of Parliament, is 
the standard of the United States Coast and Geodetic Survey, and Is recognized 
as standard throughout the country and by the Departments of the Govern- 
ment, although not so declared by Act of Congress. The yard between the 27th 
and 63d inches of a scale made for the U. S. Coast Survey by Troughton, of Lon- 
don, in 1814, is found to be of this standard length when at a temperature of 
59^.62 Fahrenheit : but at 629 is too long by 0.00083 inch, or about 1 part in 43373, 
or 1.46 inch per mile, or 0.0277 inch in 100 feet 

The Coast Survev now uses, for purposes of comparison, two measures pre- 
sented by the British Government in 1855, as copies of the Imperial fltandsrd, 
namely : 

** Bronze standard, Ko. 11 ;" of standard length at 62^.25 Fahr. 
" Malleable iron standard, No. 57 ;" " " " 62«>.io " 

See Appendix No. 12, Beport of U. S. Coast and Geodetic Survey for 1877. 
Tbe legral standard of ireielit of the United States is the Troy 

pound of tbe Mint at Philadelphia. This standard, containing 5760 

Sains, is an exact copy of the Imperial Troy pound of Grea* 
ritain. The avoirdupois or commercial pound of the United States, con- 
taining 7000 grains, and derived from the standard Troy pound of the Mint, is 
found to agree within one thousandth of a erain with the British avoirdu|M>fs 
pound. The U. S. Coast Survey therefore declares the weights of the two ooun- 
tries identical. 

Tlie Ton. In Revised Statutes of the United States, 2d Edition, 1878, Title 
XXXiy, Collection of Duties upon Imports, Chapter Six. Appraisal, says : 

"Sec. 2951. Wherever the word 'ton' is used in this cnapter, in reference to 
weight, it shall be construed as meaning twenty-hundredweight, each hundred- 
weight being one hundred and twelve pounds avoirdupois." 

This appears to be the only U. S. Government regulation on the subject. 

The ton of 2240 ft>s (often called a sross ton or Ions ton) is commonlj 
used in buying and selling iron ore, pig iron, steel rails and other manufactured 
iron and steel. . Coke and many other articles are bought and sold by the net 
ton or sliort ton of 2000 lbs. The bloom ton had 2464 ftis, = 2240 fira -^ 2 
hundredweight of 112 S>s each ; and the pig iron ton had 2268 fi>s, == 2240 lbs + a 
"sandage" of '28 fcs, or one "quarter," to allow for sand adhering to the pigs, 
but some furnace men allowed only 14 lbs. In electric traction work the ton 
means 2000 lbs. 

As a measure, the ton, or tun, is defined as 252 gallons, as 40 cubic feet of 
round or rough timber or in ship measurement, or as 60 feet of hewn timber. 252 
U. S. gallons of water weigh about 2100 Ha ; 252 Imperial gallons about 2500 lbs ; 
SO cub ft yellow pine about 2500 Sts. 

Tbe metric system * was legalised in the United States in 

* The metric system, as compared with the English, baH much the same advantagea 
and disadvantages that our American decimal coinage has in comparison witiii the 
English monetary system of pounds, shillings and pence. It will enormously facili- 
tate all calculations, but, like all other improvemeute, it will necessarily eause some 
inconvenience while the cliange is being made. The metric system has also tMa ftir- 
ther and very great advantage, that it bids fair to become univei-sal among of viliaeo 
rations. 



WEIGHTS AND MEASURES. 217 

1866, but hM not been made ot)llgfttorT. The gorernment has since ftirnished 
very exact metnc standards to the several States. The use of the metric system 
has been permitted In Great Britain, beginning with August 6, 1897. and in 
Ruflsia, beginning with 1900. I to use is now at least permissive in most civil- 
ised nations. 

Tlie laetrle nnlt of lenytb is tlie metre, er nueter, which waa 

fntended to be one ten-millionth I j of the earth's quadrant, f. c, of 

Ihat portion of a meridian embraced between either pole and the equator. This 
lengtn was measured, and a set of metrical standards of weight and measure 
were prepared in accordance with the result, and deposited among the archives 
«f France at Paris (MHre des Archives.. Kilogramme des Archives, etc.). It has 
since been discovered that errors occurred in the calculations for ascertaining 
the length of the quadrant ; but the standards nevertheless remain as originally 
preparM. 

Tlie metric measures ef surface and of capacitv^ are the squares 
and cubes of the meter and of ito (decimal) fractions and multiples. 

Tlie metric unit of welarlit is tlie grramme or grram, which is 
the weight of a milliliter or cubic centimeter * of pure water at its tempera* 
tore of maximum density, about A.5^ Gentisrade or 40^ Fahrenheit. 

By the concurrent action of the principal governments of the world, an In- 
temational Bureau of Weiyiits and MeasuriMi has been estab- 
Ushed, with its seat near Paris. It has prepared two ingots of pure platinum- 
ixidium, from one of which a number of standard kilograms (1000 grams) havf 
been made, and from the other a number of standard meter bars, both derived 
from the standards of the Archives of France. Of these copies, certain ones 
were selected as international standards, and the others were distributed to the 
different governments. Those sent to tne United States are in the keeping of 
the U. 8. Coast Survey. 

The detennination of the ei|niTalent of tbe meter in Eng^Iisii 
measure is a very difficult matter. The standard meter is measured from end 
l» sfuf of %pkUiiuan bar and at the freexbtng point ; whereas the standard yard is 
measured hehown two lines drawn on a silver seale inlaid in a brmize bar. and ai 
^aP FiihrenheU. Tbe United States Ooast Surweyf adopts, as the 
length of the meter at 62° Fahrenheit, the value determined by Capt. A. R. 
Clarke and Col. Sir Henry James, at the office of the British Ordnance Survey, 
in 1866, vis. : S9.37(M82 inches (= 8.2808666 + feet « 1.0986222 + yards) ; but the 
lawftil equiwaient, established by Congress, is 39.87 inches (=t 3.28083 feet 
= 1.098611 yards). This value is as accurate as any that can be deduced from 
existing data. 

Tbe ffram Weislis, by Prof. W. H. Miller's determination,! 15.43234874 

Sains. An examination made at the International Bureau of Weights and 
easures in 1884 makes it 15.43236639 grains. The leeal value in the United 
States is 15.432 grains. 



• 1 centimeter =» r^ meter = 0.3937 inch. 1 milliliter {^^ liter) or cubis centi- 
meter =3 0.061 + cubic inches, 
t Anpendix No. 22 to report of 1876, page 6. 
X Philosophical Transactions, 1866, pp. &3y ets. 



218 rOEBIGN COINS. 

Approximate Talses of Foreign Coins* in U. S. Honey. 
The references 0, ^, ^ and *) are to foot-notes on next page. 

From Circular of U. S. Treasury Department, Bureau of the Mint, Jan. 1, 1887; 
from " Question Mon6taire," by H. Costes, Paris, 1884; and from our 10th edition. 

Argentine Repub.— Peso = 100 Centavos, 96.5 ots.** Argentino = 5 Pesos, $4.82. 

Austria.— Florin = 100 Kieutzer,47.7 cts.,2 3o.9 cts.s Ducat, $2.29. Maria Theresa 
Thaler, or Levantin, 1780, $1.00.2 Rix Thaler, 97 cts.* Souverain, $3.57.* 

Belgium.i— Franc = 100 centimes, 17.9 ct8.,« 19.3 ots.* 

Bolivia— Boliviano = 100 Centavos, 96.5 cts.,* 72.7 cts.« Once, $14.95. Dollar, 
96 cts * 

Brazil.— Mil reis = 1000 Reis, 50.2 cts.,* 54.6 cts.3 

Canada. — English and U. S. coins. Also Pound, $4.* 

Central America.*— Doubloon, $14.50 tu $15.65. Reale, average S^ cts. See 
Honduras. 

Ceylon.— Rupee, same as India. 

Chili.— Peso = 10 Dineros or Decimos = 100 Centavos, 96.5 cts.,* 91.2 ct«.» Con- 
dor = 2 Doubloons = 5 Escudos = 10 Pesos. Dollar, 93 cts.* 

Cuba.— Peso, 93.2 cts.* Doubloon, $5.02. 

Denmark.— Crown = 100 Ore, 26.7 ct8.,« 26.8 cts.a Ducat, $1.81.* Skilling, % ct* 

Ecuador.— Sucre, 72.7 cts.» Doubloon, $3.86. Condor, $9.66. Dollar, 93 cts.* 
Eleale 9 cts * 

Egypt.— Pound = 100 Piastres :« 4000 Paras, $494,3.* 

Finland.— Markka = 100 Penni, 19.1 cts.* 10 Markkaa, $1.93. 

France.1— Franc =100 Ceniimes, 17.9 ct8.,« 19.3 cts.8 Napoleon, $3.84.* Livre, 
18.5 cts.* Sous, 1 ct.* 

Germany.— Mark = 100 Pfennigs, 21.4 cts.,2 23.8 cts.* Augustus (Saxony), $3.98.* 
Carolin (Bavaria), $4.93.* Crown (Baden, bf,varia, N. GermanyX $1.06.* 
Ducat (Hamburg, Hanover), $2.28.* Florin (Prussia, Hanover), 66 eta.* 
Groschen, 2.4 cts.* Kreutzer (Prussia), .7 ct. Maximilian (Bavaria). $3.30.* 
Rix Thaler (Hamburg, Hanover), $1.10* (Baden, Brunswick), $1.00* (Prussia, 
N. Germany, Bremen, Saxouy, Hanover), 69 cts.* 

Great Britain. — Pound Sterling or Sovereign (£) = 20 Shillings = 240 Pence, 
$4.86.65.* Guinea = 21 Shillings Crown = 6 Shillings. ShilUng (*), 22.4 
cts.,s 24.3 cts. (^ pound sterling). Penny (d), 2 cts. 

Greece.!— Drachma = 100 Lepta, 17 cts.,« 19.3 cts.* 

Hayti.— Gourde of 100 cents, 96.5 cts.s* 

Honduras.— Dollar or Piastre of 100 cents, $1.01. See Central America. 

India.— Rupee = 16 Annas, 45.9 cts.,^ 34.6 cts.* Mohur = 16 Rupees, $7.10. Star 
Pagoda (Madras), $1.81.* 

Italy, etc.i— Lira = 100 Centesimi, 17.9 cts.,2 i9.3cts.* Carlin (Sardinia), $8.21.* 
Crown (Sicily), 96 ctfi.* Livre (Sardinia), 18,6 cts.* (Tuscany, Venice), 16 
sts.* Ounce (Sicily), $2.50.* Paolo (Rome), 10 cts.* Pistola (Borne), $3.37.* 
Scudo* (Piedmont), $1.36 (Genoa), $1.28 (Rome), $1.00 (Naples, Sicily), 95 
cts. (Sardinia), 92 cts. Teston (Rome). 30 cts.* Zecchino (Rome), ^.27.* 

Japan.— Yen = 100 Sen rgold), 99.7 cts.* (silver), $1.04^, 78.4 cts.* 

Liberia.— Dollar, $1.00.* * 

Mexico.— Dollar. Peso, or Piastre = 100 Centavos (gold), 98.3 cts. (silver), $1.05,« 
79 cts.* Once or Doubloon = 16 Pesos, $15.74. 

Netherlands.— Florin of TOO cents, 40.5 cts.," 40.2 cts.« Ducatoon, $1.32.* Guilder, 
40 cts.* Rix Dollar, $1.05.* Stiver, 2 ctfl.* 

New Granada.— Doubloon, $15.34.* 

Norway.— Crown = 100 Ore = 30 Skillings, 26.7 ct8.,« 26.8 cts.« 

Parascuay .—Piastre = 8 Reals, 90 cts. 

Persia.— Thoman = 6 Sachib-Kerans = 10 Banabats = 25 Abassis — 100 Scahia, 
$2.29. 

Peru.— Sor= 10 Dineros = 100 Centavos, 96.5 cts.,a 72.7 cts.* Dollar, 93 eta.* 

Portugal.— Milreis = 10 Testoons = 1000 Reis, $1.08.* Crown = 10 Milreis. 
Moidore, $6.50.* 

Russia.- Rouble = 2 Poltinniks = 4 Tchetvertaks = 6 Abassis = 10 Griviniks = 
20 Pietaks = 100 Kopecks, 77 cts.,« 58.2 cts.* Imperial =-« 10 Roubles, $7.72. 
Ducat = 3 Roubles, $2.39. 

Sandwich Islands.- Dollar, $1.00.* 

Sicily.— See Italy. 

Spain.— Peseta or Pistareen = 100 Centimes, 17.9 cts.,* 19.3 cts.* Doubloon (new) 
= 10 Escudos = 100 Reals, $5.02. Duro = 2 Escudos,* $1.00.2 Doubloon (old), 
$15.65.* Pistole = 2 Crowns, $3.90.* Piastre, $1.04.* Reale Plate, 10 cta.^ 
Beale vellon, 6 cts.* 

1, 2, 3, 4. See foot-notes, next page. 



FOBEIGN COINS. 



219 



(Foreign Coins QnUinMd. Small flsnreft Oi *» 'i *) ^^^ ^ M^ noUs.) 

Sweden.— Crown = 100 Ore, 25.7 ct8.,« 26.8 cta.» Ducat, $2.20.* Rix Dollar, $1.05.« 

Switzerland.!— Franc = 100 Centimes, 17.9 et8.,2 19.3 ct8.« 

Tripoli.— Mahbub = 20 Piastres, 65.6 ct8.» 

Tunis.— Piastre = 16 Karobs, 12 cts.2 10 Piastres, f 1 .16.6. 

Turkey.— Piastre = 40 Paras, 4.4 cts.' Zecchin, J1.40.* 

United States of Colombia.— Peso = 10 Dineros or Decimos = 100 Centaros, 96.5 

cts.,« 72.7 ct8.3 Condor = 10 Pesos, $9.65. Dollar, 93 5 cts.* 
Uruguay.— Peso = 100 Centavos or Centesimos (goldl, $1.03 (silver^ 96.5 cts.s 
Venezuela.— Bolivar — 2 Decimos, 17.9 cta.,2 19.3 cts.* Venezolano = 5 Bolivars. 



Standard Blameiers and Welgrbte of United States 


Coins. 




Valae. 


Diam«ier. 


Wetgbt. 


€k>ld, 10 per cent, alloy : 

Double Eagle 

Eagle 

TTfLlfFagle . . 


1 

20 
10 
'5 
2.50 

1.00 
0.50 
0.25 
0.10 

0.05 
0.01 


Inches. 

1.350 
1.060 
0.848 
0.700 

1.500 
1.205 
0.955 
0.705 

0.835 
0.750 


Millimeters. 

34.29 
26.92 
21.54 
17.78 

38.10 
30.61 
24.26 
17.91 

21.20 
19.09 


Grains. 

516.00 

258.00 

129.00 

64.50 

412.60 

192.90 

96.45 

38.58 

77.16 
48.00 


Grams. 

33.436 

16.718 

8.359 


Quarter "kagle 

Silver, 10 per cent alloy : 

Standard Dollar 

TTalf Dnllfif . . 


4.180 

26.729 
12,50 


Quarter Dollar 

Dime 

JHlnor 

Five Cents, 75^^ copper, 25^« 
nickel . . .• 


6.25 
2.50 

5.00 


One Cent, 95^^ copper, 5^ tin 
and zinc 


3.11 







Perfectly pure sold is worth $1 per 28.22 grs = $20.67183 per troy oe =* 
$18.84151 per avoir oz. Bttandard (U. 8. coin) is worth $18.60465 per troy oz = 
$16.95736 per avoir oz. It consists of 9 parts by weight of pure gold, to 1 part 
alloy. Its value is that of the pure gold only ; the cost of the alloy and of the 
ooini^ being borne by Government. A cable f€»ot of pure cold irelgphs 
about 1204 avoir lbs ; and is worth $362963. A cubic ineh weighs about 11.148 
avoir oz ; and is worth $210.04. 

Pure gold is called fine, or 24 earat gold ; and when alloyed, the alloy is sup- 
posed to be divided into 24 parts by weight, and according as 10, 15, or 20, 4&c, of 
these parts are pun gold, the alloy is said to be 10, 16, or 20, Ac, carat. 

The averaipe fineness of California natlTe void, by some thou- 
sands of assays at the U. S. Mint in Philada., is 88.5 parts gold, 11.5 silver. Some 
from Georgia, 99 per cent. gold. 

•Pure sllTer fluctuates in value : thus, during 1878-1879 it ranged between 
$1.05 and $1.18 per troy oz., or $.957 and $1,076 per avoir, oz. A cubic inch weiglfs 
about 5.528 troy, or 6.065 avoir, ounces. 



1 France, Belgium, Italy, Switzerland, and Greece form the Latin Union. 
Their coins are alike in diameter, weight, and fi^ieness. 

t __ 19.3 times the value of a single coin in francs as given by Costes. 

» Par of exchange, or equivalent value in terms of U. S. gold dollar.— Treasury 
Giicalar. 

« Erom our 10th edition. 



220 WEIGHTS AND MEASURES. 

Troy Weifrbt. U. S. and British. 

24 grains 1 pennyweight, dwt. 

20 pennyweights 1 ounce = 480 grains. 

12 ounces 1 pound = 240awtB. = 5760 grains. 

Troy welcht is nsed for grold and silver. 

A carat of the jewellers, for precious stones is, in the U. S. = 3.2 grs. ; in 
London, 3.17 grs. ; in Paris, 3.18 grains., divided into 4 jewellers' grs. In troy, 
apothecaries' and avoirdupois, tbe grain is tbe same. 

Apotbecaries' Weiffbt. U. 8. and British. 

20 grains 1 scruple. 

3 scruples 1 dram = 60 grains. 

8 drams 1 ounce = 24 scruples = 480 grains. 

12 ounces 1 pound = 96 drams = 288 scruples = 5760 grains. 

In troy and apothecaries' weights, the grain, ounce and pound are the same. 

Avoirdupois or €oniniereial Weiffbt. U. 8. and British. . 

27.34875 grains - 1 dram. 

16 drams 1 ounce = 437V grains. 

16 ounces 1 pound = 256 drams = 7000 grains. 

28 pounds 1 quarter = 448 ounces. 

4 quarters ~ 1 hundredweight = 112 fl)8. 

20 hundredweights 1 ton = 80 quarters = 2240 fts. 

A stone «> 14 pounds. A quintal = 100 pounds avoir. 

Tbe standard of tbe avoirdupois pound, which is the one in 
common commercial use, is the weight of 27.7015 cub ins of pure distilled water. 
at its maximum density at about 39°.2 Fahr, in latitude of London, at the level 
of the sea ; barometer at 30 ins. But this involves an error of about 1 part in 
1362, for the IS) of water = 27.68122 cub ins. 

A troy lb = .82286 avoir ft. An avoir ft = 1.21528 troy ft, or apoth. 

A troy OS. = 1.09714 avoir, oz. An avoir, oz. = .911458 troy oz., or apotb. 

IiOn§: Measure. U. 8. and British. 

12 inches 1 foot = .3047978 metre. 

3 feet 1 yard = 36 ins = .9143919 metre. 

5^ vards 1 rod, pole, or perch =» 16U feet = 198 ins. 

40 ro^s 1 furlong = 220 yards -= 660 feet. 

Sfurlongs 1 statute, or land mile = 320 rods = 1760 y^ =.6280 ft « 63360 iiM. 

3 miles 1 league = 24 fUrlongs = 960 rods = 5280 yds = 15840 it. 

A point =y, inch. A line = 6 points =*t^ inch. ^ palm = 3 ins. A 
banS = 4ins. Aspan = 9ins. A fatbom = 6 feet. A cable's lenKtb 

= 120 fathoms = 720 feet. A Gnnter's surveying cbain is 66 feet, or 4 
rods long. It has 100 links, 7.92 inches long. 80 Gunter's chains = 1 mile. 
A nautical mile, geoffrapbical mile, sea mile, or knot, is 

variously defined as being = the length of 

metres feet statute miles 

1 min of loniritude at the equator = 1856.345 6087.16 1.15287 

1 « latitude « " = 1842.787 6045.95 1.14507 

1 ^^ lauiuu ^^ ^ 1861.655 6107.85 1.15670 

1 '« «* atlat46° = 1862.181 6076.76 1.15090 

1 "a great circle Qf a true') (value adopted .by U. S. Coa»t 
mhere whose surface area is V -=< and Geodetic Survey 
fqutl To that of the earth j ll853.248 6080.27 1.15157 
British Admiralty bnot = 1853.169 6080.00 1.15152 
The above lengths of minutes, in metres and feet, are those published by the U. S. 
CoMt and Geodetic Survey in Appendix No 12, Report for 1881, and are calculated 
from Clarke's spheroid, which is now the standard of that Survey. 

At the equator, 1° of lat =-- 68.70 land miles; at lat 20° = 68.78 ; at 40° = 
69.00 ; at 60° - 69.23 ; at 80° = 69.39 ; at 90° = 69.41. 



WBiaHT8 AKD MEASURES. 



221 



I^en^tlis of a D flg r— of Ii«B9itiide In Afferent liatltndefl, 

and at tllC level or tMke iteat The** Itngthi are In oommon land or statate mlleii, 
•r 5S80 n. SioM the flgure of the earth has nerer been prteUtli/ aaeertained, these are but oloee ap 
proximatlene. Intermediate onee may be fouid eorreettj bj simple proportion. !<> of tongituM 
* te 4 mine ef oItU or eloek tUM| 1 mln of InngltiiilB to 4 eeoi of tine. 





Degofi , 
Lat. ^ 


iilSB. 


Dec of 
Lat. 


Mike. 


Dec of 
Lit. 


MUea. 


Dec of 
Lat. 


miM. 


Dec of 
Lat. 


MUes. 


Dec of 
Lat. 


MUes. 




1 


W.16 


14 


67.12 


28 


61.11 


42 


61.47 


66 


88.76 


70 


28.72 




a 1 


ie.i2 


16 


66.50 


80 


69.94 


44 


49.88 


68 


86.74 


72 


21.43 




4 1 


M.N 


18 


65.80 


S3 


58.70 


46 


48.13 


60 


84.67 


74 


19.12 




6 ( 


B6.76 


20 


66.02 


34 


67.39 


48 


46.88 


62 


82.56 


76 


16.78 




8 


B&tt 


22 


64.16 


36 


56.01 


50 


44.54 


64 


30.40 


78 


14.42 




10 


118.12 


24 


63.21 


88 


64.56 


63 


43.67 


66 


28.21 


80. 


12.05 




13 


17.66 


96 


62.90 


40 


53.06 


54 


40.74 


68 


26.98 


82 


9.66 








InelieB redaeed to Deeimals of a 


Foot. 


Ao errors. 




Ina. ] 


root. 


las. 


Foot. 


IDI. 


Foot. 


Ins. 


Foot. 


Ins. 


Foet: 


Itti. 


Foot. 




• 


.0000 


% 


.1867 


4 


.8383 


6 


.5000 


S 


.6667 


10 


.8833 




1-SS 


.0026 




.1693 




.3359 




.5026 




.6693 




.8859 




1.16 


.0062 




.1719 




.8886 




.6052 




.6719 




.8886 




8-n 


.0078 




.1746 




.8411 




.5078 




.6746 




.8411 




Ji . 


.0104 


H 


.•771 


H 


.9488 


H 


.5104 


H 


41771 


H 


.8438 




OUO 




.1797 




a Jig 4 

mOVfn 




.6130 




.6797 




.8464 




S-16 . 


0166 




.1828 




.3480 




.5156 




41823 




.8490 




f-tt 


0182 




.1849 




.8516 




.6182 




41848 




.8616 




Ji : 


0208 


H 


.1876 


H 


.3542 


H 


.6208 


H 


.6875 


H 


.8643 




0284 




.1901 




.3568 




.5284 




.8801 




.8568 




fr-16 


0280 




.ion 




.3594 




.6200 




.6927 




.8694 




11-S9 


0286 




.1953 




.3620 




.6286 




.6953 




.8620 




H 


0313 


H 


.1979 


H 


.3646 


H 


.5313 


H 


.6879 


H 


•oDvO 




ust 


0339 




.2006 




.8672 




.5339 




.7006 




.8672 




7«1« 


086& 




.2031 




.3698 




.6866 




.7031 




.8688 




U^ 


0381 




.2067 




.3724 




.5391 




.7057 




.8724 




.^ 


0417 


H 


.2083 


H 


.3750 


H 


.6417 


H 


.7083 


H 


.8750 




17-SS 


0443 




.2109 




.8776 




.5443 




.7109 




.8776 




9-M 


0469 




.9186 




.8802 




.5469 




.7135 




.8802 




IMS 


0485 




.2161 




.8828 




.5495 




.7161 




.8828 




nji : 


0621 


H 


.2188 


H 


.3854 


H 


.5521 


H 


.7188 


H 


.8854 




0647 




.2214 




.8880 




.5647 




.7214 




.8880 




ii.i« 


0573 




.2340 




.8906 




.5573 




.7240 




.8806 




ss-ss 


0680 




.2966 




.3932 




.6599 




.7266 




.8692 




H 


0626 


H 


.2392 


H 


.8958 


H 


.5625 


h 


.7292 


H 


.8958 




Sft^ 


6661 




.2318 




.8964 




.5651 




.7818 




.8964 




lft.lC 


oon 




^2844 




.4010 




.5677 




.7344 




.9010 




S7-» 


0703 . 




.2370 




.4036 




.5703 




.7370 




J8006 




y • 


0729 


% 


.2396 


X 


.4063 


X 


.6729 


H 


.7396 


}i 


.9063 




f^ 


0765 




.2432 




.4069 




.6755 




.7422 




.9089 




mi . 


0781 




•9vfto 




.4115 




.6781 




.7448 




.9115 




• Sl-SS 


0807 




.2474 




.4141 




.6807 




.7474 




.9141 




1 


06SS 


S 


.2509 





.4167 


y 


4i688 


9 


.7500 


11 


.9167 




1« 


0869 




.2626 




.4193 




.6859 




.7526 




.9193 




1-lC 


0885 




.3563 




.4219 








.7562 




.9219 




8-S2 


0911 




.2678 




.4245 




.6911 




.7578 




.9246 




H 


0888 


H 


.2004 


H 


.4271 


H 


.5038 


H 


.7604 


H 


.9271 




5-St 


096A 




.3660 




.4297 




.5964 




.7680 




.9297 




S-I6 


0800 




.3866 




.4323 




.6990 




.7656 




.9823 




7-8i 


1016 




.3683 




.4.')49 




.6016 




.7682 




.9349 




3< • 


1042 


H 


■S& 


H 


.4876 


3i 


.6043 


H 


.7708 


H 


.9375 




9-Si 


1068 






.4401 




.8068 




.7784 




.9401 




6-16 


1684 




.2768 




.4427 




.6094 




.7760 




.9427 




11-32 


1198 




.2786 




.4453 




.6120 




.7786 




.9468 




K 


1148 


H 


.2811 


H 


.4479 


H 


.6146 


H 


.7813 


H 


.9479 




lS-3t2 


1172 




.2889 




.4505 




.6172 




.7889 




.9506 




7-16 


1198 




.2666 




.4531 




.6198 




.7865 




.9531 


« 


16-32 


1224 




.«9l 




.4567 




.6234 




.7881 




.9557 




^ 


1260 


H 


S& 


H 


.4583 


H 


.6250 


H 


.7917 


H 


.9583 




17-.% 


1276 






.4809 




.6276 




.7948 




.9609 




9-16 


UOS 




.2M9 




.4635 




.6302 




.7969 




.9636 




19-32 


1828 




:SSi 




.4661 




.6828 




.7995 




.9661 




2i.l^i : 


1864 


H 


H 


.4688 


H 


.6354 


H 


.8021 


H 


.9688 




1380 




.lOiV 




.4714 




.6380 




.8047 




.9714 




11-16 


1406 




.8978 




.4740 




.6406 




.8073 




» .9740 




SS.S2 


108 




J089 




.4766 




.6432 




.8099 




.9766 




9i 


1468 


H 


.8136 


h 


.4792 


H 


.6456 


H 


.8125 


h 


.9792 




25-S3 


1484 




.8161 




.4818 




.6484 




.8151 




.9618 




13-16 


1610 




.8177 




.4844 




.6510 




.8177 




OtlAA 
•von 




27-S2 


1686 




.8908 




.4870 




.65.<{6 




.8203 
.8229 




.9870 




H 


1668 


H 


.8228 


H 


.4896 


K 


.6bea 


X 


H 


.9896 




n.n . 


1689 




41256 




.4922 




.6589 




.8255 




.9922 




16-16 


1816 




. .8281 




.4948 




.6615 




.8281 




.9948 




n« • 


1641 




.8807 




.4974 




.6641 




.8307 




.9974 



WEIGHTS AND MEABUBBS. 



— —"■H-Ij 



» iq ill = 10a» aq tOl. 

rodi = W40 iq Ida = UMt K Ml- 









Cnblp. or Solid M^amare. 



A CBbt* a 

M Dik THd, or i.Ma» ■•knlg^ (I. 
HI iu^>llln. •> ««HHn, 



n. A tim i.iw> ai^ ci iennijtiu 



A cnbl« luch Is midaI to 

l.nuta snlllllni; e.r.ie3S«e3 arellLLnir a 

A cubic yard la emnMl l4 



1 aphere I toot In diameter, tiontnlna 



A sphere 1 Inek In diameter, eonlnlna 



WEZGHTS AUTD HBASimiiB. 



22a 



cylinder 1 foot In diameter, 

.02909 oub yard. 
.7854 cub foot. 
I35T. 1712 cub inches. 

.63112 U. S. di7 bushels. 
2.5245 U. S. dry pecks. 
a0.1958 U. S. dry quarts. 
. 40.3916 U. S. dry pints. 
5.8752 U. S. liaaid gallons. 
28.5008 U. S. liquid quarts. 



A eylinder 1 ineli in diameter, and 

.005454 cub foot. 
9.4248 cub inches. 

.2805 U. 8. dry pint. 

.3264 liquid pint. 
1.3056 U. S. gill. 



and 1 f<N>t bisrta, coui^Jiins 

47.0016 U. S. liquid piuta. 
188.0064 U. b. liquid gills. 
4.8947 Brit imp gallons. 
19.5788 Brit imp quarts. 
39.1575 Brit imp pint*. 
156.6302 Brit imp gills. 
222.S95 decilitres. 
22.2395 litres. 
2.22395 decalitres. 
.222895 hectolitre. 

1 foot liiji^li, contains 

.2719 Brit imp pint. 
1.0677 Brit imp gill. 
15.4441 centilitres. 
1.54441 decilitres. 

.164441 litres. 



I«iqald JHeasnre. u, g. only. 

The iMMda of this measure in the U. S. is the old Brit wine gallon of 231 oub ins; or 8.3S888 Ibr 
aToir of pure water, at its max dennity of about 39^.2 Fabr ; the barom at 30 ins. A cylinder 7 in» 
iiam, and 6 ins high, contains 230.904 cob ins, or almost precisely a gallon ; as does also a oube of 
t.lS68 ina on an edge. Also a gallon = .13368 of a cub ft ; and a cub ft contains 7.48052 galls ; nearly 
1H gall*-. TUs bastfl howerer InTolres ab err«r of about 1 part in 1363, for the water adtn- 

63 gallons 1 hogshead. 

2 hogsheads 1 pipe, or butt. 

2 pipes. 1 tun. 

In the U. S. and Great Brit. 1 barrel of wine or brandy = 31i^ galls ; in Pennsylvania, a half 
barrel, 16 galls; a double barrel, 64 galls; m puncheon, 84 galls; a tierce, 42 galls. A liquid 
Beasore barrel of 81^ galls contains 4.211 cub ft = a oube of 1.615 ft on an edge ; or 3.38v U. S. struck 
bosbals. A sill = 7.21875 oub ins. The followlns cyliinders contain some o.' these measure* 
very approximately. 



ally weighs 8.3450(tti tbi 

cub ins. 

4glUa Ipint =28.875. 

2 pints 1 qnart = 57.750 = 8 gills. 

4 qxaaU 1 gallon = 231 . =8 pints— 32 gills 



DIam. Height, 

enb ins. Ins. Ins. 

Omj.21875) IH 3 

><pint 2« 3« 

Pint 3« 3 

quart S^ 6 



Diam. 
Ins. 

Gallon 7 . 

2 gallons 7 . 

8 gallons 14 . 

10 gallons 11 . 



Height. 

Ins. 

6 

. 12 

. 12 

. 15 



Apotbecaries* or Wine Measure. 



1 Gallon 

mnt... 



1 Fluid ounce . . 
1 Fluid drachm. 
IMmim 



Symbol. 


Pints. 


Floid 
.ounces. 


FJoid 

draohms. 


Minims. 


Coble 

inches. 


Cong* 

m 


8 

1 

• ■ • • 

• • • • 

• • • • 


128 
16 

1 

• • • • 

• • • • 


1024 
128 

8 
1 

• • • 


61440 
7680 

480 
60 

1 


231 
28.875 

1.8047 
0.2256 
0.0088 



Weight of water4 



Pounds, av. Grains. 



8.345 

1.043 

Ounces, av. 

1.043 



68415 
7301.9 

456.4 
57.05 
0.96 



To redoce U. H» liquid measnres to Brit ones of the same denomina* 

tlon, divide by 1.30032; or near enough for common use, by 1.2; or to reduce Brit to U. S. multiply 
by 1.2. 

Dry Measure. 

U. S. only. 

Tlie basis of tliis is the old British Winchester struck bushel of 2150.42 cub 

las; or 77.627418 pounds avoir of pure water at its max density. Its dimensions by law are 18^ ins 
iaaer diam ; 19>t id> outer diam; and 8 ins deep ; and when heaped, the cone is not to be less than 6 
ins Ugh ; which makes a heaped bushel equal to 134 struck ones ; or to 1.55556 cub ft. 

Bdge of a cube of 
equal capacity. 

2 pints 1 qoart, =67.2006 cub ins = 1.16365 liquidiit 4.066 ins. 

4 quarts 1 gallon. = 8 pints, = 268.8026 cub ins, :^ 1.16:i65 liq gal 6.454 " 

2 gallons 1 peek, = 16 pints, = 8 quarts, = 537.6050 cub ins 8.131 " 

4 pe<d(s 1 stmok bushel, = 64 pinls, = 32 quarts, = 8 gals, = 2150.4200 cub ins. 12,908 " 



* Abbreviation of Latin, Congius. 
t Abbreviation of Latin, Ootarios. 

} At its maximum density, 62.426 pounds per eubio foot, correspouding to a temperature of 4° 
Ceotigrade = S9.2P Fahrenheit. 



224 



WMGH1B AKD MBA8X7BBS. 



A 9trnck bnshel =» 1.24445 cub a. A cub ft * .80356 of a struck bushel. 
Xhe dry flour barrel = 8.75 cub ft; =8 struck bushels. The dry barrel la 

not, howe%'er, n legMliied measure; and no great attention is given to its capacity; consequently, 
barrels rar^ cunsiderablT. A barrel of Qour conuins by law, liW Its. In ordering by tbe barrel, the 
amount of its contents sboald be specifled in pouods or galls. 

To reduce IJ. S. dry measures to Brit imp ones of the same name, di? 

by 1.031516 ; and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposes use 1.033. 

Brltlsb Imperial Measure, botb liquid and dry. 

This system is established throughout Great Britain, to the exclusion of tbe old ones. Its basis is 
the imperial gallon of '277.274 cub ins, or 10 lbs avoir of pure water at the temp of 62^ Fahr, when 

the barom is at 30 Ids. This basis Involves an error of about 1 part im 

18S6, for 10 lbs of the watar =:only 277.128 cab ina. 





Aroir Ihe. 
of water. 


Oob. las. 


Cab. ft. 


Edge of a cube «f 

equal capaeity. 

Inches. 


Acllla 1 pint 


1.25 

8.50 

6. 
10. 

80. -1 

80. I Dry 
820. { meaa. 


84.6688 

e».8l85 

188.687 

877.874 

554.648 

9818.188 

8878.768 

in45.686 




8.8605 


Ipinta 1 quart 

S quarts 1 pottle 

8 Dottles I Kallon 




4.1079 




6.1756 




6.6908 


S fftllODB 1 p6C!C ••••••••••• ■•• 




8. 2157 


4 Dooki 1 buhel.a.a •••••••••• 


1.8R87 

6.1847 

10.2694 


1*041? 


4 basbelsl coomb 




8 coombs 1 quarter 1 


6i0. 





TiM) imp gall = .16046 cub ft; *Dd 1 Ottb ft =<.9B918 galls. 



Measure. 


Symbol. 


Pints. 


Fluid 
ounces. 


Fluid 
drachms. 


Minims. 


Oubic 
iochM. 


Weight of watar4 




Pounds, AT. 


Graimt. 


1 Gallon 

1 piDt 


Of 

fl. OS. 

fl.dr. 
mill* 


8 

1 

• • • • 

• • • • 


160 
90 

1 

• ••• 

• • ■ • 


1280 
160 

8 

1 

• • • • 


78800 
9600 

480 

60 

1 


877.274 
86.669 

1.783 
0.217 
0.0086 


10 
1.85 
Ounces, ar. 

• ••• 


70068 
•750 

487.5 
54J875 
0.9114 


1 Fluid ounce . . . 
1 Fluid drachm.. 
1 Minim 













The weight of water aflbrds an easy way to find the cubic contents of a tressel. 



To' obtain the slae of commerelal measai^ea by means Qf tlio 
* welg^bt of water. 

At the common temperature of fh>m 70*^ to 75° Fah, a cub foot of ftesh water weighs wrr appnud> 
mately 6214 \bi avoir. A cubic half foot, (6 ius on each edge,) 7.78125 0>a. A cub quarter foo^ (8 ins 
on each edge.) .97266 n>. A cab yard, 1680.75 lbs; or .75034ton. ▲ cub half yd, (18 ins on each «das,) 
210.094 lbs ; or .0938 ton. A cub inch, .036024 0) ; or .576384 ounce ; or 9.2222 drams ; or 252.170 grama. 
An Inch square, and one foot long, .432292 Bk. Also lib = 27.76908 cab ins, or a cube of 8.096 ins on IB 
edge. An onnce, 1.785 «ub ins ; a ton, 85.964 cab ft, all near enoof h for common me. 

Original. 



Uquld Measures. i^^\^^«^- 

of Water. 

V. S.Gill 26005» 

U. 8. Pint 1.0409 

U. S. Quart 2.0804 

U.S. Gallon 8 lbs 5l 01 8.8916 

U. S. Wme Barrel, 31 H Gail 969.1810 

Dry Measures. 

U. S. Pint 1.2104 

U. 8. Quart 2.4208 

17. S. Gallon 9.6834 

V. 8. Peck 19.3668 

U. a. Bushel, struck 77.4670 

' * Or 4 ounces ; 2 drams ; 15.6625 grs. 



I«lqal€l and I>ry. Um AT*ir. 

^ ot Water. 

British Imp Gill S1914* 

*' Pint 1.94858 

" «• Quart 9.49715 

•• •' Gallon 9.9886 

" •• Peek..M. 19.9779 

" Bushel 79.9088 

* 4.9949 ; or rery nearly 5 onnoas. 

Metric Measnires. 

Centilitre .03196t 

pMilltre siMt 

Litre J.1981 

Decalitre, or Centlatere 91.9606 

Btere (eubio meter) 9198.0786 

t Or 5.6271 drams; or 153.866 gra. 
{ 3.5169 onnoes. 



* Abbreviation of Latin, Congius. 
t Abbreviation of Latin, Ootarius. 
t At the standard lemperatore, 929 Fahrenheit a 



about 16.r> Oentlf rada. 



WEIGHTS, AND UEABUBB0. 



225 



Metrle Measures of I^eni^^b. 
By U. 8. and Brltfsli StaaiUrd. 





Ins. 


Ft. 


Yds. 


Miles. 


Millimetre* 


.089370 

.89370428 

8.9370428 

89.370428 

393.70428 

Road 
measures. 


.008281 

.082809 

.8280869 

3.280869 

32.80869 

328.0869 

3280.869 

82808.69 






CeTltim«tre+t--,T,---r r ^,,r,r-,„r „f 




• 


]|[)ACini6tTA 


.1093628 
1.093628 
10.93623 
109.3623 
1093.628 
10936.23 




Metret 




Dnftiunetrft ") 




Hectometre 


.0621875 


Eflometre 


.6218760 


Kyriametre j 


6.213750 



• N«arl7 the ^ part of ao inoh. t Full K inob. 

} Yerj nearly 8 ft, 3H ioB. wbioh is too long hj onlj 1 part in 8616. 

Hetrlc Square Measure- 
By U. S. m4 British Slradard. 



8q Millimetre 

8q Centimetre 

Sq Decimetre 

Sq Metre, or Centlare., 
Sq Decametre, or Are. 

D«care (not nsed) 

Hectare 

8q Kilometre 

8q Myriametre 



Sq. Ins. 



.001550 
.155003 
1S.500B 
1550.03 
155008 



.3861090 so miles. 

38.61090 " 



Sq. Feet. 



.00001076 
.00107641 
.10764101 
10.764101 
1076.4101 
10764.101 
107641.01 
10764101 



Sq.Yd8, 



.0000012 

.0001196 

.0119601 

1.19601 

119.6011 

1196.011 

11960.11 

1196011. 



Acres. 



.000247 
.024711 
.247110 
2.47110 
247.110 
24711.0 



Metric Cubic or Solid Measure. 

Aaevrdlns to V. 8. Standard. 

Only thoM marked '« Biit" are Britiah. 



Mill1]itr«,oroab 
Centimetre.... 



Centmtre 



Decilitre 



Litis, or cubic 
Dscimetre.... 



Decidltre, or 
Coitiatere.... 



Hectolitre, or 
Decistere 



Kflolitre, or 
Cubic Metre, 
or Stere 



[friolitie, or 
Decastere 



Cub Ins. 

.0610254 

.610254 
6.10264 

61.0254 

610.254 
Cub Ft. 

.858156 

8.53156 

86.3156 
863.156 



riiiaoid. 
(Dry. 
J Liquid. 
(Dry. 
J Liquid. 
(Dry. 



.0084537 gill. 
.0070428 Brit gill. 
.0018162 dry pint 

.084537 ffUl. 
.070428 Brit gill. 
.018162 dry pint. 

.84537 gill = .21184 pint. 

.70428 Brit gill = .17607 Brit pint. 

.18162 dry pint. 



{ 



Liquid, 
Dry. 



2.1134 pints. 

fUpi 
.11351 peck = .9081 dry qt « 1.8162 dry pt 



1.05671 quart » 2.1134 pii 
.88036 Brit quart = 1.7607 Brit 



)ints. 



(Liquid. 
(Dry. 

(Dry. 
I Liquid. 
(Dry. 



2.64179 U. S. Uquid gal. 

2.20000 Brit gaL 

.283783 bush ^ 1.1851 peck « 9.061 dry qts. 

26.4179 U. S. Uquld gal. 
22.0090 Brit gal. 
2.83783 bush. 

264.179 U. S. liquid gal. 
220.090 Brit gal. 
28.3783 bush. 



Onb yds, 1.8080. 



Liquid. 2641.79 U. S. Uquid gal. 
283.783 busb. 



r Liquj 
iDry. 



} 

I Cub yds, 



18.060. 



15 



226 



WXI6H1S Ain> KBAMUMBB* 



Metric Welflrhta* redoeed to eonnnon Commercial or AtoIc 
Welfffitt of 1 poand = 16 ounces, or 7000 yralns. 



MiUigramme.. 
GentigrEunme. 
Decigramme .. 
Gramme 



Decagramme 

Hectogramme 

Kilogramme 

Mynogramme 

Quintal* 

Tonneau; Millier; or Tonne. 



Grains. 

.015432 

.15482 

1.6482 

15.432 

Pounds aT. 

.022046 
.22046 
2.2046 
22.046 
220.46 
2204.6 



The graniiM is the YtaaHa of Tr«neh wdgtatt r u>d !■ the welf ht of a cab eendmetre of ^*«^^ 
Vater at its max deniity, at lea level, la lat of Parle ; barom 29.922 ins. 

k 

Frencb Measures of tlie *' Systeme Usuel.** 

This iTstem wae In nse from about 1812 to 1840, when It was forbidden by law to nse eren its naoMB. 
This was done in order to expedite the general nse of the tables which we have before glTen. But ss 
the Systema Usnel appears In books pnbUshed daring the above interral, we add a taUa of sobw oC its 
valnes. 

Measures of liOiiflrtli* 



Ugnensml, orliae 

Pouee vsael, or inch, = 12 Ugnes. 
Pled nsnel, or foot, =12 peaces .. 

JLnne nsael, or elL 

Toise asnel,=6pieds 



Yards. 



.8M&4 

i.si2se 

2.18727 



PecC. 



.09118 
1.09862 
8.9S706 
6.M181 



.09118 

i.oasa 

U.lStM 

47.346 

78.T4in 



Weights, VsueL 



Qrala nsnel... 
GrosnsoeL... 
Once nsnel... 
Marensnd... 
Lirre nsnel, I 
1,5 



or pound, 



^75 grains. 
60.297 '• 

1.10268 arotr os. 
.66129 avoir lb. 

1.10268 avoir n>. 



Onbio, or Solid. TTsueL 




= 1.7606 British pis*. 
S.TSU British sate. 



1811, or before the '*8jsteme nend," the Old System, " Systeme Anolen," was in 

Frencli Measures of tbe '* Systeme Anden.** 



LlneaL 



Point anclen, .0148 Ins. •.....•••....., ........... 

Ligne anoien, .06881ns 

Pouoeanden. 1.06677 ins =.0888 ft 

Pled anoien, 12.76^2 ins = 1.06677 ft 

Anne anoien, 46.8989 lns=8.90782ft=l.S0261 yds 

Toise anoien. = 6.3946 ft= 2.1816 yds 

Leagne= 2282 toises= 2.7687 miles 



Sqna 



Sq. ins. 
.00789 
1.1359 



Sq.ft. 



1.1859 
40.8908 



Sq. yds. 



4.6484 



Onbio. 



0. ins. 

.0007 

1.2106 



C.ft. 

1.2106 
261.482 



G.yda. 



•.68a 



There is, however, much oonfosion about these old measures. Dliferent measnfas had the same 
same in diibreBt provinces. 

^^Ml 1 I I I .. - ^ I l ' 

• The m99tr4¥foU qniatal is 100 avelrdapois p«aui4s. 



WEIGHTS AND MEASURES. 227 

Biuwlan. 

Foot; same as U. 8. or British foot. Sacblne = 7 feet. Temi * 50C 
sachine » 3600 feet ai 116^ yards » .6629 mile. Pood » 86.114 lbs avoirdapoisi 

Spanlsb. 

Tlie eastellano of Spain and New Granada, for weighing gold, is varlouslf 
estimated, from 71.07 to 71.04 grains. At 71.0S5 grains, (the mean between th« 
two,) an avoirdapois, or common commercial oaoce contains 6.1572 castellano; 
and a lb aToirdupois contains 98.51ff. Also a troy ounce =s 6.7553 casteliano ; and 
a troy lb » 81.064 castellano. Three U.S. gold dollars weigh about 1.1 castellano. 

Tlio Spanisli nuirlL, or mareo^ for precious metals, itf South America, 
may be taken in practice, as .5065 of a lb aroirdupois. In Spain, .5076 lb. In 
other parts of Europe, it has a great number of values : most of them, however, 
being oetween JH and .54 of a pound avoirdupois. The .6065 of a lb =3 8545^ 

Sains ; and J5076 9) «■ 8553.2 grains. 1 marco = 60 castellanos = 400 tomine =» 
90 S^nish jjroM-grains. 

The arroba has various vslues in difl^rent parts of Spain. That of Cas- 
tile, or Madrid, is 25.4025 lbs avoirdupois; tlie tonolada of Castile =- 2082.2 
fts avoirdupois ; tlie quintal = 101.61 lbs avoirdupois ; the libra » 1.0161 
fta avoirdupois; tbe eantara of wine, Ac, of Castile a 4^268 U. S. gallons; 
that of Havana a 4.1 gallons. 

"nie wara of Castile =3 82.8748 inches, or almost precisely 82j^ inches; or 2 
feet 8Ji inches. Tbe iianeyada of land since 1801 » 1.5871 acres = 69134.08 
sqaare feet. Tbe ftmeffa of corn, Ac « 1.69914 U. & struck bushels. In 
California, tbe vara by law »» 88.872 U. S. inehee ; and tbe leipui - 6001 
varaa; or 2.6888 U. SL miles. 



fit iill^lfii 



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CONVERSIOIT TABLES. 



dli s Sp ill 






B| 



^i. -Ill, j«l. gigs 



1^ gsSSIU 

i 



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I i 



'°° si i' S- *" 



CX>NTBBaiOS TABLES. 



ii 



g§i ig M s III i 

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ill 



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ii 



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246 



OOXYEBSION TABLES. 



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a a 



m 



9 

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1 



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ill 



5 



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p,o 

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OONTBBSION TABLES. 



247 



• • • • • 



toao 



0<-iO C49aO 




E • • • " • 






I S 




SSI£:S 

d 



a|b 

a 

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4 f«-t»H *" 







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248 



OONVEBBION TABLES. 






<5 WJIJ 







5 & 
s a 



«-lC«fH 






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3 <* 






a 
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3 



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gOi-if-iio 
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oqIqco^ 

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fl>,«o 



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gill 
•m g;fl;fl 

9 ^ fa ^ 
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OfiOty 





^^ • • • • 

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I 




OOITYBBBION TABLES. 



249 



I 

a 



I 



o 



GOOO*^ 

<OC*3^ CO 

• • • • 

1-tOOO 

a 



^ c0(O 
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• • ■ 



7373 



B ill 



I 






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r4eoc4Nc<i 



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to 

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— • • • • 

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<ococo o 

• • • • 

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n 



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u 

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fa 

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ftOoifCO 
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(3 
« 



10900) 



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ooo 

lOOAiO 

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c 4> S 

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fioO(N 

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d 



gd ^ 



ft 



Oodd 

fa 
e8 



IB 

08 

a 
tt 

d 
■♦* 

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4> 
ft 



S 



OONTBRSION TABLES. 



s§| lip S|S^ i^l 1^1 

i 



.-if 



"s IS ,s 'I B 

|i .11 ^1 II i im III 



11 Is " t 



Hi 

si I 



jg|l| H H 8 '}pi I'l ill" ,tei 



11 ? 



i !"• f i° !|rs ^fi ff PI' I 



9ig^f:S SSsS ^S^!* ?:s SS g| ss ^= 



I i 



ti * 

g •fill Slfl 3i'i « II Jl 

B ISsE a|S| isfl S.sl-ji 

■ sisj isJE.s iS3|.a g'"?" 

S ill i life |l ill S|IS|li 



jjiss: 

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Sis III' 



"■4 ■ 






fcrtS fti^-e ^« ^a« ss« ■ 
ftoo >,** B**" ^'" a«— 

u £ I £ a 



OONVEBSIOK TABLES. 



251 




H 

H 



Soo tr- 
ie A £>« 
goo 


6375 
4721 


•^331 






coo 


a fl 



t*»-l 

l-4<0 

ooesi 
eoci 

a a 



C90 
too 

• • 




H 

p 

Hi 

o 

CO 



h- c<i»H eo'^ lOb- a>r-* 

to AC» CDQ Q«0 ^0» 
t« O0>-i «HO tOkO ^0» 



3 

CO 



o© »Hc4 









^s 



I 



o 

& 

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too 



«S «ooco 
^O .^coco 

©O OrHi-l 
O © 




to 

ao 

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iH 

o 
6 

I 

ao 
»0 

* 

o 

I 

OB 



I 



.S 



OOSVEHBION TABLBB, 



^H d rtiO t-oon .gi ^r-oa 'c^ a-« ov n^^r- lOrt 
9 3 rxr. o^S°Q I i^ k£^ °^ '^^ "" S ' v^ 

p g,;i" "«:: if!3 |-S 25 ,2« i*^ n:;d sss 



SI I 



s-. 



iTiit! ;d44iiniHii|4f| 

go Sis § i*S«gs "s «|| «la 6S8| *|s == 



SSi^ ^H 



«§ oqs »S o|8 »S§ P"sS »S2 * 



§ i i i" i 



ri 



g I I p Is 



B SS !: 



si fe^?^^&^^ s5 ii' s 

Ra at Sa £■ SB -g- ■£«* ■« 

;|i|£!-!°!l5lte| 

5 r r Is II r if 

S 3 B sw s b" 
u u u u u o 



11 ii 11 i 

« IS Ss T 



Sg Ss l| / : 
Eg s«s .Si s«s ■ 

°d5 Bb- S^^ «b^ 

w. 68g Sga *l| 
hO ft-t^ ft'^'! — 

B ;is ^Si ssi 



:ss »■ g== i== |-= 



i i s 



OOKVEBSIOK TASLBS. 



253 



Si 



hSo 

IS 
S 






Q 0» 00 •-• 00 1*» 



nO00<Hl 



O no 









5!l 



^1 



I 

3 

O 



•'•-'0000 
o oDk3 9 



9 S^^^ 3 



go! 



ffi • • • • 



S3 

CI CO 

<*o 






d 
e 

OD 



Is 



1$ 



o^- 



O^iHOOr-l 

iSooci 
ia»oo» 

• * • • • 

eoci»-'»HfH 



i^i 



o 



I 

e 



• ■ " 

ill 

5* Ml 

a?«S2 









a^SSS® 



£ 



or* 





S 

Si 

Si 



ii 



K ^ 



r 






I 

s 

o 



£lS 
5"- a 

^ • • 

« :i 

o 

1 

9 



s 

cii 

I 

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e 

Si 

On 

eg 
S 

« 
GO J, 

p8 

% 






I 

fl 
o 

OD 

u 
« 

A 

« 

5- 



& 



8 

04 



I 

•d 
fl 
o 

s| 

H : 





WEIOHTB AKD HEAHCBRS. 



TABI.E or ACRES 



■XaiJIBED pme mU*, a 

tor dimrent wldtka. 



ijk" 


jiT 


^. 


^ 


k! 


iH 


^fSi' 


E 


,H 


^ 


jk" 


.fiK 








aa 






62 


0^ 




ft 




JTO' 




w 


.006 






.002 
















Mt 


.urn 


28 


8.80 


Mi 








80 






t 


.486 




at 


8.82 


MI 


6« 


&87 


.120 


81 




JB8 


s 


.806 




so 


S.M 


JMO 








82 




.138 




Ihs 


.Olfl 


t2 


SJg 


'jaa 


K 


0.91 


.183 


83^ 


ijl 


JM 


8 


.wo 


.018 


IS 


4.00 


ssm 


88'* 


j!o3 


iB8 


84 


0.2 


JB3 


a 






















JH 








S6 


!24 








J88 












1023 












J40 




0.6 


joo 


11 


IM 


-OM 


a 


.48 


Mh 


02 


1.62 


.142 


88 


0.7 


.302 






.028 


88 


4.81 




03 




J4G 




0.8 








MO 


M 


4.TS 








.14T 




OA 








.(B2 


W 


4.81 


aa 


88 


tIbs 


J40 


K 


11. 




10 


ixt 


.OM 




.W 




80 


8. 


J6l 


91* 


LO 


.100 


u 


m* 




!j 






87 


8.13 










u 






12 


!o> 




88 


8.24 


!l60 


SS 




.213 


h'* 


%1S 


.03S 


U 


.a 

.33 


.» 


m 


8.48 


J68 


90 


lie 


.310 


u 


2.K 
















98 






9) 


a.42 


.048 


48 


:68 




TS 




!l86 






.228 










.10 


.08 




8.66 


.168 


98 




.226 












JIO 












.32T 


23 






48 


!o4 




H 






100 


13.1 


jno 


M 


2^1 


fl*6 


K 


0- 


:i4 


TS'* 


OJW 










»« 


■ 




10 








azi 


J74 

































T»bl« vr rntde" P«r mile, and per 100 fket newiaiwd hop|> 
sontallr, aad evrreapondlnr te dUferrat auBlea ot iaelb 



Ab7 bmw dl>t is — sloping dist > 

» alOBlnsdtot 19-hordlat i 

" vertltelKbt IS'hardJgt > 

or = sloping dlBl> 

A gnde of n fKt rlH per 100 f«et li uwiillf ci 



WeiUUXS AND HE&SCREa. 
H PBKT PHR 100 FT, HOROOHTAIh 

The trutlou of mfnnteg us eiren onlj la 34 feet In 100. 

A eUnonutcr gisduaud by Uie 3d column, ind numbuwd by the flnt on*, 
will gin U Hgbt tb* ilopH In feel per 11X1 Uei. So (inn. Origiunl. 



ltJ 


«"£ 




i 


Sc 


*s- 


ly 


STc' 


'.■a- 


1 

1 


ML 

Si 
si 




«. HIE. 

is! 
11 

ii 


i 


DX. UlD. 

il 

SI 

II 

is 


1 


II 
1 


1 






- UugHh 



A ; *ad thii 

)iit,^S,of Ui. 



4dlx T;iprcz<fniif«^^pn>portioaHl luS; but tbje et^gpegt 
grad^p surmounted by traction onlT^ even on olecttlo 



ft horijoiiial ' B """"*■' '" ft TanicaJ • " A" '"" 

^ Is the cotangent of the augle, a, with the horiiontil, or ths Ungenl of the 
int-le (9g°-a) wtth the >eiilcaJ. Tbus stated, a dope of 2 to 1 means t, slope of 2 



OBASBS. 



257 



Table of nrrades per mile; or per 100 feet meaaared liorl« 
■ontally. 



Grade 


Grade 


Grade 


Grade 


Grade 


Grade 


Grade 


Grade 


in ft. 


in ft. 


in ft. 


in ft. 


in ft. 


in ft. 


in ft. 


in ft. 


per mile. 


per 100 ft. 


per mile. 


per 100 ft. 


per mile. 


per 100 ft. 


per mile 


per 100 ft. 


1 


.01894 


39 


.73S64 


77 


1.46833 


115 


2.17803 


2 


.03788 


40 


.76758 


78 


1.47727 


116 


2.19607 


S 


.05682 


41 


.77652 


79 


1.49621 


117 


2.21601 


4 


.07576 


42 


.79546 


80 


1.51615 


118 


2.28486 


5 


.09470 


43 


.81439 


81 


1.53409 


119 


2,25379 


6 


-11364 


44 


.83333 


82 


1.55803 


120 


2.27278 


7 


.13258 


46 


.85227 


88 


1.67197 


1-21 


2.20167 


8 


.15152 


46 


.87121 


84 


1.59091 


122 


2.31061 


9 


.17045 


47 


.89016 


86 


1.60985 


123 


2.32966 


10 


.18939 


48 


.90900 


86 


1.62879 


124 


2.34848 


LI 


.20833 


49 


.92803 


87 


1.64773 


126 


2.36742 


12 


.22727 


60 


.94697 


88 


1.66666 


126 


2.38686 


18 


.24621 


61 


.96591 


89 


1.68661 


127 


2.40680 


14 


.26515 


52 


.98485 


90 


1.70466 


128 


2.42424 


16 


J28409 


63 


1.00379 


91 


1.72848 


129 


2.44818 


16 


.80803 


64 


1.02278 


92 


1.74212 


130 


2.46212 


17 


.32197 


65 


1.04167 


93 


1.76186 


131 


2.48106 


18 


.31001 


66 


1.06061 


94 


1.78080 


132 


2.50000 


10 


.35985 


57 


1.07956 


95 


1.79924 


133 


2.51894 


20 


.87879 


68 


1.09848 


96 


1.81818 


134 


2.53788 


21 


^773 


69 


1.11742 


97 


1.83712 


135 


2.56682 


22 


.41667 


60 


1.13636 


96 


1.85606 


136 


2.57676 


28 


.43561 


61 


1.15530 


90 


1.87500 


137 


2.59470 


24 


.45455 


62 


1.17424 


100 


1.89391 


138 


2.61364 


25 


.47348 


63 


1.19318 


101 


1.91288 


139 


2.63258 


26 • 


.49242 


64 


1.21212 


102 


1.93182 


140 


.'T.65152 


27 


.51186 


66 


1.23106 


103 


1.95076 


141 


2.67046 


28 


.53030 


66 


1.25000 


104 


1.96970 


142 


2.68039 


29 


M924 


67 


1.26894 


106 


1.98864 


143 


2.70833 


80 


.56818 


68 


1.28788 


106 


2.00768 


144 


2.72727 


81 


.58712 


69 


1.30682 


19Z 


2.02662 


145 


2.74621 


82 


.60606 


70 


1.32576 


108 


2.04646 


146 


2.76516 


83 


.62500 


71 


1.34470 


109 


2.06439 


147 


2.78409 


84 


.64804 


72 


1.36364 


110 


2.08333 


148 


2.80308 


86 


.66288 


73 


1.38258 


111 


2.10227 


149 


2.82197 


36 


.68182 


74 


1.40152 


112 


2.12121 


150 


2.84091 


87 


.70076 


75 


1.42045 
1.43939 


113 


2.14016 


151 


2.85986 


88 


.7mo 


76 


114 


2.15909 


152 


2.87870 



If the grade per mile should consist of feet and tenffuj add to tbe grade per 100 
iMt in the foregoing table, that corresponding to the number of tenths taken firom 
the tabl« below ; thus, for a grade of 48.7 feet per mile, we have .81439 -f- .01826 « 
.82766 feet per 100 feet. 



Ft. per Mile. 


Per 100 Feet. 


Ft. per Mile. 


Per 100 Feet. 


Ft. per Mile. 


Per 100 Feet. 


.06 


.00094 


.4 


.00768 


.7 


.01328 


.1 


.00189 


.46 


.00852 


.75 


.01420 


.16 


.00283 


.6 


.00947 


.8 


.01516 


J 


.00379 


.65 


,01041 


.86 


.01609 


?fi 


.00473 


.6 


.01136 


.9 


.01706 


4 


.00668 


.66 


.01230 


.95 


.0179i 


.36 


.00662 











258 



WEIGHTS AND MEASUBE8. 



TABUE OF HEADS OF WATEB COBBESPONDIHO TO 

OIYEN PBESSVBES. 

Water at maximum density, 62.425 lbs. per cubic foot ^ 1 gram per cubit 
centimeter ; corresponding to a temperature of i° Centigrade = ^.2^ Fahrenheit. 

Head in feet — 2.306768 X pressure in lbs. per square inch. 
*• ** ^ 0.0160192 X pressure in lbs. per square foot. 

Heads corresponding to pressures not given in the table can be found by theae 
formulc. or taken from the table by simple proportion. 



Premare. 


Head. 


Preaanre* 


Head. 


Preaanre. 


Head. 


lbs. pei 
■q. in. 


' lbs. per 
sq. ft 


Feet. 


lbs. pel 
sq. in. 


> lbs. per 
sq. It 


Feet 


Ibe. per 
•q. in. 


lbs. per 
sq. ft. 


Feet 


1 


144 


2.3068 


61 


7344 


117.646 


101 


14644 


282.984 


2 


288 


• 4.6135 


62 


7488 


119.952 


102 


14688 


235.290 


8 


432 


6.9203 


68 


7682 


122.259 


108 


14832 


287JS97 


4 


676 


9.2271 


64 


7776 


124.565 


104 


14976 


289.904 


5 


720 


11.6338 


65 


7920 


126.872 


106 


16120 


242.211 


6 


864 


13.8406 


66 


8064 


129.179 


106 


16264 


244.617 


7 


1008 


16.1474 


67 


8208 


181.486 


107 


16408 


246.824 


8 


1162 


18.4541 


68 


8352 


133.793 


108 


16652 


249.181 


9 


3296 


20.7609 


69 


8496 


186.099 


109 


16696 


261.488 


10 


1440 


23.0677 


60 


8640 


188.406 


110 


15840 


268.744 


11 


1684 


25.3744 


61 


8784 


140.718 


111 


16984 


256.061 


12 


1728 


27.6812 


62 


8928 


143.020 


112 


16128 


268.868 


13 


1872 


29.9880 


68 


9072 


145.326 


113 


16272 


260.666 


14 


2016 


82.2948 


64 


9216 


147.633 


114 


16416 


262.972 


16 


2160 


84.6016 


65 


9360 


149.940 


116 


16560 


266.278 


16 


2304 


86.9083 


66 


9504 


162.247 


116 


16704 


267Jm 


17 


2448 


39.2151 


67 


9648 


164.568 


117 


16848 


269.892 


18 


2692 


41.5218 


68 


9792 


156.860 


118 


16992 


272.199 


19 


2736 


43.8286 


69 


9936 


159.167 


119 


17186 


274jW 


20 


2880 


46.1354 


70 


> 10080 


161.474 


120 


17280 


276J12 


21 


8024 


48.4421 


71 


10224 


163.781 


121 


17424 


279.119 


22 


8168 


60.7489 


72 


10368 


166.087 


122 


17568 


281.426 


23 


8312 


68.0367 


78 


10512 


168.394 


128 


17712 


28S.7«2 


24 


8456 


55.3624 


74 


10656 


170.701 


124 


17856 


286.088 


26 


8600 


67.6692 


76 


10800 


173.008 


125 


18000 


288.84« 


26 


3744 


69.9760 


76 


10944 


175.814 


126 


18144 


290.698 


27 


3888 


62.2827 


77 


11088 


177.621 


127 


18288 


292.960 


28 


4032 


64.5895 


78 


11232 


179.928 


128 


18432 


295.266 


29 


4176 


66.8963 


79 


11376 


182.235 


129 


18576 


297J$7S 


80 


4320 


69.2030 


80 


11520 


184.541 


130 


18720 


299.880 


81 


4464 


71.5098 


81 


11664 


186.848 


181 


18864 


802.187 


82 


4608 


73.8166 


82 


11808 


189.166 


132 


19008 


804.498 


83 


4752 


76.1233 


88 


11952 


191.462 


138 


19162 


806.800 


84 


4896 


78.4301 


84 


12096 


193.769 


184 


19296 


809.107 


85 


5040 


80.7369 


86 


12240 


196.075 


186 


19440 


811.414 


86 


6184 


83.0436 


86 


12384 


198.382 


186 


19684 


818.720 


87 


5328 


85.3504 


87 


12528 


200.689 


187 


19728 


816.027 


88 


6472 


87.6572 


88 


12672 


202.996 


188 


19872 


8184184 


39 


6616 


89.9640 


89 


12816 


205.302 


189 


20016 


820J641 


40 


6760 


92.2707 


90 


12960 


207.609 


140 


20160 


822.946 


41 


6904 


94.5775 


91 


13104 


209.916 


141 


20804 


826.264 


42 


6048 


96.8843 


92 


13248 


212.223 


142 


20448 


827.961 


48 


6192 


99.191U 


93 


13392 


214.529 


143 


20592 


829.668 


44 


6836 


101.4978 


94 


13536 


216.836 


144 


20736 


882.175 


46 


6480 


103.8046 


96 


13680 


219.143 


145 


20880 


884.461 


46 


6624 


106.1113 


96 


13824 


221.450 


146 


21024 


886.766 


47 


6768 


108.4181 


97 


18968 


223.756 


147 


21168 




48 


6912 


110.7249 


98 


14112 


226.063 


148 


21812 


641.402 


49 


7056 


113.0:U6 


99 


14266 


228.870 


149 


21466 


846.706 


60 


7200 


115.3384 


100 


14400 


280.677 


160 


21600 


M6.016 



I 



WEIGHTS AKD MEAStTBES. 



259 



TABUB OF PRESSURES COBRESPOMDINQ TO OITEH 

HEADS OF WATER. 

Water at maximum density, 62.425 lbs. per cubic foot » 1 gram per cubio 
•tntlmeter ; eorrespondiug to a temperature of 4° Centigrade — Z9:J9 Fahrenheit. 

Pressure in lbs. per square inch = 0.433507 X head in feet. 
Pressure in lbs. per square foot = 62.425 X head in feet. 

Pressures corresponding to heads not given in the table can be found by these 
formulK, or taken from the table by simple proportion. 



Head. 


Pressure. 


Head. 

Inches. 


Pressure. 


Inches. 


lbs. per sq. in. 


lbs. per sq. ft. 


lbs. per sq. in. 


lbs. per sq. fL 




0.086126 
0.072251 
0.108377 
0.144502 
0.180628 
0.216753 


5.202083 
10.4U4167 
15.606250 
20.808333 
26.010417 
31.212500 


7 

8 

9 

10 

11 

12 


0.252879 
0.289005 
0.825130 
0.861256 
0.897381 
0.488507 


86.414583 
41.616667 
46.818750 
52.020833 
57.222917 
62.425000 




Prevnife. 



1 
2 
3 

4 
5 
6 
7 
8 
9 

tb 
11 

12 
18 

14 

10 
16 
17 
18 
19 
20 
21 



24 
2S 
26 

rf 

28 
29 
SO 
31 
88 



0.4885 
0.8670 
1.3005 
1.7340 
2.1675 
2.6010 
ZJ0S45 
3.4681 
3.9016 
4.3801 
4.7686 
5.2021 
8.6806 
6.0691 
6JM)26 
6.9361 
7.3696 
7.8031 
a2366 
8.6701 
9.1036 
9.0372 
9.9707 
10.4042 
10.8377 
11.2712 
11.7047 
12.1382 
12.6717 
18.0002 
1&48S7 
18.8722 
14.8087 
14.7392 
16.1727 
16.8008 



Pressure. 



Ibe. per 
sq. in. 



62.425 


88 


124.850 


80 


187.275 


40 


249.700 


41 


312.125 


42 


374.500 


48 


486.975 


44 


499.400 


40 


561.825 


46 


624.250 


47 


686.675 


48 


749.100 


49 


811.625 


60 


873.950 


01 


986.375 


02 


99SJBO0 


08 


1061.225 


64 


1123J650 


05 


1166i)76 


56 


1248.000 


57 


1310.925 


58 


1373.350 


09 


1435.775 


60 


1498.900 


61 


1560.626 


62 


1628.050 


63 


1685475 


64 


1747J0O 


65 


1810|25 


66 


1872.750 


67 


1935475 


68 


1997 JOO 


69 


2060105 


70 


2122J0O 
2184i70 


71 
72 


28O8J20 


78 

74 



16.4733 
16.9068 
17.3403 
17.7738 
18.2073 
18.6408 
19.0743 
19.0078 
19.9413 
20.8748 
20.8088 
21.24;8 
21.6758 
22.1089 
22.5424 
22.9759 
23.4094 
23.8429 
24.2764 
24.7099 
25.1434 
25.5769 
26.0104 
26.4439 
26.8774 
27.3109 
27.7444 
28.1780 
28.6115 
29.04.50 
29.4785 
29.9120 
30.3455 
30.7790 
31.2125 
31.6460 
32.0795 



Ibe. per 
sq. ft. 



2372.150 
2434.575 
2497.000 
2559.425 
2621.850 
2684.275 
2746.700 
2809.125 
2871.550 
2933.970 
2996.400 
3058.82.5 
3121.250 
3183.675 
3246.100 
3308.525 
3870.960 
8438.378 
3495.800 
3558.225 
3620.650 
3683.075 
3745.500 
3807.925 
3870.350 
3932.77.') 
3995.200 
4057.625 
4120.060 
4182.475 
4244.900 
4307.825 
4369.750 
4432.175 
4494.600 
4557.025 
4619.400 



Head. 

Feet. 



Pressure. 



76 

76 

77 

78 

79 

80 

81 

82 

83 

84 

80 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

110 

111 



Ibe. per 
sq. in. 



82.0130 
32.9460 
83.8800 
83.8180 
34.2471 
84.6806 
35.1141 
35.5476 
35.9811 
86.4146 
86.8481 
37.2816 
37.7151 
38.1486 
38.5821 
39.0156 
39.4491 
39.8826 
40.3162 
40.7497 
41.1832 
41.6167 
42.0502 
42.4837 
42.9172 
43.3507 
43.7842 
44.2177 
44.6512 
45.0847 
45.5182 
45.9517 
46.3852 
46.8188 
47.2523 
47.6858 
48.1193 



lbs. per 
sq. ft. 



4681.870 

4744.300 

4806.720 

4869.150 

49.^1.575 

4994.000 

5056.4^ 

5118.850 

5181.275 

5243.700 

5306.125 

5368.550 

5430.975 

5493.400 

5555.825 

5618.250 

5680.675 

5743.100 

5805.525 

5867.950 

5930.375 

5992.800 

6055.225 

6117.650 

6180.075 

6242.500 

6304.925 

6367.350 

6429.775 

6492.200 

6554.625 

6617.050 

6679.475 

6741.900 

6804.325 

6866.750 

6929.170 



260 



WEIGHTS AND MEASURES. 





TAMIiE 


OF PBESSVBES (€iMitinaed). 






Pressure. | 




Presirare. 




Presiare. 


Head. 
Feet. 






Hemd. 

Feet. 






Head. 

Feet 






lbs. per 


lbs. per 


lbs. per 


lbs. per 


lbs. per 


lbs.psr 
sq. ft 




sq. in. 


aq. ft. 




sq. in. 


sq.ft. 




sq. in. 


112 


48.5528 


6991.600 


144 


62.4260 


8989.200 


176 


76.2972 


10986.800 


lis 


48.9868 


7054.025 


145 


62.8686 


9051.626 


177 


76.7807 


11049.226 


114 


49.4198 


7116.450 


146 


63.2920 


9114.060 


178 


77.1642 


11111.660 


115 


49.8533 


7178.875 


147 


63.7266 


9176.476 


179 


77.5978 


11174.076 


116 


50.2868 


7241.300 


148 


64.1690 


9238.900 


180 


78.0313 


11236.600 


117 


50.7203 


7303.725 


149 


64.5926 


9301.825 


181 


78.4648 


11298.926 


118 


51.1538 


7366.160 


160 


65.0260 


9363.750 


182 


78.8988 


11861.360 


119 


51.5878 


7428.576 


161 


65.4596 


9426.176 


183 


79.3318 


11423.776 


.120 


52.0208 


7491.000 


162 


65.8931 


9488.600 


184 


79.7658 


11486.200 


121 


52.4543 


7653.425 


163 


66.3266 


9651.026 


186 


80.1988 


11648.626 


122 


52.8879 


7615.860 


164 


66.7601 


9613.460 


186 


80.6828 


11611.060 


123 


53.3214 


7678.275 


165 


67.1936 - 


9675.876 


187 


81.0668 


11678.475 


124 


53.7549 


7740.700 


166 


67.6271 


9738.900 


188 


81.4998 


11736.900 


126 


54.1884 


7803.125 


157 


68.0606 


9800.726 


189 


81.9328 


11798.826 


126 


54.6219 


7865.530 


168 


68.4941 


9863.150 


190 


82.8668 


11860.760 


127 


55.0554 


7927.975 


169 


68.9276 


9925.675 


191 


82.7998 


11923.176 


128 


55.4889 


7990.400 


160 


69.3611 


9988.000 


192 


83.2338 


11986.600 


129 


55.9224 


8052.825 


161 


69.7946 


10050.425 


193 


83.6669 


12048.025 


180 


56.3569 


8115.260 


162 


70.2281 


10112.850 


194 


84.1004 


12110.460 


131 


66.7894 


8177.675 


163 


70.6616 


10175.276 


196 


84.6889 


12172.876 


132 


67.2229 


8240.100 


164 


71.0951 


10237.700 


196 


84.9674 


12236JI00 


183 


57.6664 


8302.625 


165 


71.5287 


10300.125 


197 


85.4009 


12297.726 


184 


58.0899 


8364.950 


166 


71.9622 


10362.550 


198 


85.8344 


12860.150 


183 


58.5234 


8427.876 


167 


72.3957 


10424.975 


199 


86.2679 


12422JJ75 


186 


58.9570 


8489.800 


168 


72.8292 


10487.400 


200 


86.7014 


12485.000 


187 


59.3905 


8552.226 


169 


73.2627 


10549.825 


201 


87.1349 


12647.426 


188 


59.8240 


8614.650 


170 


73.6962 


10612.250 


202 


87.6684 


12609.860 


139 


60.2575 


8677.075 


171 


74.1297 


10674.675 


203 


88.0019 


12672.276 


140 


60.6910 


8739.500 


172 


74.5632 


10737.100 


204 


88.4364 


12734.700 


141 


61.1245 


8801.925 


173 


74.9967 


10799.525 


205 


88.8689 


12797.126 


142 


61.5580 


8864.350 


174 


76.4302 


10861.950 


206 


89.3024 


12869.650 


143 


61.9915 


8926.775 


176 


75.8687 


10924.375 


207 


89.7359 


12921.976 



Table sbowlnar the total pressure against a Tertleal plane 

one foot wide, extending froip the surface of the water to tJie depth named in 
the first column. 

Water at its maximum density, 62.425 lbs per cubic foot =» 1 gram p«r cubic 
centimeter, correBpondins to a temperature of 4° Cent. = 39.2° Fahr. 

Total pressure in pounds = 31.2125 X square of depth in feet. 



Depth. 


Total 
pressnre. 


Depth. 


Total 
presrare. 


Depth. 


Totol 
prewire. 


Depth. 


Total 
pt-essare 


Feet 


Pounds. 


Feet 


Pounds. 


Feet 


Pounds. 


Feet 


Pounds. 


1 


31.21 


17 


9020 


33 


38990 


49 


74941 


2 


124.85 


18 


10113 


34 


36082 


50 


78081 


3 


280.9 


19 


11268 


35 


38235 


51 


8118« 


4 


499.4 


20 


12485 


86 


40461 


62 


84899 


6 


780.3 


21 


13765 


37 


42730 


63 


87676 


6 


1124 


22 


15107 


38 


45071 


64 


9101C 


7 


1529 


23 


16511 


39 


47474 


65 


94418 


8 


1998 


24 


17978 


40 


49940 


60 


112866 


9 


2528 


25 


19508 


41 


62468 . 


65 


181878 


10 


3121 


26 


21100 


42 


55069 


70 


162941 


11 


3777 


27 


22754 


43 


57712 


76 


176570 


12 


4495 


28 


24471 


44 


60427 


80 


199760 


13 


5275 


29 


26260 


45 


63205 


86 


225610 


14 


6118 


30 


28091 


46 


66046 


90 


2S2821 


16 


7023 


31 


29995 


47 


68948 


96 


28169S 


16 


7990 


32 


31962 


48 


71914 


100 


8121S8 



WEIGHTS AND MEASUSES. 



261 



TABIiE OF 1»ISCHAB«1» Ilf CUBIC F£ET PCR SECOSTB 

coBBESPonrBiire to eiysjir DiscuABOfis in v. s. 

eAI.I.ONS P£R 24 HOVBS. 



n. S. gallon 

Discharge in cubic feet per second 



231 cubic inches. 

1.54723 X discharge in miUiwu of U. S. gal- 
lons per 24 hours. 



Millions 




Millions 




Millions 




Millions 




ofU. a 


Cubic feet 


of U.S. 


Cubic feet 


of U. 8. 


Cubic feet 


of U. S. 


Cubic feet 


gals, per 


per second. 


gals, per 


per second. 


gals, per 


per second. 


gals, per 


per second. 


24hrs. 




24hrB. 




24hr8. 




24hr8. 




.010 


.0164728 


18 


20.1140 


43 


66.6808 


72 


111.400 


.020 


.0809446 


14 


21.6612 


44 


68.0781 


73 


112.948 


.080 


.0464169 


16 


28.2084 


46 


69.6258 


74 


114.496 


.040 


.0618891 


16 


24.75P7 


46 


71.1726 


76 


116.042 


J080 


.0778614 


17 


26.8029 


47 


72.7197 


76 


117.689 


.060 


.0928837 


18 


27.8601 


48 


74J»70 


77 


119.137 


.070 


.108806 


19 


29.3978 


49 


76.8142 


78 


120.684 


.080 


.128778 


20 


80.9446 


60 


77.8614 


79 


122.281 


.000 


.189261 


21 


82.4918 


61 


78.9087 


80 


123.778 


.100 


.164728 


22 


84.0390 


52 


80.4569 


81 


126.326 


.200 


.309446 


28 


36.6868 


63 


82.0081 


82 


126.873 


.800 


.464169 


24 


87.1886 


64 


83.6508 


83 


128.420 


.400 


.618891 


26 


38.6807 


56 


86.0976 


84 


129.967 


.600 


.778614 


26 


40.2279 


66 


86.6448 


85 


131.614 


.600 


.938887 


27 


41.7752 


67 


88.1920 


86 


133.062 


.700 


1.08806 


28 


43.8224 


68 


89.7398 


87 


134.609 


.800 


1.28778 


29 


44.8696 


69 • 


91.2866 


88 


136.156 


.900 


1.89261 


80 


46.4169 


60 


92.8337 


89 


137.703 


1 


1.64728 


81 


47.9641 


61 


94.3809 


90 


139.251 


2 


8.09446 


82 


49.6118 


62 


96.9282 


91 


140.798 


3 


4.64169 


88 


61.0586 


63 


97.4764 


92 


142.345 


4 


6.18891 


84 


62.6068 


64 


99.0226 


93 


143.892 


5 


7.78614 


&9 


64.1530 


66 


100.670 


94 


145.489 


6 


9.28887 


86 


e».7002 


66 


102.117 


95 


146.987 


7 


10.8806 


87 


67.2476 


67 


103.664 


96 


148.584 


8 


12.8778 


38 


68.7947 


68 


105.212 


97 


150.081 


9 


13.9261 


89 


60.8419 


69 


106.759 


98 


151.628 


!• 


16.4728 


40 


61.8891 


70 


108.306 


99 


153.176 


n 


17.0196 


41 


68.4364 


71 


109 J68 


100 


154.728 


12 


18.6667 


42 


64.9836 











262 



WEIGHTS AND MBASURBS* 



TABIiE OF BISCHAROlMi IN CUBIC FEfiT PEB SBOOUD 
CORRESPONDING TO OITEN BISCHABOES IN IM- 
PERIAIi GAIiliONS PER 24 HOURS. 

Imperial gallon «> 277.274 cubic inches. 

Discharge in cubic feet per second = 1.85717 X discharge in Imperial gallons per 

24 hours. 



Millions 




MilUons 




Millions 




Millions 




of Imp. 


Cubic feet 


of Imp. 


Cubic feet 


of Imp. 


Cubic feet 


of Imp. 


Cubic feet 


gals, per 


per second. 


gals, per 


per second. 


gals, per 


per second. 


gals, per 


per second. 


24hrs. 




24hr8. 




24hr8. 




24hr8. 




.010 


.0185717 


13 


24.1432 


43 


79.8583 


72 


133.7162 


.020 


.0871434 


14 


26.0004 


44 


81.7155 
83.5727 


73 


135.5734 


.030 


.0557151 


15 


27.8576 


45 


74 


187.4306 


.040 


.0742868 


16 


29.7147 


46 


86.429A 


76 


139.2878 


.050 


.0928585 


17 


31.5719 


47 


87.287^1 


76 


141.1449 


.000 


.111430 


18 


33.4291 


48 


89.1442 


77 


143.0021 


.070 


.130002 


19 


35.2862 


49 


91.0013 


78 


144.8593 


.080 


.148574 


20 


37.1434 


50 


92.8585 


79 


146.7164 


.090 


.167145 


21 


39.0006 


51 


94.7157 


80 


148.6736 


.100 


.185717 


22 


40.8577 


52 


96.5728 


81 


160.4308 


.200 


.371434 


23 


42.7149 


53 


98.4300 


82 


162.11879 


.900 


.557151 


24 


44.5721 


54 


100.2872 


88 


164.1451 


.400 


.742868 


25 


46.429$ 


55 


102.1444 


84 


156.0028 


.500 


.928585 


26 


48.2864 


56 


104.0015 


86 


167.8595 


.600 


1.11430 


27 


50.1436 


67 


105.8587 


86 


169.7166 


.700 


1.30002 


28 


52.0008 


58 


107.7159 


87 


161.6738 


.800 


1.48574 


29 


53.8579 


69 


109.5730 


88 


168.4310 


.900 


1.67145 


80 


55.7151 


60 


111.4302 


89 


166.2881 


1 


1.85717 


31 


57.5728 


61 


113.2S74 


90 


167.1453 


2 


3.71434 


32 


59.4294 


62 


115.144$ 


91 


169.0025 


3 


5.57151 


33 


61.2866 


68 


117.0017 


92 


170.8696 


4 


7.42868 


34 


63.1438 


64 


118.8589 


98 


172.7168 


5 


9.28585 


35 


65.0010 


66 


120.7160 


94 


174.6740 


6 


11.1430 


36 


66.8581 


68 


122.5732 


96 


176.4S12 


7 


13.0002 


37 


68.7153 


67 


124.4304 


96 


178.2883 


8 


14.8574 


38 


70.5725 


68 


126.287$ 


97 


180.1465 


9 


16.7145 


39 


72.4296 


69 


128.1447 


98 


182.0027 


10 


18.5717 


40 


74.2868 


70 


130.0019 


99 


183.8698 


11 


20.4289 


41 


76.1440 


71 


131.8591 


100 


186.7170 


12 


22.2860 


. 42 


78.0011 











WEIGHTS AND MEASURES. 



263 



TABIiE OF DISCHAB«ES IN OAIil^OMS PER 84 HOUIIA 
COBKESPONDINO TO OITEST DISCHARGES IN CUBIC 
FEET PER SECOND. 

U. S. gallon = 231 cubic inches. Imperial gallon = 277.274 cubic inchea- 
Diaoharge in U. S. gallons per 24 hours = 646317 X discharge in cubic feet 

per second. 
Discharge in Imperial gallons per 24 hours » 538454 X discharge in cubic fe«i 

per second. 



Onb. ft. 


Millions of 


Millions of 


Cub. ft. 


Millions of 


Millions of 


U. S. gHllons 


Imperial gallons 


per sec. 


U. S. gallons 


Imperial gallons 


per 24 hours. 


per 24 hoars. 


per 24 hours. 


per 24 hours. 


1 


0.646317 


0.538454 


53 


34.254795 


,28.5880U 


2 


1.292634 


1.0769O7 


54 


34.901112 


29.076488 


8 


1.938951 


1.615361 


55 


85.547428 


29.614951 


4 


2.685268 


2.158815 


56 


36.193745 


30.153405 


i 


3.281584 


2.692266 


CT 


36.840062 


30.691859 


6 


3.877901 


3.230722 


58 


37.486379 


81.230312 


7 


4.524218 


8.769176 


59 


38.132696 


31.768766 


8 


5.170535 


4.307629 


60 


38.779013 


32.307220 


9 


5.816852 


4.846088 


61 


39.425330 


32.845678 


10 


6.463169 


5.384537 


62 


40.071647 


33.384127 


11 


7.109486 


5.922990 


68 


40.717963 


33.922581 


12 


7.755808 


6.461444 


64 


41.364280 


34.461034 


18 


8.402119 


6.999898 


65 


42.010597 


34.999488 


14 


9.0484S6 


7.538351 


66 


42.656914 


85.537942 


15 


9.694753 


8.076805 


67 


43.303231 


36.076395 


16 


10.341070 


8.615259 


68 


43.949548 


36.614849 


17 


10.987387 


9.153712 


69 


44.595865 


37.153303 


IB 


11.633704 


9.692166 


70 


45.242182 


37.691756 


19 


12.280021. 


10.230620 


71 


45.888498 


38.230210 


20 


12.926338 


10.769073 


72 


46.534815 


38.768664 


21 


13.572654 


11.307527 


78 


47.181132 


89.307117 


22 


14.218971 


11.845981 


74 


47.827449 


39.845571 


28 


14.865288 


12.384434 


75 


48.473766 


40.384025 


24 


15.511605 


12.922888 


76 


49.120083 


40.922478 


28 


16.157922 


13.461342 


77 


49.766400 


41.460932 


.28 


16.804289 


13.999795 


78 


50.412717 


41.999385 


27 


17.450556 


14.538249 


79 


51.059034 


42.537838 


28 


18.0968(73 


15.076702 


80 


51.705350 


43.076293 


.29 


18.743190 


15.615156 


81 


52.351667 


43.614746 


80 


19.889506 


16.158610 


82 


52.997984 


44.153200 


81 


20.085828 


16.692063 


83 


63.644301 


44 691654 


82 


20.682140 


17.280517 


84 


54.290618 


45.230107 


88 


21.328457 


17.768971 


85 


54.936935 


45.768561 


84 


21.974774 


18.307424 


86 


55.583252 


46.307015 


85 


22.621091 


18.845878 


87 


56.229569 


46.845468 


86 


23.267408 


19.384332 


88 


66.875885 


47..'W3922 


87 


23.913725 


19.922785 


89 


67.522202 


47.922376 


88 


24.560041 


20.461239 


90 


58.168519 


48.460829 


89 


25.206a'W 


20.999693 


91 


68.814836 


48.999283 


40 


25.852675 


21.588146 


92 


59.461153 


49.537737 


41 


26.498992 


22.076600 


93 


60.107470 


50.076190 


42 


27.145309 


22.615054 


94 


60.753787 


50.614644 


48 


27.791626 


23.158507 


95 


61.400104 


51.153098 


44 


28.487943 


23.691961 


96 


62.046420 


51.691561 


46 


29.084260 


24.280415 


97 


62.692737 


52.230006 


46 


29.730576 


24.768868 


98 


63.389054 


52.768459 


47 


30.376893 


25.307322 


99 


63.985371 


53.306912 


48 


81.028210 


25.845776 


100 


64.631688 


53.845366 


49 


81.669627 


26.384229 


101 


65.278005 


54.383820 


50 


32.315844 


26.922683 


102 


65.924322 


54.922273 


61 


32.962161 


27.461187 


103 


66.570639 


55.4WJ727 


62 


83.608476 


27.999590 


104 


67.216956 


55.999181 



264 



WEIGHTS AKD MEASURES. 



TABI4E OF BISCHABOES (Continned). 



Cub ft. 


Millions of 


Minions of 


Cub. ft. 


MilUonsof 


Millions of 


per sec. 


U. S. galloDB 


Imperial gallons 


per sec. 


U. S. gallons 


Imperial gallons 


per 24 hours. 


per 24 hours. 


per 24 hours. 


per 24 hours. 


i05 


67.863272 


66.687684 


167 


107.934919 


89.921761 


106 


68.509589 


67.076088 


168 


108.581236 


90.460215 


107 


69.155906 


57.614542 


169 


109.227553 


90.998669 


208 


69.802223 


58.162995 


170 


109.873870 


91.537122 


109 


70.448540 


68.691449 


171 


110.520186 


92.075576 


110 


71.094867 


69.229903 


172 


111.166503 


92.614030 


111 


71.741174 


69.768356 


173 


111.812820 


93.152488 


112 


72.387491 


60.306810 


174 


112.459137 


93.690937 


118 


73.033807 


60.845264 


175 


113.105454 


94.229891 


lU 


73.680124 


61.383717 


176 


113.761771 


94.767844 


115 


74.326441 


61.922171 


177 


114.898088 


95.806298 


116 


74.972768 


62.460625 


178 


115.044406 


96.844761 


117 


75.619075 


62.999078 


179 


115.690722 


96.388206 


118 


76.265392 


63.537532 


180 


116.337038 


96.921669 


119 


76.911709 


64.075986 


181 


116.983355 


97.460112 


120 


77i»8026 


64.614439 


182 


117.629672 


97.998666 


121 


78.204342 


66.152893 


188 


118.275989 


98.537020 


122 


78.850659 


66.691347 


184 


118.922306 


99.075478 


123 


79.496976 


66.229800 


185 


119.568623 


99.618927 


124 


80.143293 


66.768254 


186 


120.214940 


100.152881 


125 


80.789610 


67.306708 


187 


120.861257 


100.690684 


126 


81.435927 


67.845161 


188 


121.507578 


101.229288 


127 


82.082244 


68.383615 


189 


122.153890 


101.767742 


128 


82.728561 


68.922068 


190 


122.800207 


102.806196 


129 


83.874878 


69.460522 


191 


123.446524 


102.84464» 


180 


84.021194 


69.998976 


192 


124.092841 


108.388108 


131 


84.667511 


70.537429 


193 


124.739158 


103.921666 


132 


85.313828 


71.075883 


194 


125.»85475 


104.460010 


183 


85.960145 


71.614337 


195 


126.081792 


106.098464 


184 


86.606462 


72.162790 


196 


126.678108 


106336917 


186 


87.262779 


72.691244 


197 


127.324425 


106.076S71 


186 


87.899096 


73.229698 


198 


127.970742 


106.618825 


187 


88.545413 


73.768151 


199 


128.617059 


107.162278 


188 


89.191729 


74.306605 


200 


129.268376 


107.690782 


189 


89338046 


74.845059 


201 


129.909698 


106.229186 


140 


90.484363 


76.383612 


202 


130.566010 


108.767689 


141 


91.130680 


76.921966 


203 


131.202327 


109.306098 


142 


91.776997 


76.460420 


204 


131.848644 • 


109344647 


148 


92.423314 


76.998873 


205 


, 132.494960 


110.388000 


144 


93.069631 


77.637327 


206 


133.141277 


110.921464 


146 


93.715948 


78.075781 


207 


133.787594 


111.45990S 


146 


94.362264 


78.614234 


208 


134.433911 


111.998861 


147 


95.008581 


79.152688 


209 


135.080228 


112.536815 


148 


96.664898 


79.691142 


210 


135.726545 


113.075269 


149 


96.301215 


80.229596 


211 


136.872862 


118.618722 


150 


96.947532 


80.768049 


212 


137.019179 


114.152176 


151 


97.593849 


81.306503 


213 


137.665495 


114.690680 


152' 


98.240166 


81.844956 


214 


138.311812 


116.229088 


163 


98.886483 


82.383410 


215 


138.958129 


116.767887 


154 


99.532800 


82.921864 


216 


139.604446 


116.806891 


155 


100.179116 


83.460317 


217 


140.25U768 


116344444 


156 


100.825433 


83.998771 


218 


140.897080 


117.882898 


157 


101.471750 


84.537225 


219 


141.643397 


117321882 


158 


102.118067 


85.075678 


220 


142.189714 


118.468806 


159 


102.764384 


85.614132 


221 


142.836030 


118.998SB» 


160 


108.410701 


86.152586 


222 


143.482347 


119386n8 


161 


104.057018 


86.691039 


228 


144.128664 


120.07616ft 


162 


104.703335 


87.229498 


224 


144.774981 


120.618620 


163 


105.349651 


87.767947 


225 


146.421298 


121.163074 


164 


ia5.995968 


88.306400 


226 


146.067615 


121.600B87 


165 


106.642285 


88.844854 


227 


146.713982 


122.228881 


166 


107.288602 


89.883308 


228 


147.860249 


122.78704 



TIME. 



265 



TABIiE OF I»lS€HAReES (Contlnae^i). 



Oab. ft 


MillioDs of 


Millions uf 


Oub tt 


MilUons of 


Millious of 


per sec. 


U. S. gallons 


Imperial gallons 


per sec. 


U. S. gallons 


Imperial gallou 


per 24 hours. 


per 24 hours. 


per 24 hours. 


per 24 hours. 


229 


148.006566 


123.306888 


240 


155.116061 


129.228878 


230 


148.652882 


123.844342 


241 


155.762368 


129.767332 


231 


149.299199 


124.382795 


242 


156.408685 


180.305786 


232 


149.945516 


124.921249 


243 


157.065002 


130.844239 


233 


150.591833 


125.459703 


244 


167.701819 


131.382693 


234 


151.238150 


125.998156 


245 


158.847636 


131.921147 


235 


151.884467 


126.536610 


246 


158.993962 


132.459600 


236 


152.680784 


127.075064 


247 


169.640269 


132.998054 


237 


163.177101 


127.613517 


248 


160.286586 


133.636608 


238 


163.828417 


128.151971 


249 


160.932903 


134.074961 


239 


164.469734 


128.690426 


250 


161.579220 


134.613416 



TIME. 



60 seconds,*! marked s, =■ 

60 minutes,! *' m, = 

24 hours, " h, = 

7 days, " d, = 

Arc Time 
1° = 4 minutes 
r s= 4 seconds 
V = 0.066... second 



1 minute 
1 hour = 
1 day = 
1 week = 



3600 seconds 

1440 minutes = 86400 seconds 

168 hours = 10U80 minutes 

Time Arc 

24 hours =360° 
Ihour = 15° 
1 minute = 0° 15' 
1 second =* 0° 0' 15" 



Bletbods of reekonins time. Astronomers distinguish between mean 
solar time, true or api)arent solar time, and sidereal time. 

At a standard meridian (see page 267) mean solar time is the same at 
ordinary clock time. At any point not on a standard meridian, standard time 
is the local mean solar time of the meridian adopted as standard for such point ; 
and local time is = time at a standard meridian phu correction for longitude 
from that meridian if the place is east of the meridian, and vice versa. For the 
amount of such correction, see second table above. A true or apparent 
aolnr day is the interval of time between two successive culminations of 
the sun, «.«., between two successive transits or passages of the sun across the 
meridian of the same point ou the earth ; but, since these intervals are unequal, 
they do not correspond with the uniform movement of clock time. A fictitious 
or imaginary sun, called the "mean sun," is therefore supposed to move along 
the equator in such a way that the interval between its culminations is con- 
stant. This interval is called a day, or mean solar day, and is the average of the 
lengths of all the apparent solar days in a vear. Apparent and mean time 
agree at four points in the year, viz., about the middle of April and of June, 
September 1 and December 24. The sun is sometimes behind and sometimes 
in advance of the mean sun, and is called " slow " or " fast " accordingly. The 
sun is " slow " in winter, the maximum being about February 11, when it passes 
any standard meridian, or "souths" (making of^Mrent noon), about 14m, 28s, 
after noon by a correct clock. The sun is " fast," or in advance of the clock, in 
MJty and in the £all, with a maximum, about l^ovember 2, of about 16m, 20s. 

The difference between apparent and mean time is called the equation of 
time. It can be obtained from the Nautical Almanac, or, approximately, by 
taking the mean between the times of sunrise and sunset, as given in ordinary 
almanacs. 

As solar time is measured by the apparent daily motion of the sun, so sidereai 
time is measured by that of the fixed stars, or, more strictly speaking, by the 
motion of the vernal equinox which is the point where the sun crosses the 
equator in the spring. 

* The second was formerly divided into 60 equal parts called thirds (marked 
'") ; but it is now divided decimally. 

f The old and confusing practice of designating minutes, seconds and thirds 
of time (see footnote *) as % " and ''', is no longer in vogue. Days, hours, min- 
utes and seconds are now designated by d, h, m, and s, respectively, thus : 2d, 
20h, 48ni, 65.43 s.j and the symbols ' and " designate minutes and seconds of are. 



266 <TIMB. 

A sidereal dAy" is the interval of time between two tueeeisiye paaaages of 
the vernal equinox (or. practically, of auy star) past the meridian of a ^ven 
point on the earth. It is, practically, the time required for one complete revo- 
lution of the earth on its axi£, relatively to the stars. 

The length of the sideral day is 23 h, 56 m, 4.U9 s, of mean solar time, or S m, 
56.91 A of mean solar time less than the mean solar day of 24 hours. In other 
words, a star will, on any night, appear to set 3 m, 55.91 s earlier by a correct 
clock than it did on the preceding night. Hence, substantially, the number of 
sidereal days in a year is greater by 1 than the number of solar days. 

The sidereal day, like the solar day, is divided into 24 hours. These hours 
are. of oourse, shorter than those of tne solar day in the same proportion as the 
sidereal day is shorter than the solar day. They are counted from to 24, com- 
mencing with sidereal itoon, or the instant when the vernal equinox passes the 
ujmer meridian. 

Tlie etwil day (» 24 hours of clock or mean solar time) commences at mid- 
night ; and the astronomical solar day at noon on the civil day of the 
same date. Thus, on a standard meridian, Thursday, May 9, 2 a. m . civil time, 
is Wednesdav, May 8, 14 h, astronomical time; but Thursday, May 9, 2 p. M., 
eivil time, is Thursday, May 9, 2 h, astronomical time. 

Tbe cItII month is the ordinary and arbitrary month of the calendar, 
varying in length from 28 to 31 mean solar days. 

A sidereal montb is the time required for the moon to perform an entire 
revolution with reference to the stars. Its mean length, in mean solar time, is 
about 27 d, 7 h, 43 m, 12 s. 

A lunation, or synodic month is the time from new moon to new 
moon. Its mean length is about 29 d, 12 h, 44 m, 8 s. 

The tropical or natural year is the time during which the earth 
describes the circuit from either equinox to the same again. Its mean length, 
in mean solar time, is now about 365 d, 5 h, 48 m, 49 s. 

The sidereal year is the time during which the earth describes its orbit 
with reference to the stars. Its mean length, in mean solar time, is about 365 
d, 6 h, 9 m, 10 s. 

The elwll year is that arbitrary or conventional and variable division of 
time comprised between the 1st of January and tbe 31st of the following Decem- 
ber, both inclusive. It contains ordinarily 365 mean solar days of 24 hours, bat 
each yenr whose number is divisible by 4 contains 366 days, and is called a leap 
year, except that those years whose numbers end in 00 and are not multipMB 
of 400 are not leap years. 

To regulate a watch hy the stars. The author, after having rega- 
' lated his chronometer for a year by this method onlv,diffiereid but a few seconds 
from the actual time as deduced from careful solar observations. Select a 
window, facing west if possible, and commanding a view of a roof-crest or oth^ 
fixed horizontal line, preferably about 40^ above the horizon, in order to avoid 
disturbance due to refraction, and distant say 50 feet or more. Note the 
time when any bright fixed star (not a planet) passes the range formed between 
the roof, etc., and any fixed horizontal line about the window frame, as a pin 
fixed in <>it her Jamb. The sight in the window, and the watch, must be illumi- 
nated. The star will pass the range 3 m. 55.91 s. earlier on each suooeeding 
evening. Those stars which are nearest the equator appear to move the fastest, 
and are therefore best suited to the purpose. If the first observation of a given 
star lie made as late as midnight, that saron star will answer for about three 
months, until at last it will begin to pass the range in daylight. Before this 
happens, transfer the time to another star which sets later. By thus tabidating, 
throughout the year, about half a dozen stars which follow each other at 
nearly equal intervals of time, we may provide a standard by means of which 
correct clock time may be ascertained on any clear night. Experinfenting in 
this way with two of the best chronometers, the author found that tWr 
rates varied, at times, as much as from three to eight seconds per day. 

An average man takes two steps (one right, one left) per B c ca»d« 
Hence, march music usually takes one second per measure (or ** bar "). Modem 
watches usually tick five times, and clocks either one, two, or four tlmes^ 
per second. 



STANDARD RAILWAY TIME. 267 

STANBARD RAII.WAT TIME, ADOPTED I8SS. 

The following amtngement of standard time was recommended by the General 
and Southern Time Gonyentions of the railroads of the United States and Canada, 
held respectiyely in St. Louis, Mo., and New York city, April, 18S3, and in Chicago, 
m., and New York city, in October, 1883, and went into effect on most of the rail- 
itMMls of the United States and Canada, NoTembar 18th, 1888. Most of the principal 
cities of the United States hare made their respective local times to correspond with 
it. This system was proposed by Mr. W. V. Allen, Secretary of the Time Gonyen- 
tions, and its adoption was largely due to his efforts. We are indebted to Mr. Allen 
for documents from which the following has been condensed, five standards of time 
or five ** times," have been adopted for the United States and Canada. These are, 
respectively, the mean times of the 60th, 76th, 90th, 106th, and 120th meridians west 
of Greenwich, England. As each of these meridians, in the above order, is 16<> west 
of its predecessor, its time is one hour slower. Thus, when it is noon on the OOch 
meridian, it is 1 p.m. on the 76th, and 11 a. m. on the 106th. vThe following gives 
the name adopted for the standard time of each meridian, and the conventional 
color adopted, and uniformly adhered to, by Mr. Allen, for the purpose of designat* 
ing it and its time, Ac, on the maps published under his anspioess 



Longitude west 
from Greenwich. 


Name of 
Standard Time. 


Conventional 
color. 


W 

76P 

9(P 

106° 

laoo 


Intercolonial. 

Eastern. 

Central. 

Mountain. 

Pacific. 


Brown. 

Red. 

Blue. 

Oreen. 

Yellow. 



Theoretically, each meridian may be said to give the time for a strip of country 
ttP wide, running north and south, and having the meridian for its center. Thus 
ths meridian on which the change of time between two standard meridians is sup- 
p sssd to take place, lies half>way between them. But it would, of course, not be 
practiesble for the railroads to use an imaginary line in passing from one time 
standard to another. The changes are made at prominent stations forming the ter- 
mini of two or more lines; or, as in the case of the long Pacific roads, at the ends 
(tf divisions. As far as practicable, points at which changes uf time had previously 
basn made, were selected as the changing points under the new system. Detroit, 
Wch., Pittoburgh, Pa., Wheeling and Parkersburg, W. Va., and Augusta, Ga., al- 
though not situated upon the same meridian, are points of change between «a$tem 
and central standard times. A train arriving at Pittsburgh from the east at noon, 
and leaving for ths west 10 minutes after its arrival, leaves (by the figures shown 
npon its time-table, and by the watches of its train hands) not at 10 minutes afker 
ISjbat at 10 minntss alter 11. 

The necessity for making the changes of time at principal points, instead of on a 
true meridian line, necessitates also some "overlapping** of the times, or of their 
eolors on the map. Thus, most of the roads between Buffalo and Detroit, on the 
north side of Lake Brie, run Irf ** eastern," or **red,** time; while those on the $ouih 
side of the Ijske, between Buffalo and Toledo, immediately opposite to and directly 
south of them, run by ** central ** or " blue ** time. 

If the chauMs of time were made at ths meridians midway between the standard 
ones, it woula not be necessary for any town to change its time more than 30 min- 
utes. As it is, somewhat greater changes had to be made at a few points. Thus, 
standard time at Detroit is 32 minutes ahead, and at Savannah 86 minutes back, of 
mean locaf time. 

In most cases the necessary change was made upon the railways by simply setting 
docks and watches ahead or back the necessary number of minutes, and without 
making any change in time-tables. 

Raliux, and a few adjacent cities, use the time of the 60th meridian, that being 
the nearest one to them ; but the railroadM in the same district have adopted the 
T6th meridian, or eastern, time; so that, for railroad purposes, intercolonial time 
has never come into force. 

In 1878 there were 71 time standards in use on the railroads of the United States 

and Canada. At the time of the adoption of the present system this number had 

■been reduced, by consolidation of roads, Ac, to hS, By its adoption, the number be- 

tame 5, or, practicslly, 4, owing to the adoption of eastern time by the intercolonial 

roads; as aJrcHsdy explained. 



268 



DIAIA 



DIALLING. 




To malKe a borlxontal San^dlal, 

Draw a line a h ; and at right angles to it, draw 66. From any convenient point, bb c, 
in a fr, draw the perp c o. Make the angle cao equal to the lat of the place ; aJfo 
the angle e o « equal to the same ; Join o e. Bfake e n equal to o e; and from n as a 
center, with the rad e n, describe a quadraat e «; and div it into 6 equal parts. Draw c 
y, parallel to 6, 6; and 
firom n, through the 5 ^ DIAL ^ 

points on the quadrant, ^ 

draw lines n t^n t, ^c, 
terminating in ey. From 
a draw lines a 6, a 4, Ac, 
passing through t, i, Ac. 
From any convenient 
point, as c, describe an 
arc r nt A, as a kind of fin- 
ish or border to half the 
dial. All the lines may 
now be effaced, except 
the hour lines a 6, a 6, 
a 4, Ac, to a 12, or a A; 
unless*, as is generally 
the case, the dial is to 
be divided to quarters 
of an hour at least. In 
this case each of the 
divisions on the quad- 
rant « «, must be subdivided into 4 equal parts; and lines drawn from n, thioaf^ 
the points of subdivision, terminating in ty. The quarter-hour lines must be drawn 
from a, as were the hour Unes. Subdivisions of 6 min may be made in the same 
way ; but these, as well as single min, may usually be laid off around the border, by 
eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi- 
nary dial. To draw the other half of the Fig, make a d equal to the intended thick- 
ness of the gnomon, or style, of the dial ; and draw d 12, parallel, and equal to a 12 ; and 
draw the arc x^ to, precisely similar to the arc rmh. Between x and to, on the arc ng «0, 
space off divisions equal to those on the arc r7nh\ and number them for the hoan, 
as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be 
triangular; its thickness must throughout be equal to a<2 or &«o; its base murt 
cover the space adhv)\ its point will be at ad; and its perp height Av, over A.«^ 
must be such that lines vd^uii, drawn from its top, down to a and d, will make the 
angles u a A, « d io, each equal to the lat of the place. Its thickness, if of metal, may 
conveniently be fh>m ^ to ^ inch ; or if of stone, an inch or two, or more, aooording 
to the siie of the dial. Usually, for neatness of appearance, the back A u « to of the 
style is hollowed inward. The opper edges, ua, v d, which cast the shadows, moat 
be sharp and straight. The dial must be fixed in place hor, or perfectly level ; ah 
and dw must be placed truly north and south ; ad being south, and A«o north. Th» 
dial givee only sun or solar time ; but clock time can be found by means of the ** fiurt' 
or slow of the sun," as given by all almanacs. If by the almanac the tun is 6 miB. 
Ac, fast, the dial will be the same ; and the clock or watoh, to be correct, must be f 
Bin slower than it ; and vice versa. 

To make a Vertical Snn-Dlal. 

Proceed as directed above, except that the angles eao and eo« on the drawing, 
and the angle t«a A or v dir of the style, must lie equal to the oo-latitnde (» dif- 
ference between the latitude and 90^) of the place, and the hours must be num- 
bered the opposite wav from those in the above flgare ; i e, from A to y number 
12, 11, 10, 9, 8, 7 ; and from to tog number 12. 1, 2, 8, 4, 6. The dial plate muat be 
placed vertically, in the position shown in the figure, (kcing ezacuy south, and 
with a A and dw vertical. 



BOABD HBABCBE. 

BOABD HEASTTBE. 



■ fMlowlnv t»Me. Tha u. 



BOARD HEASURB. 
niMe at Bo»r« Mcaanre— (ConUaud.) 



£i 


-- 


„.. 


"Kr-tf 


d.M Id a 


"loriilil.) 


f 


35 


P 


IM 


THIOUt 


MM SS 


IKOIUS. 


P 


i 

i 

1 

1 

J 


.«™ 

if 

1 
1 

1 


,i 

S 
1 

1 
1 


^ 


.HSU 

i 
i 

1 


1 
i 


1 

t 

I 

1 


■a 
1 

1 

1 

s 

i 


1 
1 

i 

i 


nuK. 

1 


1 

1 

) 
J 

3 

1 

I." 

„ 

1 

1 



BOASO HEABURB. 







T»M« of Btmrd Heu 


DTC— 


(OonUn 


Md.) 






1 




a|- 


sx 


M( 


TK 


OKK> 


Ba DT 


nraa 


■S. 


'H 


IH 


1^ 


5 
i 
i 

i 

i 

i 
s 
J 

s 

It" 


i! 


i 


1 

1 


i 

is 

1 

i 


if 

i 

1 


1 

if 


IS 

sir 
;.s 

1 


1 

i 

is 
Is 

is 

!:| 

i 


1 


1 
J 
1 

i 

•- 

i 



BOARD UEASUBE. 
Table of Board Mcaaar* — (Contlnutd.} 



BOARD MEASURE. 



273 



Table of Board Biewiare~(Continued.) 



si 



H 
H 
1. 

H 

2. 
H 
H 
H 

8. 

H 
H 
H 
4. 

H 

6. 

7. 

H 

H 
8. 

H 
H 
H 

9. 

H 

10. 

H 

IS. 

IS. 

H 
u. 

16. 
It. 

17. 

18. 
10. 

ao. 

21. 
». 
IS. 

M. 



of Board Meuure oontaiBOd In on« raaning tfnA of Softotlinga 
of dilftrent dimenaiODS. < Original.) 



THIOKNEStt IK ZKOHIBS. 



10 


lOii 


io« 


lOH 


Ft Rd.M. 


FtBd.M. 


ptBa.M: 


FtBd.lC 


.1083 


.2136 


.2186 


JfM 


.4167 


.4271 


.4375 


.4479 


.6250 


.6406 


.6363 


.671* 


.8333 


.8642 


.8750 


.8956 


1.042 


1.068 


1.094 


1.120 


1.250 


1.821 


1.318 


1.344 


1.458 


1.495 


1.631 


1.568 


1.667 


1.708 


1.750 


1.793 


1.875 


1.922 


1.969 


2.016 


2.0A3 


2.135 


2.188 


2.240 


2.292 


2.349 


2.406 


2.464 


2.500 


2.563 


2.625 


2.688 


2.708 


2.776 


2.844 


2.911 


2.917 


2.990 


3.063 


3.135 


3.126 


3.208 


3.281 


3.359 


.S.333 


3.417 


3.600 


3.583 


3.542 


3.630 


8.719 


3.807 


3.750 


3.844 


3.938 


4.031 


3.958 


4.057 


4.m 


4.255 


4.167 


4.271 


4.479 


4.375 


4.484 


4.594 


4.703 


4.5a3 


4.608 


4.813 


4.927 


4.792 


4.911 


5.061 


6.161 


5.000 


5.126 


5.250 


6.376 


5.208 


5.339 


5.469 


6.599 


5.417 


5.562 


6.688 


6.898 


5.625 


6.766 


6.906 


6.047 


5.833 


5.979 


6.126 


6.271 


&04S 


6.193 


•M4 


6.486 


«.2S0 


<.406 


•.56B 


6.719 


6.468 


6690 


«TB1 


6.943 


6.667 


6.833 


T.00O 


7.167 


6.875 


tMl 


T.219 


7.391 


7.083 


T.960 


7.438 


7.616 


7.292 


1JIT4, 


7.056 


T.838 


7.500 


7J88 


7.676 


8.068 


7.708 


7.901 


8.004 


8.286 


7.917 


8.115 


8.313 


8.510 


8.125 


8.828 


8.631 


8.734 


8.3S3 


8.542 


8.760 


8.968 


8.643 


8.766 


8.960 


9.182 


8.760 


8.969 


9.188 


9.406 


8.958 


9.182 


9.406 


9.630 


9.167 


9.396 


9.626 


9.864 


9.876 


9w699 


9.844 


10 08 


9.583 


9.823 


10.06 


10.30 


9.792 


10.04 


10.28 


10.63 


10.00 


10.26 


10.50 


10.75 


10.42 


10.68 


10.94 


11.20 


10.83 


11.10 


11.88 


11.66 


11.26 


11.68 


11.81 


12.09 


11.67 


11.96 


12.26 


12.64 


12.06 


12.39 


12.69 


12.99 


12.50 


12.81 


13.13 


13.44 


12.92 


1324 


18.66 


13.89 


18.83 


13.67 


14.00 


14.33 


13.76 


14.09 


14.44 


14.78 


14.17 
14.58 


14.62 


14.88 


15.23 


14.95 


15.81 


15.77 


16.00 


15.88 


16.76 


16.13 


15.88 


16.23 


16.63 


17.02 


16.67 


17.08 


17.60 


17.92 


17.60 


17.94 


18.38 


18.81 


18.33 


18.79 


19.26 


19.71 


19.17 


19.06 


90.13 


90.60 


WJOO 


99.60 


21.00 


21.60 

1 



11 
rtBd-M. 

.9893 
.4683 
.6875 
.9167 
1.146 
1.376 
1.604 
1.833 
3.063 
2.292 
2.621 
2.750 
2.979 
8.308 
.3.438 
3.667 
8.896 
4.136 
4.354 
4.583 
4.813 
5.042 
5.271 
5.500 
5.729 
6J68 
6.188 
6.417 
6.646 
6.876 
7.104 
7.333 
7.563 
7.792 
8.021 
8.250 
8.479 
8.709 
8.939 
9.167 
9.396 
9.626 
9.854 

10.06 

10.81 

10.64 

10.77 

11.00 

11.46 

11.92 

12.38 

12.83 

13.29 

13.76 

14.21 

14.67 

16.13 

16.68 

16.04 

16.50 

17.42 

18.33 

19.25 

20.17 

21.08 

22.00 



llji 



rt.Bi.lL 

.23U 

.4688 
7031 

.9376 
1.172 
1.406 
1.641 
1.875 
2.109 
2.344 
2.578 
2.813 
3.017 
3.281 
3.516 
8.730 
3.984 
4.219 
4.453 
4.688 
4.922 
5.156 
6.391 
5.625 
5.869 
6.094 
6.328 
6.363 
6.797 
7.081 
7.366 
7.500 
7.734 
7.969 
8.303 
8.438 
8.672 
S.906 
9.141 
9.376 
9.600 

]0j08 
10.31 
10.66 
10.78 
11.02 
11.26 
11.72 
12.19 
12.66 
13.13 
13.59 
14.06 
14.63 
15.00 
16.47 
13.94 
16.41 
16.88 
17.81 
18.75 
19.69 
20.63 
21.56 
32.50 



UH 



FtBd.lC 

.9306 
.4792 
.7188 
.9688 
1.198 
1.438 
1.677 
1.917 
2.156 
2.396 
2.636 
2.876 
3.113 
3354 
.3.594 
3.833 
4.073 
4.313 
4.552 
4.791 
6.031 
5.270 
6.510 
6.750 
5.990 
6.229 
6..469 
6.708 
6.948 
7.188 
7.427 
7.667 
7.906 
8.146 
8.386 
8.625 



IIH 



9.104 
9.3a 
9.583 
9.823 
10.06 
10.30 
10.54 
10.78 
11.02 
11.36 
11.60 
11.98 
12.46 
12.94 
13.42 
13.90 
14.38 
14.85 
15.33 
13.81 
16.29 
18.77 
17.26 
18.21 
19.17 
20.13 
21.08 
22.04 
23.00 



FtBd.M. 

.2448 
.4896 
.7344 

1.224 
1.469 
1.714 
1.958 
2.203 
2.448 
2.693 
4.938 
3.182 
3.427 
3.67S 
8^17 
4.161 
4.406 
4.651 
4.896 
5.141 
5.385 

,5.680 
5.875 
6.120 
6.366 
6.609 
6.854 
7.090 
7.344 
7.589 
7.833 
8.078 
8-32B 
8.566 
8.813 
9.057 
9.302 
9.547 
9.793 

10.04 

10.28 

10.53 

10.77 

11.02 

11.36 

11.61 

11.75 

12.24 

12.73 

13.22 

13.71 

14.20 

14.69 

15.18 

15.67 

16.16 

16.65 

17.14 

17.63 

1840 

10.58 

20.56 

21.54 

32.52 

23.60 



12 



FCBd.M. 

.8600 
.5000 
.7500 
1.000 
1.250 
1.500 
1.730 
2.000 
2.250 
2.600 
2.750 
8.000 
8.250 
3.600 
8.750 
4.000 
4.250 
4.500 
4.730 
5.000 
6.250 
5.500 
6.750 
6.000 
6.250 
6.500 
6.750 
7.000 
7.250 
7.500 
7.750 
8.000 
8.250 
8.500 
8.750 
9.000 
9.250 
9.500 
9.750 

10.00 

10.26 

10.50 

10.76 

11.00 

11.25 

11.50 

11.75 

12.00 

12.50 

13.00 

13.50 

14.00 

14.50 

15.00 

15.50 

16.00 

16.50 

17.00 

17.50 

18.00 

19.00 

20.00 

21.00 

22.00 

23.00 

94.00 



*>2 
♦"9 



$ 

1. 

H 
H 

2. 

If 

H 

3. 

14 
H 
H 

4. 

H 
H 
H 

3. 

H 
H 

6. 

H 

H 

t 

• . 

H 
H 
H 

8. 

H 

9. 

H 
11. 

12. 

H 

13. 

H 

14. 

>i 
15. 

H 
16. 

H 

17. 

H 

18. 
19. 
20. 
21. 
32 
IS. 
24 



18 



274 



IiAITD SUKYBZIHa. 



LAND SURVEYING. 



In surveyliie • tnet of gimiml, the sites which eoMpose its outline are deri» 
nated by nuraben in the order in which they ocoor. Thst end of each side which 
first presents itself in theooarseof the surrey, may be called its near end ; and the 
other its /or end. The oamber of each side is plaoed at its far end. Thus, in Figr. 1, 
the sarTey being supposed to comroeDce at the corner 6, and to follow the direc- 
tion of the arrows, toe irst side is <>, 1 : and its number is placed at its far end at 1 ; 
and so of the rest. Let NS be a meridian line, that is, a north and south line; 
and EW an east and west line. Than in any side which runs northwaidly; 




Flff.1. 



whether northeast, as side 2; or north westL as sides 8 and 1; or doe north; the 
distance in a due north direction between its near end and its far end, is called 
its lunihing; thus, a 1 is the northing of side 1; Ibthe northing of side 2 ; 4e 
of idde 5. In like manner, if any side runs in a southwardly direction, whether 
southeastwardly, as side 8; or south westwardly, as sides 4 and 6; or due south ; 
the corresponding distance in a due south direction between its near end and its 
far end, is called its southing; thus, d3 is the southing of side 8; 80 of side 4; 
/6 of side 6. Both northinss and southings are included in the general term 
jD^erence of Latitude of a side ; or, more commonly but erroneously, its kUiiude, 
The distance due east, or due west, between the near and the far end of any side, 
is in like manner called the Muting^ or westing^ of that side, as the case nuy be; 
thus, 6 a is the westing of side 1; 6/ of side 6; e6 of side 5; e4 of side 4; and 
6 2 is the easting of side 2 ; 2 d of side 8. Both eastings and westing are included 
in the general term Dqaarture of a side; implying that the side d^xxrU so far 
from a north or south direction. We may say that a side norths, wests, sontheasta^ 
Ac. We shall call the northings, southings, Ac. the Ks, Ss, £b, and Ws ; the lati- 
tudes, lats; and the departures, d^. 

Perfect accuracy is unattainable in any operation inyolyinff the measur»^ 
meuts of angles and distances.* That work is accurate enough, which cannot 
be made more so without an expenditure more than commensurate with the 
object to be gained. There is no great difficulty in confining the uncertainty 
within about one-half per cent, of the content, and this probably never pre- 
▼ents a transfer in farm transactions. But errors always become apparent when 
we come to work out the field notes; and since the map or plot of the surrby, and 
the calculations for ascertaining the content, should be consistent within them- 
selres, we do what is usually called eorreding the errors, but what in fact is simply 
humoring them, in, no matter how scientific the nrocess may appear. We distrib- 
ute them all around the survey. Two methods are used for this purpose, both 
based upon precisely the same principle * one by means of drawing; the other, 
more exact but much more trouolesome. by calculation. The graphic method, in 
the hands of a correct draftsman, is sufficiently exact for all ordinarv purposes. 
Add all the sides in feet together; and divide the sum by their number, for the 
average length. IMvide this average by 8 ; the quotient will be the proper scale 
in feet per inch. In other words, take about 8 ins. to represent an average side. 
We shall take it for granted that an engineer does not consider it accurate work to 

• A 100 ft. ehalii may Tary Its length 5 feet per mile, between winter and sammer. bj m«rc 
ehange of temperature; and this alone will make a differenoe of about 1 acre in 6X1. The turn- 
dent aboald praetiao ploitlng from perfeetUr accurate dau : aa tnoL tSa ejuunpto la table. ^ 181, ot 



LAXD BUBYETINQ. 275 

■Mwatv hto MiglM t9 the nearwi qoarter of a degree, wtaieh 1« tbe atnal prMtiM amonf land'torrey 
tn. Tbey OMi, Df idmbi of tbe engineer's tmntlt, now in aniTonal ose on our pobllo works, be readfq^ 
■eMMOd within a minute or two ; and being thus nocb more accurate than the oompass oonrsee, 
(wtaiob eanoot be read off so eloselr, and which are moreover subject to many lonroes of error,) th«f 
serve to correct the Utter in the oflloe. The noting of the coarses, however, should not be confined t« 
the nearest quarters of a degree, btit should be read as closely as tbe observer oan guess at the minutes. 
The back courses also should be taken at every comer, as an additionid cheek, and for tbe deteetioa 
ef local attraction. It la 
well in taking the oom- 
pass bearings, to adopt 
as a rule, always to point 
the north of tbe compass* 
box toward tbe ohJeet 
whose bearing is to be 
taken, and to read off 
from tbe north end of the 
needle. A person who 
uses indUEerentiy th» M 
and tbe S of the box, and 
of the needle, will be very 
liable to make mistakee. 
n ie beet to measure the 
least angle (shown by 
dotted arcs, Tig 2.) at the 
sther it 




; whether it be 
exterior, ae that at oomer 
ft; or interior, as all the 
others; because it is al- 
ways less than 180° ; so ^ , . •,. ^ 
that there is less danger >; .' Fig. 8. 
ef reading it off ineor- '" 
reetly, than if Itezeeeded 
180P; tiUdBf It for grant. 

ed that the transit InstmnMDt Is graduated fhnn the same lero to 180° each way ; If it is gradnatai 
fkvm sevo to 180° tfte preeaatlon is useless. When the small angle is exterior, subtract it from SIMP 
for the interior one. 

Snppoelng the fleld work to be finished, and that we require a plot from which the oontenta may 
be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be 
measnred br scale, and thtir areas calculated one by one,} a protraction of it may be made at once 
from tbe field notes, either by uslQg tbe angles, or by first oorrtictiag the bearinga by means of the 
angles, and then nsing them. The last is tbe best, because in the first tbe protractor must be moved 
to each angle ; whereas In tbe last it will remain sUtionary while all the bearings are being pricked 
off. Kverj movement of it Inoreasea the liability to errors. The manner of oorreotlng the bewrings 
Is explained on tbe next page. 

In either case the protracted plot will oertainly not eloee precisely ; not only in oonsequence of errors in 
tbe field work, but also in the protracting itself. Thus the last side. No 6, Fig S, Instead of closing in at 
eomer 6, will end somewhere else, say, for instance, at (; the diet 1 6 being the etoting orror, which, 
however, as represented in Pig 3, is more than ten times as great, proportionally to the siie of the 
snrrey. as would be allowable in praetice. Now to hnmor-ln this error, rule through every oomer 
a short line parallel to ( d; and. in all eases, in the direetion from t (wherever it mav be) to tbm 
Btartlag point 6. Add all the sidoB together ; and measure ( fi by the scale of the plot. Then befl)i> 
BiBg at oomer 1, at the fsr end of side 1, say« as the 

Sum of aU . Total dosing . • oiii^ i • Error 

the sides • error «d •• ""^* • Ibrsidel. 

Lay eff tbia error fh>m 1 to a. Then at comer 3, say, as the 

Sum of all . Total olosing . , Sum of • Error 

the sldea • error 16 • • sides 1 and S • for side 1 

Which error lay off from 2 to 6 ; and so at each of the comers; always using, a« the third term, the 
sum of Uie sides between the starting point and the ^ven ooAier. Finally, Join the points a, b, e, 
li, e, 6 ; and the plot la finished. 

The oerreotiec has evidently changed the length of every side ; lengthening some and shortening 
others. U has also changed the angles. Tbe new lengths and angles may with tolerable accuracy 
be fonnd by means of the scale and protractor ; and be marked on the plot Instead of the old ones. 

tnm those to be fbond in books on survering. This Is the only way In which be oan learn what la 
Mt by aecorate work. His semlolrealar protractor should be about 9 to 12 Ins in diam. and gradn- 
I to 10 min. His straight edge and triangle should be of metal: we prefer (vorman silver, which 
I not rast as steel does ; and they should be made with teniptUou* aeeuraey by a skilfUl lustra- 

jt-naker. A very fine needle, with a sealing-wax beiul, should be used for pricking off disU and 

aaglcs; it mnst be held vertically ; and the eye of tbe draftsman most be directly over it. The lead 
peaeU should be hard (Paber's No. 4 is good for protracting), and must be kept to a sharp point by 
rabMiv on a fine file, after nsing a knife for removing the wood. Tbe scale should be at least as long 
aa the longest side of tbe plot, and should be made at the edge of a strip of tbe same paper as the plot 
Is drawn on. This will obviate to a considerable extent, errors arising from contracUon and expao- 
ilea. Unfortunately, a sheet of paper does not contract and expand in the same proportion length* 
•Iss and eroaswlae, thus preventing the paper scale n-om being a perfect corrective. In plots of com- 
1MB farm survi^s, iko, however, the errors rh>m this source may be neglected. For such plott as mav 
m pretraoted. divided, and computed within a time too short to admit of appreciable change, theordi- 
iarf seales of wood, ivory, or metal may be used ; but satisfHctory accuracy oannot be obtained with 
Asm on plots requiring several days, if tbe air be meanwhile alternately moist and dry, or subject to 
ssnsldarable variations in temperature. What is called parehmont paper is worae in this respect thaa 
fsed ordinary drawing-paper. 
With tba ArMoliic preoaatii«8 wa maj work tnm a drawing^ with as mnoh aoenra^ as is iwnaQf 
~i in tli« Md WW*. 



276 



LAND BUBYETINa. 




When U)« plot taM nuny sldM. tula Mlonlating the error for eaob eC tfieai _ 

4aoe, In a weU'performea aurrey and protraoUon, the entire error will be but a verj unall qoanti^, 
jjA abould not exoeed about -r^jr P^>^ of the periphery,) it may uanallj be divided among the sidee by 
merely placing about ^, ^, and H of it at oomera aboat ^ yi, and H way around the plot ; and at 

Intermediate cornera propor- 
tion It by eye. Or caloulatioB 
may be avoided Mtlrely bt 
drawing a line a 6 of a length 

Sual to the united lengtha 
all the aidea ; dividing it 
Into diatanoea a, 1 ; 1, 3 ; Sm. equal to the reapeotive aidea. Make b e equal to the entire oloaing error ; 
join a e ; and ilraw 1 , 1' ; 2. 2' , 4o, which will give the error at each oorner. 

When the plot ia thus completed, it may be divided by One pencil llnea into trianglea, whoaa 
baaea and heights may be measured by the aoale, in order to compute the oontenta. With care In 
both the anrrey and the drawing, the error ahould not exeeatf about -r-Itt V^ ot the true area. At 
leaat two distinot aeta of trianglea abould be drawn and computed, as a guard against miatakea ; and If 
the two aeta dlflbr in calculated oontenta more than about -^^ part, they have not been aa carefully 

frepared aa they abould have been. The doaing error due to imperfect fleld- work, may be accurately 
Mloulated, aa we shall ahow, and laid down on the paper before beginning the plot ; thua furnishing 
• perfect teat of the accuracy of the protraction work, which, if correctly done, will not cloae at the 
point of beginning, but at the point which indicates the error. But this calculation of the error, by 
a little additional trouble, furniahea data alao for dividing it by calculation among the diff aides; 
besides the means of drawing the plot co)-r«c(Zy at once, without the use of a protractor ; thna en»> 
bling uB to make the aubaequent meaaurementa and oomputationa of the triangles with more oar- 
tainty. 

We shall now describe thia proceaa, but would recommend that even when it la employed, and 
aapeeially in complicated surveys, a rough plot should first be made and oorreoted, by the first of the 
two mechanical methods already alluded to. It will prove to be of great service in using the method 
by oalonlation, inaamuoh aa it fumisbes an eye check to vexations mistaken which are otherwise apt 
to occur: for, although the principles involved are extremely simple, and easily remembered when 
once understood, yet the oonUnual changes in the directions of the sides will, without great ears, 
•auae na to uae Na inatead of Sa; Bs instead of Wa, Ac. 

We auppose, then, that such a rough plot has been prepared, and that the angles, bearings, and 
diatancea, aa taken ft'om the field book, are figured upon it in leadptneU. 

Add together the interior angles formed at all the cornera : call their sum a. Unit the number o* 
aidea by 1909 ; from the prod aubtract 360" : if the remainder la equal to the aum a, it ia a proof that 
the anglea have been correctly meaanred.* This, however, will rarely if ever ooeur ; there wHl 
always be aome discrepancy ; but if the field work has been performed with moderate eare, tliis wUl 
not cxcMd about two mln for each angle. In this case div it <n tqttal part* among all the anglea, 
adding or aubtracting, as the caae may be, unleaa It amounta to leaa than a min to each angle, when 
it may be entirely disregarded in common farm surveys. The corrected angles may then be marked 
0n the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be aa 
•hown in Fig S. 

Next, by meana of these 
oorreoted angles, oorreet ths 
bearings alao. thua. Fig t ; 
Select some aide (the longv 
the better) trom. the two enda 
of which the bearing and ths 
reverse bearing agreed ; thns 
showing that that bearinc 
was probably not infloenesd 
by local attraction. Let ilds 
t be the one so selected ; ••» 
sume iM bearing, N 76° ST I, 
as taken on the ground, to be 
correct; through either end 
of it, as at its far end S, draw 
the short meridian line ; par- 
allel to which draw others 
through every ooraer. Now, 
having the bearing of side S, 
M nP 8*i' B, and reqnirfaig 
that of side S, it is pltfn that 
the reverse bearing fromoor> 
ner 8 is 8 75° S2' W ; and 
that therefore the angle 1. %, 
m, is 76° 32'. Therefore, if we 
take IfP 38' trom the entire 
oorreoted angle 1, 8, S, or lUP 
67', the rem 68° 86' wiU bn 
the angle m 83 ; consequently 
the bearing of aideS mstaC be 
8 MO 86' E. For finding the bearing of aide 4, we now hare the angle 88 a of the reveraebearing af 
•Ide S, alao equal to 6»o 26' : and if we add this to the entire corrected angle 234. or tofito 88*. we havs 
theangleaS4 = «8O23'+e»°S3' = 1380 67'; which taken f^m 180°. leaveo the angle 684= il^S'; 




FI9.8. 




• BecaoM in evenr atralght*llned figure the sum of all its Interior 1 
light angles as the figure has sides, minus 4 right angles, or 300°. 



iglos Is eqnal to twlea a« 



LANS SUBYEYINa. 



277 



Mrtftal obMrrstion Is BMestaiy to B«e how tbe aereral angles are to be employed at eaeh oanmt, 
Biilea are sometimes given for this purpose, but unless frequently used, they are soon forgotten. 
The plot ueehanioally prepared obviates the necessity for such rules, inasmuch as the principle of 
proceeding thereby beoomes merely a matter of sight, and tends greatly to prarent error from asing 
the wrong bearings ; while the protractor will at onoe detect any serions mistakes as to the angles, 
and thus prevent their being carried farther along. After having obtained all the corrected bearings, 
Utev may be figured on the plot instead of those taken in the field. Thej will, however, require a 
slUi farther oorreetion after a while, since they will be affected by the adjustment of the closing error. 
We now prooeed to ealoalate the closing error <6 of Fig t, which is done on th« principle that in a 
aorreet survey the northings will be equal to the southings, and the eantings to the westings. Pre* 
pare a tabia of 7 columns, as below, and in the first S cols place the numbers of the sides, and their '^or. 
rsotedooarsee; also the diets or lengths of the Mdes, as meanured on the mugh plot, ifsnchaonQ 
has been prepared ; bnt if not, then as measured on the ground. Let them be as follows : 



Side. 


Bearing. 


Dist. Ft. 


Latitudes. 


Departures. 




N. 


8. 


£. 


W. 


1 
3 
8 

4- 
6 

• 


N10O40'W 
N 750 82' X 
8 69° 25' X 
8 41° 3' W 
N 790 40' W 
8 53030'W 


1060 

1202 

1110 

850 

802 

706 


1015.5 
300.3 

143.9 


800.2 
<U1. 

419.3 


11fl3.9 
1039.2 


804. 

658.2 

789. 
566,7 




1459.7 
1450.6 


1460.5 

Error In 
Lat. 


2203.1 

Error in 
Dep. 


2217.9 
2203.1 




9.2 


14.8 



Kow. bj means of tne Table of Sines, etc., And the N, 8, R, W, of the several sides, and place 
them in the oorrespAoding four columns. Thus, for side 1, which is 1(M0 feet long, with bearing 
N 1|0 40' W ; cos ItP 4(K &s 0.9580 ; sin 16P 40' = 0.2868. 

Hare N s 1000 x 0.9580 s 1015.5; and W s 1060 X 0.2^ = 304. Prooeed 
tbvs with all. Add vp the foor eols ; find the dllT between the N and S ools ; and also between 
the B and W ones. In this instance we find that the Ns are 0.2 feet greater than the Ss ; and that 
the Wa are 14.8 ft greater than the Is ; in other words, there is a eleslntf error which wonld cause a 
mrrtct protraotion of oar first three eels, to terminate 9.2 feet too far north of Um starting point : and 
14.8 feet too ter west of it. 80 that by placing this error npon the paper before beginning to protraet, 
We should bare a ten ftnr the aoenraoy of the protracting work ; bnt, aa before remarked, a little more 
IrenUe will now enable us to div the error proportionally amonc all the Ms, Ss, Sa, and Ws, and thereby 
give aa data for drswing the plot correctly at once, without using a protractor at all. 

To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaeea 
are farther apart : and that the addings-np ef the old N, S, B, W oolunns are omitted. The additioai 
here aotloed are made subseqaently. 

The saw table is on (ha nasi pafs. 

Bkm AKX. Tbe l>earinir And ibe reverse bearing from the two ends 
of a line will not read preciHt'ly the same argle; and the differauce varies with the 
latitode and with the length of the line, but not in the same proportion with either. 
It is, however, generally too small to be detected by the needle, bein^p, according ^o 
Gummare, only three quarters of a minute in a liue one mile long in lat 40°. In 
higher lata it is more, and in lower ones less. It is caused by the fact that meridians 
or north and soath lines are not truly parallel to each other; but would if extended 
■eet at the poles. 

Heaee tbe only bearing (bat can be run in a straigbt line, 

eilh ttrlet aocnraey, is a true N and 8 one ; except on the very equator, where alone a due E and w 

one will also be straight. But a true curved E and W line may be found 

■lywhere with suffioient accuracy for the survevor's purposes thus. Having first by means of the N 
ttMrmtUt or otherwise got a true N and 8 bearing at the starting point, lay off from it 90*, for a true 
land W DMtring at that point. This B and W bearing will be tangent to the true E and W curve. 
Baa this tangent carefully : and at intervals (say at the end of each mile) lay off ftrom it (towards 
the N If in N lat, or vice versa) an ofltet whose length in /Ml is equal to the proper one from the 
Wlowinff (able, multiplied by the sotiare of the distanee in mtlM from the star«iug point. These 
•bets will mark points in the tme K and W curve. 



10° 



lao 



SOO 



liatitade IT or H. 

250 80° 960 409 46° 



500 



550 



003 



«• 



OAieUi in ft one mile ft*oni startinfr point. 

4M .118 .179 .34S .311 .885 .467 .559 .667 .795 .952 1.15 1.43 

te, any offiiet in ft = .6666 X Total Dist in miles> X Nat Tane of Lat. 

A rtiainb line is any one that crosses a meridian obliquely, that is, ia 
■•flher d«S ir ttitf 8, nor E and W. 



278 



LAND SURVEYING. 



Side. 


Bearing. 


Dist. Ft. 


Latitudes. 


Departures. 




N. 


S. 


K. 


W. 


1 


N 16° 40^ W 
N 75° 32' E 
S 69° 25' E 

a 410 3' w 
N790 40^ W 
S 53° 30' W 

* 


1060 

1202 

1110 

850 

802 
705 


1015.6 

1.7 






3O4.0 

2.7 




1013.8... 


... 301.3 


2 


300.3 
1.9 


390.2 
1.8 


1163.9 
3.1 




3 


298.4 

143.9 
1.3 


... 1167.0 

1039.2 
2.9 




4 


392 ... 

641.0 
1.3 


... 1042.1 


558.2 
2.2 


5 


642.3... 

419.3 
1.1 


.. 556.0 

789.0 
2.1 


6 


142.6... 


... 786.9 

666.7 
1.8 




420.4... 


664.9 




5729 

Sum of 

Sides. 


1454.8 
Cor*d Na. 


1464.7 
Cor'd Ss. 


2209.1 
Cor'd Es. 


2209.1 
Cor'd Ws. 



Kow we have alrewlj foaAd by the old Uble that the Ns and th« W« are too long; oonaoquent^ 
fhey must be shortened ; while the Be, and E«, maet be lengthened ; all in the following proportieBa: 
▲•the 

Sum of all . Any given .. Total err of . Err oflat, erdep, 
the Eidee * side * * lat or dep • of giren elde. 

Thng, oommencing with the lat of side 1, we hare, as 

Sum of all the aides. . Sldel. .. Total lat err. . Lat err of side L. 
6729 • 1060 • • 9.2 • l.t 

Now as the lat of side 1 is north, It mnst be shortened ; henee tt keooma«'=:10IS.5-~l.T3dCtaj^ as 
Bgured oat in the new table. Again we hare for the departinv of side 1, 

Snm of all the aides. . Sldel. .. Total dep err. . Dep err of aide 1. 
6729 • 1060 • • 14.8 • 2.7 

Vow as the dep of side 1 is west, it most be shortaned; faenes it beaoiMB9M— S.T=^m;S, «a figvraa 
out in the new table. 

Prooeedlng thus with eaeh 
side, we obtain all the corrected 
lats and deps as shown in the 
new table : where thej are oon- 
nected wfth their reepeotlT* 
sides by dotted lines; but la 
praotioe it is better to oross oal 
the original ones when the oal" 
onlatlon is finished and proved. 
If we now add upthe 4 eols of 
oorrected N, S, S, W,w« And *^ %t 
the Ns =: the Ss ; and tha S8= 
the Ws; thus proving (hat the 
work is right. There la. It la 

5fi \ / true, a dlsorepanoy of .1 of a ft 

I \- ^^j^ — y betweentbeNs, andtheSs; bat 

tbis is owing to oar oarryiBg 
out the oemotions to only oaa 
deoimat plaoet and la too small 
to be regarded. Diaerepmnofaa 
of 8 «r 4 t^thi of a foot wtn 
sometimes ooear f^m this 
cause; but may ha n^lootad. 
The oorrsolod late and dioM 
mast ovUaatty ehaiifa tha 
bearing aad dlstanoa or a 
bnt wttheut knowing either of these, we eaa aew plot the survey by means of the 




FUr.4. 



LAND SUBTEYIMQ. 









ir.iM. 


i. iM. 


-"'"."■ 


-',-■"■ 


1 


su 


no.« 


ino.o 


i 


|g«;^5^?;^|,s-£ Stt-J'A i. 




^ 


«,d^. 


•W.i,^ 


KJKE 








USi 


i 


§ 





280 



LAND BURTEYING. 



•r the •orragr.* The oomoted northings and southian we have already found ; ae alio the eaatinfi 
and wesUngi. The middle diata are fouDd by meau of the latter, by employing their holvM ; adkUng 
hair eaatinge, and lubtraeting half wectinga. Thne it ia evident that the middle dist 2' of aide a, is 

Snal to hair the easting of side S. To this add the other half easting of side 2, and a half easang 
side S ; and the sum is plainly equal to the middle dist 8' of side 8. To this add the other half 
easting of Ride 3, and subtract a half westing of side 4. for the middle dist 4' of side 4. From this 
subtract the other half westing of side 4, and a half westing of side 6, for the middle dist 6' of side 
6i and se on. The actual calAulation mi^ be made thus : 



Half easting of side 3 = 



2 



= fi8lS.5 E £= mid dUt of side 1 
S8S.6 I 



Half easting of side 8 = 



IMll 1167.0 E 
— = 521.0 E 



1688.0 E = mid dist of side t. 
621.0 E 



■Of 



556 

ting of ride 4 = — 

2 



2209.0 E 
= 278.0 W 



19S1.0 E = mid dist Of aide 4» 

278.0 W 



786.t 166S.0 E 

Balf vesting of side 6= = 8W.5W 

2 



1259.5 E = mld«iator«ide6. 
88S.5W 



Half westing of side 6 = 



564.9 



866.0 E 
282.4 W 



688.6 EsmMdlstefiUett. 
282.4 W 



Balf veeting of side 1 = 



801.8 



801.2 E 
lfi0.6W 



160.6 Est mid dist of side 1. 



The work always proves Itself by the last two results being equal. 

Next make a table like the following, in the first 4 ools of whioh plaoe the numbers of th« sldaa, 
the middle dists. the northings, and loathings. Mult each middle dist by its corresponding northing 
or southing, and place the products in their proper col. Add up each col ; subtract the least flrom the 



Side. 

1 
2 
8 

4 
6 
6 



Middle dist. 



150.6 
583.5 

1688 

1931 

1259.5 
583.6 



Northing. 



1013.8 
298.4 



142.6 



Southing. 



392 
642.3 

420.4 



North prod. 



152678 
174116 



179605 



506390 



Sonth prod. 



661606 
1240281 

245345 



2147322 
506399 



43560)1640923(37.67 Aont. 



• Proof. To lllnatrate the principle npon whioh this 
mle is based, let a 6, be, and c a. Fig 6, represent in 
order the 8 sides of the triangular plot of a survey, with 
a meridian line <l^ drawn through the extreme west cor* 
ner, a. Let lines o d and ef be drawn from eaeh oomer, 
perp to the meridian line ; also from the middle of eaeh 
side draw lines w e, m n, « o, also perp to meridian ; and 
representing the middle dlsts of the sides. Then sinoe 
the sides are regarded in the order a 6, 5 e, e a, it is 
plain that a d represents the northing of the side a b ; 
fa the northing of ea; and d/ the southing of 6e. 
Aow if we mult the nothing ad ot the side ab, by its 
mid dist ew, the prod Is the area of the triangle abd. 
In like manner the northing fa of the side ea, mult by 
its mid dist « o, gives the area of the triangle a ef. Again, 
the $otUhing dfot the side b e, mult by lu mlddistmn, 
gives the area of the entire flg dhefd. If ftom this 
area we subtract the areas of the two triangles at tf, 
and aef, the rem is evidently the area of the plot •6«. 
^ith any other plot, however oomi^lflated. 




Fi|r.& 



IJLND SURVKTINQ. 



281 



■natMt. Th« ran will be tbe area of the rarvey in aq ft ; which, div by 4S6M, (the namber af aq ft 

la an aore,) will be tbe area in aor^a ; in this iusiauoe, 37.67 ac. 

It now remaina enly to oaloalate the eorreeted beariugs and lengptha of the sides of the sorrey, all 

of which are neceaaarUy changed by the adoption of tbe eorreeted lau and deps. To And the bearing 

of any aide, dir lu departure (K or W) by Ita 1m (N or S) ; in the table of nat tang, find (he qnot ; 

HOI 3 W 
the angle opporite It Is (he reqd angle of bearing. Thus, for the oourae of aide 1, we hare >-— ' — — 

=: .3972=rnat tang ; oppoaite which in the table is the reqd angle, l(P 8S' ; the bearing, therefore. Is 
K 1«» M' W. 

Again : fer the dial or length ef any aide, from the table of nat cosines take the cos opposite to 
tbe angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the oos. Tons 
for tlie diet of side 1, we find opposite 16° S3', the coa .9686. And 

Lat. Cos. 
1013.8 -i- .9686 » 1067.6 the reqd disk 

Tte MlaiwiBc table oontaias all the cMreotifOiis ef the foregoing snnr^y ; eonaeqaeatly, if the bear. 



Side. 


Bearing. 


Dist.7t. 


1 

S 
8 
4 
6 
6 


N 16® 33' W 
N 760 Sy E 
S e«0 23'K 
S40O63' W 
N 78«> 44' W 
8 63® 21' W 


1057.6 

12M.0 

1118.3 

849.6 

800.1 

704.3 



.*. 



tags anA dlsts are correctly plotted, they will close perfictly. The yeang asatatant Is adTised ta 
prafBtiae doing thla, as well as dtviding the plot Into triangles, and oempottng the content. In this 
manner be will soon learn what degree of care is neoeseary to insiue aocarats resalis. 

The following hlsta may often be ef serrloe. 
1st. ATold taking bearings and 
Aisle along a eirenitoas bound- a 

atyUnelikeate, Fig7;bQtma •. ......................_.._=' » .«> 

the etralght line a c ; and al - . -r* 

right anglea to It, measure ofT 
sets to tbe crooked line. 94. 
iTisblng to surrey a straight 
flna fMm a to e, bat being ana" 
ble to direct the instrument 
precisely toward e, on account 
ef iBierreainv woods, or ether 
ebattMlea; first nm atrialUnab 
as • «». as nearly in the proper 

direotlon aa can be guessed at. . 

Measure m e. and say, as a m is to in e, so ts 100 ft to T Lay off a o equal to 100 ft, and o • equal 
to r ; and run the final line a s e. Or. if m is quite small, calculate offsets like o s for erery 100 ft 
alnc a », and thus avoid the aeeesslty for running a second line. Sd. When e is Tisible from a, but 
dia uitervenlng ground dllBcnIt to measure along, on account of marshes, Ice, extend the side y a 
to good ground at t : then, making the angle ytd equal to y a o, run the line t n to that point d at 
wlaiA the ma^ ndel» found by trial to be equal to the angle atd. It will rarely be necessary to 
mmkm asore than one trial for this point d; for, suppose it to be made at x, see where it strikes a e at 
<; aioaeaw 4 e, and eontinoe ftxmi x, making a <( =< c 4th. In case of a very irregular piece of 
laad. or a lake, Fig 8, surround it by straight lines. Surrey these, and at right angles to them, 
■MMaro ofbets to the crooked boundary, ftth. SurTeyiBg a straight line from w toward y, Fig ft 





m 



Ffff.ft. 



« d 
Flff.lO. 



n 



FI9.0. 



s 



o. Is net To iMMs It, lay off aright aagletptw; measure any <«; make It* OS 
I v; make «» v < =90°; make « < = ( i»; make •<y = 90°. Then is ti = uv; and 
ly la in the straight line. Or, with less trouble, at g make I g a=aOPt measure any g a; make 
#«s3=d0O; and«s = |r0: make a«< = 60O. Then is y • = 9 a or ••; and < s, continued toward 
r. Is la the etralght Hue. fth. Being between two ol^eets, m and n. and wishing to place myself ia 
laagi with them, I lay a straight rod s b on the ground, and point it to one ef the objects m ; then 
to the end e, I And that It does not point to the otaT ofejeet. By suoeessire trials, I find tbe 
e # te vhleh H polats to both otjects, and eoaseq. wtly is ia range with them. 



282 CHAINING. 

CHAINUrO. 

Chains. EDgineers have abandoned the Gunter's chain of 6& ft, divided 
into 100 links of 7.92 ins each. They now use a chain of 100 ft^ with 100 links 
of 1 ft each, and calculate areas In sq ft, the number of which, divided by 
43,560, reduces to acres and decimals, instead of to acres, roods, and perches, 
Giinter's chain is used on U. S. Government land surveys. 

Chains are commonly made of iron or steel wire. Each link is bent &i each 
of its ends, to form an eye, by which it is connected with the adjacent linki, 
either directly, as in the Grumman patent chain, or, more commonly, by from 
1 to 3 small wire links. The wear of tnese links is a fruitAil source of inaccuracy, 
inasmuch as even a very slight wear of each link considerably increases the 
length of the chain. Hence, chains should be compared with some standard, 
sucn as a target rod, every few days while in use. For transportation, the 
lengths are folded on each other, making a compact and sheaf-like bundle. 

Tapes. With improved facilities for the manufacture of steel tape, the chain 
is going out of use. The tape, being much lighter, requires much less pull, and, 
as there are no links to wear, its length is much more nearly constant than that 
of the chain. It is replacing, to some extent, the base-measuring rod for 
accurate geodetic work. Steel tapes are made in continuous lengths up to 600, 
600, and even 1000 ft, but those of 100 ft are the most commonly used. Very 
long tapes are liable to breakage in handling. Even the shorter lengths, unless 
handled carefully^ are apt to kink and breaC Breaks are difficult to mend, and 
the repaired joint is seldom satisfactory ; whereas a kink in a wire chain seldom 
involves more than a temporary change of length. Being run over by a car or 
wagon will often kink steel tapes very badly, if it does not break them.* How* 
ever, the lightness, neatness, and reliability of the tape ofiG^et these disadvan* 
tages, which, indeed, the surveyor soon learns to overcome. 

Tapes for general field work are usually narrow (from 0.10 to 0.25 in) and 
thick (from 0.018 to 0.025 in),t and are graduated by means of small brass 
and copper rivets, spaced, in general, 6 ft apart, 1 ft apart in the 10 ft at eac^ 
end, and 0.1 ft apart in the ft at each end. They are usually mounted on reels. 

Tapes for city work are wider (from 0.25 to 0.5 in) and thinner (from 0.007 to 
0.010 in)t and are graduated (usually to 0.01 ft) throughout their length by 
means of lines and numerals etched on the steel. 

Pins are ordinarily of wire, pointed at the lower end, and bent to a ring at 
the upper end. They can be forced into almost any ground that is not exceed- 
ingly stony. A steel ring, like a large key rin^ is often used for carrying the 
pins. Each pin should have a strip of bright red flannel tied to its top, in order 
that it may be readily found, among the grass, etc., by the rear chainman. 

Corrections for Hofs and tStretcll. The following diagram ^ (seep. 
283) gives the correction for a steel tape weighing 0.75 fi> per 100 ft.t 

*The Nichols Engineering & Contracting Ck>., Chicago, guarantees that its 
tapes will not be injured by beins run over by wagons. 

fThe sizes of tapes, as made by different manufacturers, vary greatly. In 
applying the corrections, therefore, the width and thickness of the tape to be 
used should be carefully measured, and its weight per ft computed. 



X Deduced from diagrams constructed by Mr. J. O. Clarke, Proceedings Engi- 
)ers' Club of Philadelphia, April, 1901, Vol. XVIII, No. 2. from the formuU : 

Stretch, in feet 



neers' 

PS 

EA 

where 

P = pull on tape, in fl>s. 

S = span of tape, in feet. 

E = modulus of elasticity for steel = 27,600,000 flt>s per sq in. 

A = area of cross-section of tape weighing 0.76 B> per 100 ft. 

= 0.0022 square ins, 

and from the equation of the parabola, according to which 

W> S* 
shortening by sag, in feet = ^ 

where W = weight of tape, in pounds per foot. 

Except for very light pulls, this last formula gives practically the same reaalts 
as the equation of the catenary, which is absolutely correct, but much more 
cumbersome. 



, an StHi Tape Wallihing f, 



TbuA, a tupBj of uj teiigthf weiohlug 1 lb 

iDj-giTenooiiKtioD,m pull oti-^j=lHy. 

J, - r le OOfTactlan on tbaata-adard tape, weighing 0.70 

CoDveTselT : cItct a pull Qf 10 bs on a SO ft ipan of a tape wdfthlnd; D.fl Tb per 
lOOrt; requiredtheaorrectian. Ta produaelbeBameemirln tbeUpe welgbtng 



0.7S lb per 100 ft "onlii require 
the diagram at 1Z.G Bn on tb 
Tble ia thfl proper AorrAcliQi 
li^itar tape vllh 10 *■ pull. 

bB of ■Undn'd ferrglh at M^»^r. For' ordinarr eteel tape, Uie t 
MJ nperatu re it about (10000085 ft pCT ft per degree " ' 



lU of y = 10 X j;^ - 12.0 lbs. Beftrrl 
'or » » ft span, we flna comictinn " - 



□ ight, a 



Wben measuring oter slopliiL , . 

tapesbnuld beheld as Dearlf boriioutal as possible, trsnsferrlug the poaitloD of 
Ibe raised end to the ground bj means of a plumb line. Where the ground Is 
■teep, It b^xiines necessary to use a short length of tape, as the down-hul ebain- 

psraliel with theslope, and the disUncecarrecledGr the (ullowlDg form 



284 



LOCATION OF THE MERIDIAN. 



IiO€ATIOIir OF THE HERIDIAHT. 

By means of clrcampolar stars. 

(1) Seen from a point O (Figs. 1 and 2) on the earth, a circumpolar star e 
(•tar near the pole P) ap(>ears to describe daily* and counterclockwise a 
small circle, euwl, about the pole. The angle P O e, P O u, etc., subtended 
by the radius P e, P u, etc., of this circle, or the apparent distance of the 
star from the pole, is called its polar distance. The polar distances of 
stars vary sligntly from year to year. See Table 3. They vary slightly also 
during each year. In the case of Polaris this latter yariation amounts to 
about 50 seconds of arc. 

(3) The altitude of the pole is the angle N O P of the pole's elevation 
above the horizon N E S W, and Is = the latitude of the point of obser- 





FiG. 1. 



Pig. 2. 



ration. Decl Inatlon = angular distance north or south from the celestial 
equator. Thus, declination of pole = 90°. Declination of any star = 90°— its 
polar distance. 

(3) Let Z e H be an arc of a vertical circlet passing through a circumpolar 
star, e, and let H be the point where this arc meets the horizon N E S W. 
Then the angle N Z H at the zenith Z, or N O H at the point O of observa- 
tion, between the plane N Z O of the meridian and the plane H Z O of th© 
star's vertical circle (or the arc N H), is called the azlmutlkt of the star. 
If this angle N O H be laid off from O H, on the ground, the line O N will be 
in the plane of the meridian N Z S, or will be a nortb-and-sontii 
llne.|| 

(4) When a star is on the meridian Z N of the observer, above or below 
the pole P, as at u or ^, it is said to be at its upper or lower culmina- 
tion, respectively. Its azimuth is then = 0, tne line O H coinciding with 
the meridian line O N. 

(5) When the star has reached its greatest distance east or west ftom the 
pole, as at e or w, it is said to be at its eastern or western eloni^A- 
tlon.{ 



« In 23 h. 56.1 m. 

t A great circle is that section of the surface of a sphere which is formed 
by a plane passing through the center of the sphere. A vertical circle is a 
great circle passing through the zenith Z. 

I Astronomers usually reckon azimuth from the south point around 
through the west, north, and east points, to south again ; but for our pur- 
pose it is evidently much more convenient to reckon it f^om the north 
point, and either to the east or to the west, as the case may be. 

II The point N, on the horizon; is called the north point, and must not 
be confounded with the north pole P. 

g As seen ttova. the equator, a star, at either elongation, is, like the pole 
Itself, on the horizon ; and the two lines Pe,Tw, joining it with the pole, 
* — I a single straight line perpendicular to the meridian, and lying in the 



LOCATION OF THE MERIDIAN. 



285 



(6) The boar anffle of any star, at any given mconent, is the time 
which has elapsed since it was in upper culmination.'" 

(7) Evidently the azimuth of a star is continually changing. In cir- 
cumpolar stars it varies from OP to maximum (at elongation) and back to 
(P twice daily, as the star appears to revolve about the pole ; but when the 
star is near either elongation the change in azimuth takes place so slowly 
that, for some minutes, it is scarcely perceptible, the star appearing to 
travel vertically. 

(8) Given the polar distance of a star and the latitude of the point of 
observation, the aaimutli of the star, at eloiiirAtlon, may be found 
by the formula.f 



Sine of azimuth of star = 



sine of polar distance of star 

cosine of latitude of point of observation 



or see (11) and Table 3. 

(9) The following circumpolar stars are of service in connection 
obeervations for determining the meridian. See Fig. 3. 

Constellation Letter 

Ursa minor (Little bear) a (alpha) 

Ursa major (Great bear) € (epsilon) 

( " " i <(zeta) 



with 



Cassiopeia 




S (delta) 



Called 

Polaris 
Alioth 
Mizar 
Deltas 



Jfora»r^.^tet:» 








July 

Fig. 8. 




(10) Polaris^ or the nortb star, is fortunately placed for the determi- 
nation of the meridian, its polar distance being only about 1%^. See Table 
3. Fig. 3 shows the circumpolar stars as the}r appear about midnight in 
July ; inverted, as in January ; with the left side uppermost, as in April ; 
ana, with the right side uppermost, as in October. R 



horizon. The azimuth of the star is then == its polar distance. But in 
other latitudes Pc and Pit; form acute angles with the meridian, as shown, 
and these angles decrease, and the azimuth of the star at elongation in- 
creases, as the latitude increases. 

* In lat. 40° N., the hour angle, ZPc = ZP«>, of Polaris, at elongation, is 
= 5 h. 55 m. of solar time. Caation. It will be noticed that, except for 
an observer at the equator, the elongations do not occur at 90° from the 
meridian. 

t In the spherical triangle Z P «, we have : 

sin e Z P ^ sinPe 

sin Z e P ^ Bin P Z 

But, since Z « P = 90°, sin Z « P = 1. Also, sin P Z = cos (90° — P Z), and 
< Z P — azimuth of e. 

sin Pe _ sin polar d ist ance P O e 

cos latitude 



Hence, sin azimuth of e . ^ „ 

sm F Z 

1 « Cassiopeia is here called Delta, for brevity. 

I Polaris is easily fonnd by means of the two well-known stars 

called the *^ pointers '' in " the dipper," Fig. 3, which forms the binder 



286 LOCATION OF THE MERIDIAN. 

(11) Table 3 ffives the polar distances of Polaris and their log sines for 
January 1 in each third year from 1900 to 1990 inclnsive, the log cosines 
of each fifth deeree of latitude from '2/iP to 50°, and the corresponding 
azimuths of Polaris at elongation. Intermediate values may be taken by 
interpolation.* 

(12) By olMervatlon of Polaris at elonntlon. This method 
has the convenience, that at and near elongation the star appears to travel 
vertically for some minutes, its azimuth, during that time, remaining 
practically constant : but during certain parts of tne year (see Table 1;, the 
elongations of Polaris take place in daylight; so that this method cannot 
then be used. | See (18), (19), (22). Nor can it be used at any time in places 
south of about 4° N. lat., because there Polaris is not visible. 

(18) The approximate times of elongation of Polaris for certain dates, 
in 1900, are given in Table 1, with instructions for finding the times for 
other dates. Or, watch Polaris in connection with any of those stars which 
are nearly in line with it and the pole, as Delta, Mizar, and Alioth. See 
Fig. 3. The time of elongation is approximated, with sufficient clofleneas 
for the determination of the azimuth, by the cessation of apparent hori- 
zontal motion duriftg the observation. 

(14) From fifteen to thirty minutes before the time of elongation, have 
the transit, see (21). set up and carefully centered over a stake previously 
driven and marked with a center point. The transit must be in adjust- 
ment, especially in regard to the second adjustment, p. 294, or that or the 
horizontal axis, by which the line of collimation is made to describe a ver- 
tical plane when the transit is leveled and the telescope is swung upwMrd 
or downward. 

(15) Means must be provided for illuminating the cross-hairs of the tran- 
sit. X I'h^ T^^y ^ done by means of a bull's 
eye, or a dark lantern, so neld as not to throw 
its light into the eye of the observer ; or, better, 
by means of a piece of tin plate, cut and per- 
forated as in Fig. 4, bent at an angle of 45^, as 
in Fig. 5, and painted white on the surface 
next to the telescope. The ring, formed by 
bending the long sirip, is placed around the 
object end of the telescope. A li^ht, screened 
from the view of the observer^ is then held, 
at one side of the instrument, in sucb a way Fig. 4. . Fig. 6. 
that its rays, falling upon the oblique and 

whitened surface of the tin plate, are reflected directly into the telescope. 

(16) Bring the vertical hair to cut Polaris, and, bv means of the tangent 
screw, follow the star as it appears to move, to the right if approaching eoM.- 
em elongation, and mce versa, keeping the hair upon the star, as nearly as 
may be. As elongation is approached, the star will appear to move more 
and more slowly. When it appears to travel vertically along the hair, it 
has practically reached elongation, and the vertical plane of the transit, 
vriih the vertical hair cutting the star, is in the plane of the star's vertical circle. 
Depress the telescope, and fix a point in the line of sight, preferably 300 
feet or more distant from the transit.f Immediately reverse the transit, 
(swinging it horizontally through an arc of 19XP), sight to the star again. 



^ 




portion of the " great bear " (Ursa major), a line drawn through these two 
stars passing near Polaris. .\s the stars in the handle of the dipper form 
the tail of the great bear, as shown on celestial maps, so Polaris and the 
stars near it form the tail of the little bear (Ursa minor.) Polaris is also 
nearly midway and in line between Delta and Mizar. Polaris forms, with 
three other and less brilliant stars, a quite symmetrical cross, with Polaris 
at the end of the right arm. In Fig. 3 this cross is inverted. Its height is 
about 5°, or == the distance between the pointers. 

* Part of a table computed by the Surveying Class of 1882-8, School of 
Engineering, Vanderbilt University, Nashville, Tenn., and published by 
Prof Clin H. Landreth. 

t The stake must be illuminated. This may be done bv throwina' light 
upon that side of the stake which faces the transit, or, better, by holding a 
sheet of white paper behind the stake, with a lantern behind the paper. In 
the latter case, the cross-hairs of the transit, as well as the stake, and the 
knife-blade or pencil-point with which the assistant marks it, show out 
dark against the illuminated surface of the paper. 

\ See Note, page 290. 



LOCATION OF THE MERIDIAN. 287 

•gain depress, and» if the line of sight then coincides perfectly -with the 
mark first set, both are in the plane of the star's vertical circle. If not, 
note where the line of sight does strike, and make a third mark, midway 
between the two. The line of sight, when directed to this third mark, is in 
the required plane, from which the azimuth, found as in (8), has yet to be 
laid off to the meridian, to the l^ from. eaMem elongation, and vice vena, 

(17) To avoid driving the distant stake and marking it during the night, 
a fixed target at any convenient point may be used, and the horizontal 
angle formed between the line ox sight to the star and that to the target 
merely noted, for use in ascertaining and laying off the azimuth of the 
tarvet. 

(19) By otMervation of Polaris at cnlmtnaiioii. Owing to 
its greater difficulty, this method will generally be used only when that 
by elongation is impracticable. It consists in watching Polaris in connec- 
tion with another circumpolarstar (such asMizar *or Delta) until Polaris is 
seen in the same vertical ]^ane with such star, and then waiting a short and 
known time T, as follow8,t until Polaris reaches calminatlon, where- 
upon Polaris is stehted and the line of collimation is in the meridian. At 
their upper culniinations, Mizar and Delta are too near the zenith to be 
conveniently observed at latitudes north of about 25° and BOP respectively. 
At their lower culminations they are too near the horizon to be used to 
advantage at places much below about 88° of N. latitude. In general. 
Delta is conveniently obeexved at lower culmination ttom. February to 
August, and Mizar during the rest crf^kie year. 

Mizar Delta 

T= T = 

In 1900 2.6 mins 8.4 mins 

In 1910 6.5 mins 7.2 mins 

Mean annual increase, 1900-1910 . 0.39 min 0.38 min 

(19) "By obsenration of Polaris at any point In Its path* 

Table 1 gives the mean solar times of upper culmination of Polaris on the 
1st of each month in 1900, and directions ibr ascertaining the times on other 
dates ; and Table 2 gives the azimuths of Polaris corresponding to different 
values of its hour angle in civil or mean solar time, for different latitudes 
fh)m 30° to 50°, and for the years 1901 and 1906. For hour angles and lati- 
tudes intermediate of those in the table, the azimuths may be taken by 
interpolation. See Caution and formula, p. 290. 

(SO) The local time} of observation must be accurately known, and the 
time of the preceding upper culmination (as obtained from Table 1) dedu<!ted 
from it. The difference is the hour angle. If the hour angle, thus found, 
is 11 h. 58 m. or less, the star is west of the meridian. If it is greater than 
11 h. 58 m., the star is east of the meridian. In that case deduct the hour 
angle from 28 h. 56 m. and enter the table with the remaiTuier as the hour 
an^le. See Fig. 1. 

(»1) Where great accuracy is not required, Polaris may be observed by 
means of a plumb-line and sight. A brick, stone, or other heavy object 
will answer perfectly as a plumb-bob. It should hang in a pail of water. 
A compass sight, or any other device with an accurately straight slit about 
1/16 inch wide, may be used. The sight must remain always perfectly verti- 
cal, but must'be adiustable horizontally for a few feet east and west. The 
plumb-line and sight should be at least 15 feet apart, and so placed that the 
star and plumb-line can be seen together through the sight, throughout the 
observation. The plumb-line must be illuminated. It is well to arrange 
all these matters on an evening preceding that of the observation. When 
the star reaches elongation, the sight must be fastened in range with the 
plumb-line and the star. From the line thus obtained, lay off the azimuth ; 
to the toest for ea^em elongation, and vice versa. 

{fSS9) Bjr any star at eqnal altitudes. This method, applicable 
to south as well as to north latitudes, consists in observing a star when it 
is at any two equal altitudes, £. and W. of the meridian, thus locating, on the 
horizon, two points of equal and opposite aziQiuth. The meridian will 
be midway between the two points. 

• Mizar will be recognized by the small star Alcor, close to it. 

t Deduced from values calculated in astronomical time (p. 266) by the 
U. S. Ckiast and Oeodetic Survey. 

X Ijocal time agrees with standard time (p. 267) on the standard 
meridians only. For other points add to standard time 4 minutes for each 
degree of longitude east of a standard meridian, and trice versa. 



288 



LOCATION OF THE MERIDIAN. 




(as) By e^aal sliadows from the sun. Piir. 6 ADDroximAtP 
At the solstices (about June 21 and December 21) the path a b c <J traveraed 
before and after noon, by the end of «*«'<'"■ tniveraea 

the solar shadow O o, etc., of a verti- 
cal object O, or by the shadow of a 
knot tied in a plumb-line suspended 
over O, will intersect a circular arc 
a N d, described about O, at equal dis- 
tances, am^ md, from the meridian 
O N. The observations should be 
made within two hours before and 
after noon. At the vernal equinox 
(March 21) the line thus located will 
then be west, and at the autumnal 
equinox (Sept. 21) east, of the merid- 
ian, by less than 7.}4 minutes of arc. For intermediate dates the error is 
nearly proportional to the time elapsed. It is well to draw several arcs 
of different radii, O a, O 6, etc., note two points where the path of the shadow 
intersects each arc, and take the mean of all the results. A small piece of 
tin plate, with a hole pierced through it, may be placed with the hole 
vertically over O ; and the bright spot, formed by the light shining through 
the hole, used in place of the end of the shadow. 

Table 1. 

^^^S'V^^*'?"^** **'^" times of elongratlon and calmlnatlon 

muilJh hTlScX)" ■^•' ^ong. 90° W. from Greenwich, on the first of each 

The times given in this table are mean solar or local times. 
fn^ti^^^^o iS^Y.^"" 5^22i^.^i,.TJi»l^.^iL«^ i« bold-faee. 



In lattude 25^, W. elongations occur later and E. earlier K« , . , 
latitude 50°, W. " " earlier and E. later f ^^ nearly 2 mins. 

le correction fc%r Inno-iti-iHA amr\tt-n*a *ex ana■m.^^^■^ » «..• *. jfj.i , 



In 



'TK^ —-w > y. cttiijcrtiiiuji. later) * -f -•"**«'• 

For other days of the month, deduct 8.94 min. for each succeedinp fl*v 

In general, the times are a little later each vear In iSith^^S?! i^^v ^: 
b}A minutes later, but in 1905, only about 3 mlnm^s latefthan^iJT^iJ? *S2!}* 
discrepancy is due to the occurrence of leS^yeaMni^'^ ^ ^^' ^^^ 

Inasmuch as this table serves chiefly to out the obsPrvlV «« «r.,««^ ^ 
he- should be at his post from 15 to S m?nmk in advance^S^^^ ""S^^ 
the gradual increase in the times is of little conseauence Thi^oUl'^®*; 
the star at.elongation is determined by observS ^ position ot 

At culmination, where the change in azimuth is most ranid a»i o-,^. <« 

At elongation, 



an error in time of 

20 minutes 
10 minutes 

5 minutes 

1 minute 



will make an error in azimuth of 

less than 90 seconds 
less than 6 " 

less than 2 " 

about 0.06 second 



, „, ~' — aooui 0.06 second 



Jan. 1. 
12.31 A. 

July 1. 
12.51 A. 



W. 
M. 

E. 
M. 



Jan. 1. 
6.38 P. 

July 1. 
«.44 P. 



U. 
M. 

L. 



Elongratlons. (E, eastern : W, western.) 1900. 

AVp.V IfSSkli. .^SSk^. Km«: \Zl 
.»P.^«. i%-k^. .?J^kV rj.J:S: V^l 

Cnlmlnatlons. (U, upper ; L, lower.) 1900. 



E 
Mi 

w! 

M. 



Feb. 1. L. 
4.38 A. M. 

U. 
M. 



Aug. 1 
4.45 A. 



}^fn'}'h >P^- 1- ^- May 1. L. 

2.47 A. M. 12.45 A. M. 10.48 P. M. 

Sept. 1. U. Oct. 1. U. Nov. 1. U. 

2.43 A.M. 12.46 A.M. 10.40 P.M. 



Jane 1. 
S.«8P. 

Dec. 1. 
S42P. 



^ 



LOCATION OP THE MERIDIAM. 



l«.»5°4e°4IS'' HW 



410 43 47I 51 
440 47 5ll Sa 



















































m'o 58 

11 

SI i 


1 






























































































wu" uiiuillr ba 'S 



290 



LOCATION OF THE MERIDIAN. 



Table a. 

POLARIS. POLAR DISTANCES, AND AZIMUTH AT ELONGATION. 









Azimuth at Elongation, in Latitude 


u 


Polar 
Dist. of 
Polaris 


Log sin 
poldist. 




1 


S0<> 


JWO 


BOO 


85° 


40<> 


400 


50^ 




O / ft 




o / 


o t 


O f 


o / 


O f 


o / 


o / 


1900 


1 18 33 


8.38027 


1 18.8 


1 21.1 


1 24.9 


1 29.8 


1 36.1 


1 44.1 


1 64.4 


1908 


1 12 37 


8.32 472 


1 17.3 


1 20.1 


1 28.8 


1 28.7 


1 34.8 


1 42.7 


1 58.0 


1906 


1 11 41 


8.31 910 


1 16.3 


1 19.1 


1 22.8 


1 27.6 


1 33.6 


1 41.4 


1 51JS 


1909 


1 10 45 


8.31 341 


1 15.3 


1 18.1 


1 21.7 


1 26.4 


1 32.3 


1 40.1 


1 60.1 


1912 


1 9 49 


8.80 765 


] 14.3 


1 17.0 


1 20.6 


1 25.2 


1 31.1 


1 88.7 


1 48.6 


1915 


1 8 53 


8.30 181 


1 13.3 


1 16.0 


1 19.6 


1 24.1 


1 29.9 


1 37.5 


1 47.2 


1918 


1 7 58 


8.29594 


1 12.3 


1 15.0 


1 18.6 


1 28.0 


1 28.7 


1 36.1 


1 46.7 


1921 


17 2 


8.28 999 


1 11.4 


1 14.0 


1 17.4 


1 21.9 


1 27.6 


1 34.8 


1 44.8 


1924 


16 7 


8.28 401 


1 10.4 


1 13.0 


1 16.3 


1 20.7 


1 26.8 


1 33.5 


1 42.9 


1927 


1 6 12 8.27 794 


1 9.4 


1 11.9 


1 16.3 


1 19.6 


1 25.1 


1 82.2 


1 41.4 


1980 


1 4 16 8.27 169 


1 8.4 


1 10.9 


1 14.2 


1 18.5 


1 28.9 


1 30.9 


1 40.0 




Log 008 


Ut 


9.97 299 


9.95 728 


9.98 753 


9.91 337 


9.88426 


9.84949 


9.80807 



. Owing to changes in the position of Polaris during the year, the positions 
given in the table may at times be in error by as much as a minute. The 
error is greater in the nigher latitudes. 

Having the north polar distance,/), of a star, and the latitude, L, of the 
point of observation, we have, declination of star = 6 = 90° — p ; and ^e 
aslmutb, a, of the star, corresponding to any hour angle, a, may be 
found by the following formulas : 

TanM = ^ = -^. Then Tan a = <^ " ' ^° * . 
cos h cos h cos (L— M) 

The declinations, fi, of Polaris are given in the U. S. Ephemeris or Nautical 
Almanac. From these the polar distances may be obtained more accurately 
than from our Table 3. 

Caution. When it is desired to determine the meridian within one 
minute of arc, it is well to use more than one method and compare the 
results. For example, observe Polaris both E. and W. of the meridian, aitd 
a star at equal altitudes south of the zenith. 

NoTK. — Lf Polaris be found during twilight, iu the morning or evening, obsei-- 
▼atinns of it luuy be made without artificial illumiaation of the cross-haira. 
For times of elongation, see Table 1, 



CouTertiion of Arc Into Time, and vice versa. 



Arc Tike 
1° = A minutes 
1' = 4 seconds 
l» =1 0.066... second 



Time Abc 

24 hours =860° 
Ihour = 150 
1 minute = OP 16' 
1 second « (PVl^ 



1 

TBE ENGINEBB's TRANSIT. 291 

THE ENGINEER'S TRANSIT. 



292 TtTB EHQINEE&B TRAITBET. 

Thb dtMIlB of the transit, like thme of the IstgI, are dllTerCDtlT trmtgei hf 
diff nukem, and to mlt pirtkuUr purpoAU. We deocribe it In iti modern Ibrm, 
SB uude by Heller ud Brightly, arPhlladiL without the lone bBbU«-tBke 
F F, Fig 1, onder the telescope, and the BrrndBstcd an p, It la theli plklB 
teBMalt. With tb«e sHiendage*, or nther vltta a, eradusted cirde ia fite* of 
the Bra It becomea Tirtiullr s COBipl«te Ttaeodiillle. 

B D D, Fig 1, Is the tripo<l>ke«d. The Krew-tbRwds at v loeelfe the sciew 
of a wooden trlpod-head-cover vhen the inetniment Ia out of use. S B A la Qu 
l«w«r panulel |»l«t«. After the traaiit has been set tstt dmtIt oier the 
center ofa sl^e, the mlilftlns-plat«, <f d e c, enables lu, bf illabClr lonealng 
the I«TelllBK-BCrem K, to shlA the upper paiU boriiontallT a (rifle, and 
■haa bring the plumb-bob eiactlj OTer tbe center -with leaa trouUe than bf the 
elder method of puiblog one or tiro of the legs furibei Into the giouod. or apread- 
InE tbem more or leaa. Tbe acreirfl, E, are taea tightened, thereby puablDg up> 
ward the upper BBiwllel pl«M n « ni z i, and vitb It tbe balT-bkll t, ibni 
pr^alng o c llghtl; up afalnat the under lida at 8. Ths plomb-UnB paana 



throngb the yert hols in 6- Scraw-eaja, / g, protect the leTalUnMcrewi ttom. 
dual, ia The feet, i. of tbe icrewa, work In looea aocketa^^, made flat at bottom. 
to^presene S from being Indented, The paita thui far dTeKribed are guaiBUw 
left atUiched lo [he legs at all Uraea. Flj? 1 show, the method of attachmmt. 

To set (he upper puM up*a «m panllal l>l«te|k Plaoe tbe 
lowerendor UU Id 1 1, holding tbe Instrument so that the thrw bloekaonaawe 
(of which the one ahown at Fis morable) mar ^oter the three oorreapondtiiK 



THE engineer's TRANSIT. 293 

rMeeses in a, thus allowing a to bear fully on m, upon which the upper pute 
then rest. (The inner end of the spring-catch, I, in the meantime enters agroov6 
around U, Just below a, and prevents the upper parts from falling off, if the in* 
strument is now carried over the shoulder.) Kevoive the upper parts horizontaUj 
a trifle, in either direction, until thev are stopped by the striking of a small lug 
on a against one of the blocks F. Tne recesses in a are now clear of the blocks. 
Tighten g, thereby pushing inward the movable block F, which clamps the 
bevelled flange a between it and the two flxed blocks on m m, and confines the 
spindle U to the fixed parallel plates. It remains so clamped while the instrument 
is being used. 

To remoTe the upper parts ft^m theparallel plates. Loosen 
g, bring the recesses in a opposite the blocks F. Hold back I, and lift the upper 
parts, which are then held together by the broad head of the screw inserted into 
the foot of the spindle w. 

T T is the oater reTOlTlng: spindle, cast in one with the support* 
Ing^plate Z 2^, to which is fastenea the s^radnated limb O. The limb 
extends beyond the compass-box, and thus admits of larger graduations than 
would otherwise be obtainable, to wis the Inner revolving^ spindle. At 
its top it has a broad flange, to which is fastened the vernier plate P. To the 
latter are fastened the corapass-box C, the two bubble-tubes M M, the standards 
y Y, supporting the telescope, &c. Each bubble-tube is supported and adjusted 
by four capstan-head nuts, two at each end. The bent strip, curving over the 
tnbe, protects the glass from accidental blows in swinging the telescope. 

<k»iatrol of motions of ir>*»dnAl«d limb O O and wernler 
plate P. — ^The tangent-screw 6 and a spiral spring (not shown) opposite to it 
are fixed to the graduated limb 00, and hold between them a projection y from 
the loose collar t, which is thus confined to the limb and made to travel with it. 
The clamp-screw H passes through the collar t and presses against the small lug 
shown at its inner end. When H is tightened, this lug is pressed against the 
fixed spindle U U, to which the graduated limb is thus made fast. A slow mo- 
tion may, however, still be given to the limb by means of the tangent-screw G. 

The motion of the vernier plate P over the graduated limb O is simUarly 
governed by the tangent-screw 6 and its spiral spring (not shown), fixed to the 
ternier plate P, and the clamp-screw e, which passes tnrough the collar z, and 

{>re88es against the small lug shown at its inner end. In Heller and Brightly's 
nstraments, the screw b is provided with means for taking up its ** wear," or 
"lost-motion." 

There are two verniers. One is shown at ja. Fig 1. Both may be read, and 
their mean taken, when great accuracy is required. Ivory reflectors, c, facilitate 
their reading. Before the instrument is moved from one place to another, the 
eompaas-needle, ib. Fig 2, should always be pressed up against the glass cover 
of the compass-box by means of the upright miUed-head screw seen on the ver^ 
nier-plate m Fig 1, Just to the right of the nearest standard. The pivot^point is 
thus protected from injury. 

R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope 
in any required position. It is oest to let the eye-end. 1C, of the telescope revolve 
dowHiffard, as otherwise the shade on O, if in use, may fall off. The tangent-screw, 
il. moves a vert arm attached to R, and is thus used for slightly changing the 
elevation of the telescope. In the arm is a slit like that seen in the vernier-arm 
L Bt mesns of the screw D. the movable vernier-arm Y may be clamped at 
tDT desired point on the vertical limb g. When (P of the vernier is placed at 
9(Pon the arc ^, and the index of the opposite arm is placed over a small notch 
on the horizontal brace (not seen in our figs) of the standards, the two slits will 
be opposite each other, and may be used for laying off offsets, oc, at right-angles 
to the line of sight. 

One end, R, of the telescope axis rests in a movable box, under which is a screw. 
By means of the screw, the box may be raised or lowered, and the axis thus ad- 
justed for very slight derangements of the standards. For E, B, O, and A, see 
iaulf p 306. a is a dust-guard for the object-slide. 

StaaiA Kalrs. Immediately behind the capstan-screw, p. Fig 1, is seen a 
nnaller one. This and a similar one on the opposite side of the telescope, work 
in a ring inside the telescope, and hold the ring in position. Across the ring are 
itretched two additional horizontal hairs, called stadia hairs, placed at such a 
distance apart, vertically, that they will subtend say 10 divisions of a graduated rod 
placed 100 ft from the instrument, 15 divisions at 150 ft, Ac. They are thus used for 
asttsuring hor and sloping distances. 

Tbe lonff babble-tube« F F, Fig 1, enables us to use the transit as a level. 
•Ithoof h it Is not so well adaotsd as the latter to this purpose. 



294 THE engineer's transit. 

To aAinmt a plain Transit* 

When either a lerel or a transit is purchased, it is a good precaution (but one 
which the writer has never seen alluded to) to first screw the oltject-glass firmly home 
to its place ; and then make a short continuous scratch upon the ringt>f the glass, and 
upon its slide ; so as to be able to see at any time when at work, that the glass is 
always in the same position with regard to the slide. For if, after all the adjustments 
are completed, the position of the glass should become clumged, (as it is apt to be if 
unscrewed, and afterward not screwed up to the same precise spot,) the acyustments 
may thereby become materially deranged ; especially if the object-glass is eccentric, 
or not truly ground, which is often the case. Such scratches should be prepared by 
the maker. In making adjustments, as well as when using a transit or lerel, be 
careful that the eye-glass and object-glass are so drawn out that there shall be ne 
parallax. The eye-glass must first be drawn out so as to obtain perfect distinctness 
of the cross-hairs ; it must not be disturbed afterward; but the object-glass must 
be moved for different distances. 

First, to ascertain tliat tlie bnbble-tnbes, M Bf • are placed 
parallel to the vernier-plate, and that therefore when both bubbles are in 
the centers of their tubes the axis qf the inst is vert. By means of the four levelling- 
screws, K, bring both bubbles to the centers of their tubes in one position of the 
inst ; then turn the upper parts of the inst half-way round. If the Dubbles do not 
remain in the center, correct half the error by means of the two capstan-nuta 
rr; and the other half by the levelling-screws K. Repeat the trial until both 
bubbles remain in the center while the inst is being turned entirely around on 
its spindle. 

Second, to see that the standards have snfTered no deranire- 
ment ; that is, that they are of equal height and perpendicular to the vernier- 
plate, as they always are when they leave the makers hands. Level the inst 
perfectly ; then direct the intersection of the hairs to some point of a high object 
(as the top of a steeple) near by ; clamp the inst by means of screws H and e, 
and lower the telescope until the intersection strikes some point of a low object. 
(If there is none sucn drive a stake or chain-pin, Ac, in the line.) Then un- 
clamp either H or e, and turn the upper parts of the inst half-way round ; fix the 
intersection again upon the high point ; clamp ; lower the telescope to the low 
point. If the intersection still strixes the low point, the standards are in order. 
If not, correct one-ltalf of the difference by means of the adjusting-block and 
screw at the end, R, of the telescope axis. Fig. 1, and repeat the trial de novo, 
resetting the stake or chain-pin at each trial. If the inst has no adjusting-block 
for the axis, it should be returned to the maker for correction of any derange- 
ment of the standards. 

A transit may be used for running ^raight lines^ even if the standards become 
slightly bent, by the process described at the end of the fourth adjustment. 

Third, to see that the cross-hairs are traly vert and hor 
^rhen the inst is level. When the telescope inverts, the cross-hairs are 
nearer the eye-end than when it shows objects erect. The maker takes care to place 
the cross-hairs at right-angles to each other in their ring, or diaphragm ; and gene- 
rally he so places the ring in the telescope, that when levelled, they shaJl be reii 
and hor. sometimes, however, this is neglected ; or the ring may by accident be- 
come turned a little. To be certain that one hair is vert, (in which case the other 
must, by construction, be hor,) after having adjusted the bubble-tubes, level the in« 
strnment carefully, and take sight with the telescope at a plumb-line, or other yert 

straight edge. If the vert hair coincides with this object, 
it is, sofar^ in adjustment ; but if not, then loosen sKghtlv 
only two adjacent screws of the four, pp i t. Fig 1 ; and 
with a knife, key, or other small Instrument, tap verj 
gently against the screw-heads, so as to turn the rin^ » 
little in the telescope; persevering until the hair be* 
comes truly vertical. When this icr done, tighten the 
screws. In the absence of a plumb-line, or vert stsulgfat 
edge, sight the cross-hair at a Tery small distinol 
point; and see if the hair still cuts that point, when 
the telescope is raised or lowered by revolring it on 
its axis. 
The mode of performing the foregoing will be readily 
understood ft'om this Fig, which represents a section across the top part of the tele> 
acope, and at the cross-hairs. The hair-ring, or diaphragm, a; vert hair, v; tele* 
scope tube, g ; ring outside of telescope tube, d; & is one of the four capstMi-headed 
screws which hold the hair-ring, a, in its place, and also serve to a^jnst it. The 
lower ends of these screws work In the thickness of the hair-ring; so that when 
they are loosened somewhat, they do not lose their hold on the ring. Small 




THE EKOIKEES'S TIUXSIT. 



295 



mO 



washers, c, are placed under the heads h of the screws. A space ^ y is left around 
each screw where it passes through the telescope tube, to allow the screws aud ring 
together to be moved a little sideways when the screws b are slightly loosened. 

Fourth, to see tliat the wertical hair is In the line of colU- 
matlon. Flant the tripod firmly upon the ground, as at a. Level the inst ; 
clamp it; and direct the vert hair by means of tangent-screw O ffigs. 1 and 2) 
upon some convenient object h\ or if there is none such, drive a thin stake, or a 
ennin-pin. Then revolving the telescope vert on its Hxis, ^ 

observe some object, as c, where the vert hair now strikes ; ^ a ^^^ 

or if there is none, place a second pin. Uoclamp the instru- « «^^ 

ment by the clamp-screw H ; and turn the whole upper • " 

part of it around until the ven hair again strikes b. JPig, 4, 
Clamp again ; and again revolve the telescope vert on its 
axis. If the vert h»ir now strikes e, as it did before, it shows that c is really 
at ; and that 6, a, e, are in IM^ same straight line ; and therefore this adjustment 
is in order. If not, observe where it does strike, say at m, (the dist a m being 
taken equal to a c,) and place a pin there also. Measure m c ; and place a pin 
at v, in tne line m c, making m v <— one-fourth of m c. Also put a pin at 0, half- 
way between m and c, or in range with a and b. By means of the two hor 
screws that move the ring carrying the cross-hairs, adjust the vert hair until it 
euts V. Now repeat the fntire operation ; and persevere until the telescope, after 
being directed to b, shall stVike the same object 0, both Hmes, when revolved on 
its axis. See whether the movement of the ring in this 4th adjustment has dis- 
turbed the verticality of the hair. If it has, repeat the 3d adjustment. Then re- 
peat the 4th, if necessary ; and so on until both adijustments are found to be right 
at the same time. Thus a straight line mav be run, even if the hairs are out of 
adjustment ; but with somewhat more trouble. For at each station, as at a, two 
back-sights, and two fore-sights, a c and a m, may be taken, as when making the 
adjustment ; and the point 0, half-way between c and m, will be in the straight line. 
The inst may then be moved to 0, and the two back-sights be taken to a ; and so on. 

Angles measured by the transit, whether vert or hor, will evidently not be 
tifected by the hairs being out of a4justment, provided either that the vert 
liair is truly Tert. or that we use the inler^oHon of the hairs when measuring. 

The foreproiniT ^^^^ All the a^instments needed, unless the tran- 
sit is reqnlrea for levelUi^, in which case the following one muse be attended to : 




To adjust the lontr bnbble*t1Ihe« F F, Fie. l, we first place the line 
of sight of the telescope hor, and then make the bubble-tube hor, so that the 
two are parallel. Drive two pegs, a and b Fig. 5, with their tops at precisely 
the same level (see Bem. p. 296) and at least about 100 ft. apart ; 800 or more 
will be better. Plant the inst Armly, in range with them, as at c, making^ c 
an aliquot part of a b, and as short as will permit focusing on a rod at 6. The 
inst need not be leveled. Suppose the line of sight to cut e and d. Take the 
readings b e and a d. Their diff is be — ad=^an — ad=*dn\ and ah-.ac: 
dnids'i s being the height of the target at a when the readings (a «, b 0) on the 

two stakes are equal. as==ad-\-ds^ad-\ r — ' If the reading on a 

taceeeds that on b (as when the line of sight is vfg) the diff of readings is = a ^ — 

bf=sag — ai^gi\ smd as = a g — g s=aaff — ^ — j- — • Sight to «, bring the 

babble to the cen of its tube by means of the two small nuts n n at one end of the 
tube. Fig. 1, and assume that the telescope and tube are parallel.* The zeros of 

* Thla B«0eeM s mnmll «iTor due to the oarralnre of the earth ; fDr a hor line at v ia v h, tao* 
flaatiml to tlM earved (or " tofwl") torfiaoe of still water at «, whereae » • Is tangential to water aarf 
at a point midwaj between a and h. Henoe if the telesoope at « points to a li will not be parallel te 
the level bobbto-tnbe. To allow for this, and for the reftvotloa bj the air, wUeh diminUhM the 
error, rsiae the tarfet on • to a point h above a. h* — .0000000205 x square of a in (I ; bat when 
• e is S30 ft, Jk a is only aboni one tenth of an inoh and barely oovers the apparent thlekness of Um 
-bnlrlatkn ' 



296! 



THE ENaiNEER'S TRANSIT. 



the vert circle, and of its vernier, may now be aAjiitted, if they require it, by 
loosening the vernier screws and then moving the vernier until the two coin- 
cide. ^ , . - , 

Rem. If no level is at hand for levelling the two pegs o and &, it may be done 
by the transit itself, thus : Carefully level the two short bubbles, by means of the 
levelling-screws K. Drive a peg m, from 100 to 300 feet from the instrument o. 
Then placing a target-rod on m, clamp the target tight at whatever height, as sv, 

the hor hair happens to cut it ; it being of no im- 

l^ L portance whether the telescope is level or not; 

TV (J) although it might as well be as nearly so as can 

\ X conveniently be guessed at. Clamp the telescope 

g^ — JJ. in its position by the clamp-ring K, Fig. 1. Re- 

^ volve the inst a considerable way round; say 
iJifiT. 0. nearly or quite half way. Place another peg n, 

atprecUdy the same diet from the instrument that m is; and continue to drive it un- 
til the hor hair cuts the target placed on it, and still kept clamped to the rod, at the 
same height as when it was un m. When this is done, the tops of the two pegs are 
on a level with each other, and are ready to be used as before directed. 

When a transit is intended to be used for surveying farms, Ac, or for retracing 
lines of old surveys, it is very useful to set the compass so as to allow for the ** va- 
riation" during the interval between the two surveys. For this purpose a 
'' TArlatton- vernier " is added to such transiCB ; and also to the oompaos. 
When the graduations of a transit are figured, or numbered, so as to read both 

H) 10 

ways from aero, thus, i n ii 1 1 1 h i 1 1 1 1 1 1 1 1 1 1 1 1 I m the vernier also is mada 

double ; that is, it also is graduated and numbered from its sero both ways. In thia 
case, if the angle is measured from zero toward the right hand, the reading must be 
made from the right hand half of the vernier ; and vice versa. If the figuring la 
single, or only in one direction, from zero to 360^, then only the single vernier la 
necessary, as the angles are then measured only in the direction that the figuring 
counts. ICngineers differ in their preferences for various manners of figuring the 
graduations. The writer prefers from zero each way to 180^, with two double ver- 
niers. 

To replace cross-hairs in a IcTel, or transit. Take out tiie tube 
from the eye end of the telescope. Looking in, notice which side of the oroM- 
hair diaphragm is turned toward the eye end. Then loosen the four screws which 
hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new 
hairs with beeswax, varnish, glue, or gum-arabic water, Ac. This requires care. 
Then, to return the diaphi-agm to its place, press firmly into one of the screw-holes 
on the circumf of the diaphragm itself, the end of a piece, of stick, long enough to 
reach easily into the telescope as far as to where the diaphragm l^Iongs. By this 
stick, as a handle, insert the diaphragm edgewise to its place in me telescope, and hold 
it there until two cpposUe screws are put in place and screwed. Then draw the stick 
out of the hole in the diaphragm ; and with it turn the diaphragm until the same 
side presents itself toward the eye end as before ; then put in the other two screws. 

The so-called cross- hairs are actually spider-web, so fine as to be barely visible to 
the naked eye. Holler A Brightly use very fine platina wire, which is much better. 
Human hair is entirely too coarse. 

To replace a spirit-level, or bnbble^lass. Detach the level from 
the instmment; draw off its sliding ends; push out the broken glass vial, and the 
cement which held it ; insert the new one, with the proper side up (the upper side 
is always marked with a file by the maker); wrapping some paper around its ends, 
if it fits loosely. Finally, put a little putty, or melted beeswax over the ends of the 
vial, to secure it against moving in its tube. 

In purchasing instruments, especially when they are to be used far from a maker, 
it is advisable to provide extras of such parts as may be easily broken or lost ; such 
as glass compass-covers, and needles; atjQusting pins; level vials; magniflen, Ao, 



Theodolite adjustments are performed like those of the level and transit. 

let. That of the cross-hairs; the same as in the level. 

2d. The long bubble-tube of the telescope ; also as in the level. 

8d. Th^ two short bubble-tubes ; as in tne transit. 

4th The vernier of the vert limb ; as in the transit with a vert circle. 

5th. To see that the vert hair travels vertically ; as in the fourth adjustment 
of the transit. In some theodolites, no adjustment is provided for this ; but in 
Isrm onaa it is provided for by screws under the feet of the standards. 

Somttimw • second telescofKi is added ; it Is p^iic«d belov the hor limb, and to 



THE BOX OB POCKET SEXTANT. 



297 



called a toate?ur. It has its own clamp, and tangent-screw. Its use is to ascertain 
whether the sero of that limb has moved during the measurement of hor angles. 
When, previously to beginning the measurement, the zero and upper telescope are 
directed to^inund the first object, point the lower telescope to any small distant 
object, and then clamp it. During the subsequent measurement, look through i^ 
from time to time, to be sure that it still strikes that object ; thus proving that nt 
slipping has occurred. 



THE BOX OR POCKET SEXTANT. 




Ths portability of the pocket sextant, and the fact that It reads to single minutes, 
render it at times very useful to the engineer. By it, angles can be measured while 
in a boat, or on horseback ; and in many situations which preclude the use of a 
transit. It is useful for obtaining latitudes, by aid of an artificial horizon. When 
closed, it resembles a cylindrictu brass box, about 3 inches in diameter, and 1)^ 
inches deep. This box is in two parts ; 
by unscrewing which, then inverting 
one i>art,,and then screwing them to- 
gether again, the lower part becomes a 
handle for holding the instrument. 
Looking down upon its top when thus 
arranged, we see, as in this figure, a 
movable arm I C, called the index, 
which turns on a center at C, and car- 
ries the vernier Y at its other end. Q 
6 is the graduated arc or limb. It 
actually subtends about 13P, but is di- 
vided into about 146^. Its zero is at 
one end. Its graduations are not shown 
in the Fig. 

Attached to the index is a small mov- 
able lens, (not shown in the figure,) 
likewise revolving around C, for read- 
ing the flue divisions of the limb. When 
measuring an angle, the index is moved 
by turning the milled-head P of a 
pinion, which works in a rack placed within the box. The eye is applied to a eir* 
cnlar hole at the side of the box, near A. A small telescope, about 3 inches long, 
; accompanies the instrument; but may generally be dispensed with. When so, the 
eye-hole at A should be partially closed by a slide which has a very small eye-hole 
in it ; and which is moved by the pin A, moving in the curved slot. Another slide, 
at the nde of the box, carries a.dark glass for covering the eye-faole when observing 
the ran. When the telescope is used, it is fastened on by the milled-head screw T. 
The top part shown in our figure, can be separated from the cylindrical part, by 
removing 3 or 4 small screws around its edge ; and the interior can then be exam- 
ined, and cleaned if necessary. Like nautical, and other sextants, this one bm 
two principal glasses, both of them mirrors. One, the Index-fplass, is attached 
to the underside of the index, at C; its upper" edge being indicated by the 
two dotted lines. The other, the Moriaon-KliMiS) (because, when meas- 
uring the vert angles of celestial bodies, it is directed toward the horizon,) is also 
within the box; the position of its upper edge being shown by the dotted lines at 
R. The horizon-glass is silvered only half-way down ; so that one of the observed 
objects may.be seen directly through its lower half, while the image of the other 
object is seen in the upper half, reflected from the index-glass. That the instrument 
may be in a4justment, ready for use, these two glasses must be at right angles to the 
plane of the instrument ; that is, to the under side of the top of the box, to which they 
are attached; and must also be parallel to each other, when the zeros of .the vernier 
and of the Umb coincide. The index-glass is already permanently fixed by the 
ma^T, and requires no other a4ju8tment. But the horizon-glass has two adjust- 
ments, which are made by a key like that of a watch, and having a milled-head K. 
It is screwed into the top of the box, so as to be always at hand for use. When 
needad, it is unscrewed. This key fits upon two small square-heads, (like that for 



298 



THE COMPASS. 




winding a watch;) one of which is ihown at S; while the other is near it, but on the 
SIDE of the box. These squares are the heads of two small screws. Jf the 
horlEon glass H should, aa in this sketch, (where it is shown endwise,) not be at 
right angles to the top U HJ of the box, it is brought right by turning the square- 
bead S of the screw S T ; and if, after being so far rectified, it still is not parallel to 

thn index-glass when the zeros coincide, it is moved 
a little backward or forward by the mjuare head 
at the side. 

To adjust a box sextant, bring the two 
aeros to coincide precisely ; then look through the 
eye-hole, and the lower or unsilvered part of the 
horixon-glasB, at some distant object. If the instru- 
ment is in adjustment, the object thus seen directly, 
will coincide precisely with its reflected image, 
seen at the same time, at the same spot. But if it 
is not in ac^ustment, the two will appear separated 
either hor or vert, or both, thus, * • ; in which case 
apply the key E to the square-head S ; and by turning it slightly in whichever direc- 
tion may be necessary, still looking at the otjject and its ima^e^ bring the two into a hor 
position, or on a level with each other, thus, * •. Then apply the key to the square- 
head in the side of the box; and by turning it slightly, bring the two to coincide 
perfectly. The instrument is then ai^justed. 

In some instruments, the hor glass has a hinge at v, to allow it play while being 
adjusted by the single screw S T ; but others dispense with this hinge, and use two 
screws like S on top of the box, in addition to the one in the side. 

If a sextant is used for measuring vert angles by means of an artificial 
boriEon, the actual altitude wilF be but one-half of that read off on the 
limb ; because we then read at once both the actual and the reflected angle. The 
great objection to the sextant for engineering purposes, is that it does not measure 
angles horizontally, as the transit dues ; unless when the observer, and the two ob> 

jects happen to be in the same hor plane. 
Thus an observer with a sextant at A, if 
measuring the angle subtended by the 
mountain-peaks B and C, must hold the 
graduated plane of the sextant in the 
plane of A B C ; and must actually meas- 
^^ ,' ,-' I ^ ; ure the angle BAG; whereas what he 

g^k^*^':'- ' wants is the hor angle nAm. This is 

^f""" -'Wl greater than BAG, because the dista An 

A and A m, are shorter than A B and A G. 

The transit gives the hor angle n A m, be- 
<iau8e its graduated plane is first fixed hor by the levelling-screws ; and the subse- 

Suent measurement of the angle is not affected by his directing merely the line of 
[ght upward, to any extent, in order to fix it upon B and G. For more on this sub- 
ject ; and for a method of partially obviating this objection to the sextant, see the 
note to Example 2, Case 4, of " Trigonometry." 

Tbe nautical sextant, used on ships, is constructed on the same principle 
as the box sextant ; and its adjustments are very similar. In it, also, the index- 
glass is permanently fixed by the maker ; and the horizon-glass has the two adjust- 
ments of the box sextant. It also has its dark glasses for looking at the sun ; and 
a small eight-hole, td be used when the telescope is dispensed with. 




•-•- 



THE COMPASS. 



To adjust a Compass* 

The first adjustment is that of the bubbles. Plant firmly ; and level th« 

Instrument, in any position ; that is, bring the bubbles to the centers of their tubes. 
Then turn the instrument half-way round. If tlie bubbles then remain at the cen- 
ters, they are in adjustment;. but if not, correct one-half the diff" in each bubble, 
by means of the adjusting-screws of the tubes. Level the instrument again ; tun 
it half roimd ; and if the bubbles still do not remain at the center, the atiUusting- 
■crews must be again moved a little, so as to rectify half the remaining diff. Gener* 




THE COMPASS. 299 

ally several trials must be thus made, until the bubbles will remain at the oente 
while the compass is being turned entirely around. 

Seeond adjustmeiit* Level the compass, and then see that the needle it 
hor; and if not, make it so by means of the small piece of wire which is wrapped 
around it ; sliding the wire toward the high end. A needle thus horizontally ad- 
justed at one place, will not remain so if removed fietr north or south from that place. 
If carried to tiie north, the north end will dip down ; and if to the south, the soutii 
end will do so. The sliding wire is intended to counteract this. 

Tliird a^Jnatment. This is always fixed right at first by the maker; that 
is, the sights, or slits for sighting through, are placed at right angles to the compass 
plate ; so that when the latter is levelled by the bubbles, the sights 
are vert. To test whether they are so, hang up a plumb-line ; and 
having levelled the compass, take sight at the line, and see if the 
slits coincide with it. If one or both slits should prove to be 
out of plumb, as shown to an exaggerated extent in this sketch. 
It should be unscrewed from the compass, and a portion of its foot 
on the high side be filed or ground off, as per the dotted line ; or 
as a temporary expedient, a small wedge may be placed under the 
low side, so as to raise it. 

Foortb BdJaBtmeilt, to straighten the needle, if it should become bent. 
The compass being levelled, and the needle hor, and loose on its pivot, see whether 
its two ends continue to point to exactly opposite graduations, (that is, graduations 
18€P apart ;) while the compass is turned completely around. If it does, the needle 
is straight ; and its pin is in the center of the graduated circle ; bat if it does not, 
then one or both of these require adjusting. First level the compass. Then turn It 
until some graduation (say 90^) comes precisely to the north end of the needle. If 
the south end does not then point precisely to the opposite 90° division, lift off the 
needle, and bend the pivot-point until it does ; remembering that every time said 
point is bent, the compass must be turned a hairsbreadth so as to keep the north end 
of the needle at its 90^ mark. Then turn the compass half-way round, or until the 
opposite 90° mark comes precisely to the north end of the needle. Make a fine pen* 
<^ mark where the touth end of the needle now points. Then take off the needle, 
and bend it until its south end points ha^f^ay between its 90° mark and. the pencil 
mark, while its north end is kept at 90° by moving the eompass round a hairsbreadth. 
Tlie needle will then be straight, and must not be altered in making the following, 
adjostment, although it will not yet cut opposite degrees. 

Flfih a4ius^i»eiit, of the pivot-pin. After being certain that the needle is 
straight, turn the compass around until a part is arri ved at where the two ends of the 
needle happen to cut opposite degrees. Then turn the compass quarter way around, 
or through 90°. If the needle then cuts opposite degrees, the pivot-point is already 
in adjustment ; but if the needle does not so cut, bend the pivot-point until it does. 
Bapeat, if necessary, until the needle cuts opposite degrees while being turned entirely 
•round. 

Oare and nicety of observation are necessary in making these adjustments properly ; 
because the entire enor to be rectified is, in itself^ a minute quantity; and the novice 




it must be held parallel to the graduated circle. Otherwise annoying errors of 
several minutes will be made in a single observation ; and the accumulation of two 
or three such errors, arising from a cause unknown to him, may compel him to 
abandon the ac^ustments in despair. This su^estion applies also to the reaiding of 
angles taken by the transit, Ac ; although the errors are not then likely to be so 
great as in the case of the compass. In purchasing a magnifier for a compass, see 
that DO part of it, as hinges, or rivets, are made of iron ; for such would change the 
direction of the needle. 

If the sight-slits of a compass are not fixed by the maker in line with the two 
opposite zeros, the engineer cannot remedy the defect. This can be ascertained by 
passing a piece of fine thread through the slits, and observing whether it stands 
precisely over the zeros. 



THE COMPAfiS. 



THE COUFABB. 



I! 
II' 



|3|3i||| III |3 



111 



til 



1^ 



> ts 



f i 
i i 

II 
if! 

I L, 

; Hi 



bSj 



302 OOlffTOtm LIKEB. 

United StatflB, by Henry GtuuMtCi In 17th Annual Beport ef tf. 8. Geological 
Survey, 1896-^ 

Electrietty, either atmospheric, or excited by rubbing the glass ooy«r of 
the compass box, sometimes gives trouble. It may be removed by touching the 
glass with the moist tongue or finger. 

DEMAOHETIZATIOV. 

The needle, if of sqft metal, Bometimeo loses part of its magnetism, and consequently 
does not work well. It may be restored by simply drawing the north pole of a 
common magnet (either straight or horseshoe) about a dozen times, from the center 
to the end of the south half of the needle ; and the south pole, in the same way, along 
the north half; pressing the magnet gently upon the needle. After each stroke, 
remove the magnet several inches from the needle, while bringing it back to the 
center for making another stroke. Each half of the needle in turn, while being thus 
operated on, should be held flat upon a smooth hard surface. Sluggish action of the 
needle is, however, more generally produced by the dulling or other iujury of the 
point of the pivot. RemagnetiEing will throw the needle out of balance ; which must 
be counteracted by the sliding wire. 

In order to prevent mistakes by readlnn^ sometimes from one end, 
and sometimes from the other end of the needle, it is best to always point the N of 
the compass-box toward the object whose bearing is to be taken ; and to read off 
from the north end of the needle. This is also more accurate. 



OONTOUB LINES. 



A OOHTOUB um is a curved hor one, every point in which represents the same level ; 
thus each of the contour lines SSc, 91c, 94c, itc. Fig 1, indicates that every point in 
the ground through which it is traced is at the same level ; and that that level or 
height is everywhere 88, 91, or 94 ft above a certain other level or height called 
datum ; to which all others are referred. 

Frequently the level of the starting point of a survey is taken as being 0, or zero, 
or datum ; and if we are sure of meeting with no points lower than it, this answers 
every purpose. But if there is a probability of many lower points, it is better to 
assume the starting point to be so far above a certain supposed datum, that none of 
these lower points shall become minus quantities, or bdow said supposed datum or 
zero. The only object in this is to avoid the liability to error which arises when 
some of the levels are -|-» or plus ; and some — ^ or minus. Hence we may assume 
the level of the starting point to be 10, 100, 1000, Ac, ft above datum, according to 
circumstances. 

The vert dists between each two contour lines are supposed to be equal ; and in 
railroad surveys through well-known districts, where the engineer knows that his 
actual line of survey will not require to be much changed, the dist may be 1 or 2 ft 
only ; and the lines need not be laid down for widths greater than 100 or 200 ft on 
each side of his center-stakes. But in regions of which the topography is compara-' 
tively unknown ; and where consequently unexpected obstacles may occur which 
require the line to be materially changed for a considerable dist back, the observa- 
tions should extend to greater widths ; and for expedition the vertical dists apart 
may be increased to 3, 5, or even 10 ft, depending on the character of the country, 
Ac. AlsOj when a survey is made for a topographical map of a State, or of a county, 
vert dists of 5 or 10 ft will generally suffice. 

Let the line A B, Fig 1, starting from 0, represent three stations (S 1, S 2, 8 3,) of 
the center line of a railroad survey ; and let the numbers 100, 108, 101, 104, along 
that line denote the heights at the stakes above datum, as determined by levelling. 
Then the use of the contour lines is to show in the offlcH what would be the effect 
of changing the surveyed center line A B, by mrving any part of it to the right oi 



CONTOUB JUNES. 



303 



Iflft hand.* Thug, if it should be moved 100 ft to the left, the starting point wonl^ 
be on ground about 6 ft higher than at present ; inasmuch as its leyel would then 
be about 106 ft above dktum, instead of 100. Station 1 would be about 7 ft higher, 
or 110 ft instead of 103. Station 2 would be about 7 ft higher, or 108 ft instead of 
101. If the line b<« thrown to the right, it will plainly be on lower ground. 

The field obeervat^'ons for contour lines are sometimes made with the spirit-level; 
but more frequently oy a slope-man. with a straight 12-ft graduated rod, and a slope 
instniment, or clinometer. At each station he lays his rod upon the ground, as 




FIg.l. 

• 
nearbr a^ right angles to the center line A B as he can Judge by eye ; and placing 
the slope instrument upon it, he takes the angle of the slope of the ground to the 
nearest ^ of a degree. He also observes how far beyond the rod the slope continuee 
the same ; and with the rod he measures the dist. Then laying down the rod at that 

Kint also, he takes the next slope, and measures its length ; and so on as far as may 
Judged necessary. His notes are entered in Ids field-book as shown in Fig 2 ; the 
angles of the slopes being written above the lines, and their lengths below ; and 
should be accompanied by such remarks as the locality suggests ; such as woods, 
rocks, maryih. sand, field, garden, across small run, ftc, Ac. 

* la thni aiing the word* right and left wc an lUppoMd to have our baeki turned to the ■tartiog 
point of the survey. In a river, the rliplit bniik or shore is that which 
IS on the right band as we descend it, that is, in speaking of its right or left 

huk. ve are lODpoMd to hare oar backs turned toward! Ita head, or origin ; and bo with a surrey 



804- 



CONTOUR LINES, 




I- 



91 



''m^i' 



64- 70 



It is not abeolately necessary to represent the slopes roughly in the fleld-book, aa 
in Fig 2; for by usin^ the sign + to signify "up;" — "4own;" and = "'leTel,*' 
the slopes may be vrnt- 
ten in a straight line, 
as in Fig 2^. 

The notes naving been 
taken, the preparation 
of the contour lines by 
means of them, is of 
course office-work ; and 
is usually done at the 
same time as the draw- 
ing of the map, &c. The. 
field observations at each 
station are then sepa- 
rately drawn by protrac- 
tor and scale, as shown 
in Fig 3 for the starting 
point O. The scale should not be less than about -^ inch to a ft, if anything Iik« 
accuracy is aimed at. Suppose that at said station the slopes to the right, taken in 
their order, are, as in Fig 2, U°, 4°, and '26P ; and those to the left, 20°, lO^, and IQP ; 
and their lengths as in the same Fig. Draw a hor line h o. Fig 3 ; and consider the 
center of it to be the station-stake. From this point as a center, lay off these angles 
with a protractor, as shown on tho arcs in Fig 3. Then beginning say on the right 
hand, with a parallel ruler draw the first dist a c, at its proper slope of 16^ ; and of 
its proper length, 45 ft, by scale. Then the same with c y and yt.Do the same with 
those on the left hand. We then have a cross-sectitm of the ground at 8ta 0. Then 
on the map, as in Fig 1, draw a line as m n, or A 10, at right angles to the line of road, 
and passing through tha station-stake. On this line lay down nie Jior dists a d, d «, s «, 
ae^eg^gk^ marking them with a small star, as is done and lettered in Fig 1, at 8ta O. 

When extreme accuracy is pretended to, these hor dists must be found by measure 
on Fig 3 ; but as a general rule it will be near enough, when the slopes do not ex- 
ceed 10°, to assume them to be the same as the sloping diets measured in the field. 
Next ascertain how high each of the points cy tint is above datum. Thus, measure 
by scale the vert dist ae. Suppose it is found to be 5 ft ; or in other words, that e 
is 5 ft below stationHBtake 0. Then since the level at stake is 100 ft above datum, 
that at c must be 6 ft less, or 100 — 6 = 95 ft above datum ; which may be marked in 
light lead-pencU figures on the map, as at d, Fig 1. N6xt for the point y, suppose 
we find « 2/ to be 11 ft, or y to be 11 ft below stake ; then its heiglit above datum 
must be 100 — 11 =s 89 ; which also write in pencil, as at s. Proceed in the same 
way with t. Next going to the left hand of the station-stake, we find « I to be say 
2 ft ; but Z is above the level of the station-stake, therefore its height above datum is 




Biff. 8. 



100 4- 2 » 102 ft, as figured at e on the map. Let ng be 5 ft; then is n, 100 -f- ^ 
105 ft above datum, as marked at a ; and so on at eacn station. When this has been 
done at several stations, we may draw in the contour lines of that portion by hand 
thus: Suppose they are to represent vert heights of 3 ft. Beginning at Station O 
(of which the height above datum is 100 ft) to lay down a contour line 103 ft abova 
datum, we see at once that the height of 103 ft must be at ^, or at ^ the dist from « 
to g. Make a light lead-pencil dot at t ; and then go to the next fetation 1. Here 
we see that the height of 103 ft coincides with the station-stake itself; place a dot 
there, and go to Sta 2. The ^evel at this stake is 101 ; therefore the contour for lOP 



CONTOUli LDcaa. 



305 



ft mtut evidently be 2 ft higher, or at <, ^ of the dl^t fh>m Sta 2 to +104 ; theretiort 
make a dot at i. Then go to Sta 3. Here the leTel being 104 aboye datum, the con- 
tour of 103 must be at y, or i of the diet from Sta 3 to +99 ; put a dot at y. Finally 
draw by hand a curving line through ^ SI, i, and y ; and the contour line of 103 ft 
ii done. All the others are prepared in the same way, one by one. The level of each 
must be figured upon it at short intervals along the map, as at 103 c, 106 c, Ac 

Or, instead of first placing the + points on the map,l;o denote the slope dists actu- 
ally measured upon the ground, we may at once, and with lees trouble, find and show 
those only which represent the points ty S 1, t, y, Ac, of the contours themselves. 
Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec- 
tion c < of the ground has been prepared as before ; and that we want the hor dista 
from the stake, to contour linea for 94, 97, 100 ft, Ac, 3 ft apart vert. 




Draw a vert line t; 2, through the station-stake, and on it by scale mark levels 
of 94, 97, 100, dba ft. This is readily done, inasmuch as we have the level 104 of 
the stake already given. Through these levels draw the hor lines a. b, m, n, <&c. 
to the ground-slopes. Then these lines, measured by the scale, plainly give the 
requirea dists. 

When the ground is very irregular transversely, the cross-sections must be 
taken in the field nearer together than 100 ft. The preparation of contour lines 
will be greatly facilitated by the use of paper ruled into small squares of not less 
than about ^ inch to a side, for drawing the cross-sections upon. 

When the ground is very steep, it is usual to shade such portions of the map to 
represent hill-side. The closer together the contours come, the steeper of course 
is the ground between them ; and the shading should be proportionally darker 
at such portions. But for working maps it is best to omit the shading. 

In surveys of wide districts, the transit instrument with a graduated vertical 
circle or arc, g, p. 291, ia used for measuring the angles of slope, instead of 
the common slope-instrument. 

In many cases, notes similar to the following will serve the purpose of contour 
lines on railroad surveys. 



BUCO.. 
61.. 

es.. 

6S. 



... — S.1B. +S. IL. 
... + 2.2B. — 1.8L. 
... = 1. E. + 4. 1 L. 



Wblek meaai tbat at ttotlon 40, the slope of tbe groand on the right, m nearly as he can Jadge by 
0jm, •r by hi* band-lerel, is aboat S ft downward, for 1 ehain, or 100 ft ; and on the left, about 2 ft 
apward In 1 ehnln. At 61, 2 ft ap, in Zehatns t* the right; and 1 ft down in S chains to the left. 
A% tS, l«y«l for 1 ohaln to tbe right; and ascending 4 ft in 2 chains to the left. At 6S, the same as at 
n, Ai aoBie spots it will be well to add a sketch of a orons-seotion, like Fig 2 ; only, instead of the 
■agies, use ft of rise or fall, to indicate the slopes, as J udged bj eye, or by a haod-level. By this 
■ethod, the resolt at every station will be somewhat in error; bat these small errors will balance 
•aeh other m» nearly that the total may be regarded as sufBeiently correot for all the parpoees of a 
pnUmioMxy eettmate of the oost of a rood. When the final stakes for guiding the workmen are 
pioflod* the slopes should be sorefliUy taken, in order to ooloalato the qnontity of ezeavation aooa- 
ratoly for payment. 

20 



TH E LEV EL. 

Qui ptDS 1 J which coDHoe the semlclrculsr clipi 1 1, aud iheu oprnlg; the clip*. 
The pins should be tied l« the Ys, by pieces ot string, to preveut Ihofr being iom. 
(be ilide of the oMwt-gbai O, is nio>ed burkward ur rorw&rd by a rauk niid plnian, 
bf meBDB or the mlHeS hewl A. The slide of the lyt-gkia £. la moved Id [be same 
WB* br the milled head e. A cTlindrlual lube ef brass, oallal s lAmfe, is usua]]* 
hirnMied with «*eh kTel. It la Intended to bo slid on to the objeci-cnd O of the 
teleicape, to prerent Ibe ^are of the sun upua the objecl-glass, when the nun ia 
low. At Biaui outer rlDseiiclrcUag the telescope, and carrrlOE 4 small cspstan- 
beadea wrewa; tmof wfilcb.pp, are at lop and boiiom; while the other two, 
of whkh I la ODe, areittba^ea, and M right inelei top p. laslde a[ this outer 
ring la another, loaldo of the telescope, atid wblc^b bas stretched acrosa it two 



«, when cairjlQ^ 

.._ _... th'm ™r'!J'^ 

be Juii^ed'bf ejar^ls euablea the lereller lo'^see^ tbaTt/o i(^m»n hulda b"^ 

ia desired, as la KHuetimei the case, when itsking out work, ^t may ba obtained (^ 
IA* tiutrumrnt ij in perfiet a^ailmtnl, and UvUai) by tighllng at a plmnb-llne. or 
olhor »ert oltjoct, and then turning tbe tideetopo a little in Its tiw aa to bring the 

Uw teleKopa and Y, to aare that tron'cle In fiiiure. Heller & Brightly, howaTw. 



The small holes around the beadaofthe 4 small capstaii-screwsti,l,JustnrerfedU^ 
are for admitting the end of a small steel pin, or lerer, fbr tumlogtbem. If flnt 

will be lDworM^ and Iba liorltnntal hair with it. But un loiAing through the tal» I 



THE LEVBU 307 

■cope th«7 will appear to be mSsed. If first the lower one be looeened, and the npper 
one tightened, the hor hair will be Mctnally raised, but apparently lowered. This is 
because the glasses iu the eye-piece B reTerse the apparent position of objects intid€ 
cf the telescope ; which effect is obTiated, as regcurds exterior ol^Jects, by means of 
the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a 
hair appears to strike too high, it must be raised still higher ; if it appears to be 
already too &r to the right or left, it must be actually movcKl still more in the same 
direction. 

This remark, however, does not apply to teleacopn which make objects appear 
iUTerted. 

There is no danger of li^urlng the hairs by these motions, inasmuch as the four 
screws act against the ring only, and do not come in contact with the hairs them- 
t^lves. 

Under the telescope is the bubble-tube D D. One end of this tube can be raised or 
lowered slightly by means of the two capstan-headed nuts n n, one of which must 
be looeened before the other is tightened. On top of the bubblo'tube are scratches 
for showing when the bubble is central in the tube, frequently these scratches, or 
marks, are made on a strip of brass placed above the tube, as in our fig. There are 
several of them, to allow for the lengthening or shortening of the bubble by changes 
of temperatuie. At the other end of the bubble-tube are two smidl capstan-screws, 
placed on opposite sides horizontally. The circular head of one of them is shown 
near L By means of these two screws, that end of the tube can be slightly moved 
hor, or to right or left. Under the bul>ble-tube is the bak Y F ; at one end of which, 
as at y, are two large capstan-nuts to w, which operate upon a stout interior screw 
which forms a prolongation of the Y. The holes in these nuts are lai^r than the 
others, as they require a larger lever for turning them. If the lower nut is loosened 
and the upper one tightened, the Y above is raised ; and that end of the telescope 
becomes farther removed from the bar; and vice versa. Some makers place a similar 
screw and nuts under both Ys ; while others dispense with the nuts entirely, and 
substitute beneath one end of the bar a large circular milled head, to be turned by 
the fingers. This, however, is exposed to accidental alteration, which should be 
avo&ded. 

When the portions above m are put upon m. and fastened bv the screw Y, all 
the upper part may be swung round hor, in either direction, oy loosening the 
elamp-serew H ; or such motion may be prevented by tightening thatecrew. 
It frequently happens, after the telescope has been sighted very nearly upon an 
object, and then clamped by H, that we wish to bring the cross-hairs to coincide 
more precisely with the object than we can readily do by turning the telescope by 
kand: and in this case we uee the tanfrent-ticrew 5, by means of which a 
Bliffht but steady motion may be given after the instrument is clamped. For 
fuller remarks on the clamp and tangent-screws, see '* Transit." 

The parallel plates m and S are operated bv four levelllnipHierews ; 
three or which are seen in the figure, at K K. The screws work in sockets B; 
which, aa weU as the screws, extend above the upper plate. When the instrument 
is placed on the ground for levelling, it is well to set it so that the lower parallel 
plate S shall be as nearly horizontal as can be roughly judged by eye ; in order 
to avoid much turning of the levelling screws K ^ in making the upper plate 
m hor. The lower plate S, and the brass oarts below it, are together called the 
tripod-taead ; and, in connection with three wooden legs Q Q Q, constitute 
the tripod. In the figure are seen the heads of wing-nuts J which confine the 
legs to the tripod-head. Under the center of the tripod-head should always be 
placed a small ring, from which a plumb-bob may be suspended. This is not 
needed in ordinary levelling, but becomes useful when rangmg center-stakes, &c. 

To adjast a Irevel. 

This is a qnite simple operation, but requires a little patience. Be careful to avoid 
thraininff any of the screws. The large Y nuts ie w sometimes require some force to 
ttoH them ; but it should be applied by pressure, and not by blows. Before begin- 
rJDg to su^nst, attend to the o^ect-glass, as directed in the first sentence under ^^ To 
•i^nst a plain transit.** 

Three at^nstments are necessary ; and rrnist be made in the following order: 

First, that of tlie cross-bairs ; to secure that their intersection shall 
toatinue to strike the same point of a distant object, while the telescope is being 
tnnu'd round a complete revolution in its Ys. This is called ac^usting the line 
sf eolllmation, or sometimes, the line of sight; but it is not strictly the line 
of (tight until all the adjustments are finished; for until then, the line of coUimation 
vni not serve for taking levelling sights. If eross-liairs brealK* see p 296. 

Second* Miat of Uie bnbble-tnbe D D, to place it parallel to the Une 



308 TBB LBYBL. 

0f coUimatlon. preTiomly •4|asted; so that when the bahble stands at the centra o( 
ItD tube, indicating that it is lerel, we know that onr sight through the telescope is 
hor. To replace broken bubble tabe, see p 296. 

Tbird, tbat of tbe Ts, by which the telescope and bubble-tube a^re supported; 
flo that the bubble-tube, and line of sight, shall be perp to the yert axis of the instru- 
ment; so as to remain hor while the telescope is pointed to objects in diff directions, 
as when taking back and fore sights. 

To make tbe first adjastmenty or that of the cross-hairs, plant the 
tripod ^r/n2y upon the ground. In this adjustment it is not necessary to lerel the 
instrument. Open the clips of the Ys ; unclamp ; draw out the eye-glass E, until 
the cross-hairs ieure aeen perfectty cUar ; sight the telescope toward some clear dis- 
tant point of an object ; or still better, toward some straight line, whether yert or 
not. More the object-glass 0, by means of the milled head A, so that the object shsJI 
be clearly seen, wltbout parallax, that is, without any apparent dancing 
about of the cross-hairs, if the eye is moved a little up or down or sideways. To 
secure this, the object-glass alone is moved to suit different distances ; the eye-glass 
is not to be changed after it lb once properly fixed upon the cross-hairs. The neglect 
of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the 
tangent-screw d, bring either one of the cross-hairs to coincide x>reciM/y with the 
object. Then gently, and without jarring, revolve the telescope naif-way round in 
its Ys. When this is done, if the hair still coincides precisely with the object, it is 
in adjustment ; and we proceed to try tbe other hair. But if it does not coincide, 
then by means of the i screws p, t, move the ring which carries the hairs, so as to 
rectify, as nearly as can be judged hy eye, only one-fuUf of the error; remembering 
that the ring must be moved in the direction opposite to what appean to be the 
right one ; unless the telescope is an inverting one. Then turn the telescope back 
again to its former position : and again by the tangent-screw bring the cross-hair to 
coincide with the object. Then again turn the telescope half-way round as before. 
The hair will now be found to be more nearly in its right place, but, in all probabil- 
ity, not precisely so ; inasmuch as it is difficult to estimate one-half the error accu- 
rately by eye. Therefore a little more alteration of the ring must be made ; and it 
may be necessary to repeat the operation several times, before the adjustment is 
perfect. Afterward treat the other hair in precisely the same manner. When both 
are adjusted, their intersection will strike the same precise spot while the telescope 
is being turned entirely round in its Ys. This must be tried before the aci^ustment 
can be pronounced perfect; because at times the adjustment of the second hair, 
slightly deranges that of the first one ; especially if both were much out in the b» 
ginning. 

To make the second adjustment, or to place the bubble-tube paralW 
to the line of collimation. This consists of two dis> 
tinct adjustments, one vert, and one hor. The first 
of these is effected by means of the two nuts n n on 
the vert screw at one end of the tube ; and the second 
by tbe two hor screws at the other end,^, of the tube. 
Looking at the bubble-tube endwise, from t in tbe 
foregoing Fig, its two hor adjusting-screws 1 1 are 
seen as in this sketch. The larger capstan-headed 
nut helov), has nothing to do with the adjustments ; 
it merely hold^ the end of the tube in its place. 

. To make the vert adjustment of the bubble-tube, by means of the two nuts nn. Place 
the telescope over a diagonal pair of the levelling-ecreWH K. K ; and clamp it there. 
Open the clips of the Ys; and by means of the levelliug-screws bring the bubble to 
the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and 
put it back again in its reversed position. This being done, if tbe bubble still remains 
at the center of its tube, this adjustment is in order ; but if it moves toward one end, 
that end is too high, and must be lowered ; or else the other end must be rftised. 
First, correct htdf the error by means of the levelling-screws K K, and then the re- 
maining half by means of the two small capstan->headed nuts n». To roiM the end 
n, first loosen the upper nut and then tighten the lower one ; to do which, turn each 
nut so tiiat the near side moves toward your right. To louwr it, first loosen the lowei 
nut, then tighten ttie upper one, moving the lutar side of each nut toward your ^fU 
Having thus brought the bubble to the middle again, again lift the telescope out of 
its Ys ; turn it end for end, and replace it. The bubble will now settle nearer the 
center than it did before, but will probably require still further adjustment. If so, 
correct haif the remaining error by the levelling-screws, and half by the nuts, as be* 
fhre; and so continue to repeat tbe operation until the bubble remidns at the cental 
in both positions. For another method, see '* To adjust the long bubble-tube,** p 2ML 
Horizontal adjustment of bubble-tube ; to see that its axis is in the same plans 
with nhat of the telescope, as it usually is in new instruments. It is not eesily d» 




TEE LEVEL. 309 

ranged, except by blows. Have the bubble-tube, as xxearly as may be, directly under 
the telescope, or over the center of the bar T F. Bring the telescope over two of the 
leTellingHScrews K K ; clamp it there ; center the bubble with said screws ; turn the 
telescope in its Ts, say about ^ inch, bringing the bubble-tube out from over the 
center of the bar, first on one side, then on the other. If the bubble stays centered 
irhile so swung out, this adjustment is correct. It it,runs towajrd opposite ends of its 
tabe when swung out on opposite sides of the center, move the end t of the tube by 
the two horizontal screws 1 1 until the bubble stays centered when the tube is swung 
out on either side. If the bubble runs toward the same end of its tube on both sidesy 
tiie tube is not truly cylindrical, but slightly conical,* so that if the telescope is 
tamed in its Ts the bubble will leave the center, even when the horizontal a^just- 
ment is correct. It is known to be correct, in such tubes, if the bubble runs the Kune 
diikmce from the center when swung out the same distance on each side. 

Having made the horizontal adjustment, turn the telescope back in its Ys until the 
bubble-tube is over the bar. Bepeat the vertUxU adjustment (p 308), which may have 
become deranged in making this horizontal one. Persevere until both adjustments 
are found to be correct at the same time. 

To mabe tibe tliird adjustment, or to a4just the heights of th« Ts, m 
■s to make the line of coUimation parallel to the bar V F, or perp to the vert axis 
of the instrument. The other adjustments being made, fasten down the clips of the 
Ts. Make the instrument nearly level by means of all four of the levelling-screws 
K. Place the telescope over two of the levelling-screws which stand diagonally; 
and leave it there undamped. Then bring the bubble to the center of its tube, by 
the two levelling-screws. Swing the upper part of the instrument half-way around, 
BO that the telescope shall again stand over the same two screws; but end for end. 
This done, if the bubble leaves the center, bring it half-way back by the large cap- 
stan nuts to, 10 ; and the other half by the two levelling-screws. Remember that to 
raise the T, and the end of the bubble over «o, io, the lower tv must be loosened ; and 
the upper one tightened ; and vice versa. Now place the telescope over the pttier 
diagonal pair of levelling-screws; and repeat the whole operation with them, ilav- 
Ing completed it, again try with the first pair; and so keep on until the bubble re- 
mains at the center of its tube, in every position of the telescope. 

Correct levelling may be performed even if all the foregoing adjustments are 
out of order; provided each fore-sight he taken at preeiidy the tame distance from 
the instrument as the back-sight is. But a good leveller will keep his instrument always 
in acyustment; and will test the ac^ustments at least once a day when at work. As 
much, however, depends upon the rodman, or target-man, as upon the leveller. A rod- 
man who is careless about holding the rod vert, or about reading the sights correctly, 
ibould he discharged without mercy. 

The levelling-screws in many instruments become very hard to turn if dirty. Clean 
with water and a tooth-brush. Use no oil on field instruments. 

Forma for level note-books. When the distance is short, so as not to 
fsqnire two sets of books, the following is perhaps as good as any. 

I 8^olI.'S£tU^".,.| »»• |l*"l.|«»«««.| Cut. I «IL I 

Bat on pnblic works generally the original field-books have only the first five cols. 
After the grades have been determined by means of the profile drawn from these, 
the re«nlta are placed in another book, which has only the first col and the last four. 
In both cages, the right-hand page is reserved for memoranda. The writer considers 
it best, both witii the level and with the transit, to consider the term " Station " to 
apply to the whole dist between two consecutive stakes; and that its number shall 
be that vrrftten on the last stake. Thus, with the transit, Station 6 means the dist 
fin>m stake 5 to stake d; that it has a bearing or ocnirse of so and so; and its length 
is so and mo. And with the level, Station 6 also means the dist from stake 5 to stake 
6; the back-sight for that dist being taken at stake 5, and the Ibre-sight on stake 
6; and thait the level, grade, cut, or fill is that at stake 6. The starting-point of the 
nwej, wbether a stake, or any thing else, we call and mark simply 0. 

• This defect can be remedied only by removing the tube and inserting a correctly- 
ihaped one, and this is best done by an instrument-maker ; but correct work can 
be done in qpite of it, Ihus: Make all the acyustments as nearly correct as possible. 
Level the instrument. By turning the telescope in its Ts, make the vertical hair 
coincide with a plumb-line or other vertical line, and make a short continuous knife- 
Kiatch on the collar nearest the object-glass, and on the adjoining T. Lift the tele- 
Kope ont of its Ts, turn it end for end, replace it in its Ts ; again bring the upright 
hair vertical, and make on the other T a scratch coinciding with that on the collar. 
Then, in levelling or in a4justing, always see that the scratch on the collar coincides 
Mitt thai on the ac^oining T when the bubble-tube is under the telescope. 



THB HAKD-IiBVKL 
TOE BASD-LETEI. 



ffll. M arpuTged bj Prof€«ir Locke, of (;indlDll»U,l« 


SLmpljhuKIl 


.IR it in DM hind, u 


idlookinethroBgh 
'nd!^TinVni^1 








BDd Ihe oW«t . 






ebotlomof-hlA 




™,ghtl,.top 




KO.'^mmrftottl^ 






.»1dopcniog..nd 


for sijurtizig 


tie «irs, (an be 1 


loFhed hKkwird a 


ir poahod fom 


•»rdby»™al1>pri 










p1»c«J at so 1 


iiglBo''HS=,«M^ 




■h the f.>rfB.B. 


nlioned DpBniogB, 


lil'y* Jrf fa^ a^^ 


f. M shown I.) 


' tho^nlle dotl^' 


linMCMdK; Mid 



nWUi of™h8 tnbe rTi. Throup?^' 



(b« wire shull Btaoir no piinllu ; bat ■pp**' tHd; BCBinM the dIi)boI irEui Ibe <J« 
la allghll; moTod Dp or don. At «ch and oT t)i» tube B O la ■ dmdu pl«oB of 

To adlaat tbe bond-level, lint fli 
&U fbet Id 100 J'ards ^art, 'nitB beiDg done« ] 
level marlij. ud take atght a the oUisr. If, then, tbe wire does not appau- 



aleht a the oUisr. If, then, tbe wire does not appau- to be 
■ illghtly huckwiLrt or forward, M the 

hand-level tWelf, eieii If i[ la onUrely od( o( odjoil- ^ "^ f 

nhlecL u d. an that tha wire aoneui to cnl the eenter 

ro''"iVhVr°CMermito"ri^rk"f^"h»l1--w'ay"to\wee"c°i^^^^ Then (> und in will be Che 
two iBTel mirkB reijuirod. With o»re, these adjualnimti, when once msdo, will 
remain in ordet for ream. The Intlrumenl gsnenllyhas aBmall ring r, for hanging 

eiplorlng a roule. The heigh 1 of « bar* iiil I can be found bybeelnning st the ftiot. 
and ijgtiling aheed at anj little chance objei^t which the onm-wlre ma; Btrlka, ■• a 
pebble, cnlg, Ic; then going fonrard, ataud at Ibat object, and fix Che win m 

a height eqnal 10 thac of the eye, lay bK^ feet, or whateTar it may be, WheUier 
going DP or down It, If the bill la coTered with grau, bnihea, te, a target rod moR 
be need for the fore-aighw ; and the tonstant height of the eye may be reganlsdH 



IiBTXU. 



311 



To adiast a bailder*s plnmb- 
leTei, todi stand it npon any two sup- 
borta «» and it, and mark where the plumb- 
line cuts at o. Then reverse It, placing the 
foot t upon n, and d upon m, and mark where 
the line now cute at e. Half-way between o 
and e make the permuient mark. Whenerer 
the line cuts this, the fiaet t and d are on a 
level. 




To adjast a slope-lnstrament, or clinometer. As usually made, 
the bubble-tube is attached to the movable bar by a screw near each end^ and the 
head of one of the screws conceals a small slot in the bar, which allows a slight vert 
motion to the scr^w when loose, and with it to that end of the tube. Therefore, in 
order to adjust the bubble, this screw is first loosened a little, and then moved up 
«r down a trifle, as may be reqd. It is then tightened again. 



312 ZJSVBLLING BY THB BABOMETEB. 



liETEIililire BT THE BAROIHETER. 

1. Many drcnmstancM combine to render the results of this kind of WTellino^ no* 
reliable where great accuracy is required. This fact was most concluslyely proved 
by the observations made by Captain T. J. Cram, of the U. 8. Coast Stirvey. See 
Beport of U. 3. C. S., toI. for 1864. It is difficult to read oiT from an aneroid (the 
kind of barom generally employed for engineering purposes) to within from two to 
five or six ft, depending on its size. The moisture or dryness of the air aflTects the 
results; also winds, the ricinity of mountains, and the daily atmospheric tides, 
which cause incesHant and irregular fluctuations in the barom. A barom hanging 
quietly in a room will often vary -^jf of an inch within a few hours, corresponding 
to a diff of elev.ition of nearly 100 ft. No formula can posiiibly be deyised that shall 
•mbrace these sources of error. The variations dependent upon temperature, latir 
tnde, Ac, are in some measure provided for; so that with very ddicate instruments, • 
skilful observbr may measure the diff of altitude of two points dose together, such 
as the bottom and top of a steeple, with a tolerable confidence that he is within two 
or three feet of the truth. But if as short an interval as even a few hours elapses 
between his two observations, such changes may occur in the condition of the atmo- 
sphere that he may make the top of the steeple to be lower than its bottom ; or at 
least, cannot feel by any means certain that he is not ten or twenty ft in error; and 
this may occur without any perceptible change in the atmosphere. Whenever prac- 
ticable, therefore, there should be a person at each station, to observe at both points 
at the same time. Single observations at points many miles apart, and made on dif- 
ferent days, and in different states of the atmosphere, are of little value. In such 
cases the mean of many observations* extending over several days, weeks, or months, 
and made when the air is apparently undisturbeid, will give tolerable approximAtionB 
to the truth. In the tropics the rang^ of the atmospheric pres is much leas than 
in other regions, seldom exceeding ^ inch at any one spot; also more regular in 
time, and, therefore, less productive oferror. Still, the barometer, especially eitiier 
the aneroid, or Bourdon^s metallic, may be rendered highly useftil to the civil engi- 
neer, in cases where great accuracy is not demanded. By hurrying from point tO 
point, and especially by repeating, he can form a Judgment as to which of two sum- 
mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying 
party, may materially lessen their labors, especially in a rough country, by select- 
ing the general route for them in advance. The accounts of the agreement within 
a few inches, in the measurements of high mountains, by diff observers, at diff 
periods ; and those of ascertaining accurately the grades of a railroad, by means of 
an aneroid, while riding in a car, will be believed by those only who are ignonmt 
of the subject. Such results can happen only by chance. 

When possible, the observations at different places should be taken at the same 
time of day, as some check upon the effects of the daily atmospheric tides ; and In 
very important cases, a memorandum should be made of the year, month, day, and 
hour, as well as of the state of the weather, direction of the wind, latitude of the 
place, Ac, to be referred to an expert, if necessary. 

The effecto of latitade are not included in any of our formulas. When 
reqd they may be found in the table page 814. Several other corrections must be 
made when great accuracy is aimed at ; Dut they require extensive tables. 

In rapid railroad exploring, however, such refinements may be neglected, Inas- 
much as no approach to such accuracy is to be expected ; but on the contrary, errors 
01 from 1 to 10 or more feet in 100 of he^ht, wul frequently occur. 

As a very roa§rli avera^r® ^^ iQ^y assume that the barometer falls -J^ 
inch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But 
in fact its rate of tall decreases continually as we rise ; so that at one mile high it 
fiEdls ^ inch for about 106 ft rise. Table 2 shows the true rate. 



JLEVSLLING BY THE BABOM£T£B. 



813 



To «aeert«in tlie dUT of lieiirbt belweew two points. 

Jlcn^E 1. Take readings of the barom and therm (Fah) in tlie siiade at both 
stations. Add together the two readings of the barom, and div their sum bj 2, for 
their mean ; which call b. Do the same with the two readings of the thermom,*and 
call the mean t. Subtract the least reading of the barom from the greatest ; and call 
the diff d. Then mult together this diff d; the number from the next Tablt: No. 1, 
opposite ( ; and the constant number 30. Div the prod by b. Or 

Height Diff (d) of ^ Tabular number opposite v, n«„„*..„* on 
in feet "^ barom ^ mean (f) of thermom X constant du . 

mean (b) of barom. 

ExAMPLi. Beading of the barom at lower station, 26.64 ins ; and at the upper 
sta 20.82 ins. Thermom at lowest sta, 70^; at upper sta, 4^. What is the diff io 
height of the two stations? Here, 

Sarom, 26.64 Therm, 70^ 

" 20.82 *• iOP 

— — Also^ — — 

2)47.46 2)110 



23.78 mean of bar, or b. 669 mean of 

therm, or t. 
The tabular number opposite 66°, is 917.2. 

Bar. Bar. 
Again, 26.64 — 20.82 = 5.82, diff of bar ; or d. Hence, 

d. Tab No. Con. 
Height _ 5.82 X 917 Si X 30 _ 160143.12 ^^^ ^ ^^^,^ 
in feet 23.73 (or 6) "*" 23.73 

Then oorrect for latitude, if more aooaracy is reqd, by rule on next page. 

mie screw at tlie baekof an aneroid Is for adjusting the index by a stand- 
ard barom. After this has been done it must by no means be meddled with. In 
some instruments specially made to order with that intention, this screw may bo 
used also for turning the index back, after having risen to an elevation so great that 
the index has reached the extreme limit of the graduated arc. After thus turning 
it back, the indications of the index at greater heights must be added to that at- 
tained when it was turned back. 

TABIiB 1. For Rale 1. 



Mean 




Mmd 




Mean 




Mean 


• 


of 


No. 


of 


No. 


of 


No. 


of 


No. 


Ther. 




Ther. 




Ther. 




Ther. 




oo 


801.1 


80° 


864.4 


60O 


927.7 


90O 


991.0 


1 


803.2 


31 


866.6 


61 


929.8 


91 


993.1 


3 


805.3 


32 


868.6 


62 


981.9 


92 


995.2 


S 


807.4 


38 


870.7 


63 


934.0 


98 


997.3 


4 


809.6 


84 


872.8 


64 


936.1 


94 


999.4 


6 


811.7 


86 


874.9 


66 


938.2 


95 


1001.6 


. « 


818.8 


36 


817.0 


66 


940.3 


96 


1003.7 


7 


815.9 


87 


879.2 


67 


942.4 


97 


1005.8 


8 


818.0 


38 


881.8 


68 


944.6 


98 


1007.9 


9 


820.1 


80 


883.4 


69 


946.7 


96 


1010.0 


10 


822.2 


40 


886.4 


70 


948.8 


100 


1012.1 


11 


824.3 


41 


887.6 


71 


950.9 


101 


1014.2 


12 


826.4 


42 


869.6 


72 


953.0 


102 


1016.3 


13 


828.5 


48 


891.7 


73 


955.1 


103 


1018.4 


li 


880.6 


44 


893.8 


74 


967.2 


104 


1020.5 


16 


833.8 


46 


896.0 


76 


969.3 


105 


1022.7 


16 


834.9 


46 


898.1 


76 


961.4 


106 


10i4.8 


17 


887.0 


47 


900.2 


n 


968.6 


107 


1026.9 


18 


889.1 


48 


902.3 


78 


965.6 


108 


1029.0 


1» 


8«1.3 


49 


904.6 


79 


967.7 


lOB 


1031.1 


20 


84SJI 


60 


906.6 


80 


860.9 


110 


ia'M.2 


21 


8A5.4 


61 


908.7 


81 


972.0 


111 


1035.3 


23 


847.6 


63 


910.8 


82 


974.1 


112 


1037.4 


28 


848.6 


63 


913.0 


83 


976.2 


118 


1039.5 


3i 


861.8 


64 


916.1 


84 


978.3 


lU 


1041.6 


25 


853.9 


66 


917.2 


86 


980.4 


116 


1043.8 


96 


8G6.0 


66 


919.3 


86 


982.6 


116 


1045.9 


27 


868.1 


67 


921.4 


87 


964.7 


117 


1048.0 


28 


800.2 


68 


923.6 


88 


966.8 


118 


1050.1 


» 


863.8 


69 


925.6 


89 


988.9 


119 


1052.2 



314 



LEVSLLINO BT THE BAROMETEB. 



RuLi 2. BelTlUe's short approx rale is the one beit adapted to rapid 
Aeld use, namely, add together the two readings of the barom only. Also find the 
diir between said two readings; then, as tbe sam of the two readlnffs 
is to tbelr dlff, so Is 55000 feet to the reqd altitude. 

<3orreetion for latitude is usually omitted where great accuracy is not 
required. To apply it, first find the altitude by the rule, as before. Then divide it 
by the number in the following table opposite the latitude of the place. (If the two 
places are in different latitudes, use their mean.) Add the quotient to the altitude 
if the latitude is leea than 45°. Subtract it if the Utitude is more than 45°. No cor- 
rection required for latitude 45°. 







Table of corrections 


for latitude. 






Lat. 




Lat. 




Lat. 




Lat. 




Lat. 




Lat. 




0° 


S52 


14° 


890 


280 


630 


420 


8867 


640 


1140 


680 


490 


a 


S54 


16 


416 


80 


706 


44 


10101 


66 


941 


70 


460 


4 


856 


18 


486 


82 . 


804 


46 


00 


68 


804 


72 


486 


6 


860 


ao 


460 


U 


941 


46 


10101 


60 


705 


74 


416 


8 


867 


22 


490 


86 


1140 


48 


8867 


62 


680 


76 


990 


10 


8T5 


M 


527 


88 


1468 


60 


9028 


64 


572 


78 


886 


IS 


886 


26 


672 40 


9038 1 63 


1458 


66 


527 


80 


876 



lieTCllins by Barometer; or bjr the bollini^ point. 

Rule 3. The following table. No. 2, enables us to measure heights either by means 
of boiling water, or by the barom. The third column shows the approximate alti- 
tude above sea-level corresponding to diif heights, or readings of the barom ; and to 
the diif degrees of Fahrenheit's thermom,at which water boils in the open air. Thus 
when the barom, under undisturbed conditions of the atmosphere, stands at 24.08 
inches, or when pure rain or distilled water boils at the t«mp of 201° Fah ; the place 
is about 5764 ft above the level of the sea, as shown by the table. It is therefore 
rery easy to find the diffoi altitude of two places. Thus : take out from table No 2, 
the altitudes opposite to the two boiling temperatures ; or to the two barom readings. 
Subtract the one opposite the lower reading, from that opposite tbe upper reading. 
The rem will be the reqd height, as a rough approximation. To correct this, add 
together the two therm readings ; and div the sum by 2, for their mean. From teble 
for temperature, p 816, take out the number opposite this mean. Mult the ap- 
proximate height just found, by this tabular number. Then correct for lat if reqd. 

Ex. The same as preceding ; namely, barom at lower sta, 26.64 ; and at npper ata, 
20.82. Thermom at lower sta, 70° Fnh ; and at the npper one, 40°. What is the diff 
of height of the two stations ? 

Alt. 

Here the tabular altitudes are, for 20.82 9579 

and for 26.64 3115 



To correct this, we have 



70° + 40° 110° 



6464 ft, approx height. . 
65° mean ; and in table p 816, opp to 



55°, we find 1.048. Therefore 6464 X 1.048 = 6774 ft, the reqd height. 
This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft In each 100 ft. 

At 70° Fah, pure water will boil at 1° less of temp, for an average of about 660 ft 
of elevation above sea-level, up to a height of U a mile. At the height of 1 mile, V* 
of boiling temp will correspond to about 560 ft of elevation. In table p315 the 
mean of the temps at the two stations is assumed to be 32° Fah ; at which no correc- 
tion for temp is necessary in using the table ; hence the tabular number opposite 
32°, in table p 316, is 1. 

This diff produced in the temp of the hailing pointy by change of elevation, most 
not be confounded with that of the atmotpherej due to the same cause. The air be- 
comes cooler as we ascend above sea-level, at the rate (very roughly) of about 1^ Fah 
for every 200 ft near sea-level, to 350 ft at the height of 1 mile. 

The followingr table, "So. 2, (so tar as it relates to the barom^ was da^ 
dncnd by the wnter from the standard worU on the barom 'by Lieut.-Ool. R. S. Wil- 
liamson, U. S. army."* 

• FablUbed by penaiMton of OoTernmeni In 1868 by Vao Koetraod. N. T- 



lAVELLINQ BT THE BABOKBTEB, ETC. 315 



TABI.E 9. 
I.«ivellliifc by Bfkrometer ; or by the bnllliift p»liil. 

imed templn theebide 32° Full. JI pot S2°, mult harnni sk us per TBbIe,p 



316 



SOUND. 



Corre«il«iis f«r temperatare; to be used in eonnecUon wltb 
Bule 3, wlien irreater aecuracy is necessary. Also in con- 
nection witli TaMe 2 wlien tlie temp is not 33°. 



Mean 


• 


Mean 




Mean 




Mean 


* 


temp 


Malt 


temp 


Mult 


temp 


Mnlt 


temp 


Mult 


in the 


by 


in the 


by 


In the 


by in the 


by 


shade. 




shade. 


» 


ihade. 


shade. 




Zero. 


.933 


28° 


.992 


5«o 


1.050 


84° 


1.108 


20 


.937 


30 


.996 


68 


1.064 


86 


1.112 


4 


.942 


32 


1.000 


60 


1.058 


88 


1.117 


6 


.946 


34 


1.004 


62 


1.062 


90 


1.121 


8 


.960 


36 


1.008 


64 


1.066 


92 


1.126 


10 


.954 


38 


1.012 


66 


1.071 


94 


1.129 


12 


.958 


40 


1.016 


68 


1.076 


96 


1.133 


14 


.962 


42 


1.020 


70 


1.079 


98 


1.138 


16 


.967 


44 


1.024 


72 


1.083 


100 


1.142 


18 


.971 


46 


1.028 


74 


1.087 


102 


1.146 


20 


.976 


48 


1.032 


76 


1.091 


104 


1.150 


22 


.979 


60 


1.036 


78 


1.096 


1U6 


1.154 


U 


.983 


62 


1.041 


80 


1.100 


108 


1.168 


» 


.987 


64 


1.046 


82 


1.104 


110 


1.163 



SOUND. 



u 


— 20° 


M 


1040 


« 


— 10° 


u 


1060 


it 





u 


1060 


it 


10° 


« 


1070 


U 


20° 


u 


1080 


M 


• 32° 


u 


1092 


« 


40° 


u 


1100 


M 


50° 


u 


1110 


t( 


60° 


it 


1120 


H 


70° 


M 


1130 


U 


80° 


U 


1140 


(« 


90° 


U 


1160 


K 


100° 


t( 


1160 


M 


110° 


<( 


1170 


« 


120° 


a 


1180 



(( 



(( 



t: 



u 






u 



it 



It 
It 



it 



t( 



M 



U 



M 

tt 



It 
tt 



tt 



U 

tt 



tt 
4( 



<t 

tt 



it 
it 



tt 
tt 



U 
tt 



u 

M 



(« 



« 

tt 



«. 1 


tt 


6.08 


.. 1 


u 


5.03 


■B 1 


tt 


4.98 


*■ 1 


« 


4.93 


^ 1 


(( 


4.8S 


IBS X 


u 


4.83 


■> I 


« 


4.80 


^ 1 


« 


4.78 


^ 1 


H 


4.73 


m^ 1 


U 


4.68 


m= 1 


It 


4.63 


*B 1 


l( 


4.69 


■B X 


u 


4.65 


IM \ 


tt 


4.61 


— 1 


tt 


4.47 



« 



(« 



-reloeitjr at sound in quiet open air, haa been experimentally deter> 
mined to be very approximately 1090 feet per second, when the temperature is at 
freezing point, or 32° Fahienheit. For every degree Fahrenheit uf increase of 
temperature, the velocity increases by from V^ foot to 1^ feet per second, according 
to different authorities. Taking the iucreasu at 1 foot per second for each degree 
(which agrjBes closely with theoretical calculations), we have 

at ^ 30° Fahr 1030 feet per sec '^ 0.1951 mile per sec — 1 mile in 6.13 seconds. 

— 0.1970 

— 0.1989 

— 0.2008 
» 0.2027 

— 0.2045 

— 0.2068 

— 0.2083 

— 0.2102 

— 0.2121 

— 0.2140 

— 0.2169 

— 0.2178 

— 0.2197 

— 0.2216 

— 0.2236 

If the air is calm, fog or rain does not appreciably affect the retult ; but wisds do. 
Very loud sounds appear to travel somewhat faster than low ones. The watchword 
of sentinels has been heard across still water, on a calm night, 10^ miles ; and a 
cannon 20 miles. Separate sounds, at intsrvals of ^ of a second, cannot be distin- 
guished, but appear to be connected. The distances at whieh a speaker can be 
understood, in front, on one side, and behiud him, are about ab 4, S, and 1. 

Dr. Charles M. Cresson informs the writer tliat, by repeated trials, he found that 
in a Philadelphia gas main 20 inches diameter and 16000 feet long, laid and covered 
in the earth, but empty of gas, and having one horizontal bend of 90^, and of 40 fast 
reuUus, the sound of a pistol-shot travelled 16000 feet in precisely 16 seconds, or 1000 
feet per second. The arrival of the sound was barely audible ; but was rendered 
very apparent to the eye by its blowing off a diaphragm of tissue-paper placed over 
the end of the main. 

Turo bosits anchored some distance apart may serve as a base line for 
triangulating objects along the coast; the distance between them being first found 
by firing guns on board one of them. 

In ivater tliie velocity is about 4708 feet pef second, or about 4 times that 
in air. In iwroodsy it is from 10 to 16 times ; and in metalSf fh>m 4 to 10 times 
greater than in air, according to some authorities. 



w 



t« 



g!^;.'.: 



Eaeb 13^ M IS" of bekt prodncaln wr*t Ir^ i 






°* ^ I"" '° "" '^t^' Id Iki no «!• niv U(>. n IM Iha I«(Ilu i 



, „ ezpsnalon of HtVD* wUlfmi* 

TinM mcRlnS points «re qnlM lUiccrtatM. W« ^n (be miu of 

•atanwIborMH. iMoWlii tUiMlJi »(aiMjtor Bnool^tioiit l«l»,»t«|Mlro» "HlilHriH 

■aiBnUHT wUh Uu H« olulsiuil • «'n n» will diiim lu l>ii|lk [au? of in Iniik. 



THEBUOIfETEBS. 



T« «liBnc« derreea of Fitlirenbelt 1« Ike eorrMipOBdIns de- 
■re«a •rc«ntl)?«de) l&kBiir>)irBidliiK32°liivn-ihitnih« ilit|aoiis: mnlt 
— lD°IMiit Agali,— 190F>b = r— II— a])>cCi-t = — •&XC'i-?=— l^Oaal. ~ ~ 

To cli»iiKe P>h taMBOi uks & Fati rudlug 32^ Iswrthu ihs (Ina 

• ti^-eoBi.o. linln,— IPlVlisI— ls-ijfx'*+»='--«SXt-H'=— *1°B*M. 
ToeliAnce £«ntto l'nb|niiill ihe Oni nwllDg b> 0; dirlds by ! 



eIlAn» O 



5l*i>^'=r^^?"° 



>. Tkkaft 



i^taii.—tfOmn=i--xx'-T6>+ai=—*''r^. 



shannlMml to Fohri- 

niiu:Tir'i«i™=("X9*«)+»t 



,^'-4i« 



'^^oekanreBtenMCentiiDnltbyH; div by 4. Thna: -fB°R«u — + 8° 
TABI>E1> FHtarenhelteomiMredwItliCeiitlBr^deaiidK^a* 



THERHOUETERS. Zl^ 

TABLE 3. CcntlrnMie eom|»a>«d with rahrcnbclt a>« 



C F. K. C. F. R. C. F. I K. C F. R. 



TABLE S. Ktaaoiar coiapitrfld wltb FabMiBbelt luul 



tlTradc. 



K. F. C. B. F. C. R. F. C. R. F. C. 

M III.M KnOO 4< 4119 tl.iS I* TS Jt.n —1 1.16 — I1.1S 

II M!ffi K.a M ullS SI.M It 00 wiot — < O^UI — IiIm 

ig 303.01 (MM 11 lU.ie M.it IS lb e.11 — s —1.75 — i8.7e 

Ts via.Ti 0.11 H 9ij» u.m 11 u 1.U — s -4.in _».og 

TS IK.r, •l!«i •> MM i'lUI II M s!lM - 8 -H 60 -h'm 

II I "" — SLIl (U n.oi> uiloo ID M LU — » — isigo ~ia!oi> 

a Bi.n 91 ot!eo iiiw i ti.ui l.oo ~h ~ie'^ -si'm 

a i nils n mIn leM i km i.m - la ~-9i.oo — SSM 

« 17.50 II 01,11 a.ib I U.1S ].^ —a -u.a -x.u, 

11 iZn MM 10 J7!oo »!oo -10 g'so '\-ii.sb -« -».a> -ooiot 



320 Aia 

AIR-ATMOSPHERE. 

The atmospliere is known to extend to at least 4S miles 

abore the earth. It is a mixture of about 79 measures of nitrogen gas and 21 
of oxygen gas ; or about 77 nitrogen, 23 oxygen, br weight. It generallr con- 
tains, however, a trace of water, and of carbonic acid and carbu retted hydrogen 
gaaes, and still less ammonia. 

Density of air. Under *' normal ** or " standard " conditions (sea level, 
lat 45^, barometer 760 mm => 29.922 ins, temperature O^C^ZTP F) dry air 
weljirhs 1.292673 kilograms per cubic meter * = 2.17888 fi>s avoir per cubic yard. 
For other lats and elevations — 

Density, in kg per cu m, =i 1.292673 X j^^^A ^ ^^ —0.002837 oos 2 lat) • 

where B = earth's mean radius =» 6,366,198 meters ; A >« eleTation above aea 
level, in meters. For other temperatures, see below. 

Under normal conditions, but with 0.04 parts carbonic acid (0 O,) in 100 parts 
of air, density = 1.293052 kg per cu m.f » 2.17952 fi» avoir per cu yd.^ 

The atmospherie pressure, at any given place, may yarr 2 inches or 
more from day to day. 'rhe averagr® pressure, at sea level ^ varies from 
about 745 to 770 millimeters of mercury according to the latitude and locality. 
760 millimeters * is generally accepted as the mean atmospheric pressure, and 
called an atmosphere. The '* metrie atmosphere,** taken arbitrarily 
at 1 kilogram per square centimeter, is in general use in Continental Europe. 
The pressure diminishes as the altitude increases.f Therefore, a pump in a high 
region will not lift water to as great a height as in a low one. The pressure of 
air, like that of water, is, at any given point, equal in all directions. 

It is often stated that the temperature of the atmosphere lowers at 
the rate of 1<^ Fah for each 300 feet of ascent above the earth's snrfhees 
but this is liable to many exceptions, and varies much with local causes. Actual 
observation in balloons seems to show that, up to the first 1000 feet, 1^ in aboat 
200 feet is nearer the truth ; at 2000 feet, 1° in 250 feet ; at 4000 feet, 1° in 300 feet; 
and, at a mile, 1° in 350 feet. 

In breathingr, a grown person at rest requires from 0.25^ to 0.35 of a cubic 
foot of air per minute : which, when breathed, vitiates from 8.5 to 5 cubic feet. 
When walking, or hard at work, he breathes and vitiates two or three times as 
much. About 5 cubic feet of fresh air per person per minute are required for the 
perfect ventilation of rooms in winter; 8 in summer. Hospitals M to 80. 

Beneath the ipeneral level of the surface of the earth, in temperate 
regions, a tolerably uniform temperature of about 50° to 60^ Fah exists at 
the depth of about 50 to 60 feet ; and inereases about 1° for each additional 50 to 
60 feet ; all subject, however, to considerable deviations owing to many local 
causes. In the Rose Bridge Colliery, England, at the depth of 2424 feet, the 
temperature of the coal is 93.5° Fah ; and at the bottom of a boring 4169 feet 
d eep, near Berlin, the temperature is 119°. 

The air is a werjr slow eondnetor of heat; hence hollow walls 
serre to retain the heat in dwellings ; besides keeping them dry. It mahea 
into a waeunm near sea level with a velocity of about 1157 feet per second ; 
or 13.8 miles per minute ; or about as fast as sound ordinarily travels through 
quiet air. See Sound. ^ 

Iiike all other elastie fluids, air expands eoually witik 
e^ual increases of temperature. Every increase of o° Fah, expands 
the bulk of any of them slightly more than 1 per cent of that which it has at 0^ 
Fah ; or 500° about doubles its bulk at xero. The bulk of anv of them diminishes 
inversely in proportion to the total pressure to which it is subjected. 

This holds good with air at least up to pressures of about 750 fte per sqnare 
inch, or 50 times its natural pressure ; the air in this case occupying one-flxtietii 
of its natural bulk. In like manner the bulk will increase as the total preasuiv 
is diminished. Substances which follow these laws, are said to be perCeetiy 

* H. V. Regnault, M6moires de 1* Acaddmie Royale des Sciences de Plnstitiit de 
France, Tome XXI, 1847. Translation in abstract. Journal Of Franklin Insti- 
tute, Phila., June, 1848. 

fTravaux et M6moire8 du Bureau International.desPoidset Mesnres, Tomel 

£age A 54. Smithsonian Meteorological Tables, 1898, publiabed In Smithsooian 
[iscellaneous Collections, Vol. XXXV, 1897. 
I See Conversion Tables. 
f See Leveling by the Barometer. 



WIND. 



321 



1 



elAstle. Under apressure of about 6^ tons persqiiaie Indi, air would become 
as dense as wa^er. Since the air at the surface of the earth is pressed 14^ !ba per 
square inch by the.atmosphere above it, and since this is equal to the we^ht of a 
oolumn of water 1 inch square and 34 feet high, it follows that at the depths of 
84, 68, 102 feet, &4i, below water, air will be compressed into ^, 3^, 3^ Ac, 
01 its bulk at the surface. 

In a divliiK-bell, men, after some experience, can readily work for seyeral 
hours at a depth of 51 feet, or under a pressure of 2^ atmospheres ; or 37^ ftis 

Kir square inch. But at 90 feet deep, or under 3.64 atmospheres, or nearly 55 
8 per square inch, they can work for but about an hour, without serious suffer^ 
ing from paralvsis. or even danger of death. Still, at the St Louis bridge, work 
was done at a deptn of 1103>^ feet ; pressure 63.7 9>8 per square inch. 

The dew point is that temp (varying) at which the air deposits its vapor. 

Tlie gnreatest beat of tlie air in the sun probably never exeeeds 
145° Fah J nor the greatest cold — 74P at u ight. About 130° above, and 40° below 
zero, are the extremes in the U. S. east of the Mississippi ; and 65^ below in the 
N. W.; all at common ground level. It is stated, however, that —81° has been 
observed in N. E. Siberia: and +10lo Fah in the shade in Paris; and +153° in 
the sun at Greenwich Observatory, both in July, 1881. It has frequently ex- 
'beaded -i-l(XP Fah in the shade in Philadelphia during recent years. 



WIND. 

The relation between the weloeity of wind, and its preas* 
lire against an obstacle placed either at right angles to its course, or inclined 
to it, has not been well determined ; and still less so its pressure against curved 
surfaces. The pressure against a laige surface is probably proportionally greater 
than gainst a small one. It is generally supposed to vary nearly as the squares 
of the velocities; and when the obstacle is at right angles to its direction, the 

Sressure in lbs per square foot of exposed surface is considered to be equal to 
lie square jof the velocity in miles per hour, divided by 200. On this basis, 
which is probably quite aefective, the following table, as given by Smeaton, is 
prepared. 



YeiL in MUes 


Vei. m Ft. 


Frea. in Lbs. 


Remarks. 


per Hour. 


per Sec. 


per Sq. Ft. 


1 


1.467 


.005 


Hardlj perceptible. ^..^^ 
PleMsnt. ^C~J>g 


s 


2.933 


.020 


8 


4.400 


.045 


^ 


4 . 


5.867 


.OBO 


^ 


5 
10 


7.38 
14.67 


.125 
.5 


zJo/rt 


12H 


18.S3 


.781 


Fresh breexe. O 


lb 


n. 


1.125 




20 


S9.33 


8. 


^ . Th« prei acainit 


25 


86.67 


3.125 


Brlakwind. « iiemioylindrioal 


so 


44. 


4.5 


Strong wind. sarfftoe ac&nom 


40 


S6.67 


8. 


High wind. ig about half that 


60 


73.88 


12.5 


Storm. against the flat 


60 


88. 


18. 


Violent storm. gnrf abnni. 


SO 


117.3 


32. 


Hurricane. 


100 


146.7 


60. 


Violent hunieane, uprooting large trees. 



TreddTOld reeommends to allow 40 lbs per sq ft of roof for the 

pras of wind against it ; but aa roob are oonstruoted with a slope, and oonsequentty do not receive 
<ke ftill foree or the wind, this is plainly too much.* Moreover, only one>half of a roof is usually ex- 
I, even thas partially, to the wind. Probably the force in suoh cases varies approximately as the 
of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per 
prodmsed a pre* of 14 lbs per sq ft againut an object perp to it: and one of 70 miles, per hour, 
(the Mvterect gale on reoord at that city.) 43 lbs per sq foot. These would make the ores per sq ft, 
More nearly equal to the ■qoAre of the vel iq miles per hour, dlv by 100 ; or nearly twice as great as 
glvea in Smaaton's table, we should ourselves give the preference to the Liverpool observations. A 
very violent gale in Scotland, registered by an excellent anemometer, or wind-gauge, 45 lbs per sq 
ft. It la stated that aa high as 55 lbs has been observed at Glasgow. High winds often l^ roots. 

The gaoge at Oirard Coliese, Fhilada, broke onder a strain of 43 lbs per sq ft ; a tornado passing 
St the moment, within a mils. 
By inrersion of SoMaton's rule, if the force in Iba per sq ft, be mult by 200, the sq rt of the prod 
Igive the vel in milec per hoar. Smeaton's rule is used by the U. S. Signal Service. 



«i/c 



• The writer thinks 8 lbs per sq foot of mrdinarn doubte-aloping roofi, or 10 lbs for •Ked-rooft, suffl ■ 
«imt allowanee for prea of wind. 

21 



322 



RAIN AND SNOW. 



RAIN AND SNOW. 



The annaal preelpitatlon * at any giyen place varies greatly from 
year to year, the ratio between maximum and minimum being frequently greater 
than 2 : 1. Beware of averai^es. In estimating ^oo^«, take the maximum 
falls, and in estimating water supply, the mtnimttm, not only per annum, but for 
short periods. In estimating water supply, make deductions for evaporatios 
and leakage. 

Maxima and minima deduced fh>m observations covering only 4 or 5 years are 
apt to be misleading. Data covering even 10 or more years may just miss includ- 
ing a very severe flood or drought. Becords of from 15 to 20 years may usually 
be accepted as sufficient. 

Table 1. Averafre Preelpltatlon * In tbe United States, in ins. 
(Frmn Bulletin C of U. S. Department of Agriculture, compiled to end of 1891.) 



Steto. Spr. 

Alabama 14.9 

Ariisona 1.3 

Arkansas 14.8 

California. 6.2 

Colorado 42 

Connecticut 11.1 

Delaware 10.2 

Dist. Columbia.11.0 

Florida 10.2 

Georgia 12.4 

Idaho 4.4 

Illinois 10.2 

Indiana 11.0 

Indian T'y 10.6 

Iowa 8.3 

Kansas 8.9 

Kentucky 12.4 

Louisiana 13.7 

Maine 11.1 

Maryland 11.4 

Massachusetts. ..11.6 

Michigan 7.9 

Minnesota 6.5 

Mississippi 14.9 

Missouri 10.0 



8am. Aat. Win. Atxn'l 



13.8 


10.0 


149 


53.6 


43 


2.2 


3.1 


10.9 


12.5 


11.0 


12.8 


50.6 


0.3 


3.5 


11.9 


21.9 


5.5 


2.8 


2.3 


148 


12.5 


11.7 


11.5 


46.8 


11.0 


10.0 


9.6 


40.8 


12.4 


9.4 


9.0 


41.8 


21.4 


14.2 


9.1 


549 


15.6 


10.7 


12.7 


51.4 


2.1 


3.6 


7.0 


17.1 


11.2 


9.0 


7.7 


38.1 


11.7 


9.7 


10.3 


42.7 


11.0 


8.9 


6.7 


36.2 


12.4 


8.1 


41 


32.9 


11.9 


6.7 


3.5 


31.0 


12.5 


9.7 


11.8 


46.4 


15.0 


10.8 


144 


53.9 


10.5 


12.3 


11.1 


45.0 


12.4 


10.7 


9.5 


440 


11.4 


11.9 


11.7 


46.6 


9.7 


9.2 


7.0 83.8 


10.8 


5.8 


8.1 


26.2 


12.6 


10.1 


15.4 


53.0 


12.4 


9.1 


6.5 


38.0 



SUte. 



Spr. Sum. Aat. Win. Annl 



Montana 4.2 

Nebraska S.9 

Nevada 2.3 

N. Hampshire. 9.8 

New Jersey 11.7 

New Mexico..... 1.4 

New York 8.5 

N. Carolina 12.9 

N. Dakota 46 

Ohio 10.0 

Oregon 9.8 

Pennsylvania...l0.3 
Rhode Island. ..11.9 

S. Carolina 9.8 

S. Dakota 7.2 

Tennessee 18.6 

Texas 8.1 

Utah 3.4 

Vermont 9.2 

Virginia 10.9 

Washington 8.6 

W. Virginia 10.9 

Wisconsin 7.8 

Wyoming 4.8 

United States... 9.2 



49 


2.6 


2.8 


140 


10.9 


49 


2.2 


26.9 


0.8 


1.3 


3.2 


7.6 


12.2 


11.4 


10.7 


44.1 


13.3 


11.2 


11.1 


47.8 


5.8 


8.5 


2.0 


12.7 


10.4 


9.7 


7.9 


86.5 


16.6 


12.0 


12.2 


68.7 


8.0 


2.8 


1.7 


17.1 


11.9 


9.0 


9.1 


40.0 


2.7 


10.5 


21.0 


440 


12.7 


10.0 


».6 


42.6 


10.7 


11.7 


12.4 


46.7 


16.2 


9.7 


9.7 


46.4 


9.7 


8.5 


2.5 


22.9 


12.5 


10.2 


145 


60.7 


8.6 


7.6 


6.0 


80.3 


1.5 


2.2 


8.5 


lao 


12.2 


11.4 


9.8 


42.1 


12.5 


9.5 


9.7 


42.6 


3.9 


10.5 


16.8 


89.8 


12.9 


9.0 


10.0 


42.8 


11.6 


7.8 


6.2 


82.6 


8.5 


2.2 


1.6 


11.0 


10.3 


8.3 


8.6 


80.8 



At Philadelphia, in 1869, during which occurred the greatest drought known 
there for at least 50 years, 43.21 inches fell ; August 13, 1873, 7.3 inches in 1 day ; 
August, 1867, 15.8 inches in 1 month ; July, 1842, 6 inches in 2 hours ; 9 inches 
per month not more than 7 or 8 times in 25 years. From 1825 to 1893, greatest 
in one year, 61 inches, in 1867 \ least, 30 inches, in 1826 and 1880. At Norristown, 
Pennsylvania, in 1865, the writer ^aw evidence that at least 9 inches fell within 

5 hours. At Genoa, Italy, on one occasion, 32 inches fell in 24 hours ; at Geneva, 
Switzerland, 6 inches in 3 hours ; at Marseilles, France, 13 inches in 14 hoars; 
in Chicago, Sept., 1878, .97 inch in 7 minutes. 

Near iJondon, Eng^land, the mean total fall for many years is 28 inches. 
On one occasion, 6 inches fell in 1% hours! In the mountain districts of the 
English lakes, the fall is enormous: reaching in some years to 180 or 240 inches; 
or from 15 to 20 feet ! while, in tne adjacent neighborhood, it is but 40 to 00 
inches. At Liverpool, the average is 34 inches ; at Ckiinburgh, 30 : Glasgow, 22; 
Ireland, 36; Madras, 47; Calcutta, 60; maximum for 16 years, 82; Delhi, Si; 
Gibraltar, 80 ; Adelaide, Australia, 23 ; West Indies, 36 to 96 ; Rome, 89. On the 
Khassya hills north of Calcutta, 500 inches, or 41 feet 8 inches, have Allien in the 

6 rainy months I In other mountainous districts of India, annual falls of 10 to 
20 feet are common. 

A moderate steady rain , continuing 24 hours, will yield a depth of about an indu 

As a seneral rule, more rain fhlls in warm tban in 99MA 

€SonntrIes; and more in elevated regions than in low ones. Local pecuUaxw 



* Precipitation includes snow, hail, and sleet, melted, 
estimated at 10 inches snow » 1 inch rain. 



Unmelted snow ia 



BAIV AND SNOW. 



323 



KieB, howerer, sometimeB reyerae this : and also oanse great differences in the 
amounts in places quite near each other ; as in the English lake districts Just 
alluded to. It is sometimes difficult to account for these variations. In some 
lagoons in New Granada, South. America, the writer has known three or four 
heavy raiiio to occur weekly for some months, during which not a drop fell on 
hills about 1000 feet high, within ten miles' distance, and within full sight. At 
another locality, almost a dead-level plain, fully three-quarters of the rains that 
fell for two years, at a spot two miles from his residence, occurred in the morn- 
ing ; while those which fell about three miles from it, in an opposite direction, 
were in the afternoon. 

Tlie relation between precipitation and stream»0ow is greatly 
ai^cted by the existence of forests or crops, by the slope and character of ground 
on the water-shed, especially as to rate of absorption, by the season of the year, 
the frost in the ground, etc. The stream-flow may ordinarily be taken as vary* 
ing between 0.2 and 0.8 of the rainfall. Streams in limestone regions frequently 
loee a very large proportion of their flow through subterranean caverns. 

Aasnminff a fall of 2 feet in 1 year (=3 76,379 cubic feet per square mile per 
day), that half the rainfall is available for water supply, and that a per capita 
consumption of 4 cubic feet (^t 30 gallons) per day is sufficient, one square mile 
will supply 19,095 persons ; or a square of 88.26 feet on a side will supply one 
person. 



Ineb of rain amonnto to 3630 enble fiBet; or 27156 U. SL 

EkUonB ; or 101.3 tone per acre ; or to 2323200 cubic feet ; or 17378743 U. S. gal^ 
ns ; or 64821 tons per squ&re mile at 62^^ fts per cubic foot. • ^ 
The most destructive rains are usualhr those which fall upon snow, nnder 
which the ground is frozen, so as not to absorb water. 

Table 2. Kaxlmnm intensify of rainlMl for periods of 5, 10, and 

60 minutes at Weather Bureau stations equipped with self-registering 

gauges, compiled from all available records to the end of 1896. 

(From Balletin D of U. S. Department of Agriculture.) 



Stations. 


Rate per hour for— 


Stations. 


Rate per hour for— 


6min. 


lOmins. 


60 mins. 


6min. 


10 mins. 


60 mins. 


Bismarck. 


Ins. 

9.00 
8.40 
8.16 
7.80 
7.80 
7.50 
7.44 
7.20 
7.20 
6.72 
6.60 
6.60 
6.60 


Inches. 

6.00 
6.00 
4.86 
4.20 
6.60 
6.10 
7.08 
6.00 
4.92 
4.98 
6.00 
3.90 
4.80 


Inches. 

2.00 
1.30 
2.18 
1.25 
2.40 
1.78 
2.20 
2.15 
1.60 
1.68 
2.21 
L60 
1.86 


Chicago 


Ins. 

6.60 
6.48 
6.00 
6.00 
5.76 
6.64 
6.46 
5.40 
6.40 
4.80 
4.56 
8.60 
3.60 


Inches. 

6.92 
6.58 
4.80 
4.20 
6.46 
3.66 
5.46 
4.80 
4.02 
3.84 
4.20 
3.30 
240 


Inches. 
1 60 


St. Paul 


Galveston... 

Omaha 


2 55 


Kew Orleans 


1.65 


Milwaukee 


Dodge City 

Norfolk 


1.84 
1 55 


Washington 

Jacksonville 


Cleveland 

'Atlanta. 


1.12 
1.50 


Detroit. 


Key West 

Philadelphia... 

St Louis... 

Cincinnati 

Denver... 


2.26 
1.60 
2.25 


New York aty> 
Boston 


Savannah 


1.70 
1 18 


Indianapolis...... 


Memphis.. 


Duluth— 


1.35 









The welirbt of firesbly flallen snow, as measured by the author, 
varies from aoont 5 to 12 lbs per cubic foot ; apparently depending chiefly upon 
the degree of humidity of the air through whicn it had passed. On one occasion, 
when minsled snow and hail had fallen to the depth of 6 inches, he found its 
weight to Be 81 fbs per cubic foot. It was very dry and incoherent. A cubic foot 
of heavy snow mav, by a gentle sprinkling of water, be converted into abont 
half a cubic foot of slush, weighing 20 9>s.; which will not slide or mn oflf 
from a shingled roof sloping 30^, if the weather is cold. A cubic block of snow 
•atorated with water until it weighed 45 Tba per cubic foot, Just slid on a rough 
board inclined at 45''; on a smoothly planed one at 30^ ; and on slate at 18° : all 
ipproximate. A prism of snow, saturated to 62 lbs per cubic foot, one inch 
square, and 4 inches high, bore a weif^t of 7 fi»s ; which at first compressed 
it abont one-quarter part of its length. European engineers consider 6 n>s per 
square foot of roof to oe snffielent allowance for the weight of snow; 



324 RAIN AND 6NOW. 

and 8 lbs for the pressare of wind ; total. 14 lbs. The writer thinks that in the 
U. S. the allowance for snow should not be taken at leu than 12 fi>8 ; or the total 
for snow and wind, at 20 Bm. There is no danger that snow on a roof will 
become saturated to the extent Just alluded to ; because a rain that would supply 
the necessary q^uantity of water would also by its violence wash away the snow ; 
but we entertain no doubt whatever that the united pressures from snow and 
wind, in our Northern States, do actually at times reach, and even surpass. 
20 fbs per square foot of root The limit of 

perpetnal snow at the equator is at the height of about 16000 feet, or say 
3 miles above sea-level; in lat 45° north or south, It is libout half that neight; 
while near the poles it is about at searleveL 

Rain Oaoi^es. Plain cylindrical vessels are ill adapted to service as rain 
gauges ; because moderate rains, even though sufficient to yield a large run-off 
from a moderate area, are not of sufficient depth to be satisfactorily measured 
unless the depth be exaggerated. The inaccuracy of measurement, always con- 
siderable, is too great relatively to the depth. 

In its simplest and most usual form, the gauge (see Fig.) consists essentially 
of a funnel. A, which receives the rain and leads it into a measuring 
tube, B, of smaller cross-section. The funnel should have a verticci 



and fairly sharp edge, and, in order to minimize the loss through xA/ 
evaporation, it should fit closely over the tube, and its lower end ^ 

diould be of small diameter. ' 



The depth of water in the tube is ascertained by inserting, to the 
bottom of the tube, a measuring stick of some unpolished wood 
which will readily show to what aepth it has been wet. The stick 
may be permanently graduated, or it may be compared with an ordi- 
nary scale at each observation. The tube is usually of such diameter 
that the area of its cross-section, minus that of the btick, is one-tenth 
of the area of the funnel month. The depth of rainiaU is then one- 
tenth of the depth as measured by the stick. 



B 



DiiCENsiONS OP Standard U. S. Wbathbb Bubbau Bain Gauge. Ins. 

A. Beceiver or funnel. Diameter 8 

B. Measuring tube. Height 20 ins. " 2.53 
C C. Overflow attachment and snow gauge. " 9 

Such gauges, with the tubes carefully made from seamless drawn brass tubing, 
«08t about $5.00 each ; but an intelligent and careftil tinsmith, given the dimen- 
sions accurately, can construct, of galvanized iron, for about ^.00 a gauge that 
will answer every purpose of the engineer. 

Tbe exposure has a very marked effect upon the results obtained. The 
funnel should be elevated about 3 ft, in order to prevent rain from splashing back 
into it from the ground or roof. If on a roof, the latter shoald be nat, and pref- 
erably 50 ft wide or wider, and the gause should be placed as far as possible 
from tbe edges ■ else the air currents, produced by the wind striking the side of 
the building, will carry some of the rain over the gauge. No objects much higher 
than the gauge should be near it, as they produce variable air currents which 
•may seriously affect its indications. 

An overflow tank, G, should be provided, for cases of overfilling the tube. 

Water, freezing in the gauge, may burst it, or force the bottom off, or at least 
<ao deform the gauge as to destroy its accuracy. 

To measure snow, the funnel is removed, and the snow is collected in 
the overflow attachment or other cylindrical vessel deep enough to prevent the 
snow from being blown out, and the cross-sectional area of which is accurately 
known. The snow is then melted, either by allowing it to stand in a warm 
place, or, with less loss through evaporation, by adding an accurately known 
quantity of luke-warm water. In the latter case, the volume of the added water 
must of course be deducted from tbe measurement. 

Rainfall equivalent of snow. Ten inches of snow are usually taken 
as equivalent to 1 in of rain ; but, according to various authorities, the equiva- 
lent may vary between 2>^ and 34; i. e., between 25 and 1.84 &». per cubic foot. 

Self-reeordinir g^ngr^s, of which several forms are on the market, are 

Jiuite expensive, and, even when purchased from regular makers, seldom per- 
ectly reliable. Gauges using a small tipping bucket register inaccurately la 
heavy rains ; those using a float are limitea as to the total depth which they c 
xegister ; while those which weigh,tbe rain, if exposed, are aflbcted by wind. 



BAIir AND 81I0V. 



Bulletia Cot U.S. DeparUuent of AftlcaUura, IBM.) 



•F0riinlaiu)^Abbmmm,UablIa,tiioata0.2Mniih,34.S mauuthBton!4,3per 
pt. of the dnjA ombrsc^d ivltfain tb* 30 yean, ram fall to a depth of from a 

tFMaiOiitobarU7Bcail;. t Fnm Juaarr 1S14 oeOj. iFiomliUy ISTi odI;. 



326 WATEB. 



WATER. 

Pure water, as boiled and distilled, Is eomposed of the tiro gases, hydro- 
gen and oxygen ; in the proportions of 2 measures hydrogen to 1 of oxygen ; 
or 1 weight of hydrogen to 8 of oxvgen. Ordinarily, however, it contains sev- 
erid foreign ingredients, as carbonic and other acids ; and soluble mineral, or 
organic substances. When it contains mirch lime,- it is said to be h€a^; and will 
not make a good lather with soap. Tbe air in its ordinary state conlwiiis 
about 4 grains of water per cubic foot. 

The average pressure of tlie air at sea level, will balamee a 
colamn of water 34 feet high ; or about 30 inches of mercury. At its boil- 
ing point of 212° Fah, its bulk is about one twenty-third greater than at IQP. 

Its welg^lit per cubic foot is taken at 62^ fi>fl,or 1000 ounces avoir; but 62}^ 
lbs would be nearer the truth, as per table beh>w. It is about 816 times hearier 
than air, when both are at the temperature of 62°; and the barometer at 80 
inches. With barometer at 30 inches the weight of perfectlv pure water is as 
follows. At about 39*^ it has its maximum density of 62.425 ros per cubic foot. 



Temp, Fah. Lbs per Cub Ft. 

929 62.417 

40° 62.423 

50° 62.409 

60° 62^7 



Temp, Fah. Lbs per Cub Ft. 

70° 62.302 

80® - 62.218 

90°- 62.119 

212°- « 69.7 



Weifflil; of sea ivater 64.00 to 64.27 B>s per cubic foot, or say 1.6 to 1.9 9>8 

per cubic foot more than fresh water. See also p 328. 

Water has its maxlmnm density when its temperature is a littler above 
89° Fah ; or about 7^ above the freezing point. By best authorities 39.2°. From 
about 39° it expands either by cold, or by heat. When the temperature of 320 
reduces it to ice, its weight is but about 57.2 lbs. per cubic foot ; and its specific 
gravity about .9176, according to the investigations of L. Dufour. Hence, as 
ice, it has expanded one- twelfth of its original bulk as water; and the sadcleii 
expansive force exerted at the moment of freezing, is sufficiently great to 
split iron water-pipes; being probably not less than 30000 lbs per square inch. 
Instances have occurred of its splitting cast tubular posts of iron bridges, and 
of ordinary buildings, when full of rain water Arom exposure. It also loosens 
and throws down masses of rock, through the Joints or which rain or spring 
water has found its way. Retaining- walls also are sometimes overthrown, or 
at least bulged, by the freezing of water which has settled between their backs 
and the earth filling which they sustain ; and walls which are not founded at a 
sufficient depth, are often lifted upward by the same process. 

It is said that in a irlass tube ^ Incli in diameter, water will not 
freeze until the temperature is reduced to 23°; and in tubes of less than^ 
inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in 
rapid running streams. Ancbor lee, sometimes found at depths as great as 
26 feet, consists of an aggregation of small crystals or needles of ice frosen s* 
the surface of rapid open water ; and probably carried below by the fbroe of ths 
stream. It does not form under frozen water. 

Since ice floats in waters and a floatinff body displaces a weight of the 
liquid equal to its own weight, it follows that a cubic foot of floating ice weighing 
57.2 lbs, must displace 57.2 fSs of water. But 67.2 lbs of water, one foot square, is 11 
inches deep: therefore, floating ice of a cubical or paralleloplpedal shape, will 
have \^ of its volume under water; and only ^ above: and a square foot of ice 
of any thickness, will require a weiffht equal to ^ of its own weight to sink it 
to the surface of the water. In practice, however, this must be regarded merely 
as a close approxima}iion, since the weight of ice is somewhat iSfocted by en- 
closed air-bubbles. 

Pure water is usually assumed to boll at 212° Fah In the open air, at the 
level of the sea ; the barometer being at SO inches ; and at about 1^ less for every 
620 feet above sea level, for heights within 1 mile. In fsct, its boiling point 
Varies like its freezing point, with its purity, the density of the air, the material 
4>f the vessel, dbc. In a metallio vessel, it may boil at 210°; and in a glass one, 
at from 212° to 220°; and it is stated that if all air be previously extracted, it 
requires 275°. 

It evaporates at all temperatures; dissolves more substances than any 
other agent : and has a greater capacity for heat than any other known substanosi 

It is eomjpressfHl at the rate of about one-21740th. (or about ^^ of an 
inch in 18^ feet,) by each atmosphere or pressure of 16 lbs per square Inclk 
When the pressure is removed, it* »>\»uHniXj restores its orisinal boUk 



J 



WATER. .327 

Effeet on metals. The lime contained in many waters, forms deposits In 
metallic water-pipes^ and in channels of earthenware, or of masonry ; especially 
if the current oe slow. Some other substances do the same ; obstructing the 
flow of the water to such an extent, that it is always expedient to use pipes of 
diameters larger than would otherwise be necessary. The lime also forms very 
hard inemstatioiis at tbe bottoms of boilers^ very much impair- 
ing their efficiency ; and rendering them more liable to burst. Such water is 
unfit for locomotives. We have seen it stated that the Southwestern B R Ck>, 
England, prevent this lime deposit, along their limestone sections, by dissolving 
1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms 
similar deposits in boilers; as uso does mud, and other impurities. 

Water, either when very pure, as rain water; or when it contains carbonic 
acid, (as most water does,) produees carbonate of lead in lead 
pip^ ; and as this is an active poison, such pipes should not be used for such 
waters. Tinned lead pipes may be substituted for them. If, however, sulphate 
of lime also be present, as is very frequently the case, this effect is not always 
produced; and several other substances usually found in spring and river 
water, also diminish it to a greater or less degree. Fresh uraier corrodes 
vrronslit Iron more rapidly tban cast; but the reverse appears to 
be the case with sea water; although it also affects wrought iron very 
quickly ; so that thick flakes may be detached from it with case. The corrosion 
of iron or steel by sea water increases with the carbon. Cast-iron cannons 
from a vessel which had been sunk in the fresh water of the Delaware River 
for more than 40 years, were perfectly free from rust. Gen. Pasley, who had 
examined the metals found in the ships Royal George, and Edgar, the first of 
which had remained sunk in the sea for 62 years, and the last for 133 years, 
"stated that the cast iron had generally become quite soft; and in some cases 
resembled plumbago. Some of the shot when exposed to the air became hot; 
and burst into many pieces. The wrought iron was not so much injured, 
except when in eantaet vkth copper, or brcus gun^metal. Neither of these last was 
much affected, except when in contact with iron. Some of the wrought iron 
was reworked by a blacksmith, and pronounced superior to modern iron." **Mr. 
Cottam stated that some of the guns had been carefully removed in their soft 
state, to the Tower of London : and in time (within 4 years) returned their orig^ 
inal hardneu. Brass cannons rrom the Mary Rose, which had been sunk in the 
sea for 292 years, were considerably honevcombed in spots only ; (perhaps where 
iron had been in contact with them.) The old cannons, of wrought-iron bars 
hooped together, were corroded about }^ inch deep; but had probskoly been pro- 
tected bv mud. The cast-iron shot became redhot on exposure to the air; and 
fell to pieces like dry clay I" 

** Unprotected parts of cast-iron sluice-valves, on the sea gates of the Cale- 
donian canal, were converted into a soft plumbaginous substance, to a depth 
of % of an inch, within 4 years; but where they had been coated with common 
Swedish tar, they were entirely uninjured. This softening effect on cast iron 
appears to be as rapid even when the water is but slightly orackish ; and that 
only at intervals, it also takes place on cast iron imbedded in salt earth. Some 
water pipes thus laid near the Liverpool docks, at the expiration of 20 years 
were soft enough to be cut by a knife ; while the same kind, on higher ground 
beyond the influence of the sea water, were as good as new at tne end of 60 years." 

Observation has, however, shown that the rapidity of this action 
depends ntncn on the quality of the Iron ; that which is dark- 
colored, and contains much carbon mechanically combined with it, corrodes 
most rapidly : while hard white, or light-gray castings remain secure for a long 
time. Some cast-iron sea-piles of this character, showed no deterioration in 40 
years. 

Contact wltli brass or copper is said to induce a galvanic action 
which greatly hastens decay in either fresh or salt water. Some muskets were 
recovered from a wreck which had been submerged in sea water for 70 years 
near New York. The brass parts were in perfect condition ; but the iron parts 
had entirely disappeared. Galwanlstng: (coating with zinc) acts as a pre* 
serrative to the iron, but at the expense of the sine, which soon disappears. 
The iren then corrodes. If iron be well heated, and then coated with toot 
coal-tar, it will resist the action of either salt or freshwater for many years. 
It is very important that the tar be perfectly purified. Sucji a coat« 

ing, or one of paint, will not prevent barnacles and other shells from 
attaching themselves to the iron. Asphaltum, if pure, answers as well aa 
4M>a]->tar. 

Copper and bronse are very little affected by sea water. 

Ko galvanic action has. been detected where bnun leroles are inserted intt 
the water-pipea in Philadelphia. 



328 TIDES. 

Tbe most prejudicial exposure for Iron, as well as for wood, is 
that to alternate wet and dry. At some dangerous spots In Long Island Sound, 
it has heen the practice to drive round bars of rolled iron about 4 inches diam- 
eter, for supporting signals. These wear away most rapidly between high and 
low water; at the rate of about an inch in depth in 20 years ; in which time the 
4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under 
frenh water especially, or under ground, a thin coating of coal-pitch vamishi 
carefully applied, will protect iron, such as water-pipes, Ac, for a long time. 
See page 655. *The sulphuric acid contained in the water from coal minei 
corrodes iron pipes rapidly. In tbe ft'esli water of canals, iron boata 
have continued In service from 20 to 40 years. Wood remains sound for 
centuries under either fresh or salt water, if not exposed to be worn away by 
the action of currents : or to be destroyea by marine insects. 

fitea urater welgrns from 64 to 64.27 ft>s per cubic foot, or say from 1.6 to 
1.9 ft)s per cubic foot more than fresh water, varying with the locality, and not 
appreciably with the depth. Theexcess, over the weight of fresh water, is chiefly 
common salt. At 64 lbs per cubic foot, 35 cubic feet weigh 2240 fi>s. Sea water 
freezes at about 27° Fahr. The ice is fresh ; but (especially at low tempera- 
tures) brine may be entrapped In the ice. 

A teaspoonful of powdered alum, well stirred into a bucket of dirty w^ater, 
will generally purify it sufficiently within a few hours to be drinkable. If « 
hole 3 or 4 feet deep be dug in the sand of the sea-shore, the infiltrating watei 
will usually be sumciently fresh for washing with soap; or even for drinking. 
It is also stated that water may be preserved sweet for many years by placing 
in the containing vessel 1 ounce of black oxide of manganese for each gallon 
of water. 

It is said that water kept in zinc tanks ; or flowing through iron 
tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc 
formed thereby; and it is recommended to coat zinc surfaces with asphalt 
varnish to prevent this. Yet, in the city of Hartford, Conn, service pipes of 
iron, galvanized inside and out, were adopted in 1855, at the recommendation 
of the water commissioners ; and have been in use ever since. They are like- 
wise used in Philadelphia and other cities to a considerable extent. In many 
hotels and other builaings in Boston, the *' Seamless Drawn Brass Tube" of the 
American Tube Works at Boston, has for many years been in use for service 

Eipe ; and has given great satisfaction. It is stated that the softest water may 
e kept in brass vessels for years without any deleterious result. 

Tlie action of lead upon some waters (even pure ones) is highlr poison- 
ous. The subject, however, is a complicated one. An injurious ingredient may 
be attended by another which neutralizes its action. Organic matter, whether 
vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm- 
less. 

Ice may be so impure that its water is dangerous to drink. 

Tke popular notion tbat hot water freezes more qniclLljr 
than cold, with air at the same temperature, is erroneous. 

TIDES. . 

The tides are those well-known rises and falls of the surface of the sea 
and of some rivers, caused by the attraction of the sun and moon. There are 
two rises, floods, or high tides ; and two falls, ebbs, or low tides, every 24 hoars 
and 50 minutes (a lunar day) ; making the average of S hours 12^^ minutes 
between high and low water. These intervals are, however, subject to 
fpreat variations; as are also the heights of the tides; and this not only 
at different places, but at the same place. These irregularities are owing to the 
shape of the coast line, the depth of water, winds, ana other causes. ImuMy at 
new and full moon, or rather a day or two after, (or twice in each lunar month, 
at intervals of two weeks,) the tiaes rise higher, and fall lower than at other 
times; and these are called spring tides. Also, one or two days after the 
moon is iu her quarters^ twice in a lunar month, they both rise and fall less than 
at other times ; and are then called neap tides. From neap to spring they 
rise and fall more daily ; and vice versa. The time of hifrii water at any 
place, is generally two or three hours after the moon has passed over either 
the upper or lower meridian ; and is called the establishment of that 
place; because, when this time is established, the time of high water on any 
other day may be found from it in most cases. The total height of spring tides 
is generally from 1}^ to 2 times as great as that of neaps. The great ii<t*I 
wave is merely an undulation, unattended by any current, or progressive motion 
of the particles of water. Each successive hijgh tide occurs STOUt 24 mlnatei 
later than the preceding one ; anil so with the Um tides- 



EVAPOBATIOK AND LEAKAGE. 329 

EVAPOEATION, F UTBATIO N, AND LEAKAGE. 

Tbe amount of evaporation from surfaces of water exposed to 

tlM natural effect* of the open air, is of cooree greater in aammer than in winter ; althoagh It is quite 
perceptible in even, the coldest weather. It is greater in ahalloir water than in deep, inasmuch aa th* 
bottom also beoomes heated by the sun. It is greater in running, than in standing water ; on much 
the same principle that it is greater daring winds than calms. It is probable that the average dailj 
loss from ^ reservoir of moderate depth, m>m evaporation alone, throughout the 3 warmer months 

of the year, (June, Jniy, Aagust,) rarely exceeds about -^ inch, in any part of the United States. Or 

JL inch daring the 9 colder months ; except in the Soathem States. These two averages would give 

adaily one of .16 inch ; or a total annual loss of $6 ins, or 4 ft 7 ins. It probably is S.5 to 4 ft. 

By some trials by the writer. In the tropics, ponds of pure water 

8 ft deep, in a stiff retentive day, and ftally exposed to a very hot san all day, lost during the dry sea- 
son, preoijiely 2 ins in 16 days ; or H ^^oh per day ; while the evaporation from a glass tumbler was 
V inch per day. The air in that region is highly charged with moisture ; and the dews are heavy. 
Every day during the trial the thermometer reached ftt>m 115° to 126° in the sun. 

The total annual evaporation in several parts of England and Scotland is stated to average fhmi 22 
to 38 ins ; at Paris, 84; Boston, Mass, 32 ; many places in the U. 8.,' SO to 36 ins. This last would give 

a dailj average of -aA^ ineh for the whole year. Such statements, ho.wever, are of very little value, 

nnless accompanied by memoranda of the circumstances of the case ; such as the depth, exposure, 
sixe and nature of the vessel, pond. Ac, which contains the water, Ac. Sometimes the total annua) 
evaporation from a district of country exceeds the rain fall ; and vice versa. 

On canals, reservoirs, Ac, it is usual to combine the lofis bj eyaporation* 

with that by filtration. The last is that which soaks into the earth ; and of which some portion 
passes entirely through the banks, (when in embankt;) and if in very small quantity, may be dried 
up by the son and air as fast as it reaches the outside ; so as not to exhibit itself as water ; but if is 
greater quantity, it becomes apparent, as leakage. 

E. H. Gill, € E, stat^ the average evaporation and filtra- 
tion on tlie Sandy and Beaver canal, Oliio, (38 ft wide at ^ater snr- 

Cmo; 26 ft at bottom ; and 4 ft deep.) to be but IS cub ft per mile per minute, in a dry secuon. Here 
the exposed water surf in one mile is 200640 sq ft; and in order, with this surf, to lose 13 cub ft per 

mln, or 18720 cub ft per day of 34 hours, the quantity lost must be innjWV ~ '^'^^ f^> — ^H loch fa 
depth per day. Moreover, one mile of the canal contains 675840 cab ft ; therefore, the number of days 
teqd for the combined evaporation and filtration to amount to as mach as all the water in the canal, is 

^-I^ ^J^ = 36 days. Observations in warm weather on. a 22'mile reaeh of the Chenango canal, N 
18720 
York, (40; 28 ; and 4 ft,) gave 9SH cub ft per mile per min ; or 6 times as much aa in the preceding 
ease. This rate would empty the canal in about 8 days. Besides this there was an excessive leakage 
at the gates of a look, (of only bH ft lift,) of 479 cub ft per min, 22 cub ft per mile per min ; and at 
aqnedneta, and waste-weirs, others amounting to 19 cub ft per mileper min. The leakage at other 
locks with lifts of 8 ft, or. less, did not excMa about 350 cub ft per min, at each. On other canals, it 
has been found to be fhom 60, to 500 ft per min. On the Chesapeake and Ohio canal, (where 60, 82, 
and 6 ft.) Mr. Fisk, C E, estimated the loss by evap and filtration in 2 weeks of warm weather, to be 

Moai 10 all the water in the canal. Professor Baublue assumes 2 Ins per 
day, for leafcaffe of canal bed, and evaporation, on Eni^llsb 

canals* i. B. Jervls, B, estimated the loss trom evap, filtration, and leakage through lock' 
gates, on the original Erie canal, (40, 28, and 4 ft.) at 100 eub ft per mile per min; or 144000 cub It 
per day. The water surf in a mile Is 211200 sq ft ; therefore, the daily loss would be equal to a dsjpth of 

<Hi tbe Belaware division of tbe Pennsylvania canals, when 

the sapply is temporarily shut off f^m any long reach, tbe water falls from 4 to 8 ins per day. The 
filtration will of course be muoh greater on embankta, than in eota. In some of our canals, the depth 
at high embankta beoomes quite considerable ; the earth, from motives of economy, not being filled in 
level under the bottom of the canal ; but merely left to form its own natural slopes. At one spot at 
least, on tbe Ches and Ohio canal, where one side Is a natural face of vertical rock, this depth is 46 
ft. Sooh depths increase the leakage very greatly ; especially when, as is frequently the case, the em- 
baakta are not paddled; and the practice Is not to be commended, for other reasons also. 

Tbe total averaire loss from reservoli^ of moderate deptbs. 

In ease tbe earthen dams be constmeted with proper oare, and well settled bv time, will not exoeed 
ahont f^om ^ to 1 inch per day ; Imt in new ones, it will usually be oonsiderabiy greater. 

Tbe loss flrom dltcbes, or cbannels of small area, is much 

greater than that from navigable canals ; so that long canal feeders usually deliver but a small pre* 
psrtion of the water which enters them at their heads. 



330 FORGE IN RIGID BODIES. 



MECHANICS. FORCE IN BIQID BODIES. 



In the following pages we endeavor to make clear a few elementary prinoiples 
of Mechanics. The opening articles are devoted chiefly to the subject of matter m 
motion; for, while an acquaintance with this is perhaps not absolutely required in 
obtaining a loorking Itnowledge of those principles of Statics which enter so largely 
into the computations of the civil engineer, yet it must be an Important aid to their 
intelligent appreciation. 

Art. 1 (a). Meolianlcs may b« dellned as that branch of science which 
treats of the effects of force upon matter. 

This broad definition of the word *' Mechanics" includes hvdroetatics, hydraulics, 
pneumatics, etc., if not also electricity, optics, acoustics, and indeed all branches of 
physics ; but we f^hall here confine ourselves chiefly to the consideration of the action 
of extraneous forces upon bodies supposed to be rigid, or incapable of change of shape. 

S) Mechanics is divided into two branches, namely : 
Inematlos $ or the study of the moliona of bodies, without reference to the 
causei of motion ; and 
Dynamlesy or the study of force and its efiiects. 
The latter is sob-divided into 

Kinetics; which treats of the relations between force and motion; and 
Statics t which considers those special, but very numerous, cases, where etpui 
and opporite forces counteract each other and thus destroy each other's motions. 

Art. $8 (a). Matter, or substaitoey may be defined as whatever occupies spao^ 
as metal, stone, wood, water, air, steam, gas, etc. 

(b) A iMKly is any portion of matter which is either more or less completely 
separated in fact from all other matter, or which we take into consideration by itself 
and as if it were so separated. Thus, a stone is a body, whethsr it be falling thronngh 
the air or lying detached upon the ground, or built up into a wall. Alao^ the wall is 
a body ; or, if we wish, we may consider any portion of the wall, as any particulsr 
cubic foot or inch in it, as a body. The earth and the other planets are bodies, and 
their smallest atoms are bodies. 

A train of cars may be regarded as a body; as may also each car, each wheel or 
axle or other part of the car, each passenger, etc., etc 

Similarly, the ocean is a body, or we may take as a body any portion of it at plsss- 
nre, such as a cubic foot, a certain bay, a drop, etc. 

(c) But in what follows we shall (as already stated) consider chiefly rigid bodies: 
i. «., bodies which undergo no change in shape^ such as by being crushed or str^chea 
or pulled apart, or penetrated by another body. AH actual bodies are of course more 
or less subject to some such changes of shape ; t. «., no body i* in fact absolutely 
rigid; but we may properly, for convenience, suppose such bodies to exist, because 
many bodies are so nearly rigid that under ordinary circumstances they undergo 
little or no change of shape, and because such change as does occur may be con- 
sidered under the distinct head of Strength of Materials. 

(d) But while bodies are thns to be regarded as incapable of change at form, it Is 
squally important that we regard them as smeeplihle to change of p^ititm as wholm. 
Thus, they may be upset or turned around horizontally or in any other direction, or 
moved along in any straight or curved line, with or without turning around a point 
within themselves. In short they are capable of moHon, as wholes. 



FORCE IN RIGID BODIES. 331 

A.ictm 3 (a). Motion of a body is change of its poeitton fn relation to another 
body or to some real or imaginary point, which (for conyenieiice) we regard as fixed, 
or at rest. Thns, while a stone &11b from a roof to the ground, its position, relatively 
to the roof, is constantly changing, as is also that relatively to the ground and that 
relatively to any given point in the wall ; and we say that the stone is in motion relor 
tively to either of tkote bodies, or to any point in them. But if two stones, A and B, 
flail from the roof at the same instant and reach the jironnd at the same (subsequent) 
instant, we say that although each moves, relatively to roof and ground, yet they 
have no fi^otum rebxtivdy to each other; or, they are at rest relatively 1o each other; 
for their position in regard to each other does not change ; i. e., in whatever direction 
and at whatever distance stone A may be from stone B at the time of starting, it 
remains in that same direction, and at that same distance from B during the whole 
time of the fall. Similarly, the roof, the wall and the ground are at rest relatively 
to each other, yet they are in motion relatively to a falling stone. They are also in 
motion relatively to the sun, owing to the earth's daily rotation about its axis, and 
iti annual movement around the sun. 

(b) If a train-man walks toward the rear along the top of a freight train Just as 
flwt as the train moves forward, he is in motion relatively to the train; but, as a 
whole, be is at real relatively to ImUdingSf etc. near by ; for a spectator, standing at 
a little distance from the track, sees him continually opposite the same part of such 
building, etc. If the man on the train now stops walking, he comes to rest relatively 
to the irotn, but at the same time comes into motion relatively to the surrounding 
bnHdinffSt etc., for the spectator sees him begin to move along with the train. 

(c) Since we know of no absolutely fixed point in space, we cannot say, of any 
body, what its absoltUe motion is. Consequently, we do not know of such a thing as 
absolute re«^ and are si^e in saying that all bodies are in motion. 

Art* 4 (u). The ▼eloetty of a moving body is its rate of motion. A body (as a 
railroad train) is said to move with uniform -velocttFy or constant velooit^y 
when the distancee moved over in equal times are equal to each other^ no matter how 
tmall those times may be taken. 

(b) The -velocity la cxprcsacd by stating the dittance passed over during some 
giv0n feme, or which tBovid be passed over during that time if the uniform motion 
continued so long Thus, if a railroad train, moving with constant velocity, passes 
over 10 miles in half an hour, we may say that its velocity, during that time, is 
(». «., that it moves at (he rate of) 20 miles per hour, or 105,600 feet per hour, or 1780 
feet per minute, or 2Si^ feet per second. Or, we may, if desirable, say that it moves 
at the rate of 10 miles in half an hour, or 8R feet in three seconds, etc. ; but it is 
generally more convenient to Htate the distance passed over in a unit of time, as in 
one day. one hour, one second, etc. 

(c) I^ of two trains, A and B. moving ^ith constant velocity, 

A moves 10 miles in half an hour, 

B moves 10 miles in quarter of an hour, 

then the veloeitieB are, 

A, SX) miles per hour, 

B, 40 miles per hour. 

In other words, the velocity of a body (which may be defined as the distance passed 
over in a given time) is inversely as the time required to pafis over a given distance. 

(d) By nnlt velocity is meant that velocity whieh, by common consent, is taken 
as equal to unity or one. Where English measures are used, the unit velocity gen- 
erally adopted in the study of Mechanics is 1 foot per second. 

(e) When we say that a body has a velocity of 20 miles per hour, or 10 feet per 
second, etc.. we do not imply that it will necessarily travel 20 miles, or 10 feet, etc. ; 
for it may nc^ have snfBcient time for tbat. We mean merely that it is traveling at 
the rate of 20 miles per hour, or 10 feet per second, etc. ; so that if it coniimied to move 
at that same rate for an hour, or a second, etc., it would travel 20 miles, or 10 feet. etc. 

(t) When velocity inereaget. it is said to be accelerated. When it decreases. 
It is said to be retarded. If the acceleration or retardation is in exact proportion 
to the time ; that is, when during any and every equal interval of time, the same degree 
of change takes place, it is uniformly accelerated, or retarded. When otherwise, the 
words vcuriahle and variaMy are used. 

(s) A body may have, at the same time, tivro qr more Independent veloel- 
requlring to be considered. For instance, a ball fired vertically upward from a 



J 



332 FOBOE IK RIGID BODIES. 

Sn, and then falling again to the earth, has, daring the whole time of its rise and 
1, (iBt) the tmiform vptoard Telocity with which it leaves the muzzle, and (2nd) the 
continually acceUrated dovmward Telocity given to it by gravity, which acts upon it 
daring the whole time. Its remUant (or apparent) velocity at any moment is the 
d^ertnoe between these two. 

Thus, immediately after learlng the gun, the downward velocity given by 
gravity is very small, and the resultant velocity is therrfore npwanl and Teiy 
nearly equal to the whole upward velocity due to the powder. But after awhila 
the downward velocity (by constantly increasing) beoomes equal to the upward 
velocity ; i. «., their difference, or the resultant velocity, becomes nothing ; the ball 
at that instant stands still ; but its downward velocity continues to increase, and 
immediately becomes a little greater than the upward velocity ; then greater and 
greater, until the ball strikes the ground. At that instant its resultant velocity is 

rthe downward Telocity which it would ) , ( the uniform upward 
*» •€ have acquired by falling dwring the V — < velocity given by the 
(, vahoU tivM of its rite and faU. ) ( powder. 

We have here neglected the resistance of the air, which of course retards botb 
flie ascent and the descent of the ball. 

(li) As a further illustration, regard a b n c as a raft drifting in the direction 
ca ox nh. A man on the rait walks with uniform velocity from comer n ta 
corner c while the raft drifts (with a uniform velocity a 
little greater than that of the man) through the distance n b. /^a\ 

Therefore, when the man reaches corner c, that comer has v'H^vVs^ 

moved to the point which, when he started, was occupied by xTff-"-^^ 

a. The man's resultant motion, relatively to the bed of the / ; / 

river or to a point on shore, has therefore been » a. His / j. / 

motion at right angles to n a, due to his walking, is t c, but ^<" — -fi / 
that due to the drifting of the raft is o 6. These two are ***--.i'''* 

equal and opposite. Hence his resultant motion <U right il 

angles to n a is nothing ; he does not move from the line n a. 
His walking moves him through a distance equal to n i, in the direction n a; 
and the drifting through a distance equal to t a, and the sum of these two is n a. 

(i) All the motions which we see given to bodies are but €hang«a in their unknown 
absolute motions. For convenience, we may conflne our attention to some one or 
more of these changes, neglecting others. 

Thus, in the case of the ball fired upward from a gun (see. (9) above) we may 
neglect Its uniform upward motion and consider only its constantly accelerated 
downward motion under the action of gravity ; or, as is more usual, we may oonaldar 
only the retuUard or appatrmi motion, which is first upward and then downvrard. In 
both cases we neglect the motions of the ball caused by the several motions of 
the earth in spaed. 

Art. 5 (•)• Forcoy <be «Miiu« of change of motion. Suppose • 
perfectly smooth ball resting upon a perfectly hard, frictionless and level surfiMS^ 
and suppose the resistance of the air to be removed. In erder to merely move the 
ball horizontally (i. e., to set it in motion — ^to change its state of motion) some /orc« 
must act upon it. Or, if such a ball were already in motion, we could not retard 
or hasten it, or turn it from its path without exerting force upon it. For, as stated 
in Neiirton's flrat \wk-v¥ of motion, ''•-rerjr body continues In its 
•tstte of rest or of motion in a straight line, except in so far as it may be com- 
pelled by impressed forces to change that state." On the other hand, if a force act* 
upon a body, the motion of the body must undergo change. 

(b) Force Is an action betifreen t-wo bodies, fending eitber i» 
separate them or to bring them closer togeU&er. For Instance, when 
a stone falls to the ground, we explain the Csct by saying that a force (the attnction 
of gravitation) tends to draw the earth and the stone together. 

Magnetic and electric attraction, and the cohesive force between the particles of a 
body, are other instances of ottmcttee force. 

(c) Force applied by contsMst. In practice we apply force to a body (B) 
by causing contact between it and another body (A) which has a tendency to motwti 
toward B. A repulsive force is tljus called into action between the two bodies (io 
■omo way which we cannot understand), and this force pushes B forward (or in the 



FOBCE IN RIGID BODIEBL 333 

direction of A's tendency to move) and pushes A backward, thus diminishii^ its for- 
ward tendency * 

If, for instance, a stone be laid npon the ground, it tends to moTe downward, bat 
does not do so, because a repulsive force pushes it and the earth apart Just as hard as 
the force of gravity tends to draw them together. 

Similarly, when we attempt to lift a moderate weight with our hand, we do so by 
giving the hand a tendency to move upward. If the hand slips from the weight 
this tendency moves the hand rapidly upward before our will force can dieck it. 
But otherwise, the repulsive force, generated by contact between the hand (tending 
upward) and the weight, moves the latter upward in spite of the force of gravity, 
and pushes the hand downward, depriving it of much of the upward velocity which 
it would otherwise have. It is perhaps chiefly fh>m the eftortf of Vhich we are 
conscious in such cases, that we derive our notions of <Yorce." 

When a moving billiard ball. A, strikes another one, B, at rest, the tendency 
of A to continue moving forward is resisted by a repulsive force acting between it 
and B. This force pushes B forward, and A backward, retarding its former velocity. 
As explained in Art. 23 (a), ' the repulsive force does not exist in either body 

ontil the two meet 

(d) The repulsive force thus generated by contact between two bodies, continues to 
act only so long as they remain in contact, and only so long as they tend (from * 
■ome extraneous cause) to come closer together. But it is genenJly or always 
accompanied by an additional repulsive force, due to the compreuion of the particles 
of the bodies and their tendency to return to their original positions. This eUutic 
repulsive force may continue to act after the tendency to compression has ceased. 

(e) Force acts either sui a P^I or sui a puatai. Thus, when a weight 
Is susiMnded by a hook at the end of a rope, gravity jmU« the weight downward, the 
weifrht ptuhn the hook, and the hook puUi the rope, each of these actions being 
accompanied, of course, by its corresponding and opposite "reaction.** When two 
bodies collide, each pushps the other, generally for a very short time. 

(1) EjqiiaUtjr of actloni aad reaction. A force always exerts itself equally 
upon the two bodies between which it acts. Thus, the force (or attraction) of 
gravitation, acting between the earth and a stone, draws the earth upward just as 
hard as it draws the stone downward ; and the repulsive force, acting between a 
table and a stone resting upon it, pushes the table and the earth downward just as 
bard as it pushes the stone upward.' This is the fact expressed by Ne'vrton's 
tl&lrd lainr ot motloiiy that **to every action there is always an equal and 
contrary reaction.*' For measures of force, see Arts. 11, 12, 13. 

If a cannonbidl in its flight cuts a leaf from a tree, we say that the leaf has reacted 
against the batt with precisely the same force with which the ball acted against the 
leaf. That degree of force was sufficient to cut off a leaf, but not to arrest the ball. 
A ship of war, in running against a canoe, or the fist of a pugilist strikint; his 
opponent in the foce, receives as violent a blow as it gives ; but the same blow that 
will upset or sink a canoe, will not opprecto&Iy affect the motion of a ship, and the 
blow which may seriously damage a nose, mouth, or eyes, may have no such effect 
upon hard knuckles. 

The resistance which an abutment opposes to the pressure of an arch ; or a retain- 
ing^wall to the pressure of the earth behind it, is no greater than those pressures 
themselves ; but the abutment and the wall are, for the sake of safety, made capable 
of sustaining much greater pressures, in case accidental circumstances should pro- 
duce such. 

(§p) In most practical cases 'we liaT-e to consider only one of the two bodies 
between which a force acts. Hence, for convenience, we commonly speak as if the 
force were divided into two equal and opposite forces, one for each of the two bodies, 
and confine our attention to one of the bodies and the force acting upon it, neglect- 
ing the other. Thus we may speak of the force of steam in an engine as acting 
upon the pitton, and neglect its equal and opposite pressure against the head of 
the founder. 

(h) That point of a body to which, theoretically, a force is applied, is called the 
pplnt ot application. In practice we cannot apply force to a point, according 
to the seientlflo meaning of that word ; but have to apply it distributed over an ap- 
preciable area (sometimes very large) of the surface of the body. 

* We ordinarily express all this by saying simply that A pushes B forward, and this 
is sufficiently exact for practical purpoees ; but it is well to recognize that it iH merely 
a convenient expression and does not fully state the facts, and that every force neees- 
aarUff consists of two equal^and opposite pulls or pushes exerted between two bodiai. 



334 FORCE IN RIGID BODIBS. 

For the present we shall aasume that the line of action of the force passes 
through toe center of gravitf of the body and forms a right angle with the sur- 
face at the point of application. 

Art* 7 (a). Acoeleratlon. When an unresisted force, acting upon a body, 
sets it in motion (i. «., gi^es it Telocity) in the direction of the force, this velocity 
increases as the force continues to act; each equal interval of time (if the force 
remains constant) bringing its own equal increase of velocity. 

Thus, if a stone bu let full, the furce of gravity gives to it, in the first in- 
conceivably short interval of time, a small velocity downward. In the next equal 
interval of time, it adds a second equal velocity, and so on, so that at the end of 
the second interval the velocity of the stone is twice as great, at the end of the 
third interval three times as great, as at the end of the first one, and so on. We 
may divide the time into as small equal intervals as we please. In each such 
interval the constant* force of gravity gives to the stone an equal increase of 
velocity. 

Such increase of velocity is called accelerstion-f When a body is thrown verticaUy 
uptoardy the downward acceleration of gravity appears as a retardation of the upward 
motion. When a force thus acu offaitut the motion under consideration, its acceleivp 
tion is called negatim. 



Art* 8 (a). Tbe rate of aujoeleratloiii is the acceleration which takes 
place in a given Hmsj as one second. 

rb) The unit rate of acceleration is that which adds unit of velocity in a 
unit of time ; or, where Bnglish measures are used, one foot per second, per geeond, 

(o) For a given rate of acceleratioo, the total accelerations are of course propor- 
tional to the HmsM during which the velocity increases at that r&te. 

Art. (tt), Iia^rs of acceleration* Suppose two blocks of iron, one fwhich 
we will call A^ twice as large as the other (a), placed each upon a perfectly fricnonless 
and horizontal plane, so that in moving them horizontally we are opposed by no force 
tending to hold them still. Now apply to each block, 

through a spring balance, a pull such as will keep the pointer of each balance always 
at the same mark, as, for instance, constantly at 2 in both balances. We thus have 
equal forces acting upon unequ^ masses.^ Here the rate of acceleration of a Is 
double that of A ; for nrlien the forces are equal tbe rates of auseelera- 
ration are Inversely as tike masses* 

In other words, in one second (or in any other given time) the small block of iron, 
a, will acquire twice the increase of velocity that A (twice as lai^e) vdll acquire ; so 
that if both blocks start at the same time from a state of rest, the smaller one, a, will 
have, at the end of any given time, twice the velooitff of A, which has twice its mass. 

(b) Again, let the two masses, A and a, be equal, but let the foree exerted upon a 
be twice that exerted upon A. Then the rate of acceleration of a will (as before) be 
twice that of A ; for, 'vrl&en tl&e masses are cqnatly tbe rates of aoeelera* 
ration are alrectljr as the forces* 

(e) We thus arrive at the principle that, in any case, the rate of acceleration 
Is dlreotljr proportlonsil to the force and Inwerselw proportional 
to the n&ass* 



* We here speak of the force of gravity, exerted in a given place, as constant, 
because it is so for all practical purposes. Strictly speaking, it increases a very little 
as the stone approaches the earth. 

t Since the rtUe of acceleration is generally of frreater conseq-qence. in Meohanies, 
than the total acceleration, or the "acceleration" proper, srienttfic writers (for the 
sake of brevity) use the term "acceleration" to denote that rate, and the term 
"total acceleration" to denote the total increase or decrease of velocity occnrrinK 
during any given time. Thus, the rate of acceleration of gravity (about 32.2 ft. per 
second per second) is called, simply, the "acceleration of gravity.'* As we shall not 
have to use either expression very frequently, we shall, generally, to avoid misappre' 
hension, give to each idea its full name ; thus, <* total acceleration " for the whoU 
change of velocUy in a given case, and " rate of acceleration " for the rate of that 
change. 

t The mass of a body Is the quantity of matter that it contains. 



FOBCB IN BIOID BODIES. 335 

* 

(d) Htticat if we make the two forces propmrtloiial to the two maases, tbe rases 
of aoceleratioQ will be equal ; or, t»r m fpiwea vmtm of acosleimtloii^ tbe 
forces most be dlrectljr as the masses. 

(e) Hence, also, a greater force is required to Impart a g^Ten Teloci^ to a girea 
body in a short time than to impart the same Telocity in a longer time. For instance, 
the forward coupling links of a long train of cam wonld snap instantly nnder a pull 
safBoIent to give to the train in two seconds a Telocity of twenty miles per hour, sup- 

Ciing a suflBoiently powerftil looomotiTe to exist In many such cases, therefore) we 
Te to be contented with a slow, instead of a rapid acceleration. 
A string may safely sustain a^ weight of one pound suspended from our hand. If 
we wish to impart a great upward Telocity to the weight in a very sJiort timey we eTi- 
dently can do so only by exerting upon it a great force; in other words, by Jerking 
the sMng Tiolently upward. But if the string has not tensile strength sufficient to 
transmit this force from our hand to the weight, it will break. We might safely 
giTe to the weight the desired Telocity by applying a le$» Jbre^ during a longer time. 

{t) When a stone falls, the fi>rce pulling the earth upward is (as remarked aboye) 
equal to that which pulls the stone downward, but the tncun of tne earth is so Tastly 
greater than that of the stone that its motion is totally imperctptible to us, and 
would still be so. eTen if it were not counteracted by motions in other directions 
in other parts of the earth. Hence we are pracHcaUy^ though not abtolutelif, right 
when we say that the earth remains at rest while the stone fiJls. 

(§;) Bat in the case of the two billiard balls (Art 5e. p. 388), we can dearly see 

the result of die action of the force upon each of the two bodies; for tbe second 

ball, B» which was at rest, now moTes forward, while the forward Telocity of tbs 

lint OB», A, is dimiidshed or destroyed, its backward mention thus appearing as a 

. ntenlaMBa of ita forward motion. And, (since the same force acts upon both balls) 

mass . mass . rate of acceleration . rate of negatlTe acceleration 
ofA'ofB'* ofB ofA 

or (siaee the ibrce acts Ibr tbe saate time upon both balls) 

miuMy mass forward Telodty . loss of forward Telocity 
oTA * ofB *' OfB OfA 

' Ok) "RgwAng. A man oaamot 1^ a weight of 20 tons; but if it be placed upon 
prv^r friction rollers^ he can move it horisontally, as we sea in some drawbridges, 
tumtablea, Ac. ; and if friction and the resistance of the air could be entirely remoTed, 
he could BOTC it by a ringle breath ; and it would continue to uoto forerer after the 
foiee of the brecrth had ceased to act upon it. It would, howsTer, moTe Tory slowly, 
because the force of tbe single breath would hsTe to diffuse itself among 20 tons of 
matter. He can more it, if it be placed in a suitable Teasel in water, or if snqiended 
from a long rope. A powerful locomotlTe that may moTe 2000 tons, cannot lift 10 tons 

Terticaltar. 

If we imagine two bodies, each as large and heaTy as the earth, to be precisely 
balanced in a pair of scales without friction, a single grain of sand added to either 
icale'paa, would giTe motion to both bodies. ' 

Art. lO (a). The constant force of gravity is a uniformly accelerating force 
when it acts upon a body falling freely ; for it then Increases the Telocity at uie uni- 
form rate of .322 of a foot per second during every hundredth part of a second, or 32,2 
feet per second in eTery second. Also when it acts upon a body moving down an in- 
clined plane; although in this case the increase is not so rapid, becatise it is caused 
l^ only a part of the graTity, while another pert preeses the body to the plane, and a 
third part OTercomes the friction. It is a uniformly retarding force, upon a body 
thrown Tertically upward; for no matter what may be the Telocity of the body 
when projected upward, it will be diminished .322 of a foot per second in each 
hundredth part of a second during its rise, or 82.2 feet per second during each 
entire second. At least, such would be the case were it not for the varying resistance 
of the air at difforent Telocities. It is a uniformly straining force when it causes a 
body at rest, to press ux)on another body ; or to pull upon a strinfi; by which it is 
suspended. The foregoing expressions, like those of momentum, strain, push, pull, 
lift, work, &c., do not indicate different hinde of force ; but merely different kinds of 
^eets producM by the one grand principle, force. 

(b) The aboTe 82.2 feet per second is called the aeceleratton otgrm.'vltr f and 
by scientiile writers is conTcntlonally denoted by a small g % or, more correctly qieak- 



336 FOBOB IN BIGTD BODIEa. 

tag, ifnce the aoc«l«ratloii li not precisely the eame at ftll parti of Che Mrtb, g 
denoteethe aooeleratloii ptf aeoond, whateTer it may be, at Any particular idaoe. 

. Art. 11 (a). Ralatton b«tw«eit force and nuuM* The mass of a body 
is the quantity of matter which It contains. Ons cubic foot of water has ttotei 
AS great a mass as /la^ a cubic foot of water, but a lesi mass than one cubic 
foot of iron. Thus, the n'Mof a body is a measure of mass between bodies 
of the tame material, but not between bodies of different materials. 

(b) When bodies are allowed to fall freely in a racuum at a given place, 
4hey are found to acquire equal velocities in any eiven time, of whatever 
different materials they may be Qomposed. From ibis we know (Art. 9 (dV, 
p. 335). that the forcee moving them downward, viz. : their respective tMighu 
at that place, must be proportional to their maeaee. 

Thus, in any given placet the weight of a bodv is a perfect measure of its mam. 
But the weight of a given bodv changes when the body is moved from one level 
above the sea to another, or from one latitude to another; while the mass of 
the body of course remains t/ie same in all places. ThuSga piece of iron which 
weighs a pound at the level of the sea, will weigh leee than a pound by a spring 
balance, upon the top of a mountain close by. because the attraction between 
the earth and a eiven mass diminishes when tne latter recedes from the earth's 
center. Or if tne piece of iron weighs one pound near the North or South 
Pole, it will, for the same reason, weigh leet toan a pound by a spring balaliioe 
if weighed nearer to the equator and at the same level above the sea. 

The difference in the weight of a body in different localities is so slight as 
io be of no account in questions of ordinary practical Mechanics ;• bat 
scientific exactness requires a measure of mass which will give the same 
expression for the quantity of matter in a given body, wherever it may 
be; and, since weighing Is a verv convenient way of arriving at the quantity 
of matter in a body, it is desirable that we should still be able to express tiie 
mass in terms of the weight. Now, when a given body is carried to a hieher 
level, or to a lower latitude, its loss of weight is simply a decrease in the Jores 
with which gravity draws it downward, and this same decrease also causes 
a decrease of the velocitu which the body acquires in falling during any 
given time. The change m velocity, by Art. (6), p. 884, is necessarily propor* - 
nonal to the change in weight 

Therefore, if the weight of a body at any place be divided by the velocity 
which gravity imparts in one second at the same place (and called sr^ or the 
aeceUrcttion of gravity for that place), the quotient will be tne same at aU plaoei^ 
and therefore serves as an invariable mei^ure of the mass. 

(c) By common consent, the m&it ot mass, in scientific Mechanics, is said 
io be that quantity of matter to which a unit of force can give unit rate of 
acceleration. This unit rate, in countries where English measures are used, 
is one foot per second, per second. It remains then to adjust the units offeree 
and of maee. Two methods (an old and a new one) are in use for doing this. 
We shall refer to them here as methods A and B respectively. 

fd) In metl&od A, still generally used in questions of etatics^ the untt 
ox n»roe is fixed as that force which is equal to the weiaht of one pound in a 
certain place; i.e.. the force with which the earth at that place attracts a 
certain standard piece of platinum called a pound: and the unit of maee is 
not this standard piece of metal, but, as stated in (c)^ that mass to which this 
unit force of one pound gives, in one second, a velocity of one foot per second. 
Now the one pound attraction of the earth upon a mass of one pound will 
(Art. 1, p. 330) in one second give to that mass a velocity — (/ or about 32 feet 
per second; and (Art 9 (a), p. 834), for a given force the masses are inversely as 
the velocities imparted in a given time. Therefore, to give in one second a 
velocity of only one foot per second (instead of g or about 32) the one pound 
unit of force would have to act upon a mass g times (or about 82 times) that 
which weighs one pound. 

This could be accomplished, with an Attwood*s machine. Art 16 (e), p. 889, 
by making the two equsU weights each «- 15 ^ lbs. and the third weight *■ 1 Ibw 

*The greatest discrepancy that can occur at various heights and latitadeS| 
by adopting weight as the measure of quantity, would not oe likely to s x ess a 
1 in 300; or. under ordinary circumstances, 1 in 1000. 






FOBOE IN BiaiD BODIES. 337 

By method A, therefore, the unit of masn is g times (or about 33 times) the 
mass of the standard piece of metal called a pound; i. e., a body containing 
one such unit of mass wei^s g lbs. or about 32 lbs.; or, tijr method A, 

the weight of any given body ^ ^ y the mass of the body, 
in lt«. — Sf A jjj oaitg Qf mass. 

file moss of a body, in units of mass - l^.^ ^^^g^^ ^^ ^^^ ^Q^y> ^^ POP°<^ 

g 
For instance: 

in a body weighing the mass is about 

y^ pound ^ unit of mass 

1 •* 

2 « 

82 " i 

64 « 2 *• •• 

It has been suggested to call this unit of mass a *' Matt/* 

(•) In naetlMPd By the moM of the standard pound piece of platinum is taken 
as toe unit of nuum and is called a pound} and the force which will give 
to it in one second a velocity of one foot per second is taken as the unit of force. 
This small unit of force is called a ponndal* In order that it may in one 
second give to the mass of one pouna a velocity of only one foot per second, it 

must (by Art f b), be -i. f or about Jjj of the weight of said pound mass. 

Hence, "by m«tl&od By 

the ma8$ of any given body, in jxrunds - the weight of the body in poundaU 

and 

the weiifht of a body, in potmdale — gr X the maea of the body in pound§. 

Forinstar'jce: 

in a body weighing the mass of the body is about 

^ pouitdal — > JL pound JL pound 

82 " — 1 " 1 •• 

64 " — 2 « 2 " 

tty VoT coi&'renlenoey we sometimes disregard the scientific require- 
ment that the unit of force must be that which will give unit rate of accele- 
ration to nnit mass, and take a pound of matter as our unit of maes^ and a 
pound weight as our unit of force. Our unit of force will then in one second 
give ft velocity of g (or about 32.2 feet per second) to our unit of mass. In 
Sicties^ we are not concerned with the masses of bodies, but only with the 
fijrees acting upon them, including their weights. 

Art* 13 (a). Impnlse. By taking, as the unit of force, that force which, in 
one second, will give to unit mass a velocity of one foot per second, we have 
(by Art. 9, p. 334), in any case of unbalanced /orc« acting upon a mass during a 
iSvon timei 

Velocity - force X time ^^ 

mass 

Force - volocity X mass ^gj 

time 
Mass - . force X time ^3^ 

velocity 
Time - niass Xvelocity ^^ 

force 

Force X time — mass X velocity. . . . (5> * 

25 



338 FORCE IN RIQID BODIES. 

Tb the prodnoft^ force X time, in equation (5), writers now give the name 
Impolee^ which was formerly given to eoUUion (now called liiip««t}* See 
Art 24 (a). The term impuUe, as now used^ conveys merely the idea 

of force acting through a certain length of time. Equation (5) tells us that an 
impulse (the product of a force by the time of its action) is numerically equal 
to the momentum* which it produces. Eqilation (2) tells us that any force is 
numerically equal to the momentum which it can produce in one second. In 
other words, the monftentmn of a body moving with a given veloci^ is 
numerically equal to the force which in one second can produce or destroy 
that velocity in that body; or, a force is numerically equal to the rate pw 
second at which it can produce momentum. Thus, forces are proportional to 
the momentums which they can produce in a given time; or, in a given time^ 
equal forces produce equal momentums. Therefore a force must always give 
equal and opposite momentums to the two bodies between which it acts. 

Art* 13 (»)• Tlk* luiiial -wajr of meaanrln^ a fbrce is by ascertaining 
the amount of some other force which it can counteract. Thus we may meas- 
ure the weight of a body by hanging it to a spring balance. The scale of the 
balance then indicates the amount of tension m the spring: and we know thai 
the weight of the body is equal to the tension, because the weight just pre 
vents the tension firom drawing the hook upward. 

Thus, fbremm are conveniently expressed In -vrelfpltt|i9 as in pounds, 
tons, &c., and they are generally so measured in Statics, and in our following 
articles. 

(b) A fbroe mav' be aonstant or Tarlable* When a stone rests upon 
the ground, the pull of gravity upon it (i. e., its weight) remains constanlh 
neither increasing nor decreasing. But when a stone is thrown upward its 
weight decreases very slightly as it recedes from the earth, and again increases 
as it approaches it during its fall. In this case, the force of gravity, acting 
upon the stone, decreases or increases eteadUy, But a force may change 
euddenlyf or irregtUariyf or may be intermittent $ as when a series of uneqiul 
blows are struck by a nammer. In what follows we shall have to do only with 
forces supposed to be conatarU, 

Art. 14b («)• "DonuUjr* The deneitiea of materials are proportional to the 
mauee contained in a given volume, as a cubio inch j or inoerselff as the volume 
required to contain a given mass. Or^ since the weights at a given place are 
proportional to the masses, the densities are proportional to the weights per 
unit of volume (or ** specific gravities **) of the materials. Thus, a body weigh* 
inff 100 lbs. per cubio foot is twice as dense as one weighing only 60 Iba. per 
cuoic foot ait the same place. 

Art. 15 (a). Inertia. The inability of matter to set itself in motion^ or ta 
change the rato or direction of its motion, is called its inertia, or inertneeaL 
\Blien we say that a certain body has twice the inertia (inertness) of a smaller 
one, we mean that twice the /or<:0 is required to give it an equal ratoof acoete* 
ration ; and that, since all force (Art. 5f)t acts equally in both direo* 

tions, we experience twice as great a reaction (or so-called ** resistance*^ from 
the larger body as from the smaller one. The ** inertia** of a body is therefora 
a measure of the/ore« required to produce in it a given rato of acceleration; Ob 
which is the same thing, it is a measure of the mass of the body. We mi^ 
therefore consider ** inertia'* and **mass** as identical. 



(b) What is called the ** resistance of inertia** of a body, ia simply 
reaction, (i s., one of . the two equal and opposite actions) of whatever 
force we apply to the body. Hence, its amount depends not only upon tiia 
mass of the Dody, but also upon the rato of acceleration which we choose to 

*The momentam of a body (sometimes called its ** quantity of motion") 
is equal to the product obtained by multiplying its ma»9 by its velocity. If «• 
adopt the pound as the unit of mass, as in '* method B/* Art. 11 («), tha 

proauct, voeight in pownds X velocity, is numerically either exactly or neurly 
the same as the product, m(M8 in pounds X velocity, depending upon whether 
or not the body is in that latitude and at that level where a ma8$ of one poufltdl 
is said to weigh one pound. But the product, weight in poundats X velocity; It 
exactly a times (about 82.2 times) the product, mass in pounds X velocity; afeo^ 
k^ ** menod A,** iMi^M in |K>uncte X velocity — y X «MM in ** matte '* X 



FORCE IN RIGID BODIES. 339 

giTe to it. Therefore we cannot tell, from the mass or weight of a bodj alone, 
what its " reeistance of inertia " in any given case will be. 

Art. 16 (a). Forees In opposite directions. When two equal and 
opposite forces act upon a body at the same time, and in the same straight line, 
we say that they destroy each other's tendencies to more the body, and it remains 
at rest. If two unequal forces thus act in opposition, the smaller force and an 
equal portion of the greater one are said to counteract each other in the same 
way, but the remainder of the greater force, acting as an unbalanced or unresisted 
force, moves the body in its own direction, as it would do if it were the only 
force acting upon it. 

Thus, when we move bodies, in practice, we encounter not only the " resist- 
ance of inertia" (i. e., we not only have to exert force in order to move inert 
matter), but we are also opposed by other /otom, acting against us, as friction, 
the resistance of the air. and, often, all or a part of the wHght of the body. By 
'' resistances," in the following, we mean such resisting /oroM, and do not include 
in the term the " resistance of xHertia," 

(b) If separated, the two bodies, A and B, of 8 &m and 2 lbs respectively, would 

fidl with equal accelerations = g ; each unit, — , of mass being acted upon by its 

own weight, W, Bat. connected as they are, A will 

move downward, and B upward, with an acceler- T^2A 

ation =» only f ; for now an unbalanced force of 

5 

only 8 — 2 = lfb must give acceleration to a mass 



(J'*^0 



T*2j4 



ation =» only | ; for now an unbalanced force of 

5 

,8 + 2 6 « " 2 T-2.4 

of = -. But, to give to a mass, B, of -, an 

aoeelof |, requires a force of -. I »^]b=»a4 lb. 3|liji EjI^ 

This, plus 2 lbs (required to balance the weight of Al 

B) is the tension, 2.4 lbs existing throughout the 

cord. Exerted at A, this tension balances 2.4 of 

the 3 lbs weight of A. The remainder (8 — 2.4 = 0.6 !b) of the weight, acting 

downward upon the mass, -, of A, gives to it the required acceleration of ^; 

, - force .- 8 0.6 g .. g 

for here = 0.6 -«- - = -r-2 = 0.2 g = f . 

mass g 8 * 6 ^ 

Or we may regard the total tension, 2.4 lbs, in the cord at A, as acting upon A 

O 

and giving to it a negative or upward acceleration of 2.4 -t- - = 0.8 g, which, 

g 

dedacted from g (the acceleration which A would otherwise have) leaves 

Acceleration = g — 0.8 g = 0.2 g = |. 

Let W = weight of A 
w ss weight of B 
F a> net force available for acceleration » W — w 

mr -4- w 
M =3 combined mass of both bodies = — — — 

g 

m => mass of B » - 

g 
a => acceleration 
T a tension in cord. 

Then: a = ^ = (W - w) ^ :5^^ = «-^£i:^ 
M ^ g W + w 

m . . ^ , ^ g(W — W) / W — W\ 

T = w + ma = w +— a==wH ^-vi?— ; = w 1 1 + ^5,— — |. 

g 'gW + w \Wh-w/ 

(e) An ** Ati¥00«l*s Machine'* consists essentially of a pulley, a flexible 
cord passing over the pulley, two equal weights (one suspended at each end of 
the cord), and a third weigfnt, generally much lighter than either of the other 
two. The two equal weights balance each other by means of the pulley and 
cord. The third weight is laid upon one of the other two weights. The force 
of gravity, acting upon the third weight, then sets the masses of the three 
weights in motion at a small but constantly increasing velocity. In order to do 
this it mast also overcome the friction of the pulley and cord, and the rigidity 



340 FORCE IN RIGID BODIES. 

of the latter ; bat, as these are made as slight as possible, they are, fbr ooo- 
venience, neglected. The machine is used for Illustrating the acceleration given 
to inert matter by unbalanced force, and forma an excel^nt example of the two 
distinct duties which a moving force generally has to perform, vis: (1st) the 
balancing of resistance, and (2nd) acceleration. 

(d) In the case of a lo«oiiiotlire. drawfngr a train on a leTel, fHc- 
tioa and the resistance of the air are the only resistances to be balanced ; for Uie 
weight of the train here opposes no resistance. Unless the force of the steam is 
more than sufficient to balance the resistances, it cannot mote the train. If it 
exceeds the resistances, the excess, however slight, gives motion to the inert 
matter of the train. If, at any moment while the train is moving, the force of 
the steam becomes jtut equal to the resUtcmces (whether by an increase of the 
latter or by diminishing the force) the train will move on at a uniform velocity 
equal to that which it had at the moment when the force and resistance were 
equalized ; and, if these could always be kept equal, it would so move on forever. 

But so lone as the excess of steam pressure over the resistances continues to act, 
the velocity Is increased at each instant ; for during eaeh such instant liie excess 
of force gives a small velocity in addition to that already existing. 

On a level railroad, let 
P »- the total tractive force of the locomotive = say 13 tons <• 
W sa weight of locomotive = 50 tons 
w sa weight of train = 336 tons 

R *= resistance of locomotive (including internal fHction, etc.) «> 8 tona 
r a resistance of train =■ 1 ton 
F » net force available for acceleration — P — R — r-s9 toms 

M « mass of engine and train =■ — — ,. _ -* 12 

* g 82.2 

- ^ , w 8S6 ^^ , . 
m «■ mass of train = - — —- = 10.44 

g 32.2 

a = acceleration 

T = tension on draw-bar. 

F 9 
Then : Acceleration at a » ^ — r^ = 0.75 ft per second per second. 

The tension T on the draw-bar « resistance of train + force causing accel- 
eration a, orT=r + ma — 1 + 10.44 X 0.76 = 1 + 7.83 = 8.83 tons. 
This tension, T, pulling backward against the locomotive, causes there a 

T a aa «r 

retardatim, or negative acceleration, of masa of tocomoUve = -go- = »» « 

per sec per sec, and thus reduces, by that amount, the acceleration which the 

(P ■■'' r) s 10 X 8S.8 
locomotive would otherwise have, and which would be — ^ — ka ■■ ,. — 

oO 60 

~ 6.44. This, less 6.69, » 0.75 ft per sec per sec — acceleration of train. 

(e) If the tractive force of a locomotive exceeds the resistances, due to friction, 
grades, and air, the velocity will be accelerated ; but it then heoomeB more dilB- 
cult to maintain the excess of force, for the pistons must travel fast«r through 
the cylinders, and the boiler can no longer supply steam fast enough to maintain 
the original cylinder pressure Besides, some of the resistances increase with 
increase of velocity. We thus reach a speed at which the engine, alUiough 
exerting its utmost force, can do no more than balance the resistances. T^e 
train then moves with a uniform velocity equal to that which it had when thia 
condition was reached. 

When it becomes necessary to stop at a station some distance ahead, steam It 
shut off, so that the steam force of the engine shall no longer counterbalance or 
destroy the resisting forces; and the number of the resistances themselves is in- 
creased by adding to them the friction of the brakes. The reBistanoea, thus 
incneased, are now the only forces acting npon the train, and their acoeleration 
is negative, or a retardation. Hence, the train moves more and more slowly, and 
must eventually stop. 

(f) Caution. When two opposite forces are in equilibrium, an addition to 
one of the forces does not always form an unbalanced force ; for in many cases 
the other force increases eguallyy up to a certain point. For Instance, when we 
attempt to lift a weight, W, its downward resistance^ R, remains constantly Just 
equal to our upward pull, P, however P may vary, until P exceeds W. Thas, R 
can never exceed W, but may be much less than It. Indeed, when we atop pull- 
ing, R ceases, although W (the attraction between the eartn and the weight) of 



FORCE IN RIGID BODIES. 841 

eoarse remnins unchanged throaghout. Such Tariation of resisting force, to meet 
varying demands, occurs in all those innumerable cases where structures sustain 
varying loads within their ultimate strength. 

Art. 17 (a). Work. Force, when it moves a body,* is said to do " work " 
upon it. The whole work done by the force in moving the body through any dis- 
tance is measured by multiplying the force by thedutance; or: Work = Force 
X distance. If the force is taken in pounds, and the distance in feet, the product 
([or the work done) will be in foot-pounde ; if the force is in tons and the distance 
in inches, the product will be in inch-tons ; and so on.f 

Thus, if a force of moves a body through we have work = 

1 pound 10,000 feet 10,000 foot-pounds 

100 pounds 100 '* 10,000 " 

10,000 " 1 foot 10,000 " 

or, in any case, if the fiprce be F pounds, the whole work done by it in moving a 
body through s feet, is F « foot-pounds. 

(I») The foot-pound, the foot-ton, the inch-pound, the inch-ton, etc., etc., are 
called unlto oi wwrfc.f 

For practical purposes, in this country, forces are most frequently stated in 
pounds, and the distances (through which they act) in feet. Hence tbe ordi- 
nary anii of work, is the foot-pound. The metric nnit of work 
is the klloflrram-meter, i e. l Kilogram raised 1 meter = 2.2046 pounds 
raiaed 3.2800 feet, = 7.23S1 foot-pounds. 1 foot-pound = 0.13825 kilogram-meter. 

(«) In most cases, a portion at least of the work done by a force is ex- 
pended in owereomlnv reflistiunees. Thus, when a locomotive begins 
to move a train, a portion of its force works against, and balances, the resist- 
anoM of friction or of an up-grade, while the remainder, acting as unbalanced 
toroe upon the inert mass of the train, increases its velocity. 

An upward pull of exactly one pound will not raise a one pound weight, but 
will merely biuanoe the downward force of gravity. If we increase the upward 
pail from one pound (=» 16 ounces) to 17 ounces, the ounce so added, being 
unbalanced foroe, will give motion to the mass, and will acceleirate its upward 
velocity as long as it continues to act. If we now reduce the upward pull to 1 
pound, thus miking it just equal to the downward pull of gravity, the body will 
move on upward with a uniform velocity : but if we reduce the upward force to 

15 ounces (= || pound), then there will be anjunbalanced dovmward force of 1 
ounce acting upon the body, and this downward force will generate in the body 
a downward or negative acceleration or retardation, and will destroy the upward 
velocity in the same time aa the upward excess of 1 ounce required to produce it. 

Daring any time, while the 17 ounces upward ** force" were acting against the 

16 ounces downward " resistance," the product of total upward force X distance 
mast be gre<Uer than that of resistance X distance. The excess is the work done 
in accelerating the velocity, by virtue of which the body has acquired kinetic 
energy or capacity for doing work in coming to rest. 

On the other hand, while the npward velocity was being retarded, the product 
of total upward force X dist was less than that of resistance X dist, the difference 
being the work done by the kinetic energy against the resistance of gravity. 

In practice, the term " work" is usually restricted to that j9or<ion of the work 
which a force performs in balancing the resistances which act against it ; in other 
words, to the work done by so much of the force as is equal to the resistance. 

With this restriction, we have work ^ force X dist, = resistance X dist. 

Thus, if the resistance be a friction of 4 ft>s., overcome at every point along a 
distance of 8 feet; or if it be a weight of 4 S>s., lifted 3 feet high, then the work 
done amounts to 4 X 8 » 12 foot-9>8, provided the initial and the final velocities 
are equal. 

(d) In cases wbere tbe weloeity Is nnlform, as in a steadily running 
macbine, tbe force is necessarily equal to the resistance ; and where the velocities 
at the beginning and end of any work are equal (as where the machine starts 
from rest and conies to rest again) the mean force is equal to the mean resistance. 
In such cases, therefore, the two products, mean force X distance, and mean 
resistance X distance, are equal, and we have, as before, 

Work =^ force X dist = resistance X dist. 

♦ A man who Is standing still is not considered to be working, any more than 
is a post or a rope when sustaining a heavy load ; although he may be support- 
ing an oppressive burden, or holding a car-brake with all his strength ; for his 
force moves nothing in either case. 

t These products must not be confounded with momerUs, — force X leverage. 



342 FOKCE IN RIGID BODIES. 



(f ) In calculating the work done by machinery, etc., allowance must be made for 
this expenditure of a portion of the work in overcoming resistances. Thus, in pump- 
ing water, part of the applied force is required to balance the friction of the different 
parts of the pump; so that a steam or water "power,** exerting a force of 1(H) &8., 
and moving 6 feet per second, cannot raise 100 fi>8. of water to a height of 6 feet 
per second. Therefore machines, so far from gaining power ^ according to the popular 
idea, actually lose it in one sense of the word. In Uarting a piece of machinery, the 
forces employed have (1st) to balance, react a^rainst, or destroy the resisting force 
of friction and the cohesive forces of the material which is to be operated on ; and 
(2d) to give motion to the unresisting matter of the machine and of the material 
operated on, after the resisting forces which had acted upon them have thus been 
rendered ineffective. But after the desired velocity has been established, the forces 
have merely to bcUance the resistances in order that the velocity may continue uniform. 

(g) That portion of the work of a machine, etc., which is expended against fric- 
tion is sometimes called <* lost -work " or ** prejudicial ^rorky" M'hile only 
that portion is called " useful -vrork " which renders visible and tangible service 
in the shape of output, etc. Thus, in pumping water, the work done in overcoming 
the friction of the inimp and of the water is said to be lost or prejudicial, while the 
useful work would be represented by the product, weight of water deliverwl X height 
to which it is lifted. 

The distinction, although artiflcial| and somewhat arbitrary, is often a very con- 
venient one ; but the work is of course not actually ** lost," and still less is it ** pre- 
judicial ;" for the water could not be delirered without first overcoming the resist- 
ances. A merchant might as well call that portion of bis money lost which he 
expends for clerk-hire, etc. 

(it) For a given force and distance^ tlie i^ork done is independent of the 

time $ for the product, force X distance, then remains the same, whatever the time 
may be. But the distance through which a given force will work at a given velocity 
is of course proportional to the time during which it is allowed to work. Thus, in 
order to lift 50 pounds 100 feet, a man must do the same work, (= 6000 foot-pounds) 
whether he do it in one hour or in ten ; but, if he exerts constantly the scrnie foroey 
he will lift 50 &>s. ten times as high in ten hours as in one, and thus will do ten times 
the work. Thus, for a given force, the vrork is proportional to the tinte* 

Art. 18 (a), Poorer. The quantity of any work may evidently be considered 
without regard to the time required to perform it ; but we often require to know the 
rate at which work can be done ; that is, how much can be done within a certain 
time. 

The rate at which a machine, etc. can work is called its -power. Thus, in selecting 
a steam-engine, it is important to know how much it can do per minute, hour, or dag. 
We therefore stipulate that it shall be of so many horse-powers; which means nothing 
more than that it shall be capable of overcoming resisting forces at the rate of so 
many times 33,000 foot-pounds per minute when running at a uniform velocity, i. e., 
when force X distance = resistance X distance. 

(b) The liorse-poiver, 33,000 foot-pounds per minute, or 550 foot-pounds per 
second, is the unit of ponrer, or of rate of ivork, commonly used in connec- 
tion with engines. The metric horse-poorer, called "force dt 
cheval," " cheval-vapeur," or (German) " Pferdekraft," is 75 kilogram-meters pel 
second = 542.48 ft-ibs. per sec. = 32,549 ft.-ft>s. per minute = 0.9863 horse-power. 1 
horse-power = 1.0138 " force de cheval." In theoretical Mechanics the foot-ponud 
per second is used in English measure ; and the lUlo§;ram-meter per ceo- 
ond in metric measure, 

1 foot-pound per second =» 0.13826 kilogram-meter per second. 
1 kilogram-meter per second = 7.2331 foot-pounds per second. 

(c) Up to the time when the velocity becomes uniform, the po-wer, or rate 9t 
vrork, of the train, in Art. 16 (d), is variable, being gradually axelerated. 
For in each second it overcomes its resistances (and moves its point of application) 
through a greater distance than during the preceding second. Also, after the steam is 
shut off, the rate of work is variable, being gradually retarded. When the force of 
the steam just balances the resistances, the rate of work is uniform. 

(d) Po-«rer = force X velocity. Since the rate of work is equal to the work 

done in a given Hrne, as so m&xxy foot-pounds per second, we may find it by dividing the 

work in foot-pounds done during any given time by the number of seconds in tkst 

time. Thus 

_ ^ * , force in pounds X distance In feet 

Power =■ rate of work = \. , ; • 

time in seconds 



FOBGE IN RIGIB BODIES. 



343 



Bat this is eqaivalent to 

- . * . J V ^ distance in feet 

Power -rate of work -force in pounds X time in seconds 

— -orce in lbs. X velocity in feet per second. 

Or if we treat only of the work of that force which overcomes resUtancea: or i« 
eawes where the velocity is either uniform throughout or the same at the 
beginning and end of the work; 

Power rate of work _ resistance, w velocity, 

in ft-lbs. per sec " in ft-lbs. per sec in lbs. ^ in ft per sec. 

Thus if the resistance is 3300 lbs. and is overcome thrpugh a distance of 10 
feet in every minute; or if the resistance is 33 lbs. and is overcome through 
?di8tonce of 1000 f4et per minute, the rate. of the work i^J^J^^^J'^ 
the same, namely, 33,000 foot-pounds per mmute, or one horso-power; Sat 

lbs. vel. lbs. vel. . 

8300 X 10 — 33 X 1000 — 33,000 foot-pounds per mmute. 

M The same "power" which will overcome a given resistance through* 
riven distance, in agiven time, will also overcome any other resistance through 
Wiy other distance, in that same time, provided the «:<»w**^°®**°^.*^^®.5S!? 
when multiplied together give the same amount as m the first case. Thus. 
the power that will lift 60 pounds through 10 feet in asecond, will m a second 
Hft 600 pounds, 1 foot; or 25 pounds. 20 feet; or 6000 pounds ^ oi a foot. 
El practice, the adjustment of the speed to suit different resistances, is usually 
effected by the medium of cog-wheels, belts^or lever.. By "^eans of 
these the engine, watei>wheel, horse, or other motive power, exerting a given 
force and ruhning at a given velocity, may be made to overcome small resist* 
ances rapidly, or great ones slowly, as desired. 

Art. 19 (a). The 'vrork 'vrhldi a bodjr ean do hy -rlrtiie ot its 
motion j or (which is the same thing) the 'vrorh reonircd to brins 
the body to rest. Kinetic energy* -vim -viTa^ or "living ttorce.'* 

As already remarked, a force equal to the weight of any body, at any place, 
will, in one second, give to the mass or matter of the body a velocity — g, or 
(on the earth's surface) about 32.2 feet per second. Or if a body be thrown 
\Lpward with a velocity — ■ g, its weight will stop it in one second. 

Since, in the latter case, the velocity at the beeinning and at the end of the 
■econd are, respectively,— g feet per second, ana — 0, the mean velocity of the 

iody is -£- feet per second. Therefore, during the second it will rise _^ feeC^ 

2 2 

or about 16 feet. In other words, the work which any body can do, by virtue 
of being thrown vertically upward with an initial velocity (velocity at the 
gtart) otg feet per second, is equal to the product of its weight multiplied of 

-J- feet Or, 

work in foot-pounds — weight X -^ 

Ifotioe that in this ease (since the initial velocity v Is equal to jy), JL. — 1. 

^ 9 

Smppose now that the same body be thrown upward with double the former 

velocity; i. e., with an initisd velocity equal to.2 g (or about 64 feet per seconds 
dince gravity requires (Art 8 c), two seconds to impart or destroy this 

velocity, the body will now move upward during two seconds, or twice as long 
a Urns as before. But its mean velocity now is p. or twice as great as before. 
Therefore, moving for double the time and with double the velocity, it will 
teavel /our times as far, overcoming the same resistance as before (viz. : its 
own weight) through /our times the distance. 

Thus, by making its initial velocity v — 2 p, {. «., by doubling its -L-. making 

g 
it — 2, we have enabled the body to do four times the work which it could 

io when its — !L was 1; so that the work in the second case is equal to the 

9 



344 FOBOE IN RIGID BODIE& 

product of that in the first case multiplied by the 8quar$ of -2L( Qg^ 

- weight X -2- X ^ 

— weight X — 

And it is plain that this would be ithe case for any other velocity. Now the 
total amount of the work which the body can do, is independent of the 
amount of the resistance against which it is done; for if we increase the 
resistance we diminish the distance in the same proportion, so that their 
product, or the amount of work, remains the same. The above formula^ 
therefore, applies to all cases ; i. 6., the total amoiuit ot 'vrorfc, in fo^ 
pounds, whicn any body will do, f^ainst any resistance, by virtue of its motioii 
Alone, in coming to rest, is 

Work - weight of moving body, in lbs. X square of its velocity in ft per sec^d 

f/ 
— weight of moving body, in lbs. X fall in ft required to give the velocity 

_ weight of moving body, in lbs, y square of its velocity in ft per second 
g 2 

In these equations, the weight is that which the body has in any given plaoe^ 
and g is the acceleration of gravity at that same place. 

(b) Since the weight of a body j^ j^^ ^^^ ^^^^ 1^^ ^ 336), the last formula 
becomes, by "method A,^* Art. 11 (d). 



mass of moving body w square of its velocity in ft per second 
in foot^ot^mb " in "matU^' '^ 2 

and by "method B," Art. U (e), 



mass of moving body v> square of its velocity in ft per eeobad 

infoo^poundato" in potmdij ^ 2 

(c) In the above equations the left hand side represents the work (or resis- 
tance overcome through a. distaiice) in any given case, while the right hand 
side represents the Unetlo energy of the body, by which it is enabled to do 
that work. Some writers call this energy "via ▼!▼»,»» or " living force" a 
name formerly given (for convenience) to a quantity just double the energy, 
or — mass X velocity*. 

(d) As an illustration of the foregoing, take a train weighing 1,120,008 
pounds, and moving at the rato of 22 feet per second. The kinetic energy 
ef such a train is 

energy - weight X I5!2^; or. 

1,120,000 lbs. X — — 8,400,000 ft.-lbs. 
64.4 

That is, if steam be shut off, the train will perform a work of 8,400^000 fL-lba. 
in coming to rest. Thus, if the sum of all the resistances (of friction, air, 
grades, curves, ete.) remained constantly — 6000 lbs.,* the train would travel 

8,400,000 ft.-lb8. _ lesott, 
5000 lbs. 

(e) We thus see that the total quantity of work which a body can do by virtua 
of its motion alone, and without assistance ft-om extraneous forces, is in pi^ 
portion to the weight of the body and to the square of its velocity when it 
begins to do the work. For example, suppose that a train, at the momaDft 
when steam is shut off, has a velocity of 10 miles an hour and that the kinetio 
energy, which that velocity gives it, will by itself carry the tram against th» 

•In practice, this would not be the case. 



9OB0B IK RIGID BODIES. 345 

CMistances of Che road, etc^ for it distance of ons quarter of a mile before it 
stops. Then, if steam be shut off while the train is moTing at 5, 20, 30 or 40 
miles per hour (t. e^ with ^^ 2, 8 or 4 times 10 miles per hour) the train will 
tiavel JL, 1, 2 ^ or 4 miles (or ^ 4, 9 or 10 times ^ mile) before coming to 

rest* 

Bat the rate of work done is proportional simply to the resistance and the 
ntoeity (Art* IBd, p. 842). Therefore, the locomotive whose steam is shat oft 
at 20, 80 or 40 miles per honr, will require, for running its 4. 9 or 16 quarters 
tf a mile, but 2, 3 or 4 times as many seconds ae it required at 10 miles per hour. 
The same principle applies to all cases of acceleration or of retardation.f 
For instance, in the case of a falling body, the distance through which it 
mnst fall in order to acquire any giren velocity is as the square of that 
Telocity, but the time required is simply as the velocity. Also, if a body is 
ttirown Terticanlly upward with any given velocity, the height to which it will 
rise bvh the time gravitv destroys that velocity, will be as the square of the 
Yelooity,but the time wiU be simply as the velocify. 

Art. SO (a). The momentnin of a moving body (or the product of its 
mass by its velocity) is the rate, in foot-pounds per second, at which it works 
against a resisting force equal to its own weighty as in the case of a body thrown 
vertically upward. At the instant when it comes to rest, its momentum, or rate 
of work, is of course = nothing. Therefore its mean rate of work, or mean 
momentum, is one-half of that which it has at the moment of startiug. 

Thus, suppose such a body to weigh 5 lbs. Then, whatever its velocity may 
be, 6 pounds is the resisting force, against which it must work while coming 
to resL Let the initial velocity be 96 feet per second. Then its 

momentum ■• mass X velocity «— 6 X 96 — 480 foot-pounds per second? 

Mid, while ooming to rest, its 

•Moa momentum -» mass X T . ^r^ ■« 240 foot-pounds per second. 

Now, in falling, the weight of the body (5 lbs.), would ^ve it a velocity of 96 
foet per second in about three seconds. Consequently, in rising, it will destroy im 

lelooity in the tame time. In other words, the time — ,. velocity ^ velocity 

•^ acceleration g 

M £| 1. 3. Three seconds, therefore, is the time during which it can work. 

How, if the mean rate of work in foot-pounds per seeond (at which a body 
ean work against a resistance) be multiplied by the time during which it can 
ooBtinue so to work, the product must be the total work done. Or, in this case^ 

work mean rate of work v^ time, oji* v <» ion *r^* »wvn»^. 

to IWbe, - in flrlbs. per sec. X or No. of sees. - 240 X 3 - 720 footrpounds. 

-weight X 12}2£ife X ^^l^^ifc 
2 g 

.weight X y^'?^ ,asinAjt.l9(o),-6 X ^ - 720 ft.ponnda 

(b) We may notice also that since, in the case of a falling body, or of one 

ihixywn upward, . ^"^^ is the time during which it must fall in order to 


acquire a given velocify, or during which it must rise in order to lose it^ 
therefore, 

Telocity ^ reloaiij ^ ^ ^^^ velocily X time — distance traversed; 

so that 

weight X 1212215? - weight X H^SpLx I2!22!5 ^ 
weight X dislanee traversed -« the work. 

— - ^' ' ™" ■^-- l■^■■^■ — ■-■I --■-■■■■■■ ■-■ — ■ . ■■ 1^ ■■ ■■■III, ■■■■■■■■■ _^ ■ ■ I I ■^■^■^M— — i— ^M^ 

• This sappofes, for oonvenience, that the resistances remain uniform through* 
out, and are the same in all the cases, which, however, would not hold good in 
praotioe. 

t Retardation is merely acceleration in a direction opposite to that of the 
motion which we happen to be coasidering. 




346 fOBCE IN BIOIB BODIES. 

Art. 91 (a)* Bnawrf to toJ — irucU blc. Energy, expended In wortt, to 
not destroyed. It is either transterred to other bodies, or eue stored ap in the 
body itself; or part may be ithua transferred, and the re^t thus stored. Bnt^ 
althoagh ener^ cannot be destroyed, it may be rendered useless to us. Thn^ 
amoTing train, in coming to rest on » level track, transfers its kinetic enei 
into other kinetio energy: namely, the useless heat due tofidctioo at the r 
brakesand Journals ; and this heat, although none of itiadeatrayed, is disai] 
Jed the earth and air so as to be practicallyoeyond our recovery. 

Alt. sa (a). Potential •nergy* or possible energy, may be defined as 
•toted-np energy. We lift a one-pound body one-foot oy expending upon it 
one foot-pound of energy. But this foot-pound is stored up in the **sy8tem ** 
(composed of the earth and the body) as an addition to its stock of potential 
energy. For, while the stone falls through one foot, the system wilt acquire 
a kinetic energy of one foot-pound, and will part with one foot-pound of its 
potential energy. • 

(b) The potentiai energy of a ''system*' of bodies (such as the earth and a 
weight raised above it, or the atoms of a mass of powder, or those of 
a bent spring) depends upon the relative poaitiona of those bodies, and 
upon their tendencies to change those positions. The kinetie energy of a 
system (such as the earth and a moving train of cars) depends upon the tnaM«6 
m its bodies and upon their motion relatively to each other. 

Familiar instances of potential energy are— the weight or spring of a clock 
When fully or partly wound up, and whether moving or not; the pent-up water 
In a reservoir; the steam pressure in a boiler; and the explosive energy of 
powder. We have mechanical energy in the case of the weight or springs or 
water; heat energy in the case of the steam, and obemica! energy in that 
df the powder. 

(o) In many oases we ma3r conveniently estimate the total potential enei^ 
of a systenu Thus (neglecting the resistance of Uie air) the explosive energy 
of a pound of powder is » the weight of any given cannon ball X the height 
to which the force of that powder could throw it. •» the weight of the ball X 
(the square of the initial velocity given to it by the explosion) -i- 20. But in 
other cases we care to find only a certain definite portion of the total potential 
energy. Thus, the toM potential energy of a olock-weight* would not be 
exhausted until the weight reached the center of the earth: but we generally 
deal only with that portion which was stored In it by winding-up. and which 
tt will give out again as kinetio energy in running down. This portion is -• th^ 
weight X the height which it has to run down -• the weight X (the square of 
the velocity which it would acquire in fallin^/V>oe{y through that height) -i- 2if. 

(d) There are many cases of energy in which we may hesitate as to whether 
the term "kinetic" or "potential** Is the more appropriate. Thus, the pres- 
sure of steam in a boiler is believed to be due to tne violent motion of the 
particles of steam, which bombard the inner surface of the boiler-shell; so 
that, from this point of view, we should call the energy of steam kinetie. But, 
on tne other hand, the shell itself remains stationary; and, until the steam is 
permitted to escape from the boiler, there fs no outward evidence of energy 
in the shape of work. The energy remains stored up in the boiler ready kt 
nse. From this point of view, we may call th e energy of steam potential energy. 

(e) It seems reasonable to suppose that further knowledge as to the nature 
of other forms of energy, apparently potential (as is that of steam), might 
reveal the fact that all energy is ultimatiely kinetio. 

Art. 23 (a). There is much confusion of ideas in regard to those 
actions to which, in Mechanics, we give the names, *' force," *• enerfry«'* 
** power," etc. This arises from i he fact that in every-day language these 
terms are used indiscriminately to express the sime ideas. 

Thus, we commonly speak of the " force " of a cannon-ball flying through the 
air, meaning, however, the repulsive force which would be exerted between the 
ball and a building, etc. with which it might come into contact. This force 
would tend to move a part of the building along in the direction of the flight 
of the ball, and would move the ball backward ; (i. e., would retard Its forward 
motion). But this great repulsive "force" does not exist until the ball strikes 
the building. Indeed, we cannot even tell, from the velocity and weight of the 
ball, what tne amount of the force will be, for this depends upon the strength, 
etc., of the building. If the building is of glass, the foroe mav be so slight as 
scarcely to retard the motion of the ball perceptibly, while,'if the building is an 

* For convenience we may thus speak of the energy of a mdem of bodies (the 
earth and the clock-weight) as resiaing in only one of the bodies. 



FORCE IN RIOIB BODIES. 347 

earth embankment, the force will be much greater, and may retard the motion 
oX the ball so rapidly as to entirely stop it before it has gone a foot farther. 

The moving ball has great (kinetic) energy; but the only force that it exerts 
during its flij^ht is the comparatively very slight one required to push aside the 
particles of air. 

The energy of the ball, and therefore the total work which it can do, are inde^ 
pendent of the nature or the obstruction which it meets ; but since the work is 
the product of the resistance oifered and the distance throu^^h which it can be 
overcome, the distance must be inversely as the resistance offered ; or (which is 
the same thing) inversely as the force required of, and exerted by, the ball in 
balancing that resistance. 

Since work, in ft.-lb8. => force, in &>s., X distance traversed, in feet, we have 

force in lbs. = work, in ft.-lbs. _ rate of work, 

' distance traversed, in feet in ft.-lbs. per fool. 

Art. S4 (a). An impact, blow, stroke or collision takes place when a 
moving body encounters another body. The peculiarity of such cases is that 
the time of adion of the repulsive force due to the collision Is so short that een- 
erally it is impossible to measure it, and we therefore cannot calculate the force 
ttovsx the momentum produced by it in either of the two bodies : but since both 
bodies undergo a great change of velocity (i.e., a great acceleration) during this 
Short time, we know that the repulsive force acting between them must be very 
great. 

We shall consider only cases of direet Impact, or impact where the centers 
of gravity of the two bodies approach each other in one straieht line, and where 
the nature of the surfaces of contact is such that the repulsive 

force caused by the impact also acts through those centers and in their line of 
approach. 

(b) This forcCj acting equally upon the two bodies (Art. fi/), for the 
same length of time (namely, tne time during which they are in contact), neces- 
sarily produces equal and opposite changes in their momentums (Art. 12, p. 888). 
Hence, the total momentum (or product, mass X velocity) of the ttoo bodies is 
always the same after impact as it was before. 

(c) But the relative behavior of the two bodies, after collision, depends upon 
their elasticity. If they could be perfectly inelastic, their velocities, after im- 
pact, would be equal. In other words, they would move on together. If they 
could be perfectly elastic, they would separate from each other, after collision, 
with the same velocity with which they approached each other before collision. 

(d) Between these two extremes, neither of which is ever perfectly realized in 

Enictice, there are all possible degrees of elasticity , with corresponding differences 
1 the behavior of the bodies. The subject, especially that of indirect impact, is 
a very complex one, but seldom comes up in practical civil engineering. 

(e) " In some careful experiments made at Portsmouth dock-yard, England, a 
man of medium strength, and striking with a maul weighing 18 lbs., the handle 
•f which was 44 inches long, barely started a bolt about '% of an inch at each 
blow ; and it required a quiet pressure of 107 tons to press the bolt down the 
same quantity ; but a smsQl additional weight pressed it completely home." 



348 



GRA.VITY — ^PALLING BODI£S. 



«RATITT. FAIililire BOBIK8. 

Bodies flAlllngr Tertleally. A body, falling freely in racuo 
from a state of rest, acquires, by the end of tbe first second, a Telocity of about 
32.2 feet per second ; and, in each succeeding second, an cuidition of velocity, or 
aoceleratiod, of about 82.2 feet per second. In other worda^ tbe Telocity receivM in 
each second an acceleration of about '62.2 feet per second, or is accelerated at tbe 
raU of about 32.2 feet per second, per B^cond. This rate ie generally called (fbr 
brerity, see foot-Bote,t p. 334), simply the sM)oeleratloia of gravity (bat see * 
below), and is denoted by |p« It increases ftx>m about 82.1 f«et per second, par 
second, at the equator, to about 32.5 at the poles. In the latitude of London it if 
82.19. These are its values at sea-level ; but at a height of 6 miles above that level 
it is diminished by only about 1 part in iOO. For most practical purpoeee it may be 
taken at 32.2. 

Caution. Owlnar to tbe resistance of the air none of the follow- 
ing rules give perfectly accurate results in practice, especially at great vela. 
The greater the specific gravity of the body the better will oe the rMnlt. The air 
ffeelets botn rislnir and fklllnir bodies. 

If a body be tbrown vertically upwards with a given vel, it will 
rise to the same height from wiiich it must have fallen in order to acquire said 
vel; and its vel will be retarded in each second 32.2 It per lec* Its average ascend' 
ing velocity will be half of that with which it startled ; as in all other cases of 
uniformly retarded vel. In falling it will acquire the same vel that it started 
up with, and in the same time. See above Caution. 



Acceleration acquired* 

in a given time = ff X time 

in a given fall from rest = \^ 2 g X fall. 

in a given fall from rest ) __ twice the fall 

and given time j *~ time 

Time required 

- , , x» acceleration 

for a giyen acceleration >- — 



9 



for a given fall flrom rest 






fall 



fall 



3^ final velocity 
fall 



for a given fall from rest i _^ 

or otherwise / ™ mean vel ~~ J^ (initial vel + final vel) 

FaU 

In a given time (starting from rest) — time X H ^^"^ ^^ ^ timeS X ^iff 
in . giren time (.t«:ttagi _ inltl.1 t«1 + ftn.l r«| 

from rest or otherwise) J 2 

reqd for a given acceleration "i __ acceleration^ 

(starting from rest) ) 2g 

during any one given second (counting from rest) 

■» ^ X (number of the second (Ist, 2d, Ac) — \\ 

during any equal consecutive times (starting from rest) « 1, 3, 5, 7, 9, Ae. 



wfti^e f ^^*- 2d. 3d. 4th. 6th. 6th. 7th. 8th. 9th. 10th. 
' seconds 



Velocity; ft per sec. 
Dist fallen since end 
of preceding sec ; ft. 

Total diet fallen; ft. 



32.2 
16.1 



64.4 
48.3 



96.6 
80.5 



16.1 1 64.4 144.9 



128.8 
112.7 
257.6 



161.0 
144.9 
402.6 



193.2 
177.1 
679.6 



225.4 
209.3 
788.9 



267.6 

241.6 

1090.4 



289.8 

278.7 

1904.1 



822.0 

805.9 

1610.0 



* By " acceleration,** in thi» article, we mean the total aooelerstion ; C «., tbe whole 
change of velocity occarring in the givwi tins or fUl. For the raft oC *rflHtwrtn> 
we use simnly the letter g. 



DESCENT ON INCLINED PLANES. 



349 




I^escent on Inclined plirnes. When a body, U. is placed 
upon an inclined plane, AC, its whole weight W is not employed m giviug it 
▼elocity (as in the case of bodies falling vertically) 
but a portion, P, of it (= W X cosine of o = W X 
cosine of a*) is expended in perpendicular pressure 
against the plane; while only S, (= W X sine of o 
— W X sine of a*) acts upon U in a direction parallel 
to the surface AC of the plane, and tends to slide it 
down that surf. 

The acceleration, generated in a given body in a 
given time, is proportional to the force acting upon 
the body in the direction of the acceleration 

Hence If we make W to represent bv scale 
tbe ttccfeleration g (say 32.2 ft per l*c) which gray 
would give to U in a sec if falling freely, then S will 
give, by the same scale, the acceleration in ft per 
sec which the actual sliding force 8 would give to U in one sec if there were 
no friction between U and the plane. We have therefore 

theontio^ acceleration down the plane = gr x siae of a. 

Therefore we have only to substitute "^. sin a" in place of "flr;" and the 
</omn^ distance or "slide" AC in place of the corresponding vertical distance 
or " fall " A £ in the equations, in order to obtain the acceleratioos etc as 

follows : 

on an inclined plane witbont friction. 

In the foUowingr* tbe slides A € are in feet, tbe times in 
seconds, and tbe velocities and accelerations in feet per 
second.t 

Accelerationfof sliding velocity 

i« - -j««« n^^ "^^rt accel acquired in falling) w i^ _ 
in a given time = ^^^^ ^^^j^^ the same time / X sin a 

B g. sin a X time 



in agiven slide, as AC,> slide 
from rest i 14 ti°>e 



f vert accel acquired in falling) 
=< freely thro the corresponding >•■ 
{ verthtAE J 

» y' 2 ^. sin a X slide 



V^7^'^ 



for a given sliding acceleration 



Time required 
sliding acceleration 



ff, sin a 

for a given slide, as A C, lirom _^ slide 

wst "" y^ final sliding veloc 



« /__8lid 
iity "" V H flf. si 



sl ide 
sin a 



time reqd to fall freely thro the correspond- 
ing verthtAE 



sin a 



slide 



slide 



for a given slide, from > ^ ^ 

rest or otherwise J "* mean sliding vel "" H (initial + final sliding vels) 



Cosine a 



Sine a 



horizontal stretch, as E C» 

base EC of any length, aa A C ^ l/AC» — Al? 

length A C ^ that length A C 

height A E _ fall, A E. in any given length, A C ^ T/AC2 — te<> 

length AC"" "^ 



that length 



AC 



* Because o and a are equal. 
tHr acceleration,*! flW» cartielet we mean the total acceleration, t. «., the whok 
eha&flle in telodty occurring in the given time or slide, for ttie rate of acceleration 
ire nse tiaiolT the letter a. 



350 GRAVITY — PENDULUMS. 

Slide, u A C 

in a glyen time, starting from rest = time X }4 final sliding Tel 

= time *X}iff. sin a. 

in a given time, s<«rting from rest ., ,, 

or otherwise — ""*® X mean sliding Tel 

- time X H (initial + final, sliding rels) 

required for a ffiren sliding accel- „ sliding acceleration* 
oration (starting from rest) *" 2 p. sin a 

But in praetlce the sUdlmr on the plane ts always on- 

£!?;^ ^X ™««»- To Inclnde the emJt of Metionrwe hJ^ 

only to substitute 

sin a - (cos a. ooeff fric)] " in place of « g. sin a " in the abore eqoatlona. 
Lse 

Friction = Perpendicnlar pressure P X coefficient of friction 

= weight W X cosine a X coefficient of friction 
and 

retardation of firletlon '^gX cosine a X coefficient of friction. 

Besnitant slidinir acceleration 

« theoretical sliding accel (due to the sliding force, S) — retardation of fHo 
=- iff. sin a) — (g. cosine a. coeff fric) 

= ffX fsin a — (cosine a. coelTfrlc) j 

If the retardation of friction (•= y. cos a X coeff fVic) is not leu than the total 
•r theoretical accel ("^. sin a") the body cannot slide down the plane. 



"PX 
Because 



• ^ 



PENDULUMS. 



Tex numbers of ribrations which diff pendulums will make in any ^Ten place la 
a giren time, are inversely as the square roots of their lengths : thus, if one of them 
Is 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; but 
its number of vibrations will be but ^ /^i or i^ as great. The times in which diff 
pendulums will make a yibration, are directly as the sq rts of their lengths. Thna, 
if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, S, or 4 times aa 
great ; and so also will be the time occupied in one of its vibrations. 

The length of a pendulum vibrating seconds at the level of the sea, in a Taonmn. 
in the lat of London (51^ North) is 39.1393 ins ; and in the lat of N. York (409^ 

North) 39.1013 ins. At the equator about ^ inch shorter ; and at the poles, about -ffg 
Inch longer. Approximately enough for experiments which occupy but a few sec, 
we may at any place call the length of a seconds pendulum in the open air, 89 ins ; 
half sec, fl^^ ins ; and may assume that long and short ribrations of the same pen- 
dulum are made in the same time ; which they actually are, very nearly. For mea»- 
nring depths, or dists by sound, a sufficiently good sec pendulum may be* made of a 
pebble (a small piece of metal is better) and a piecfi of thread, suspended fh>m a 
common pin. The length of 39 ins should be measured from the centre of the pebble. 



PXBJSTDULUMS, ETC. 351 

In Btartliig tlie Tibratlons, the pebble, or boby must not be thrown into motion, but 
meroly lei drop^ after extending the string at the proper height.. 

To find the lenKrtb of a pendalam read to make a given number of 
vibrations in a min, divide 375 by said reqd number. The square of the quot will bo 
the length in ins, near enough for such temporary purposes as the foregoing. Thus, 
for a pendulum to make 100 vibrations per min, we have |^^ =» 3.75 ; and the square 
of 3.75 = 14.06 ins, the reqd length. 

To find (lie namber of ▼ibrationti per min for a pendulum of 
given length, in ins, take the sq rt of said length, and div 375 by said sq rt. Thus, 

for a pendulum 14.06 ins long, the sq rt is 3.75 ; and z-=i » 100, the reqd number. 

Rkk. 1. By practising before the sec pendulum of a dock, or one prepared as Just 
stated, a person will soon learn to ooant 5 in a sec, for a few sec in succession ; and will 
thus be able to divide a sec into 5 equal parts ; and this may at times be oseftil for 
▼ery rough estimating when he has no pendalam. 

Oentre of Oscillation and Pereusslon* 

Bsv. 2. When a pendulum, or any other suspended body, is vibrating or oscillating 
backward and forward, it is plain that those particles of it which are far front the 
point of suspension move faster than those which are near it. But there is always 
a certain point in the body, such that if all the particles were concentrated at it, so 
that all should move with the same actual vel, neither the number of oscillations, 
nor their angular vel, would be changed. This point is called the center of oKiUa- 
Hon. It is not the same as the cen of grav, and is always farther than it fh)m the 
point of suspension. It is also the cerUre of percussion of the suspended vibrating 
body. The dist of this point fh>m the point of snap is found thus : Suppose the body 
to be divided into many (the more the better) small parts ; the smaller the better. 
Find the wc^gnt of each part. Also find the cen of grav of each part ; also the dist 
firom each such con of gray to the point of susp. Square each of these diets, and 
mult each square by the wt of the corresponding small part of the body. Add the 
products together, and call their sum p. Next mult the weight of the entire body 
by the dist of its cen of grav from the point of susp. Gall the prod p. Divide p hyg* 
Thinp is the moment of inertia of the body, and if divided by the wt of the 
body the sq rt of the quotient will be the Radius of Gyration. 

Angrnlar Telocity. 

When a body revolves around any axis, the parts which are farther from that 
axis move faster than those nearer to it. Therefore we cannot assign a stated 
linear velocity in feet per second, or miles per hour etc, that shall apply to every 
patriot it. But every part of the body revolves around an entire circle, or 
through an angle of 860P, in the same time. Hence, all the part« have the same 
▼elocity in deare^i per second, or in revolutions per seoond. This is called the 
angular velocity. Scientific writers measure it by the length of the arc de- 
scribed by any point In the body in a given time, as a second, the length of the 
arc being measured by the number of times the length of Us ottn radius la con- 
tained in it. When so measured, 

Angular velocity __ liaear velocity (in feet etc) per sec 
in radU per second - length of radius (in feet etc) 

Here, as before, the angular velocity is the same for all the points in the body, 
because the velocities of the several points are directly as their radii or dis- 
tances from the axis of revolution. 

In each revolution, each point describes the circumference of the circle in 
which ft revolves =» 2 v r (ir = 3.1416 etc ; r = radius of said circle). 0>nse- 
qaently, if the body makes n revolutions per second, the length of the arc de- 
scribed by each point in one second is 2irrn; and the angular velocity of the 
body, or linear velocity of any point measured in its own radii, is . 



2irr» 



2 w » «= say 6.2832 X revs per second = say .1047 X revs pe» f^inute. 



Moment of Inertia. 



Sappose a body revolving around an axis, as a grindstone; or oscillating, like 
apeodnlum. Suppose that the distance from the axis of revolution (which, in 
the pendulum, Is the point of suspension) to each individual particle of the 
body, has been measured; and that the square of each such distance has been 
multiplied by the weight of that particle to which said distance was measured. 



352 



MOMENT OF INERTIA. 



The sum of all these products is the moment of inertia of the bodf . Or 



Moment 
of Inertia 



-{ 



the sum, 
lor all the particles 



}•' 



r weight square of dist 

-< of X of particle from 
(.particle axis of revolution 



or, I = 'S,<fiw. 
Scientific writers frequently use the mass of each particle ; 



ie, 



its weight 



instead of its weight, in calcnlatiug 



acceleration (g) of gravity, or about 32.2 
the moiueiit of inertia. 

Ill practice we may suppose the body to be divided into portions measuring 
a cubic inch (or some other small size) each : and use these insteaO of the theo- 
retical infioitely small particles. The smaller these portions are taken, the 
more nearly correct will be the result. 

When the moment of inertia of a mere surface is wanted (instead of that of a 
body), we suppose the surface to be divided into a numl)er of small areaSf and 
use them instead of the weights of the small portions of the body. 

weight of body, Muare of 



Table of Radii of Clyratlon. 



Body 



Any body or 
fig^nre 



Solid cylin- 
der 



ditto 

ditto, infinitely 

short (circular 

surface) 

Hollow cyl* 
inder 

ditto, infinitely 
thin 

ditto, of any 
thickness 

ditto, infinitely 
thin 

ditto, infinitely 
thin and infinitely 
short (circumfer- 
ence of a circle) 

Solid spbere 



Rewolwinff 
around 



any given axis 



Its longitudinal 
axis 



adiam, midway 
between Its enas 



a diameter 



its longitudinal 
axis 



ditto 

a diam midway 
between its ends 

ditto 



a diameter 
a diameter 



V 



Badlas of Gyratioii 

' moment of inertia around the given axis 
weight of body, or area of surface 

radius of cylinder X 'Xj-k- 
* radius of cylinder X aboat .7071 



V 



' length' radiu8> of cylinder 
12 "*" 4 



V 



radius of oylinder 



i nner rad» + outer radi 
2 

nidlUB of eylfnder 



V 



din ner rad' + outer rad* ^ length^ 
4 ■*" 12 



V 



radius^ of cylinder length* 



12 



radias of cylinder X 
■at radius of cylinder X 



about .7071 



V 



radius* of sphere 
2.5 



= radfus of sphere X V'Ti" 

= rndins of sphere X about .68246 



BADU OP GYRATION. 



353 



Table of Radii of Oyration,— CoimiruBD. 



Hollow 
•plioro of any 

thickness 

ditto, thin 

ditto, inflnitelT 

thin (spherical 

surface) 

8traiirl>t line, 

ab 



Solid eone 

Circular 
plate, of rect- 
angular cross seo- 
tion 

Circular 
ring^, of rectan- 
$oIftr cross section 

Square, rect- 
angrle and 
otlier snr* 



RevolviniT 
aroand 



a diameter 
ditto 

ditto 

any point, £, In its 
length 

either end, a or 6 

Its center, e 
its axis 



S«e Solid cylin- 
der 



See Hollow cylin- 
der 



Badins of Oyration 



V 



2 (outer rad* — inner rad») 



5 (outer rad* — inner rad') 
approz (outer rad + inner rad) x ^065 



radius of sphere 
■■ radios of sphere X al>but .8166 



Sab 



length aft X "\-^ 
— length abX about JB776 

•- length abX about .2887 

radius of base of cone X '\/~S' 
M radius of base of cone x .5477 



For the thidcnest of plate or ring, 
measured perpendicularly to the plane 
of the circumference, take the length of 
the cylinder. 



For Uasl radius of gyration, or that around the longe$t aafs, 
see p 496 and 497. 



2a 



364 CENTRIFUGAL FORCE. 



GEBTTRIFVOAI. FORCE. 

When a body a, Fig. 1 , moves in a circular path abd^ it tends, at each point, as 
a or 6^ to move in a tangent at or bif to the circle at that point. But at each 
point, as a, etc., in the path, it is <ie;fiected from the tangent by a force acting 
toward the center, c, of tlie circle. This force may be the tension of a string, ca, 
or the attraction between a planet at e and its moon a^ or the inward pressure 
of the rails, ah,OB & curve, etc., etc. Like all force, it is an action between two 
bodies, tending either to separate them or to draw them closer together, and act- 
ing equally upon both. (See Art. 5 (6), p. 882). In the case of the string, itpnlU 
the body a, Un/xird the center, r, and the nail or hand, etc., at c, toward the body 
at a or 6, etc. ; i. e.^from the center. In the case of a oar on a curve it pushes the 
car toward the center, and the rails from the center. The pull or push on the 
revolving body toward the center is-called the eentripetttl forc«; while the 
pull or push tending to move the defecting body from the center is called the 
cenArira^al force. These two *^ forces," being merely the two ** sides " ^as 
it were) of the same stress, are necessarily equal and opposite, and can onlv exist 
toffether. The moment the stress or tension exceeds ihe strength (or inherent 
conesive force) of the string, etc., the latter breaks. The centripetal and centrif- 
ugal forces therefore instantly cease ; and the body, no longer disturbed by a 
deflecting force, moves on, at a uniform velocity,* in a tangent, at or M', etc., to 
its circular path*; «. «., at right angles to the direction whloh the centrifugal force 
had at the moment it ceased. 



4 



(a). A singrle revoliringr body, a, Fig. 1. Let 
= the centrifugal or centripetal force, in pounds. 



= the weight of the body a, in pounds, 
= the radius ca of the path of the center of arc 
V = the uniform velocity of the body a in ftt circular path dbd,'iu feet per 



= the radius ca of the path of the center ofgraviiy of the body a, in feet. 



second, 

ft — the number of revolutions per minute, 
^ a the acceleration of gravity = say 32.2 feet per second per second, 

900 ^ = about 28980. 
«■ = circumference -^ diameter » say 3.1416. ir* « about 9.869ft. 

Then, for the centriAigal force, /: 

If we have the velocity v in feet per second : / = W ^ t • • • (1) 

If we have the number n of revolutions per minute : / = W ' t • • • (2) 

9U0 g 

/» about .0008406 WB»* 2 ... (8). 



* Neglecting friction, gravity, the resistance of the air, etc. 

t For let a/. Fig. 1, represent the amount and direction of the velocity • of the body 
at a in feet per second. Then at the end of one second the body will have reached 
the point b (the arc ab being made = a/), and the amount and direction of its 
velocity at b will then be represented by the line bt' = a< in lengtli, but differing; in 
direction. Drawing cu and cm' at the center, equal and parallel respectively t<i at 
and bt'y we find that the change in the direction of the motion (».«., the acceleration 
towaid the center) during the second is represented by the arc mm' ; and, since angle 
aeb = angle ttcu', we have the proportion, radius H or m : ab or at :: cu or at: arc 
Mt\ In other words, the acceleration tnt' in one second, or rate of acceleration, is ^ 

aC V* 

■o '^ ^* ^"^> ^^^ ^^® f<°^^ causing that acceleration, we have 

/ = mass of body X rate of acceleration =» mass of body X v "■ ^ ^S~' 
JByformula(l),/ = W . But»=— — — :andv* 



^g 60 * 3600 900 ' 

It X „, »r*R'n« _.ir«Rn» 

$ Formula (3) is obtained from (2) by substituting the values 9.8696 and 2898U for 
v" and 900 g respectively. 



CENTRIFUGAL POftCE. 



355 



(b) Wbecls and dines. Suppose the rim of a wheel to be cut into verj short 
dices, as shown (much exaggerated) at a, Fig. 2. Then for each slice, as a, by 

formula (1): /= weight W of slice X ^ ;* and if each slice were connected 




ti m o 



with tb« eenter by a separate string, the mini of the titresses in all the strings 
(taeglectlhg friction between adjacent slices) would be: 



F — sum of centrifugal forces of all the slices f = weight of riin X 



'Big' 



(4). 



But the stress with which we ure usually concerned in such cases (viz. : the 
tension f n tbe rim Itself in the direction of a taiugent to its own cir- 
cumference) is much Uss than the theoretical quantity F obtained from formula 

(4), being in fact only T^j^n ^^ ^^* ^^^ suppose first that the same thin rim is 

cut only at two opposite points m and n, Fig. 3, and that its two halves are held 
together only by toe string S. 



* If the rim is very thin in proportion to itB diameter mn^ we may take the center 

•f gravity of each slice as bein^ io a circle mn midway between the inner and outer 

M * A*. _» ^ Ai..^-.. inner radius 4- outer radius - - - . ^, 

edges of the rim, so thAt K = ^ . In a rim of appreciable 

thidnMSS, this is not the case, because each slice is a little thicker at its outer than at 
its inner end. See Fig. 6. Hence its center of grHvity is a little outside of the curved 
line AMI, Fig. 2. 

t In a perfectly balanced rim (». «., a rim whose center of gravity coincides with its 
eenter of rotation, as in Fif;. 3) the centrifugal forces of tbe particles on one side of c 
counterbalance those on the opposite side. Here, too, K = 0. Hence, as a lehole^ 
nich a rim hss lu> centrifugal force ; i. «., no tendency to leave the center in any one 
Abrection by rirtue of its rotation. But if the two centers do not coincide (Fig. 4), 
then the rim is a single revolving body, and its centrifugal force is : / = weight 

•f entire rim X ^~ ; where B is the distance between the two centers, and v the 

&g 
velocity of the center of grarity a. The force / acts in the line Joining the two 
wnters. 



356 CENTRIFUGAL FORCE. 

Then : • 

F 
semi-circumference mzn : diameter tnn : '• 'tt ' pull on the string S ; 

80 that 

pull on half weight ^ i^ ^ _2 weight t>* F F 

■trtny S "^ of rim '^ R^ ^ ir "" of rim '^ R ^ir"" ir"" 3.1416 ' ' * ^ '' 

and if the rim is now made complete bv joining the ends at m and n, and if the 
string S is removed, then the pull on the string by formula (5) will be equallv 
iLivided between m and n. Hence each cross-section, as m orn, of the rim, will 
sustain a tensile stress equal to half the pull on the string; or 

«^».«^« *- «««. il ?- weight of rim Xt>' .-. 

tension m rim »- - ^28^ ^ 6.2882 Ry ' <®>' 

The centripetal force,/, Fig. 2, holding any part o of the rim to its circular path, 
is the resultant of the two equal tensions at the ends of that part. 
For the streu per square inch of cross-section of rim, we have : 

_ ten sion in rim 

~ area A of cross-eection of rim, in square inches 

F _ weight of rim X v* ,-. 

""6.2832A~ 6.2882 A R^ ^ '* 

We shall arrive at the same result if we reflect that the pull in the string S 
or the sum of the two tensions at m and n, is equal to the centrifugal force /of 
either half of the rim, revolving, as a sinsle body, about the center e. Find the 
center of gravity G of the half rim, and then, in formula (1), use the velocity of 
that point, and the radius cG instead of velocity at g and radius cz reepectively ; 
thus: 

«.,ii *« »*^«« — / ^ centrifugal force_ „^i„ut «<• u„i/ ^„ >• (velocity at G)« . 
pull in string == / = ^^ haff-rim = '^^'^^^ ^f half-rim X ^^^ ; 

and half of this Is the tension in each cross-section of the rim.t 

If the rim were Infinitely thin, cG, Fig. 3, would be 0.6366 ex. 
If Its thickness must be taken Into consideration, and If it is of rectangular 
crofls-section, find the centers of gravity g and jK, Fig. 6, of the whole semiolrcular 
segment cz and of the small segment c6 respectively (eg *» 0.4244 oe, and eg' = 
0.4244 eb. Then 

. , area of entir e segment cz 

g'^ = gg'X area~of halTrim * 

For rims of other than rectangular cross-secticHi, use formulae (4), (5) and r6). 

In a disc, sncli as a irrlndstone, the tension In each full cross-section 
mn. Fig. 7, is equal to the centrifugal force / of ha^ the disc. Let W » weight 
of half disc. The distance cG from the center c to the center of gravity G of 
the half disc, Is cG = 0.4244 cz ; and the 

* In Fig. 2, let the centrifugal force of any slice, o, be represented by the diagonal, 
/, of a rectangle, whose sides. H and Y, are respectively parallel and perpendicular 
to the given diameter mn. Tnen H and V represent the components of / in those 
two directions. The equal and opposite horizontal components H, of o and of th<* 
corresponding slice o', being parallel to mn, have no tendency to pull the rim ^^art at 
m or n. Hence, the pull on a string S, Fig. 3, perpendicular to mn, is the sum of the 
components Y of all the slices. For each very thin slice. Fig. 6 (greatly exaggerated) 
we have (since angle A = angle A') : 

Length I . its horizontal . . centrifugal force , its vertical 
of slice • projection, p ' ' /, of slice * component V. 

Hence, for the entire half-rim mn^ Fig. 3 (made up of such slices), we have: 

«rh.lf.rim • prelection «, ' ' sT^or^. llfiim' ' f*^ J,,^'. •>' 

which is identical with the proportion at top of page. 

t The rim* of revolving wheels are usually made strong enough to resist the tension 
due to the centrifugal force, without aid from the apohe$^ which thus have merely to 
support the weight of the wheel. But if the rim breaks, the centrifugal forces of its 
fragments come entirely upon the spokes; and, since the breakage is always irregn- 
lar, some of the spokes will always receive more than their share. 



CENTRIFUGAL FORCE. 



357 



rad. cGXfl' 0.4244 czx^ 

(8). 

(»). 



= W 

= w 



0.4244 (vel. at g)« 

czXg 
0.4244 g« n» cz 

900^ 



The stress per square inch in any full section mn is 

tension in mn 



unit stress => 



area of cross-section in square inches 
0.4244 (velocity at g)' 



= W 



-W 



diam. mm, ins. X thickness, ins. XczXff 

0.4244 ir« w« cz 

diam. mn, ins. X thickness, ins. X 900 ^ 



. .(10)1 
. . (11). 




Fig. 5 



n m c n 
Fig. 7 



f= the centripetal force, in pounds, acting uvon a single revolving body, a, 
Figs. 1, 2, 4 and 5, or upon the halt-rim or half-disc, Figs. 8, 6 and 7 
= the centrifugal force exerted by such body.' 

TP = the sura of the centrifugal forces f, of all the particles of a rim, Fig. 3. 

W = the weight of the body, in pounds. 

R = the radius c<iy Figs. 1, 4 aud 5, of the path of the center of gravity of the 
body. 

V = the uniform velocity of the body in its circular path, in feet per second. 

n = tlie number of revolutions per minute. 

g = the acceleration of gravity = say 32.2 feet per second. 900 g = about 
28980. 

oircumfereuce 



w = 



= say 3.1416. ir* = about 9.8696. 



diameter 

In m rolling wlieel, each point in the rim, during the moment when it 
touches the ground, is stationary tpith respect to the earth; but each particle has 
the same velocity abont the center as if the latter were stationary, and hence the 
•entrifugal force has no effect upon the weight. 



368 8T4TICS. 



STATICS. 

FORCES. 

!• Statics Defined. The science of 3tatics, or of equilibrium of forces; 
takes account of those very numerous cases where the forces under con- 
sideration are in equilibrium, or balanced. It embraces, therefore, all cases 
of bodies which are said to be "at rest."* 

2. In the problems usually presented in civil engineering, a certain 
given force, or certain given forces, applied to a stationary* body (as a bridge 
or building) tend to produce motion, either in the structure as a whole or in 
one or more of its members; and it is required to find and to apply another 
force or other forces which will balance the tendency to motion, and thus 
permit the structure and its members to remain at rest. See If 33, below. 

3. Equilibrium* Suppose a body to be acted upon by certain forces. 
Then those forces are said to be in equilibrium, when, as a whole, they pro- 
duce no change in the body's state of rest or of motion, either as regards its 
motion as a whole along any particular line (motion of translation), or as 
regards its rotation about any point, either within or without the body. 
In such cases the body also is said to be in equilibrium. See % 84, below. 

4* A body may be in equilibrium as regards the forces imder consideration, 
even -though not in equilibrium as regards other forces. Thus, a. stone, held 
between the thumb and finger, is in equilibrium as regards their two equal 
pressures, even though it may be lifted upward by the excess of the muscular 
force of the arm over the attraction between the earth and the stone. Simi- 
larly, on a level railroad, a car is in equilibrium as regards gravity and the 
upward resistance of the rails, although the horizontal pull of the locomotive 
may exceed the resistance to traction. 

5. molecular Action. Any force, applied to a body, is in fact made 
up of a system of forces, often parallel or nearly so, applied to the several 
particles of the body. Thus, the attraction exerted by the earth upon a 
grain of sand or upon the moon is, strictly speaking, a cluster of nearly par- 
allel forces exerted upon the several particles of those bodies ; but, for con- 
venience, and so far only as concerns their tendency to move the body as a 
whole, we conceive of such forces as replaced by a single force, equal to 
their sum and acting in one line. In thus considering the forces, we as- 
^me that the bodies are absolutely rigid, so that each of them acts as a 
angle " material particle" or " material point." 

6. Transmission of Force. The upward pressure of the ground, upon 
a stone resting upon it, acts directly only upon those particles which are 
nearest to the ground. These, in turn, exert a (practically) equal upward 
force upon those immediately above them, and so on; and the i<^rce is thus 
transmitted throughout the stone. 

7. Rigid Bodies. In treating of bodies as rigid, we assume that the 
intermolecular forces hold the several particles absolutely in their original 
relative positions. 

It is not the material that resists being broken, but the forces which hold its 
particles in their places. Thus, a cake of ice may sustain a great pressure; 
but its particles yield readily when its cohesive forces are destroyed by a 
melting temperature. 

8. Force Units. The force units generally used in statics are those of 
weight, as the pound and the kilogram. See Conversion Tables, p. 235. 

In statics we have no occasion to consider the masses of bodies (except 



* Strictly speaking, absolute rest is scarcely conceivable, since all bodies 
are actually in motion (see Art. 3, p. 331). so that unbalanced forces produce 
merely changes in the states of motion oi bodies. Yet, for a body to be at 
rest, relative to other bodies, is a very common condition, and, in practical 
statics, we usually regard the body under consideration as being at rest 
relatively to the earth or to some other large body, so that the oaange of 
state of motion, due to the action of unbalanced force upon it, consists in a 
change from relative rest to relative motion. See % 33, below. 



FORCES. 359 

in so far as these determine their weights, or the force of gravity exerted upon 
them), bodies being regarded merely as the media upon and through which 
the forces under consideration are exerted. Hence we require, in statics, 
no units of mass; and, as the bodies are regarded as being "at rest," no upits 
of time, velocity, acceleration, momentum, or energy. 

0. Forces, how Petermlned. A force is fully determined when we 
know (1) its amount (as in pounds, or in some other weight unit), (2) its 
direction, (3) its sense (see % 10), and (4) its position or its point of applica- 
tion. 

10. When a force is represented by a line, the length of the line 
mav be made to represent by scale the amount of the force, and its direction 
and position may often be made to indicate those of the force, while the sense 
of the force may be shown by arrows or letters affixed to the lines, or by the 
signs, + and — . 

Thus, the directuma of the forces represented by lines a and 6, Fig. 1, are 
vertical, and those of e and d are horizontal. The sense of a is upward, of b 
downward, of c right-handed, of d left-handed. Thus, a and b are of like 
direction, but of opposite sense; and so with c and d. In treating of vertical 
or horieontal forces, we usually call upward or right-handed forces posi- 
tive, and downward or left-handed forces nef^rative, as indicated by the 
signs^ 4- and — , in Fig. 1.^ When a force is designated by two letters, at- 
tached to the line representing it, one at each end of the line, the sense of the 
force may be indicated by the order in which the letters are taken. Thus, in 
Fig. 1, having regard to the directions of the arrows, we have forces, ef, ha* 
k Cand n m, 

11. Hfine of Action, etc. The point (see ^ 6) at which a force P, Fig. 2, 
is supposed to be applied, as a, is called its point -of application. But 
the force is transmitted, by the particles, throughout the body (see ^ 6), and 



:t:i 



k I *n n 

— y ■< — 





g 

ri». 1. 

the e€fect of the force, as regards the body as a whole, is not changed if it 
be re^rded as acting at any other ix>int, as 6, in its line of action. We 
may therefore regard any point in that line as a point of application of the 
force. For instance, the tendency to move the stone, Fig. 2, as a whole, will 
not bo changed if, instead of pushing it, at a, we apply a puU (in the same 
direction and in the same sense) at b; and if a weight, P, be laid upon the 
top of the hook, at b. Fig. 3, it will have the same tendency, to move the 
hook as a whole, as it has when suspended from the hook as in the Fig. 

A force cannot actually be applied to a body at a point outside of the sub- 
stance of the body, as between the upper and lower portions of the hook in 
Fig. 3, yet this portion also of the line a 6 is a part of the line of action of the 
force. The vertical force, exerted by the weight, P, is transmitted to b by 
means c^ bending moments in the bent portion of the hook. 

12. Stress. (See Art. 1, Strength of Materials, p. 454.) Opposing 
forces, applied to a body by contact (see Art. 5 c, p. 332), cause stress, or the 
exertion of intermolecular force, within it, or between its particles, tending 
to pull them apart (tension) or to press them closer together (compression). 
The stress, due to two equal opposing forces, is equal to one of them. 

Tension and Compression. Ties, Struts, etc. If the action of 
the forces tends to pull farther apart the particles of the body upon which 
they act, the stress is called a tension or pull, or a tensile stress. If it 
tends to press them claser together, the stress is called a pressure, com- 
tvession or push, or a compressive stress. A long slender piece sustaining 
tension is called a tie. One sustaining compression is called a strut or 
|X)8t. One capable of sustaining either tension or compression is called a 
tie-strut or strut-tie. 



360 STATICS. 

MOMENTS. 

13. Moments. If, from any point, o, or </, Fig. 4, a line, o c or o' «, be 
drawn normally to the line of action, n m, of a force. Pi, whether the point, o 
or o\ be within or outside of the body upon which the force, Pi, is acting, said 
line, ocot </ «, is called the arm or leverage of the force about such point; 
and if the amount of the force, in lbs., eto., be multiplied by the length of the 
arm, in ft., etc., then the product, in ft.-lbs., etc., is called the moment of 
the force about that point.* The moment represents the total tendency of 
the force to produce rotation about the given point. A force has evidently 
no moment about any point in its line of action. 

14. Sense of Moments. Since the moment of Pi about o. Fig. 4, 
tends to cause rotation (about that point) in the direction of the motion of 
the hands of a clock, as we look at the clock and at the figure, or from left to 
right, as indicated by the arrow on the circle around o, it is called a clock- 
wise or right-hand moment ; but the moment of the same force about </ 
tends to produce rotation from right to left. Hence it is called a counter- 
clockwise or left-hand moment, as is also that of P« about o. Right- 
hand or clockwise moments are conventionally considered as positive, 
or +t and left-hand or counter-clockwise moments as negativet or — ; 

15. The pl£ine of a moment is that plane in which lie both the line 
of action and the arm of the force. 

16. The resultant or combined tendencv of two or more moments in 
the same plane is equal to the algebraic sum ox the several moments. Thus, 
Fig. 4, if the forces, Pi, P2, and Pa, are respectively 6, 5, and 3 lbs., and if 
the arms, oc, oy, and o 0, of their moments about o are respectively 7* 6, and 
8 ft., we have 

Pi . c — Pi .0 y 4- Ps . o « 
-6X7—6X6 + 3X3 
- 42 — 30 + 9 =21 ft.-lbs. 



^Zy'm. 





I K — n — • 



©i^^o 



5*— IF— ^ 
k- — f ^ 



FIgr. S. Figr. 6. 

17. If the algebraic sum of the moments is zero, they are in equilibrium 
and tend to cause no rotation of the body about the given point. 

Thus, in Fig. 6, where W is the weight, and G the center of ^pavity of the 

body, and R the upward reaction of the left support, a, taking moments 

about the right support, b, we have R / — W a; — zero ; or R i — W «. Hence, 

W X 
having W, x and Z, to find R, we have R — - .- . 

Similarly, in Fig. 6, where W — weight of beam alone, and g^ the center of 
gravity of W, is at the center of the span /, so that the leverage b g of the 

weight of the beam about h, is -■ - -, we take moments about &, thus: 

R Z i- O o — W- - — Mm — N n — zero; or 

Mm + Nn + W — — Oo 

R- 2 . 

I 



'*'Note that a very small force may have a great moment about a point, 
while a much greater force, passing nearer to the same point, may have a 
smaller moment about it ; or, passing through the jwint, no moment at all. 



MOMENTS. 



861 



In Fig. 7, where W is the weight of the beam itself, and w its leverage, tak- 
ing moments about b, we have 

+ RZ + O0 — Nn — Ww-|-Mm = 0; 

Wi£> + Nn — Mm — Oo 



Hence, 



Reaction at a 



R 



I 



In any case, if W be the combined weight and G the common center of 
gravity, of the beam and its several loads, and x the horizontal distance of 
that center from the right support, h\ and if I be the span, R the reaction of 
the left support, a, and R' that of the right support, 6, we have 



R - 



Wx 



I W 

Ifx-|-, Ria-^ -R'. 



I 



and R' - W — R. 




Flff. 7. 

Note that the moments^ of two or more forces, about a given point, 
may be in equilibrium, while the forces themselves are not in equilibrium. 
See 1 84, below. 

18. Center of Moments* So far as concerns equilibrium of moments, 
it is immaterial what point is selected as a center of moments ; but it is gen- 
erally convenient to take the .center of moments in the line of action of 
one (or more, if there be concurrent forces, see ^ 19) of the unknown forces, 
for we thus eliminate that force or those forces from the equation. 



CLASSIFICATION OF FORCES. 

19. Classification of Forces, 
and Parallel Forces. 



Concurrent, Colin ear, Coplanar. 

Forces are called concurrent when their lines of 





Figr* s. 



Figr. 9. 



action meet at one point, as a, b, c, d, e and /, or / and g. Fig. 8 ; non-concur- 
rent when they do not so meet, as c and g; colinear when their lines of action 
coincide, as a and b. or c and d; non-colinear when they do not coincide, as 
b and /; coplanar when their lines of action lie in one plane,* as a, b, c, d and 
c, or b, f and (7, etc. ; non-coplanar, as c and g, or 6, / and d, when they do not 
he in one plane; parallel wnen their lines of action are parallel, as and g\ 
non-parallel when those lines are not parallel, as b and /. 

*Acting wpon a plane, as in Fig. 9, must not be confounded with acting in 
that plane, as in Figs. 70, etc. 



862 



STATICS. 



Any two parallel forces must be coplanar. Three or more parallel forces 
may or may not be coplanar. Any two concurrent forces must be coplanar. 
Three or more concurrent forces mav or may not be coplanar. Any two 
ooplanar forces must be either parallel or concurrent. 

COMPOSITION AND BESOLUTION OF FORCES. 

SO. Kesultant. A single force, which can produce, upon a body con« 
sidered as a whole, the same effect as two or more given forces combined, is 
called the resultant of those forces. Thus, in Fig. 10 (b), a downward pres- 
sure, G, ■= to + W, is the resultant of the downward pressures w and W; 
and, in Fig. 11 (6), a downward pressure, =■ W — tr, is the resultant of the 
downward pressure W and the upward pull w of the leit-hand string.* 

31. Component.- Any two or more forces which, together, produce, 
upon a body considered as a whole, the same effect as one given force, are 
called the components of that force, which thus' becomes their resultant. 
Thus, in Fig. 10 (6), w and W are the components of the total force, G, = 
«; + W. In Fig. 1 1 (6), + W ( = 5) and m) ( - — 3) are the components of G.* 

22. If we take into account the resultant of any given forces, those forces 
(components) themselves must of course be left out of account, as regards 
their action upon the body as a whole; although we may still have to con- 
eider their effect upon its particles. Vice versa, if the forces (components) 
are considered, their resultant must be neglected. 




Fflff. 10. 




6 



(C) 



3 



S^ 



» 



FI9. 11. 



23. Anti-resultant. The anti-resultant of one or more forces is a sinsle 
force which, acting upon any body or system of bodies considered as a wh(ue, 
produces an effect eoual, but opposite, to that of their resultant. In other 
words, the anti-resultant is the force reouired to hold the given force or 
forces in equilibrium. Thus, in Fig. 10 (o), the upward reaction, G, of the 
sround, is the anti-resultant of the two downward forces, w and W ; and the 
downward resultant, W 4- to, of W and to, is the anti-resultant of G. In 
Fig. 11 (6), G (upward) is the anti-resultant of W (downward) and to (acting 
upward through the left-hand string). Similarly, this upward pull of tff is 
the anti-resultant of W and G. 

24. In any system of balanced forces (forces in equilibrium), any one of 
the forces is the anti-resultant of all the rest ; and any two or more of them 
have, for their resultaht, the anti-resultant of all the rest. In such a system, 
the resultant (and the anti-resultant) of all the (balanced) forces is zero. 

25. Anti-component. The anti-components of a given force, or of a 
given system of forces, are any two or more forces whose resultant is the anti- 
resultant of the given force or of the given system of forces. 

26. Composition and Resolution of Forces. The operation of 
finding the resultant of any given system of forces is called the composition of 
forces; while that of finding any desired components of a given force is called 
the resolution of the force. 



♦ For convenience, we here reverse the convention of H 10. 



COLINEAR FORCES. 



363 



Colinear Forces. 

27* Let the vertical line, w. Fig. 10 (6), represent, by any oonyenient 
scale, the weight of the upper stone in Fi^. 10 (a), and W that of the lower 
stone. Then, w + W, ■". G, ~ the combined length of the two lines, gives, 
by the same scale, the combined weight of the two stones, and a verticu line 
G, coincident with them, equal to tneir sum, and pointing upward, would 
represent their anti-resultuit, or the reaotioii of the ground. 



(a) 






\ 


\ 


V 


/ 


/ 




a 


\ 


\ 


A 


/ 


/ 


6 . 


W^ 












w* 



lb) 



(C) 



z=io< 



JUve to to to to to to 
Tauat» t9 t9 t» t9 t» ^ 



r^) 






^ J 



::i 



B'-'Sei 



1>« 

>t» 



94 



\ J 



Fi«. 13. 



!88. Similarly, if, at each panel point of the lower chord in the bridgo 
truss in Fig. 12 (a), we have 2 tons dead load (weight of bridge and floor, 
etc.*) axul 10 tons live load (train, vehicles, cattle, passengers, etc.), the com* 
bined length of the two lines in Fig. 12 (b), L - 10, and D - 2, gives the tota*. 
panel load of 12 tons. 

29. In Fig. 11 the prenure, 5 lbs., of W upon the ground, is diminished by 
the 3 lbs. upward pull of the cord, transmitted from the smaller weight i9, 
leaving 2 lbs. upwara pressure to be exerted bv the ground in order to main- 
tun equilibrium. The upward reaction, R, of the pulley is — w + W — G 
-■8 + 6 — 2 -* 6. This is represented graphically in Fig. 11 (c). 

30. In the truss shown in Fig. 12 (a), the total dead and live load is — 6 
X 12—72 tons, and half this total load, or 36 tons, rests upon each abut- 
ment. Hence, to preserve equilibrium, each abutment must exert an up- 
ward reaction of 36 tons; but, in order to ascertain how much of these 36 
tons is iranamiUed through the end-pott, a e, we must deduct from it the 12 
tons which we assume to be originally concentrated, as dead and live load* 
at the panel jpoint a; for this portion is evidently not transmitted through 
a e. Accordingly, in Fig. 12 (c), we draw R upward, and equal by scale to 
36 tons: and, from its upper end, draw p downward and — 12 tons. The 
remainder of R, — R — p -• 36 ^ 12 — 24 tons, is then the pressure trans- 
mitted through a e. 

31* Golinear forces are called similar when they are of like sense, and 
opposite when of opposite sense. The tame distinction applies to result* 
ants. 



b 

h 



a 



■^— f 



o c 

Figr. 13. 



d 



• 



33* For equilibrium, under the action of colinear forces, itia, 
ci oo^irse, necessary that toe sum of the forces acting in one sense be equal to 
the sum of those acting in the oppomte sense, or, in other words, that the 
algebraic sum of ail the forces be zero. Thus, in Fig. 13, if the forces are in 
equilibrium, the sum, b a ■{• a o, ot the two right-handed forces must be 
equal to the sum, ed + dc + co, of the three left-handed forces. Or, con- 
sidering the right-handed forces, b a and a o, as positive, and the left-handed 
forces, e dj d c and c o, as negative, as in ^ 10, we have, as the condition of 
equilibrium of colinear forces : 

ba •{- ao — oc—'cd — de — O. 



*The dead load is, of course, never actually concentrated upon one chord, 
as here indicated ; but It is often assumed, for convenience, that it is so 
concentrated. 



364 STATICS. 

In other words, the algebraic sum of all the forces must be zero; or, more 
briefly, 

2 forces — 0, 

where the Greek letter S (sigma), or sign of summation, is to be read "The 
sum of — ." 

33* Two equal and opposite forces, acting upon a body, are com- 
monly said to keep it at rest ; but, strictly speaking, they merely prevent each 
other from moving the body, and thus permit it to remain at rest, so far as 
they are concerned ; for they cannot keep it at rest against the action of any 
third force, however slight and in whatever direction it may act; and the 
body itself has no tendency to move. 

34. Unequal Opposite Forces. If two opposite forces, acting upon 
a body, are unequal, the smaller one, and an equal portion of the greater 
one, act against each other, producing no effect Uf^n the body as a whole; 
while the remainder, the resultant, moves the body in its own direction. 



Concurrent Coplanar Forces. The Force Parallelogram. 

35. Composition. Let the two lines, ao,bo, in any of the diagrams of 
Fig. 14, represent, in magnitude, direction and sense, concurrent forces 
whose lines of action meet at the point o. Then, in the parallelogram, acbo, 
formed upon the lines a o^ b o, the resultant of those two forces is repre- 
sented, in magnitude and in direction, by that diagonal, R, which passes 
through the point, o, ci concurrence. The parallelogram, a c & o, is called a 
force parallelogram. 



a' (a) 

o 





"^^^^V* 




ligr. 14. 

36. Resolution. Conversely, to find the components of a given force, 
o c, Fig. 14, when it is resolved in any two ^ven directions, o a, o 6, draw the 
lines, o a\ o b\ in those directions and of mdefinite length, and upon these 
lines, with the diagonal R » o c, construct the force parallelogram a ch o» 
The sides, o a, ob, of the parallelogram then represent the required compo- 
nents in amotmt and in direction. ^ 

37. Caution. The two forces, a o and b o. Fig. 14, may act either toward 
or from the point o; or, in other words^ they may act either as pulls or as 
pushes ; but the lines representing them m the parallelogram, and meeting at 
the point, o, must be drawn, either both as pushes or both as pulls; and the 
resultant, R, as represented by the diagonal of the pandlelogram, will be a 
pull or a push, according as the two forces are represented as pulls or as 
pushes. 

38. Thus, in Fig. 15 (^a), the inclined end-post of the truss pushes obliouely 
downward toward o, with a force represented by a' o, while the lower chord 
pulls away from o, toward the ri^ht, with a force represented by o V, If, 
now, we were to construct, in Fig. 15 (a), the parallelogram o a' cf V^ we 
should obtain the diagonal o cf or c' o, which does not represent the true re- 
sultant. In fact, as one of the two forces acts toward, and the other from, 
the point, o, we could not tell (even if R' were the direction of the resultant) 
in which sense its arrow should point. 

We must first either suppose the push, a' o, in the end-post, toward o,^ to be 
carried on beyond o, so as to act as a pull, o a. Fig. 16 (o) (of course, in the 
same direction and sense as before), thus treating both forces as pulls; or 



FOBCE PABALIiELOOKAH. 



366 



rise we must similarly suppose the pull, o V, in the chord, to be transformed 
into the push, 6 o, of Fig. 15 (c), thus treating both forces as pushes. In 
either case we obtain the true resultant, R ( » a' 5', Fig. 15a), which, in this 
ease, represents the vertical downward pressure of the end of the truss upon 
the abutment. 




FtfT- IS- 

Caution. The tensile force, exerted at the end of a flexible tie, neces- 
sarily acts in the line of the tie; but, in general, the pressure, exerted at 
the end of a strut, acts in the line of the axis of the strut only when all 
the forces producing it are applied at the other end of the strut. Thus, 
in Fig. 15 id), the components, R and H, of the weight, W, do not coin- 
cide with the axis of the beam which supports the Toad; but in Fig. 15 
(e), where the weight acts at the intersection of the two struts, its com- 
ponents, R and H, do coincide with the axes of the struts. See idso Figs. 
143 and 145 (b). 

39. Demonstration. The rational demonstration of the principle of 
the force parallelogram is given in treatises on Mechanics. (See Bioliog- 
raphy.) It may be established experimentally as indicated in Fig. 16, 
where c o represents by scale the pull shown by the spring balance C, while 
o a and o h represent those shown by A and B respectively. 




40.- Equations for Components and Resultant. Given the 
amounts of the forces, a and c, or of the resultant, R, and the angles formed 
between them. Fig. 17 (a), we have'*': 



♦ See dotted lines, Fig. 17 (a), noting that c* ^ c; c. sin (x + i/) -» R. sin 
X, and a. sin (x + y) >« K. sin y. 



366 



STATICS. 



rt = c 



sin (x 4- y) 



R^ 



Bin X 
sin X 



— a 



si n (x + y) 
sin y 



„ = R »>» » 



sin (x + y)* 



sin (X -i- 1/)* , . ^, 

If the angle between the two forces is 90**, Fig. 17 (b), these formulas be< 
me: 



come: 



cos y cos X 
c — R cos y; a =» R cos x. 





FiiT- 17. 

41. Position and Sense of Resultant. Figs. 18. If the lines 
representing the components be drawn in accordance with Iff 37 and 38, 
and if a straight line, m n or m' n', be drawn through the point, o. of concur- 
rence, in such a way that both forces are on one side of that line, then the line 
representing the resultant will be found upon the same side of that line with 
the components, and between them ; and it will act toward the line, m n or 
tn* n'r ii the components act toward it, and vice versa. The resultant is 
necessarily in the same plane with its two components. 



tm 




/^^MiosS^ 



nC^T^^^' 



^r 

-s/' 



Fiir. 18. 



Fis. 19. 



42* If one of the components is colinear with the force, it is the force itself, 
and the other component is zero. In other words, a force cannot be resolved 
into two non-colinear components, one of which is^n the line of action of the 
force. Thus the rope, o e. Fig. 19, may receive assistance from tu}o ad- 
ditional ropes, pulling in the directions a c, and c b; for the resultant of their 
pulls may coincide with o c; but, so long as o c remains vertical, no aingU 
force, as c a or c b, can relieve it, imless acting in its own direction c o. 

43. In Fig. 20, the load, P, placed at C, ia suspended entirely by the verti- 
cal member B C, and exerts directly no pull along the horizontal member, 
C £. Neither does a puU in the latter exert any eneot upon the force acting 
in B C, so long as B C remains vertical. But the tension in B C, acting 
at B, does exert a thrust o a along B D, although that member is at right 
angles to B C; for B G meets there also the inclined member A B; and 
the tension o d \3 thus resolved into o a and o 6, along B D and B A 
respectively. The horizontal thrust, o a, in B D, is really the anti-resultant 
of the horizontal comp>onent, db, of the oblique thrust in the end-poet B A, 
at its head, B, which thrust is — the pull in A £, due to P. 



FORCE TRIAKGLE. 



367 



44. In Fig. 21, the tension, o e, in the inclined tie, D G, is resolved, at D, 
into o a and o b, acting at right angles to each other along D F and D £ re- 
spectively. 

45. A resultant may be either greater or less than either one of its two 
oblique components, but it is always less than their sum. If the components 
are equal, and if the angle between them » 120^, the resultant is eaueii to one 
of them. Therefore the same weight which would break a single vertical 
rope or post, would break two such ropes or posts, each inclined 60° to the 
vertical. 





Fly. 91. 



The Force Triangle. 

46. The Force Triangle. Inasmuch as the two triangles, into which a 
paralldogram is divided by its diagonal, are similar and equal. It is suffi- 
cient to cu^w either one of these triangles, aoc or h oc. Figs. 14, 16, 18, in- 
stead of the entire parallelogram. 

47. If three concurrent coplanar forces are in equilibrium, the lines repH 
resenting them form a triangle; and the arrows, indicating their senses, 
foUow each other around the triangle. Thus, in Fi^. 22 (a), we have, acting 
at o and balancing each other there, three forces: vu., (1) the vertical down- 
ward force o c of the weight, acting as a pull through the rope o c, (2) the 
horizontal thrust a o through the oeam a o, and (3) the upward inclined 
thrust 6 of the strut o b, all acting in the senses (o c,ao,b o) in which the 
letters are taken, and as indicated by the arrows. 

48. Each of the forces in Fig. 22 (&) and (c) is th6 anti-resultant of the 
other two in the same triangle ; and, if its sense be reversed, it becomes their 
resultant. Thus, o c, Fisj. 22 (b), is the anti-resultant, and c o the resultant, 
ofea and a o; and o c. Fig. 22 (c), is the anti-resultant, and c o the resultant 
of e & and bo,cb being parallel to a o. Fig (b), and representing the thruflt 
exerted by the horizontal beam against the joint o, Fig. (a).* 



ib) (c) id) (e) 




^^ • 



c e^t, 



Flff. 33. 





*Fig. 22 (tO and (e), representing the same two forces, a o, b o, of Fig. 
22 (a), show the erroneous resultant (a b) obtained if the lines are drawn 
with their arrows pointing both toward or both from the meeting-point of the 
lines. See ^1f 37, 38. A comparison of any force parallelogram, as that 
in Fig. 18, with either of the two force triangles composing it, will show 
that this, while apparently contradicting Ht 37 and 38, is merely another 
statement of the same fact. The apparent contradiction is due to the 
fact that, in the force triangle, the lines representing the forces do not 
meet at the point, o, of concurrence of the forces.' 



368 



STATICS. 



40. Converselsr, if the three sides of a trian^e be taken as representing, 
in direction and in amount, three concurrent forces whose senses are such 
that arrows, representing them and affixed to their respective sides in the 
triangle, follow each other around it, then those forces are in equilibrium. 

50. The three forcest Fig. 23, are proportional, respectively, to the 
sines of their opposite angles. Thus: 



Force a : force b : force e 
— Sin A : sin B : sin C. Fly. 2S. 



51* Example. In Fig. 24, the half arch and its spandrel, acting as a 
nngle rigid bodv, are assumed to be held in equilibrium by their combined 
weight, W, the horizontal pressure h at the crown, and the reaction R of the 
skewback, which is assumed to act through the center of the skewback. In 
the force triangle c « t, e «, acting through the center of gravity of the half 
arch and spandrel, represents the known weight W, and 8 t ia drawn hori- 
sontal, or parallel to h . From c, where h, produced, meets the line of ac- 
tion of W, draw c t through the center of the skewback. Then • t and e I 
give us the amounts of h and R respectively. 






Fig. 24. 



Figr- 9Xi, 



52. Example. Let Fig. 25 represent a roof truss, resting upon its abut- 
ments and carrying three loads, as shown by the arrows. Draw a R ver- 
tically, to represent the proportion of the loads carried by the left abut- 
ment, a, or, which is the same thing, the vertical upward reaction of that 
abutment. Then, drawing R c, parallel to the chord member, a <2, to inter- 
sect a 6 in c, we have, for the stresses in a e and a d, due to the three loads: 



Stress in a « s a e 
" od = Re 



It 




%4 ^'^bA 





63. While any two or more given forces, as o 6 and h c. Fig. 26 (a) (arrows 
reversed), or o b' and b' c,oroa and a c, or o a' and o' c, can nave but one re- 
sultant c; a sinffle force, as o c. may be resolved into two or more concur- 
rent components in any desired directions. In other words, there is an 
infinite number of possible systems of concurrent forces which have o c for 
their resultant. 



SECTANOULAB COMPONENTS. 



869 



Bectangular Gomponenti. 

54. ResoluteSt or Rectangular Components. A very common case 
of resolution of forces is that where a force, as the pressure, c n, of the post, 
fig. 27, is to be resolved into components at rieht angles to each other, as are 
the vertical and horizontal components c t and tn in Fig. 27 (a). Two such 
components, taken together, are called the resolutes or rectangular compo- 
nents of ibjb force. The joint, o d, in Fig. 27 (a), is properly placed at right 
angles to e n; but the joint c ib. Fig. 27 (5), provides also against accidental 
changes in the direction of c n. In Fig. 27 (6), the surfaces, c i and i b, are 
preferably {proportioned as the components, c i and t ih Fig. 27 (a), respec- 
tively, by simUarity of triangles, ctb, ctn^ 





Tig. 27. 




Fl«r. 28. 



55* Example. In bridge and roof trusses it is often required to find the 
vertical and horizontal resolutes of the stress in an inclined member, or to 
find the stress brought ui>on an inclined member by a given vertical or hori- 
zontal stress applied at one of its ends, in conjunction with another stress 
(whose amount may or may not be given) at right angles to it. 

Thus, in Fig. 28, the tension C p in the diagonal C d is resolved into a com- 
pression e p along the upper chord member CD* and a compression C e in the 
• post Cc.*^ Addmg to C c the load at c, and representing their sum by / c, we 
nave tension f g in chord member e d, and tension c g in the diagonal B c. 
Making B A = c g,-we have i A, compression in B C, and B j, compression in 
the end-post or batter poet B A. But the load at b also sends to B, through 
the hip vertical B 6, a load (tension) equal to itself. Representing this by 
B ;fc, we have ( A; as its component along the chord member B O, and B I as its 
oom]M)nent along the end-post B A. Now, making A *» = the sum of B; 
and B /, we find the vertical resolute A » = so much of the vertical reaction 
of the abutment as is due to the three loads only, and the horizontal resolute 
mn '^ the corresponding stress in the chord member, A c. 






'\ 




>a 



Flff. 80. 

56. Example. Inclined Plane. Again, in Fig. 29, let it be required 
to find the two resolutes of P (the weight of the ball) respectively parallel and 
perpendicular to the inclined plane. The former is the tendency of the ball 
to move down the plane, and is called the tangential component. The 



^'Ilie stress^ thus found is not necessarily the total stress in the member. 
The compression in C c (neglecting its own weight and that of the top chord) 
fe due entirely to the tension C p in C <i, acting at its top, and hence C e rep- 
tmenta the total compression in C c; but e p ia only a portion of the com- 
pression sustained by C D ; for B C also contributes its share toward this. 

24 



btl«rut 




ofthebBUaCUDBt 


h« plui«, aod ia (wlled the noimal 


compon 


enl. 












Herew 




to draw 


the triftogl 


of for. 




OEOC-Pto 

JdiiMtkma. 


S£ 


Iho weight 


of the b 


•11. Uld » 


undo 




s^/F- 




ely the do 


mid wid the taaeeatia 




87. If 




the ineli 


ed plane g 


m, Fig 


29, to be fri«t 


o»le«, and if 


the body 


018 to be prevented t 


rom sliding 


down 


'o're^J^^'g."- 


umotafone 


■.ppliedm&direotiDTi 


parsJIel 


to the plwi 
the pluu 


e. that 




thua, 


n Fig. 30, B 


ihe stoi 




be friotionlees, 


mhave a e 




™ agamsl 










SS. Table, of a 






tordiffenot 



T>Ft. H«. 

i: i' 
1: t 

Id Il.t 


\>»« 


Dl-K 



• Or * c. It both triangles are drawn, we have the foro 
trhs line a « (or c a) is called the prolecllon of o c ui 



BTBE8S CX>HFONENT8. 



371 



59. Equations. In Fig. 29. 

o a »- P . cos e o a 
a c ■" P . sin e o a 

and, since the angle eoa between the vertical o e and the normal component 
o a is equal to the angle A of inclination between the plane g m and the hori- 
sontal a n, we have : 

Normal component, o a » P . cos A. 
Tangential component, a c » P . sin A. 

60. When a force is resolved into rectangular components, as in Figs. 29 
and 30, each of these components represents the total effort or tendency which 
that force alone ean exert in that direction. 




FI9. 81. 

Thus, in Fig. 31, the utmost force which the weight o e alone can exert 
perpendicularly againat tke plane is that represented by the component o a. 
iVue, if, in order to prevent the bo<ly from sliding down the plane, we apply 
a force in some other direction, such as the horisontal one, h o, instead of the 
tangential one h o, and find the components of o c in the directions h o and o a, 
weuiall find the normal component o d greater than before; but the increase 
a d is due entirely to the normal component, h fr, of the horisontal force h o. 
Thus, the only effect upon the body o, and upon the plane, of substituting 
h o for b o, is to add the normal component, h 2>, of the former, to that (o a) 
doe. 

Stress Components. 

61. Stress Components. In Fig. 32, let a o and & o be any two forces, 
and c o their resultant. From a and 6 draw a a' and h 6' at right angles to 
the diagonal o c of the force parallelogram a o b c^ and construct the sub-> 
parallelograms (rectangles), oa' a a" and oVh If'. Each of the original com- 
ponents, o a, o- h, is thus resolved into two sub-components, perpendicular to 
each other, one of which is perpendicular also to the resultant, o c, while the 
other coincides with o c in position and in sense. Now, perpendiculars, let 
fall from the opposite angles of a paralldogram upon its diagonal, are equaL 



.// (a) 



.// 



,, (6) 





^ 




/ 


4 


< / 


f< 


"-, 


/ 




6 -X 



Flff. 32. 

Bence the two colinear forces, o a'\ and o 6", acting upon the body at o, are 
equal and opposite (although the lines, a' a and h' 6, representing them, are 
not opposite). Hence also they are in equilibrium, and their only effect 
upon the body is a stress of compression in Fig. 32 (a), and of tension 
in Fig. 32 (6). They may therefore be called the stress components. The 
other two sub-components (o a' of o a, and o 5' of o h) combine to form the 
resultant o e, which is equal to their sum, and which tends to move the body 
in its own dir&ction. 



372 



STATICS. 



62. The two great forces, o a, ob, in Fig. 33 (6) have the same reeultant, 
oc, = o c', as the two small forces, o a' o b\ in Fig. 33 (a), although their 
stress components, a" a, = V b, are much greater. 

63. It often happens that one of the components is itself normal to the 
resultant. Thus, in Fig. 22, where o c is vertical, its component, o a, is hori- 
zontal, and the perpendicular, let fall from a upon o c, represents its hori- 
zontal anti-component, a o. Here the horizontal and the inclined beam 
sustain equal horizontal pressures; but the vertical pressure, o c, "^ the 
weight, W, is borne entirely by the inclined beam. 





Flip. 33. 



Flip. 34. 



64. When, as in Fig. 34, the resultant, o c, forms, with one of the original 
components, o a and o b, an angle, aoc, greater than 90^, the perpendicularB, 
a a', b 6', from a and &, must be let fall upon the line of the resiiltant produced. 
Here, however, as before, the two equal and opposite sub-components, o a" 
and o b"j are in equilibrium at o, while the other two sub-components, o b* and 
o a', go to make up the resultant o c; which, however (since o 6' and o a* here 
act in oppottte senses) is equal to their difference, and not to their sum, as in 
Fig. 32. 

Fig. 34 shows that a dowrvward force, o e, may be so resolved that one of its 
components is an upward force, o a, greater than the original downward force, 
and that the pressure, o 6, has a component, o b* or V &, parallel to o c, and 
greater than o c itself; for b" b — o 6' '^ o c -\- cV. 

Applied and Imparted Forces. 

65. Applied and Imparted Forces* In Fig. 29, the ball is free to 
roll down the inclined plane. Hence, although the entire weight P of the 
ball is applied to the body g mn, only the normal component o a is imparted 
to it or exerts any pressure upon it, and this pressure is in tlie direction o a. 

But in Fig. 30, the body g mn ceeeives and resists not only the normal 
component o a, but also (by means of the stop «) the tanaential component 
o b; and the entire force P, or o c, is thus imparted to the body g mn, pres»- 
ing it in the direction o c. 



Comiposition and Kesolution of Concurrent Forces by Means 

of Co-ordinates. 

66. In Fig. 35 (a) let the three coplanar forces E, F and G act through 
the point x. Draw two lines, H H, and V V, Fig. 35 (b), crossing each 
other at right angles, as at o.* These lines are called rectangular co-ordin- 
ates. From o, draw lines E o, F o, G o, parallel to E a:, F x, (jrx, Fig. 35 (o), 
and equal respectively to the forces E, F, and G by any convenient scale. Re- 
solve each of these forces, Fig. 35 (6), into two components, parallel to H H 
and V V respectively. Thus, E o is resolved into t o and n o, F o into u o 
and e o, G o into i o and m o. Then, summing up the resolutes, we have: 

Sum of horizontal resolutes = u o — io — to — — so, and 

Sum of vertical resolutes == 



no + e o — m,o 



— ao, 
ao; 



*It is only for convenience that the co-ordinates are usually drawn (as in 
Fig. 35) at right angles. They may be drawn at any other angle (see Fig. 
36) ; but. in any case, the forces must of course be resolved into components 

EaraUel to the co-ordinaUa, whatever the directions of those co-ordinatee may 
e. 



COMPOSITION AND RESOLUTION. 



373 



and — 9 and a o are the resolutes of the resultant, R, of the three forces, E, 
F and G. 

67. When a system of (concurrent) forces is in equilibrium, the algebraic* 
sum of the components of all the forces, along either of the two co-ordinates, 
is zero. Thus, in Fig. 35 (6) or 36, if the sense of R be such that it shall act as 
the anti-resultant of the other three forces E, F and G, its component, o « or 
o a, along either co-ordinate, will be found to balance those of the other 
forces along the same co-ordinate. 




Flff. 35. 

Henoe we have the very important proposition that : When a system of 
ooncurrent coplanar forces is in equilibrium, the algebraic sums of their com- 
ponents, in any two directions, are each equal to zero. 




Fig:. S6. 

68. Conversely, in a system of concurrent forces, if the algebraic sums of 
the components in any two directions are each jequal to zero, the forces are 
in equilibrium. 

If the sum of the components in one of anv two directions is not equal to 
zero, the forces cannot be in equilibrium. Thus, in Fig. 35 (6) or 36 (b), the 
sum of the components, along either one (as VV) of the two co-ordinates, 
may be zero; and yet, if the sum of those along the other co-ordinate is 
not zero, their resultant, or algebraic sum, will move the body, on which 
they act, in the direction of that resultant. 



♦The components being taken as + or — , according to the sense of each. 



374 



STATICS. 



69. With Tertical and horizontal co-ordinates, the condition of 
equilibrium* becomes: 

. The sum of the horizontal resolutes must be equal to zero ; 

The sum of the vertical resolutes must be equal to sero; 

or, more briefly: 

2 horizontal resolutes ■- 

2 vertical resolutes ■» 

Conversely, if these conditions are fulfilled, the forces are in equilibrium. 






Tig. 37. 



Tig. 3S. 



Flip. 39. 



70. Resultant of More than Two Coplanar Forces. Where it 
is required to find the resultant of more than two concurrent and coplanar 
forces, as in Fig. 37, we may first find the resultant Ri of any two of them, 
as of P] and Ps; then the resultant, R^, of Ri and a third force, as Pa; and so 
on, until we finally obtain the resultant R of all the forces. This resultant is 
evidently concurrent and coplanar with the given forces. 

71. It is quite immaterial in what order the forces are taken. 

Thus, we may, as in Fig. 38, first combine Pi and Ps; then their resultant Ri 
with Ps, obtaining R2; and, finally, R^ with P4, obtaining R;or, as in Fig. 39, 
we may first combine any two of the forces, as Pi and Ps, obtaining their 
resultant Ri ; then proceed to any other two forces, as Ps and P4, and obtain 
their resultant R^; and finally combine the two resultants, Ri and R^, ob- 
taining the resultant R. 



The Force Polygon. 

73. The Force Polygon. Comparing Figs. 37 and 38 with Figs. 40 
and 41, respectively, we see that we may arrive at the same resultant R by 
simply drawing, as in Fig. 41, lines representing the several forces in any 
order, but following each other according to their senses. It will be noticed 
that this is merely an abbreviation of the process of drawing the several force 
parallelograms. 

73. Resultant and Anti-resultant. The line, — R, required to com- 
plete the polygon, represents the an<i-resultant of the other forces if its sense 
IS such that it follows them around the polygon, as in Fig. 40. If its sense is 
opposed to theirs, as in Fig. 41, it is their reavUant, R. 

74. In other words, if any number of concurrent forces, as Pj, Pj, Pj, P* 
and R, Figs. 37 and 38, f are in equilibrium, the lines representing them, if 
drawn in any order, but so that tneir senses follow each other, will form a 
closed F>olygon, as in Fig. 40 (or in Fig. 41 if the sense of R be reversed). 

75. Conversely, if the lines representing any system of concurrent 
coplanar forces, when drawn with their senses following each other, form a 
closed polygon, as in Fig. 40, those forces are in equilibrium. 



*With non-concurrent forces, another condition must be satisfied. See ^ 83. 

tR is here regarded as tending upward, so as to form the anft-resultant of 
the other forces. 



FORCE POLYGON. 



375 



It will be noticed that the force triangle, and the straight line representing 
a system of colinear forces, Figs. 10 and 11. Hlf 20, etc., or a system of 
parallel forces, Figs. 55, etc., tf 111, etc., are merely special cases of the 
force polygon. 

76. In a force polygon. Fig. 42, any one of the forces is the anti-resultant 
of all the rest. Any two or more of the forces balance all the rest ; or, their 
resultant is the anti-resultant of all the rest. 

If a line a c or 6 d, Fig. 42, be drawn, connecting any two comers of a force 




rig. 40. 





Fiff. 49. 



polygon, that line represents the resultant, or the anti-resultant (according i 
its arrow is drawn) of all the forces on either side of it. Thus : 



a c is the resultant of Pi Ps and the anti-resultant of P3 P4 Pg 
<5 a " " " Ps P4 Ps * " '* Pi Fa 

6 rf " " " Ps Ps " " " P4 P6 Pi 

d b " •* " P4 Pft Pi " " " Pi P3 

77* Knowing the directions of all the forces of a system, as Pi P5, 

Fig. 42, and the am&unta of all but two of them, as Ps and P3, we may find the 
amounts of those two by first drawing the others, P4, Ps and Pi, as in the 
figure. Then two lines b c and c d, drawn in the directions of the other two 
and dosing the polygon, will necessarily give their amounts. 





Tig. 48. 



Tig. 44. 



78. If any two points, as o and c. Fig. 43. be taken, then the force or forces 
represented by any line or system of lines joining those two points will be 
equivalent to o c. Thus :oe''oabc'^ode^onpc''ohkmc = 
on mc '^ o fc " o gc, etc., etc. 

Similarly, in Fig. 42, the force polygon abe deais equivalent to the force 
polygon ab fdea, and to the force triangle, abca, eacn being = zero. 

• 
Non-concuirent Coplanar Forces. 

79. Non-concurrent Coplanar Forces. Fig. 44. The process of 
finding the resultant of three or more coplanar but non-concurrent forces is 
the same as if they were concurrent. Thus, let Pi, Ps and Ps represent three 
sueh forces.* We may first find the resultant Ri of any two of them, as Ps 

*Apy two coplanar non-parallel forces, as P; and P2, or P^ and Ps are 
necessarily concurrent (see % 19); but there is no single pomt in which 
the three forces meet. 



376 



STATICS. 



and P3; and then, by combining Ri with the remaining force Pi, we find the 
resultant R of the three forces. Here the line R represents the resultant, not 
only in amount and in direction, but also in position. That ls, the line of 
action of the resultant coincides with R. 

80. The resultant R is the same, in amount and in direction, as if the 
forces were concurrent, and its position is the same as it would have been if 
their point of concurrence were m the line of R. If there are more than three 
forces, we proceed in the same waj'. 

81. Conversely, the resultant R, or any other force, may be resolved 
into a system of any number of concurrent or nonconcurrent coplanar forces, 
in any direction^, at pleasure. Thus, we may first resolve R into Pi and Ri; 
then either of these into two other forces, as Ri into P2 and P3, and so on. 

83. If a system of non-concurrent coplanar forces is in equilibrium, the 
forces will still be in equilibrium if they are so placed as to be concurrent; 
provided, of course, that their directions, senses and amounts remain un- 
changed ; but it does not follow that a system of forces, whicl> is in equilib- 
rium when concurrent, will remain in equilibrium when so placed as to be 
non-concurrent. 

Thus, the five forces, Pt Pr„ Fig. 45 (a), may be so placed, as in Fig. 

45 (6), that the resultant a c, of Pi and Pa, does not coincide with the re- 
sultant c a of P3, P4 and Ps. but is panUlel to it. These two resultants -then 
form a couple. (See tlf 155, etc.) 





Fig. 45. 



83. Third Condition of Equilibrium. Hence, 
equilibrium for concurrent forces, stated in \ 69, 



the oondHiona of 



2 
2 



vertical 
horizontal 



components ■■ 
components -= 



do not suffice for non-concurrent forces, and a third condition must be added, 
viz. : — 

2 moments « 0; 

t. e., the moments of the forces, taken about any point, must be in equilib- 
rium. 

A system of forces in equilibrium has no resultant ; hence it has no moment 
about any point. In other words, the moments of the forces, as well as the 
forces themselves, are in equilibrium. 

84. The resultant of a system of unbalanced non-concuireiit 
forces, acting upon a body, may be either 

(1) a single force, acting through the center of gravity of the body; or 

(2) a couple; t. e., two equal and parallel forces of opposite sense (see 
m 155, etc.) ; or 

(3) either (a) a single force, acting through the center of gravity of the 
body, and a couple ; or (b) a single forca acting elsewhera th»r throu^k ( he 
center of gravity of the body. 

^ In Case (3), the two alternative resultants are interchangeable; t. e.. a 
single force, acting elsewhere than through the center of gravity of the body, 
may always be replaced by an equivalent combination consisting of an eqijuu 



CORD POLYGON. 



377 



parallel force*, acting through the center of gravity of the body, and a couple, 
and vice versa. See HI 161', etc. 

The resultant gives to the body, in Case (1), motion of translation in a 
straight line, without rotation; in Case (2), rotation without translation; 
and m Case (3), both translation and rotation. See foot-note (*), t !• 

85. The force polygon, ^ 72, Figs. 40, etc., and the method by co- 
ordinates. H 66, Fig. 35, therefore, give us only the amount, direction and 
sense of the resultant of non-ooncurrent forces, and not its position. To find 
the position of the resultant of non-concurrent forces, we may have recourse 
to a figure, like Fig. 44. where the forces are represented ib their actual posi- 
tions, or to the cord polygon, H1[ 86, etc., Fig. 46. 

The Cord Polygon. 

86. In the force triangle any two of the three lines may be regarded as 
representing, by their directions, the positions of two members (two struts 
or two ties, or one strut and one tie) of indefinite length, resisting the third 
force ; while their lengths give the amounts of the forces which those mem- 
bers must exert in oTaer to maintain equilibrium. 




FlfT* 20 (repeatefl). 

87. Thus, in Fig. 26 (6), are shown four different systems, of two mem- 
bers each, inclined respectively like the forces c h and b o in Fig. 26 (a) and 
balancing the third force o c. The stresses in these two members are given 
by the lengths of the lines e b and b o in Fig. 26 (a). 

Tlie members acting as struts are represented, in Fig. 20 (b), as abutting 
against flat surfaces, while those acting as ties are represented as attached 
to hooks, against which they pull. 

In Fig. 26 (c) and (d) are indicated systems of members, inclined like the 
forces c a' and a' o, ca and a o, respectively, of Fig. 26 (a), by which the third 
force o c might be supported. 

88. In the force polygon abed ea. Fig. 46 (6), representing the four 
forces, Pi, Ps, P3, Pj, of Fig. 46 (a), if we select, at pleasiue, any point o 
(called the pole) and draw from it a series of straight lines oa,ob, etc. (called 
mys), radiatinff to the ends, a, b, c, etc., of the lines Pi, Ps, etc., representing 
the forces, we snatl form a series of force triangles, aobthoc, etc. 

Thus, in the triangle d b o we have the force Pi, or a b, balanced by the two 
forces o a and b o; m the triangle b c o, the force P2, or b c, balanced by the 
two forces o b and c o; and so on. 

89. The Cord Polygon. If, now, in Fig. 46 (a), we draw the lines a 
and b, parallel respectively to the rays o a and o b of Fig. 46 (b) and meeting 
in the une representing the force Pi, they will represent the positions of two 
tension members of indefinite length, which will balance the force Pi by ex- 
erting forces represented, in amount as well as in direction, by the rayS a 
and b o, Fi^. 46 (b). Again, taking pol? o'. Fig. 46 (b), instead of o, we have 
a' and b*. Fig. 46 (aO, parallel respectively to the rays, o' a and 0' b, and rep- 
resenting a pair of struts performing the same duty. 

90. Similarly, the lines b and e. Fig. 46 (a), parallel respectively to rays o h 
and o c, represent two tension members, which, with stresses equal respec- 
tively to o b and c 0, Fig. 46 (b), balance the force Pg. 



378 



STATICS. 



01. We thus obtain, finally, a system of five tension members, ab e de. 
Fig. 46 (a), which, if properly fastened at the ends a and e respectively, will^ 
by exerting forces represented respectively by the rays, o a, ob, oc, etc.. Fig. 
46 (6), balance the four given forces Pi, P{, Ps and P4. 

92. The figure abode. Fig. 46 (a), is called a cord polygon, funicular 
polygon, or equilibrium polygon. 

03. Resultant, Anti-resultant. Amount and Direction. In the 

force polygon, Fig. 46 (6) or (d), the line e a, joining the end of the last force- 
line d e with the beginning of the first one a b, represents the anti-resultant of 
the given system of four forces, and a e their resultant. Evidently, there- 
fore, the rays^ a o and o e, which represent two components of a e, represent 
also, in direction and in amount, two forces which would balance e a, or which 
would be equivalent to the given system of (four) forces. 




Flffs. 46 (a), (a') and (fr). 



04. Position of Resultant. Hence, in the cord |>olygon. Fig. 46 (a)^ 
the intersection, i, of the cords a and e, parallel respectiv^y to the rays o a 
and e o, is a point in the line of action of the resultant R; and. if we imasine 
a i and e i to be rigid rods, and apply, at t, a force, — R, equal and parallel to 
a e, but of opposite sense, that force will be the anti-resultant of the (four) 
given forces, and we shall have a frame-work be di of cords and rods, kept in 
equilibrium by the action of the five forces, Pi, Pg, Pg, P4 and — R. 

06. By choosing other positions of the pole, as o\ Fig. 46 (fi), or by differ* 
ently arranging the given forces, as in Fig. 46 (c), we merely change the 
shape of the cord polygon, and (in some cases) reverse the sense of the 
stresses in the members. Thus, in Fig. 46 (a), all the stresses are tensions, or 
pulls : while in Fig. 46 (c) a, b, d and e are tensions or pulls, and c is a com- 
pression or push. 

06. In constructing the cord polygon, Fig. 46 (a), (aO. (c), and (e), car* 
must be taken to draw the cords m their proper places ; and for this it is neo- 
essary to remember, simply, that the two rays pertaining to any particular 
force line in the force polygon. Fig. 46 (6), represent those members which, 
in the cord polygon. Fig. 46 (a), take the components of that force. 



CORD POLYGON. 



379 



Thus, o a and h o» Fig. 46 (6), pertain to the force Pi ; o b and e o to the 
force Pj. Hence, in Fig. 46 (a") or {c) we draw a and h (parallel respectively 
to o a and 6 o) meeting in the line of action of Pi : h and c (parallel respect- 
ively to o 6 and c o) meeting in the line of action of Ps, etc., etc. 

97. Each ray in the force polygon. Fig. 46 (6), including the outside ones, 
is thus seen to pertain to two force?, and each force has two rays. The two 
oords,^ parallel respectively to the two rays of any force, must be drawn to 
meet in the line of^aetion of that force; and each cord must join the lines of 
action of the two forces to which its parallel ray pertains. The lines, a, &, c. 
etc., in the cord polygon. Fig. 46 (a) and (c), give merely the incLinatwM oi 
members which, as there arran^d, would sustain the given forces. ' The 
lengths of these lines have nothing to do with the amounts of the stretaes. 
These are given by the lengths of the corresponding raya in the force polygon, 
Fig. 46 (6). 




Flffs. 46 (o), {d) and («). 

08« If the anti-resultant force, — R, is not applied, the cords a and e may 
be supposed fastened to firm supports, against which they exert stresses rep- 
resented, in amount and in direction, by the rays a o and o e respectively. 
But the resistances of those two supports are plainly equal and opposite to 
those stresses, or equal to o a and e o respectively. Hence, their resultant is 
the anti-resultant, — R, of the foiu> origmal forces. 

99* If, Fig. 46 («), the two end members a and e were attached merely to 
two ties, V and V, parallel to the anti-resultant, — R, they would evidently 
draw the ends of those ties inward toward each other. To prevent this, let 
the strut k be inserted, making it of such length that the ties V and V may 
remain parallel to — R, and draw o k, Fig. 46 (6), parallel to k. Then a k 
and k e give the stresses in V and V respectively. 

ipO. If the anti-resultant, — R, found by means of the force pply^n, be 
applied in a line passing through the intersection of the outer (initial and 
final) members in the cord polvgon, all the forces, includinff of course the 
aati-resuliant, will be in equilibrium. In other words, coplanar forces are 
in equilibrium if they may be so drawn as to form a dosed force polygon, and 
if a closed cord polygon may be drawn between them. But if the anti-re- 
soltant be applied elsewhere, we shall have a couple, composed of the anti- 
rwnltant, — K, and the resultant R of the forces. 



380 



STATICS. 



Concurrent Xon-eoplanar Forces. . 

101. Any two of the concurrent forces, as o o and o c. Fig. 47 (a) or (6), are 
necessarily coplanar. Find their resultant, o r, which must be coplanar with 
them and witn a third force o h. Then the resultant, R, of o r and o 6 is the 
resultant of the three forces. If there are other forces, proceed in the same 
way. 

102. No three non-coplanar forces, whether concurrent or not, can be in 
equilibrium. 

103. Force Parallelopiped. The resultant of any three concurrent 
non-4ioplanar forces, o a, o\ o c. Figs. 47, will be represented by the diagonal 
a R, of a parallelopiped, of which three converging edges represent the three 
forces. 

104. Methods by Models, (a) For three forces. , Construct a 
box, Kg. 47 (a) or (6), with three conver^nt edges representing the three- 
forces in position and amount. Then a stryig o R, joining the proper corners, 
will represent the resultant. 





Fig. 47. 

Or, let ao,ho, c o. Fig. 48 (o), be three forces, meeting at o, "DnM on 
pasteboard the three forces a o, b o, e o, as in Fig. 48 (6), with their actual 
angles aob, boc, coa, and find the resultant wooi the middle pair, b o and 
c o. Cut out neatly the whole figure, a o a c w b a. Make deep knife- 
scratches along o 6, o c, so that the two outer triangles may be more readily 
turned at angles to the middle one. Turn them until the two edges o ci^oa 
meet, and then paste a piece of thin paper along the meeting joint to keep 




\ \ / 






w 



(«) 



(ft) 

Fig:. 48. 



(«) 



them in place. Stand the model upon its side o & tp c as a base, and we aball 
have the slipper shape a ob w. Fig. 48 ic)\o w being the sole, and aob the 
hollow foot. In the model, the force a o and the resultant to o of the other 
two forces, are now in their actual relative positions. To find their resultant, 
cut out a separate piece of pasteboard, R a o to, with R a and R w parallel 
respectively to w o and a o. Draw upon each side of it the diagonal R o. 
Paste this piece inside the model, with its lower edge tt; o on the line to o. Fig. 
48 (6), and its edge a o in the comer a o. This done, R o represents the re« 
sultant oia o,b o, c o, Fig. 48 (a), in its actual position relative to them. 

105. (b) For four forces, aaa o,bo,co,d o, in Fig. 49. Draw them as in 
Fig. 40 (a), with their angles aob, boc, etc. Draw also the resultants « o, of 
c o and b o; and wo,oico and d o. Then out out the entire figiire, as before, 
and paste together the two edges a o, a o. Hold the model in such a way 
that two of its jylanea (as a o 6 and boc) form the same angle with each other 



NON-COPLAXAR FOBCES. 



881 



as do the two corresponding; planes between the forces. Then we have the 
two resultants vo^wo, Fig. 49 (6), in their ctctiMl relative poeitiona. Cut out a 
separate piece of pasteboard R v o w, Fig. 40 (&), draw the diagonal R o on 
each side of it, and paste it inside the model, with o v and o to on the oorre* 
sponding lixteB of the model. Then R o will represent the resultant of the 
four forces, ao^bo,cOtdo, in its actiial position relative to them. 

The model may be made ol wood, the triangles aobth oc, etc., being cut 
out separately, the joining edges bevelled, and then glued to«^ther. 




(«) 




FUr. 49. 



(*) 



Non-concurrent Non-coplanar Forces. 

lOG. Non-concurrent Non-coplanar Forces. Fig. 50 (a). (For par- 
allel non-coplanar forces, see ^'^ 110, etc.) Resolve each force mto two rec- 
tangular components, one normal to an assumed plane, the other coin- 
ciding with the plane.* Find the resultant of the (coplanar) components 
coinciding with the plane, by methods already given, and that of the normal 
(parallel) components, by 1ft 110, etc. If these two resultants are coplanar, 
tney are also concurrent, and their resultant (which is the resultant of the 
system) is readily found. 

107. If not, let V, Fig. 50 (6), be the resultant normal to the plane, and H 
the resultant lying in tl^e plane. By If 162, substitute, for H, the eqtial and 
parallel force H', meeting V at O, and the couple H . O a, and find the result- 
ant, R', of V and H'. The system of forces is thus reduced to the single force 
R' and the couple H . O a. For Couples, see If 155. 




108. Moments of Non-coplanar Forces. Th<% action of the weight 
W of the wall. Fig. 51 (a), and of the non-coplanar forces Pi and Pe, may be 
represented as in Fig. 51 (&), where the axle a* cf represents the edge a c 
about which the wall tends to turn, while the bars or levers represent the 
leverages of the forces. So far as regards the overturning stability of the 
wall, regarded as a rigid body and as capable of turning only about the edge 
a e, it is immaterial whether an extraneous force, as Pi, is applied at p or at 
g; but it is plainly not immaterial as regards a tendency to swing the wall 
around horizontally, or to fracture it; or as regards pressures (and conse- 
quent friction) between the axle a' <f and its bearings. For equilibrium. Pi vik 

■- Pc A + W. — . Here a torsional or twisting stress is exerted in the axle. 



*Wires, stuck in a board representing the plane, will facilitate this. 



382 



STATICS. 



and the presBures of its ends in the bearings are more or less modified ; bui, 
so far as merely the equilibrium of the moments is oonoerned, we may sup- 
pose all of the forces and their moments to be shifted into one and the same 
plane, as in Fig. 51 (c). 

109* In oases like that represented in Fig. 51, it is usual, for convenience, 
to restrict ourselves to a supposed vertical alice, «, 1 foot thick, and to the 
forces acting upon such slice ; supposing the weight of the slice to be concen- 
trated at its center of cavity, and the extraneous forces to be applied in the 
same vertical plane with gravity. In eflfect, we are then dealing with a 
slice indefinitely thin, but luiving the weight of the 1-ft. slice. 




Flff. 51. 



PARALLEL FORCES. 

110. The resultant of any number of parallel forces, whether 
they are in the same plane or not, and whether in the same direction or not, 
is parallel to them and — their algebraic sum. 

Coplanar Parallel Forces. 

111. The resultant of any number of coplanar parallel forces 

is in the same plane with them, whether the forces are of the same or 
of opposite sense; and the leverages, or arms, of such forces, and of their 
resultant, about any given point in the same plane, are in one straight line. 
Thus, in Fig. 56 (a), where the five forces, a, b, c, d and e are in one plane, their 
resultant, R, is in that same plane; and tne levera^ of the forces, and 
of R, about any point, as 6 or v, in the same plane, are in the straight line R v. 




Fig. 02. 




113. The resultant, R, or anti-resultant, Q, Fig. 52, of two parallel 
forces, a and b, intersects any straight line, u v, joining the directions of 
the two forces. Hence, if three parallel forces are in equilibrium, they ara 
m the same plane. In Fig. 62 (a), the two forces, o and 6, are of like 
sense. R is then between a and b, and R = 6 -H a. In Fig. 52 (6), a and 
b are of opposite sense. R is then not between a and 6, and R — fr — a. 



PABALL£1< FORCES. 



383 



113* To find the position of the resultant, draw and measure any straight 
line* u V, joininjs the lines of action of the forces. It is immaterieil whether 
u « is perpendicular to said directions, or not. The line representing the 
resultant cuts u v, and its position is found thus: 

M i — tt « X -p- ; and v i = u v X -^. 




FliT* 93. 




*-i 



114. This may be conveniently done by making u v equal, by any conve- 
nient scale, to the sum of the forces, as in Fig. 53, where uv^ 42. Then 
make u i equal, by the same scale, to the force at v,oxvi equal to the force at 
u. Then a line, R, Fig. 52 (a), drawn through t parallel to a and h, gives the 
position and direction of their resultant ; and its amount is equal to the sum 
of a and h; or R =- a + 6. In other words, if a force, Q, parallel to a and 6, 
and equal to their sum, but of opposite sense, be applied to the body any- 
where in a line passing through i, it will balance a and 6, or will be their anti- 
resultant. 



\~ — 







.-.^1^ 



/^ 



I 



x^l 



y 




Ftgr. 55. 



(ft) 



115. The position of the resultant, so found, satisfies the condition of 
equilibrium of moments : thus, h.vi — a.ui « zero. 

If the two forces are equal, their resultant R is evidently midway between 
them. 

116. In the common steelyard, Fig. 54, the two forces a and &, of 
Fig. 52 (a), are represented by the two weights, a == 3 pounds at i«, and h =• 
1 pound at v, with leverages ui and vi respectively, as 2 : 6, or as 1 : 3. 



384 



STATICS. 



It will be noticed that in Fig. 56 (a) the resultant, R, owing to the posi-* 
tions and amounts of the several forces, falls outside of the system of given 
forces. 

117. Figs. 65 to 58 illustrate the application of the cord polygon (^^ 86 
to 100) to coplanar parallel forces. Here the force polygon is necessarily a 
straight line. 



Jt 



a Jft \d _^__y 




(«) 



Tig, 56. 




118. Resolution. Let Fig. 57 (a) represent a beam bearine a single 
concentrated load* a, elsewhere than at its center; and let it be required 
to find the pressure on each of the two supports, w and x. 





FIgr. 57. 



(6) 



Draw X a. Fig. 57 <fe), to represent the load a by scale, and rays X O, a O, 
to any point O not in the line X a. In Fig. (a), from any point, t. in the 
vertical through the point, a, where the load is applied, draw t • and t r, 
parallel respectively to O X and O a. Join r «, and in Fig. (h) draw O to par- 
allel to r 8. Then the two segments, w a and X w,^ of X a, give by scale the 
pressures upon the two supFK>rts, w and x respectivelv. ,The greater pres- 
sure will of course be upon the support nearest to the load; but we may 
be guided also by remembering that the segment X w, adjoining the radiv 
line O X in Fig. (6) represents the pressure on that supi>ort, x, Pig. (a), 
which pertains to the line i 8 parallel to O X; and vice versa. 

119. Fig. 58 represents a case where there are several loads on the 
beam. Here the intersection, i, of the lines h a and k r, Fig. (a), drawn 
parallel respectively to O X ana c O, Fig. (6) shows the^ i>osition of the 
resultant of the three loads. Here, as in Fig. 57. we join r «, ftc. (a), 



PARALLEL FORCES. 



385 



and draw O w, Fig. (b), parallel to r ». Then X to, Fig. (5), gives the 
pressure upon x^ and w c that upon w. 





(a) Fl|r. SS. 

Non-coplanar Parallel Forces. 



(&) 



120. Non-coplanar Parallel Forces. Fig. 59 (a). Between the 
lines of action of any two of the forces, as a and b, draw any straight line, u v, 
and make 

u i = u V X 1 r ; or v i — uv X 



a ^ b * 



a + 6 * 



Through i draw R', parallel to a and ft, and equal to their sum. Then 
is R' the resultant of a and 6. Then, from any pomt, t, in the line of action 
of R', draw i z to any point, z, in the line of action of c, and make 

c R' 

ik — i z X p^ ; or « A; = t z X —3757 • Through k «draw R parallel to 

a, ft and c, and equal by scale to their sum. Then is R the resultant of the 
three forces, a, b and c. If there are other forces, proceed in the same way 
with them. 





c-tf 



(«) 



Fiir« 99. 



(6) 



131. In Fig. 59 (a) we have shown the forces, a and c, acting upon surfaces 
raised above the general plane, merely in order to illustrate the fact that it is 
not at all necessary that the forces be supposed to act upon or against a plane 
surface. 

122. Although Fig. 59 (o) illustrates the method of finding the resultant 
of non-coplanar parallel forces, yet it plainly does not give the actual relative 
positions of the forces and their resultant * because it is necessarily drawn in a 
jcind of perspective, and therefore all the parts cannot be measured by a 
scale. The true relative positions may of course be represented in plan, as 
by the five stars, a, ft, c, % and k, Fig. 59 (6), corresponding to the points where 

26 



386 



STATICS. 



the forces and resultants intersect some one chosen plane. But it is now 
impossible to represent the forces themselves by lines. They must there- 
fore be stated in figures, as is here done. It is then easy to find the positions 
of the resultants, as before. 

1^. If there are also forces acting: In the opposite dfrection, as 

d and e. Fig. 59 (a), find their resultant separately. We thus obtain, finally, 
two resultants of opposite sense. These resultants may be equal or unequal, 
and colinear or non-colinear. If they are non-colinear, see ^ 84, and Couples, 
nil 155, etc. 

134:. Metliod by projections. Fig. 60. First find the projections. 
a\ h* and cf of the forces, a, h and c, upon any plane, bs x y, parallel to 
them; and then their projections, a", 6", and c'', upon a second plane, x r, 
parallel to them and normal to the first. Find the position, R', of the re- 
sultant of a^ h' and c^, in plane x y, and that R'', of a'^ 6" and c", in plane 
X V. Now, as the lines, a', b\ c', and o", b", c", are projections of the forces, 
a, b and c, so R', R'', are projections of the resultant, R, of the forces. The 
position of R is therefore at the intersection of two planes, R R' and R R'', 
perpendicular \o the planes, x y and x v, and standing upon the projections 
K' and R", of the resultant, R. R = o + 6 + c. 




CENTER OF GRAVITY. 

195. If a body. Fig. 1,* or a system of bodies. Fig. 2, be held successively 
in different positions, (a), b), etc., the resultant of the parallel forces of grav- 
ity, acting upon its particles and indicated by the arrows m the figures, will 
occupy different positions, relatively to the figure of the body or system. 
That point, where all these positions, or lines of gravity, meet, is called the 
center of gravity of the body or system. Thus, if a homogeneous cylinder 
be stood vertically upon either end, the line of gravity will coincide with 
the axis of the cylinder; but if the cylinder be then laid upon its side, the 
line of gravity will intersect the axis at right angles ana will bisect it. 
Hence, in the cylinder, the center of gravity is at the center of the axis. 

1|26. About the center of gravity the moments of all the forces of mivity 
are in equilibrium, in whatever pK)sition the body or system may be. Hence, 
the body, or system, if suspended by this point, and acted upon by gravity 
alone, will balance itself; t. e., if at rest it will remain at rest;^ or, if set ib 
motion revolving about its center of ^avity, and then left to itself, it wilt 
continue to revolve about that center indefinitely and with uniform anfculaf 
velocity. Or, if suspended freely from any point, it will oscillate until the 
center of gravity comes to rest vertically under such point. 



* Figs. 1 to 45, relating to Center of Gravity, are numbered independently 

of the rest of the series oi figures relating to Statics. 



CENTER OF GRAVITY. 



387 



127* In some bodies, such as the cube, or other parallelopiped, the sphere, 
etc., the center of gravity is also the center of the v>eigfU of the body; but 
very frequently this is not the case. Thus, in a body a b. Fig. 2, with its 
center of gravity at G, there is more weight on the side a G, than on the side 
G6. 







Tig. 1, 



Stablet Unstable, and Indifferent Equlllbrluni. 

128* A body is said to be in stable equilibrium when, as in the pendulum, 
it is so suspended that, if swung a little to either side, it tends to oscillate 
until it comes to rest again, with its center of gravity vertically under the 
point of su8f>en8ion. 

129» It is said to be in unstable equilibrium when, as in the case of an 
efx, stood upon its point, it is so supported that, if swung a little to either 
side, and left to itself, it swings farther out from the vertical and eventually 
falls. 

130. It is said to be in indifferent equilibrium when, as in the case of a 
grindstone, supported by its horizontal axis, or of a sphere resting upon a 
horizontal table, it is so suspended or supported that, if made to rotate about 
ita center of gravity and then left to itself, it will continue in that state ol rest 
or of angpilar motion in which it is left. 




(a) (b) 

Wig. s. 



General Rules, 

131. The following general rules (I) to (6), form the basis of the special 
rules, (7) to (39). 

In speaking of the center of gravity of one or more bodies, we shall assume, 
for simplicity, that they are homogeneous (i. e., of uniform density through- 
out) and of the same density with each other. The center of gravity is then 
the same as the center of volume, and we may use the volumes of the bodies 
(as in cubic feet, etc.) in the rules, instead of their weigfda (as in pounds, etc.). 

In applying these general rules to surfaces, use the area^ of the surfaces, 
and' in applymg them to lines, use the lengths of the lines, in place of the 
weights or volumes of the bodies. 

In all of the rules and figures, pp. 388 to 398, G represents the center of 
gravity, except where otherwise stated. 



388 



FORCE IN RIGID BODIES. 



(1). Anjr turo Itodies, Fig. 3. Havine found the center of gravity, g^ ^^ 
of each body, by means of the rules given oelow: then O is in the line joining 
yand^S' and 






weight of flf^ 



sum of weights of g and g^ 



weight of g 



sum of weights of g and g^ 




G 





iris.3 



(5S). An^ number ot bodies^ as «, & and e, Fig. 4, whether their centen 
of gravity are in the same plane or not. 

First, by means of rule (l> find the center of gravitv, gr, of any two of th« 
bodies, as a and 6. Then the center of gravity, Gp of the three bodies, a, b 
and c, 18 in the line ^p' joining g with the center of gravity, g^ of e; and 



gG=^ gg' X 



weight of e 



sum of weights of a, & and e * 

</ 6 — -^ w sum of weights of o and b , 
sum of weights of a, 6 and c 
and so on, if there are other bodies. 

(3)* In many cases, a slnffle eomplex bodjr may be supposed to be diylded 
into parts whose several centers of gravity can be readily fonnd. Then the 
center of gravity of the whole may be found by the foregoing and following 
rules. Thus, in Fig. 6, we may find separately the centers of gravity of the 





two parallelopipeds and of the cylinder between them (each in the center of 
its respective portion of the whole solid) ; and in Fig. 6 the centers of gravity 
of the square prism and the square pyramid (the latter by rule (36), 
and then, knowing in either case the weightis of the several parts, find their 
common center oi gravity as directed in rules (1) and (2). 



CICNTEB OF GRAVITY. 889 

(4^, Anjr l&olloiw- iMdy, or body Gontaining one or more openings. Fig.T 
fmd the oommon*center of gravity, g'^ of the openings by role (1) or (2X anc, 




the center ot gmyit^, p, of the entire figure, as though it had no opeatega 
Then G is in the line £ry'» extended, and 

aQ ^ Off V snm of volumes of openings 

Tolome of entire body — yolumes of openings 



t^-nifx 



volume of entire body 



volume of entire body — volumes of openings 



Bbmabk. For convenience, we have shown the several centers of gravity, 
9t ?/ ^9 upon the awrface of the figure. In the real solid (supposed to be <» 
uniform tniokness) they would of course be in the middle oi its thickness- 
and immediately under the positions shown in the figure. 

(•). In any line, figure or body, or iaany system of lines, figuresorbodie8,any 

Slane passing through the center of gravitv is called a " puuse «f §;r»vlt]r ^ 
>r said line, etc., or system of lines, ete. The intersection of two such planes 
of gravity is called a '< line ot ^pcmrvU^** The center of gravity is (Ist) the 
intersection of two lines of gravity; (2nd) the intersection of three planes 
of gravity, or (3rd) the intersection of a plane of gravity with a line, of gravity 
not lying m sMd plane. 

If a figure or body has an axis or plane of swn&aaetrjr {i. a., a Mne or plane 
dividing it into two equal and similar portions) said axis or plane ia a line of 
place of gravity. If a figure or body has a central point, said point is the 
center of gravity. 

In Fig. 1, the string represents a line of gravity; and any piano with 

which the string coincides Is a plane of gravity. Thus Qt may often be con- 
veniently found, especially in the case of a flat body, by allowing it to hang 
freely from a string attached alternately at different comers of it, or by bal- 
ancing it in two or more positions over a fanife-edge, etc., and finding G in 
either case by the intersecnon of the lines or planes of gravity thus found. 

(6). Tlie irraplilc metliod of finding the resultant of parallel forces 
may often be advantageously used for finding the center of gravity of a com- 
pound body or figure, or of a system of bodies or figures, when the centers of 
gravity of the several parts are xnown. 

Thus, in Fig. 8, let a^ 6 and e represent three figures or bodies whose centers 
of gravity are in one plane. Draw vertical lines through said centers, and 
construct the polygon of forces, xa 6e, Fig. 9. making the lin's xa^ahy etc., 
proportional to the weights of a, 6 and e; and from any convenient point O 
draw radial lines Ox, Oa, etc. In Fig. 8, draw m ti^mrunp, and p A;, parallel 
IMpectively to O a;, O a, O 6, e. Then a vertical line, i Gy drawn through the 
Intersection, i, of m A and pkAs a line of gravity of me system or figure. If 
the body or figure is symmetriealf as in the cross section of a T rail, I oeam or 
deck beam, etc., the axis of symmetry, dividing the figure, etc. into two simi- 
lar and equal paxts, is also a line of gravity, and its in^rsection with the line 
<0 already f