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THE
CIVIL ENGINEER'S
POCKETBOOK
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 Pocketbook, 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 seeeaw, let them see (and measure for them
•elTbs) that the lighter one is farther from the fencerail 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 seesaws 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 seesaw, ikey balance
each other. Let them see that the weight of the heavy hoy, when multiplied hy
his distance in feet from the fencerail 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 footpounds. 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 earpulling, 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, cuberoots, &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 pocketsize, 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 pocketbooks, 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 masterminds,
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.
TWENTYSECOND THOUSAND, 1885.
CI INCE the appearance of its last edition (ihe twentieth thousand)
'^ in 1883, the " PpcketBoo]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 rearrangement, 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 steambammer 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 wellboring 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 retouched and relettered, 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 recast into the same form.
ix
X FB£:fAC£.
The addition of new matter, and a number of blank spaces
necessarily left in making the rearrangement, have increased the
number of pages about onefifth.
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 FerrisPitot 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.
9961007 Business Directory.
10081023 BibUography.
The following articles have been almost or entirely rewritten:
Nkw Pages Old Pages
35 47 Arithmetic 3337
210211 Specific Gravity 380381
265266 Time 395
282283 Chains and Chaining 176
284290 Location of the Meridian 177179
322325 Rain and Snow 220221
358453 Statics 318 f361, 370375
xi
• •
XU PREFACE.
New Pages Old Pages
466494 Strength of Beams 478520, 528536
499 Shearing Strength 476
499500 Torsional Strength 476477
501503 Opening Remarks on Hydrostatics 222224
537538 Effect of Curves and Bends on Flow in Pipes 255256
689744 Trusses 647614
856864 Locomotives 805810
865866 Cars 811813
867869 Railroad Statistics 814818
892899 I Beams, Channels, Angles and T Shapes 521527
930942 Cement 673678
943947 Concrete 678682
954956 Timber Preservation 425425 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 fiveplace 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
fiveplace 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 hourangles, 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 " scissorsand
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. . .44r46
Duodenal Notation 47
Reciprocals 4852
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
Twopage Table 78
Twelvepage 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
Rightangled — 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 — 182185
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 — 205207
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 212216
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 — 255257
Heads and Pressures of
Water; Tables of — 258260
Discharges in Gals, per Day
and Cu. Ft. per Second;
Tables 261265
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
Croashairs; 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 Streamflow 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
Foice Triangle 367
Rectangular Components 369
Inclined Plane 369
Stress Components 371
Applied and Imparted Forces . . 372
Resolution, etc., by means
of Coordinates 372
Force Polygon 374
NonconeurrentCopUnarForoes 375
Equilibrium of Moments 376
Cord Polygon 377
Concurrent Non  coplanar
Forces 380
Nonconcurrent Noncoplanar
Forces 381
Parallel Forces 382
Coplanar 382
Noncoplanar 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
Crossshaped 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
FerrisPitot 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
Broadcrested 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 Drainpipes 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
Riprap 583
Protection from Scour 683
Timber Cribs 684
Caissons 685
Cofferdams 686
Earth Banks 686
Crib Cofferdams 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
SandPiles 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 Archstones 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, Standpipes 663
Service Pipes 664
Tapping Machines 664
Antibursting 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
Locknut 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
ZBar 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«lldliir 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
*
Coo'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
Coordinates.
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 semicircumference. 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
semiclrcumferenoe.
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. €• 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*
4X4
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^ = 5f ; I X f = ;
ofofof^ = 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
o7g — oXir — 7V — 5S.
(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 . Af B
Composition. —  — = — ^ — ; — g—
., ,, AB CD 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 = ls^'.
(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 mandays, and 12,000
12 000
bricks will require 6 X XaSa = 6 X 3 = 18 mandays ; 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 ^, 1J^, 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) Onefiftietli, 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 = 1J^ 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 misread six onehundredths of one per cent, or six ouehun
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 onesixth 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, semiannually, 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 lifetime, 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 = ^^^^:^^^: iiW. Hence. W = )ff^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 lifetime 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 sidetracks 80 8.11
Steel pipe, valves, blowo£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, sandwasher, 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«bKOMiiMiin£> 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 — ai — = an.
fieiice, a f Becip a = a * — =»aX'7~ = a*.
Thus. Eedp2 = 0.6.andg^ = ^ = 42..
(: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
. 81854
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
1566066B
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
218849M
2140U7W
210000080
237U066O
338188000
389488001
240B1S8
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
8H?o
6.6816
8.BV3
8J
8i
oj
8.C
8J
8^
8.1
8.1
84773
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, S4nare 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
4897
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
UOl
1736
41.65
12.02
2135
46.21
.12.88
2620
51.19
1179
1M#
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 anoIHir 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.ao6i
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
530
121665
10.4
3363232
20.2
24300000
30.0
714924299
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
600
228776
11.8
4701850
21.6
42191410
33.5
1252332576
66.
286.292
3.10
8445.96
6.10
248832
12.0
4923597
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
680
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
123096020
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
184528125
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
380204032
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»lini»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 yl = ^^
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^^ =ni =  , and noa = n«
= j it follows that log wi = 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 89053 = 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 neyatlve 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 crosssection 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 12, 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 reentered 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 = 11.5 + 12 = log 8 = distance 1M. But
log 15 + log 20 = (11.5 + 110) + (12 + 110), 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,
1X* 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 11
losaiithxn (0.30 103) of 2. If now we add these two dis
by laylnflT off 12 ttom 1.5 on 1X of the chart, or by placl
In the figure, we obtain the distance 13 = .47 712 = the m
or of log (2 X 1.5).* Conversely, to divide 3 by 2, we graphica
cally subtract 12 fh>m 13.
(•) In tbe l4»9Arftliinlc chart, the scales of both axes,
1Y, being equal, a line 1H, 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 coordinates. Thus, points In the line x, imm
over 2, 3, 4, etc.. in 1X, are also opposite 2, 3, 4, etc., respect!'
1Y. 8ee (4).
g*) If lines 2A\ SK, etc. (marked 2x, 8a;, etc.), parallel to m
, be drawn through 2, 8, etc., on 1Y, then points in such li
mediately over any number, x, in 1X, 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 1B, 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 1H and through 2, S, etc., on 1X, give. values of ^^ ?, etc.,
respectively. If these lines ^^ «• etc., be produced downward, they will
cut 1Y (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 1X, whose tangent, f^ is 2, it will give values of z*. Thus, the ver
tical through 3, on 1X, cuts the line x* opposite 9 (= 3*) on 1Y. 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 1X is = n,
and intersecting 1Y 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 1Y, 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 1X, should be perma
nently ruled across the sheet at short intervals.
(11) For any log, as 18 (= log 3), we may substitute its equal. MN
or 3N, extending to the central diagonal line 1H, marked x; and then,
since, for instance, 11.2 = NQ, 13 = NK, etc., we may add any log
(as 13) by moving upward from line x (as from N to K) or to the right,
and siw^act any log (as 11.2) 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).
FG = 1F = log 1.5. Add GJ = 12 = log 2 ; sum = FJ = log 3 = 13 =
MN. Add NK = 13 = log 3 ; sum = MK = log 9 = 19 = 9R. Subtract
R_T = 12 = log 2 ; remainder = 9T = 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 42
leaving 1Y, at 4, and forming, with 1X, 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 22. Lines (C x) giving multi
ples and submultiples 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 2A. 2 X 5 = 10 ; in 3K, 8 X 3.38 ... = 10, etc. The
chart thus furnishes a table of reciprocals. . ,
(t) Even with fullsize 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 8091. 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 5place logs, it is better to find difTer
enoes by subtraction : but where a twopage 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
k403,
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
8a60322
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 rlpb 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 (apellne, ehain, or a
■Masuringrod 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 onehalf 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 onefourth 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 straightlined 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 straightlined 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 straightlined 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 slraightlined fig,
no matter how many sides it may have, amount to 860° ; but, In (he case of
a reentering 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 + 6ctofca« = 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 fouraided BtraightUBed 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^aboi
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 twofoot rnle, either on a drawing or between dis
tant objects in the field. If the inner edges of a common twofoot 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 2foot 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
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(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 footnote, 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
tobe 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 eosine, corersed sine ; cotangent; 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 wellknown 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, coversed 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.
Ooverted 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 coversedsine), take the angle trota 180° : if between 180° and 370° take 180° fkom
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.
I
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s
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 radiusscale ; 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 Indiarubber, and dulling of the pencilpoint. 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 radiusscale) 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 .866025 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 rightangled
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.
Bifflitaiiirle^ Tri»iiirlefl«
4.U the foregoing appw also to rightangled 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 ttrBszOoeBs/ 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>
frrj *'*'*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 plumbline 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 reftaction 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 ftom 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 tapeline 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 rehaction, 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 MiraUelotrani 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 Maare^
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 6sided
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 lidedeptha 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 aidedepths 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 atraightalded 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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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.
164
.004091
732
.057269
2764
132
.008181
1564
.061359
716
864
.012272
ili
.065450
2964
116
.016362
.069640
1632
«64
.020463
082
.073631
8164
8^
.024644
1064
.077722
8^
7^
.028634
616
.081812
Hu
.032726
2164
.086908
1732
U)36816
1132
.089994
8664
632
.040908
2364
.094084
916
1164
.044997
1^
.098176
8764
816
.049087
.102266
1932
IM4
.068178
1332
.106366
3964
Giroamr,
_lbat._
.110447
.114637
.118628
.122718
026809
030900
034990
039081
048172
047262
061868
056448
059534
Diam.
68
4164
2132
4364
1116
4664
2882
4764
Jii
2632
6164
1316
Ciroamf,
.163626
067715
.171806
.176896
079987
084078
.188168
092269
.196360
.200440
.204531
.208621
.212712
Diam,
Inelu
6364
2732
6564
78
6764
2932
6964
1516
6164
8132
6864
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 Bemicircle.
** " Fig. i «« *• «• or greater than, a ■emicirelt^
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 semicircle.
M u pig^ 2 ** ^ ** or greater than a 8emi<clrclo.
R adimiy eOfC^pi or cbp
. (H «<>)« + ri»e« ^ ijjga. 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 dlrtders.
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 onehalf 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
beamcompass, 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
160
o /
9 9.76
313.
1.00107
.2601
u
o /
66 8.70
6^
1.04116
•
.2688
146
10 10.76
263.626
1.00132
.2501
63 46.90
6.626
1.06366
.2649
140
11 26.98
200.6
1.00167
.2602
.165
68 63.63
6.70291
1.06288
.2667
136
13 4.92
163.625
1.00219
.2502
16
73 44.89
6.
1.07260
.26t6
130
15 16.38
113.
1.00296
.2503
.18
79 11.73
4.36803
1.08428
.2676
126
18 17.74
78.626
1.00426
.2504
16
87 12.34
3.626
1.10847
.2693
120
22 60.54
60.6
1.00666
.2506
.207107
90
3.41422
1.11072
.2699
119
24 2.16
46.026
1.00737
.2607
.226
96 64.67
2.96913
1.12997
.2616
118
26 21.65
41.
1.00821
.2508
.2^6
106 16.61
2.6
1.16912
.2639
117
26 60.36
36.626
1.00920
.2609
116 14.69
2.15289
1. J 9083
.2666
116
28 30.00
82.6
1.01088
.2510
.3
123 6130
1.88889
1.22496
.2692
116
30 22.71
28.626
1.01181
.2611
^
134 46.62
1.626
1.27401
.2729
114
32 31.22
26.
1.01366
.2613
144 30.08
1.43827
1.32413
.2766
113
34 69.08
21.626
1.01671
.2516
.4
154 38.35
1.28125
1.^322
.2808
112
37 60.»6
18.6
1.01842
.2517
.426
161 27.52
1.10204
1.42764
.2838
111
41 13.16
16.626
1.02189
.2620
.45
167 66.93
1.11728
1.47377
.2868
110
46 14.38
•18.
1.02646
.2625
.476
174 7.49
1.06402
1.62162
.2899
10
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 crescentshaped
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 SBONEVTStCoHTHiDH:
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 rightangles 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 onehaif 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
•eealled 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 diamsbe 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 onehalf 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?ig4.
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 onehalf
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 onehalf 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 pocketbooks, is as IbUows t
Length — 2 X V(H '^>^e)a + \% Umes the (Height^
(g Where the height does not exceed 1lOth 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 ithe 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 tapeline 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 4sided, 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
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in deci
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TaMo oontlniied, bat wtth tbe dlanui In feet.
Gab.
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198 CONTENTS AND LININ08 OF WELI*.
COSTENT8 AKD LIJriHeB OF VELIA.
For lIuH WlBe u irul u IkaH In Ih. UUt. Ibr »>• n» JiU iC Unliil. Uli ml tbm onuM
OM kaf 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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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.
FigDH>">(>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
90k 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 sizeided 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 onechain 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
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UENBUBATION.
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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 sriiere. 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 XIMIWB.
__.__ _ 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 sawdust, 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 ** wenseabbled 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 onefifth to nearly onehalf more than
4fT;and ordinary building timbers when tolerably seasoned about onesixth 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 longleafed 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 Steaite 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 onefifth to nearly ODehalf more than
dry ; and ordinary building timbers when tulerably seasoned about cuesixth 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
SMiee oeenpied by eoal. In cubic feet per ton of 2240 pounds.
PennsylTanla Anthracite.
Hard white ash*
Freeburning white ash *.
Shamokin * ,
Schuylkill white ash *.
" red " *.
Lykens Valley *
Wyoming freebumingf *
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 avoirduM>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 twentyhundredweight, 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 univeisal 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 tenmillionth 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 einiTalent 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 footnotes 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 SachibKerans = 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 footnotes, 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 18781879 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
1SS
.0026
.1693
.3359
.5026
.6693
.8859
1.16
.0062
.1719
.8886
.6052
.6719
.8886
8n
.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
S16 .
0166
.1828
.3480
.5156
41823
.8490
ftt
0182
.1849
.8516
.6182
41848
.8616
Ji :
0208
H
.1876
H
.3542
H
.6208
H
.6875
H
.8643
0284
.1901
.3568
.5284
.8801
.8568
fr16
0280
.ion
.3594
.6200
.6927
.8694
11S9
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
17SS
0443
.2109
.8776
.5443
.7109
.8776
9M
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
ssss
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
• SlSS
0807
.2474
.4141
.6807
.7474
.9141
1
06SS
S
.2509
.4167
y
4i688
9
.7500
11
.9167
1«
0869
.2626
.4193
.6859
.7526
.9193
1lC
0885
.3563
.4219
.7562
.9219
8S2
0911
.2678
.4245
.6911
.7578
.9246
H
0888
H
.2004
H
.4271
H
.5038
H
.7604
H
.9271
5St
096A
.3660
.4297
.5964
.7680
.9297
SI6
0800
.3866
.4323
.6990
.7656
.9823
78i
1016
.3683
.4.')49
.6016
.7682
.9349
3< •
1042
H
■S&
H
.4876
3i
.6043
H
.7708
H
.9375
9Si
1068
.4401
.8068
.7784
.9401
616
1684
.2768
.4427
.6094
.7760
.9427
1132
1198
.2786
.4453
.6120
.7786
.9468
K
1148
H
.2811
H
.4479
H
.6146
H
.7813
H
.9479
lS3t2
1172
.2889
.4505
.6172
.7889
.9506
716
1198
.2666
.4531
.6198
.7865
.9531
«
1632
1224
.«9l
.4567
.6234
.7881
.9557
^
1260
H
S&
H
.4583
H
.6250
H
.7917
H
.9583
17.%
1276
.4809
.6276
.7948
.9609
916
UOS
.2M9
.4635
.6302
.7969
.9636
1932
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
1116
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
25S3
1484
.8161
.4818
.6484
.8151
.9618
1316
1610
.8177
.4844
.6510
.8177
OtlAA
•von
27S2
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
1616
1816
. .8281
.4948
.6615
.8281
.9948
n« •
1641
.8807
.4974
.6641
.8307
.9974
WEIGHTS AND MEABUBBS.
— —"■HIj
» 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 jjroMgrains.
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
I III 1 Slj,
Ppini.1T!
mm ^i'
11
11
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i I !
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i , I
i J
m
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239
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OONTBRSION TABLES.
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OONVEBSIOK TABLES.
251
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w. 68g Sga *l
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OOKVEBSIOK TASLBS.
253
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WEIOHTB AKD HEAHCBRS.
TABI.E or ACRES
■XaiJIBED pme mU*, a
tor dimrent wldtka.
ijk"
jiT
^.
^
k!
iH
^fSi'
E
,H
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aa
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ft
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w
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28
8.80
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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
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.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'*
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.03S
U
.a
.33
.»
m
8.48
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90
lie
.310
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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
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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 19hordlat 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
intle (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
121
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
181025
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
ptessare
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 roofcrest 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 timetable, 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 timetables.
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 quarterhour 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 SnnDlal.
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 oolatitnde (» 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

„..
"Krtf
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
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TK
OKK>
Ba DT
nraa
■S.
'H
IH
1^
5
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sir
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i
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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
rtBdM.
.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
832B
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 onehalf 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 hnmorln 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
jtnaker. A very fine needle, with a sealingwax 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 nom 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 drawingpaper.
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 rItt 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 10 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 addingsnp 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£ SttJ'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.
Fir.&
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 sheaflike 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 basemeasuring 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 crosssection 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
iDjgiTenooiiKtioD,m pull oti^j=lHy.
J,  r le OOfTactlan on tbaataadard 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 downhul 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 nortbandsontii
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 wellknown 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 crosshairs 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 18828, 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 crosshairs of the transit, as well as the stake, and the
knifeblade or pencilpoint 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, 19001910 . 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 plumbline and sight. A brick, stone, or other heavy object
will answer perfectly as a plumbbob. 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
plumbline and sight should be at least 15 feet apart, and so placed that the
star and plumbline can be seen together through the sight, throughout the
observation. The plumbline 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
plumbline 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 plumbline 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« boldfaee.
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 InnoitiiHA amr\ttn*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 crosshaira.
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 KrewtbRwds at v loeelfe the sciew
of a wooden trlpodheadcover 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 mlilftlnsplat«, <f d e c, enables lu, bf illabClr lonealng
the I«TelllBKBCrem K, to shlA the upper paiU boriiontallT a (rifle, and
■haa bring the plumbbob 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 balTbkll t, ibni
pr^alng o c llghtl; up afalnat the under lida at 8. Ths plombUnB paana
throngb the yert hols in 6 Scraweaja, / 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«tek 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 springcatch, 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 compassbox, 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 corapassbox C, the two bubbletubes M M, the standards
y Y, supporting the telescope, &c. Each bubbletube is supported and adjusted
by four capstanhead 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 tangentscrew 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 clampscrew 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 tangentscrew G.
The motion of the vernier plate P over the graduated limb O is simUarly
governed by the tangentscrew 6 and its spiral spring (not shown), fixed to the
ternier plate P, and the clampscrew 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
"lostmotion."
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
eompaasneedle, ib. Fig 2, should always be pressed up against the glass cover
of the compassbox by means of the upright miUedhead screw seen on the ver^
nierplate 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 eyeend. 1C, of the telescope revolve
dowHiffard, as otherwise the shade on O, if in use, may fall off. The tangentscrew,
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 vernierarm
L Bt mesns of the screw D. the movable vernierarm 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 rightangles
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 dustguard for the objectslide.
StaaiA Kalrs. Immediately behind the capstanscrew, 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 babbletube« 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 oltjectglass 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 objectglass 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 eyeglass and objectglass are so drawn out that there shall be ne
parallax. The eyeglass must first be drawn out so as to obtain perfect distinctness
of the crosshairs ; it must not be disturbed afterward; but the objectglass must
be moved for different distances.
First, to ascertain tliat tlie bnbbletnbes, M Bf • are placed
parallel to the vernierplate, 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 halfway round. If the Dubbles do not
remain in the center, correct half the error by means of the two capstannuta
rr; and the other half by the levellingscrews 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 chainpin, Ac, in the line.) Then un
clamp either H or e, and turn the upper parts of the inst halfway 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 oneltalf of the difference by means of the adjustingblock and
screw at the end, R, of the telescope axis. Fig. 1, and repeat the trial de novo,
resetting the stake or chainpin at each trial. If the inst has no adjustingblock
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 crosshairs are traly vert and hor
^rhen the inst is level. When the telescope inverts, the crosshairs are
nearer the eyeend than when it shows objects erect. The maker takes care to place
the crosshairs at rightangles 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 bubbletubes, level the in«
strnment carefully, and take sight with the telescope at a plumbline, 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 screwheads, 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 plumbline, or vert stsulgfat
edge, sight the crosshair 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 crosshairs. The hairring, or diaphragm, a; vert hair, v; tele*
scope tube, g ; ring outside of telescope tube, d; & is one of the four capstMiheaded
screws which hold the hairring, a, in its place, and also serve to a^jnst it. The
lower ends of these screws work In the thickness of the hairring; 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 tangentscrew O ffigs. 1 and 2)
upon some convenient object h\ or if there is none such, drive a thin stake, or a
enninpin. 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 clampscrew 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 <— onefourth 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 crosshairs, 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
backsights, and two foresights, a c and a m, may be taken, as when making the
adjustment ; and the point 0, halfway between c and m, will be in the straight line.
The inst may then be moved to 0, and the two backsights 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 bubbletube 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 bobbtotnbe. 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
levellingscrews K. Drive a peg m, from 100 to 300 feet from the instrument o.
Then placing a targetrod 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 clampring 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 crosshairs 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 gumarabic water, Ac. This requires care.
Then, to return the diaphiagm to its place, press firmly into one of the screwholes
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 socalled cross hairs are actually spiderweb, 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 spiritlevel, 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 compasscovers, and needles; atjQusting pins; level vials; magniflen, Ao,
Theodolite adjustments are performed like those of the level and transit.
let. That of the crosshairs; the same as in the level.
2d. The long bubbletube of the telescope ; also as in the level.
8d. Th^ two short bubbletubes ; 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 tangentscrew. 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 milledhead 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
eyehole at A should be partially closed by a slide which has a very small eyehole
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 eyefaole when observing
the ran. When the telescope is used, it is fastened on by the milledhead 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 Indexfplass, is attached
to the underside of the index, at C; its upper" edge being indicated by the
two dotted lines. The other, the MoriaonKliMiS) (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 horizonglass is silvered only halfway 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 indexglass. 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 indexglass is already permanently fixed by the
ma^T, and requires no other a4ju8tment. But the horizonglass has two adjust
ments, which are made by a key like that of a watch, and having a milledhead 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 squareheads, (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 indexglass 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
eyehole, and the lower or unsilvered part of the
horixonglasB, 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 squarehead 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 onehalf 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
mountainpeaks 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 levellingscrews ; 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 horizonglass has the two adjust
ments of the box sextant. It also has its dark glasses for looking at the sun ; and
a small eighthole, 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 halfway round. If tlie bubbles then remain at the cen
ters, they are in adjustment;. but if not, correct onehalf the diff" in each bubble,
by means of the adjustingscrews 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 plumbline ; 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 pivotpoint 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 halfway 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 pivotpin. 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 pivotpoint is already
in adjustment ; but if the needle does not so cut, bend the pivotpoint 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 sightslits 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'
33i 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 compassbox 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 wellknown 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 centerstakes. 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 spiritlevel;
but more frequently oy a slopeman. with a straight 12ft 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 fieldbook 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 fleldbook, 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 officework ; 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 stationstake. 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 crosssectitm 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 stationstake. 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 leadpencU 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 stationstake, we find « I to be say
2 ft ; but Z is above the level of the stationstake, 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 leadpencil dot at t ; and then go to the next fetation 1. Here
we see that the height of 103 ft coincides with the stationstake 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 stationstake, Fig 4, the level is 104; that the crosssec
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 stationstake, 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 groundslopes. Then these lines, measured by the scale, plainly give the
requirea dists.
When the ground is very irregular transversely, the crosssections 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 crosssections upon.
When the ground is very steep, it is usual to shade such portions of the map to
represent hillside. 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 slopeinstrument.
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* bandlerel, 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 oronsseotion, 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 haodlevel. 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 oMwtgbai O, is nio>ed burkward ur rorw&rd by a rauk niid plnian,
bf meBDB or the mlHeS hewl A. The slide of the lytgkia £. 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 objecicnd O of the
teleicape, to prerent Ibe ^are of the sun upua the objeclglass, 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 plmnbllne. 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 capstaiiscrewsti,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 eyepiece B reTerse the apparent position of objects intid€
cf the telescope ; which effect is obTiated, as regcurds exterior ol^Jects, by means of
the objectglass 0. This must be remembered when adjusting the crosshairs ; 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 bubbletube D D. One end of this tube can be raised or
lowered slightly by means of the two capstanheaded 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 bubbletube are two smidl capstanscrews,
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>bletube is the bak Y F ; at one end of which,
as at y, are two large capstannuts 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
elampserew 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 crosshairs 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 tanfrentticrew 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 tangentscrews, 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
tripodtaead ; and, in connection with three wooden legs Q Q Q, constitute
the tripod. In the figure are seen the heads of wingnuts J which confine the
legs to the tripodhead. Under the center of the tripodhead should always be
placed a small ring, from which a plumbbob may be suspended. This is not
needed in ordinary levelling, but becomes useful when rangmg centerstakes, &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^ectglass, 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 crossbairs ; 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 erossliairs brealK* see p 296.
Second* Miat of Uie bnbbletnbe D D, to place it parallel to the Une
308 TBB LBYBL.
0f coUimatlon. preTiomly •4asted; 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 bubbletube a^re supported;
flo that the bubbletube, 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 crosshairs, 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 eyeglass E, until
the crosshairs 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 objectglass 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 crosshairs, if the eye is moved a little up or down or sideways. To
secure this, the objectglass alone is moved to suit different distances ; the eyeglass
is not to be changed after it lb once properly fixed upon the crosshairs. The neglect
of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the
tangentscrew d, bring either one of the crosshairs to coincide x>reciM/y with the
object. Then gently, and without jarring, revolve the telescope naifway 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 onefuUf 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 tangentscrew bring the crosshair to
coincide with the object. Then again turn the telescope halfway 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 onehalf 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 bubbletube 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 bubbletube endwise, from t in tbe
foregoing Fig, its two hor adjustingscrews 1 1 are
seen as in this sketch. The larger capstanheaded
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 bubbletube, by means of the two nuts nn. Place
the telescope over a diagonal pair of the levellingecreWH K. K ; and clamp it there.
Open the clips of the Ys; and by means of the levelliugscrews 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 levellingscrews 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 levellingscrews, 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 bubbletube,** p 2ML
Horizontal adjustment of bubbletube ; 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 bubbletube, 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 bubbletube 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
bubbletube 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 levellingscrews
K. Place the telescope over two of the levellingscrews which stand diagonally;
and leave it there undamped. Then bring the bubble to the center of its tube, by
the two levellingscrews. Swing the upper part of the instrument halfway 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 halfway back by the large cap
stan nuts to, 10 ; and the other half by the two levellingscrews. 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 levellingscrews; 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 foresight he taken at preeiidy the tame distance from
the instrument as the backsight 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 targetman, 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 levellingscrews in many instruments become very hard to turn if dirty. Clean
with water and a toothbrush. Use no oil on field instruments.
Forma for level notebooks. 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 fieldbooks 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 righthand 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 backsight for that dist being taken at stake 5, and the Ibresight on stake
6; and thait the level, grade, cut, or fill is that at stake 6. The startingpoint 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 instrumentmaker ; 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 plumbline or other vertical line, and make a short continuous knife
Kiatch on the collar nearest the objectglass, 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 bubbletube is under the telescope.
THB HAKDIiBVKL
TOE BASDLETEI.
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 .
ebotlomofhlA
™,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 bondlevel, 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
handlevel 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»l1w'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 onmwlre 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 foreaighw ; 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. Halfway 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 slopelnstrament, or clinometer. As usually made,
the bubbletube 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 sealevel 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 sealevel, 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 sealevel, at the rate (very roughly) of about 1^ Fah
for every 200 ft near sealevel, 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 pistolshot 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 tissuepaper 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>iilk [au? of in Iniik.
THEBUOIfETEBS.
T« «liBnc« derreea of Fitlirenbelt 1« Ike eorrMipOBdIns de
■re«a •rc«ntl)?«de) l&kBiir>)irBidliiK32°liivnihitnih« ilitaoiis: mnlt
— lD°IMiit Agali,— 190F>b = r— II— a])>cCit = — •&XC'i?=— l^Oaal. ~ ~
To cli»iiKe P>h taMBOi uks & Fati rudlug 32^ Iswrthu ihs (Ina
• ti^eoBi.o. linln,— IPlVlisI— lsijfx'*+»='«SXtH'=— *1°B*M.
ToeliAnce £«ntto l'nbniiill ihe Oni nwllDg b> 0; dirlds by !
eIlAn» O
5l*i>^'=r^^?"°
>. Tkkaft
i^taii.—tfOmn=ixx'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
AIRATMOSPHERE.
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 oneflxtietii
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 divliiKbell, 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 il(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 windgauge, 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 doubtealoping roofi, or 10 lbs for •Kedrooft, 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 deadlevel plain, fully threequarters 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 watershed, especially as to rate of absorption, by the season of the year,
the frost in the ground, etc. The streamflow 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 selfregistering
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 onequarter 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 sealevel; 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 runoff
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 crosssection. 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 crosssection, minus that of the btick, is onetenth
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 crosssectional 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 lukewarm 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.
Selfreeordinir 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 twentythird 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 waterpipes; 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 airbubbles.
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 one21740th. (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 waterpipes^ 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 salammoniac 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. Castiron 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 wroughtiron bars
hooped together, were corroded about }^ inch deep; but had probskoly been pro
tected bv mud. The castiron shot became redhot on exposure to the air; and
fell to pieces like dry clay I"
** Unprotected parts of castiron sluicevalves, 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 lightgray castings remain secure for a long
time. Some castiron seapiles 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
coaltar, 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 waterpipea 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
4inch bar becomes reduced to a 2inch one, along that portion of it. Under
frenh water especially, or under ground, a thin coating of coalpitch vamishi
carefully applied, will protect iron, such as waterpipes, 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 seashore, 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 wellknown 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 wasteweirs, 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 sobdivided 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 trainman 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 \wkv¥ 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 two 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 liaTe 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 accelerstionf 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 conseqqence. 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 vrelfpltti9 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 socalled ** 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 T2.4
of = . But, to give to a mass, B, of , an
aoeelof , requires a force of . I »^]b=»a4 lb. 3liji 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 «  = r2 = 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 \Whw/
(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 — rs9 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 drawbar.
F 9
Then : Acceleration at a » ^ — r^ = 0.75 ft per second per second.
The tension T on the drawbar « 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 footpounde ; if the force is in tons and the distance
in inches, the product will be in inchtons ; and so on.f
Thus, if a force of moves a body through we have work =
1 pound 10,000 feet 10,000 footpounds
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 « footpounds.
(I») The footpound, the footton, the inchpound, the inchton, 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 footpound. The metric nnit of work
is the klloflrrammeter, i e. l Kilogram raised 1 meter = 2.2046 pounds
raiaed 3.2800 feet, = 7.23S1 footpounds. 1 footpound = 0.13825 kilogrammeter.
(«) 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 upgrade, 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 foot9>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 carbrake 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 clerkhire, 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 footpounds)
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 steamengine, 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 horsepowers; which means nothing
more than that it shall be capable of overcoming resisting forces at the rate of so
many times 33,000 footpounds per minute when running at a uniform velocity, i. e.,
when force X distance = resistance X distance.
(b) The liorsepoiver, 33,000 footpounds per minute, or 550 footpounds per
second, is the unit of ponrer, or of rate of ivork, commonly used in connec
tion with engines. The metric horsepoorer, called "force dt
cheval," " chevalvapeur," or (German) " Pferdekraft," is 75 kilogrammeters pel
second = 542.48 ftibs. per sec. = 32,549 ft.ft>s. per minute = 0.9863 horsepower. 1
horsepower = 1.0138 " force de cheval." In theoretical Mechanics the footponud
per second is used in English measure ; and the lUlo§;rammeter per ceo
ond in metric measure,
1 footpound per second =» 0.13826 kilogrammeter per second.
1 kilogrammeter per second = 7.2331 footpounds per second.
(c) Up to the time when the velocity becomes uniform, the power, 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 footpounds per second, we may find it by dividing the
work in footpounds 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 ftlbs. per sec " in ftlbs. 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 footpounds per mmute, or one horsopower; Sat
lbs. vel. lbs. vel. .
8300 X 10 — 33 X 1000 — 33,000 footpounds 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 cogwheels, 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 footpounds — 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 fLlba.
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 ftom 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 footpounds 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 onehalf 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 footpounds per second?
Mid, while ooming to rest, its
•Moa momentum » mass X T . ^r^ ■« 240 footpounds 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 footpounds 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
•totednp energy. We lift a onepound body onefoot oy expending upon it
one footpound of energy. But this footpound 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 footpound, and will part with one footpound 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 pentup 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 olockweight* 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 windingup. 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 boilershell; 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 everyday language these
terms are used indiscriminately to express the sime ideas.
Thus, we commonly speak of the " force " of a cannonball 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 clockweight) 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 dockyard, 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 footBote,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 sealevel ; 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 'Xjk
* 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 iscalled 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
semicircumference 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 crosssection, 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 crosssection of rim, we have :
_ ten sion in rim
~ area A of crosseection 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 == / = ^^ haffrim = '^^'^^^ ^f halfrim X ^^^ ;
and half of this Is the tension in each crosssection 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
croflssection, 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 crosssecticHi, use formulae (4), (5) and r6).
In a disc, sncli as a irrlndstone, the tension In each full crosssection
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 halfrim 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 crosssection 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 haltrim or halfdisc, 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 righthanded, of d lefthanded. 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 righthanded forces posi
tive, and downward or lefthanded 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
tiestrut or struttie.
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 righthand 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 lefthand moment, as is also that of P« about o. Right
hand or clockwise moments are conventionally considered as positive,
or +t and lefthand or counterclockwise 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 — WwMm = 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 ; nonconcur
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; noncolinear 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. ; noncoplanar, 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\
nonparallel 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 leithand 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. Antiresultant. The antiresultant 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 antiresultant 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 antiresultant of the two downward forces, w and W ; and the
downward resultant, W 4 to, of W and to, is the antiresultant of G. In
Fig. 11 (6), G (upward) is the antiresultant of W (downward) and to (acting
upward through the lefthand string). Similarly, this upward pull of tff is
the antiresultant of W and G.
24. In any system of balanced forces (forces in equilibrium), any one of
the forces is the antiresultant of all the rest ; and any two or more of them
have, for their resultaht, the antiresultant of all the rest. In such a system,
the resultant (and the antiresultant) of all the (balanced) forces is zero.
25. Anticomponent. The anticomponents 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 antiresultuit, 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 endpott, 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 righthanded forces must be
equal to the sum, ed + dc + co, of the three lefthanded forces. Or, con
sidering the righthanded forces, b a and a o, as positive, and the lefthanded
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 endpost 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 endpost, 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 noncolinear 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 antiresultant
of the horizontal comp>onent, db, of the oblique thrust in the endpoet 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 antiresultant 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 antiresultant, and c o the resultant,
ofea and a o; and o c. Fig. 22 (c), is the antiresultant, 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 meetingpoint 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 endpost 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 endpost 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.
OEOCPto
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
\>»«
DlK
• 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 subcomponents, 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 subcomponents (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 anticomponent, 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 subcomponents, o a"
and o b"j are in equilibrium at o, while the other two subcomponents, 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 Coordinates.
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 coordin
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 coordinates 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 coordinaUa, whatever the directions of those coordinatee 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 coordinates,
is zero. Thus, in Fig. 35 (6) or 36, if the sense of R be such that it shall act as
the antiresultant of the other three forces E, F and G, its component, o « or
o a, along either coordinate, will be found to balance those of the other
forces along the same coordinate.
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 coordinates,
may be zero; and yet, if the sum of those along the other coordinate 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 coordinates, 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 Antiresultant. The line, — R, required to com
plete the polygon, represents the an<iresultant 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 nonconcurrent forces, another condition must be satisfied. See ^ 83.
tR is here regarded as tending upward, so as to form the anftresultant 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 antiresultant
of all the rest. Any two or more of the forces balance all the rest ; or, their
resultant is the antiresultant 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 antiresultant (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 antiresultant 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.
•
Nonconcuirent Coplanar Forces.
79. Nonconcurrent Coplanar Forces. Fig. 44. The process of
finding the resultant of three or more coplanar but nonconcurrent 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 nonparallel 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 nonconcurrent 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
nonconcurrent.
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 nonconcurrent 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 nonconcuireiit
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 footnote (*), 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 nonooncurrent forces, and not its position. To find
the position of the resultant of nonconcurrent 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, Antiresultant. 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 antiresultant 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 antiresultant of the (four)
given forces, and we shall have a framework 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 antiresultant 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 antiresultant, — 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 antiresultant, — 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 antiresultant, — 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
aatiresuliant, 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 antire
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 Xoneoplanar 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 noncoplanar forces, whether concurrent or not, can be in
equilibrium.
103. Force Parallelopiped. The resultant of any three concurrent
non4ioplanar 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
NONCOPLAXAR 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.
(*)
Nonconcurrent Noncoplanar Forces.
lOG. Nonconcurrent Noncoplanar Forces. Fig. 50 (a). (For par
allel noncoplanar 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 Noncoplanar Forces. Th<% action of the weight
W of the wall. Fig. 51 (a), and of the noncoplanar 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 1ft. 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 antiresultant, 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) Flr. SS.
Noncoplanar Parallel Forces.
(&)
120. Noncoplanar 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.
ctf
(«)
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 noncoplanar 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 noncolinear. If they are noncolinear, 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.
126. 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 fanifeedge, 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 found is the required center of gravity G. In such cases it is
generally most convenient to draw the lines through the several centers of
gravity perpendicular to the axis of symmetry, so that the line of gravity
found will also be perpendicular to it.
But if. as in Fig. 8, the body or figure, etc.. is not symmetrical, we must find
a second line of gravity, the intersection of which with the first will give the
• center of gravity, G* To do this, repeat the process, drawing another set of
parallel lines through the several centers of gravity, Fig. 8. It will be most
convenient to draw them horizontally, or at right angles to those already drawn,
and in the following instructions we suppose 'this to be done.
890
FORCE Ilf BIGID BODIES.
Then draw a seoond funicular polygon, m'n'p'i^ Fie. 8, making the line^
w^n,' etc., pdrp<mdteutar (instead of parallel) to the radial lines O z, etc.. Fig^9;
and draw the second line of gravity, t' Q. tnroueh %\ perpendicular to the nxBt
Then Gt is at (he intersection of the two lines of gravi^.
The drawing of the second ftinicular polygon is ofben less simple than thift
of the first, because in the second thepaiBlIellines through the several centers
of gravity do not necessarily follow each other in the same order as in the first
Bear in mind that the two lines (as n'j/f n' m') meeting in the parallel line
(as hnf) pertaining to any given pait, 6, of the figure, must be perpendicnlar
respectively to those radial lines (O a, O b) which meet the ends of the line,
a 6, that represents that same part.
Figs. 10 and 11 show the application of the same process to an irregalar fi^
nre composea of three rectangles, a, 6 and c The lettering is the same as m
FiKS. 8 and 9; but in Fig. 10 it happens that ^ and j/ of the seoond ftiniculac
poqrgon iiedl upon the same point.
EUs. lO
BU8.1X
If the centers of gravity of the several bodies, or of the several parts of the
body, etc., are in more than one plane, we must find their projections upon
certain planes, and apply the process to those prujections.
OEMTBK OF GRAVITY.
891
Speeial Rules*
132. Special Rnles, derived from the general rules, (1) to («).
Ijinea.
(T). Stnilffl&t line. O is in the line, and at the middle of tts length.
(8). Circular »r©,* aob. Figs. 12 and 13 (center of circle at e). « is in tt»
line CO joining the center of the circle with the middle of the arc, and
cO *— radius a e X
chord a b
lengthof arcao6 *
<8a). If the arc is a aemieirelef*
eQ — radius a X —
vr
— radius a e X 0^80.
(85). Approximate rules for distanee aG, Fig. 12, from chord to center d
giuviiy.
If rise a o a* .01 chord a 6; «0 ^ .666 8 9
«••«= .10 *' •* ; « ,=» .665 • o
s> .6A3«o
a« .660 « o
^M7 to
a
«
a
«
a
« — .15
« —.20
•«— .26
M
«
•«
U
If rise «o — .80 chord ah; «Q— .663 8 o
" — .35 ** " ; " — .648 «
U M
M
a
u
« —.40
« —.46
— .60
ft
u
**
M
— .645ao
— .641 « 3
— .6»7 «
(9). Triancle, a be. Fig. 14. The center of
gravity, O, of its three sides* is the center of the
circle inscribed by a triangle, d ef, whose corners
are in the centers of the sides of the given triangle.
(10). Parallelofprana (square, rectangle,
rhombus or rhomboid^ The center of gravily
of the four sides* is at the intersection of the
diagonals.
(11). CJirele, •llipse. or regular polygon.
The center of gravity of the outline or circumfer
ence* is the center of the figure.
(1»). Ragiili
or ftmatniii. The center orgravity of "the edgt
In the prism, the position of G is not affected by either including or ezcludio^
the sides of b<^h oi the polygons forming the ends.
(12a). Cycloid.* Seep. 194.
prtana, right or oblique, and riglkt regular pjramidf
e center of eravity of the edees* is the center of the axis.
Sarfaees*
A. Plane anrflao^a.
We now treat of the csenters of gravity of plane turfaces, which may be
regarded as infinitely thin flat bodies. The rules for surfaces inay be used
also for actual flat bodies, in which, however, the center of gravity ts m the
middle of the thickness, immediately under the points tound by the rules.
(13) Parallelogram (square, rectangle, rhombus or rhomboid), clrole»
•llipae or regular polygon. G is the center of the figure; or the inter
section of any two diameters, or the middle of any diameter. In a ParallelO"
gram, G is the intersection of the two diagonals.
(14). Triangle, Fig. 15. G is at the intersection of lines (as a e and c d)
drawn from any two angles, a and c, to the centers, e and d, of the s ides, ot
•We are now treating of lines only; not of the surfeu^es bounded by theAi
F6r surfaces, see rules (13), etc.
FOBCB Df BIOID BODIES.
(1*6), Kb, IB, i(.»,«u t .U.U
Ders aad of 6 tKm uiy MrHlgbt 11ns m iiibuti u i^ i ^im
OG' JiCoC + ftfr* + B«0.
niJH give* Ds tbe pcaltlDU of the Una of gnTitr Q O". Id (he same nj m
Bod the dlBtanos Qfy'tit Q from uiy teiMnd line or plane, ft" c*. ThlB ^Tea
aejtie ^aiUoa of ■ secoDd lloe of gnTit^OG'. O is at the interaectoa ot
GO'
•ss
G*.
It foHoWB trom thb that the alkorfHl dIetanoe,eD,af O from aiiTalde (ai
■c) Is — k tbaahortsM distance, o* A, m<m the pune side to ICa oppoelle angle b.
rt rblloA's alBO that pQ''%pi,iula Rule 1<)
r). Trapeilua or lispculd, Pig. IS. For tiapeaolds, see also Bale
Draw the two diaEonB!B,se and bd. Dlvlda either ol tJiem. as o e. Into
'0 equal parte. anandttn. Ftomb,oBbd,lKr<dIb»—itiottromiH^ta
16 is tbe center of gravis of tbotriaasl