The civil engineer's pocket-book

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THE

CIVIL ENGINEER'S
POCKET-BOOK

JOHN 0. TRAUTWINE

CIVIL ENGINEER

EKVISBD BY
JOHN C. TKAUTWINE, Jb.

AND

JOHN C. TRAUTWINE, 3d.

CIVIL ENGINEERS

EIGHTEENTH EDITION, NINETIETH THOUSAND

NEW YORK

JOHN WILEY A SONS
LovDoir: CHAPMAN & HALL, Limited

1907

\.

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

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

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

>

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

ELECTROTYPERS AND MIINTKRS BINDERS

PHILADELPHIA PHILADELPHIA

THE AUTHOR

DEDICATES THIS BOOK

TO THE MEMORY OF HIS FRIEND,

THE LATS

BENJAMIN H. LATROBE, Esq.,

CITIL ENQINEXK.

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

JOHN 'WILEY & SONS,

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

PREFACE

TO FIRST EDITION, 1872.

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

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

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

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

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

V

VI PREFACE.

^^^ •

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

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

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

PREFACE. Vll

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

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

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

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

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

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

Viii PREFACE.

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

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

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

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

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

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

probably escape notice.

John C. Tbautwhi*.

Philadelphia, November 13th, 1871.

PREFACE TO NINTH EDITION.

TWENTY-SECOND THOUSAND, 1885.

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

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

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

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

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

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

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

ix

X FB£:fAC£.

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

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

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

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

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

PREFACE TO EIGHTEENTH EDITION.

(SEVENTIETH THOUSAND, 1902.)

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

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

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

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

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

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

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

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

35- 47 Arithmetic 33-37

210-211 Specific Gravity 380-381

265-266 Time 395

282-283 Chains and Chaining 176

284-290 Location of the Meridian 177-179

322-325 Rain and Snow 220-221

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

xi

• •

XU PREFACE.

New Pages Old Pages

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

499 Shearing Strength 476

499-500 Torsional Strength 476-477

501-503 Opening Remarks on Hydrostatics 222-224

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

689-744 Trusses 647-614

856-864 Locomotives 805-810

865-866 Cars 811-813

867-869 Railroad Statistics 814-818

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

930-942 Cement 673-678

943-947 Concrete 678-682

954-956 Timber Preservation 425-425 a

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

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

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

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

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

FBEFAOS. xiii

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

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

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

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

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

XIV PREFACE.

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

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

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

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

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

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

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

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

PREFACE. XV

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

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

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

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

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

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

Cement, Mr. Richard L. Humphrey.

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

Preservation of Timber, Mr. O. Chanute.

Building Material, Mr. John T. Willis.

John C. Trautwine, Jr.,

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

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

XTi

CONTENTS.

MATHEMATICS, paob

Mathematical Ssnnbote 33

Greek Alphabet 34

Aritliinetie.

Factors and Multiples 35

Fractions 35

Decimals 37

Ratio and Proportion 38

Progression 39

Permutation, Combination, Al-
ligation 40

Percentage, Interest, Annuities 40

Simple Interest 41

Equation of Payments 42

Compound Interest 42

Annuity^ Sinking Fund, De-
preciation, etc 43

Equations and Tables. . .44r-46

Duodenal Notation 47

Reciprocals 48-52

Roots and Powers.
Square and cube.

Tables ; 64

Rules 66

Fifth Roots and Powers .... 67

LoKarithms 70

Rules 70

Logarithmic Chart and Slide

Rule 73

Two-page Table 78

Twelve-page Table 80

Geometry. Alensiiration,
and Tnyonometrjr.

liines.

Definitions 02

Angles-
Definitions 92

Construction 93

Bisection 94

Inscribed 94

Complement and Supplement . 94

In a Parallelogram 95

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

Angles 96

Sine, Tangent, etc 97

Definitions 97

Table 98

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

PAOB

Polygons.

R^^ular — , Tables, etc.. of — 148

Triangles.

Dennitions. Properties 148

Right-angled — 150

Trigonometrical Problems . . 150

Parallelogram '. 157

Trapezoid. Trapezium 158

Polygons 159

Regular 159

Reduction of Figures. . .159, 160

Circle 161

Radius. Diameter 161

Area, Center, to Find — ... 161

Problems 161. 162

Tables of — .

Diameter in Units, Eighths,

etc 163

Diameters in Units and

Tenths 166

Diameters in Units and

Twelfths 172

Arc. Circular.

Chord, Length 179

Radius, Rise, and Ordinates. 180

Of Large Radius, to Draw — 181

Tables of — 182-185

Circular Sector, Ring, Zone,

and Lune 186

Circular Segment.

Area of — ; to Find 186

Area of — ; Table 187

Ellipse.

Properties of ^ 189

Ordinates and Circumference

of —; to Find — 189

Elliptic Arc 189

Tables of Lengths of — ... 190

Area of; to Find — 190

Construction. Tangents. . . 190

Oval or False — 191

C^ma Recta, Cyma Reversa,

Ogee 191

Parabola.

Properties of — 192

Parabolic Curve. Length of-^- 192

Area 192

Parabolic Zone or Frustum . 192

Construction 193

Cycloid 194

Solids.

Regular Bodies. Tetiahedron,

Hexahedron, etc 194

Guldinus Theorem 194

Parallelopiped, Properties 105

XXV

XXVI

CONTENTS.

PAGE

Priam .- 195

Frustum 195

Cylinder.

Volume and Surface of — . . 196

Volume. Table of — , in Cu.

Ft. and U. S. Gala 197

Wella; Contenta of — and

Masonry in Walla of — ... 198

Cylindrio Ungula 199

Pyramid and Cone 200

Frustums of 201

Prismoid 202

Wedge 203

Sphere.

Properties 204

Volume, Surface, etc.

Formulas for — 204

Tables of — 205-207

Segment and Zone of — . . . . 208

Spherical Shell 208

Spheroid or Ellipsoid 208

Paraboloid 209

Frustum of — 209

Circular Spindle 209

Circular Ring 209

Specific OraTity.

Principles 210

Table 212-216

Welgrbts and Measures.

U. S., British and Metric — ,

Units of — 216

Coins; Foreign and U.S. — 218

Gold and Silver 219

Weights; Troy, Apothecaries'

and Avoirdupois — 220

Long Measure 220

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

a Foot. Table 221

Square or Land Measure 222

Cubic or Solid Measure 222

Liquid Measures 223

Diy Meaaure 223

British Imperial Measures 224

Volumes and Weights of Water 224

Metric Units 226

Systfeme Usuel, — Ancien 226

Russian 227

Spanish 227

Conversion Tables 228

Introduction and Explana-
tion 228

List of Tables 229

Fundamental Equivalents . . 230

Abbreviations 230

Equivalents and Numbers in

Common Use 231

Metric Prefixes 231

Tables 232

Aorea per Mile and per 100 feet.

Table 254

PAGE

Grades, Tables of — 255-257

Heads and Pressures of

Water; Tables of — 258-260

Discharges in Gals, per Day

and Cu. Ft. per Second;

Tables 261-265

Time. Definitions, etc 265

Standard Railway — 267

Dialing 268

Board Measure. Table 269

Survey infT.

Testa of Accuracy, Distribution

of Error, etc 274

Chaining 282

Location of Meridian 284

By Circumpolar Stars 284

Definitiona 284

By Meana of Polaris 285

By Means of Any Star at

Equal Altitudes 287

Times of Elonflnition and Cul-
mination of Polaris 288

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

Engineer's Transit 291

Adjustment and Repairs. . . . 294

Vernier 296

Croas-hairs; to Replace 296

Bubble Glasa; to Replace. . . 296

Theodolite . . ; 296

Pocket Sextant 297

Compaaa.

Adjustment 298

Magnetic Declination and
Variation.

Isogenic Chart of U. S 300

Declination 301

Variation 301

Demagnetization 302

Leveling.

Contour Lines 302

Y Level 306

Adjustment 307

Forms for Notes 309

Hand Level, Adjustment . . . 310

Builder's Plumb Level 311

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

or Boiling Point 312

Table 316

NATrRAI. PHENOMENA.

Sound.

Volocity of 316

Heat.

Expansion and Melting Points.
Table 317

Thermometer.

Conversion of Scales 318

Tables 318, 319

CJ0NTENT8.

XXVll

Air. Atmospliere. page

Properties 320

Pressure in Diving Bells, etc. . . 321

Dew Point 321

Heat and Cold, Records of ... . 321

Wind.

Velocity and Pressure. Table. 321

Bain »nd Snow.

Precipitation.

Average 322

Effect of Climate on — 322

and Stream-flow 323

Maximum Rates of — 323

Weight of Snow 323

Rain Gau^ 324

Precipitation, Details of — in

U.S., Table 325

Water.

Composition, Properties 326

Ice 326

Effects of Water on Metals, etc. 327

Tides 328

KTaporatlon, ratration,

lieakai^e 329

MECHANICS, FOBCE IN

RieiD BOBIES.

Definitions 330

Matter; Body 330

Djmaiiiies.

Motion, Velocity 331

Force 332

Action and Reactioti 333

Acceleration 334

Mass 336

Impulse 337

Density; Inertia 338

Opposite Forces 339

Work :. 341

Power 842

Kinetic Energy 343

Momentum 345

Potential Energy 346

Impact 347

Gravity, Falling Bodies 34$

Descent on Inouned Planes . . . 349

Pendulums 350

Center of Oscillation 351

Center of Percussion 351

Angular Velocity 351

Moment of Inertia 351

Radius of Gyration 352

OnthfuffBd Force 354

StatlctB. PAoa

Forces .• 358

Line of Action 359

Stress 359

Moments 360

Classification of Forces 361

Composition and Resolution

of Forces 362

Force Parallelogram 364

Foi-ce Triangle 367

Rectangular Components 369

Inclined Plane 369

Stress Components 371

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

of Co-ordinates 372

Force Polygon 374

Non-coneurrentCopUnarForoes 375

Equilibrium of Moments 376

Cord Polygon 377

Concurrent Non - coplanar

Forces 380

Non-concurrent Non-coplanar

Forces 381

Parallel Forces 382

Coplanar 382

Non-coplanar 385

Center of Gravitv 386

Stable, Unstable, and Indif-
ferent Equilibrium 387

General Rules 387

Special Rules 391

Line of Pressure. Center of

Force or of Pressure 399

Position of Resultant 399

Distribution of Pressure .... 400

"Middle Third" 402

Couples 404

Friction 407

Coefficient ' 408

Morin's Laws 410

Table of Coefficients 411

Other Experiments 412

Rolling Friction 414

Lubricated Surfaces 415

Friction Rollers 417

Resistance of Trains 417

Workof Overcoming Friction 418

Natural Slope 419

Friction of Revolving Shaft 419

Levers 419

StabUity 422

Work of Overturning 422

On Inclined Planes 424

The Cord 425

Funicular Machine 427

Toggle Joint 427

PuHey , . . . 428

Loaded Cord or Chain 428

Arches, Dams, etc. Thrust

and Resistance Linec .... 430

Arches 430

Graphic Method 430

Practical Considerations. . 432

Masonrv Dam 433

Graphic Method 435

Practical Considerations. . 436

The Rcrew 436

zxviii

OONTBKTB.

PAOB

Forces Acting upon Beams and

Trusses 437

Conditions of Equilibrium. . 437

End Reactions 439

Moments 440

In Cantilevers 442

In Beams 443

Inclined Beams 445

Curved Beams 446

Shear 446

Influence Diagrams 449

For Moments 449

For Shear 460

Relation between Moment
and Shear 452

STREHGTS OF HATE-
1IIAI.S.

Ctoneral Principles. 454

Stretch, Stress and Strain .... 455

Modulus of Elasticity 456

Limit of Elasticity 458

Yield Point 459

Resilience 460

Suddenly Applied Loads 460

Elastic Ratio 461

Strengths of Sections 462

Fatigue of Materials 465

TransTerae Streng^tb

Conditions of Equilibrium .... 466

Neutral Axis 466

Resisting Moment 467

Modulus of Rupture 468

Moment M Inertia. 468

Table 469

Section Modulus 473

Loading. Strength 473

Table 474

Beam of Unit Dimensions .... 475

Coefficients, Table 476

Weight of Beam as Load 477

Comparison of Similar Beams. 478

Horizontal Shear 478

Deflections 480

Elastic Limit 482

Elastic Curve •. . 482

Deflection Coeffioi^it 483

Eccentric Loads 484

Uniform Loads • 486

Inclined Beams ....'. 485

Sirlindrical Beams 485
aximum Permissible — . . . . 485

Suddenly Applied Loads . . . 486
Uniform Strength 486

Cantilevers. Table 487

Beams. Table 488

Continuous Beams 489

Table 490

Cross-shaped Beam 492

Plates 492

Transverse and Longitudinal

Stresses Combined 493

PAoa
Strengrtb of Piilam. 496

Radius of Gyration 496

Table 496

Remarks 40S

Slieariiiff Strentrtli . 499

ToMtanal 8ir«iivtli. 490

HTDBOSTATICfiL

Principles 601

Center of Pressure 601

Air Pressure 602

Horisontal and Vertical

Components 603

Pressure in Vessels 503

Opposite Pressures 503

Rules 604

Transmission of Pressure 606

Center of Pressure 609

Walls to Resist Pressure 608

Thickness at Base 609

Stability 510

Contents 510

Liability to Crush 51Q

Thickness for Cylinders 511

Iron Pipes 512

Lead Pipes 513

Buovancy 513

dotation. Metaeenter 614

Draught of Vessels 515

HTDRAUI«ICS.

Flow Of W«ter tbrouffb

Pipes 610

Head of Water 616

Velocity Head 616

Entry Head 616

Friction Head 616

Pressure Head 618

Piezometers 618

Hydraulic Grade Line 519

Siphon 620

Velocity Formulae 622

Kutter's Formukk 523

Weight of Water in Pipes 526

Areas and Contents of Pipes . . . 526

Total Head Required 627

Table of Velocity and Friction

Heads and Discharge 628

Compound Pipe 631

Venturi Meter.

Theory 632

Tube 634

Register 536

Ferris-Pitot Meter 53ft

Curves and Bends 637

OONnsStB,

PAOK

Flow thronff li Ortflees

Tbeoretical Velocities £39

With Short Tubes 640

Through Thin Partition 641

Discharge from One Reservoir

to Another 643

Rectangular Openings 644

Time of Emptying Pond. . . . 646

Miner's Inch 646

Flow OTor Wolrs

End Contractions 647

Measiu«ment of Head 648

Formulae , 649

Francis 660

Table of Discharges 561

Basin 662

Values of m 663

Submerged Weirs 664

Velocity of Approach 666

Iztelined Weirs 668

Broad-crested Orerf all 669

Triangular Notch 669

Trapezoidal Notch 669

Flow In Open Channels

ligations of Velocities 660

Steam Gauging 660

Pitot Tube, etc 661

Wheel Meter 662

Abrasion of Channel 663

Theory of Flow 663

Kutter's Formula 664

Coefficient of Roughness 664

Coeffs of Roughness. Table 666

Coefficient, e. Table 666

To Draw Kutter Diagram. 670

Flow in Sewers 674

Flow to Sewers 676

Flow in Drain-pipes 676

Constriction of Channel 676

Scour 677

Obstruction's in Streams 677

Power of Falling Water 678

Water Wheels. 678

Hydraulic Ram 678

Power of Running Stream .... 678

COVSTBVCnONS, ETC.

"Dredging*

Cost of Dredging 680

Horse Dredges 681

Weight of Material 681

Foundations.

Foundations 682

Borings in Common Soils 682

Unreliable Soils 683

Resistanoe of Soils. . , 688

PAOB

Rip-rap 583

Protection from Scour 683

Timber Cribs 684

Caissons 685

Coffer-dams 686

Earth Banks 686

Crib Coffer-dams 687

Mooring Caissons or Cribs 689

Sinking through Soft Soil 689

PUes 689

Sheet Piles 690

Grillage 690

Pile Drivers 690

Resistance of Piles 592

Penetrability of Soils 693

Driving 693

Screw Piles 694

Drivin/s by Water Jet 695

Hollow Iron Cylinders 696

Pneumatic Process 696

Timber Caisson 598

Masonry Cylinders 699

Fascines 699

Sand-Piles 699

Stonework.

Cost, etc 600

Retaining Walls.

General Remarks 603

Theory 606

Surcharged Walls 609

Wharf Wails 611

Transformation of profile 611

Sliding, etc 612

Stone Bridg^es.

Definitions 613

Depth of Keystone 613

Pressures on Arch-stones 614

Table of Arches 615

Abutments 617

Abutment Pi^s 619

Inclination of Courses 620

Culverts 622

Wing Walls s, 624

Foundations 627

Drains 627

Drainage of Roadway 62S

Contents of Piers 62$

Brick Arches 62P

Centers 631

Timber Bams.

Primary Requisites 642

Examples 642

Abutments. Sluices, Ground

Plan, Cost 645

Measuring Weirs 64i

Trembling 648

Thickness of Planking Re-
quired 648

CONTENTS.

WATER SUPPI<T. PAGE

Consumption, Use and Waste. 649
Waste Restriction ; Water

Meters 649

Water for Fire Protection . . . 650

Reservoirs 650

Leakage through — , Mud

in— 651

Storage Reservoirs 652

Valve Towers, etc 652

Comj^ensation 653

Distributing Reservoirs .... 653

Water Pipes 653

Concretions in — , preven-
tion of — 655

Weights of Cast Iron Pipes . . 666

Wrought Iron Pipes 656

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

Pipe Joints 660

Pipe Jointer 660

Flexible Joints.. 661

Special Castings 661

Repairs and Connections. . . 662

Air Valves 662

Air Vessels, Stand-pipes 663

Service Pipes 664

Tapping Machines 664

Anti-bursting Device 665

Valves, Gates 666

Fire Hydrants 668

TEST AND WEI^Ii BORING.

Test Boring Tools 670

Artesian Well Drilling 671

ROCK DRII4I1S.

Diamond . Drills 675

Percussion DrjUs 676

Hand Drills 681

Channeling 681

Air Compressors 681

TRACTION, ANIMAIi
POWER.

On Roads, Canals, etc 683

TRUSSES.

Introdnetion.

General Principles 689

Loading, Counterbraoing 690

Cross bracing 691

Types of Trusses 691

Camber 696

Cantilevers 696

Movable Bridges 696

Skew Bridges 697

Koof Trusses 698

Stresses in Trnss Mem-
bers

€(eneral Principles 698

Method by Sections 700

Chord Stresses, Moments,

Chord Increments 701

FACIB

Shear 702

Influence Diagram 702

Dead Load Stresses 703

Live Load Stresses 705

Typical Wheel Loads 706

Cooper's 706

Live Load Web Stresses 706

Live Load Chord Stresses. . . 709

Wind Loads 710

Impact, etc 711

Maximum and Minimum

Stresses 712

Effect of Curves 712

Counterbracing 713

Stresses in Roof Trusses 713

Weights and Loads 713

Wind Pressures 714

Graphic Method 715

Timber Roof Trusses 716

Deflections 718

Redundant Members 720

Brtdg^e I>etalls and Con-
struction

General Principles 720

Floor System and Bearings. . 720

Design 721

Flexible and Rigid Tension

Members 721

Compression Members 721

Pin and Riveted Connec-
tions 721

Floor Beam Connections 721

Tension Members, Detail . . . 722

Compression Members, De-
tail 722

End Post and Portal Bracing 723

Joints 724

Pin Plates 724

Pins 725

Expansion Bearings 725

Loads, Clearance, etc., for

Highway Bridges 726

Camber 726

Examples 726

Weights of Steel Railroad

Bridges 731

List of Large Bridges 732

Timber Trusses 732

Joints 733

Howe Truss Bridges 736

Examples 738

Metal Roof Trusses 740

Broad Street Station, Phila. . 740

List of Large Arched Roofs. 742

Timber Roof Trusses 742

Transportation and Erection . . 743

Digests of Speelfleations for
Brldgres and Buildings.

For Steel Railroad and
Highway Bridges.

General Design 745

Material 751

i Loads 755

C0NTEKT8.

PAOB

Btreeses and Dimensioos 759

Protection 763

Erection 763

For Combination Railroad
Bridyes.

General Design 763

Material 763

Loads 764

Stresses and Dimensions 764

Protection 764

For Roofli, Bulldlngns* etc.

General Design, Material, etc.. 764

Sl^SPENSIOM BRIDOIS.

Data Required 765

- Formulas 766

Anchorages 770

RITETS AND RITETINe.

Rules and Tables 772

RAIIiROADS.

Carves.

Definitions 780

Tables, etc 784

EartliworlK.

Table of Level Cuttings 790

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

Tunnels.

Coostruction 812

Trestles.

Construction 813

Track.

Ballast 815

Ties 816

Tie Plates 816

Rails 817

Spikes 818

Rail Joints 819

Turnouts 824

Eqnlpment.

Turntables I 845

Water Stations 851

Track Tanks 853

Track Scales, Fences, etc 854

Cost of Mile of Track 855

Rolling Stoe

J?

XXXI

PASS

Locomotives.

Dimensions. Weights, etc. . . 856

Performance 860

Tonnage Rating 862

Fast Runs 863

Running Expenses 864

Cars 865

Statistics.

Earnings, Expenses, etc.

867

MATERIAUS).

Metals.

Iron and Steel.

'Requirements. International

Ass'n for Testing Materials. 870

Cast Iron 874

Weight 875

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

Steel 877

Roofing Iron 880

Corrugated Iron 881

Wrought Iron Pipes and Fit-
tings 882

Screw Threads, Bolts, Nuts

and Washers 883

Lock-nut Washers 885

Buckle Plates 885

Bolts. Weight and Strength,

Table 886

Wire Gauges 887

Circular Measure 889

Wire, Table , 891

Structural Shapes.

I Beams 892

Channels 894

Angles and T Shapes 896

Separators for I Beams 900

Z-Bar Columns 901

Phcenix Segment Columns . . 904

Gray Column 906

Strengths of Iron Pillars,

Tables 907

Floor Sections 914

Chains 915

0kber Metals.

Tin and Zinc 916

Copper,' Lead, etc 918

Tensile Strengths, Table 920

Compressive Strengths, Table. 921

Stone, etc.

Tensile Strengths. Table 922

Compressive Strengths, Table 923

Transverse Strength, ^^able. . . 924

XXXll

CONTENTS.

lIortov,Briclu»efe. page

Lime Mortar 926

Bricks 927

Cement 930

Cement Mortar 931

Sand 935

Effects on Metab 936

Efflorescence 936

Silica Cement 937

Recommendations, Am. Soc.

C. E 937

Tests 938

Report of Board of U. S A.

Engineer Officers 940

' Tests 941

Requirements 942

Concrete 943

Properties 943

Handling 946

Explosives.

Nitro^ycerine and Dynamite. 948

Blasting Powders 951

Firing 962

Gunpowder 963

Timber.

Decay and Preservation 954

Tensile Strength 957

Compressive Strength 958

Transverse Strength 959

Strength as Pillars 963

B«lldlii|r Materials and

Op^V^^OXS. PAOS

Plastering 966

Slating 969

Shingtes 971

Painting 971

Glass and Glasing 973

Sundry Materials.

Rope 976

Wire Ropes 976

Paper 978

Blue Prints, etc 979

Price lilst and Business Bt-
rectoiry.

Prieelist 984

Business Directory 996

Biblioffrapiiy.

List of Engineering Books 1008

GLOSSARY 1026

INDEX 10»

KATHEMATIGS.

MATHEMATICAI. STMBOIA.

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

— Minas, nejg^ative, subtract.

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

7 Minus or plus.

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

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

y) ^

: : : Proportion. Thus, a : b : : c : <2, as a is to 6, so is 0 to <<.
-= Equals, is equal to.
> Is ffreater than. Thus, 6 > 5.
< Is less than. Thus, 5 < 6.

'^ Is not equal to.

:^ Is greater or less than.

j^ Is not greater than.

^ Is not less than.

;^ Is equal to or greater than.

^ Is equal to or less than.

oc la proportional to, varies with.

00 Innnity.

J. Is perpendicular to.

^ \ Angla

'v Is similar ta

I la parallel to.

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

** J a s— nth root of a.
Parenthesis.

11

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

-Vinculum. J

*.* Since, because.

.*. Hence, therefore.

o Degrees.

' Minutes of arc,* feet.

" Seconds of arc,* inches. *

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

a prime ; of', a second, etc.

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

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

Diameter '

E, Modulus of elasticity.

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

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

approximately 9.81 meters per second per second.

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

3 33

34

OBEEK ALPHABET.

THE eREEK AI.PHABET.

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

Greek letters.

Name.

Approximate
equivalent.

Commonly used to designate

Capital.

Small.

*

A

a

Alpha

a

Angles, Coefficients.

B

^

Beta

b

it u

r

y

Gamma

g

" " Specific gravity.

A

i

Delta

d

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

s

«

Epsilon

e (short)

-j 2.7182818.

V Eccentricity in conic sections.

z

<

Zeta

*

Co-o'rdinates, Coefficients.

H

n

Eta

e (long)

ii (I

e

9&

Theta

th

Angles.

I

I

loU

i

K

iC

Kappa

k

A

A

Lambda

1

Angles, Coefficients, Latitude.

M

Jtt

Mu

m

tt t<

N

V

Nn

B

t(

B

f

Xi

X

Co-ordinates.

O

o

Omicron

0 (short)

n

w

Pi

P

Circumference -i- radios.*

p

p

Bho

r

Badius, Batio.

2

o-«

Sigma

•

Distance (space).t

T

T

Tau

t

Temperature, Time.

Y

V

Upsilon

u or y

«

*

Phi

ph

Angles, Coefficients.

X

X

Chi

ch

♦

^

Psi

P8

Angles.

o

w

Omega

o (lon«)

Angular velocities.

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

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

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

AssTButata. ' 35

ABITHMETIO.

FACTORS AND MVI4TIPI1ES.

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

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

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

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

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

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

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

675)782(1
575

A 207)575(2
414

B 161)207(1
161

G 46)161(8
188

D 28)46(2 H. G. F. =- D » 2&

46

0

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

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

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

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

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

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

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

FRACnOBTS.

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

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

36 ARITHMETia

(8) To rednce to a oommoii denominfttor. Let

N °. the new numerator of any fraction

n = its old numerator

d a its old denominator

C » the common denominator

Then _, C

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

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

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

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

Thu8,|,l^,f c = 84

2 _ 2X4X7 se. 1 _ 1X3X7 _ 21. 5 _gX3X4 eo
t ~ 84 T¥' T ~ 84 ^^' 7" 84 TT*

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

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

3_l7_64.7_13_,6

f_l— 2_1.8_5_2_7_20_jr.7_3_7 6_1

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

lvl_l. 8vl S_« 3 V 5 v^ 2 _ 6 .

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

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

l^l=,lv2_2_,. 3^1_8v4_8„-.

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

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

1^ simplified = |-.

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

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

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

85 85 + 17 S

ARITHMETIO.

87

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

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

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

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

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

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

2.679454 26,794.54 .^ ,_

0.0062 62

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

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

Table 1. Decimal eqniTalents

Of common

fractions.

8thB

16tha

SMi

64t]u

,

8ths

lethg

82dB

64tlis

1

:015625

S3

.515625

1

2
3

.03125
.046875

17

34 .
35

.53125
.546875

1

2

*4
5

.0625
.078125

9

18

36
,37

.5625
.578125

8

6

7

.09375
.109375

19

38
39

.59375
.609875

1

2

4

8
9

.126
.140625

5

10

20

40
41

.625
.640625

5

10
11

.15625
.171876

21

42
48

.65625
.671875

' 8

6

12
13

.1875
.203125

11

22

44
45

.6875
.708125

7

14
15

.21875
.234375

23

46
47

.71875
.734375

2

4

8

16
17

.25
.265625

6

12

24

48
49

.75
.765625

9

18
19

.28125
.296875

25

50
51

.78126
.796875

5

10.

20
21

.3125
.328126

•

13

26

52
63

.8126-
.828125

11

22
23

.34375
.359375

27

54
55

.84376
.859375

8

6

12

24
25

.375
.390625

7

14

28

56
57

.875
.890625

13

26

27

.40625
.421875

29

58
59

.90625
.921875

7

14

28
29

.4375
.453125

15

30

60
61

.9375
.958125

15

80
31

.46875
.484375

31

62
63

.96875
.984375

4

8

16

82

.5

8

16

82

64

1.

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

0.25

0.25
1.00

25
100

1
4'

= . ; 0.75 =

To
100

3

4'

^ : 0.800626 =

890626
1000000

57
64*

38 ABITHMETIO,

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

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

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

RATIO AND PlU^PORTIOir*

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

A*

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

is the duplicate ratio of A and B.

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

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

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

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

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

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

_ , B D B A D C

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

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

Composition. — - — = — ^ — ; — g—

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

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

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

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

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

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

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

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

In continued proportion,

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

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

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

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

A_AB AAC

C ' B • C' *°** B " C • B"

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

ABITHMETIC. 39

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

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

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

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

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

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

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

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

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

bricks X -g- = 20,000 bricks.

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

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

(12) Double rule of tbree.

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

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

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

PROGRESSION.

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

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

i = a + (n — 1) d

Required
I

Given
a d n

I

ads

s

a d n

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

dls o=»-|-(f± l/(/-|-^d)8 — 2d*

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

2d

n dls

21 + d ± >/(2/ + d)2— 8dj
2d

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

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

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

iatio = 2.

40 AKITHMETIO.

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

Bequired
I

Given
a r n

I

art

1

r H *

^^g + (r- 1)*
r

^ r" — 1

a n Z «=>

r n I * =

r«-.r*~*

«»-i

PiaKMVTATIOH, Ete.

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

1X2X3X4X6X6X7X8X9 = 362880 positions. Ans.

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

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

9X8X7X6 3024 ,„^ ., ^, .

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

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

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

fts. cts. cts.

20 X 15 = 800 _, - 1050 „, ,

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

60 lbs. 1050 cts.

PEBCENTAOE, INTEREST, ANNUITIES.

Percentagre*

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

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

AMTtBUmiC. 41

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

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

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

Stone =|. = 0.626= 62.6 " "

Concrete = f = 1000 = 100.0 " "

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

Interest.

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

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

Simple Interest.

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

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

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

A = P + P rn =» P (1 + rn).

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

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

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

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

the amount, A, we have

p

1 + rn

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

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

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

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

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

42

ARITHMETIC.

Principal .tl264.85

Interest, 2 mos, 1 per cent 12.64

2mo8, 1 " 12.64

" Imo, h " 6.82

" 20 days, I " 4.21

" 6 days, 0.1 " 1.26

" 2 days, ^ " 0.42

Interest at 6 per cent $37.^

Deduct one-sixth 6.25

Interest at 5 per cent $31.24

Equation of Paymente.

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

$ months.

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

300 X 6 = 1800

1200 5000

Compound Interest.

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

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

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

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

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

p = (l+r)«

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

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

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

<i

M

((

«(

i

k /

z.

21

y

x*-

rTv

<^

F(l + r)n ^

5

^

r

J

?r

i

».

^

^

«

^

w^^

<M

_r^^

at

^^

>;

\

r^^

*—

\

'

•8^'

^

1

I

^L

0

t

J

3

o

•

J

>

>

' >

'

>

r < '

i

> J

\ A

; ii

r 4

\ i

F <

:;

1 t

r 1

9 »

I

Years

Figr. 1.

ABITHHETIC.

43

Ttible S« CompouiMl Interest.

Amount of 81 at Compoand Interest.

8

»H

4

^

6

6H

6

«H

Yean.

per

per

per

per

per

per

per

per

cent.

oent.

cent.

cent

cent

oent

cent.

cent

1

1.030

1.035

1.040

1.045

1.060

1.066

1.060

1.065

2

1.061

1.071

1.082

1.002

1.103

1.118

1.124

1.134

8

1.098

1.109

1.126

1.141

1.168

1.174

1.191

l.i08

4

1.126

1.148

1.170

1.193

1.216

1.239

1.262

l.f86

5

1.159

1.188

1.217

1.246

1.276

L807

1.338

1.870

6

1.194

1.229

1.265

1.302

1.340

1.379

1.419

1.459

7

1.230

1.272

1.316

1.361

1.407

1.455

1.504

1.654

8

1.267

1.817

1.869

1.422

1.477

1.635

1.594

1.655

9

1.805

1.863

1.423

1.486

1.651

1.619

1.689

1.763

10

1.844

1.411

1.480

1.553

1.629

1.708

1.791

1.877

11

1.384

1.460

1.539

1.623

1.710

1.802

1.898

1.999

18

1.426

1.511

1.601

1.696

1.796

1.901

2.012

2.129

18

1.469

1.564

1.665

1.772

1.886

2.006

2.133

2.267

14

1.518

1.619

1.732

1.852

1.980

2.116

2.261

2.415

15

1.558

1.675

1.801

1.935

2.079

2.282

2.397

2.672

16

1.606

1.734

1.878

2.022

2.188

2.355

2.540

2.739

17

1.653

1.795

1.948

2.113

2.292

2.486

2.693

2.917

18

1.702

1.868

2.026

2.208

2.407

2.621

2.854

3.107

19

1.754

1.923

2.107

2.308

2.527

2.766

3.026

3.309

98

1.806

1.990

2.191

2.412

2.653

2.918

3.207

3.524

91

1.860

2.069

•2279

2.520

2.786

8.078

8.400

3.753

92

1.916

2.132

2.370

2.634

2.925

3.248

3.604

3.997

98

1.974

2.206

2.465

2.752

3.072

3.426

8.820

4.256

94

2.033

2.283

2.563

2.876

3.225

3.615

4.049

4.533

95

2.004

2.863

2.666

3.005

3.386

a8i8

4.292

4.828

98

2.157

2.446

2.772

3.141

3.556

4.023

4.549

5.141

97

2.221

2.532

2.883

3.282

3.733

4.244

4.822

5.476

98

2.288

2.620

2.999

3.430

3.920

4.478

6.112

5.832

98

2.857

2.712

3.119

3.584

4.116

4.724

5.418

6.211

80

2.427

2.807

3.243

3.745

4.822

4.984

&743

6.614

81

2.500

2.905

3.373

3.914

4.538

6.268

6.088

7.044

89

2.575

3.007

3.508

4.090

4.765

6.547

6.453

7.502

88

2.652

8.112

3.648

4.274

5.008

5.852

6.841

7.990

84

2.732

8.221

3.794

4.466

5.253

6.174

7.251

8.509

85

2.814

8.834

3.946

4.667

5.516

6.514

7.686

9.062

88

2.898

3.450

4.104

4.877

6.792

6.872

8.147

9.651

87

2.985

3.671

4.268

5.097

6.081

7.250

8.636

10.279

88

8.075

3.696

4.439

5.826

6.385

7.649

9.154

10.947

89

3.167

3.825

4.616

6.566

6.706

8.069

9.704

11.658

40

8.262

3.959

4.801

6.816

7.040

8.613

10.286

12.416

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

AnBnity, Sinkinir Fand, Amortisatloii, ]>epreeiaftloii.

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

44

ARITHMEnO.

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

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

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

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

i

> 1

1

z

•

1

I

(*+'>'■ ^

^

^

^

^

1 1 i

.^

^

J

V

t

1

1
L^

a

,^

f

>

r

>

f

J

y

f \

f

1

r \ r

<

> J

[ J

\ a

( 4

\ I

S i

i :

r «

r I

i »

%

Years
Flff.l.

O X 2 3 4 s a
Year*

FlV. 2.

7 S 9 n

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

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

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

Compound Interest.

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

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

fol

isW =

(1 + r)'

= (1 + r)

Annuities.

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

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

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

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

ARITUMETIG.

45

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

1 i—

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

r (1 + r) r

See Table 3.

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

1* r
P='W^ i —

1 —

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

jf = p -T

W

ft

1 —

(l + r)*-l

See Table 4. (1 + r) *

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

Bate of Interest (Compound).

2^

8

8H

4

4^

6

6Ji

6

Tears.

per

per

per

per

per

per

per

per

•

cent.

cent.

cent

•

cent.

cent.

oent.

cent.

cent.

6

4,646

4,580

4,515

4,452

4,390

4,829

4,268

4,212

10

8,752

8,580

8,816

8,111

7,913

7,722

7,688

7,360

16

12,381

11,938

11,517

11,118

10,740

10,380

10,037

9,712

ao

15,589

14,877

14,212

13,590

18,008

12,462

11,950

11,470

26

18,424

17,413

16,482

15,622

14,828

14,094

13,414

12,783

80

20,930

19,600

18,392

17,292

16,289

15,372

14,534

13,765

S6

23,145

21,487

20,000

18,664

17,461

16,374

15,391

14,498

40

25,103

23,115

21,865

19,793

18,401

17,159

16,045

16,046

46

26,833

24,519

22,495

20,720

19,156

17,774

16,648

15,456

60

28,362

25,730

23,456

21,482

19,762

18,256

16,982

15,762

100

36,614

31,599

27,655

24,505

21,950

19,848

18,096

16,618

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

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

Present worth Annuity
* Because, W $1.00

Equation (4)

Present worth Annuity

Sl.OO : p. Hence, j9

Equation (5)

1

Annuity .Amount Annuity Amount ^

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

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

46

▲BITHMEnO.

BTBUCVDBm, Apparatus, etc. Lvb, Ahkoitt

in years f

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

Permanent buildings 100 1.65

Cast iron pipe, railroad side-tracks 80 8.11

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

Engines and pumps 30 21.02

Boilers, electric light plants, tramways and equipment,

iron 'fences 20 87.22

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

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

BiVER Watkb, takkn within City
Ldcixb and Filtbbkd.

Unfiltbbed Watbb, by Aqubduct.

First Out.

8toraffe leservoirs. 930,900,000

Aqueducts 47,730,000

Distribution 8,655,000

Distributing reservoir 1,000,000

Total $88,185,000

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

I%rstCbH.

Filter plants 828,174,680

Mains ^ » 10,980,000

Depreciation

Operation and Maintenance.

Analyses and inspec-
tion 841,620

Ordinary repairs ^,150

Pumping and wages 140,770

198,640

281,540

Total $84,154,68^

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

Depreciation

206,540

Operaiion and MaMenanee.

Pumping 81,216,021

Filtration 525,600

82,925,780

1,741,621
82,971,801

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

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

Bate of Interest (Compound).

1

2

2^

t

«K

4

5

6

Years.

per

per

per

per

per

per

per

per

cent.

cenL

cent.

cent

cent.

cent.

cent.

cent.

5

196.04

192.16

190.24

188.36

186.49

184.63

180.98

177.80

10

95.58

91.33

89.25

87.23

85.24

83.29

79.60

75.87

15

62.12

57.83

66.77

53.77

61.82

49.94

46.34

42.90

20

45.42

41.16

89.14

37.22

85.36

33.58

30.24

27.18

85

85.41

31.22

29.27

27.43

25.67

24.01

20.96

18.28

SO

28.75

24.65

22.78

21.02

19.37

17.83

15.05

12.65

S5

24.00

20.00

18.20

16.54

15.00

13.68

11.07

8.97

40

20.46

16.55

14.84

13.26

11.88

10.62

8.28

6.46

45

17.71

13.91

12.27

10.79

9.45

8.26

6.26

4.70

50

15.51

11.82

10.26

8.87

7.63

6.55

4.78

8.44

60

12.24

8.77

7.35

6.18

6.09

420

2.83

1.88

70

9.93

6.67

5.40

434

3.46

2."74

1.70

1.08

80

8.22

5.16

4.03

8.11

2:88

1.81

1.08

0.578

90

6.91

405

8.04

2.26

1.66

1.21

0.627

0.318

100

5.87

3.20

2.31

1.65

1.16

0.808

0.383

0.177

ARITHMETio. 47

I>rODENAI« OB BUOBBNART NOTATION.*

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

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

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

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

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

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

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

48

RECIPBOCALS OP NUMBERS.

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

No.

Reciprocal.

No.

Reciprocal.

No.

fleciprocal.

No.

ReciprocaL

1

1.000000000

56

.017857148

Ill

.009009009

166

.006024096

2

0.500000000

57

.017543860

112

.008928571

167

.005988024

3

.333333333

58

.017241379

113

.008849558

168

.005952381

4

.250000000

59

.016949153

114

.008771930

169

.005917160

5

.200000000

60

.016666667

115

.006695652

170

.005882353

6

.166666667

61

.016393443

116

.008620690

171

.005847953

7

.142857143

62

.016129032

117

.008547009

172

.005813953

8

.125000000

63

.015873016

118

.008474576

173

.005780347

9

.111111111

64

.015625000

119

.006403361

174

.005747126

10

.100000000

65

.015384615

120

..008333333

175

.005714286

11

.090909091

66

.015151515

121

.008264463

176

.005681818

12

.083333333

67

.014925373

122

-.008196721

177

.006649718

18

.076923077

68

.014705882

123

.008130081

178

.005617978

14

.071428571

69

.014492754

124

.008064516

179

.005586592

15

.066666667

70

.014285714

125

.008000000

180

.005565656

16

.062500000

71

.014084507

126

.007936608

181

.005524862

17

.058828529

72

.013888889

127

.007874016

182

.005494505

18

.055555556

73

.013698630

128

.007812500

183

.005464481

19

.052631579

74

.013513514

129

.007751988

184

X)05434788

20

.050000000

75

.013333333

130

.007692308

185

.005405405

a

.047619048

76

.013157895

181

.007633588

186

.005876844

•22

.045454545

77

.012987013

132

.007575758

187

.005347594

28

.043478261

78

.012820513

138

.007518797

188

. .005319149

24

.041666667

79

.012658228

134

.007462687

189

.005291005

25

.040000000

80

.012500000

135

.007407407

190

.005263158

26

.038461538

81

.012345679

136

.007352941

191

.005235602

: 27

.037037037

82

.012195122

137

.007299270

192

.005208333

28

.035714286

83

.012048193

138

.007246377

198

.005181847

29

.034482759

84

.011904762

139

.007194245

194

.005154639

30

.033333333

85

.011764706

140

.007142857

195

.005128205

31

.032258065

86

.011627907

141

.007092199

196

.005102041

32

.031250000

87

.011494253

142

.007042254

197

.005076142

33

.030303030

88

.011363636

143

.006993007

198

.005050505

34

.029411765

89

.011235955

144

•UUOU's^.'z 4fB

199

.005025126

35

.028571429

90

.011111111

145

.006896552

200

.005000000

36

.027777778

91

.010989011

146

.006849815

201

.004975124

37

.027027027

92

.010869565

147

.006802721

202

.004950495

38

.026315789

93

.010752688

148

.006756757

203

.004926108

39

.025641026

94

.010638298

149

.006711409

204

.004901961

40

.025000000

95

.010526316

150

.006666667

205

.004878049

41

.024390244

96

.010416667

151

.006622517

206

.004854369

42

.023809524

97

.010309278

152

.006578947.

207

.004830918

43

.023255814

98

.010204082

153

.006535948

208

004807692

44

.022727273

99

.010101010

154

.006493506

209

.004784689

45

.022222222

100

.010000000

155

.006451613

210

.004761905

46

.021739130

101

.009900990

156

.006410256

211

.004739336

47

.021276600

102

.009803922

157

.006369427

212

.004716981

48

.020833333

103

.009708738

158

.006329114

213

.004694836

49

.020408163

104

.009615385

159

.006289308

214

.004672897

60

.020000000

105

.009523810

160

.006250000

215

.004651168

bl

.019607843

106

.009433962

161

.006211180

216

.004629680

52

.019230769

107

.009345794

162

.006172840

217

.004608295

£3

.018867925

108

.009259259

163

.006134969

218

.004587156

£4

.01851&'>19

109

.009174312

164

.006097561

219-

.004566210

Sb

.018181818

110

.009090909

165

.006060606

220

.004545455

BECIPROCALS OF NUMBEBS.

49

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

Ka

BedprooaL

No.

Reciprocal.

Na

Beciprooal.

No.

BeciprocaL

221

.004524887

276

.003623188

831

.008021148

886

.002590674

222

.004504505

277

.008610108

832

.008012048

887

.002588979

228

.0044848a'>

278

.003597122

888

.003008003

888

.002577320

224

.004464286
.004444444

279

.008584229

834

.002994012

889

.002570694

225

280

.008571429

83l>

.002965075

890

.002564103

226

.004424779

281

.003558719

836

.002976190

901

.002557545

227

.004405286

282

.008546099

887

.002967859

892

.002551020

228

.0048a5965

283

.003533569

838

.002958580

893

.002544529

229

.004366812

284

.003521127

339

.002949853

394

.002538071

280

.004347826

285

.008508772

340

.002941176

895

.002531646

231

.004329004

286

.003496503

341

.002982551

396

.002525258

232

.004310345

287

.003484321

842

.002923977

897

.002518892

238

.004291845

288

.003472222

343

.002915452

896

.002512568

234

.004278504

289

.008460208

344

.002906977

399

.002506266

235

.004255819

290

.008448276

845

.002898551

400

.002600000

236

.004237288

291

.003436426

846

.002890173

401

.002493766

237

.004219409

292

.003424658

347

.002881844

402

.002487562

238

.004201681

293

.008412969

818

.002873563

408

.002481890

289

.004184100

294

.003401861

349

.002865330

404

.002475248

240

.004166667

295

.003888831

350

.002857143

405

.002469186

241

.004149878

296

.003378378

351

.002849008

406

.002463054

242

.004132231

297

.008367003

352

.002840909

407

.002457002

243

.004115226'

298

.003855705

858

.002832861

408

.002450960

244

.004098861

299

.008344482

354

.002824859

409

.002444988

245

.0040K1638

800

.008338833

355

.002816901

410

.002439024

246

.004065041

301

.003322259

856

.002808989

411

.002438090

247

.004048583

802

.008811258

857

.002801120

412

.002427184

a<8

.004082258

308

.003300830

358

.002798296

418

.002421808

249

.004016064

804

.008289474

3591 .002785515

414

.002415459

250

.004000000

805

.008278689

360

.002777778

415

.00240968t

251

.008984064

306

.003267974

361

.002770088

416

.002408846

252

.003968254

307

.003257829

362

.002762431

417

.002398062

258

.003952569

308

.003246753

363

.002754821

418

.002392344

254

.003987008

809

.003236246

364

.002747253

419

.002386685

255

.003921569

810

.003225806

365

.002739726

420

.002380962

256

.003906250

811

.008215434

866

.002782240

421

.002375297

267

.003891051

812

.003205128

867

.002724796

422

.002369668

258

.003875969

813

.003194888

868

.002717391

428

.002864066

259

.003861004

314

.008184718

869

.002710027

424

.002358491

260

X)08846154

81d

.003174603

370

.002702703

425

.002352941

261

.008881418

816

.008164557

371

.002695418

426

.002347418

2G2

.003816794

817

.003154574

872

.002688172

427

.002341920

268

.008802281

818

.003144654

873

.002680965

428

.002336449

264

.003787879

319

.003134796

374

.00267^97

429

.002381002

265

.003778585

320

.003125000

375

.002666667

430

.002825561

266

.008759398

321

.003115266

376

.002659574

431

.002320186

267

.003745318

322

.008105590

377

.002652520

432

.002314^5

268

.003731348

323

.008095975

378

.002645503

433

.002309469

269

.003717472

324

.008086420

379

.002638522

484

.002304147

270

.003703704

325

.008076923

380

.002681579

485

.002298851

271

.003690037

826

.008067485

381

.002624672

436

.0022Sfe578

272

.003676471

327

.008058104

882

.002617801

437

.002288380

273

.003668004

328

.003048780

888

.002610966

438

.002283105

274

.000649685

329

.008039514

384

.002604167

489

.002277904

275

.003636864

830

.008080808

885

.002597408

440

.002272727

50

BEOIPROCALS OF mTHBXltS.

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

Kc

Recipi^ooal.

N<y.

Beeipvocal.

No.

Reoiprocul.

No.

Recipf^dal

441
44JJ
443
444
445

.002267574
.002262443
.002237836
.0022622B2
.002247191

496

m

496
499
500

.002016129
.002012072
.002000032
.002004008
.002600000

651
5S2
668
564

565

.001814882
.001811594
.001806818
.001806054
.001801802

606
697

668
609
610

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

446
447
446
441>
450

.002242152
.002287136
.002282143
.002227171

.002222222

601
602
503
501
505

.001996008
.001992032
.001988072
.001984127
.001980198

556
657
558
559
660

.001798561
.001795332
.001792115
.001768909
.0Q198S714

611
612
618
614
615

.001686661
.001633887
.00169Uei
.001628664
.001626016

m

492
458
464
469

.002217295
.002212889
.002207506
.002202643
.002197802

506
507
506
509
510

.001976285
.001972387
.001968504
.001964637
.001960784

561
562
568
564
665

.001782581
.001770859
.001776199
.001773050
.001769912

616
617
618
619
620

.001628877
.001620746
.001618128
.001615909
.00161f990S

46^
467
4$^
«9

4m

.002192982 '

.0021881S4

.002188406

.002178649

.002178918

511
512
618
514
515

.001966047
.001958125
.001949818
.001945629
.001941748

566

5<fr

568
569
570

.001766784

.001769661

.0O176056J

.001767469

.00175488^

621
622

m
ess

.001610806
.001607717
.00160006
.00160SN4
.001600000

m
4m

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

516
517
518
519
620

.001937984-
.001934236

.0019805021
.001926782
.001928077

571
572
573
574
676

.001761813
.001748252
.001745201
.001742160
.001789130

626
637
628
629
680

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

46$
467
468
469,
470

.002145923
.002141828
.002136752
.002182i96
.002127660

521
522
528
524
525

.001919886
.0019157091
.001912046
.001908897
.001904762

676
577
578
579
680

.0017961111
.001788102
.001799104
.001727116
.001724138

681
682
688
694
685

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

471
472

474
476

.002128142
.002118644
.002114165
.002109705
.002106263

526
527
528
529
530

.001901141
.001897533
.001898939
.001890859
.0018867921

681
582
588

584
885

.001721170
.001718213
.001716266
.001712329
.001709403

636
637'
638
639
640

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

476
477
478
479
480

.002100840
.002096436
.002092050
.002087683
.002088833

531
532
598
534
585

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

586
587
588
589
590

.00170648.1
.001703678
.001700680
.001697793
.001694913

641
642
643
644
645

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

461
482
408
464
48&

.002079002
.002074689
.002070393
.002066116
.002061856

536
537
538
589
540

.001865672
.001862197
.001858736
,001855288
.001851852

591
592
598
694
595

.001692047
.001689189
.001686841
.001683502
.001680672

646
647
648
649
650

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

486
487
468
489
490

.002057613
.002053888
.002049180
.002044990
.002040816

541
542
•543
544
545

.001848429
.001846018
.001841621
.001838235
.001834862

596
597
598
599
600

.001677852
.001676042
.001672241
.001669449
.001666667

651
652
653
654
665

.0015360^
.001538742
.001531894
.001529062
.001526718

491
492
493
494
495

.002036660
.002032520
.002028398
.002024291
.002020202

546
547
548
549
550

.001831502
.001828154
.001824818
.001821494
.001818182

601
602
608
604
605

.001663894
,001661130
.001658875
.001655629
.001662893

666
667
668
669
660

.001524890
.001622070
.001619787
.001517461
.001515162

RECIPROCALS OP NUMBERS.

51

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

WtJL

Beciprotat

668
66S
664

000-

667
668
668
«70

671
672
67B
674
«7fi?

€fS

ersf

«S2
088
«M

<I85

685
697
688

689

691

6M

695

696.

697

698

699

700

701
702
7€8 i

706

25§

707
708
709
710

711
712
713
714
715

.001512869
.601510874
;*0015e8996
.001506024
i0015aS759

;OOi5erso2.

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

.001490813

.001488095

.001488884:

.001498660;

.00148M81,

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

.00146M29.

.001466976.

.00146«129

.001461988

.001459654

.00145V726

.0014S6604,

.001488488;

.00146*979'

.001449075

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

.001438849

.091436382
)1484720

.001426534
.0OT424501
.001422475
J001420455
.001418440

:06l4i643i

.001414427
.001412429
.001410437
.001408451

.001406470
.001404494
.001402525
.001400560
.001398601

M«.

716
717
718
719
720

721

72»
72»
724
726

726
727
728
729
730

73t
732
738
734
786

796
7S7
788
789
740

741

742
748
744
745

746
747
74B
749
750

761
j2

754
766

756
757
758
759
760

7^
762
763
764
765

766
767
7^8
769
770

Beciporocal.

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

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

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

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

i001358696
.001866852
.001865014
i001$58186
• .001351351

.00134952

.0019477C

.00184689(

.001844086

.00184228^

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

.001331558
.001329787
.001328021
.001326260
.001324508

.001322751
.001321004
.001819261
.001317523
.001315789

.0013l4d60
.001812386
.001310616
.001308901
.001307190

.001305483
.001303781
.001302083
.001300390
.001298701

No.

771
772
778
774
775

776

777
778
779
780

781
782
788

784
785

786
787
788
789
790

791
792
793
794
795

796

797
798
799
800

^

802

•«03

804

805

806
807
808
809
810

811
812
813
814
815

816
817
818
819
820

821
822
823
824
825

Becipracal.

.001297617
.001296337
.001296661

.001291990
J0O129O323

J00128866Q
J0O12870O1
J001283847
.001283697
.001282051

.001280410
.001278772
.00127713^
.001276610
.001278686

.001272265
.001270648
.001269036
.001267427
.001265828

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

.001266281
.001254705
.00126813$
.001261564
.001260000

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

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

.001233046
.001231527
.001230012
.001228501
.001226994

.001228990
.(X)1222494
.001221001
.001219512

.001218027
".001216545
.001215067
.001213592
.001212121

No.

826

827

829
880

891

832
838
834
885

886
887
838
839
640

841
842
843
844
846

846
847
848
849
850

861
862
863
854
855

856
857
889
869
860

861
862
863
864
865

866
867
868
869
870

871
872
873
874

875

B^ciproOftL

.001210664
.001200190
.001207729
.0012062^
.001204819

.001208869
.001201928
.001200480
.0011990a
.001197605

.001196172
.001194743
.001193817
.001191885
.001190476

.001189061
.001187648
.001186240
.001184884
,001183432

.001182083
.001180688
.001179245
.001177866
X)01176471

.001176088
.001178709
.001172883
.001170960
.00116S691

.001168224
.001166861
'.O0I168GO1
.001164144
.001162791

.001161440
.001160093
.001158749
.001157407
.001156060

.001154734
.0011584031
,001152074
.001100748
.001149426

.00114fflOe
.001146789
.001145475
.001144165
.001142857

876 .001141558

877 .001140251

878 .001138952

879 .001137656

880 t .001136864

62

BECIPROCALS OF NUMBEB8.

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

No.

Reciprocal.

No.

Reciprocal.

No.

941

Reciprocal

No.

Redproesl

881

.001138074

911

.001097695

.001062699

971

.001029666

882

.001133787

912

.001096491

942

.001061571

972

.001028807

888

.001132nO3

913

.001005290

943

.001060445

973

.001027749

884

.001131222

914

.001094092

944

.001059322

974

.001026694

886

.001129944

915

.001092896

945

.001058201

975

.001025641

886

.001128668

916

.001091703

946

.001057062

976

.001024590

887

.001127396

917

.001090513

947

.001065966

977

.001023541

888

.001126126

918

.001089325

948

.001054852

978

.001022495

889

.001124859

919

.001088139

949

.001063741

979

.001021450

880

.001123596

920

.001086957

950

.001052632

960

.001020408

891

.001122334

921

.001065776

951

.001051526

961

.001019368

892

.001121076

922

.001084599

952

.001060420

982

.001018380

893

.001119821

923

.001068424

953

.001049318

983

.001017294

894

.001118568

924

.001062251

954

.001048218

964

.001016200

895

.001117318

925

.001061081

955

.001047120

966

.001015228

896

.001116071

926

.001079914

956

.001046026

986

.001014199

897

.001114827

927

.001078749

957

.001044932

987

.001018171

898

.001113586

928

.001077586

958

.001043841

988

.001012146

899

.001112347

929

.001076426

959

.001042753

989

.001011122

900

.001111111

930

.001075269

960

.001041667

990

.001010101

901

.001109878

931

.001074114

961

.001040588

991

.001009062

902

.001108647

932

.001072961

962

.001039501

992

.001008065

903

.001107420

933

.001071811

963

.001038422

998

.001007049

904

.001106195

934

.001070664

964

.001087344

994

.001006086

905

.001104972

935

.001069519

965

.001036269

995

.001005026

906

.001103753

936

.001068376

966

.001035197

996

.001004016

W7

.001102536

937

.001067236

967

.001034126

997

.001003009

908

.001101322

988

.001066098

968

.001083058

998

.001002004

909

.001100110

939

.001064963

969

.001031992

999

.001001001

«10

.001098901

940

.001063830

970

.001030928

1000

.001000000

BECIPBOCAIiS.

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

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

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

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

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

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

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

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

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

BEGIPROOALS OF NI7MBEBS. 53

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

•r, n X — =r — = 1.
* n n

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

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

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

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

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

Becip 390 = .002564103 ;

Becip 3900 = .0002564103;

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

Becip 227 = .004405286;

Becip 22.7 = .04405286;

Becip 2.27 » .4405286;

Becip .227 = 4.405286;

Becip .0227 = 44.05286.

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

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

Becip 236 =.004237288
Becip 237 =■.004219409

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

and Becip 236 =.004237288

minus .000007152

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

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

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

Log 1 =0.000000
Subtract Log 236.4 = 2.373647

'7.626353 = liOg 0.00428012

Becip 286.4 — 0.00423012.

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

Log 236.4 = 2.378647
Subtract Log 8424 = 8.925518

"5.448129 = Log 0.0280627.

fU24
Re«ip-^^ = 0.0280627.

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

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

N

1

i
It

8qui.be and cube boots.

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

1
i

Ml

■i"

a

S

1

1

1

i
i

i

"1

i

1

i
s

i

i

i

i

J

is

ii

Is
IS

i|

Is

IS

i

ii

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

5d

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

of Vumbers f^om 1 to lOOO.

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

Bqpamf.

Cpbe.

6<l. Rt.

C. Rt.

No.

Sqnave.

Cube.

6q. Kt.

CBt.

1

1

1.0000

1.0000

61

.9721

220981

7.8102

3.9965

4

8

1.4i«3

1.3999

68

9844

238328

7.8740

3.9679

•

2T

J.79»l

l.(«29

J98

9999

250047

7.9873
8.0000

3.9791

U

64

9.0990

isooi

1.5874

«4

4096

262144

4.

15

1»

I.TI90

65

4226

274625

8.0623

4.0207

96

316

^4495

1.8171

66

4356

287496

8.1240

4.0411

1

S

SM

9.M58

1.9129

67

4480 ■

300789

. 8-1854

4.0615

#

51S

2.8384

2.9890

68

4624

31449S

8.2462

4.9617

(a

7fl»

8.0909

2.0691

69

4761

328909

8.3016

4.1016

10

IM

1600

9.1939

9.1544

70

4899

843090

8.3996

4.1218

B

m

1331

9.S166

2.2240

71

5041

357911

8.4261

4.1406

144 *

1738

8.4941

2.2804

72

5184

373248
389017

8.4866

4.1602

19

3I6T

8.6966

2.3513

78

5476

8.5440

4.1798

n

^^

3944

8.741t

2.4101

74

495224

8.6028

4.1989

ii

0n

8.8^90

2.4692

75

5625

421975

8.6696

4.2179

•»

^

4096

40000

2.5198

76

5776

498076
466593

8.7178

4.2858

^9

Svn

*^& 1

2.5713

77

5999

8.7760

4.25a

19

32

56^

2.9907

78

6094

474652

8.8318 :

4.9717

»

Ml

fMiO

2.9684

79

6341

498080

8.8883

4.2996

'»

«D

MM

iiSsi

i

2.7144

80

6400

512009

8.9448

4.8099

n

m

1^107

2.7589

81

6561

591441

9.

4.3267

.»

2^

H.mNM '

fi.9920

«2

6794

551868

9.0554

4.8445

9

^S

4.7^58

2.9489

88

6889

571787

9.1104 ;

4.3631

M

. i

5!^

9.9945

84

7066

592T04

9.1952

4.8796

»

2.9940

85

7395

614196

9.2195

4.8968

J7S7fi

siSS-

B.9625
8.9900

86
87

7396
7569

686056
658608

9.2736
9.3274

4.4140
4.4810

.19

M|

3H9I

5.'^ ]

8.0866
8.6783

88

7744

681472

9.3808

4.4480

»

«g[

B4Mfr

80

7931

704999

9.4840

4.4647

»

^

MWJ

^!4772

8.1072

90 '

8100 1

729000

9.4868

4.4814

n

w

bSos

5.5678

8.1414 '

91

8381

758571

9.5394

4.4879

s>

'iSi

5.6599 •

8.1748

99

8464

778688

9.5017

4.6144

ft

iohII

piBST

6.7446 <

8.ao75

99

8649

804857

9.6487 ,

4.5307

n

ySH

IN804

5.8610

B.S896'
B.8711

94

8886

880584

9.6954

4.5480

•?

in»

«af5

5.9161

95

9025

857875

9.7466

44690

M

1U0

44556

6.0000

8.3019

96

9316

884796

9.7980

4.5780

n

Xjg^

fi^^

6.08eB

8J822'

VJ

0499

912678

0.8488

4.6047

»

144ft

oSSui

6.1644

3.8620

«8

9604

941192

9.8895

4.0104

SI

isn

6961^

6.3469

S.8S12

90

0891

970299

. 9.9499

4jKt61

40

1600

64000

6.3946

8.i«a00

100

10000

1000900

10.

4.6416

41

1Q61

tSS

B.4081

8.4482

101

10201

1080301

10.0499

4.6570

tt

17M

8.4160

192

10404

1061206

10.0095

4.6728

48

1M»

'TttitfT

Jgg

3.5034

169

10609

1002727

10.1480

4.0875

44

^

. m

3.5903

104

10816

1124864

10.1980

4.7037'

a

6!T98a

BJ>5«9

105

U025

1157925

10.2470

4.7li77

«

2U6

iSSS

6.7828

3.5830

106

11236

1191016

10.2956

4]7336

4r

a)()^

6.885T

S.9068

107

114l9f

1225048

10.3441

4.7475

48

11«SM

6.9989

8.0842

106

11604

1359712

10.3933

4.7023

II

^

1350W

7.0000

3.6663

109

]!l881

1295029

10.4408

4.7760

*

7.0^11

8.8840

110

12109

lasiooo

10.4881

4.7914

51

3Q01

1183661

7.1414

8.7084

111

12321

1867631

10.5357

4.8080

b

2T04

140986

7.aHii

8.7S25

112

12644

1404928 .

10..'i880

4.8988

tt

aaoir

149B77

7.aB01

3.7B63

118

12789

1442897

10.9801

'4.8B46

&

mn

167464

7.3485

3.n98

114

12908

1481544

10.6771

4.84B8

3035

166875

7.4162

3.8030

115

13225

1520875

10.7238

4.8639

M

3136

175616

7.4883

3.8259

116

13456

1560896

10.7703

4.8770

0

SM9

185198

7.5496

3.8485

117

18689

1601613

10.8187

4.8910

tt

8«64

19511S

7.6158

3.8709

118

13924

1643082

10.8628

4.9040

g

S4S1
S600

305979

7.6811

S.88S0

110

14161

1686169

10.9087

4.9187

o

%k9m

7.7460

8.0149

120

14400

1728000

10.9545

4.9896

56

SQUARES, CUBES, AND ROOTS.

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

JTo.

m
m

123
134

i»

IM
137
128
139
180

ISl
1S3
1S8
1S4
IM

1»6
18T

in

IM

140

141
148
148
144
146

14«
14T
148
149
IM

ISl
US
168
164
166

16«
16T
168
16»
100

181
108
188
164
166

166
167
168
160
170

m

173
178
174
176

176
177
178
170
180

181
183
188
184
186

Sqnmre.

14641
14864
15139
16876
16636

16876
16138
16884

16641
16800

17161
17434
17680
17866
18336

18486
18710
19044
19831
19600

19681
80164
80440

30780
81086

81816
31600
31904
33301
38600

33801
23104
23400
23716
34026

84886
84640
34864
86381
25600

35931
36344
36660

37226

37656
37880
28324
28561
S8900

892^
29684
39939
30276
80626

80976
81839
81664
82041
83400

32761
33124
33488
33856
84226

Cube.

1771561
1815848
1860867
1906624
1868136

8000876

3048388
3007153
3146680
3197000

3348091
3399968

3853687
3406104
3460376

3616466
3671868
3638073
3886619
8744000

3808331

3034907

3886964
8048636

8113186
8176638
3341793
8307949
8875000

8442961
3511806
8581577
3663364

8733875

8796416
8869898
3944813
4019679
4096000

4173381
4351528
4830747
4410944
4483125

4674396
4657463
4741633
4836800
4913000

6000811
5068448
617ni7
5268034
5359375

5461776
5645233
6639752
5736339
6832000

5029741
6028668
6128487
6229604
6831625

8q. Bt.

1.

1.0464

1.0906

1.1366

1.1808

1.2260
1.2694
1.8187
1.8578
1.4018

1.4466
1.4891
1.5326
1.6758
1.6190

1.6619
1.7047
1.7478
1.7806
1.8833

1.8748

1.9164

1.9588

3.

3.0416

3.0680
3.1344
3.1666
3.3066

3.3474

3.2883

3.8388
3.8688

3.4007
3.4499

3.4900
3.6300
2.5696
2.6006
3.6491

2.6866
2.7279
3.7671
2.8062
3.8452

2.8841

2.9238

2.9616

8.

3.0884

3.0767
8.1148
3.1529
3.1909
8.2288

3.2666
8.3041
3.3417
8.3791
3.4164

3.4586
3.4907
3.5277
3.5647
3.6015

O.Bt.

KO.

4.9461

186

4.9687

187

4.9783

188

4.9866

189

6.

190

6.0188

191

6.0365

193

6.0897

198

6.0638

194

6.0658

196

6.0788

196

6.0016

197

6.1045

196

6.1172

199

6.1399

300

6.1436

301

6.1551

303

6.1676

808

6.1801

304

6.1936

306

6.3048

306

6.3171

307

6.3398

306

6.3416

309

6.2636

310

6.3666

311

6.3776

313

6.3886

318

6.3015

814

6.3183

316

6.8351

316

6.8868

317

6.8486

318

6.8601

319

601717

220

6.8882

331

6.8947

333

6.4061

338

6.4176

324

6.4288

336

6.4401

236

6.4614

837

6.4626

338

6.4787

338

6.4848

330

6.4860

381

6.5068

233

6.5178

338

6.52R8

334

6.5397

286

5.5605

386

6.5613

287

6.5721

238

5.5828

280

6.5934

240

6.6041

341

6.6147

242

6.6252

248

5.6357

244

6.6462

246

5.6567

346

5.6671

247

5.6774

248

5.6877

249

5.6880

250

Sqiure.

84596
84869
86344
35721
86100

86481
86864
87249
87686

88026

88416
88809

89304
88601
40000

40401
40804
41300
41616
43036

43486

43848
43364
43681
44100

44621

45869

45796

46226

46666

47069
47524
47961
48400

48841
49384
49729
60176
60625

61076
61629
61984
63441
63800

68861
68824
64289
64756
65225

65696
56169
56644
57121
67600

bdOei
68564
69049
69686
60025

60516
61009
61604
62001
62500

Cube.

6434856
6539308
6644672
6751269
6659000

6867871
7077888
7189067
7301884
7414875

7629586
7646873
7762892
7880599
8000000

8120601

8842408
8366427
8489664
8615136

8741816
8869743
8998913
9129829
9961000

9893931
9628138
9663597
9800844
9938376

0077696
0318318
0360333
0503459
0648000

0793861
0941048
1089567
1339434
1890625

1543176
1697083
1852852
3008969
2167000

2826391
2487168
2649887
2813904
3877875

8144266
3313053
3481272
16651919
8824000

3997521
4172488
4348907
4526784
4706125

4886936
5069223
5252992
5438248
5625000

Bq. Ht.

13.6383
13.6748
13.7113
13.7477
18.7840

18.8308
18.8564
18.8934
13.9284
13.9643

14.

14.0067

14.0713

14.1067

14.1431

14.1774
14.3137
14.3478
14.3898
14.8178

14.8637
14.3876
14.4333
14.4568
14.4914

14.6258
14.5603
14.5046
14.6387
14.6628

14.6069
14.7800
14.7648
14.7866
14.8834

U.8661
14.8887
14.9832
14.9666
16.

15.0333
16.0665
16.0907
16.1327
16.1668

16.1987
15.2316
16.2643
16.2971
16.8297

16.3638
16.3948
16.4272
16.4686
16.4819

16.5243
15.5663
15.5885
16.6306
15.6626

15.6844
15.7162
15.7480
16.7797
15.8114

as*^

6.7088
6.7186
6.7287
6.738a
6.7480

6.760(>
6.7680
6.7790
6.789a
6.7989

6.8e8»
6.8189
6.838S
6.8808

6.8486

6.8678
6.8676

6.8771

6.8ei»

6.8864

6.9089
6.9156
6.9360
6.9846
6.9489

6.9689
6.S6ST
6.9731
6.9614
6.990T

6.

6.0008

6.0186

6.037T

•.0869

6.04i»
6.066«
6.0641
6.0782
6.0633

6.0912
6.1002
6.1091
6.1180
6.1369

6.1368
6.1440
6.1634
6.1633
6.1710

6.1797
6.1885
6.1972
6.3068
6.2145

6.2281
6.2817
6.3408
6.3488
6.367»

6.3668
8.2743
6.868»
6.3912
6.2996

SQUABES, CUBES, AND BOOTS.

67

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

No.

BqxiAra.

Cabe.

Sq. Bt.

C. Bt.

No.

Square.

Cube.

Sq. Bt.

CBt.

S61

68001

15818251

15.8480

6.3060

316

99856

31554496

17.7764

6.8113

3tU

68904

16003006

15.8745

6.3164

817

100489

31R55018

17.8045

6.8185

253

«4UW

16194277

15.9060

6.3847

818

101124

32157432

17.8886

6.8256

S64

64516

16387064

15.9374

&3880

819

101761

32461759

17.8606

6.8328

355

65035

16581875

15.9687

6.3413

830

102400

32768000

17.8886

6.8899

S66

665S6

16777216

16.

6.3486

821

103041

83076161

17.9165

6.8470

957

06048

16974598

16.0313

6.3579

822

103664

33386248

17.9444

6.8541

SS8

66564

17178512

16.0624

6.8661

828

104829

33698267

17.9722

6.8612

»e

67061

17378879

16.0885

6.3748

824

. 104976

34013824

18.

6.8683

160

67600

17576000

16.1245

6.8835

825

105625

34328185

18.0278

6.8763

381

68121

17779561

16.1555

6.3907

826

106276

84645976

18.0555

6.8824

363

6R644

17984728

16.1864

6.3968

827

106929

34965783

18.0831

6.8884

968

68169

18191447

16.2178

6.4070

828

107584

35287562

18.1108

6.8964

964

OTUlfD

18390744

16.2461

6.4151

829

106241

35611380

18.1884

6.9684

966

70325

18608625

16.2788

6.4282

880

108900

35987000

18.1659

6.91M

966

70766

18821096

16.3085

6.4312

881

109561

36264691

18.1934

6.9174

967

712H0

19084163

16.3401

6.4898

832

110224

36594868

18.2209

6.9944

968

71824

19248882

16.3707

6.4478

888

110889

36926087

18.2483

6.9813

960

72361

19465109

16.4012

6.4558

884

111556

37259704

18.2757

6.9382

970

72900

19688000

16.4317

6.4688

885

112225

37695375

18.3080

6.9461

871

73441

19902511

16.4621

6.4718

886

112886

37938056

18.8308

6.9531

973

73964

20129648

16.4824

6.4792

887

113669

88272758

18.3576

6.958»

973

74528

20346417

16.5227

6.4872

888

114244

38614472

18.3848

6»9668

974

76076

20570824

16.5529

6.4851

888

114921

38958219

18.4120

6.9727

375

75625

20796875

16.5881

6.5080

840

115600

89304000

18.4391

6.9796

976

76176

21024576

16.6132

6.5106

841

116881

39651821

18.4662

6.9664

977

76729

21258883

16.6488

6.5187

842

116864

40001686

18.4832

6.9088

378

77384

21484852

16.6788

6.5265

843

. 117649

40858607

18.5203

7.

379

77841

21717688

16.7063

6.5843

844

118886

40707584

18.5472

7.0068

360

78400

21952000

16.7832

6.5421

345

119035

41068635

18.5742

7.0136

981

78861

22188041

16.7681

6.5499

846

119716

41421786

18.6011

7.0206

383

79524

22435768

16.7829

6.5577

847

120409

41781928

18.6279

7.0271

3RS

80689

22665187

16.8226

6.5654

848

121104

4214492

18.6648

7.0838

984

80656

22906804

16.8528

6.5731

849

121801

42508549

18.6815

7.0406

885

81285

28146135

16.8819

6.5806

850

122500

42875000

18.7088

7.0478

3R6

81796

23398666

16.9116

6.9865

851

123801

43248651

18.7350

7.0540

987

82369

23688908

16.9411

6.5962

862

123904

43614206

18.7617

7.0607

«8

82944

23887872

16.9706

6.6089

858

124609

48966977

18.7888

7.0674

968

83521

24187568

17.

6.6115

854

125316

44361864

18.8149

7.0740

»0

saoo

24S8800O

17.0394

6.0191

855

126025

44788R75

18.8414

7.0807

sn

84681

24643171

17.0587

6.6267

856

126736

45118016

18.8680

7.0878

983

85264

24897068

17.0680

6.6348

857

127449

45499298

18.8944

7.0940

386

85848

25158757

17.1172

6.6419

858

128164

45882712

18.9209

7.1006

984

86488

25412184

17.1464

6.6494

859

128881

46268379

18.9473

7.1072

985

87025

25673875

17.1756

6.6569

860

129600

46656000

18.9737

7.1138

986

87616

25884886

17.2047

6.6644

861

130821

4704S881

19.

7.1804

987

88209

36198078

17.2887

6.6719

862

131044

47437828

19.0263

7.1260

986

88804

36468582

17.2627

6.6794

863

131760

47882147

19.0526

7.1386

988

88401

26780699

17.2916

6.6869

864

132496

48228544

19.0788

7.1400

800

90000

27000800

17.3305

6.6948

865

133225

48627125

19.1050

7.1466

801

90601

27276801

17.8484

6.7018

866

. 183956

48027896

19.1811

7.1531

803

91304

27548806

17.3781

6.7092

867

134689

49439863

19.1572

7.1586

806

91809

27818137

17.4068

6.7166

868

135424

49836082

19.1888

7.1661

804

93416

28084M4

17.4856

6.7240

369

136161

50248409

19.2094

7.1726

806

83025

28373635

17.4643

6.7818

870

136900

50658000

19.2354

7.1791

806

9S686

38659816

17.4929

6.7887

871

137641

51064811

19.2614

7.1866

«rr

94348

28884448

17.5214

6.7460

872

188884

51478848

19.2873

7.1920

806

94864

29316112

17.5499

6.7588

878

188129

51895117

19.3132

7.1984

808

85481

apfiowM

17.5784

6.7606

874

189676

53818624

19.3391

7.2048

AO

86100

29791000

17.6068

6.7678

875

140635

52784875

19.3649

7.2112

811

86731

80080881

17.6868

6.ni2

876

141876

58157876

19.8907

7.2177

813

87844

808n828

17.6685

6.7834

877

142129

53583688

19.4165

7.2240

818

87888

80684987

17.6818

6.7887

878

142884

54010152

19.4422

7.2804

814

86686

80860144

17.7300

6.7968

879

143641

54439969

19.4679

7.2368

816

96996

81366676

17.T48S

6.8041

880

144400

5487360(r

19.4986

7J4S2

58

SQUARES, •CUBES^ Ain> BOOTS.

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

STo.

S81
M3
188
884
886

887

•qiuura.

itfin

140M

147456
148»5

S»7
806

401
408
408
404
405

40T
408
400
•410

411
418

418
414
4U

43M
41T
419
4It

411
41t

4»
484

435

496
487
488
498
480

■ai

4S2

488
434
436

486
487
488
480
440

441
448
'448
444
446

140768
150644
161S21
158100

153881
15S664

154440
155386
156035

156816
157600
158404
150301
160000

160601
161604
163400
168316
164036

164886

166640
166464
167881
168100

168031
160744
170660
171886
172335

173066
178889
174734
175661
176400

in341
178064
178038
179776
180686

181476
183880
18S184
184041
184900

185761
186624
187480
188S66

189386

190096
190060
191844
192721
193600

194481
195804
190349
1971S0
196026

Gab«.

8q. Bt.

65306841
65748068
66181887
66623104
67066616

67513«<
67960608
68411073
68868860
69318000

69770471
60236388
60696467
61169804
61639075

63099186
62570778
63044793
63531199
64000000

64481901
64864800
65460087
65880864
66480196

06038410
674101tt 1
67917818
68417929
68891000

09436681
68034638
70444007
7096V0a
71478875

71991386
72511718
73034603
73660660
74088000

74610101
75161440
75686007
76338094
76766096

7780en6
77854488
78403763
78058680
79507000

80003001
80631566

81183T87
81746604
83319876

82881866
83458458

84027673
84604619
85184000

85766131
86350888
869S8807
87528884
\ 881211<t5

19.5192
19.5448
19.5704
19.6950
19.6314

19.6468
19.6738
19.6977
19.7381
19.7484

19.7787
19.7990
19.8343
19.8494
19.8746

19.8887
18.9349
10.9499
19.8760
80.

30.0860
30.0480
30.0740
90.0908
a0.18«6

30.1494
30.1743
30.1990
20.2387
90.3485

90.2731
90.9078
90.3834
90.8470
90.8716

20.8881
30.4806
20.4460
20.4096
90.4088

90.5183
30.5496
20.5670
90.5918
90.6166

20.6396
20.6640
20.6882
20.7138
20.7864

20.7606
20.7846
20.8087
20.8387
20.8607

20.8806
20.9046
20.9884
20.9638
20.9708

21.

21.0838

21.0476

21.0713

21.0050

cut.

7.2495
7.2558
7.2028
7.2086
7.2748

7.2811

7.2874
7.J
7.3
7.8061

7.3134
7.8186
7.8946
7JS10
TJ87B

7.3484
7.3490
7.8660

7.3610
7.8681

7.8743

7.1

7.J

7.

7.

7.4047
7.4100
7.4100
7.4299
7.4990

7.4860

7.4410

7.4470

1.4

1.4

K«.

7.4660
7.4710
7.47T0
7.4899
1.4

T.4048
7.5007
7.5067
7.5190
7.6186

7.5944
7.5800
7.58a
7.5410

7.5478

7.5587
7.5585
7.5664
7.6711
7.5770

7.5898
7.5886
7.5044
7.6001
7.6060

7.6117
7.6174
7.6Sa2
7.6289
7.6846

446
447
448

460

451
469
468

454

466

466

457
468
460
460

401
463
468
484
486

SqtuuM.

467
468
460

470

471
473
473
474
475

476
477
478
479
480

481
483
488

484
485

486
487
488
480
480

491
493
483
494
485

496
tf7
498
499
fiOO

601
50S
SOS
804
606

6DB

607
SOB
609
610

196916
190800
300704
301601

tM500

S0B4O1
304904
3a30O

306110

307085

307866
308840

300764
310681
311600

313531
318444
314368
315296
316396

117166
318080
310034
310061
330900

221841
222784
223720
224676
235636

nssfm

237590
928484
2204^
280400

231361
232304
233280
234256
235235

236106
237109
238144
239121
240100

241081
242064
249040
244036
245035

346016
347000
348004
249001
260000

261001
252004
253600
254016
255026

256086
257040
258064
259081
860100

Gubo.

Bq.Bt.

88716586
89614688
88015803
90516649
911360004

91788861
93845406
93968077
9S57W64
94186675

94816616
95448088

9a(moi8

96703570

21.1107
81.1404
21.1080
81.1880
11.2183

97071181
98611118
99361847
9988T844
100644096

101194880

103608883
103161760
10380000

104487111
105164048
105818817
106486434
107in676

107850176
106581888

108315861
108001180
llOSOMOO

111384041
1I1980168
113078567
113370804
114OOC10

114701360
115601808
116214372
116880160
117040800

iioswrri

119006488

iioeiffcr

120651104
121381tf»

122098806
123708478
123506082
124211408
125000000

123761601
126506089
127288617
128014804
128787086

128564116
13QS13M8
131006618

isisfrsaso

133651000

CL&U

11.

11.

31.!

91.3078

81.8987

31.3543
21.3776
21.4000
21.4248
31.4476

11.4709
11.4043
31.5174
31.5407
31.5680

11.5890
21.6103
31.6888
21.6664
21.6705

31.7025
31.7366
31.7486
31.7n6
31.7045

81.8174
31.8408
31.8681
21.8061
11.0000

11.9317

11.9545

31.9778

32.

33.0337

32.0454
32.0661
83.0907
22.1188
22.1860

32.1585
32.1811
22.3080
32.2381
12.3486

22.2711
23.3085
33.3160
32.8883
32.3607

23.3880
22.4054
23.4877
22.4480
28.4732

23.4844 4
22.5181 <
22.5680 '
23.5610
28.6881^

7.6400
7.0480
7.6517

7.6574
7j

7.

7.6744

7.6801

7.6867

7.6014

7.6870

7.74

7.71

7.718B

7.7U6

7.7860
7.7800
7.7801
7.7410
7.747t

7.7610
7.7604
7.7030
7.708t
7.7750

7.780i
7.7800
7.7015
7.7070
7.8005

7.80IO
7.8184
7.8180

7.8140
7.

7.
7.8100
7.8400
7.8U4

7.J

7.f

7.80M

7.8110

7.8784

7.8

7.8801

l.t

1.1

7.0051

7.M06

7.0108
7.0811

7.0184
7.90lt

7.9870

i:

7.9tTi
l.i
1.1
7.S

l.i

7.9700

7.9701

7.1

1.1

BqUAK£8» CUBES, AKD BOOX8.

5ft

HI
•18

•18

«n

618
St8

Mi

8ifr

584

•8T

Bfl

i4«

Mi

««r

C«8

6tT

»

WO

m

IT8

Sqoas*.

Oab*.

Sq. Bk

CBt.

K«.

961111
983144
388168
364188

1SS4S38S1
184317198
185006687
M6788744
186B86B26

83.6068

82.6374
33.6485
83.6710
33.6660

7.9048

8.

&0062

&01O4

8.0166

670
677
678
670
600

387388
36BS84

3m4B8

1S78B6B88
138168fi8

isaoHoao

13910686*
140686888

83.7160
32.7870
32.7908
n.7810
33.6006

8.0060
8.QBtl
8.0008
8.0416

6»3

6tt4
666

sn4a

373184
318S88
374898
375895

1414MI61
143g664B

1438TSaM
144761185

33.8354
32.8418
32.8003
33.8810
22.9130

8.04(86
8.0617
8.0609
8.00aD
8.0671

688

667
680
600
600

3166n

37770
378784

378641

1456SIBK
146860188

147181668
148085668
148811088

22.9015
23.0780
33.
33.0817

8.0723
8.0774
8.6036
8.0876
8.0037

001

608
608
604
605

iiil!

148mS81

15050aM8
16141M87
163818684
168188875

SS.0404
23.0651
23.00n
23.1004
3S.I801

8.6878
8.1038
8.1019
8.1180
8.U80

686
607
100
660
000

387n8
388868
388444
380131
381680

ISItWHB
164854158
155780018
166688018

167461680

ss.iosr

23.1738
23.1048
23.2M4
2S.281t

8.1331
8.1381
8.U32
8.M82
8.1683

001
003
008
004
085

383681
98784
384648
386886
387895-

168840481

15|66066B
lflM06688
166668184
161M60a5

23.3804
28.3010
28.S0U
28.3388
28.8458

8.1083
8.1533
8.1568
8.U8S
8.1)603

000
007
006
000
•10

386118
388988
800684
801401
809500

103191006

10808038
164806068
1666681188
166891000

28.8666
28.3688
23.4004
28.4807
2S.4ft8t

8.1783
8.1783
8.1883
8.1883
6.1983

Oil
013
018
014
015

808661

904704
806608
800818
806686

10f80tl51

168080868
16BUlBn
170681464
170666835

28.48U
28.4061
28.6100
23.6888
23.5604

8.8862
8.3881
8.8861
8.8180
8.0180

OM

017
010

a.*

808186
810848
811864
813481
818680

171878818

1787I1UI
1740M8IO

28.6709
28.0808
23.0080
28.8888
28.0840

8.3839
8.8378
8.3837
8.3377
8.3436

031

688
038
034
OK

814181

816844
816888
818086
818935

178506681
171004838
178U0649
17MI0lt4
180069186

2S.6854
28.7066
28.7376
28.7487
28.7607

6.9475
6.3684
6.3673
6.8681
8.«70

686

687
638
680
680

890658

831488
823634
838701
894800

181831406
183984868
188880483
1841890000
1861800QO

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

8.8719

8.W66
84il6
8.0106
84^3

mt

088

884
036

ilii!

186100411

187140948

laouasu

188110894
ll8M60875

23.8066
33.0166
33.8074
33.9688
88.8798

8.89B2

6.aoao

6JW59
8.8107
8.U66

080
087
688

089
040

3S1770
333900
884004

886341

888714

34238(

843300
S446a

346744
3488101
S4810O

840981
860404
85104O
862880
864035

866310
860400

S6700i
868001

870681
8T3M»

873331
874644
876700
876600
8783)16

879450
880680
381984

883101
884400

8918X0
8081^
884884
385041
886000

S86101
309444
400660
4OUIS1O
403836

404480
406700
407044
400881
409600

19U0887O
1931000SS
193100663
10a04680
196118000

311708786
8137U178
218849|M
2140U7W
210000080

237U066O

338188000

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

247Mp
260O430QO

267360466
266474868
2&80940;3
200017110

sraSior

24.

34.0306
34.0410
34.0034
34.0688

84.1060
34.1847
34.104
34.1081
34.1868

94.SM5
34.88U
34.S5U
34.Sm

34.4181
34.4836
24.4640
24.47I&
24.4040

34.0171

34!8n
34.0t70
34.0068

C.

34.8tt6
S4.8m
34.8000

35.2190
25.2380
35.3687
35.3384
36.3M8

8.3208

8.sasi

8.3900

8.SS48
0.S386

0.8448
8.3401
8.3^
8.3607
8.8084

8.4

8.:

m

8.4104

tt&

8.i
8.
8.^
8.
8.4

8.4
8.4
8.4
8.4
0.^

8.4688

1:1

0.400
ImIO

8.1

ei
8.r
8.1
8-H?o

6.6816
8.BV3
8J
8-i

oj

8.C
8J
8^
8.1
8.1

84|773
eSts

6.6807
6.6048
8.00B8

60

SQUARES, CUBES, AND BOOTS.

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

No.

SQnmre.

Cube.

Bq. Bt.

O.Bt.

No.

Square.

Cube.

Bq.&t.

CBU

Ml

410881

363374721

25.3180

8.6233

706

498436

861886816

86.5707

8.90tt

643

412164

264609388

35.3377

6.6267

707

498848

353996343

86.6896

8.9686

643

413449

265847707

25.3574

8.6818

708

601364

854894818

96.6068

8.9187

644

414TS6

267089984

25.3773

8.6867

709

602681

856400838

66.6371

8.9168

646

416026

368336135

25.3969

8.6401

710

604100

867911000

96.6468

8.99U

646

417316

269586136

35.4165

8.6446

711

606681

859436481

96.6646

8.996S

647

418608

270840033

35.4363

8.6480

713

606844

860844138

86.6889

8.9386

648

419904

372097793

85.4558

8.66S6

m

608969

362467097

86.7081

8.9897

649

431201

273359449

86.4766

8.6679

714

6097W

863884844

86i7a06

8.9878

6&0

432500

274625000

85.4961

8.6684

716

611336

366686876

96.7996

8.8498

661

483801

375894451

85.5147

8.6668

716

61366«

967061686

96.7689

8.8489

652

48S104

377167808

35.5343

8.6718

717

614080

368601813

36.7769

8.9609

65S

426409

378445077

25.5539

8.6767

718

615584

370146888

36.7966

8.9646

«4

427716

279736364

35.5734

8.6801

719

616961

371684869

36.8148

8.9687

•66

439025

381011375

25.5990 .

8.6846

730

618400

973348000

36.8838

8.9899

666

430336

383900416

35.6135

8.6890

721

519641

374806961

36.8514

8.9nt

<6T

431649

283593893

35.6330

8.6834

733

631884

376967048

36.8701

8.8711

668

432964

284890313

35.6515

8.6878

733

633789

377999067

36.8887

8.876?

66»

434281

286191179

36.6710

8.7033

724

634176

379609434

36.9073

8.8T84

660

436600

387496000

86.6906

8.7066

736

635686

981078136

36.9868

8.8696

661

436931

388804781

35.7099

8.7110

786

637076

883667176

^m*w%Aw

8.987«

662

438244

390117538

35.7394

8.7164

737

638639

884840689

26.9689

8.981S

«3

439569

391494347

85.7488

8.7196

738

689984

386888963

36.9816

8.986»

664

440896

393754944

25.7683

8.7341

739

631441

387480489

37.

8.

665

443326

394079636

85.7876

8.7886

730

633800

389017000

37.0186

8.00a

666

4436S6

395406396

85.8070

8.7389

731

6S496I

890617891

37.0970

8.6089

667

444889

396740968

85.8363

8.7373

733

636884

893888168

37.0666

8.0199

668

446324

298077633

25.8457

8.7416

733

637888

393833897

37.0740

8.01«4

668

447661

399418309

25.8650

8.7460

784

638766

395448804

37.0884

8.8906

mo

448900

600763000

36.8844

8.7606

736

640886

397066976

87.1108

8.6a4S

671

460241

303111711

36.9037

8.7647

736

641686

99e688SS«
400816669

87.1896

8.09W

673

461684

303464448

26.9830

8.7690

787

6491«8

87.1477

8.689B

67S

463939

304831317

35.9433

8.7634

738

644644

401947878

87.1668

8.Q868

674

464376

306183034

35.9616

8.7677

739

646181

409689418

87.1846

9.0410

676

465626

30764687S

86.9806

8.7791

740

647600

406884000

37.9088

9.04SO

676

466876

308915776

26.

8.7764

741

649081

406868081

87.8819

9.04n

677

468339

310288733

26.0198

8.7807

743

650664

4086IS488

87.3997

9.0699

678

469684

311665752

36.0384

8.7860

743

668048

410179407

87.8680

8.06T9

679

461041

313046839

36.0676

8.789S

744

663696

411890784

87.8764

9.90M

680

463400

314433000

36.0768

8.7987

746

666086

418486696

37.8847

8.06M

661

463761

815831341

36.0960

8.7960

746

666616

416160886

37.9190

8.0604

663

465134

317314568

26.1151

8.8083

747

668008

416689789

37.9913

8.0796

683

466489

318611967

36.1343

8.8066

748

669604

418608893

37.3486

8.0n6

684

467856

330013504

36.1534

8.8109

749

(^1001
563600

420189748

37.8679

8.06U

686

469335

331419136

26.1735

8.8163

750

421876000

37.9861

8.QM6

686

470596

333818866

36.1916

8.8194

751

564001

488664761

37.4044

8.068S

687

471969

S3434370S

36.3107

8.8887

763

665604

ITSXMK

87.4886

8.0987

688

473344

336660673

36.3896

8.8380

753

667008

430967777

37.4406

8.0877

689

474721

337083769

26.3488

8.8S2i

764

668616

428661664

37.4691

8.1017

600

476100

338609000

86.3679

8.8866

766

670036

430968876

37.4773

8.1067

691

477481

339939S71

36.3869

8.8406

756

671686

432061816

37.4966

8.1086

692

478864

SS1373888

36.3069

8.8461

757

573048

433796098

37.5136

8.118ft

698

480249

333813657

86.3349

8.8483

758

574664

436619619

37.6318

8.117ft

694

481636

334365384

26.3439

8.8686

758

676081

43784647*

37.6600

8.191ft

•96

483035

S36703S75

26.3639

8.8678

760

577600

438976000

37.6681

8.196ft

696

484416

SS7158586

26.3818

8.8621

761

579181

440711061

37.5868

8.138ft

•97

4A5809

338608873

26.4008

8.8663

763

580644

449460788

37.6043

8.199ft

•98

4B7204

340068392

88.4197

8.8706

768

583168

444194847

37.6326

8.197S

•99

488601

841533099

26.4886

8.8748

764

688686

446948744

27.6405

8.141ft

700

490000

848000000

26.4575

8.8790

766

685885

447697136

37.6686

8.146ft

701

491401

344472101

26.4764

8.8883

766

686756

448466006

37.6767

9.1488

702

492804

346948408

26.4963

8.8875

767

688989

461817668

37.6848

8.16ST

70S

494309

347438837

26.5141

8.8817

768

689634

468864888

37.7188

8.1677

T04

495616

348913664

36.5380

8.8868

769

681361

4547&6B08

37.7806

8.1ttT

t06

497026

850403635

36.6618

8.8001

770

683800

466539000

37.7489

8.1861

SQUARES, CUBES, AKD BOOTS.

61

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

No.

807

810

811
812
818
814
815

810
817
818
810
830

821

814
836

888
827
838

880

881
888

884

fl86

Square.

50M41
S9S884
&075W
599076
600625

602176
603720
606284
606841
608400

609861
611534
613060
614666
616225

617796
619369
620944
622521
624100

625681
627264
628848
630436
632025

688616
635209
636804
638401
640000

641601
643204
644809
646416
648035

648636
651248
662864
664481
6S6100

657731
668344

662586
664235

665856

667480
669134
670761
672400

674041
676684
677339
678976
680625

682276

685584
687241
688800

600561
602234

MUUUM
OMKKIO

007225

Onbe.

458314011
460090648
461889917
463684824
465484375

467388576
469007433
470010962
473730139
474653000

476870541
478311768
480048687
481800304
483736635

485687666

487443403
488803873
491168060
493039000

494913671
496798088
4086n257
600606184
602459675

604858336
506361573
608160603
510063399
512000000

613822401
515849608
617781637
619718464
521660135

523606616
535667948
637514113
529476130
681441000

533411731
635387838
637367797
538353144
541343375

643338486
545338513
647343433
649358259
661368000

653387661
666412248
657441767
550476224
561616625

563559076
565600388
567663552
568723780
571787000

573866191
675930868
678009537
580093704
683183875

8q. Bt.

27.7660
27.7848
27.8029
27.8300
27.8388

27.8568
27.8747
27.8927
27.9106
27.9285

27.9464

27.9643

27.9831

28.

38.0179

28.0367
28.0535
28.0713
28.0691
28.1069

28.1847
28.1425
28.1608
28.1780
28.1957

38.2185
28.2312
38.2488
28.2666
28.2848

28.8019
28.3106
38.3378
38.3540
28.3735

38.3901
28.4077
28.4353
38.4439
38.4606

38.4781
28.4956
38.5182
38.5307
38.5482

88.5667
38.5832
28.6007
38.6183
28.6366

38.6581
38.6705
88.6880

38.7054
38.7338

38.7402

28.7576
28.7760
28.7924
28.8097

28.8271
28.8444
28.8617
38.8791
38.8864

C. Bt.

No.

9.1696

886

9.1736

837

9.1775

838

9.1815

839

9.1855

840

9.1894

841

9.1933

843

9.1973

843

9.2013

844

9.3063

845

8.2091

846

9.3130

847

9.3170

848

9.3300

849

0.3348

850

9.3287

851

9.2326

863

9.2365

853

9.2404

854

9.2443

855

9.2482

856

8.2521

867

9.2560

858

9.3599

850

9.3638

860

9.2677

861

9.27ie

862

9.2754

863

9.2793

864

9.3832

865

9.3870

866

9.3900

867

9.3948

868

9.2986

868

9.3025

870

0.3063

871

9.3102

872

9.3140

873

9.3179

874

9.8217

875

9.3355

876

9.3394

877

9.8332

878

9.3370

879

9.3406

880

9.3447

881

9.8486

882

9.3533

883

9.8561

884

9.8599

885

9.8637

886

9.3675

887

9.3713

888

9.3751

889

9.8789

890

9.3827

891

9.3865

893

9.S902

893

9.3940

894

9.3978

895

9.4016

896

9.4053

897

9.4091

898

8.4129

899

9.4166

900

Square.

700569
702344
703921
705600

707381
706964
710649
.712336
714025

716716
717409
719104
730801
732500

724201
735904
737609
739316
731025

782736
734449
736164
737881
738600

741321
743044
744760
746486
748225

748966
751689
758434
755161
756900

758641
760384
762129
763876
766625

767376
769139
770884
772641
774400

776161
777924
779689
781456
783225

784996
786760
788544
790S2I
792100

793881
796664
797449
799236
801025

802816
804609
806404
808201
810000

Cube.

8q. Bt.

584277056
586376258
588480(72
590688719
592704000

594828321
596947688
599077107
601311584
603851135

606496786
607646423
600800193
611900049
614135000

616395061
618470308
6206504n
622835864
626026875

637233016
629432793
631628713
633839779
636056000

638377381
640608938
642735647
644872644
647214626

648461896
651714368
658972032
656284809
658608000

660n6311
668064848
665838617

667627624
660921875

678321376
674636183
676836152
679151439
681473000

683797841
686128868
688465387
690807104
698154125

695506456
697864106

700227072
702595369
704969000

707847971
709732288
712121957
714516984
716817375

719323136
721734273
724150792
726673699
739000000

28.9137
28.9310
28.9483
28.9655
28.9828

29.

29.0172

29.0345

29.0517

29.0689

19.0861
29.1083
29.1204
29.1376
28.1648

29.in9
28.1890
29.2062
29.2283
29.2404

29.2675
29.2746
39.2916
29.3087
29.3258

89.8488
29.3598
29.8769
88.8939
29.4109

89.4279
29.4448
29.4618
29.4788
29.4958

29.5137
29.5296
29.5466
29.5635
29.5804

29.5973
29.6142
29.6311
29.6479
29.6648

29.6816
29.6985
29.7153
29.7321
29.7489

29.7668
29.7825
29.7993
29.8161
29.8329

39JB496
29.8664
29.8881
29.8998
29.9166

29.9383
29.9500
29.9666
29.9833
SO.

O.BK

8.4204
8.4241
9.4279
9.4316
8.4854

8.4391
9.4429
9.4466
8.4503
9.4541

8.4578
9.4615
8.4652
9.4690
9.4727

8.4764
0.4801
9.4888

9.4875
9.4918

9.4948
9.4968
9.5028
9.5000
9.508T

9.5184
9.5171
9.5207
9.5244
8.5281

9.5817
9.6864
8.5S9I
8.5427
9.6464

8.6601
9.6537
9.5574
9.5610
9.5647

9.5688
9.5719
9.5756
9.5792
9.5828

9.5865
9.5901
9Ji937
9.5973
9.6010

9.6046
9.6062
9.6118
8.6154
8.6190

9.6226
9.6262
9.6298
9.6S34
9.6370

9.6406
9.6442
9.6477
9.6513
9.6549

62

8QUABEB, OUBB8, ANI> ROOXfiL

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

ITa

Sqiuun.

901

m

903
904
905

906
*W ,
908 *
900
910

911
912
9IS
9U
916

tie

917
•18

m

M0
Ml

&7
931^
999

Mo

961
913
913
9M

m

966,

9«r'

9S8

941
94S
94S
944
943

946
W
948
948
950

811801
813604
815409
817316
816036

830836
833648
834464
838381
838100

839931
831744
833569
835386
837335

839Q6ft
840889
84273

8464d0

84B3a
8500M

851939
85B776
856625

857476
859339
861184
868041
664900

866761
868624
870489
87235i
874335

876086
877968
8798U
881731
383600

885481
887364
889249
891136
893025

894916
896800
898704
900601
903500

Cul>«. , 8q. &t.

731433701
733870808
736314337
738763364
741217635

743677416
746143643
7486LS312
751089439
753571000

75606808]
758550638
761048497
763661944
766060875

768676386
7710063X3
773^32
77616I56»
778688000

781338861
78S777448
78633(Mff7
788888034

791^25

79402976
796597983
799178762
801765089
8O436710OO

806964481
8O95&7608
8131607
814780604
817400876

8200;

82

825283612

8279S60I9

830584000

833337621

8S5886tt8
838561 W7
Ml 233384
843908625

846590536
849278123
851971392
854670349
857375000

30.0167
30.0333
30.0500
80.0666
30.0832

30.0998
80.1164
30.1330
30.1486
30.1663

30.1838

so.iwi

30.2159
30.2334
30.3490

30.2666
30.2830
30.2986
30.3160
30.3316

30.3480
30.3«I5
3a3809
30.3974
30.4138

3O.4S0A
30.4467
30.4631
30.4796
30.4959

30.5133
30.5287
30.5460
30.5614
30.5778

30.5941
30.610$
30.62i8
30.6431
30.6594

30.70<

30.7246

30.7409

S0.75T1
80,7734
30.7896
30.8058
30.8221

cut.

Ko.

fkiaave.

9.6586
9.6630
9.6666
9.6693
9.6737

9.6763
9.6799
9.6834
9.6970
9.6906

9.^1

9.7013
9.7047
9.7083

9.7118
9.7153
9.7188
9.7334
9.7359

9.738i
9.7338
9.73W
9.7400
9.7436

9.7470
9.7505
9.7540
9.7575
9.7610

9.7645
9.7680
9.7716
9.7750
9.7785

9.7819
9.7864
9.7889
9.7934
9.7959

9.7983
9.8038
9.8063
9.8087
9.8132

9.8167
9.8201
9.8236
9.8270
9.8305

951
962
963
954
956

956
967
968
969
960

%

9B3
964

966

%

966

909

970

971
973
973
974
976

976

977
rf78
9T9
980

981

d83
984
965

986
987

9e9

990

991
993
993
991

996

•fvQ

998

999

1000

904401
906304
908309
9L0U6
913036

918986
91689
917784

933LeW

938166
936088
83T034
838861
840800

943841
944784
946739
94867«
956636

953576
954539
956484
958441
968400

963361
964334
96628»
968256
9702^

972196
974169
976144
978121
980100

9830^
984094
986049
988036
990036

993016
994009
99000%
998001
1000000

CulM.

Sq.su.

860085351
862801408
865633177
868360664
870083876

873733816
876467^
87921^913
881874579
884736000

887603681
89037t(28
dKOBHsm

89684IS64
898683135

901438686
9O4S3t06S
907089333
909863309
913679000

91

91

931

t2401«i434

9368S98T6

939714176
9336Y4883
9S5«US63j
9383;
941 ll

944076141
946966168
949863087
952763904
966671635

958686356

961604803<

964430373

9678616m

970299000

973248371

976191488'

979146667

982107784

985074875

968047936
991030*73
994011992
997008999
1000000000

30.8383
30.8545
80.8707
30.8869
30.9031

30.9192

sasaN

8O.8$0
80.8877
80.8838

31.

81.0161

81.0333

81.04B

31.0644

31.0806
31.0893
81.113T
31.1288
81.1448

31.1608
31.176»
31.183»
31.2090
31.3360

31.3410
31.36T0
31.3730
81.3890
31.3060

31.3208
31.3369
81.3538
31.3688
31.8847

31.4006
31.41«
31.4335
31.4484
31.4643

31.4803
31.4966
31.6lf9
S1.627«
31.5438

31.5595
31.5753
31 .5911
81.6070
31.6228

CS^.

9.8339
9.8374
9.8408
9.8443
9.84!t

9.86U
9.8M«
9.8660
9.8614
9.8848

•••^

9.9631
9.96M

ISZ

9.9698
9.9738
9.97W

9.98»
9«V^Bo

.9666

.9m

9.
9.1

9.9988
9.99Wr
10.

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

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

SQUABi: AND GITBB BOOTS.

63

No CTTora.

Num.

Sq. Rt.

Ca. Rt.

Num.

Sq. Rt.

Ca. Rt.
11.20

Nam.

Sq. Rt.

Cu. Rt.

Nam.

Sq. Rt.

Cu.Rt.

ido&

81.70

10.02

1405

87.48

1805

42.49

12.18

2205

46.96

1102

XOlO

31.78

10.03

1410

87.56

11.21

1810

42.54

12.19

2210

47.01

1?«

1015.

91.86

10.05

1416

87.62

U.23

1815

42.60

12.20

2216

47.00

19.04

low

31 .04

10.07

1420

87.68

11.24

1820

42.66

12.21

2220

47.12

1«.05

10»

82.0S

' 10.06

1426

87.76

11.26

1826

42.72

12.22

2226

47.17

I9.0ft

U»0

82.oe

10.10

1430

87.82

11.27

1830

42.78

12.23

2230

47,22
47.28

i$.oe

1036.

32.17

10.12

1436

87.88

11.28

1836

42.84

12.24

2236

19.07
19.08

1040

82.25

10.13

1440

87.96

11.29

1840

42.90

12.25

2240

47.99

lOtf
106O

38.88

10.15

1446

88.01

lUl

1845

42.96

12.20

2246

47.98

19.00

82.40

10.16

1450

88.08

11.32

1850

43.01

12.28

2250

47.43

13.10

iioo

32.48

10.18

1456

38.14

U.33

1856

43.07

12.29

2256

47.^

19.11

82.56

10.20

1460

88.21
88.21

11.34

1860

43.13

12.30
12.81

8260

47.64

14.12

106&

$2.68

10.21

1466

11.36

1866

43.19

2266

47.89

1^13

I074»

82.71

10,23

1470

38.34
88.41

14.37

1870

1876

4S.2i

12.32

2270

47.64^

lil4

^

82.70

10.24

. 1476

11.38

43.30

12.33

2876

47.70^

l£lS

$2.86

10.26

1480

38.47

U.40

1860

43.36

12.34

2280

47.75

19.10

1066

82.04
83.08

10.28

' 1486

98.60
88.6t

U.41
11.42

1886

43.42

12J5

2286

47,80
47.86

i9.n

109V

10.29

1490

1890

.43.47

18.36

2290

19!S

1 06

83.00

10.31

1496

11.43

1896

43.53

12.37

2296

47.91

l<N>^

83.17

10.82

1500

38.73

U.46

1900

43.50

12.30

2300

47.0^

19.20

101^

89.34

10.84

1506

38.79

U.46

1906

43.3

12.40

2906

48.01

19.21

Ul«

33.8S

10,36

1510

88.86
98.99

11.47

1910

43.7)1

12.41

3310

48.00

19.22

uw

88.30

is.47

10.87

1516

U.49

1916

43.71

12.42

' 2315

49.11

li29

UM

10.38
10.40
10.42

1520

89.12

11.50

19«

43.8!

12.43

zS20

49.17

isjit

88.54
88.68

1526
■ 1530

11.51
li.63

1926
1930

49.8!
43.9:

12.44
12.46

2330

49.22
49.92

19:25

19.26

1 sfr

3^.60

10.43

1536

98.18

U.54

1936

43.9)

12.40

. 2336

19.27

1 40

83.76

UL46

1540

38.24

U.56

1940

44.06

lt47

2940

48-97

19.28

83.84

10.46

1646

S.'S

11.66

1946

44.10

12.48

2945

48.43

19.29

liso

83.01

10.48

1550

11.57

1950

44.16

12.19

2950

48.48

19.90

1^6

83.00

10.40

. 1656

89.49

U.59

1956

44.23

12.60

2856

48.63

19.90

ifiS

84.06

10.51

1560

S9.g

11.60

1960

44.27

12.51

2360

48.58

19*91

fj/i^

84.18

10.63

1566

99!62

11.61

1966

US

44.U

12.63

2366

48.69

19.92

^

84.21
84.26

10.64
1(^65

1570
1575

11.62
11.69

1970
1976

ll54
12.66

2970

2376

48.68

48.79

19.98

19.94

UJBO

84.36

10.57

1680
16^

S9.7&

ll.((5

1980

44.50
44.56

12,80

2380

48.70

li.3S

n^K

84.43

10.58

ov.u

11.66

1986

12-§T

3986

48.84

13.98

iSo

«4.5<)

10.60

1690

^.87

11.67

1990

^•^

lite

liM
12.00

2S9D

48.89

iljst

nj6

84.57

10.61
10.63

1696

g.9i

11.66

1996

44.fl»

2995

48.94

iSJiS
13.89

Qoo

U.U

1600
1606

4o!m

11.70

2000

44.72

MOO

48.99

U06

84.71
34.70

10.04

11.71
11.72

2006

44.78

12.61

2106

49.04

13.40

uso

10.60

1610

40.12

2010

44.83

12.62

»10

49L<[»

18.41

♦jll^

94.80

10.67

1616

40.li
40.25

11.19

2016

44.n

12.09

i&5

4a.u

19.42

y<£3i

84.08

10,69

1620

11.74

2020

44.94

12.64

2480

tt.24

19.48

196

35.00

10.70

1626

40.31
40.St

11.76

2025

45.0D

12.66

2485

19.«

y^

36.21

10.71

1630

11.77

2030

45.0B

12.60

2430

40.ao

18.44

S£

10.73
10.74

1636
1640

40.44
40.60

11.78

i;.7d

2036
2040

45.11
45.17

12.67
12.68
12.«

2436
2440

4».&

1I45
ll4ft

15.20

10.76

1646

40.59

11.80

2046

45.22

2445

4^*45

19.47

;Ei6d

85.30

10.77

1650

40.62

11.82

2050

45.28

12.70

2460

4S.8O

19.48

466

95.43

10.79

1656

40.68

11.83

2055

45.33

12.71

2460

«^.60

19J2

85.50

10.80

1660

40.7i

11.84

2060

45.39

12.72

2470

49.70

!M6

35.67

10.82

1066

10.80

11.83

2066

45.44

12.73

2480

48.80

19.64

S9»

86.64

10.83

1670

40.87

11.86

2070

45.50
45.55

12.74

2490

49.90

19.66

U76

86.71

10.84

1675

40.99

11.88

2075

12.75

2500

60.00

19.67

85.78

10.86

1680

40.99

11.89

208O

45.61

12.77

2610

90.10

19.59

3B6

35.86

10.87

1686

41.06

11.90
11.91

2086

45.66

12.78

2520

60.20

13.61

aoo

85.92

10.89

1690

41.11

2090

46.72

12.79

26SO
2540

80.30

19.63

85.90

10.90

1695

41.17

11.92

2095

45.77

12.M

50.40

19.64

s

36.06

10.91

1700

41.23

11.93

2100

45.89

12.8T

2650

60.30

1166

80.13

10.99

1705

41.29

11.93

2105

43.88

12.82

2560

60.60

19.68

^DO

86.10

10.94

1710

41.36

11.96

2110

45.93

12.83

2570

50.70

1170

lljiy

ioiS

10.96

1715

41.41

11.97

2116

45.99

12.84

2580

50.79

1172

ICW

58S

1720

41.47

11.98

2120

46.04

12.83

2590

50.89

19.79

S5

96.40

1726

41.63

11.99

2125

46.10

12.86

2600

60.99

1175

S5o

36.47

11.00

1730

42.59

12.00

2130

46.15

12.87

2610

61.09

19.7T

x56

96.54

U-Ol

1736

41.65

12.02

2135

46.21

.12.88

2620

51.19

1179

1|M#

90.61

11.02

1740

41.71

12.03

2140

46.26

12.89

2630

51.28

19.80

iMft

36.67

11.04

1746

41.77

12.04

2145

46.31

12.90

2640

51.38

1182

itso

96.74

11.06

1750

41.83

12.05

2150

48.37

12.91

2650

61.48

1184

S{

96.81

11.07

1755

41.89

12.06

2155

46.42

12.92

2660

61.58

1I86

SS

90.88

11.08

1760

41.96

12.07

2160

46.48

12.93

2670

51.67

1187

Mt

90.96

11.09

1765

42.01

12.09

2165

46.53

12.94

2680

61.77

ll89

SM

97.01

11.11

1770

42.07

12.10

2170

46.58

12.95

2690

51.87

ll91

Bo

97.08

11.12

1776

42.13

12.11

2175

46.64

12.96

2700

61.96

18.92

97.U

11.13

1780

42.19

12.12

2180

46.69

12.97

2710

52.06

18.94

IW

97.82

1U6

lOo

1786

42.23

12.13

2185

46.74

12.98

2720

52.15

18.90

m

97.28

1790

4i.U

12.14

2190

46.80

12.99

2730

52.25

19.98

m

97.86

11.17
11.10

1795

42.37

12.15

2195

46.85

13.00

2740

52..35

19.99

um

87.42

1800

42.43

12.10

2200

46.90

13.01

2730

62.44

14.01

8QUAKE A.ND CUBE £

SQUABB AND CUBB BOOTS.

66

SQUARE AND CUBE ROOTS.

Square Boots and Cube Roots oflf nmbem fWmi 1000 to lOOM

— (GONTIirUXD.)

Hun.

Sq.Bt.

Co. Bt.

Nora.

Sq.Bt.

Od. Bt.

Nam.

8q. Bt.

Ca.Bt.

Num.

Bq.Bi.

01I.B4

tow

W.29

».M

0990

M.64

21.04

9660

97.79

21.22

97M

96.M

I1J»

MM

96.S4

ao.87

OSM

M.6e

91.06

96W

97.78

21.22

97M

.96.94

S1.8t

91M

96.89

ao.ae

9S40

M.04

91.M

9670

97.88

21.28

9eM

W.M

31.M

9110

96.46

30.89

9060

M.70

91.07

9680

97.88

21.24

9810

M.06

si.a

9iao

95.60

ao.H9

99M

M.76

91.07

96M

97.M

21.26

9820

M.10

si.a

91M

96.66

M.M

n7o

M.M

U.W

96M

97.98

21.26

98W

M.16

tl.4t

9140

96.M

90.91

OSM

M.86

Sl.M

WIO

W.M

21.26

9840

M.20

tLU

9160

96.M

90.09

99M

M.M

91.10

WJO

96.08

21.27

9660

M.26

81.44

91M

96.71

90.99

9400

M.M

91.10

96M

W.1S

21.28

OSM

M.M

31.44

9170

96.7C

90.M

9410

97.01

91.11

9840

06.18

21.28

W70

M.85

S1.4ft

•IM

96.81

90.94

94M

97.M

91.12

9850

96.38

21.29

96M

M.40

21.46

91M

96.W

90.96

94M

97.11

91.1S

98M

W.39

21.M

9eM

M.45

S1.4T

9»0

96.92

90.W

9440

97.18

91.18

9870

W.84

21.80

99m

M.60

21.4T

9910

96.97

90.M

9460

97.91

91.14

9880

98.89

21.81

MIO

M.66

21.48

9no

M.03

90.97

94M

97.96

91.15

98W

86.44

21.82

M20

M.M

21.49

9B0

90.07

90.98

9470

97.81

91.16

9700

96.48

21.88

99M

M.86

21.4S

9140

W.13

90.M

94M

97.8T

91.16

9710

96.64

21.88

9940

M.70

UM

91M

M.18

90.M

94M

97.49

91.17

9720

W.69

21.84

9960

M.76

tLM

tMO

W.23

31.M

96M

97.47

21.18

97M

96.84

31.36

90M

M.M

tun

9970

W.»

91.01

9610

97.69

91.19

9740

96.W

21.88

9970

M.86

S1.6S

9180

W.SS

91.01

9690

97.57

91.19

9750

98.74

21 J6

99M

M.M

21.6t

99M

M.38

91.09

9680

97.83

91.90

97M

98.79

21.87

99M

M.M

S1.64

9iW

M.U

91 .OS

9640

97.87

31.31

9770

98.84

21 JK

lOOM

1M.00

1144

HIO

M.49

91.04

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

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

For the squfufe root.

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

A : B : : Tabular root : Beqd root.

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

947 = ub nam
8

3841
940.68 = gl

8787.68 = ▲.

and

948.58 = given num.
8

2889.58
947 = tab nam.

.8786.59 ^^ B.

A.

S787.5I

B. Tab root. Beqd root.

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

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

For the cube root.

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

A : B : : Tabular root : Reqd root.

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

= tab nam.

18718 y and

7868 = given nam.

310Mr:A.

B. Tab Boot. BeqdBt.
21696 —

7868 = given num.

2

14788
8859 = Ub nam.

. 21696 =:B.

Then, as 210M 21696 19 19.4585

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

BQCABE AND CUBE BOOTS. 67

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

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

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

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

Tsrj ibiipla, ud SDmn u Ua OltA mmanl loolHlia.

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

Fin

b roo

tr

,.„,

Sir

,™

!

j

1

1

i

1

1

i

3

1
1

ill
11

68

ROOTS AND POWEBB.

Fiftli roots and flftb powero— (Continued).

Power.

No. Ot
Boot.

Power.

Rio^j f o'«r-

No. or p„_^
Boot. ^«'«'*

No. Of
Boot.

Power.

No. or
Boot.

Power.

No. Of
Root.

88.2735

2.45

2824.75

4.90

86873

9.70

2609193

19.2

20511149

^.0

459165034

54.

V1.ao6-i

2.5U

2y71.84

4.95

9U392

9.80

2747949

19.4

21228258

29.2

508284376

56.

107.b20

2.55

3125.00

3.00

95099

9.90

2892547

19.6

21965275

'29.4

550731776

66.

118 bl4

2.60

3450.25

5.10

100000

10.0

3043168

19.8

22722628

29.6

601693067

57.

130.(>d«

2.65

3802.04

5.20

110408

10.2

3200000

20.0

23500728

29.8

656356768

68.

lU.MIt

2.70

4181.95

5-30

121665

10.4

3363232

20.2

24300000

30.0

7149-24299

69.

167.276

2.73

4591.65

5.40

133823

10.6

3533059

20.4

26393634

30.5

777600000

60.

172.104

2.80

5032.84

5.50

146933

10.8

3709677

20.6

28629151

81.0

844696301

61.

188.(Md

2.85

5507.32

5.60

161051

11.0

3893289

20.8

31013642

31.5

916132832

62.

203.111

2.90

6016.92

5.70

176234

11.2

4084101

21.0

33554432

32.0

992436543

63.

U9.4U

2.95

6563.57

5.80

192541

11.4

4282322

21.2

36259082

32.5

1073741824

64.

243.000

3.00

7149.24

5.90

210034

11.6

4488166

21.4

39135393

33.0

1160290625

66.

263.936

3.0a

7776.00

6-00

228776

11.8

4701850

21.6

42191410

33.5

1252332576

66.

286.292

3.10

8445.96

6.10

248832

12.0

49-23597

21.8

45435424

84.0

1850125107

67.

810.136

3.15

9161.33

6.20

270271

12.2

515.3632

22.0

48875980

34.5

1463933568

68.

835.54i

3.20

9924.37

6.30

298163

12.4

5392186

22.2

52521875

35.0

1564031349

69.

962.391

3.25

10737

6.40

317580

12.6

5639493

22.4

56382167

35.5

1680700000

70.

891.334

3.30

11603

650

343597

12.8

5895793

22.6

60466176

360

1804229361

7L

421.419

3.35

12523

6.60

371293

13.0

6161327

22.8

647&3487

365

19S49176B2

7*.

454.354

3.40

13501

6.70

400746

13.2

6436343

23.0

69343957

37.0

2073071593

7i

488.760

3.45

145.39

6-80

432040

13.4

6721093

23.2

74167715

37.5

2219006624

74.

525.219

3.50

15640

6.90

465259

13.6

7015834

23.4

79235168

38.0

2373046876

76-

563.822

8.55

16807

7.00

500490

13.8

7320825

23.6

84587005

36.5

7535525376

76.

604.662

3.60

18042

7.10

537824

14.0

7636332

23.8

90224199

39.0

2706784157

77.

647.835

3.65

19319

7.20

577353

14.2

7962624

24.0

96158012

39.5

-2887174368

781

693.440

3.70

20731

7.30

619174

14.4

8299976

24.2

102400000

40.0

3077056399

79.

T41,577

3.75

22190

7.40

663383

14.6

8648666

24.4

108962013

40.5

3276800000

80l

792.352

3.80

23730

7.60

710082

14.8

9008978

24.6

115856201

41.0

3486784401

81.

845.870

3.85

25355

7.60

759375

15.0

9381200

24.8

1-23096020

41.5

3707398432

83.

902.242

3.90

27068

7.70

811368

15.2

9765625

25.0

130691232

42.0

3939040643

83.

961.380

3.95

28872

7.80

866171

15.4

10162550

25.2

138657910

42.5

4182119424

84.

1024.00

4.00

30771

7.90

923896

15.6

10572278

25.4

147008443

43.0

4437053125

86.

1089.62

4.05

32768

8.00

984658

15.8

10995116

25.6

155756538

48.5

4704270176

86.

1158.56

4.10

34868

8.10

1048576

16.0

11431377

25.8

164916224

44 0

4984209207

87.

1230.95

4.15

37074

8.20

1115771

16.2

11881376

26.0

174501858

44.5

5277319168

88.

1306.91

4.20

39.390

.8.30

1186367

16.4

12345437

26.2

1845281-25

45.0

5584059449

89.

1386.58

4.25

41821

8.40

1260493

16.6

12823886

26.4

195010045

45.5

5904900000

90.

1470.08

4.30

44371

8.50

1.338278

16.8

13317055

26.6

205962976

46.0

6240321451

91.

1557.57

4.35

47043

8.60

1419857

17.0

1.3825281

26.8

217402615

46.5

6590815232

92.

1649.16

4.40

49842

8.70

1505366

17.2

14348907

27.0

229345007

47 0

6956883693

93.

1745.02

4.45

52773

8.80

1594947

17.4

14888280

27.2

241806543

47.5

7.339040224

94.

1845.28

4.50

55841

8.90

1688742

17.6

15443752

27.4

254803968

48.0

7737809375

96.

1950.10

4.55

59049

9.00

1786899

17.8

16015681

27.6

J68.354383

48.5

8153726976

96.

2059.63

4.60

62403

9.10

1889568

18.0

16604430

27.8

^>8'2475249

49.0

8587340257

97.

2174.03

4.65

65908

9.20

1996903

18.2

17210368

2M.0

•297184.391

49.5

9039207968

• 98.

2293.45

4.70

69569

9.30

2109061

1H.+

17833868

28.2

U2500000

50.0

9509900499

99.

2418.07

4.75

73390

9.40

•2'2?«203

18 6

1 8475:^09

28.4

345025251

51.

2548.04

4.80

77378

9.aO

234«493

18.8

19135075

28.6

380-204032

62.

9683.54

4.85

81537

9.60

2476099

19..0

19813557

28.8

418195493

63.

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

or % powers of numbers, n^^.

See table, page 69.

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

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

Values of n, ending in 0 or in 5, are exact values. All others end in repeat-
ing decimals. Thus: n = 0.052083 signifies n«» 0.052083333

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

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

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

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

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

or loEB of tecton.

.,„jt dividend -log of

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

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

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

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

" 287 = 2.45 788 " 0.0287 - 2.15 788

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

2.87 = 0.46 788 " 0.0002B7 = 4.45 788

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

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

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

t Log 1 = log 18 - log 10— log 10 = 1— 1 = 0 ; ot 1 - 10».
Log 0. 1 = log A = tog 1 — 'og 10 = 0 — 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 = 0 — log n.
Similarly, log recip — = log — — — = o — log .

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

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

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

xr w 1 xr y ^e 290.19 =2.46 269

Multiply Nos, J i* 1260.7 =3.10 062

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

{

Divide Nos. f Log 6.3023 = 0.79 950

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, = 0 — log divisor = 10 — log divisor — 10. Thus :.

1523 _

3.382 X 8.655

Log 1523 = 3.18 270

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

Sum of logs and cologs = 21.72 273 — 20

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

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

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

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

s

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

72

LOGARITHMS.

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

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

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

Multiplying. Dividing,

either log 7425 = 3.87 070 = 8.87 070

log 0.25 = 1.39 794 = 1.39 794

3.26 864 4.47 276

or log 7425 = 3.87 070 = 3.87 070

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

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

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

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

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

2.89 375. See (15). •

Log 0.00048= 4.68 124

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

Dividing by 3) 26.68 124 — 30

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

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

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

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

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

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

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

LOGARITHMIC CITART AND SLIDE RULE.

73

1,1-

JLog».
l.O-

OJO-

OJS

0.7-

0.0-

OJi

oa

OJO-

IJDr-

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

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

I

I

0,9 0,4

t L_

o.e 0,8

—J L_

1,0

L_

1.9

I

1.4

I

2.0

— I

1.9

9.0

1__

9.9

jro«.

Mo9*

E

2 3 4 S 97801

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

2

r
Bl

C

T

9

1 — I 1 M I
4 5 0 7891

3 4 5 07891J\

I I I I .1 I r I I

+

Dl

i ri^-^

T 1 1 ■ I 'I I I 1 I

2 3 4 S G78»l
S e 7 8 9

lA.

9

3

-T

5

e

-T
7

T — 1 I ■ f
8 9 lU]

J»L

i3

-« ' 1 1 1 1 1 r—

ij} 0.0 0.1 0.9 0.3 0.4 ojs o,e

Log»»

0.7

— I r-

0:8 0.9

1.0

— I
1.1

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

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

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

74

LOGARITHMIC CHART AND SLIDE RULE.

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

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

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

A single scale would therefore serve for all values, from 0 to infinity ;
but for convenience several contiguous scales are sometimes added, as in
the log chart*.

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

TiiTiP Trifi.i*1rAi^

between

•

correspondi to values of

xH

xttom

xH from

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

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

Ito 10
10 to 31.62
81.02 to 100
100 to 1000

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

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

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

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

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

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

LOGABrrHMIC CHABT AND SUBB BI

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

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

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

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

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

C B 10 li

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

I C 0.38 :

2.

76 LOGARITHMIC CHART AND SLIDE RULE.

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

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

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

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

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

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

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

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

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

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

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

Thus:

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

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

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

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

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

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

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

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

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

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

LOOARITHHIO CBABT AND SLIDE RULE.

77

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

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

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

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

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

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

ZtogB.

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

J I I I I I 1 t I I ' I «

J«0«. Cfi 5 8 4 H €7891 3 3

-U » I . I I .1 I It .... 1 ■ I

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

7. ^

4 J 078»ljA

'.' ■i''i'r'i

JBl » 8 dS87891 2 8 4S87891M

r^ U — ,"^ f , ?, f ,^ f J.Mfg)

^00. tPJ IJf 9 3 4 5 0 7 8 0 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.

0

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

0

82477

82542

83123

83187

83758

83821

84385

84447

86003

85064

86612

85672

86213

86272

86806

86864

87890

87448

87966

88024

88636

88692

89098

89163

89662

89707

90200

90264

90741

90794

01276

91328

01808

91866

92324

92376

92839

92890

93848

93399

93862

93902

94840

94398

94841

94890

95327

06376

95808

96866

06284

96331

96754

96801

97220

97266

97680

97726

08136

98181

98587

98632

99033

99078

09475

99619

99913

99966

Prop*

66
66
64
63
62

61
60
60
68
67
66
66
•66
64
64

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

48
47
47
46
46
46
46
44
44

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

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

Log 2870

-8.45788

u

287

= 2.46788

«l

28.7

» 1.45788

u

2.87

»= 0.45788

n
sristtes.

Log 0.287

= 0.45788 -

1

= 1.46788

" 0.0287

= 0.46788 -

2

= 2.45788

" 0.00287

= 0.45788 -

8

= 8.45788

" 0.000287

= 0.46788 -

■4

= 4.4578^

80

LOQARITHMS.

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

90. Log.

,1000

01
02
03
04
09
06
07
08
09

1010

11

12
13
14
15
16
17
18
19

1020

21
22
23
•• 24
25
26
27
28
29

1030

31
32
33
34
36
36
37
88
89

1040

41
42
43
44
45
46
47
48
49

00000

043

— Q87

130

173

—217

—260

—303

346

389

432

475

518
—561
—604
—647

689

732
—775

817

860
—903

945
—988
01030

072

1571^2
199:^2

— 242,t^

42

43
44
43
43
44
43
43
43
43
43

43
43
43
43
43
42
43
43
42
43

43

42
43
42

42
43

—284

—326

—368

410

452

494

—536

—578

—620

-«62

703

745

—787

828

870

—912

953

—995

02036

—078

42
42
42
42
42
42
42
42
42
41

42

42
41
42
42
41
42
41
42
41

No.

Log.

1090 02119 7:

160 J*

—202 ;f

53 — 243j}

51
52

54
55
56
57
58
59

1060

61
62
63
64
65
66
67
68
, 69

1070

71
72
73
74
75
76
77
78
79

1080

81
82
83
84
85
86
87
88
89

1090

91
92
93
94
95
96
97
98
99

284
325
366
407

■■^'1 45/

—490

-^31

-572

612

653

694

-735

—776

816

857

—898

938
—979
03019
—060

100
—141

181
—222
—262

302

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

—743
782
822
862

-902
941
981

04021
060

—100

41
41
41
42
41
41

41
40
41
41
41
41
40
41
41
41

41
40
41
40
41
40
41
40
40
40

41

40
40
40
40
40
40
40
40
40

39
40
40
40
39
40
40

39
40
39

No.

1100

01
02
03

Log.

^

04139

—179

218

—258

04 —297

05

336

06

—376

07

—415

08

—454

09

493

1110

532

11

571

12

610

13

—650

14

—689

15

727

16

766

17

805

18

844

19

883

1120

—922

21

—961

22

999

23

05038

24

-077

25

116

26

—154

27

192

28

—231

29

269

1130

—308

31

346

32

—385

33

—423

34

461

35

—500

86

—538

37

576

38

614

39

652

1140

690

41

—729

42

—767

43

—805

44

—843

45

—881

46

918

47

956

48

994

49

06032

40
39
40
39
39
40
39
39
89
39

39
39
40
39
38
39
39
39
39
39

39
38
39

39
38
39
38
39
38
39

38
39
38
38
39
38
38
38
38
38

89
38
38
38
38
37
38
38
38
38

No.

Log.

IISO 06070

51 —108

52 145

53 1—183

54 —221

56
56
57
58
59

1160

61
62
63
64
65
66
67
68
69

1170

. 71
72
73
74
75
76
77
78
79

1180

81
82
83
84
85
86
87
88
89

1190

91
92
93
94
95
96
97
98
99

258
—296

333
—371

408

—446

483

—521

—558

595

—633

—670

707

744

781

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

188

—225

—262

298

335

—372

408

445

—482

518

—660
591

—628
664
700

—737
773
809

—846

—882

5

38
37
38
38
37
38
37
88
87
88

87
38
37
37
38
37
87
37
87
38

37
37
37
37
37

37
37
37
36
37

37
37
36
37
37
36
87
87
36
37

86
87
36
36
37
86
36
37
36
36

No.

1200

01
02
08
04
05
06
07
* 08
09

1210

11

It

13

14

15

16

17

18

19^

1220

21
22
23
24
25
26
27
28
29

1230

31
32
33
34
35
36
37
.38
39

1240

41
42
43
44
45
46

liOg. s

07918 36

954 36
990 37

36

08027;;^

-099!^
— 135 on
—171^

48
49

—207
—243

—279
314
350
386

—422

—468
493
629

—565
600

—636
—672

707
—743

778
—814

849

884
—920

965

—991

09026

061

096

—132

—167

—202

—237

272

307

342
377
412
447
482
—517

.47^-687

621
656

36
36
36

85
36
36
86
86
36
86
86
86
86

86
86
36
35
86
36
36
36
35
86

35

85
85
36
85
85
85
35
35
35

35
85
35
35
86
36
35
84
35
85

Example:

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

Log. 11820 = 07262

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

= 22
Log. 11826 =

07262 + 22 = 07284

1
2
3

4
5
6

7
8
9

44

4
9
13
18
22
26
31
Z5
40

43

4
9
13
17
22
26
30
34
39

42

4
8
13
17
21
25
29
84
88

41

4
8
12
16
21
25
29
33
87

40

4
8
12
16
20
24
28
32
36

39

4
8
12
16
20
23
27
31
35

38

4
8
11
15
19
23
27
30
84

37

4
7
11
15
19
22
26
80
33

36

•4
7
11
14
18
22
25
29
32

35

4
7
11
14
18
.21
25
28
32

84

3

7

10
14
17
20
24
27
81

1
2
3

4
5
6

7

8

9

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

LOQAKITHU8
CMnnB*n •r Brigita LoynrlMiiii

liOOABITHHS.

83

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

9o.

1790

62
64
66
68

1760

62
64
66
68

1770

72
74
76
78

1780

82

84
86
88

1790

92
94
96
98

1800

02
04
06
08

1810

12
14
16
18

1820

22
24

26
28

18S0

32
34
86
88

1840

42
. 44

46

48

Log.

24304
853

k-403,
462'

—602

551

—601

—650

699

748

797
846
895
944
993

26042

—091

139

188

—237

286
—334

382
—431
—479

627
675

—624
-672

—720

—768
—816
—864
—912
969

26007
—055
102
150
—198

245
—293
—340

387
—436

—482

—629

676

623

670

S3

49
50
49
50
49

60
49
49
49
49

49
49
49
49
49

49

48
49
49
48

49
48
49

48
48

48
49
48
48
48

48
48
48
47
48

48
47
48
48
47

48
47
47
48
47

47
47
47
47
47

Ko.

1850

52
64
56
58

1800

62
64
66
68

1870

72
74
76
78

1880

82
84
86
88

1800

92
94
96
98

1900

02
04
06
08

1910

12
14
16
18

1920

22
24
26
28

1930

32
34
36
38

1940

42
44
46

48

Log.

26717
764
—811
—868
—905

951
—998
27045

091
—138

184
—231
—277

323
—370

-416

—462

508

564

600

646

692

—738

—784

—830

875

921

—967

28012

—068

103
—149

194
—240
—285

330

375

—421

—466

—511

-656
—601
—646
—691
735

780
—825
—870

914
—959

S3
O

47

47
47
47
46

47
47

46

47
46

47
46
46
47
46

46
46
46
46
46

46
46
46
46
46

46
46
46
46
45

46
45
46
45
45

45
46
45
45
45

45
45
45
44

45

45
45
44
46
44

ToflDdLog. 18117:

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

u

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

No.

1
2
8

4
6
6
7
8
9
10

00

3
5
8

10

13

16

18

20

23

26

49

2
6
7
10
12
15
17
20
22
26

1900

52
54
66
58

1960

62
64
66
68

1970

72
74
76
78

1980

82
84
86
88

1990

92
94
96
98

2000

02
04
06
06

2010

12
14
16
18

2020

22
24
26

28

2030

32

Log.

29008
—048

092
—187

181

—226

—270

314

358

—403

—447
—491
—636
—679
—628

—667
710
754

—798
—842

886
—929
—973
30016
—060

—108
146

—190
233
276

—820

—363

—406

449

492

635

678

621

—664

—707

—750
792

34

8a5

36

—878

38

920

2040

963

42

31006

44

048

46

—091

48

—183

45
44
46
44

46

44
44
44
46
44

44
44
44
44
44

43
44
44

44

43

44
44

48
44

48

48
44

43
43
44

48
48
43
43
43

43
43
43
43
43

42
43
43
42
43

43

55
43
42
42

No.

2000

62
54
66
68

2060

62
64
66
68

2070

72
74
76
78

2080

82
84
86
88,

2090

92
94
96
98

2100

02
04
06
08

2110

12
14
16
18

2120

22
24
26
28

2130

32
34
36
38

2140

42
44
46

48

Log.

31176

—218

260

802

—846

—887

—429

-471

618

665

697
—689
—681
—723
—765

806
848
—890
981
973

32016 41

I

43
42
42
48
42

42
42
42
42
42

42
42
42
42
41

42
42
41
42
42

056
—098

189
—181

—222
263
—306
—846
887

428
469
610
—652
—593

—684

—675

715

756
797

-888

—879

919

960

33001

041
—082

122
—163

203

42
41
42
41

41
42
41
41
41

41
41
42
41
41

41
40
41
41

41

41
40
41
41

40

41
40
41
40
41

No.

Log.

88244

2100

62

284

64

—825

66

—866

68

406

2160

445

62

—486

64

—626

66

—666

68

—606

2170

--646

72

—686

•74

—726

76

—766

78

-806

2180

-«46

82

886

84

926

86

966

88

84005

2190

044

92

084

94

—124

96

168

98

—203

2200

242

02

—282

04

821

06

—861

08

—400

2210

489

12

—479

14

—618

16

—667

18

696

2220

686

22

674

24

718

26

—768

28

—792

2230

880

32

869

34

908

86

947

38

986

2240

85026

42

—064

44

102

46

—141

48

—180

15

41
40
40
40

41
40
40
40
40

40
40
40
40
40

89
40
40
40

89

40
40
89
40
89

40
89
40
89
89

40
89
89
39
39

39
89
40
8f
38

39
89
89
39
39

39
38
39
39
38

48

2

5

7
10
12
14
17
19
22
24

47

2
5

7
9

12

14

16

19

21

24

46

2

6

7

9
12
14
16
18
21
23

40

2
5
7
9

11

14

16

18

20

23

44

2

4

7

9
11
13
16
18
20
22

43

2

4
6
9

11

13

15

17

19

22

42

2
4
6
8

11

18

16

17

19

21

41

2

4

6

8

10

12

14

16

18

21

40

2

4

6

8

10

12

14

16

18

20

39

2

4

6

8

10

12

14

16

18

20

88

2

4

6

8

10

11

18

16

17

19

1
2
S

4
5
6
7
8
9
10

84

LOOABITHMB.

CommoB or Brlns I«oirftiltli;

10.

Ho.

Log.

3200

85218

02

—267

64

295

56

—834

68

372

2360

—411

62

449

64

—488

66

—526

68

564

3370

-603

72

—641

74

679

76

717

78

765

33S0

793

82

—832

84

—870

86

—908

88

—946

3390

—984

d2

36021

M

059

m

097

98

185

3300

—173

02

—211

04

248

Ort

—286

08

—324

3310

361

12

—399

14

436

16

—474

18

511

3320,

22 1
24
26 ;
28

3330

32

34
36
88

2340

42
44
46

48

—549

686
—624
—661

698

—786

—773

810

847

884

—822
—959
—996
37033
—070

89
38
39
38
39

38
39
38
38
39

38
38
38
38
38

39
38
38
38
38

37
38
38
38
38

38
37
38
38
37

38
37
38
37
38

37
38
37
37
38

37
37
37
37
38

37
37
87

S7
87

No.

3850

62
64
66
68

3360

62
64
66

68

3370

72

74
76

78

3380

82
84
86
88

3390

92
94
96
98

3400

02
04
06
08

3410

12
14
16
18

3430

22
24
26

28

3480

32
34
36
88

3440

42
44

46

48

Lof.

87107
—144
—181
—218
264

291

—828

—366

401

488

—476
611
648

—585
621

—658
694

—731
767
803

—840

876

912

—949

—986

38021

057

093

—130

—166

-202
—238
—274
—810
—346

—382
417
453
489

—625

—661
596
632

—668
703

—739

—775
810

—846
881

87
37
37
36
37

87
37
36
37
37

36
37
37
36
37

36
37
36
36
37

36
37
36
36

86
36
37
36
36

36
36
36
36
36

35
36
36
36
36

a5

36
36
35
36

36
35
36
35
36

No.

3450

62
64
66

68

3460

62
64
66
68

3470

72
74
76
78

3480

82
84
86

88

3490

92
94
96
98

3500

02
04
06
08

3510

12
14
16
18

3530

22
24

26
28

3530

32
34
36
38

3540

42
44
46

48

L09.

38917
962
987

39023
068

—094

—129

164

199

—236

—270

—805

—840

875

410

446
480
615

660
685

—620

—666

—690

724

769

794

—829
863
898

—933

967
40002
—037

071
—106

140
—176
—209

243
—278

312
346

—381

—415

449

483
—518
—562
—586
—620

86
86
86

35
36

86
35
35
36
35

35
35
35
»5
36

35
35
35
35
35

35
35
34
36
35

86
34
85
35
34

35

35
34
35
34

35
34
34
35
34

34
35
34
34
34

35
34
34
34
34

No.

3550

62
64
56
68

3560

62
64
66
68

3570

72
74
76

78

3580

82
84

36
88

3590

92
94
96
98

3600

02
04
06
08

3610

12
14
16
18

3630

22
24
26
28

3630

32
34
36
38

3640

42
44
46

48

Lof.

40654
688
722
766
790

—824
—868
—892
—926
—960

993
41027
—061
—096

128

—162
—196

229
—263

296

-4J30
363

—897
430

—464

497
—631
664
697
—631

664
697
—731
—764
—797

830
863
896
929
—963

—996
42029
—062
—095
127

160

193

226

—269

—292

84
84
84
34
34

34
34
34
34
33

34

34
34
38
34

34
83
34
33
34

33
34
33
84
33

34
33
33
34
33

38
34
33
33
33

33
33
33
34
33

33

33
32
33

33
33
33
33
33

No.

3650

62
64

66
68

3660

62
64
66
68

3670

72
74
76
78

3680

82
84
86
88

3690

92
94
96
98

3700

02
04'
06
08

3710

12
14
16
18

3730

22
24
26
28

3730

32
34
36
38

3740

42
44
46
48

Log.

42826
867
890

—423
456

488
—621
663
686
-619

661
—684

716
—749

781

813
—846

878
-911

-943

976
43008
—040
—072

104

136
—169
—201
-233
—265

—297
—829
—861
—393
—426

—467
-489
—521
—653
684

616

648

—680

—712

743

776

—807

888

870

—902

32
33
33
32
33

33
32
33
33
32

83
82
33
32
32

38
32
38

32
32

^
32
82
32
32

33
32
32
32
32

32
32
32
32
32

32
32
32
31
32

32
32
32
31
32

32
31
82
82
81

To find Log. 23335 :

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

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

-78 + 28 = 36801.

39

38

37

36

85

34

38

83

31

1

2

2

2

2

2

2

2

2

2

3

4

4

4

4

4

8

8

3

8

3

6

6

6

6

6

6

6

6

6

4

8

8

7

7

7

7

7

6

6

5

10

10

9

9

9

9

8

8

8

6

12

11

11

11

11

10

10

10

9

7

14

18

13

13

12

12

12

11

11

8

16

15

15

14

14

14

13

18

IS

9l 18

17

17

16

16

16

16

14

14

10

\70

19

19

18-

18

17

17

16

16

1

2

8

4
5
6
7
8
9
19

LOOABTTHMS.

85

Common or Brigrips I«ograrittams. Base « 10.

No.

»750

52
54
56
58

62
64
66
68

«770

72
74
76

78

»780

82
84
86
88

9790

92
94

96
98

98O0

02
04
06
08

12
14
16

16

98»0

22
24
26

28

»S30

32
84
36
38

42
44
46
48

Log.

43933
—965

996
44028

059

-091
122

—154
185

—217

—248

279

—311

—342

373

404
—436
—467
498
529

560
—692
—623
—664
—686

—716

—747
—778
—809
—840

—871

—902

982

963

994

45025

—056

086

117

—148

—179

209

—240

—271

301

—332
362

—393
423

—454

S3

32
31
d2

31
32

31
32
31
32
31

31

32

31-

31

31

32
31
31
31
31

32
31
31
31
31

31
31
31
31
81

31
30
31
31
31

31
30
31
31
31

30
31
31
30
81

30
31
30
31
30

No.

$8850

52
54
56

58

2860

62
64
66
68

2870

72

74
76

78

2880

82
84
86
88

2800

92
94
96
98

2900

02
04
06
08

2910

12
14
16
18

2920

22
24
26

28

2930

32
34
36
38

2940

42
44
46

48

Log.

45484
—515

545
—576

606

—637
—667
697
—728
—758

788
818
—849
—879
909

939
969
46000
—030
—060

—090
—120
—150
—180
—210

—240
—270
—300
—330
359

389
419

—449
-479

—509

538
568
^598
627
657

—687
716
746

—776
805

—835
864

—894
923

—953

31
30
31
80
31

30
30
31
30
30

30
31
30
30
30

30
31

30
30
30

30
30
30
30
30

30
30
30
29
30

30
30
30
30
29

30
30
29
30
80

29
30
30
29
30

29
30
29
30
29

No.

2950

52
54
56
58

2060

62
64
66
68

2970

72

74
76
78

2980

82
84
86
88

2990

92
94
96

98

3000

02
04
06
08

3010

12
14
16
18

802O

22
24
26

28

3030

32
34
86
38

3040

42
44
46

48

Log.

46982

47012

041

070

—100

129

-159

—188

217

246

—276
—305
334
363
392

—422

—451

-480

—509

538

567
596
625
654
683

712

741

—770

—799

—828

—857
885
914
943

—972

48001

029

058

-087

—116

144
—173
—202

230
—259

287
-316

344
—373

401

Cm

30

29
29
30
29

30
29
29
29
30

29
29
29
29
30

29
29
29
29
29

29
29
29
29
29

29
29
29
29
29

28
29
29
29
29

28
29
29
29

28

29
29
28
29
28

29
28
29
28
29

No.

3050

52
54
56
58

3060

62
64
66

68

3070

72
74
76
78

3080

82
84
86
88

3090

92
94
96
98

3100

02
04
06
08

3110

12
14
16
18

3120

22
24
26
28

3130

32
34
36

38

3140

42
44
46

48

Log.

48430
458

—487
515

,-^44

572
—601
—629

657
—686

—714
742
770
—799
—827

855
883
911
—940
—968

—996
49024
052
080
108

136
164
192
220
248

276
—304
—332
—360
—388

415

443

471

-499

—527

554
582
—610
—638
665

—€93
—721

748
—776

803

(M

28
29
28
29
28

29
28
28
29
28

28
28
29
28
28

28
28
29
28
28

28

28
28
28
28

28
28
28
28
28

28
28
28
28
27

28
28
28
28
27

28
28
28
27
28

28
27
28
27
28

No.

3150

52
54
56
58

3160

62
64
66
68

3170

72
74
76
78

3180

82
84
86
88

3190

92
94
96

98

3200

02
04
06
08

3210

12
14
16

18

Log.

49831 28

—859
886

—914
941

—969
996

50024
051

—079

—106
133

—161
188
215

—243
270
297
—325
—352

379
406
433
—461

—488

—515
542
669
596
623

—651
—678
—705
—732
—759

3220 —786

22

—813

24

—840

26

866

28

893

3230

920

32

947

34

974

36

51001

38

—028

3240

—055

42

081

44

108

46

—135

48

—162

27
28
27
28

27
28

27
28
27

27
28
27
27
28

27
27

28
27
27

27
27
28
27
27

27
27
27
27
28

27
27
27
27
27

27
27
26
27
27

27
27
27

27
27

26
27
27
27
26

To find Log. 29019:

Log. 29020 == 46270
Dil 20 30

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

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

|3S
1 2

2

3

4
5
6

7

^
9

10

32

2

8

5

6

8

10

11

18

14

16

31 30 29 28

2

3

6

6

8

9

11

12

14

16

2

3

5

6

8

9

11

12

14

16

1
3
4
6

7

9

10

12

13

1

3

4

6.

7

8
10
11
13

15 14

27

1

3

4

5

7

8

9

11

12

14

26

1

3

4

5

7

8

9

10

12

13

1
2
3
4
5
6
7
8
9
10

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

*^ 3130=4956448

86

IX)GARITHMS.

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

No.

39150

62
M
56
58

00 290

8a60--322
62 348

Log.

51188! „-

215, 27

-2421 S

268; 26

295 27

26
27

348
375

428 26

d/o
—402

j 428

—455
481
-508
534

62
64
66

68

8»70

72 -^M.
74 —508
76 534
78 —661

8980 587

82 —614
84 640
86 —667
88 693

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

8800

02
04
06

24
26
28

8880

32
84

I 26
27
26
27
26

27
26
27
2C
27

*51 27
■»'« 2fi

930 ^
08 -957 27

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

8880 —1141 ^
22 —140 -6
166 26

^26

270 26
-297 -'

36 —323 26
38 —349 ^^

38 —349 ^

8840 —375 „-
42 —401 26

46 -453! 26

48 -H179| 5°

No.

3300

52
54
56
68

3360

62
64
66
68

3370

72
74
76
78

3380

82
84
86
88

8300

92
94
96
98

8400

02
04
06
08

8410

12
14
16
18

8490

22
24
26
28

3430

32
84
86
38

3440

42
44
46

48

Log

52504
530
556
582
608

—634

—660

—686

711

737

—763

— 7«i9

—815

840

—866

—892
917
943

-969
994

58020
—046

071
—097

122

—148
173

—199
224

—260

275
—801

326
—852

877

—403
428
453

—479
504

529

—655

586

605

-631

—666

681

706

—782

-757

26
26
26
26
26

26
26
25
26
26

26
26
25
26
26

25
26
26
25
26

26
25
26
25
26

25
26
25
26
25

26

25
26
25
26

25
2o
26
25
26

26
26
25
26
25

25
25
26
25
25

3740

42
44

46

48

Log.

56229
263
—277
—301
824

348
—372
—396

419
-443

—467
490
—614
—538
661

—585
608
—632
—656
679

—703
726

—750
773

—797

820
—844

867
—891
—914

937
—961

984
67008
-031

054
—078
—101

124
—148

—171
194
217

—241

—264

287
810
—834
—857
—880

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

Dif. 20 94
3B114 — 36100=»14

Under 24
Dif. for 10 = 12

(> (I

4= 6

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

27

1

1

2

3

3

4

4

5

5

T

6

8

7

10

8

11

9

12

10

14

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

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

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

LOGARITHHB.

LOOARlTHMa.

8S

90

10
20

n

SB

40
15

flO
SB

70
79

ao
le

' 90

OB

MO*

OB
»

30

■X

«o

4S

«0
6B

70

811
SB

w

OS

;|

s u

- i

- >

- n

i

35
85

31

3!
I

35

S4
39
34
»
34
36
SI
34

S.
M

Si

Nb.
•BOO

IS
80

40

45

SKIM

«0

70
7S
80

w

6«00

10

so

40
46

«a

«0

M

S700

15

. 85

40

Log.

4M

-sst

723

— 76T

-m

~MI

—086

—119

161

249
2N

-«18

880
441

«H
— S4(

—70;
-73;

-8oi
-ss.

891

1

38
S8
S3
S

S
33

s;

Ne.'
•7BO

76
10

ss

45
•BSO

SB

95
10

ao

SB

30

M
4B

«»se

80

1

Lw.

-9«

- K

i

n

w

85^

-11 ■

5

33
K

3;

32
32
32
32
31

32
3

32
32

Ho.

40
4fi

™.

60

80

90

9fi

7100

06
10

20
2fi
30
8B

*^
71S«

60

90

Taoo

16

80
35

45

Log

-54
—57

—63

7S
SB

-94
SSOO

09

~11
-24

33

40

-46

-82
-66

61

IS
-82

-«a

-91
—94.
-91

S

81
30

31

30

30

No.

raao

«o

7300

10

so

80

40

46

7SS0

60
S6

80
86
90
95
7400

EO
3S
SO

40

45

74<tO

Log.

86wi
—064
— OM

-37!
361

4S1

~6»!
SB)

7i:
—74:

-801
— 95J

on

-216
-24!

1

30
M

90
80
30
9>
30
30

SO

so

29

20
30

20

39
30

39
30
39
30

29
20
39

so

39

30

39

29

29
39
29
29
20

MOD

s
Its

MS

-as

J»BSO.

3

S

i

7
8
9
■0

39

':

s:

3a

0.6

31

0.(

8

)

i
3

6.8

3

i

•testbi
ue Is
e Ubu

iHBth

ta be
Bnh:

1"

ioa

but

i61

LOOABTTHUS.

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

92

eSOXBTBT.

QEOMETBI.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

AJlGIaES.

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

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

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

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

laanon o n

Z

QEOHETBY.

93

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

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

An interior angle*

•

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

An exterior angle

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

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

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

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

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

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

Or use the last foregoing process with

rada 8, 4, and 5.

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

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

b>ft^c

P 0

^^ftK

V^e

94

eXOMETBT.

If thm line, a b, !■ on tbe rronnd,

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

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

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

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

6 n

10

y 8

rsTi— W

«/.

to anotber line.

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

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

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

7^^^

e

n

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

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

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

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

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

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

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

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

AITGLES.

95

Aayles fln a ParaUeloffimm.

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

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

T« reduce Minutes and Seconds to Beyrees and decimals

of a Degree, etc.

In any given angle —

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

SB Kumber of seconds -^ 3600.

»

Hnmber of mlnntes = Number of degrees x 60.

= Number of seconds -^ 60.

H'nniber of seconds

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

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

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

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

Sees. Deg.

Sees. Deg. Sees. Deg,

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

1

0.016

21

0.350

41

0.683

1

0.00027

21

0.00583

41

0.01138

2

0.033

22

0.866

42

0.700

2

0.00055

22

0.00611

42

0.01166

8

0.060

23

0.383'

43

0.716

3

0.00083

23

0.00638

43

0.01194

4

0.066

24

0.400

44

0.733

4

0.00111

24

0.00666 ; 44

0.01222

5

0.083

25

0.416

45

0.750

5

0.00138

25

0.00694 45

0.01250

6

0.100

26

0.433

4e

0.766

6

0.00166

26

0.00722 46

0.01277

7

0.116

27

0.450

47

0.783

7

0.00194 «

27

0.00750 47

0.01305

8

0.133

28

0.466

48

0.800

8

0.00222

28

0.00777 48

0.01333

9

0.150

29

0.483

49

0.816

9

0.00260

29

0.00805 49

0.01361

10

0.166

30

0.500

50

0.833

10

0.00277

30

0.00833 , 60

0.01388

11

0.183

31

0.516

51

0.850

11

0.00305

31

0.00861 ! 51

0.01416

12

0.200

32

0.533

52

0.866

12

0.00333

32

0.00888 I 52

0.01444

13

0.216

33

0.550

53

0.883

13

0.00361

33

0.00916 53

0.01472

14

0.233

34

0.566

54

0.900.

14

0.00388

34

0.00944

54

0.01600

15

0.250

85

0.583

55

0.916

15

0.00416

35

0.00972

55

0.01527

16

0.266

36

0.600

56

0.933

16

0.00444

36

0.01000

66

0.01555

17

0.283

87

0.616

57

0.950

17

0.00472

37

0.01027

67

0.01583

18

0.300

88

0.633

58

0.966

18

0.00500

38

0.01055

58

0.01611

19

0.816

39

0.650

59

0.983

19

0.00527

39

0.01083 59

0.01638

20

0.383

40

0.66G

60

1.000

20

0.00555

40

0.01111

60

0.01666

-
Sees. Mio.

Sees

. Min.

Sees,

Min.

Sees

. Deg.

Sees. Deg.

Sees. Deg.

* Each equivalent is a repeating decimal, thus :

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

12 seconds = 0.2000000

1 second = 0.0002777

50 seconds = 0.0138888

minute
degree

96

ANGLES.

Approzimate Measurement of Angrles.

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

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

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

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

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

(Original).

Correet.

Ini.

Deg. mio.|

lD>.

Deg. mln.|

Ins.

Deg. min.]

Ids.

Dsg.min.]

Ins.

Deg.mln.]

Ins.

Dag. min.

H

1

12

<y*

20

24

8M

40

IS

l2Ji

61

23

16K

85

14

20 Ji

115 6

1

48

21

40

61

62

5

86

S

116 »

H

2

24

H

21

37

H

41

29

H

62

47

H

86

52

H

117 »

8

00

22

13

42

7

«3

28

87

41

118 30

H

8

86

H

22

60

H

42

46

H

64

11

H

88

81

H

119 40

4

11

23

27

43

24

04

58

89

21

120 52

1

4

47

5

24

3

9

44

t

13

66

35

17

90

12

21

122 •

6

33

24

39

44

42

66

18

91

8

123 20

H

6

58

H

25

16

H

45

21

y*

67

1

H

91

64

H

124 ZS

«

34

25

53

45

59

67

44

92

46

125 64

H

7

10

H

26

90

H

46

88

H

68

28

H

96

88* H i

127 14

7

46

27

7

47

17

69

12

94

81

128 36

H

8

22

H

27

44

H

47

66

H

69

55

H

95

24

H

129 59

8

58

28

21

48

35

70

38

96

17

131 2ft

s

9

34

6

28

58

10 .

49

15

14

71

22

18

97

11

22

132 ftS

10

10

29

35

49

54

72

6

96

6

184 M

H

10

46

H

30

11

H

60

34

H

72

61

H

99

00

H

135 6S

11

22

30

49

51

13

78

86

99

65

187 36

H

11

58

Vi

31

26

H

61

63

H

74

21

H

100

61

H

189 1%

12

34

32

8

62

83

75

6

101

48

141 1

H

18

10

H

32

40

H

53

13

H

75

51

H

102

45

H

142 51

IS

46

83

17

63

63

76

86

103

48

lU 4f

1

14

22

7

33

54

11

64

34

15

77

22

19

104

41

28

146 46

14

68

34

83

55

14

78

8

106

40

148 6B

34

16

34

H

35

10

Vi

65

65

}i

78

54

H'

106

89

H

151 ir

16

10

85

47

56

35

79

40

107

40

153 41

H

16

46

H

36

25

H

57

16

H

80

27

H

106

41

H

156 Si

17

22

37

8

67

57

81

14

109

48

159 41

H

17

59

H

37

41

H

58

38

H

82

2

H

110

46

H

168 27

18

35

38

19

59

19

82

49

111

49

168 18

4

19

12

8

38

67*

12

60

00

16

83

37

20

112

53

24

180 00

19

46

39

86

tiU

41

84

26

118

58

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

^ foot = 1.5 inches.

1 in. « .083 ft.

4 ins. = .333 ft.

7 ins. -= .583 ft.

10 ins. « .833 ft.

2 ins. — .167 ft.

5 ins. = .416 ft.

8 ins. = .667 ft.

Hins. =» .917 ft.

3 ins. = .25 ft.

6 ins. >« .5 ft.

9 ins. — .76 ft.

12 ins. = l.O ft.

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

Sine of half _ half the distance between the marks

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

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

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

SnfTBS, TAKQENTS, B70.

97

Sines, Tans^nta, Ac.

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

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

Cosine e < of an angle acb^

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

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

Tanicen tbworad, of any angle

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

W, is the secant of the angle

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

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

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

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

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

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

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

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

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 radius-scale ; and the lint
et drawn through the point t will form the reqd angle act of 46^. And so at each
angle. The degree of accuracy attained will evidently depend on the length of the
rad, and the neatness of the drafting. The method becomes preferable to the com-
mon protractor in proportion as the lengths of the sides of the angles exceed the rad
of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not
much exceeding the same limits, the protractor will usually be preferable. The di-
viders in boxes of instruments are rarely fit for accurate arcs of more than about 6
ins diam. In practice it is not necessary to actually describe the whole arc, but
merely the portion near t, as well as can be Judged by eye. We thus avoid much use
of the India-rubber, and dulling of the pencil-point. For larger radii we may dis-
pense with the dividers, and use a straight strip of paper with the length of the rad
marked on one edge ; and by laying it from c toward «, and at the same time placing
another Jtrip (witii one edge divided to a radius-scale) from n toward t, we can
by trial find their exact point of intersection at the required point t. In such mat*
ters, practice and some Ingenuity are very essentlial to satisfactory results. We can'
not devote more space to the subject.

m' »

CHORDS TO A RADIUS 1.

M.

OO

1°

HP

SO

4°

5°

e°

70

80

90

10°

M.

0'

.0000

.0175

.0849

.0584

.0098

.0872"

.1047

.1221

.1395

.1669

.1748

0'

2

.0000

.0180

.0855

.0589

.0704

.0878

.1063

.1227

.1401

.1675

.1749

8

4

.0012

.0186

.0061

.0585

.0710

.0884

.1068

.1288

.1407

.1581

.1756

4

6

.0017

.0192

.0M6

.0541

.0715

.0890

.1064

.1288

.1418

.1587

.1761

6

8

.0023

.0198

.0872

.0547

.0721

.0896

.1070

.1244

.1418

.1692

.1766

6

10

.0029

.0204

.0878

.0668

.0727

.0901

.1076

.1860

.1484

.1596

.1772

10

IS

.0035

.0200

.0884

.0558

.0738

.0907

.1082

.1256

.1430

.1604

.1778

18

14

.0041

.0215

.0890

.0564

.0739

.0913

.1087

.1262

.1436

.1610

.1784

14

16

.0047

.0221

.0896

.0570

.0745

.0919

.1093

.1267

.1442

.1616

.1789

16

18

.0052

.0227

.0401

.0576

.0750

.0025

.1009

.1278

.1447

.1621

.1706

18

SO

2S

] .0058

.0238

.0407

.0682

.0766

.0981

.1105

.1279

.1468

.1627

.1801

20

-.0004

.0239

.0413

.0588

.0762

.0936

.1111

.1285

.1459

.1683

.1807

22

24

.0070

.0244

.0419

.0598

.0768

.0942

.1116

.1291

.1465

.1639

.1813

24

ss

.0076

.0250

.0426

.0599

.0774

.0948

.1182

.1296

.1471

.1645

.1818

26

28

.0081

.0256

.0430

.0605

.0779

.0954

.1128

.1302

.1476

.1660

.1824

28

80

.0087

.0262

.0486

.0611

.0785

.0960

.1134

.1808

.1482

.1666

.1830

30

S2

.0008

.0268

.0442

.0617

.0791

.0965

.1140

.1314

.1488

.1662

.1836

32

94

.0000

.0273

.0448

.0622

.0797

.0671

.1145

.1320

.1494

.1668

.1842

34

SS

.0105

.0279

.0464

.0628

.0808

.0977

.1151

.1325

.1500

.1874

.1847

86

88

.0111

.0285

.0460

.0684

.0808

.0983

.1157

.1831

.1505

.1679

.1858

36

40

.0116

.0291

.0465

.0640

.0614

.0989

.1168

.1887

.1511

.1685

.1859

40

42

.0122

.0297

.0471

.0646

.0620

.0994

.1169

.1343

.1517

.1691

.1865

42

44 .0128

.0303

.04n

.0651

.0826

.1000

.1175

.1349

.1523

.1697

.1871

44

48

.0134

.0308

.0483

.0857

.0832

.1006

.1180

.1366

.1529

.1703

.1876

46

48

.0140

.0314

•fPMHp

.0463

.0838

.1012

.1186

.1360

.1534

.1708

.1882

48

M)

.0146

.0320

.0404

.0660

.0843

.1018

.1192

.1366

.1540

.1714

.1888

50

(2

.0151

.0386

.0500

,0675

.0849

.1023

.1198

.1372

.1546

.1720

.1894

5S

64

.0157

.0382

.0606

.0681

.0856

.1029

.1204

.1.378

.1552

.1726

.1900

54

M

.0163

.0387

.0512

.0686

.0861

.1035

-.1209

.1384

.1658

.nn

.1905

56

68

.0160

.0848

.6618

.0092

.0867

.1041

.1215

.1389

.1561

.1737

.1911

58

•D

join

U»49

.0524

.0096

.0872

.1047

.1221

.1396

.1569

.1743

.1917

60

144

TABLE OF CHORDS.

Table of Cbords, In

parte

of aradl;

for protractlng^-Gontinued.

M.

11°

12°

13°

14°

15°

1«°

17°

18°

1©°

20°

M.

0*

.1917

.2091

.2264

.2437

.2611

.278S

.2966

.3129

.3301

.3478

0'

2

.1»2S

.2096

.2270

.2443

.2616

.2789

.2961

.8134

.8807

.8479

2

4

.1928

.2102

.2276

.2449

.2622

.2796

.2968

.8140

.8812

•••Cm

4

6

.19S4

.2108

.2281

.2455

.2628

.2801

.2973

.3146

.3318

.8480

6

8

.1940

.2114

.2287

.2460

.2634

.2807

.2979

.8152

.3824

Jt496

8

10

.1946

.2119

.2293

.2466

.2639

.2812

.2986

.3167

.8330

.8502

20

n

.1962

.2125

.2299

.2472

.2645

.2818

.2901

.S16S

.8386

.8607

12

u

.1957

.2131

.2305

.2478

.2651

.2834

mMK^9

.S169

.8341

.8618

14

16

.1963

.2137

.2310

.2484

.2657

.2830

.8002

.8176

JU47

.S6I»

16

18

.1960

.2143

.2316

.2489

.2662

.2836

.3008

.8180

.3353

.8526

18

20

.1975

.2148

.2322

.2495

.2668

.2841

.SOU

.3186

.8366

.8630

20

22

.1981

.2154

.2328

.2501

.2674

.284T

.9019

.8192

.8364

.3536

22

M

.1986

.2160

.2333

.2507

.2680

.2853

.3026

.8198

.8370

.3542

34

26

.1992

.2166

.2339

.2512

.2685

.2858

.3081

.8208

.3376

.3547

36

28

.1998

.2172

.2345

.2518

.2691

.2864

.3087

.8200

.8381

.3553

38

SO

.2004

.2177

.2351

.2524
.2530

.2697

.2870

.8042

.8215

.3387

.3659

80

32

.2010

.2183

.2357

.2703

.2876

.3048

.3221

.3398

.3565

S3'

Si

.2015

.2189

.2362

.2536

.2709

.2881

.3054

.3226

.3398

.3570

84

36

.2021

.2195

.2368

.2541

.2714

.2887

.3060

.3233

.3404

.3576

86

38

.2027

.2200

.2374

.2547

.2720

.289S

.3065

.8288

.3410

.3688

88

40

.2033

.2206

.2380

.2553

.2726

.2890

.3071

.8244

.3416

.3587

40

42

.2038

.2212

.2385

.2559

.2732

.2904

.3077

.8249

.3421

.3693

43

44

.20U

.2218

.2391

.2564

.27.'57

.2910

.3088

.8255

.8427

•oOW

44

46

.2050

.2224

.2397

.2570

.2743

.2916

.3088

.8261

.3433

.8606

4C

48

.2056

.2229

.2403

.2576

.2749

.2922

.3094

.3267

.8439

.8610

48

60

.2062

.2235

.2409

.2582

.2755

.2927

.3100

.3272

.3444
.3450

.3616

60

52

.2067

.2241

.2414

.2587

.2760

.2933

.3106

.3278

.3622

63

54

.2073

.2247

.2420

.2593

.2766

.2989

.3111

.8284

.3456

.8626

66

56

.2079

.2253

.2426

.2599

.2772

.2945

.3117

.3289

.8462

.3633

5ft

58

.2085

.2258

.2432

.2605

.2778

.2950

.3123

.3295

.3467

..H639

58

60

.2091

.2264

.2487

.2611

.2783

.2956

.3129

.8801

.3473

.3645

60

M.

21°

22°

28°

24°

25°

26°

27°

28°

2»°

so°

"-.

0'

.3645

.3816

.3967

.4158

.4329

.4489

.4609

.4838

.5008

.5176

0'

3

.3650

.3822

.3898

.4164

.43^4

.4606

.4675

.4844

.5013

.5182

2

4

.3656

.3828

.3999

.4170

.4340

.4510

.4680

.4850

.5019

.5188

i

6

.3662

.3833

.4004

.4175

.4346

.4616

.4686

.4855

.6034

.5193

•

8

.3668

.3839

.4010

.4181

.4352

.4523

.4608

.4861

.5030

.5199

8

10

.8673

.3845

.4016

.4187

.4357

.4527

.4697

.4867

.6036

.5204

10

12

.3679

.3850

.4022

.4192

.4363

.4538

.4703

.4872

.5041

.5210

12

14

.3686

.3856

.4027

.4198

.4369

.4539

.4708

.4878

.6047

.5816

14

16

.3690

.3862

.4033

.4204

.4374

.4544

.4714

.4884

.5063

.5221

16

18

.3696

.3868

.4039

.4209

.4.180

.4550

.4720

.4888

.5058

.6227

18

ao

.3702

.3873

.4044

.4215

.4386

.4556

.4725

.4885

.6064

.5233

30

22

.3708

.8879

.4050

1
.4221

.4391

.4661

.4731

.4901

.5070

.5238

22

24

.3713

.3885

.4056

.4226

.4397

.4567

.4787

.4906

.5075

.5244

34

26

.8719

.3890

.4061

.4232

.4403

.4573

.4742

.4812

.5081

.5249

36

28

.3725

.3886

.4067

.4238

.4408

.4578

.4748

.4917

.5086

.5255

38

SO

.3730

.8902

.4073

.4244

.4414

.4584

.4754

.4923

.6092

.5261

SO

32

.3736

.3908

.4070

.4249

.U20

.4590

.4759

.4929

.5098

.5266

S3

34

.3742

.3913

.4084

.4255

.4425

.4595

.4765

.4934

.5108

.6272

34

36

.3748

.3919

.4090

.4261

.4431

.4601

.4771

.4940

.5109

.5277

36

88

.3753

.3936

.4096

.4266

.4487

.4607

.4776

.4946

.5115

.52b3

80

40

.3759

.3980

.4101

.4272

.4442

.4612

.4782 .

.4061

.5120

.5269

40

42

.3765

.3936

.4107

.4278

.4448

.4618

.4788

.4957

.5126

.5294

48

44

.8770

.3942

.4113

.4283

.4454

.4624

.4793

.4963

.6131

.5300

44

46

.3776

.3947

.4118

.4289

.4459

.4629

.4799

.4<M8

.5137

.5306

40

48

.3782

.8953

.4124

.4295

.4465

.4635

.4805

.4974

.5143

.5311

40

.1.

.3788

.3959

.4130

.4800

.4471

.4641

.4810

.4979

.5148

.5317

60

52

.3798

.3065

.4135

.4.HG6

.4476

.4646

.4816

.4985

..M54

..^322

fit

54

.3799

.3970

.4141

.4312

.4482

.4652

.4822

.4991

.5100

.58?8

M

56

.9806

.8976

.4147

.4317

.4488

.46.')8

.4827

.4996

.6166

.5834

60

58

.3810

.3982

.4153

.4323

.4493

.4663

.4888

.6003

.6171

.5839

60

00

.3816

.3987

.4158

.4329

.4499

.4669

.4888

.5008

.6176

.5846

00

TABLE OF CHOBDB.

145

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

M.

81°

as*"

Sso

Z4P

99°

86°

87°

88°

89°

40°

M.

••

.5846

.5613

.5680

.5847

.6014

.6180

.6346

.6511

.6676

.6840

0'

3

.5850

.5618

.5686

.5868

.6030

.6186

.6363

.6517

.MH'X

•OBVQ

2

A

.5856

.5534

.5601

.6868

.6035

.6191

.6357

.6633

.6687

.6851

4

«

.5868

.5630

.5697

.5864

.6081

.6197

.6363

.6538

.6693

.6867

6

8

.5867

.6685

.5708

.5870

.6036

.6303

.6368

.6633

.6606

.6863

8

M

.5878

.5541

.5706

.5676

.6042

.6306

.6874

.6630

.6704

•0888

10

13

.5878

.5646

'.5714

.5881

.6047

.6314

.6379

.6544

.6709

.6873

12

14

.5884

.5562

.5719

.6886

.6063

.6310

.6385

.6560

.6715

.6879

14

U

.5880

.5667

.5786

.5893

.6058

.6335

.6390

.6730

16

18

.5395

.5568

.5780

.5897

.6064

.6280

.6396

.6661

.6725

.6890

18

90

.5401

.5569

.6796

.5803

.0070

6236

.6401

.6666

.6731

.6895

20

S

.5406

.S6T4

.5743

.5600

.6075

.6241

.6407

.6673

.6736

.6901

22

M

.5413

.5580

.5747

.5814

.6081

.6247

.6412

.6677

.6743

.6906

24

»

.5418

.5686

.6758

.5830

.0086

.6353

.6418

.6683

.6747

.6911

26

»

.M2S

.5501

.6768

.5936

.6002

.6258

.6438

.6588

.6763

.6917

28

JO

.54*29

.5507

.6764

.6981

.6097

.6263

.6439

.6694

.6758

.6923

80

n

.5484

.5608

.6769

5986

.6103

.6260

.6484

.6589

.6764

.6838

82

a

.5440

.6606

.6775

.5843

.6108

.6374

.6440

.6605

.6769

.6933

81

»

.5446

.5613

.6781

.5047

.6114

.6280

.6445

6610

.6775

.6039

M

18

.5451

.5619

.6786

.6963

.6119

.6386

.6451

.6616

.6780

.J944

38

40

.5457

.5625

.6793

mngg\
•OWOV

.6135

.6391

.6456

.6631

.6786

.6950

40

43

.5463

.5630

.6797

.5964

.6130

.6396

.6463

.6637

.6791

.6955

42

44

•9voO

.6686

.6806

.6870

.6136

.6303

.6467

.6632

.6797

.6061

44

46

.5474

.5641

.6808

.5075

.6143

.6307

.6473

.6638

.6803

.q8od

46

48

.5479

.5647

.5814

.5061

.6147

.6313

.8476

.6643

.6806

.6971

48

fiO

.5485

.5653

.5820

.5866

.6153

.6318

.6484

.6649
.6654

.6613

.6977

50

51

.5490

.5668

.6826

.5983

.6158

.6334

.6489

.6619

.6983

52

64

.5486

.5664

.6861

.5087

.6164

.6330

.6495

.6660

.6824

.6988

54

M

.5502

.5660

.5886

.6006

.6169

.6336

.6600

.6665

.6829

.6993

56

W

.5507

.5675

.6648

.6000

.6175

.6841

.6606

.6671

.6835

.6999

66

40

.5513

.5680

.6847

.6014

.6160

.6846

.6611

.6676

.6840

.7064

60

0'
3
4
6
8
10

i7

14
16
U

21
24

28
28
10

HT

J4

16
18
40

46
46
50

IS'

54

M

41°

48°

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

, .7167
I .7171
I .7176

.7184
I .7188

.7186

.7200
.7206
.7211
.7216
.7222

.7227
.7232
.7238
.7343
.7249

.7081

.7254

.7097

.7280

.7102

.7265

.7106

.7270

.7113

.7276

.7118
.7124
.7129
.7135
.7140

.7281
.7387
.7282
.7388
.7803

.7146
.7151
.7156
.7162
.7187

.7806
.7314
.7819
.7126
.TIM

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

44'

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

.7362
.7368
.7878
.7379
.7384

.7390
.7385
.7400
.7406
.7411

.7417
.7432
.7427
.7433

.7488

.7524
.7580
.7536
.7541
.7546

.7551
.7557
.7562
.7568
.7573

.7578
.7584
.7588
.7596
.7600

.7443
.7448
.7464
.7460

.7466

.7471
.7476
.7481
.7487
.7493

.7605
.7611
.7616
.7631
.7637

.7683
.7638
.7648
.7648
.7664

45°

48°

.7664

.7816

.7659

.7820

.7664

.7826

.7670

.7831

.7675

.7836

.7681

.7841

.7686

.7847

.7691

.7852

.7687

.7857

.7703

.7868

.7707

.7868

.7713

.7873

.7718

.7879

.7733

.7884

.7739

.7890

.7784

.7895

.7740

.7900

.7746

.7906

.7750

.7911

.7756

.7916

.7761

.7933

.n66

.7987

.7773

.7933

.7777

.7938

.7783

.7948

.7788

.7948

.7793

.7954

.7799

.7959

.7804

.7964

.7809

.7970

.7815

.7975

47° 48'=

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

.8007
.8013

.8018
.8033
.8028

.8084

.8030
.8044
.8050
.8065

.8060

.8071
.8076
.8083

.8067
.8093
.8098
.8103
.8108

.8118
.8119
.8134
.8139
.8136

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

.8167
.8173
.8177
.8183
.8188

.8193
8198
.8204
.8209
.8314

.8320
.8235
.8230
.6236
.8341

.8246
.8351
.8257
.8263
.8367

.8273
.8278
.8383
.8389
.8394

49°

59°

.8394

.8453

.8299

.8458

.8304

.8463

.8310

•o40d

.8315

.8473

.8320

.8479

.8336

fUAL

.8331

.8489

.8336

.8495

.8341

.8500

.8347

.8505

.8353

.8510

.8357

.8516

.8363

.8521

.8368

.8526

.8373

.8531

.8378

.8537

.8.^84

.8543

.8389

.8547

.8394

.8552

.8400

.8558

.8405

.8563

.8410

■8668

.8415

.8573

.6431

.8579

.8436

.8584

.8431

.8589

.8437

.8694

.8443

.8600

.8447

.8605

.8453

.8610

V
3
4
6

8
10

13
14
16
IB
20

33
34
36
38
30

82
34
36

98
40

42
44

48

48
50

~M
54
68

58

10

146

TABLE OF CHORDS.

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

M.

n°

6SO

MP

54''

Ofto

56°

57«

Sfio

59°

•o°

0'

MIO

.8767

.8934

.9060

.9286

.9889

.9648

.9696

1.0000

3

.8615

.8778

.8939

.9066

.9340

.9396

.9648

.9701

.9864

1.0006

4

.8621

.8778

.8984

.9090

.9345

.9400

.9568

.9706

.9860

1.0010

«

.8636

.8783

.8940

.9096

.9260

.9405

.9569

.9711

UCMBJ

1JW16

8

.8681

.8788

.8946

.9101

.9256

.9410

.9564

.9717

•vonP

1.0030

10

.8686

.8794

.8960

.9106

.9281

.9416

.9669

.9733

.9674

1.0036

13

.8642

8790

.8966

.9111

.9266

.9430

.9674

.9737

.9879

1.0060

14

.8647

.8804

.8960

.9116

.9271

.9436

.9679

.9782

.9884

1.0066

16

.8662

.8809

•8D0D

.9131

.9276

.9480

.9684

.9737

•VSoV

1.0040

18

.8667

.8814

.8971

.9136

.9281

.9486

.9689

.9742

■INNM

1.0046

30

.8668

.8830

.6976

.9183

.9287

.9441

.9694

.9747

.9899

1.0060

38

■8Od0

.8836

.8961

.9187

.9292

OMf

.9763

.9904

1.0065

34

.867S

.8880

•cWBo

.9143

.9297

.9461

•9604

.9767

.9909

1.0060

as

.8678

.8885

.8993

.9147

.9302

.9466

.9610

.9763

.9914

1.0065

38

.8684

.8841

.8897

.9163

.9807

.9461

.9616

.9767

.9919

1.0070

M

.8688

.8846

.9003

.9167

.9312

.9466

.9630

.9773

.9934

1.0076

83

•OWv

.8851

.9007

.9168

.9817

.9473

.9626

.9778

.9939

1.0060

M

.8690

.8866

.9013

.9168

.9823

.9477

.9680

.9788

■VvV*

1.0066

86

.8706

.8861

.9018

.9178

.9828

.9483

.9685

.9788

.9989

1.0061

88

.8710

.8867

.9038

.9178

.9833

.9487

.9640

.9798

.9946

1.0096

40

.8716

.8872

.9038

.9183

.9888

.9493

.9646

.9798

.9960

1.0101

43

.8720

.8877

.9088

.9188

.9843

.9497

.9660

.9808

.9955

1.0106

44

.8736

.8882

.9088

.9194

.9348

.9503

.9666

.9608

.9060

1.0111

46

.8781

.8887

.9044

.9199

.9853

.9607

.9661

.9618

.9965

1.0116

48

.8786

.8888

.9049

.9304

.9869

.9512

•VOBo

.9818

.9970

1.0131

60

.8741

ftflOfi

.9064

.9309

.9364

.9518

.9671

.96X8

.9976

1.0136

63

.8747

.8908

.9069

.9314

.9869

.9623

.9676

J638

.9980

1.0181

64

.8762

.8908

.9064

.9319

.9874

.9638

.9681

.9668

.9986

1.0186

66

.8757

.8914

.9069

.9335

.9379

.9638

.9686

.9888

.9990

1.0141

68

.8762

.8019

.9076

.9330

.9884

.96a6

.9691

.9648

.9996

1.0146

60

.8767

.8924

.9080

.9336

.9880

.9548

•VQVD

.9648

1.0000

1.0161

9

3
4
6
8
10

13

14
1«

18

IS

94
16
SB
10

ss

84

44

6S
64

M.

en.o

62°

•8°

64°

65°

e^°

•7°

•SO

er>

700

M.

0'

1.0151

1.0801

1.0450

1.0698

1.0746

1.0693

1.1089

1.1184

1.1838

1.1473

0-

3

1.0156

1.0306

1.0455

1.0608

1.0761

1.0898

1.1044

1.1189

1.1888

1.1476

s

4

1.0161

1.0811

1.0460

1.0608

1.0756

1.0903

1.1048

1.1194

1.1888

1.1481

4

6

1.0166

1.0316

1.0466

1.0613

1.0761

1.0907

1.1063

1.1198

1.1S43

1.1486

e

8

1.0171

1.0321

1.0470

1.0618

1.0766

1.0912

1.1068

1.1203

1.1S47

1.1491

s

10

1.0176

1.0826

1.0475

1.0623

1.0771

1.0917

1.1063

1.1208

1.U63

1.1496

M

13

. 0181

1.0331

1.0480

1.0628

1.0775

1.0923

1.1068

1.1213

1.IS67

1.1500

IS

14

1.0186

1.0336

1.0485

1.0683

1.0780

1.0927

1.1073

1.1218

1.1963

1.1606

U

16

1.0191

1.0841

1.0490

1.0688

1.0785

1.0982

1.1078

1.1222

1.1866

1.1610

16

18

1.0196

1.0346

1.0495

1.0643

1.0790

1.0937

1.1082

1.1227

1.1371

1.1614

U

20

1.0301

1.0361

1.0500

1.0648

1.0795

1.0942

1.1067

1.1232

1.1876

1.1619

33

1.0206

1.0356

1.0504

1.0653

1.0800

1.0946

1.1093

1.1237

1.1381

1.1634

S8

34

1.0211

1.0361

1.0609

1.0658

1.0605

1.0951

1.1097

1.1242

1.1386

1.1529

S4

26

1.0216

1.0866

1.0614

1.0662

1.0810

1.0956

1.1102

1.1246

1.1390

1.1683

38

1.0221

1.0870

1.0619

1.0667

1.0615

1.0961

1.1107

1.1351

1.1395

1.1538

36

80

1.0236

1.0876

1.0534

1.0672

1.0620

1.0966

1.1111

1.1366

1.1400

1.1643

80

83

1.0231

1.0380

1.0529

1.0677

1.0824

1.0971

1.1116

1.1261

1.1406

1.1548

83

84

1.02S6

1.0385

1.0534

1.0682

1.0829

1.0976

1.1121

1.1266

1.1409

1.1562

84

86

1.0241

1.0390

1.0539

1.0687

1.0834

1.0980

1.1126

1.1271

1.1414

1.1667

86

88

1.0246

1.0896

1.0644

1.0692

1.0839

1.0985

1.1131

1.1275

1.1419

1.1662

Si

40

1.0251

1.0400

1.0648

1.0697

1.0644

1.0990

1.1136

1.1280

1.1434

1.1567

46

43

1.0256

1.0406

1.0554

1.0702

1.0649

1.0995

1.1140

1.1285

1.1439

1.1571

4S

44

1.0361

1.0410

1.0659

1.0707

1.0654

1.1000

1.1145

1.1290

1.1433

1.1576

44

46

1.0266

1.0416

10664

1.0712

1.0859

1.1006

1.1150

1.1295

1.1438

1.1681

4ft

48

1.0271

1.0420

1.0568

1.0717

1.0863

1.1010

1.1165

1.1299

1.1443

1.1586

4ft

60

1.0376

1.0425

1.0574

1.0721

1.0868

1.1014

1.1160

1.1304

1.1448

1.1690

60

63

1.0281

1.0430

1.0579

1.0726

1.0673

1.1019

1.1165

1.1309

1.1453

1.1506

63

64

1.0286

1.0435

1.0584

1.0781

1.0678

1.1024

1.1169

1.1314

1.1467

1.1600

64

66

1.0391

1.0440

1.0589

1.0736

1.0683

1.1029

1.1174

1.1319

1.1462

1.1606

M

16

1.0396

1.0445

1.0598

L0741

1.0888

1.1034

1.1179

1.1833

1.1467

1.1600

6B

•0

1.0801

1.0460

1.0666

1.0746

•-

1.0693

1.1039

1.1184

1.1828

1.1473

l.ljSU

•ft

TABLE OF CHORDS.

147

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

for protractlnfT—

-Continued

M.

71°

TSB®

7SO

740

750

7«o

770

78°

7V>

80°

ML

0'

1.1614

1.1756

1.1896

1.2036

1.2175

1.2313

1.2450

1.9586

1.2722

1.2856

»

•i

1.1619

1.1700

1.1901

1.2041

1.2180

1.2318

1.2455

1.2691

1.27a

1.2860

3

i

1.1624

1.17«

1.1906

1.2046

1.2184

1.2322

1.2459

1.2505

1.2731

1.2865

4

•

1.1628

1.1770

1.1910

1.2050

1.2188

1.2327

1.2464

1.2600

1.2735

1.2869

«

•

1.163S

1.1775

1.1916

1.2056

1.2194

1.2882

1.2468

1.2604

1.2740

1.2874

8

10

1.1638
1.1642

L1770
1.1704

1.1920

1.9060

1.2198

1.2886

1.2473

1.2609

1.2744

1.2878

10

u

1.1934

1.9004

1.3303

1.2841

1.3478

1.M14

1.2748

1.2882

IS

14

1.1647

1.1T80

1.1939

1.2060

1.2208

1.2346

1.24«i

1.2618

1.2763

1.2887

14

U

1.1663

1.170S

1.19S4

1.3073

1.2212

1.2360

1.24H7

1.2623

1.2757

1.2891

10

18

i.nsT

1.1706

1.1IS8

1.3078

1.2217

1.3364

1.2491

1.9627

1.2763

1.2896

18

30

1.1661

i.isa

1.1948

1.3086

1.9991

1.3869

1.9496

1.9I83

1.27M

1.2900

n

n

1.1666

1.1807

1.1949

1.3067

1.3236

1.2364

1.2500

1.2636

1.2771

1.2905

39

34

1.1671

1.1813

l.MM

1.2003

1.9381

1.3368

1.2506

1.9641

1.2776

1.2909

84

3S

1.1676

1.1817

1.1W7

1.9007

1.3236

1.2873

1.2600

1.3646

1.3780

l.»I4

a

a

1.1680

lun

11063

1.2101

1.8340

1.2377

1.2514

1.2660

1.2784

l.»18

a

»

1.U86

1.18M

1.1866

1.3106

1.32a

1.2389

1.K18

1.9664

1.1789

1.2933

so

n

l.ltM

1.1BS1

1.1971

1.3111

1.93a

1.2886

1.2523

1.2659

1.3798

1.2937

88

M.

1.1604

LUM

1.1976

1.3116

1.2254

1.2891

1.2528

1.2663

1.2798

1.2931

84

M

LU99

1.1840

1.1980

1.2120

1.2268

1.2896

1.2539

1.2668

1.2802

1.2936

M

M L1T04

1.1846

1.1986

1.2124

1.22tt

1.2400

1.2687

l.a72

1.2807

1.29a

M

40 i.no0

1.1860

1.1990

1.2129

1.3967

1.2406

1.2641

i.an

1.2811

1.2945

a

43

LHU

1.1864

1.1994

1.2134

1.2272

1.2409

1.2546

1.9B8I

i.aie

1.2949

48

44

1.1718

1.1659

1.1900

1.2138

1.2277

1.2414

1.2550

1.2686

1.2820

1.2954

44

a

Ln23

1.1864

1.9004

1.2143

1.2281

1.2418

1.2555

1.2690

1.2825

1.2958

a

a

1.1727

1.1868

1.3006

1.2148

1.2286

1.2428

1.2559

1.2695

I.28»

1.2962

a

w

1.17S2

1.187S

1.201S

1.2152

1.2290

1.2428

1.2564

1.2690

1.3838

1.2967

60

it

1.1TS7

1.1878

I.90I8

1.2157

1.2296

1.2432

1.3668

1.2704

1.2838

1.2971

68

u

1.174S

1.188t

1.9022

1.2161

1.2299

1.2437

1.2573

1.2706

1.3842

1.2976

64

M

1.1746

1.1887

1.9037

1.2166

1.3304

1.2441

1.2577

1.2713

1.2847

1.2980

66

M

1.1761

1.1803

l.aOS2

1.2171

1.2309

1.2446

1.2582

1.2717

1.2861

1.2985

68

m

L1756

1.1896

1.3066

1.2176

1.3311

1.2450

1.2586

1.2722

1.2866

1.2989

M

0'

9
4
•
8
10

18
14
16
18
80

"m"

84

a
a

M

38
84

48
44

a
a

w

IT

H

16

«1«

.3903

1.

1.

1.

1.9008

1.M07

1.9011

1.8015
1.3030
1.S024

i.soa
i.soa

i.9oa

1.3048

i.soa

1.8061
1.3056

1.3060
1.3064
1.3068
1.8073
1.S0T7

1.3068

1.8086

1.3000

1.!

1.

1.3104
1.3106
1.8118
1.1117
1.S181

1.8181
1.3ta
1.3ia
1.3134
1.3ia

i.8ia

1.3147
1.3158
1.3156
1.3161

i.3ia

i.3ia

1.9174
1.3178
IJia

1J187

1.3191
1.S1M
1.3800
1.3904
1.

1.8213
1.3318
1.8828
1.33a
1.8881

1.3336
1.3239
1.9844

1.83a
i.3a8

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

1.32tt
1.8270
1.3274

1.3379
1.3388
1.3287
1.3293

Lsao

1.3800
1.8306
1.38a
l.ai8
l.ttl8

l.a28

i.aa

1.3W1
1J886
1.

1.044
1.83a
1.3868
1.3367
1.3M1

1.3865

1.3370
l.a74
1.3878
1.8383

840

1.1

1.3387

i.8ai

18896
1.3400
1.3404

1.3409
1.3413
1.3417
1.3481
1.84a

1.3430
1.8484
l.S4a

i.Ma

1.3U7

1.3468
1.84S6

1.8460
1.8466
1.

1.8473
1.3477
1.3a8
1.34a
1.3490

1.3486
1.3499
1.8608
1.3508
l.ai2

85^

1.3612
1.3516
1.85W
1.3525
l.a29
1.3533

1.3538
1.3542
1.3546
1.3560
1.8665

1.85a

1.3663
1.3667
l.a72
l.tt76

1.8580
1.8586
1.85a
1.86a
1.3697

1.8a2
1.3606
1.K10
1.3614
1.3619

1.8623

1.3627
1.3631
1.3636
1.3640

8«°

1.86W
1.3644
l.S6a
1.3668
1.3657
1.3M1

1.3665
1.M70
l.a74
1.M78
1.3682

1.3687

Lsai

1.36M
IJMW
1.3704

1.37a
1.3712
1.3716
1.8721
1.87a

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

1.8750

1.3754
1.37a
1.3783
1.3767

870

880

880

1.8767

1.88a

i.ai8

1.3771

1.3897

1.4028

1.3776

1.3902

i.4oa

1.37a

1.39M

1.4031

1.3784

i.aio

1.4035

1.87a

i.ai4

i.4oa

1.8792

i.ai8

1.4043

1.3797

1.3922

1.4047

1.3801

i.a27

1.4051

1.3806

i.3ai

1.4055

1.8800

1.3966

i.4oa

i.ais

i.aso

1.4064

i.ai8

1.3943

1.4068

i.Mn

1.3947

1.4072

i.a26

1.3952

1.4076

1.8830

1.3966

1.4080

1.3»4

l.S9a

1.4084

1.38a

1.3964

1.4089

1.38a

1.39a

1.4O03

1.8847

l.a72

1.4097

1.3861

l.a77

1.4101

1.3855

i.3ai

1.4105

1.3860

1.3985

1.4109

1.8864

1.38a

1.4113

1.3868

1.3993

1.4U7

1.3872

i.sa7

1.4122

l.a76

1.4002

i.4ia

1.3881

1.4006

1.41M

1.3885

1.4010

1.4134

1.3889

1.4014

i.4ia

1.8808

1.4018

1.4148

0'

8
4
6
8
10

18
14
16
16
M

22
24

a
a
a

88
34
M

a

40

48
44

a

a
a

la
54
a
a

a

F0LYG0N8.

m. HfiuaH. BipUoam. Dctiun.

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

T»I>I« orBeroluP Polygons,

X

■.itTk-

*^ar

M

tiiugla.

Deongon,
UndKBgon.

J .«.„

Ji77»M

eo°

ISO"

Ii;i96152

:»so6si

108°

isn°
H7° is.sese'

180°

90"
60°

.,.».„

40°
32°43.«3M'

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

nx.«,^«,l.

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

■■Ur = iaa°x

TBIASTOIiES.

*A »/K /K* h\i> IV^

E

7

f\ /^

\/r \i^ ^

^ r\

\

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

TBIANaLES.

149

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

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

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

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

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

therefore* ^j s= 397988.6 aqnare ft area.

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

Find tbe nat sines of the three angles ; mult together the sines of the anglae
d and 0 : dlT the sine of the angle h by tbe prod ; mult the qnot by the squari
of the perp height a & ; dlr by 2.

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

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

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

n

C a B

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

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

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

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

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

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

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

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

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

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

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

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

Q allel to e a.

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

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

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

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

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

% t 1% drawn in the triangle. *

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

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

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

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

an r

;\5(\«5-«;''t.^-

150

FLANE TBIGOKOMETBT.

Bifflit-aiiirle^ Tri»iiirlefl«

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

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

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

them only.
>e sidM nwMtlfelf

e h

Also Sine of 0 = -; OoeO = ~/
0 a

Tang I
TangOi

h § 5

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

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

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

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

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

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

' «\

h

^^

•■

;\

^ u

/.

9

►-•-

PLANE TEIGONOMETEY.

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

■ ' C pH<» va

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

to find the oilier sides and an^le.

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

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

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

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

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

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

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

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

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

PLJLKE TRIOONOMETBT.

161

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

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

between tbem.

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

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

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

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

€ase 4. Kaviuir ^b® tbree sides*

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

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

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

The base

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

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

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

-«>

Fiir.i.

Tig.fi.

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

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

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

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

i dlsts on sloping ^ronnd must be measured hor-

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

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

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

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

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

be no accordance between the two. Tba«

PLANE TBIOONOMETKY.

PLANE TRIGONOMBTBY.

153

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

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

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

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

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

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

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

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

and for refraction ; combined.

Fig.7.

Dist.

Allow.

Dist.

Allow.

Dist.

AUow.

Dist.

Allow.

inyarda.

feet.

in miles.

feet.

in miles.

feet.

in milee.

feet.

100

.002

.036

6

20.6

20

229

150

.004

xt

.143

7

28.0

22

277

200

.007

y^

.321

8

86.6

25

357

800

.017

1

.572

9

46.3

30

614

400

.080

11^

.803

10

57.2

35

700

500

.046

\Xc

1.29

11

69.2

40

916

600

.066 '

1%

1.75

12

82.3

45

1168

700

.090

2

2.29

13

96.6

60

1429

800

.118

2H

3.67

14

112

55

1729

goo

.140

3

5.14

15

129

60

2058

1000

.185

3K

7.00

16

140

70

2801

1200

.266

4

9.15

17

165

80

3659

1500

.415

4^

11.6

18

185

90

4631

2000

.738

6

14.3

19

206

100

5717

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

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

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

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

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

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

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

• — I — ' ■ — ' ~i

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

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

154

PLANE TBIQOKOMETBY.

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

tract tbe angle < o e ftom
180° ; tbe rem will be tbe
angle e • n. Cenaeqaent-
ly in tbe triangle nee, we
bare one side n o, and two
angles, «no and e o n, to
find by Case 1 tbe aide o e.
Again, take tbe angle iod
from 180° ; tbe remainder
will be tbe angle n o d, ao
that in tbe triangle dno
we bare one side n o, and
tbe two angle* dno and
» 0 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 0 d, tbe reqd rerfe
bfligbt.

Figr.a.

Figr.9.

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

£x. 5. See Sx 10.

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

oliJecUi,

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

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

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

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

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

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

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

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

bill, above a i^iven point i.

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

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

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

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

Case 1. So in Ex. 2, if the height na be known, the angles measd in that example, wfU enable «k
to compute the dist a 0 ; 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 0 c. Fig 12K

by an assistant. T

"•"fir.. Pifir.l2>

-ksbe I 6L-->*

sn •m^.Mex::^ — ' 1,

flff.lS.

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

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

Is accesftlble.

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

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

Fiff.lfi.

Or tbnss

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

n]f:tt»»t:nn»:nA.

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

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

will ben A.

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

Or tbns, without measurlnir
any ang^le ;

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

Or tbna, witbont meiisarlnir ^ny anffle.

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

vpifxvaBx xett xt0.

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

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

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

PLANE TRIGONOMETRY.

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

lu dlr«ei

FABALLELOGBAHB.

167

Square.

PARAI«IiEI.OOBAMB.

Rectangle. Bhombus.

Rhomboid.

]^""*--,

8

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

To flnd tbe area of any parallelosram.

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

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

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

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

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

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

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

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

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

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

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

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

the

158

TRAPEZOIDS AND TBAPBZIUM8.

TBAPEZOmS.

fi t m n

a « e at

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

To And tbe area of any trapoaold.

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

TRAPCZIUMS.

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

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

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

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

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

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

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

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

Havinff three sides ; and the two included anfrles.

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

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

intersection.

Sea Bamarka affear Parmllalegrams.
In railroad measurements

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

The aame general rule for area appliea to all three
of theae flga ; namely, mult the extreme hor width
« « by ko^ the center depth r «, r e, or r t. an the
oaae may be. Also molt one fawih of the width of
roadway m n, by the mm of tbe two aide-depths I u
and 0 «. 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 «»iT«*> area as a/« d e 6 a; but it will have but five sides, while the other has six. Again,
tt«s Indrawn parallel to 4/, and d* joined, we have upon the same base es, aud between the same
mut^MM e a aadd/. the two equal triangles e • d. and e •/. with the part eot common to both ; and
iMmMBay the rewaintaig part e o d or one. and o «/ of the other, are equal. Therefore, if o «/ be
AaffftMn the polygon, and so d be taken into it, the new polygon gad eg, Fig 2. will have the same
Mas a/ e d eo ; but It has but fbor sides, while the other has five. Finally, if g t, Fig 2, be
ttmZJl u>wa«d)t: aad d » drawn parallel to c s : and c n joined, we have on the same base c «, and
tSMsa lAe aaMt paraMtlt e s and d n, the two equal triangles etn, and ttd, with the part c s I
MHaM le hoth. Tberefore, If we leave out c d (, and take ltt.s f n, we have tbe triangle gne equal
•theaolfBOBjadcy.Pigi; orto o/«dc6a, FIgl. , , „
TM/ffT^P'* method it applicable to polygons of any number of aides.

Wtel

160

POLYGOKS.

IU^hede fg, to a ■mailer

To reduee a larire

nlmllar one.

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

To redaee a map to one on a smaller seale.

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

In a reetanfpnlar tk§;^ ghsd,

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

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

purposes, by means of transparent paper.

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

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

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

mc:r

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

To find tbe area of any irre^nlar fiynre.

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

cn

dBCLESb

161

CIBCIiES.

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

T* And tbe circnmrerenee.

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

To find tbe diam.

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

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

To find tbe area of a cflrele.

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

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

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

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

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

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

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

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

Malt the diam by .8863S.

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

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

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

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

11

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

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

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

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

162

OEMXJLBBm

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

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

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

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

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

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

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

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

To draw a tani; t6 two circles.

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

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

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

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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 0 ' 15.70796

19.63495

10 0

31.41593

78.53982

0 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 0

3.14159

.785398

6 0

18.84956

28.27433

11 0

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 0

6.28319

3.14159

7 0

21.99115

38.48451

12 0

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 0

9.42478

7.06858

8 0

25.13274

50.26548

18 0

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 0

12.56637

12.56637

• 0

28.27433

68.61725

14 0

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 0

47.12389

170.7146

20 0

62.88185

314.1598

25 0

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 0

50.26548

201.0619

21 0

66.97345

346.3606

28 0

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 0

58.40708

226.9801

22 0

69.11504 ' 380.1327 1

27 0

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 0

56.54867

254.4690

28 0

72.25663

415.4756

28 0

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» 0

50.69026

288.5287

24 0

75.39822

452.3893

29 0

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 0

94.24778

706.a'j88

t6 0

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 0

97.38937

764.7676

M 0

118.0973

1017.8760

41 0

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 0

100.5310

804.2477

87 0

116.2389

1075.2101

42 0

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 0

103.6726

865.2986

88 0

119.3805

1134.1149

48 0

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 0

106.8142

907.9203

89 0

122;5221

1194.5906

44 0

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 0

141.8717

1590.4313

50 0

157.0796

1968.4964

56 0

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 0

144.5133

1661.9025

51 0

160.2212

2042.8206

66 0

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 0

147.6649

17849445

63 0

163.3628

2123.7166

67 0

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 0

150.7964

1809.5574

68 0

166.5044

2206.1834

58 0

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

0

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 0

153.9380

1885.7410

64 0

169.6460

2290.2210

69 0

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 0

188.4956

2827.4334

65 0

204.2085

8318.3072

70 0

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 0

191.6372 2922.4666

66 0

207.3451

3421.1944

71 0

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 0

1 94.7787 i 3019.0705

67 0

210.4867

8525.6524

72 0

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 0

197.9203 3117.2453

68 0

213.6283

3631.6811

78 0

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 0

201.0619

3216.9909

60 0

216.7699 3739.2807

74 0

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 0

235.6194

4417.8647

80 0

251.8274

5026J>482

86 0

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 0

238.7610

4586.4598

81 0

254.4690

5152.9974

86 0

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 0

241.9026

4656.6257

81 0

257.6106

5281.0178

87 0

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 0

246.0442

477&3624

88 0

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 0

24&1868

4901.6699

84 0

263.8938

5541.7694

89 0

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 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 0

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 0

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 0

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 0

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 0

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 0

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 0

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 0

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 0

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 0

314.1593

7858.9816

4

293.2153

6841.6907

Diam.

Ciroamf,

Diam,

Ciroamf,

Diam,

iMk.

Ibot. 1

Ineh.

■ Ibot.

Ineh.

1-64

.004091

7-32

.057269

27-64

1-32

.008181

15-64

.061359

7-16

8-64

.012272

ili

.065450

29-64

1-16

.016362

.069640

16-32

«-64

.020463

0-82

.073631

81-64

8^

.024644

10-64

.077722

8^

7^

.028634

6-16

.081812

Hu

.032726

21-64

.086908

17-32

U)36816

11-32

.089994

86-64

6-32

.040908

23-64

.094084

9-16

11-64

.044997

1^

.098176

87-64

8-16

.049087

.102266

19-32

IM4

.068178

13-32

.106366

39-64

Giroamr,
_lbat._

.110447
.114637
.118628
.122718
026809
030900
034990
039081
048172
047262
061868
056448
059534

Diam.

6-8
41-64
21-32
43-64
11-16
46-64
28-82
47-64

Jii

26-32
61-64
13-16

Ciroamf,

.163626

067715

.171806

.176896

079987

084078

.188168

092269

.196360

.200440

.204531

.208621

.212712

Diam,
Inelu

63-64
27-32
65-64
7-8
67-64
29-32
69-64
15-16
61-64
81-32
68-64
1

Cireomr*

.216808
.220808
.224064
.229074
.238161
.237266
.241346
.246487
.249688
.263618
.267700
.261799

dBCULAB ARCS.
CIBCVI.AB ARCS.

179

S^itf.l

BnlM for Fig. 1 apply to all arei •qnal to, or l€w than, a Bemi-circle.
** " Fig. i «« *• «• or greater than, a ■emi-cirelt^

Cltordy a b, ot -vrlfcole aircy mdb,

2 X \/raditi«s — (radiua — rise)^. Fig. 1.

2 X \/iadia«> — (rise — radiiis)^. Fig. 2.

2 X \/rise X (2 X radius — rise). Figs. 1 and 2.
2 X radius X >ine cf}4acb. Figs. 1 and 2.
rise

— 2 X

Figs. 1 and 2.

tangent of a b d.*
2 X dbl X cosine of a&d.* Figs. 1 and 2.

2 X >/db9 — rise*. Figs. 1 and 2.§

approximately 8 X db^ — 3 X Length of arc adb^. Fig. 1.

— 2 « radius X

JjmiMjgOkf adb,

arc a d 5 in degrees

360

. Figs. 1 and 2;

•^ .01746 X radius X arc a d b in degrees. Figs. 1 and 2.

drenmference of circle — length of mnaU arc subtending angle aeb. Fig. 2.

. 8 X d&§ — ohordaft.** ^ ,
approximately 5 Fig. 1.

•abdis — ^ofttie angle a 0 b, subtended by the arc. In Fig. 2 the latter angle
exceeds 180°.

2<I6 — chord of dib^ or of half ad&— \/rlBe« + (i^ab)*. Figs. 1 and 2,

flf rise —
^ chord,
.4 «

..833 «
.8 «

••If rise —
.6 chord
.4 «
.833 **
.8 •*

multiply the rsaolt l^
1.036
1.0196
1.0114
l.t083

multiply the rasnli by

1.012
1.0066
1.00B8
1.0t28

If rise —
.26 chord,
.2 «
.126 «
.1 «

If lisa —
.26 chord
.2 «
.126 «
.1 «

multiply the result by
1.0044
1.0021
1.00036
1.00016

multiply the result Hr
1.0015
1.0007
1.00012
1.00006

180

OIBGULAB ABGB.

Ooattnwd from p. 179.

Bolts for Fig. 1 appij to all arcs equal to or less than a semi-circle.
M u pig^ 2 ** ^ ** or greater than a 8emi<clrclo.

Radimiy eOfC^pi or cbp

. (H «<>)« + ri»e« ^ ij-jga. 1 and 2.
2 X rise

. ^^§_ , Pigs. 1 and 2.
2 X rise

%ab

, Figs. 1 and 2.

sine of ^ a e 6

1 — cosine of ^ a e 6

- ^<^^? , ngs. 1 audi,
sineof >^6e<i|

risedc

1 4- cosine of ^ a o d f

, FIg.x

Rifle* or middle ordliisite» d9p

radius — \/radius« — Q^ab]^, Fig. 1.

radius + \/ndiwfl — Q^aS^, Fig. 2.
radius X (1 — cosine of 6 e d ||), Fig. 1.
radius X (1 + cosine of b e d ||),t Fi^. 2,

^^^ , Figs. 1 and 2.
2 X radius

liab X tangent cf abd,* Figs. 1 and S.

approximately ^^^^ ' '*«• 1-
2 X radius

When radius — chord a b, the resftit is 6.7 parts In lUO too shwrt.
** *^ — 3X chord a b, the result is 0.7 parts in 100 too ahoft;

Side ordimatey as n <»

= >/radiu8> —en* + rise — radlni, Figfc 1 and S.
= proximately /^ ^^. Fig. l.t

* a b d is s 3>^ of the angle acb^ subtended by the arc.

t Strictly, this should read 1 mimu cosine; but the ooslBes of angles between 90*
and 270^ must then be regarded as mimu or negative. Our rule, therefore, amonnta
to the same thing.

^db '^ chord of dib, or of half adb, — \/rUe» + (^a^)'- Xig>- 1 and 2.

I be d — half the angle eob subtended by the are, la Fig. 2, the latter angle
exceeds 180°.
\ When radius = chord a b, this makes de 6.7 parts in 100 too short

'< «< = 3 X chord a b, this makes d e 0.7 parts in 100 too short
The proportionate error is greater with the side ordinates.

CflBCDLAB ABGB.

181

Angley acb, sabtended lay Arc* adb.

An angle and its supplement (as 5 e « and bed, Fig. 2) have the same «ine, the
same cosine and the same tangmU.

CAUtlon. The following sines, etc., are those of only half aob.

fflneof J^oc6 — H?^ . Figs.land2.

radius
radius — rise

rise — radius
radius

, ng.2.

Cosineof Jiac6 J^aST" *^«-^*

Tangent of >^ a c6 ^,^"^^ , Tig.l; - ^ ^**^^. , Fig. i

^* radius ~ rise ® * rise — radius '

Versed sine of ^ a« 6 ■—

rise
radius

, Figs. 1 and 2.

Vo dMwrilM ttie mve sf m elrde too Isury* ftnr Um dl-rtders.

Let a c 1m the choordy and o b the height, of the required arc, as

laid down om the drawing. On a separate sMp af paper, «• m n, drawa c. o h. and aft.
•Ibo b e, parallel to the chord a c. It Is well to make b«,and b e, each a little longer
than a b. Then cut off the paper earefhUy along the lines 8 h and 6 «, so as to leare
renaaining only the strip tabemn. Now, if the straight sides s b and 6 e be applied
to tlie drawing, so that any narts of them shall touch at the same time the points a
and 6, or b and e, the point h on the strip will be in ttie circumference of the arc,
and may be prldced off. Thus, any number of points in the arc may be found, and
afterward united to form the corre.

31d Hi ottiodt Draw tteOMn a b; the rise re; and a 0^6 a From c with radios

e r describe a drele. Make each of the arcs o I and i I equal to ro or r i; and draw
c C cL DiTide eC, eZ, er, each into half as many equal parts as the curre is to be divided
into. Draw the lines 61, 52, 2>3; and a4, a5, a6, extended to meet the first ones at
e, «, A. Then e, «, A, are points in one half the curve. Then for the other half, draw
simUar lines flrom a to 7» 8, 9; and others from b to meet them, as before. Trace
tte ennro by hand.

182

CIRCULAR ARCS.

^It DMj firaquentlj b* of um to

'afhattaiABjMedoi^nol '

azMeding 29<*, or in o<:her wordi, whou cluyrd be it of Uad tiadUm Umm iUriM, th*
nUddle oratnate a o, will be one-half of a c, quite near enovgh fbr manj pap*
poses; b c and < e boinir tangenta to the arc.f And Tica Tena, if in tnch an arc we
make o c equal a o, then will o be, rwj nearly, the point at which tangents fh>m th«
ends of the arc will meet. Also the muUlle oxdlnate n, ot thm ikmlt uno ob,or
ott will be approximately 3^ of a ft, the middle ordinate of the whole arc. Indeed,
this last obserTadon will apply near enough for many approximate uses even if the
arc be as great as 46°; for if in that case we take ^ of o a fbr the ordinate n, n wlU
then be but 1 part in 1U3 too small; and therefore the principle may often be used
in drawings, for finding points in a curve of too great radius to be drawn by the
diTiders ; for in the same manner, V^ of n will be the middle ordinate for the arc n h
or n o; and so on to any extent. Below will be f>uud a table bjr nrldelk tbe
rlae or middle ordliuite ot a ludf mrc can be obtained with greater
accuracy when required for more exact drawings.

CIRCUIjAR arcs in FBSMiUKlIT ITSIB.

The fifth column is of use for finding points for drawing arcs too \argB fbr tiM
beam-compass, on the principle giren above. In even the largest cfllce drawings it
will not be necessary to use more than the first three decimals of the fifth column ;
and after the arc is subdirided into parts smaller than about 86° each, the first two
decimals .25 will generally su£Bce. OriginaL

BlM

For

ForriM

BiM

For

Fer

in

De(r«ei

For nA

length of

of half

In

Dogreei

For rad

length of

rlMoff

paru

in whole

mult rise

aro malt

aro

paru

in whole

multrlM

aro nalt

halfara

of

•ro.

by

oborA

mnltriM

of

are.

iv

ehord

bibIS

dioid.

•

by

by

sherd.

bj

ti—hf

1-60

o /
9 9.76

313.

1.00107

.2601

u

o /
66 8.70

6^

1.04116

•

.2688

1-46

10 10.76

263.626

1.00132

.2501

63 46.90

6.626

1.06366

.2649

1-40

11 26.98

200.6

1.00167

.2602

.165

68 63.63

6.70291

1.06288

.2667

1-36

13 4.92

163.625

1.00219

.2502

1-6

73 44.89

6.

1.07260

.26t6

1-30

15 16.38

113.

1.00296

.2503

.18

79 11.73

4.36803

1.08428

.2676

1-26

18 17.74

78.626

1.00426

.2504

1-6

87 12.34

3.626

1.10847

.2693

1-20

22 60.54

60.6

1.00666

.2506

.207107

90

3.41422

1.11072

.2699

1-19

24 2.16

46.026

1.00737

.2607

.226

96 64.67

2.96913

1.12997

.2616

1-18

26 21.65

41.

1.00821

.2508

.2^6

106 16.61

2.6

1.16912

.2639

1-17

26 60.36

36.626

1.00920

.2609

116 14.69

2.15289

1. J 9083

.2666

1-16

28 30.00

82.6

1.01088

.2510

.3

123 6130

1.88889

1.22496

.2692

1-16

30 22.71

28.626

1.01181

.2611

^

134 46.62

1.626

1.27401

.2729

1-14

32 31.22

26.

1.01366

.2613

144 30.08

1.43827

1.32413

.2766

1-13

34 69.08

21.626

1.01671

.2516

.4

154 38.35

1.28125

1.^322

.2808

1-12

37 60.»6

18.6

1.01842

.2517

.426

161 27.52

1.10204

1.42764

.2838

1-11

41 13.16

16.626

1.02189

.2620

.45

167 66.93

1.11728

1.47377

.2868

1-10

46 14.38

•18.

1.02646

.2625

.476

174 7.49

1.06402

1.62162

.2899

1-0

60 6.9II

10.625

1.03260

.2630

.6

180

1.

1.67080

.2929

V At 29° o • thus fbond will be bat about 8 parti too tiiort in 100.

MENaUKATION, 183

bniStbB af elpenlH »f«s. If itrc«zce«da aaeialelrel«,H*p IS4

riMolii lu obon) ud bdibb dlrtd> Iha fal«tt bj lb« Uud. Ttaa In Uu MoBn dT balibli Iki

MiUpIj llu Uit EiBbir bj ili> Itatlh of U> Jru lEonL * « omn <> Dt>U

TABLE OF CIKOVLAB ABCS. H«nn».

Uvi^i. P'lbu. I'Oiiftb*. H'ibli. l^nctbB. B'lliUr L«DBI^

184 MESSUBATIOH.

TABI.B «F CIKCIJI.AB ARCH—

n arc of 1° if tbe eartb's Krent circle Is but 4.3354 feet loBcrr tbni lt>
1. lu lsiijiUiiiO.lt lindi>riiuniumnn. ■ulli'i«|»virli>lnil:^>HI.b10Siiill«. Polir 3*«><fT.

MENBUKATIOI'. 185

T« Bad tbe Ie>|rUi of > circular src br tbe followliic teUe-

I'EireTBS or circdi.ak abcs to bad i

mi

186

MENSURATION,

CmCVttAR BBCTORSy BINGMS, SBOmSRVS, SSTCX

^ * Area of a eironiar ■eetor, adbe^ Fig. A,

arc adh

X radlua o a.
— area of entire drole X

Fig. B.

aro g d 6 In degrees;
S60

Area of a clrc«lar ving. Fig. B,
.—1 area of larger circle, 0 d, — area of smaller one, a b.
1^ — .7854 X (sam of diams. cd + ah)X (cUfil of diams. e d^a 6.)
— 1.5708 X thickness e « X *<i°^ <^ diameters « d and a h.

To And. the rmdi«a of a clrele -vrhleli aliall have the aanie
as a giyrevk elrciilar rln|^ c» dab. Fig. B,

Draw any radius n r of the outer circle ; and from where said radins cuts tht
bner circle at t, drew < « at right angles to it. Then will t « be the required ladins.

Bresultl&y ea^mbd, of a circular rl»|ft Flf. ^

iM. V^ difference of diameters e d and a &.

« ^ (diameter ed—w 1.2732 area of circle a 6.)

Area of a eirenlar xone abed^

0m area of circle m n — areas of segments am 5 and end,
(for areas of segments, see below.)

A circular Inne is a crescent-shaped
figure, comprised between two arcs abe
. and a o e of circles of different radii, a d
and AM.

of a drcvlar lume uheo

^ area of segment ahe — area of segment a oc^
(fix arcM of segments sea bcloir.)

Pig.D.

V»flndflio

«f »olreiilMP

it^mbodf Figi.O^Di.

Area of Segment adbn, Fig. A (at top of page)
■■ Area of Sector a d 5 e — Am of Triangle a 5 0.
•^^iiAroadb X tadinaa* — en X cbordafty.

Vmwinff the area of a aeKment required to bo ent mtt
gkvewk clrelcy €0 flnd tta chord suad rise.

^ IHTide the area hj the square of the diameter of the clrele : look for the qnotleot
In th9 column of areas in the table of areas, opposite; taice out from the table
Che corresponding number In the column of risei. Mnltipljr this nninbar bgr the
diameter. The product will be the required rise, Thea

ahord — 2 X V^ (dUmeter — rte) X

MENSURATION.

187

TABUB OP AREAS OF CIB€UI«AR SEOlIEjnni, Fiffi C, Dl
' If the seyment exeeeda a semieirelef it* are« i* = %nm <a eireie— i

of • aegmant whose riie Is = (dUm of eirelt — rise of giren segment). Dlaai of eird* * (eqiian
ef hair ohord t> rise) 4* rise, whether the segment exeeeds a eemieirole or not.

Rise

Area=

Rise

.Areas

Rise

Areas

Rise

Area»

Rise

Areap*

dlrhf

(sqnare

diYby

(sqaare
of diam)

dlTby

(Bonare
of diam)

diT by^

(square
of diam)

dirby

(sqnare

diamef

ef diam)

diam of

diam of

liaaof

diam of

of diam

•irele.

malt by

oirole.

moltbj

eiioto.

moltby

einia.

mult by

oirole.
.25^

BMritby

.001

..000042

.064

.021168

.127

.057991

.190

.103900

.166149

.002

.000119

.065

.021660

.128

.058658

.191

.104686

.254

.157019

.003

.000219

.066

.02'2;55

.129

.059328

.192

• .106472

.255

467891

.004

.000337

.067

.022663

.130

.059999

.193

.106261

.256

.168768

.005

.000471

.068

.023156

.131

.060673

.194

.107051

.257

469686

.006

.000619

.060

.023660

.132

.061349

.196

.107843

.258

460511

.007

.000779

.070

.024168

.133

.062027

.196

.108636

.269

461386

.008

.000952

.071

.024680

.134

.062707

.197

.109431

.260

462268

.009

.OOllSft

.072

.025196

.136

.063389

.198

.110227

iS61

.168141

.010

.001329

.073

.025714

.136

.064074

.199

.111025

.262

464020

.011

.001633

.074

.026236

.137

.064761

.200

.111824

.263

464900

.012

.001746

.076

.026761

.138

.065449

.201

.112626

i264

.166781

JQIS

.001969

.076

.027290

.139

.066140

.202

.113427

.266

.166688

mt

.002199

.077

.027821

J40

.066833

.203

.114231

.266

487646

XH6

.002438

.078

.028356

.141

.067528

.204

.115036

.267

.188481

Me

.002685

.079

.028894

.142

.068225

.205

.115842

.268

.109816

.017

.002940

.080

.029435

.143

.068924

.206

.116651

.260

.170202

.018

.008202

.081

.029979

.144

.069626

.207

.117460

.270

471090

.019

.003472

.082

.030526

.146

.070329

.208

.118271

.271

.171978

.020

.003749

.083

.031077

.146

.071034

JHOd

419084

.272

.172868

joai

.004032

.084

.031630

.147

.071741

.210

419898

.273

.173768

JOZ

.004322

.086

.032186

.148

.072450

.211

420718

.274

474660

JOSS

.004619

.086

.032746

.149

J073162

.212

.121530

.276

.176542

J024

.004922

.087

.033308

.160

.073876

.213

422348

.276

476486

J0fi6

.005231

.088

.033873

.181

.074590

.214

423167

.277

477830

JM

.005546

.089

.034441

.152

.076307

.216

.123988

.278

478226

Ml

.005807

.090

.035012

.163

.076026

.216

424811

.279

479122

xas

.006194

.091

.035586

.164

Wfl747

.217

.126634

.280

480020

M9

.006627

.092

.036162

.165

.077470

.218

426469

.281

.180918

J06O

.006866

.003

.036742

.166

.078194

.210

.127286

.282

481818

JOSL

.007209

.094

.037824

.157

.078921

.220

428114

.283

482718

M2

.007660

.096

.037909

.168

.079660

.221

428948

.284

488619

JOBS

.007913

.006

.038497

.169

.080380

.222".

.129778

.286

484622

J084

.008273

.097

.039087

.160

.081112

.223

430606

.286

.186426

j066

.008638

.098

.039681

.161

.081847

.224

431488

.287

486329

JOM

.009006

.099

.040277

.162

.062682

.225

.132278

.288

487236

.037

.009388

.100

.040875

.163

.088320

.226

483109

.289

488141

JOSS

.009764

.101

.041477

.164

.084060

.227

.133946

.290

489048

.080

.010148

.102

.042081

.165

.084801

.228

434784

.291

.189956

J040

.010638

.103

.042687

.166

.085545

.229

.136624

.292

.190866

041

.010932

.104

.043296

.167

.086290

.230

.136466

.293

.191774

J042

.011831

.106

.043908

.168

.087037

.231

.137307

.294

492685

.048

^11734

.106

.044623

.169

.087785

.232

.138151

.296

493597

J044

.012142

.107

.045140

.170

.088536

.233

438996

.296

494509

.046

.012555

.108

.045759

.171

.089288

.234

439842

.287

.196428

.046

.012971

.109

.046381

.172

.090042

.235

140689

.298

496337

j047

.013303

.110

.047006

.173

.090797

.236

.141538

.299

497262

.048

.013818

.111

.047633

.174

.091556

.237

.142388

.300

.198168

J04»

.014248

.112

.048262

.175

.092314

.238

443239

.301

.199086

.060

.014681

.118

.048894

.176

4»8074

.239

.144091

«302

.200008

JO&I

.016110

.114

.049529

.177

.093837

.240

.144945

.308

.200922

M>2

.016661

.115

.060165

.178

.094601

.241

.145800

.804

.201841

J06»

.016008

ai6

.060805

.179

.095367

.242

446656

.366

.202762

J064

.016468

J17

.061446

.180

.096135

.243

.147513

.306

.203688

j06§

.016013

.118

.062090

.181

•090804

.244

448371

.307

.204606

iNM

M79n

419

.062737

.182

.097675

.246

.149231

.308

.206628

jm

.017881

.120

.063886

.183

.098447

.246

.160091

.309

.206462

MB

.018907

.121

.0640:7

.184

.099221

.247

460953

.310

.207376

JOM

.018766

.122

.064690

.186

.099997

.248

461816

.311

.208302

JIMO

.019188

428

.066846

.186

.100774

.249

452681

.312

.209228

jOd

.oime

J24

.066004

.187

.101553

.250

463546

.313

.210166

jm

/mm

096

.066664

.188

.102.334

.261

.154413

.314

.211083

Ml

iMMBI

Jfl6

.087827

.189

J03116

1 .262

.166281

.816

.212011

188 MENSURATION.

TABLK OF AKEAS OF CIRCDE.AK SBONEVTS-tCoHTHiDH:

Urn

1«

A««_

dl.BT

irdi™

«lui<

orai>~i

^nli.

"^imi

.363

i!73«

.380

.383603

'427

mint.

Biujt

56e730

.284H9

.320940

406

5677 2S

£li

.314S0'J

.302

JiU

.369723

isieeea

.3607a

il7«»

.!BaB3.;

.SOS

4«9

.3fll7M

xa

^86W

XSi

^wee

isao

.ae;

!363IU

Mt

33SMM

.Ml

su

SMHli

-390

592390

M»

iaZMTS

iVi

!36fl711

.tzg^S

.8M

.168385

.330S6fe

.367710

MS

.2M1M

.366

JJB9SM

•402

.331861

jm

,236094

too

.ismi

!38383«

.?I070S

^1

.2«S249

■406

.834829

JS2

iersi

406

.33S

.300238

!4S1

571

.301221

.837810

sitm

.S3B

'409

xe

JSl

!»l^04e

504171

^143

5T7T01

jaa

M17SB

xss

MO
Ml

liwai

is78

.i710Bl

A\t

1307126
JOSllD

462

I4477S

4m1

asijoo

MS

.affi3«9

.309096

M673S

.382700

Ma

.238319

.380

464

M*

jawss

.381

.ii*sa

-418

aiiow

466

!492

:384eM

.M0Z19

.848766

,3S68»

!3S3

.38Se»0

Ml

.3S4

^77748

J21

468

.381390

1213074

.38*

.316017

■.3fil7«.

5<W«90

MS

.383

562142

.389300

Xba

.24U80

jHoero

J»i

3MaS6

.28HM3

426

.317981

!4«2

.364736

!499

.S913W

Mi

.MflSM

.380

.118970

4«3

566733

JiOO

xMm

ELLIPSE (page 139).

Focal dlBMiice^/0 =

HENSURATIOir.

189

THK BIiI.IPSfi»

An «B!tM« Is m enrra, • «««, Fig I. formed by an obllqae Mctioa of eltlMr • oone or s eylinder, paaa*
Ins throngh Ita ourred Mrfaee, withoat cattiog the base, lu nature la luoh that if t«o linei, aa
n/ and n g. Fig. 8, be drawn from any point n in Ita periphery or etraamf, to two oertain points/
nnd g, in iu long diam o w, (and called the foei of the eiUpie,) their ram will be eqnal to that of any
other two lines, as i/, and b g, drawn from any other point. a« 6, in the clreumf, to the fooi/aad ^j
slao the snm of any two snch lines will be equal to the long diam « w. The line e w diriding the ellioso
Into two eqnal parte lengthwise, is oalled its transverse, or major axis, or long diam ; and • i, whieh
dirtdee it equally at right-angles to e io, is called the oonjogate, or minor axis, or diort 41ain. To
find the position of the tool of an ellipse, from either end, as 6, of the short diam, memsnre olf the
diets ft /and 6 g. Fig S, each equal to o c, or one-haif the long diam.

The parameter of an ellipse is a oertain length obtained thus ; as the long diam i short diam : :
short diam : parameter. Any line r v, or • d, Fig S, drawn from the eireamf* to, and at right angloa
to, eliher diam, is ealled an ortUnau; and the parts e v and 9W,b» and • «, of that diam* between
the ord and the eiroumf, are oalled al^teUam, or a&seiseei^

To flnil tlie leufftli of any ordinate, rvovsd, drawn to eitbetf

dianif e W or h a* Knowing (h« ahecisa, « • or « a, and tiM two diams, e w, ft •{

ew*:fta<::cvXvwiFA

ftd^i««!*::fr« X « a:g<i>.

To lind the elreumf of an elHpse.

Mathe— HelnM have fhmisked praodeal men with no simple working rale Ibr this pvrpoae. The
•e-ealled appvMdmate mlea do not deserre the name. They are as foUowa, D being the long diam ;
4 the aiiorteino.

RvLB 1. Circamf =8.141« R±A. • Rvlb S. S.M16 / f^^^-\ • Buu t. «.2ai6y' DS^hP:

thte if tiie nme aa Bnle 2, bnt In a dllT shape. Sou4.2X|/ DS+ 1.1874 A Now, in an elUpse

vhoae long and short dlams are 10 and S, the oirenmf Is MtnaUy 11, very approximately; bnt rule 1
(ires it = 18.85 ; rale 2, or 3, == 22.65 ; and rule 4. =: 30.68. Again, if the diams-be 10 and 6, the dr.
•omf aotnallT = 25.50; but rule 4 gives 24.72. These examples show that none of the rales nsnaUy
SiT0n are reliable. The following one by the writer, is snfflclently exact for ordinary pnrpoaes; Ml
Mag iasrrer probably more than 1 part in 1000. When D la not more than 6 ttaass as long as 4,

If D ezeeeda 5 times if, then in- fr

stead of dividing (D — d^ by 8.8, div i^ by Si m

the number in ibis table. o

The following rule originated with Mr. M.
Arnold Pears, of New South Wales, Australia,

s;«S«««SSm68SSS!:fl«

stetSkeisteCaieiSeisieiee^ee

and was by him kindly communicated to the author. Although not more accu*
rate than our own, it is much neater.

3.1416 d + 2(D — d) — d(D — d)

Circumf

V<(D -f d) X (D + 2d)

The following table of senii»elllptle arcs was prepwvd by oar niik

To nse this table, div the height or rise of the are, by its span or ehord. The qnet
will be the height of an are whose span is 1. Find this quot in the oolnmn of
heights ; and Uke out the oorresponding number ft*om the ool. of lengths. Halt this
number by the actoal span. The prod will be thereqd lenRth.

When the height becomes .500 of the chord fas at the end of the table) the ellipse
beeomee a eirole. When the height exceeds .500 of the chord, as in a b e, then take
a o, or half the ehord, as the rise ; and dir this rise by the long diam 6 d, for the
qnot to be looked ror in the ool of heights ; and to be mult by long diam. We tfens
get the aro had, which is evidently equal to a 6 c

190

MENSUIUTIO>.

TABI.E OF I^ENOTHB OF 8EMI.EI«I«IPTI€ ABCB.

ftnrlglnal4

Height

Lengtl^a

Hdght

Lengths

Height

Length v

Height

Lengths

•I'SlAn.

spanxby

. •A'lpftn.

■pan X by

•fr span.

■pan X by

4- ■pan.

■pan X by

JOOb

1.000

.130

1.079

.266

1.219

.880

1.390

M

1.001

.136

1.084

.260

1.226

.385

1.897

.015

1.002

.140

1.089 .

.266

1.233

.890

1.404

.02

1.003

.145

1.094

^0

1.239

.396

1.412

026

1.004

.160

1.099

.276

1.245

.400

1.419

.03

1.006

.166

1.104

.280

1.262

.406

1.425

.036

1.008

.160

1.109

.286

1.259

AIO

1.434

X)4

1.011

.166

1.116

.290

1.265

.416

1.441

X)46

1.014

.170

1.120

.295

1.272

.420

1.44P

.06

1.017

.176

1.126

.300

1.279

.425

1.456

.066

1.020

.180

1.131

.306

1.286

.430

1.464

.06

1.023

.186

1.137

.310

1.292

.436

1.471

.066

1.026

.190

1.142

.316

1.298

.440

1.47»

..07

1.029

.196

1147

.320

1.306

.446

1.486

.076

1.032

.200

1.153

.326

1.312

.460

1.494

.08

1.036

.206

1.169

.330

1.319

.455

1.50i

.086

1.039

.210

1.166

JXif>

1.325

.460

1.509

.09

1.043

.216

1.171

.340

1.332

.465

1.517

.096

1.046

.220

1.177

•346

1.339

.470

1.624

.100

1.061

.226

1.183

.350

1.346

476

1.582

.105

1.066

.230

1.189

.365

1.368

.480

1.540

aio

1.069.

.236

1.196

.360

1.361

.486

1.547

J16

1.064

.240

1.202

.365

1.368

.490

1.556

.120

1.069

.1?45

1.207

.370

1.376

.495

1.568

.126

1.074

.260

1.213

.375

1.382

.500

1.571

Area of an ellipse = prod of dlam^ X .78M. Bz. D = lO ; d = «. Then 10 X 6 X .T§6«
c 47.124 area. The area of an elUpiie la a mean proportional between the areae of two cirelae, d«*
■eribed on its two dlama ; therefore it may be found by mult together the areaii of.thote two -eirolaa ^
and taking the aq rt of the prod. The area of ah ellipse ii therefore always greater than that of th«
eircolar seotion of the cylinder f^om which it may be supposed to be derived.

Dlam of circ of same area as a given ellipse = i^Long diam x ahort diaml
To find tbe area of an elliptic segment wbose iNwe is paral.

lei to eitlier dlam. DIt the height of the segment, bT that diam of which wid height
!■ a part. From the table of circular segments take out the tabular area opposite the qnot. If nil
together this area, the long diam, and the short diam.

To drair an ellipse. Having its long and short dtaas a b and e d, Pig. 4.

BoLB 1. From either end of the short
diam., as c, lay off the dists. ef, ef, each
equal to « a, or to one-half of the loug diam.
The points/, /' are the foci of the ellipse.
• Prepare a string, fn/.orfgf. with a loop
at each end ; the total length of string from
end to end of loop, being equal to the long
diam. Place pins at /and/'; and placing
the lloops over them, trace the curve by a
pencil, which in every position, as at n, org,
keeps the string/' n /, or /' gf stretched all
the time.

Note. Owing to the diflDoulty of keeping
the string equally stretched, this method is
not as satisfactory as the following.

Bulb 2. On the edge of a strip of paper
«0 «, mark w I equal to half the short diam. ;
and IS a equal half the long diam. Then in
whatever position this strip be placed, keep-
ing I on the long diam., and s on the short
diam., te will mark a point in the eircumf. of the ellipse. We may thna obtain at many each polnu
as we please ; and then draw the curve through them by hand.

Bdlb 8. From the two foci / and /', Fig. 4, with a rad. equal to any part whatever of the long
diam. describe 4 short arcs, o o o o; also with a rad. equal to the remaining part of the lon^ diam.,
describe 4 other arcs, iiii. The intersections of these four pairs of ares, will give four points in tha
eircumf. In this manner any number of such pointt may be found, and the curve be drawn by hand.

To draw a tanarent 1 1, at any point n of an ellipse. Draw n /

and n /', to the foci ; bisect the angle / n /' by the line xp ; draw < n ( at right angles to xp.

To draw a Joint n p^ of an elliptic arcli, f^om any point a, im

tbe arcb. Proceed as ic the foregoing rule for a tangent, only omitting (I; np will be
required joiac

I?ig-4.

IfBNSUBATHnr.

191

To draw an OTal, or felse ellipse.

When only tbo long diam a b It given, tbe fbllowing
will give agreeable caires, of wbicb tbe span a h wiU
not exceed abont tbree times tbe riie e o. On a & d»>
■eribe two Intersecting circle* of any rad; through
their Interseetiona t, 9, draw ay; make • g and r •
each eqnal to tbe dtam of one of the eirelea. Tbrongb
the center* of tbe circles, draw «f,*h,gd,gU FroB
edeioribeA<y; and from y dMoribe d o I.

"Wiieii the span, «nn^ and tlio
rise* s t, are boUi yliren.

Make any f w and mr, eqnal to each otbei;^
but each less than t ». Draw r w; and throngn
its center o draw tbe perp toy. Draw y r «•
Make n « equal mr, and draw tfxb. From sand
r describe n e and m m; and fh>m y describ*
ate. By making « d eaaal to « y, we obtain
the center Ibr tbe other side of the oral.

Tbe beaaty of tbe canre will depend npon
what portion of I « is taken for m r and t m.
When OB oval le verf flat, more than three cen-
ters are reqnired for drawing a gracefbl enrre ;
bat the flnflng of these centers Is qaite aa tron*
bleseme as to draw tbe oorrect ellipse.

€tai the §:!▼«>■ line, a 9, to draw a
cyma reeta^ aes.

Find the eenter e, of a ». From «, e, and $, with one-half
ef • • aa rad, draw the fonr small arcs ato. o. The inter*
o, «, are the oenters Ibr drawing the oyma, with
I ra4. By rerersing the position of the ares, w«
oreyee, 4 </.

192

MENSURATION.

THB PAIIABOI.A*

The eommoii or eonle iiarabola,

o b e. Fif 1, is a onrre formed by oatting • oone in a dlreetlon b a, parallel to ita lida.
•arred line obe itself is called theptrimt«r of the parabola ; the line o e is called ita bcwe ; ft • iti
height or axta ; b its apex or vertex i any line e s, or o a. Fig S, drawn from theonrve, to, and at right
angles to, the axis, is an ordinate ; and the part s 6, or a i, of the axis, between the ordinate and the
apax b, is an abscissa. The /ooms of a parabola is that point in the axis, where the abaoisaa 6 «, is
oqual to one-half of the ord e ». The dist from apex to focus, called the focal diet, is found thus:
square auy oid, as o a; div this sauare br the abscissa i a of that ord; diy the quot by 4. The

Cature of the parabola is such that its absoiBsas, as 6 s, 6 a, fto, are to each other as, or in proportion
», the sanares of their respective ords s s, o a, Ac; that is, as i s : ba : : ss* :o<i>;orbs:ss>::b«:
• a* . If the square of any ord be divided by iu abscissa, the qnot will ho a constant qnantltj ; that
Is, it wHl bo equal to the sqoaro of any other ord dlTlded by Its abscissa. This qnot or oonstantqaan*
tfty Is also equal to a eertsln quantity oallod the pmrameter of the parabola. Thersfbra tho p^'^nwtsr
may be found by squaring s s, or e a, (one>ha^ of the base,) and dividing said square bv tho height
i s. or b a, as the case may be. If the square of any ord be divided by tho panoMtar, tbt qnot wff
he the abscissa of that ord.

To And (lio lenyth of a parabolle enrre.

The approximate rule given by various pocket-books, is as IbUows t

Length — 2 X V(H '^>^e)a + \% Umes the (Height^

(g Where the height does not exceed 1-lOth of the base, thls'mle may, for praetlMi

purposes, be called exact. With ht = )^ base, it gives about H par oeat tos
Bueh; ht s M base, about 3^ percent; htsbase, about 8K per coot; ht =
%«tee the base, about 11% percent; ht= 10 X base, or more, about 15)t( per oeat

The flillewlas \ij the writer U eorreel
within perhaps 1 part in aOO, in all eases ; and will
therefore answer for many purposes.

Let a d b. Fig S, orik a d. Fig 4, be the parabola.
In whioh are given the base abvtndt and tte
height c li or c a. Imagine the eonpleteflg ad bs,
or » a 4< b, to be drawn ; and in sttAsr ease, aaanms
Us loMi^ dlam a b to be the chord or base; and one-
half the short diam, or e <i, to be the heightt of a
circular arc. Find the length of this circular are,
by means of the rule and table given for that pur*
pose. Then div the chord or Immo a b, or n d of
the parabola, by its height c d or e a. Look for
the qnot in the column of bases in the following
table, and take from the table the correspondiag
multiplier. Mult the length of the eireolar aro by
this ; the prod will be the length of are a d b, or
n a cl, as the case may be. For bases of parabolas
less than .05 of the hdght, or greater than lOtimea
the height, the multiplier is 1, and is very approx>
imate; or in other words, the parabola will be
of almost exactly the same length as the eiroular
are.

To find the area of a |»arabola ta a n l^.

Mult iU base m n, Fig 5, by its height a h ; and Uke %^^M of the prod.
The area of any segment, as « b v, whose base tt v is parallel to as n, is
found in the same way, using u « and s b, instead of iw i» and a b.

To find the area of a parabolic aone, or fl^as-

tam, as t>» n t« V.

RuLx 1. First find by the preceding rule the area of the whole pambola
m b n ; then that of the segment « b « ; and subtract the last mm the
flmt.

RuLK 1. From the cube of m n, take the eubo of « v; eall the difP %,
From the square of m n, take the square of m « ; eall the dlff «. Div e bf
«. Mult the quot by ^ds of the height • s.

MENSURATION,

193

1

Table lor I^enytlis off Parabolic Curves. See opp page. (Original.)

Baa«.

Mole

BM6.

Molt.

Bue.

Molt.

, Base.

Molt.

.05

1.000

1.10

.999

2.15

.949

8.20

.983

.10

1.001

1.16

.997

2.20

.951

3.30

.984

J6

1.002

1.20

.995

2.25

.954

3.40

.986

.20

1.004

1.25

.993

2.30

.956

3.50

.986

.25

1.006

1.80

.990

2.S5

.958

3.60

.987

JSO

1.007

1.35

.987

2.40

JMM)

8.70

.988

JB6

1.007

1.40

.984

2.45

.002

3.80

.989

AO

1.008

1.45

.980

2.50

.963

3.90

.990

.45

1.009

1.50

.977

2.55

.965

4.00

.991

.60

1.010

1.55

.974

2.60

.967

4.25

.992

.65

IMO

1.60

J>70

2.65

.969

4.50

.993

jOO

1.010

1.65

.966

2.70

.970

4.75

.994

.66

1.011

1.70

.963

2.75

.972

6.00

.996

.70

1.011

1.75

.960

2.80

.973

5.25

.996

.76

1.010

1.80

.957

2.85

.975

6.50

.997

.80

1.009

1.85

.953

2.90

.976

6.76

.908

.85

1.008

1.90

.950

2.95

.978

6.00

.998

.00

1.006

1.95

.946

3.00

.979

7.00

.999

.96

1.004

2.00

.942

306

.980

8.00

1.000

1.00

1.002

2.05

.944

3.10

.981

10.00

1.000

1.05

1.001

2.10

.946

3.15

.982

To draw a parabola) having base o t and height « o.

••«, Flc6. Make e I eqoal to the height «e. DraweCand
• I; and dlride each ofthem into aoT number of equal parte;
BmnberlDg them as in the Fig. Join 1,1; 2, 2 ; 3, 3, Ao ;
then draw the oorve by hand. It will be obeenred that Um
itttereeetions of the lines 1,1; 1, 3, &o, do not give pointi in
the eurre ; but a portion of each of those lines forms a tan.
gent to the eurre. By increasing the number of diri^iona
on e < and « t, an almost perfect oorre is formed, scaroelj
teqnlring to be tooohed up by hand. In practice it is best
first to draw onlr the center portions of the two lines whioh
•rasa eaeh other Just aboTO o ; and trom them to work down*
ward; aetnally drawing oalj that small portion of eaeh
low« Une, whioh is neoessary to indioate th«

bo drawn

Fifir.tt.

Or the i»araboIa ma

tbasx

Let ft «, Fig T, be the base ; and a d the height. Draw th»
leetangie hnine; dir each half of the base into an j nom.
ber of equal parts, and number them ftom the center each
vmT. DIt n h, and m e into the same number of equal parts ;
■ad number them from the top, downward. From the points
on b e draw rert lines ; and trom those at the sides draw lines
to d. Then the interseetions of lines 1,1; 2. 3, ke,
will form points in the parabola. As in the pre-
esding ease. It is not necessary to draw the entire
lines ; but merely portions of them, as shown be.
teeeu d and c.

Or a parabola may be drawn by first dlT the
height a h. Fig 5, into any number of parts, either
equal or unequal; and then ealoulating the ordi.
aatea u»,Ao; thus, as the height a h : square of
half base am : : any absciss b s : square of iu
erd « «. Take the sq rt for ««.

I. —When the height of a parabola is not
ir than 1.10th part iu base, the eurre eoin-
■o very eloeely with jlhat of a drcntar are,
that in the preparation of drawings for suspen>
rieo bridges. Ac., the eironlar are may be em.
ployed ; or if no groat aoenraoy is veqd, the olrole
■ay be need eren when the hMghfe la aa great ••
«e^«igfath of the base.

To dra^w a tangr^nt w v, TIk- 5, to a parabola, from any point v.

Draw V » perp to axis a h ; prolong a h until b w equals s b. Join v> v.

13

194

MENSURATION.

a

Tlie Cycloid,

^^h i-the curve deacribed by a point a in the circumference of a circle,
.^'d'ix^fonr^^^S^^o.uLn'S the clro.e.^roU^^^

d h cycloid.

Tlie vertex of the cycloid is at e.
Base, a 6, =s circumference of generat-
ing circle a u
=s diameter, cd, of generat-
ing circleXir = 3.1416«i.

Axis, or taeli^lit, cd=^an.
lieuiTtli, oc6, = 4cd.

I, a c 6 d = 3 X area of generating circle, o n
= 3?^ = ca8 X 3ir = cci« X 2.3562.
Center of sravity of surface at g. cg = t\ c d. Center of gravity oi
cydoid (curved line a c 6) in axis c d at a point (as ») distant J c d ttom c.

To draw a tangent, «o, from any point e in a cjrcloid; draw « » at right
anTlM to the axScd; one d describe the generatingcircle dc<; join /c; from
J draw CO parallS to / c. The cycloid is the curve of a uickest descent ;

So thit a ESdy would fall from"^ h to c along the curvelm c, in less time than
along the inclined plane 6 ic, or any other line.

TKE REGVIiAB BOBIES.

A revnlar body, or reffular polyhedron, is one which has all its
dies, and its solid angles, resnectively similar and equal to each other. There
'e but five such bodies, as follows :

■ides
are

Name.

Tetrahedron .........

Hexahedron or cube

Octahedron

Dodecahedron

Icosahedron •

Bounded by

4 equilateral triangles.
6 squares.

8 equilateral triangles,
12 " pentagons,
|20 " triangles.

Surface

(—sum of surfaces
of all the faces).

Multiply the square
of the length of
one edge by

1.7320
6.

3.4641

20.6458

8.6602

Tolnme.

Multiply the

cube of the

length of one

edge by

.1178
1.

.4714
7.6681
2.1817

Ouldinus' Tbeorem.

Fig. A. Fig. B.

I

To find the volume of any body <as the

irregular mass a 6 c w. Fig A, or the rinft
abom^ Fig B), generated by a complete
or partial reyofution of any figure (as
_ ahca) around one of its sides (as/ie,

Fig A), or around any other axis (as
a;v,FigB).

volume =3 surface ahcaY. length
of arc described by its center of grar^
ity G.

If the revolution is complete, the arc
described is = circumference = radius
0 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.

0 fl

0 ■

# I

dL g

Fig. 10J4

This rule may be used for aacertainins beforehand, the Quantity of earth to
be removed from a "borrow pit." The irregular surface of the ground is first
staked out in squares; (the tape-line being stretched horizontally ^ when meas*

uring o£f their sides). These squares should be of such
a suse that without material error each of them may be
considered to be a plane surface, either horizontal or in-
clined. The depth of the horizontal bottom of the pit
being determined on, and the levels being taken at every
^b corner of the squares, we Hre thereby furnished with the
lengths of the four parallel vertical edges of each of the
resulting Arnstums of earth. In Figs 10^ y may be sup-
posed to represent one of these Arustums.
If the frustdm is that of an irregular 4-sided, or polyg-
onal prism, first divide its cross section perpendicular to \ts sides, into tri-
angles, by lines drawn frpm any one of its angles, as a, Fisr 10^. Calculate the
area of each of these triangles separately ; then consider the entire frustum to
be made up of so many triangular ones; calculate the volume
(•;\ of each of these by the preceding rule for triangular frustnms;

and add them together, for the volume of the entire frustum.

Tolnme of any frnstam of any prism.

Or of a cylinder. Consider either end to be the base ; and find its
area. Also fipd the center of gravity c of the other end, and the
perpendictUar distance n c, from the base to said center of g^ravity.

Then Volame of frnstam = area of base X»«, Fig 10^.

The slant end, c, is an ellipse. Its area is greater than that of the circular end.
Snrfaee of any prism. Figs 5 to 10, whether right or oblique, regular
or irregular

/ circumference measured s^ «-*„-i iA«»ti, >. A i »tt™ of the areas
" Vperpendicular to the sides ^ *®^"" lengin, a <> j + of the two enda.

CTIilHTBERS.

. If A cylinder is any solid whose ends are

^h^-^_^ jC ^ parallel, similar, and equal curved fignires ;

and whose sections parallel to the ends
are everywhere the same as the ends.
Hence there are circular cylinders, ellip-
P tic cylinders (or cylindroids) and many
others ; but when not otherwise expressea,
the circular one is understood. A right
cylinder is one whose ends are perpen-
dicular to its sides, as Fig. 11 ; when otner-
Fig. 11. Fig. 12. wise, it is oblique, as Fig 12. If the ends

of a right circular cylinder be cut so as to
make it oblique, it becomes an elliptic one ; oecause then both its ends, and aJl
sections parallel to them, are ellipses. An oblique circular cylinder seldom
occurs ; it may be conceived of by imagining the two ends of Fig 12 to be circlet^
united by straight lines forming its curved sides.
A cylinder is a prism having an infinite number of sides.

Volume of any cylinder (whether circular or elliptic, Ac, right or obliqa^
= area of one end X perpendicular distance, j9, to the other end,

-{rJZ^^^Zi^ X actual length, « 6. Figs U and 12.

^ 3 X volume of a cone whose base and height are » those of the cylinder.
Snrface of any cylinder (whether circular or elliptic, &c, right or oblioue)

(circumference ^ g^m ^f ^^jje areas

measured perpendicularly X actual length, o 6 1 + ^f the two ends
to the sides, as at c o. Fig 12, f

RIfirlit circular cylinder whose lieiirb^ " diameter.

Volume = H X volume of inscribed sphere.

Curved surface = surface of inscribed spltere.

Area of one end == \ surface of inscribed sphere =«= \ curved surface.

Entire surface = U X surface of inscribed sphere =« IJ X curved surfkee.

CJONTENTB OP CTUNDBRS, OB PIPEa.

197

ContentB for one fi»ot tn lenstti, in Cub Ft, and in U. 8. Gallons of

Ml oab ins, or 7.4806 Galls to a Cub Ft. A e«1» Rof water wei«lu aboat 62M lbs ; and a gallon
altoat 6H IlM. IHaaw »• 8» or 10 Hmm m svMt* «iTe i, 9. or 100 times tbe (Mutant.

For in. in

For I ft in

For 1 ft. im

length.

lengtH.

length.

Dlam.

Dlam.
in deoi-

Diam.
in

Dlam.
in deci-

Diam.

Dlam.
in deci-

in

-•3

•

^ ■

-5

*s 2

^ v^^

^ 9

Ins.

malsof

• H *

o a

Ins.

mals of

^a^

o a

in

mal* of

8?*i

0 a

afoot.

h

afoot.

ii

Ins.

afoot.

^s^

5 .

•§sS'

5"

■pg-

=50

•Ss^

^0

t

"3

^n

i

«3

5s

«3

*a

^Yt

.0206

.0003*

.0025

%

.5625

.2485

1.869

19.

1.683

1.969

14.73

.0260

.0005

.0040

7.

.6833

.2673

1.999

H

1.626

2.074

16.61

,-!i

.0313

.0008

.0057

' ^

.6042

.2867

2.146

20.

1.667

2.182

16.32

.0366

.0010

.0078

.6260

.3068

2.296

34

1.708

2.292

17.15

».^

0417

.0014

.0102

yi

.6466

.3276

2.460

21.

1.750

2.406

17.99

.0409

.0017

.0129

8.

.6667

.3491

2.611

H

1.792

2.621

16.86

nM

.0521

.0021

.0180

§

.6876

.3712

2.777

22.

1.833

2.640

19.76

.0673

.0026

.0193

.7083

.3941

2.948

H

1.875

2.761

20.66

4

.0625

.0031

.0230

%

.7292

.4176

3.125

23.

1.917

2.885

21.68

.0677

.0036

.0209

9.

.7500

.4418

3.306

}4

1.968

3.012

22.63

is-fi

.0729

.0042

.0312

H

.7708

.4667

8.491

24.

2.000

3.142

23.60

.0781

.0048

.0359

.7917

.4922

3.682

25.

2.083

3.400

25.60

1.

.0633

.0065

.0408

74

.8126

.5185

3.879

26.

2.167

3.687

27.66

8

.1042

.0085

.0638

10.

.8333

.5464

4.060

27.

2.260

3.976

29.74

.1260

.0123

.0918

i

.8542

.5730

4.266

26.

2.333

4.276

31.90

H

.1458

.0167

.1240

.8760

.6018

4.498

29.

2.417

4.687

34.31

2. ^*

.1667

.0218

.1632

Z4

.8968

.6303

4.716

30.

2.600

4.009

36.72

/4

.1876

.0276

.2066

11.

.9167

.6600

4.937

31.

2.683

6.241

39.21

.2063

.0841

.2650

H

.9375

.0903

5.164

32.

2.667

6.585

41.78

5i

.2-292

.0412

.3085

.9683

.7213

6.S96

33.

2.760

6.940

44.43

a. ^*

.2500

.0491

.3612

7*

.9792

.7680

5.638

34.

2.833

6.306

47.15

.2708

.0670

.4300

12.

1 Foot.

.7854

5.876

36.

2.917

6.681

49.98

.2917

.0668

.4906

H

1.042

.6522

6.376

36.

3.000

7.060

62.68

5k

.3125

.0767

.5738

18.^

1.083

.9216

6.896

37.

3.068

7.46T

66.86

i. *

.3333

.0873

.6628

u^

1.126

.9940

7.436

36.

3.167

7.876

68.92

.8542

.0986

.7360

1.167

1.069

7.997

39.

3.260

8.206

62.06

.3750

.1104

.8263

H

1.208

1.147

8.678

40.

3AS3

8.727

65.28

5i

.9958

.1231

.9206

15.

1.250

1.227

9.180

41.

3.417

9.168

68.68

5. ^*

^167

.1864

1.020

H

1.292

1.310

9.801

42.

3.600

9.621

71.97

.4375

.1508

U26

16.^

1.383

1.396

10.44

43.

3.683

10.085

76.44

.4583

.1650

1.234

H

1.375

1.485

11.11

44.

3.667

10.659

76.99

X*

.4792

.1808

1.340

17.

1.417

1.576

11.79

46.

8.760

11.046

82.62

«.

.5000

.1903

1.469

u

t.458

1.670

12.49

46.

3.833

11.641

86.33

.5208

.2131

1.594 18.' "

1.600

1.767

13.22

47.

3.917

12.046

90.13

.5417

.2804

1.724 }4

1.642

1.867

13.96

48.

4.000

12.666

94.00

TaMo oontlniied, bat wtth tbe dlanui In feet.

Gab.

U.S.

Dlam.

Onb.

U.S.

DU.

Gab.

U.S.

Dia.

«ab.

U.S.

Feet.

Feet.

Gallfl.

Feet.

Feet.

Oallfc

Feet.

Feet.

Galls.

Feet.

Feet.

Galla.

4

12^

04.0

7

S8.48

287.9

12

113.1

846.0

24

452.4

8884

1^

14.19

106.1

41.28

808.8

18

132.7

992.9

25

490.9

8672

xt

10.90

119U)

23

44.18

330.5

14

153.9

1152.

26

530.9

3972

/i

17.72

182.6

&

47.17

852.9

16

176.7

1822.

27

672.6

4288

%

19.0S

146.0

8

60.27

876.0

16

201.1

1604.

28

616.8

4606

W

21.66

161.9

M

66.75

424.5

17

227.0

1698.

29

660i»

4941

I4

28.76

177.7

0^

63.62

475.9

18

254.6

1904.

30

706.9

5288

/•

25.97

1912

K

70.88

580.2

19

283U$

2121.

31

764.8

6646

f

2&27

211A.

10

78.54

687.5

20

814.2

2850.

32

8012

6018

l^

80.68

22BA

K

86.59

647.7

21

346.4

2591.

33

865.3

6398

xc

88.18

248^

11

06.08

710.9

22

380.1

2844.

84

907.9

6792

%

8&78

287.7

%

108.87

777.0

28

415.5

8108.

85

962.1

7197

198 CONTENTS AND LININ08 OF WELI*.

COSTENT8 AKD LIJriHeB OF VELIA.

For lIuH WlBe u irul u IkaH In Ih. U-Ut. Ibr »>• n» JiU iC Unliil. Uli ml tbm onuM
OM ka|f dT lh> inuH dim ; u< khU IMM In 4, Tkm, iH- Un gDl ill [u (loli tvA of d>pi£ id •
vallfll r««tlB«w, llmUbautrniiaillauW*lbo«a«»a<U«lli*dtioar]A4fcfaK; nu»J/, A.Hi.
Tbn t.Mi X t ~ n.aU anb Jill ngd tor UnHlft^Um. BHItattlK uoni llnlDf •rnUlu HliU
ar pUiHiiH, BiU tbg laJmUr gguilu •hkhIH half U» ininr OiH. br 1. TllL thi HnMi tf
IKH nlUMf Ht aKik HM gf «iplk ors will of II 6 diuD. wlU la LOn X I = I.IM. Ir tht nil li

r™

eMh"(D«

pm-

»lu»«.

"inf

Diull.

Hllll(.

s;

#i

-..

f

l.Wl

s

'.(Mi

Inn

K

i.Ba

JIUl

W

Pt

I.BU

s

iuM

3IU

«

:<!

"■b

;lm

In

M

K

■^!

J

ii:

■i

JM

:!>9H

,*W

:»!»

5

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iT

e-ioi

K

.«B

1

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i

i

1

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ioK

& ™

.MT

')

a

■»

"lis

[JSJ

s

"'a

'1

410

tu

H

'a

jg

m

J»

^

'1

IS

rm

M6

!S

!:3S

K

.._.

S

M

!«

nS

31

N

.VH

«J

M

^1

■»

^ i

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■^

Si

«^

|^«

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icSi

K

J

i

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i

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ew

A

at>7d

^«oai

.b"U

!«■

ItV«r«bea u« named In ■ '

CYLINDRIC CNOULAB, ETC. 199

CIBCUI^B CVLIITDBIC UNQITI^B.

■ the enttlng plaoe dvea mot eat tbe baae. Flp l^ 14

1 -m

]ft J I perp u ildw, u z, '^ jr'n,al,nieai<Jili>iieChB>id«.

Add arena *t ends If required.

r«r area* of SAetlaiis perpendioulir to tbe ildM, see GIrelH.

r«r areaa of aecUoaa oblfqu* to tbs ildea, •»• Tbu ElltpN.

II. Wb«n the enttlns plane to>«taea tbe baae. Flgi A l« IX

.^--*

Talome FlgA-(^at

X*ra*a<l«t otbau)-

(«ta.ih«

-«»WX«ii.

FigD-H>">(>f (sIrcIayM X""
~ ^ TolDins of cfUiider c y m n.

Kg A - foi X "in - o< X length of are imh ) |^.
(^,, FisB-»,X-.

nBgoU FlgC — (lift Xn>n + 00 X length of «jc dm* )— —.

oflj) "-

200

PYRAMIDS AND COVES,

PTH^ttlDft AND COITEB.

4 5

A pjrainid, Fin. 1, 2, 8. Is any solid which has, for Its base, a plane figure
of any number of sides, ana, for its sides, plane triangles all terminating at one
point d, called its apex, or top. When the base is a regular figure, the pyramid
Is regular ; otherwise irr^uCar.

A cone, Figs. 4 and 5, is a solid, of which th6 base is a curved figure; and
which may be considered as made or generated by a line, of which one end is
stationary at a certain point d, called the apex or top, while the line is being
carried around the circumference of the base, which may be a circle, ellipse,
or other curve. A cone may also be regarded as a pyramid with an infinite
numoer of sides.

The axis of a pyramid or cone. Is a straight line eZ o In Figs. 1, 2, 4 ; and diiA
Fi^s. 8 and 5, from the apex e2, to the center of gravity of the base. When the
aj^s is perpendicular to the base, as In Figs. 1, 2, 4, the solid is said to be a right
one ; when otherwise, as Figs. 3. 5, an oblique one. When the word cone is used
alone, the right circular cone. Fig. 4, is understood. If such a cone be cut, as at
1 1, obliquely to its base, the new base 1 1 will be an ellipse; and the cone dtt
becomes an oblique elliptic one. Fig. 6 will represent either an obUtiue <^ular
eoiie, or an oblique elliptic one, according as its base Is a circle or an ellipse.

V oliune or pyramtd or co£e, regular or Irregnlai^ right or obliqu«.

Volume mm ^ «rea of base X perpendicular height d o. Figs. 1 to i.

-» ^ volume of prism or cylinder having same area of base and
same perpendicular height.

— K volume of hemlsphJBre of same base and same height

Or, a oone. hemisphere and cylinder, of the same base and same height, havt
volumes as 1, 2 and 3.

Area of anrlkec of sides of right regular pyramid or right dicular ooiM.

Area — J^ circumference of base X slant height.*^

In the cone, this becomes I Add area of bass

Area of sarfoce of oblique elliptic eone, dtt,

Fig. 6i, cut from a rieht circular cone, dss. From the point
c where the axis d o of the right circular cone cuts the elliptio
base t L measure a perpendicular, r, in any direction, to the
curved surface of the cone. Let v = the volume of oblique
elliptio cone, dti; let a — the area of its elliptic base t (.and
let A = the height d u measured perpendicularly to said nase.
Then

Carved snrlkiee = = .

r r

Add area of base if required

No measurement, has been devised for the surface of an
oblique circular cone.

*In the pyramid, this slant height must be measured along the middle of one
of the sides, and not along one of the edges.

PYRAMIDS AND CONES. 201

To And thm surfiwe of mat IrvcffiKlar p jramld.

Whether right or oblique, each side must be calculated as a separate triangle (i
p. 148); and we several areas added together. Add the area of base if required.

FRUSTUMS OF PYRAMIDS AND CONES.

Flff.0. Fig. 7.

Frastam at pjnunld (Fig. 6) or of oono (Fig. 7) with haw aad Uff
pnaUeL

Tolmne (regular or irregular, right or oblique)

my ^, perpendicular v- / area i area i / area v/ area \

— >* P^ height oo ^ ^of top •" ©f base t" V of top -^ of base/

^ w vr perpendicular w / area' i "«» i * ^ areaof aBection \

— X X *helght oo X V of top + of base + l^^^ to, and midway I

>» ' between, base and top /

»^ (for ffmstam of right or oblique circular cone only; Hee Fig. 7)
« X "^SSS^ X M4M X (•<• 4 •»* + •« . o.)

of frustum of righi fgiAjur pjmunid or ooue, with top And base paiallelt
9|0k 6 and 7.

J. /diemnferenoe _i oirouinlbrenoeX v^ dant •
>^\ oftop T ofbaM y X iMigiitfC

Aid MiM «f top and Inuo If nq«li«4.

Im tlM finuitoaA of a vl|^t etreolar oono^ tUibMoaat

"^ Vof top T^ of basej X hdght f f
(ir * 8.1416) . Add areas of top and base !f reqafawd.

of IwegiUsur or o1»liq«« pjnroiBld or ooim. Sorlhee ••
■an of smrfiwes of sldsi, each of which must be treated as a trapeasoid.

•In the frustum of the jpframld (fig 8), this slant height must be measured along
of Ite MM (M at <s), Mid net along one of tha edgsib

202

PBI6HOID&

PBIBHOIDB.

Flff.L

VtK.2.

A prUnnoUl is sometimM d<iHwtl M AfBlid bttdng Ibr Hi ends two paralWI
plane figures, connected by other plane flfiuns on which* and through every point
of which, a straight line may be drawn nom one of tho two parallel ends to ^s
other. These connecting planes msj bo parallelograms or not. and parallel to each
other or not.

Tbla doflnltlon iroiild Imolndo the cube and all other parallelopipeds;
the prism : the cylinder (considered as a prism baring an infinite namber of sides);
the pyramid and cone (in whieb one of the two parallel endl^ i« theonelbiminiftiio
apex, is considered to be infinitely small), and their frnstams with top and boso
parallel ; and the wedge.

But the use of the term prlanaold is frequently restxietod to siz-eided aolidd,
in which the two parallel ends are unequal quadrangles; and the connecting plane^
trapezoids; as in Figs. 1 and 2; and, by soma writers, to cases where the patalkl
quadrangular ends are rtetatiffies.

The following •'prlsmoldal fbrmnla** i^Uas to all tbo ftregolng •olidi^
and to others, as noted below.

Let A — the area of one of the two parallal ends.

a — <* ** the other of the two panUlel ends.
M — « *< a cross section midway between, and panllil to^ Hm tm

parallel ends.
L — the peipendicnlar distance between tfao two psnlU <

Then

Tolmiae — L X

0

^ L X mean area of enm section.

The following six flgnrss repvstent a few of the irregular solids which ftlltBderlht
aboye broad definition of '< prismoid,*' and to which the prismoidal formnla appUiC
They may be regarded as one-chain lengths of raihroad cutttnga; a o being^the loogUv
sr perpendicular (horiaontal) distance between the two parallel (Tertloal) '

WEDGES.

203

The prismoldal ft»rmii]» applies also to the qihere) hemiiphere, and
ether qpE«rlcel segmeiite; also to any aeotlf joe each aeafroi^aiid onidbct ai the

In which the ddee ad^ ae, or od, <<i^ are itraiffhii tM ttuj are onty when the
•atttng plane ade paaaes Umugh ike apes or top a. Also to ih» cylliiidev

when a plane paraUd to the tides passes through both ends; but not if the plane
«s is obHquet as in the fig., though never erring more than 1 in 142. In tl&la last
case we must imagine the plane to be extended until it cuts the side of the cylinder
likewise extended ; and then by page 199 find the solidity of the uegnlathus formed.
Then find the solidity of the snuUl nngnla above to, also thus formed, and subtract
it fh>m the large one.

This very extended applicability of the prismoidal formula was first discorered,
and made known* hy KUwood Morris, a B., of Philadelphia, in 1840.

WEDGES*

m n m

SI m m

Fiff.]a

m

Fiff.n.

b neaally defined to be a solid. Figs. 8 and OjjKenerated by a plane triangle, anei,
moving; parallel to itself; In a straight line. This definition requires that the twe
triamgnlar ends of the wedge should be parallel; but a wedge may be shaped as in
ng. 10 or 11. We wouid therefore propose the following definition, which embraces
sll the figs.; besides vuious modifications of them. A solid of five plane faces ; one
sf which is a parallelogram abed, two opposite sides of which, as a e and h d, are
onlted by means of two triangular foces aen, and frdm, to an edge or line « m,
parallel to the other opposite sides ab and ed. The parallelogram abed maj be
eitlier rectangular, or not ; the two triangular Ikces may be similar, or npt ; and the
with r^ard to the other two fhces. The following rale appUss equally to all :

SunoTleDgths
— K X oftheSedges

peiphtj^from
edgetobaok

width of

back {abed^

massed neip to « it

204

lOENBURATIOV.

SPHERES OR GLOBES.

A Sphere

Is a solid generated by the revolation of a semicircle around its diameter. E^ery
point in the surface of a sphere is equidistant (h)m a certain point called the center.
Any line passing entirely throns;h a sphere, and through its center, is called its axis,
or diameter. Any circle described on tlie surface of a sphere, fh>m the center ol
the sphere as the center of the circle, is called a great eirde of that sphere i in other
words any entire circumference of a sphere is a great circl«f. A «phere has a greatei
content or solidity 'than any other solid with the same amount of surface ^so that i|
the riiape of a sphere be any way changed, its content will be reduced. The inter-
section of a sphere with any plane is a circle.

Tohune of sphere

— J TT radios*

— )^ TT diameter*

, ^ circumference *
■" •« zr5

— 4.1888

— 0.5236

radius'
diameter*

-» 0.01689 circomferenoe*

— 3^ diameter X area of surface
"" ^ diameter X area of great circle
«- % Tdlume of circumscribing cylinder
^ 0.6236 Tolnme of circumscribing cube.

ot avtrfiace of sphere

— 4 TT radius*

— w diameter*
circumference*

— 12.6664 radius*

— 8.1416 diameter*

•^ 0.8183 circumference*

— diameter X circumference
■- 4 X area of great circle

^ area of circle whose diameter is equal to twloe diameter of

— curved surfkce of circumscribing cylinder
6 X volume

diameter.

Badlw of sphere
s t

= * f

volume

= O.e2036 'v^volna*

= /

Area of surface

47r

= ^.07968 X anaof Boxflwe

Gireiinalbrenee of sphere
=s \/6 TT* volume

a« ^TT Area of surfisoe

_^ area of snrikoe
~^ diameter.

=r '^/59.2176 VolWM

=s ^8.1416 are* of ioifiMe

MEKBUBATION.

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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 s|riiere. Bin^ S. Molt the oiroumf of the splierB by the height e •

of the sag.

To find tbe solidity of * spberieal cone.

Add together the square of the rad • d, the square of rad o &,
and H<1 of ^^^ square of the perp height «o; mult the earn by
1.&706; and mult this prod by the height «•. •

To find the carved snrflsee off a spiier-

ical sone.

BvLB 1. Mult together the diam m n of the sphere ; the height
e 0 of tbe sone, and the number S.U16. Or nee preoeding Rule t
tor surf of segmenta. Bale S. Mult the etroamf of the sphere, by
the bf^lghtof the zone. •

To find the solidity off a hollow spher-
ical shell.

Take f^m the fbregolng table the loltditlee of two aphorae haTlBf
the diams a &, and e <L Snbtraot the least fhmi the grMtMi. B«i«
a c or » 4 U the Ihiokneie «r tha ahMtt.

THE ElililPSOID, OR SPHEROID,

Is a solid generated by the rerolution of an ellipse around either Ita long or ita short dlam. When
around the long (or transverse) diam, as at a. Fig 1, it is an oblons" or pr<
late spheroid; when around the short (or co^Jaffate) one, as at m, in Fig %
it is oblate.

Fiir.i.

Flg.2.

For the solidity in either case, mult the fixed diam or ^tU br the •quare

of the revolving one ; and mult the prod by .5336. ^

—— ■ — — . . » ___^_^

*This rule applies, whether the zone includes the equator (as in our figure) or
not, as in the earth's temperate zoues.

PAAABOliOWa,

THE PAHABOLOID, OH PARABOI.IC COKOID,

r lU ■olldltj' mult the ires of Ita bue, bj batr lu belaht, re. Oi

Pop tbe •olldlt^ af a (TiutaB,

I. (bl mil ot wlloll ««r.r]J Hi IMUjiril IiMHIIMIwlM

To and the anraMm ofK pBraboloM,

To And IM sarlkee.

1b« dirt a« rroBi tbe«at4r ortha oIhU w tb« cvrtUr ar Eb« iplndle. CkQ

To Nnd tbe ■oIMIly of » mldfUe cone ofn elrnilsp Bplndlo,

((—'?)«.•)-(••«—'■■■))"-

Tvlnme- of ,),iei, rfng ia mads '< dlamBlflm, ooand 6t XI-MIWB.
-__-.__ _ drcumfflivrMfl of bar ^ 1 Bum of Inner HOd out^r w a tjifau

210 SPECIFIC GRATITT.

SPEOmO GEAVm.

1. The specific gravity, or relative density, D*. of a sabstancei

is the ratio between the weight, W, of any given volume of that substance and
the weight, A, of an equal volume of some substance adopted as a standard of

w

comparison. Or: D = -^.

2» For ffaseous substances, the standard substance is air, at a temper-
ature of 0° Cent. =s ^29 Fahr., with barometer at 760 millimeters = 29.922 incnes.

3. For solids and liquids, the standard substance is distilled water, at its
temperature (4^ Cent = 39.2° Fahr.) of maximum density.

4. For all ordinary purposes of civil engineering, any clear fresh
water, at any ordinary temperature, may be used. Even with water at
SOP Cent., = 86^ Fahr., the result is only 4 parts in 1000 too great.

5. When a body is immersed in water, the upward forc& or '* buoyancy***
exerted upon it by the water, or the **loss of weight " of the body, due to its
immersion, is equal to the weight of the water displaced by the immersion of
the body f ; or, if

W = the weiffht of the body in air,

u; = its weight in water,

D ■= its relative density or specific gravity,

A = the weight of water displaced ;

then A =- W — ic ; and D = -r- — tT? •

' A W — w

6. Since the volume, V, of a body, of given weight, W, is inverselv aa iti
density, or specific gravity, D ; the specific gravity is equal also to the ratio
between the volume V, of an equal weight of the standard substance, to the

volume, V, of the body in question ; or D = ^'.

7. The specific gravities of substances heavier than water are ordi-
narily determined by weighing a mass of the substance, first in air (obtain-
ing its weight, W), and then when the mass is completely submerged in water

W

(obtaining its diminished weight, w). Then D = -^ , as in If 5.

8. If the body Is lighter than water » it must be entirely immersed,
and held down against its tendency to rise. Its weight, «>, in water, or ita
upward tendency, is then a negative quantity, and means must beprovided for
measuring it, as by making it act upward against the scale pan. We then have^
A = W — (— w) = W + M? ; or

Loss due to immersion = weight of body in air, pltis its buoyancy.

9. Or, first allow the body to float upon the water, and note the resulting di»-
placeraent, t>, of water, as by the rise of its surface level in a prismatic vessel.
Then immerse the body completely, and again note the displacement, V. Now
V, the volume displaced by the body when floating, and V, tne volume displaced
by the body when completely immersed, are proportional respectively to the
weight, W, of the body, and to the weight, W — tr, of a mass of water of equal

volume with the body. Hence D == = ^^.

W — w V

10. Or, attach to the light body, b, a heavier body, or sinker, S, of such den*
sity and mass that both bodies together will sink in water. Let W be the
weight of the light body, 6, in air ; Q the weight of both bodies in air, and q
their combined weight in water. Then Q — ^ = the weight of a mass of water
of equal volume with the two bodies, and Q — W =». the weight, S, of the sinker
in air. By immersing the sinker alone, find the weight, fc, of water equal in
volume to the sinker alone, — loss of weight in sinker, due to immersion.
Then, for the weight, A, of water of equal volume with the light body, fr, or tor

* Strictly speaking, " specific gravity " refers to weight, and " relative density »»
to mcus (see Mechanics, Art. 14 a); but, as specific gravity and density
numerically equal, they are often treated as identicaL

t See Hydrostatics, Art. 18.

1

SFifiOIPIC GRAVITY. 211

the low of weight of b, due to immersion, we have A = Q --0 — k ; and, for
the specific gravity, D, of the light body, 6, we have D =- a_ ^. =" ^_^
where to — the (unknown) buoyancy of b.

11. A granular body, as a mass of saw-dust, gravel, sand, cement, etc.,
or a porous body, as a maas of wood, cinder, concrete, sandstone, etc., is a com-
posite body, consisting partly of solid matter and partly of air. Thus, a cubic
foot of quartz sand weighs about 100 fi>s.; while a cubic foot of quartz weighs
about 165 lbs.

12. The specific sraTltj of porous substances is usually taken
as that of the composite mass of solid and air. Thus, a wood, weighing (with
its contained air) 62.5 ttw. per cubic foot, or the same as water, is said to have a
specific gravity of 1. The absorption or water, when such bodies are immersed
forthe purpose of determining their specific gravities, may be prevented by a
thin coat of varnish.

13. The specific grravity of granular substances is sometimes taken
as that of the solid part alone. Thus, Portland cements ordinarily weigh (in
air) from 75 to 90 fts. per cubic foot, which would correspond to specific gravities
of from 1.20 to 1.44 ; out the specific gravity of the solid portion ranges from
8.00 to 3.25 ; and the latter figures are usually taken as representing the speciflo
gravities.

14. In determining the specific gravities of substances (such as cement)
which are soluble in water or otherwise affected by it, the substances are
weighed in some liquid (such as benzine, turpentine or alcohol) which will not
affect them, instead of in water. The result, so obtained, must then be multi'
plied by the ratio between the density of the liquid and that of water.

15. The specific ifpavity of a liquid is most directly determined by
weighing equal volumes of the liquid and of water.

16. Or weigh, in the liquid, some body, whose weight, W, in air, and whose
specific gravity, d, are known. Let u/ = its weight m the liquid. Then, for
the specific gravity, D, of the liquid, we have

d(W— «/)
W:W — «/ = d:D; or D = -^— ^^^ — •'.

17. Or, let the body, in f 16 (weighing W in air), weigh %o in water, and (as
before) to' in the liquid in question. Then, since specific gravity of water =s 1,
we have

W — m;:W — u/^lrD; orD = 3~"^»

w — to

18. The specific gravities of liquids are commonly obtained by observing the
depth to which some standard instrument (called a hydrometer) sinks when
allowed to float upon the surface of the liquid. The greater the depth, the less
the specific gravity of the liquid. In Beaum4('s hydrometer tne depth
of immersion is shown by a scale upon the instrument. The graduations of the
scale are arbitrary. For liquids heavier than water, 0^ corresponds to a specific
gravity of 1, and 76^ to a specific gravity of 2. For liquids lighter than water,
10° correspond to a specific gravity of 1, and 60° to a specific gravity of 0.745.

19. In Twaddell's hydrometer, used for liquids heavier than water,

.- ,^ 6 X No. of degrees + 1,000
specific gravity = -^ J —

Thus, if the reading be 90°,

,- 4* 5 X 90 -f 1,000 1,450 , ^„
specific gravity j^^^^-i- ^ ^^ 1.46.

20. In Nicholson's hydrometer, largely used also for solids, the specific
gravity is deduced from the weights required to produce a standard depth of
immersion. It consists of a hollow metal float, trom which rises a thin but stiff
▼ire carrying a shallow dish, which always remains above water. From the
float is suspended a loaded dish, which, like the float, is always submerged. On
tile wire supporting the upper dish is a standard mark, which, in observations,
is alwavB brought to the surface of the water. The specific gravity is then deter-
miiied by means of the weights carried in the two dishes respectively.

21. The determination of the specific graYltles of ffaseous sub-
■tanees requires the skill of expert chemists.

212

8PE0IFIC GRAVITY.

Table of speelfle ^mvitiefl, and w«lirlita*

In this table, the sp gr of air, and gases also, are oompared with that of watec
instead of that of air ; which last is usual.

Th« specific gravity of any substance Is « its weicht
in fframs per enbie «eiitlmetre.

»•••••••

<«
«

Air, atmoapbario ; MfiO° Fkh, and ander tbe pnMve ef oat atmMph«r> or

14.7 Afl per aq iaoh, weigh* j\j part as mooh aa water at 00°

Aleobol, pure

" of oommeroe

" proof spirit ^ * ',

▲ab, perfcotly dry. V.V.V.V.V.V.'.'aTe'iife. *

1000 ft board meaaore weighs 1.748 tons.

Aab, American white, dry "

1000 ft board meaaare weigha 1.414 tona.
Alabaater, fklaely ao ealled; bat reaUy MarUea

" real; a eonpaot white plaster of Paria aTerage..

Alamlnlom

Antimony, caat,'6.86 to 6.74 averace ..

" natlTe •• ..

Anibraoite. See Coal, below.

Aaphaltom, 1 to 1.8

Baaalt. See Limeatonea, qnarrled

Bath Btone, Oolite. .....................................

lUamoth, oast. Alaonatlre

gltamen, aolid. See Aaphaltom.
rasa, (Copper and Zinc,) oast, 7.8 to 8.4 "

" rolled ««

Brooie. Copper 8 parte; Tin 1. (Gun metal.) 8.4 to 8.6 '•

Brick, beat pressed

" common hard

** ■eft, inferior

Brickwork. See Masonry.

Boxwood, dry

Oaloite, transparent.. ,

Carbonic Acid Oas. is IM times as beary aa air " ..

Cement. (See T IS.)

•• Portland, 8.00 to J.tft.-. »

•• Natural, 2.75 to 8.00

Chalk, S.'i te 3.8. Bee Limestones, quarried m

Charcoal, of pines andoak«.~

Cherty, perfectly dry

Chestnut, perfectly dry ......^

Goal. See also page S15.

Anthracite, 1.8 to 1.7

" piled loose

Biinmlnons, 1.8 to 1.4 m««*.««m.....

" piled loose ....M...M

€oke

** piled loose

In ooUag, coals swell from 86 to 60 per sent.

Copper, oast, 8.6 to 8.6 ,

" roUed, S.8to».0

Crystal, pure Qnarti. See Quartz.

.1

4»

.*••••••*...«•.•..•

Cork.

Diamond, 8.44 to 8.66 ; asaaUy8.61 to 3.66

■arth ; common loam, perfectly dry, loose

" •' " shaken

" ** " moderately rammed....

*' slightly moist, looae.

" more moiat, " •

" •« ahaken

" *' moderately packed • •

" aa a aofl flewinr. mad

" aaaaoftmud, well preaaed into a box........

il

4<

4<

M

«4

M

M

M

U

Ether

Blm, peribctly 4rr.

1000 ft board measnre weicbs 1.803 teas.

Bbeny, dry

Emerald, 3.68 to 2.76

Fat.

.average.

flint ••

Feldspar, i.5tot.8 •*

Qarnet,8.5to4.8; Preoions, 4.1 to 4.8 **

Qtaas, 8.6 to 8.46 «

" oommon window .' *'

" Mill viUe, Kew Jersey. Thiek flooring glass "

Oranite, 8.66 to 8.88. See Limestooe. 160 to 180 "

ATerage
BpOr.

.00188

.798

.884

.916
.768

.61

8.7

8.81

2.6

6.70

6.67

1.4
8.9

8.1

9.74

8.1
8.4

8.6

J6
8.788

.00187

8.19

2.87
840

0.67
0.66

1.60

1.80

1.00

S.T
&9

.96
8.68

•••• •»••!

.716
.66

1.28
8.7
.08
9.6

9.66

9.W
9.69
946
9.79

ATerage

Wtof a

Cab Ft.

Lbs.

.6766
4i>.48
63.1
67.8
4T.

88.

US.

14i.
163.
418.
41«.

87.8
181.
181.
607.

804.
694.
629.
160.
196.
100.

60
169.9

TSteSO
60 to 66

1B6»
16 to 90
48.
4L

nteVM
4TtoM

Tswaa

«4toU
CiJ6

79 to 80

89to 99

90 to 100

70 to 76

66 to 68

76 to 90

00 to 100

104 to 119

UOtoUO

44.6

86.

W.1
0B.

tot.

10k

106.

167.

160.

t ITOw

8PEOIFIO GBAVITY.

213

T»¥le of speelflc frnkvttlea, mnA welffbtfi— (Ooutiiiiifld.)

The specific gravity of any anbatance is » ttm welfllt
in grains per cubic eentiHietre.

(I
«<

ftneiMt oommoa* t.68 to 2.76

** In looM piles

" Hornbtondlo

'* " quarried, in loose piles.

Oyponia, Plaster of Paris, 2.24 to 2.80

** in irregular lamps "

'* gronnd, loose, per straok Iraahel, 70 "

M «• well shaken '* *' 80.... •• "

'• " Oaloined, loose, per stniokbaVhVtt to ral *.'.*. II " 1!

GtMnstone. trarr *>8 to 3.2 «' ,.

'* " fnarried, in loose piles **

Oravel, abont the same as sand, which see.

Gold, oast, ptixv, or 34 earau '*

" native, pure, 19.3 to 10.34 '* ..

" *' freqaentiy oontaining silrer, 15.6 to 19.8 *'

" pure, hammered, 19.4 to 19.6. > "

OnttaPeroha ** ..

HomUende, blaok, 8.1 to 8.4 '*

Hydrofm G«s, is 14)^ times lighter than air ; and 16 times lighter than

o^gea average..

Hendoek, perfeotljdrr. "

1000 reet board measure weighs .930 ton.

Hlekorj, perfeotly dry. "

1000 feet board measure weighs 1.971 tons.
Inn, and steel.
•• Pig and oast iron and cast steel

•* Wvoaght iron and steel, and wire, 7.6 to 7.9 •..••..

Ivory '

lee, .911 to .922

fiidiarobber '*

Lignum vita, dry *<

Lard " ..

Lead, of eoBaMree,U.80ta 11. 4T; either rolled or east '•

UmMtanee and Marbles, 3.4 to 2.M,U0 to 17&8

" " •* ordinarily about

** ** ** quarried in irregular fragments. 1 oub yard solid,

makes abont 1.9 cab yds perfeotly loose : or about
1^ yds piled. In this last oase. 571 of the pile
is solid; aod the Nmaining .429 part of it is

voids piled..

UmBt qafBk, ground, loose, per straok bushel 62 to 70 lbs

•• •• " well shaken. •• »• ....80 "

♦• " " thoroughly shaken, '* ...MH "

ICahogaay, Spanish, dry*..... ....« «•.•... ...average. •

** Honduras, dry "

Ibpte, di7« ♦' ..

MarMei, sea Limestones.

Maaoiuy, of granite or limestones, well dressed throngheal.

** *' " weU>scabbled mortar rubble. About 4 of the mass

will be mortar

- f ** wen-seabbled drr nibble

M •< M roughly soabbled morur rabble. About H to Mi P^^ri

will be mortar

M M M rsntfily soabbled drv rabble

▲t 156 lbs per eub n, a cub y trd weighs 1.868 tons ; and 14.46 oub ft,
1 ton.
Masouy of sandstone ; about H part less than the fbregolnf .

*' briokwork, pressed briok, fine Joints average. .

medlam quality •«

(1 M <t

•• " •* eoarse; infbrlor soft bricks "

At 135 fl>s per eub ft, a oub yard weighs 1.607 tons; and 17.98 eub

fl. 1 ton.

IbraaiT.atSSOFah

» 60° "

•< tijo •«

llka.2.75toS.l....

Mortar, hardened, 1.4 to 1.9k..
Mad, dry, close

moderately pressed.

fluid

Average
BpGr.

3.69

'2.8*"
*2.'27'

8.

19.268
19.32

19.6
.96
3.35

A

.86

7.2
7.76
1.82
.92
.98
1.38
.96
11.88
3.6
2.7

.86

.66
.79

19.62

13.58

13.88

2.93

1.66

Average

Wt of a

Cub Ft.

Lbs.

168.

96.
176.
100.
141.6

8Z

66.

64.

52 to 66
187.
107.

1204.

1206.

1217.
61.1
203.

.00531
25.

53.

450.
4T6ta4

114.

57.4

58.

83.

59.3
709.6
164.4
168.

96.
61.

64.
76.
63.
86.
40.

166.

154.
138.

150.
135.

140.
135.
100.

849.
.846.

8S6.

183.

103.

80 to 110
110 to 130
104 to 120

• Green timbers asually weigh from one-fifth to nearly one-half more than

4fT;and ordinary building timbers when tolerably seasoned about one-sixth morethao perfectly dry

214

SPEOIFIO GRAVITY.

Table of speelflc ffravitleB, and wetybUi— (Oontinaed.)

The specific gravity of any sQbfltance is » its weiifllt
in i^rams per cnbie centimetre.

ATenge
. Sp Or.

Naphtlia

Viirog«D Gas is about -^ part lighter than air

Oak. live, perfeotly dry, .88 to 1.02* averafQ..

" r«d. blacli, 4o« " ..

Oils, irhale; olive ••

" oftarpentine "

Oolites, or Boestones, 1.9 to 2.6 "

Ozygeu Oas, a little more than JL part heavier thau air

Petroleum

Peat, dry, unpressed

Pine, white, perfectly dry, .86 to .46*

1000 ft board measure weighs .080 ton.*

" yellow, Northern, .48 to .62

1000 ft board measure weighs 1.276 tons.*

• " Southern, .64 to .80

1000 ft board mean u re weighs 1.674 tons.*
Pine, heart of long-leafed Southern yellow, luueai. ...

1000 ft board measure weighs 2.418 tens.

Pitch

Plaster of Paris ; see Gypsum.

Powder, slightly shaken

Porphyry, 2.66 to 2.8

Platinum 21 to 22

" native, in grains 16 to 19

Qnarti, common, pure 2.64 to 2.67

*' " finely pulveriied, loose

** *' " " well shaken

" " " " well packed

" quarried, loose. One measure solid, makes full IK broken and

piled

Baby and Sapphire, 8.8 to 4.0b^

Bosin

8alt......

Sand, pure quarts, perfectly dry, loose

•* <• ** •* •* slightly shaken

•« «« rammed, dry....

Natural sand consists of grains of differeat sixes, and weighs more, per
unit of volume, than a sand sifted from it and having grains of
uniform site. Sharp sand with very large and rery ■mall grains

may weigh as much as <

Sand is very retentive of moisture, and, when in large bulk, its natural
moisture may diminish its weight from 6 to 10 per eent.

** perfectly wet, voids full of water ->»-

Sandstones, fit for building, drv, 2.1 to l.YS 131 to 171.

'* quarried, and piled, 1 measure solid, makes about IH piled...

Serpentines, good 2.5 to 2.66

Bnow, fresh fallen

** moistened, and compacted by rain...

Sycamore, perfectly dry.

1000 ft board measure weighs 1.S76 tons.

Shales, red or black 2.4 to 2.8 average..

** quarried, in piles " ..

Slate t.Tto2.9 • ** ..

Silver " ..

Soapstone, or Stea|ite 2.66 to 2.8 *' ..

Steel, 7. T to 7.9. The heaviest oon tains least earbon " ..

Steel is not heavier than the iron from which it is made; onless the
iron had impurities which were expelled daring its oonversion into
steel.

Svlphur ...., •.••.....•...•«••■■.•.....•..••..••..... average..

Spruce, perftiotly drr. • " .«

1000 ft board raeasore weighs .990 ton.

Spelter, or Zinc 6.8 to 7.2.

Sapphire; and Ruby, 3.8 tQ i...«

Tallow "

Tar "

Trap, compact, 2.8 to 3.2 **

" quarried; in piles "

Topaz. 8.46 to 8.66 "

.95

.77

.92
.87
2.2

.00186
.678
•"•'-'•■

.65

.72
1.04
1.16

1.

2.78
21.6
17.5

8.66

S.9
1.1

M
U

t.41

• • • •

S.6

••••••

.60
S.6

2.8
10.6
S.TS
7.66

1.
.4

7.00
8.9
.94
1.
8.

*S!65*

Average

Wt of a

Cob Ft.

Lbs.

6X.9

.0741
50.3
48.
82 to4B
57.3
54.8
137.
.0648

54.8
20 to SO
25.

34.8

45.

66.

Tl.T

82.8
170L
1343.

185.

90.

105.

112.

94.

88.t

60 to 70

90 to 108

92 to 110

100 to 180

117.

U8 to ISO
111.

88.
182.

6 to IS
16 to 50

87.

161.
92.
17i.
656.
17QL
480.

1S5.

487.6

68.8
02.4

187.
107.

* Green timbers usually weigh from one-fifth to nearly ODe-half more than

dry ; and ordinary building timbers when tulerably seasoned about cue-sixth more than perfectly dry.

WEIGHT OF COAL.

215

Table of apeclfle gravities, and weiffbta— (Continued.)

The specific gravity of any substance is » its welgrli^
In yrams per eubie eentimetre.

Tin, oast, 7.2 to 7.5 arerage.

Turf, or Peal, dry, unpreaaed

Water. Sm pagA 3*i6.

Wax. bees average.

Wine*. .993 to 1.04 •» ,

WalDOt, blaok, perfectly dry. " .

1000 ft board measnre weighs 1.414 tons.

Zlno, or Spelter, 6.8 to 7.2.... < «« .

Zirooo, 4.0 to 4.9 ** .

Average

8pOr.

7.35

OQR
•wo

.61

7.00
4.45

Averace

Wt of a

Cub Ft.

Lbs.

459.

90 to 80
68.417

eo.5

63.8

38.

4S7.6

S|Miee oeenpied by eoal. In cubic feet per ton of 2240 pounds.

PennsylTanla Anthracite.

Hard white ash*

Free-burning white ash *.
Shamokin * ,

Schuylkill white ash *.
" red " *.

Lykens Valley *

Wyoming free-bumingf *

Lehigh t

Lehigh ; Reading C. & I. Co. *...
Lehigh : f Lump, 40.5 ; cupola, 40

Bro-
ken.

Egg.

Stove.

Nut.

Pea.

Buck-
wheat.

f

38.6

39.2

39.8

40.5

41.1

' 39.4

39.6

39.6

39.6

89.8

39.8

39.0

39.6

40.2

40.8

41.6

' 39.6

39.6

39.6

41.2

41.9

42.4

39.3

39.9

40.5

41.2

41.9

39.0

39.9

42.6

45.7

46.5

47.7

39.6

40.3

40.9

41.6

42.3

40.0

40.5

41.1

41.7

42.3

{44.2

44.8

45.2

45.7

46.2

46.7

443

44.3

45.0

46.1

46.5

40.0

39.8

39.4

39.4

38.8

38.5

38.4

42.1

41.4

38.5

38.8

40.1

40.3

40.3

40.5

0.3; du

Lst, 39.]

.•

Aver-
age.

39.8
39.6
40.2
40.7
40.6
43.6
40.9
41.1
45.7
45.1
39.7
40.0
39.7

3itaininoas«

From Coxe Bros. & Co. f

Pittsburg 48.2

Erie 46.6

Hocking Valley 45.4

Ohio Cannel 45.5

Indiana Block 51.1

Dlinols 47.4

From Jour. XJ. S. Ass'n Charcoal Iron Workers.
Vol. Ill, 1882.g

Pittsburg 47.1

Cumberland, max 42.3

min 41.2

Blossburg, Pa 42.2

Clover Hill, Va 49.0

Richmond, Va.

(Midlothian) 41.0

Caunelton, Ind ,....47.0

Pictou,N. S 45.0

Sydney, Cape Bretou.47.0

Logarithm.
1 cubic foot per ton of 2240 pounds =

0.89286 cubic foot per ton of 2000 pounds 1.950 7820

2240 (exact) pounds per cubic foot 3.350 2480

1 cubic foot per ton of 2000 pounds =

1.12 (exact) cubic reet per ton of 2240 pounds 0.049 2180

aOOO (exact) pounds per cubic foot .3.801 0800

1 pound per cubic foot =

2240 (exact) cubic feet per ton of 2240 pounds 8.850 2480

2000 " " " 2000 " 3.301 0300

•From Edwin F. Smith, Sup't A Eng'r, Canal Div., Phila. and Reading R. R.

fFrom very oarefiil weighings in the Chicago yards of Coxe Bros. & Co.
Kote the irregular variation with size of anthracite In Coxe Bros.' figures.

^Quoted from ITie Mining Record. On the authority of *• many years' experi-
ence" of "a prominent retail dealer in Philadelphia," the Journal gives also
figures requiring from 4 to 13 per cent, less volume per ton than those here
quoted from the Journal and from other authorities.

216 WEIGHTS AND KEASITKE8.

WEIGHTS AND MEASURES.

United States and Brttisb measures of lengrtli and weiirbt»

of the same denomination, may, /or all ordinarp pttrposeSf be ooncidered as equal ;
but the liquid and dry measures of the same denomination differ widely
in the two countries. Ttaie standard measure of leng^tb of both coun-
tries is theoretically that of a pendulum vibratiiig seconds at the level of the
sea, in the latitude of Loudon, in a vacuum, with Fahrenheit's thermometer at
629. The length of such a pendulum is supposed to be divided into S9.1393
equal parts, called inches ; and 36 of these inches were adopted as the standard
yard of both countries. But the Parliamentary standard having been destroyed
by fire, in 1834, it was found to be impossible to restore it by measarement of a
pendulum. The present British Imperial yard, as determined, at a temperature
of 629 Fahrenheit, by the standard preserved in the Houses of Parliament, is
the standard of the United States Coast and Geodetic Survey, and Is recognized
as standard throughout the country and by the Departments of the Govern-
ment, although not so declared by Act of Congress. The yard between the 27th
and 63d inches of a scale made for the U. S. Coast Survey by Troughton, of Lon-
don, in 1814, is found to be of this standard length when at a temperature of
59^.62 Fahrenheit : but at 629 is too long by 0.00083 inch, or about 1 part in 43373,
or 1.46 inch per mile, or 0.0277 inch in 100 feet

The Coast Survev now uses, for purposes of comparison, two measures pre-
sented by the British Government in 1855, as copies of the Imperial fltandsrd,
namely :

** Bronze standard, Ko. 11 ;" of standard length at 62^.25 Fahr.
" Malleable iron standard, No. 57 ;" " " " 62«>.io "

See Appendix No. 12, Beport of U. S. Coast and Geodetic Survey for 1877.
Tbe legral standard of ireielit of the United States is the Troy

pound of tbe Mint at Philadelphia. This standard, containing 5760

Sains, is an exact copy of the Imperial Troy pound of Grea*
ritain. The avoirdupois or commercial pound of the United States, con-
taining 7000 grains, and derived from the standard Troy pound of the Mint, is
found to agree within one thousandth of a erain with the British avoirdu|M>fs
pound. The U. S. Coast Survey therefore declares the weights of the two ooun-
tries identical.

Tlie Ton. In Revised Statutes of the United States, 2d Edition, 1878, Title
XXXiy, Collection of Duties upon Imports, Chapter Six. Appraisal, says :

"Sec. 2951. Wherever the word 'ton' is used in this cnapter, in reference to
weight, it shall be construed as meaning twenty-hundredweight, each hundred-
weight being one hundred and twelve pounds avoirdupois."

This appears to be the only U. S. Government regulation on the subject.

The ton of 2240 ft>s (often called a sross ton or Ions ton) is commonlj
used in buying and selling iron ore, pig iron, steel rails and other manufactured
iron and steel. . Coke and many other articles are bought and sold by the net
ton or sliort ton of 2000 lbs. The bloom ton had 2464 ftis, = 2240 fira -^ 2
hundredweight of 112 S>s each ; and the pig iron ton had 2268 fi>s, == 2240 lbs + a
"sandage" of '28 fcs, or one "quarter," to allow for sand adhering to the pigs,
but some furnace men allowed only 14 lbs. In electric traction work the ton
means 2000 lbs.

As a measure, the ton, or tun, is defined as 252 gallons, as 40 cubic feet of
round or rough timber or in ship measurement, or as 60 feet of hewn timber. 252
U. S. gallons of water weigh about 2100 Ha ; 252 Imperial gallons about 2500 lbs ;
SO cub ft yellow pine about 2500 Sts.

Tbe metric system * was legalised in the United States in

* The metric system, as compared with the English, baH much the same advantagea
and disadvantages that our American decimal coinage has in comparison witiii the
English monetary system of pounds, shillings and pence. It will enormously facili-
tate all calculations, but, like all other improvemeute, it will necessarily eause some
inconvenience while the cliange is being made. The metric system has also tMa ftir-
ther and very great advantage, that it bids fair to become univei-sal among of viliaeo
rations.

WEIGHTS AND MEASURES. 217

1866, but hM not been made ot)llgfttorT. The gorernment has since ftirnished
very exact metnc standards to the several States. The use of the metric system
has been permitted In Great Britain, beginning with August 6, 1897. and in
Ruflsia, beginning with 1900. I to use is now at least permissive in most civil-
ised nations.

Tlie laetrle nnlt of lenytb is tlie metre, er nueter, which waa

fntended to be one ten-millionth I j of the earth's quadrant, f. c, of

Ihat portion of a meridian embraced between either pole and the equator. This
lengtn was measured, and a set of metrical standards of weight and measure
were prepared in accordance with the result, and deposited among the archives
«f France at Paris (MHre des Archives.. Kilogramme des Archives, etc.). It has
since been discovered that errors occurred in the calculations for ascertaining
the length of the quadrant ; but the standards nevertheless remain as originally
preparM.

Tlie metric measures ef surface and of capacitv^ are the squares
and cubes of the meter and of ito (decimal) fractions and multiples.

Tlie metric unit of welarlit is tlie grramme or grram, which is
the weight of a milliliter or cubic centimeter * of pure water at its tempera*
tore of maximum density, about A.5^ Gentisrade or 40^ Fahrenheit.

By the concurrent action of the principal governments of the world, an In-
temational Bureau of Weiyiits and MeasuriMi has been estab-
Ushed, with its seat near Paris. It has prepared two ingots of pure platinum-
ixidium, from one of which a number of standard kilograms (1000 grams) havf
been made, and from the other a number of standard meter bars, both derived
from the standards of the Archives of France. Of these copies, certain ones
were selected as international standards, and the others were distributed to the
different governments. Those sent to tne United States are in the keeping of
the U. 8. Coast Survey.

The detennination of the ei|niTalent of tbe meter in Eng^Iisii
measure is a very difficult matter. The standard meter is measured from end
l» sfuf of %pkUiiuan bar and at the freexbtng point ; whereas the standard yard is
measured hehown two lines drawn on a silver seale inlaid in a brmize bar. and ai
^aP FiihrenheU. Tbe United States Ooast Surweyf adopts, as the
length of the meter at 62° Fahrenheit, the value determined by Capt. A. R.
Clarke and Col. Sir Henry James, at the office of the British Ordnance Survey,
in 1866, vis. : S9.37(M82 inches (= 8.2808666 + feet « 1.0986222 + yards) ; but the
lawftil equiwaient, established by Congress, is 39.87 inches (=t 3.28083 feet
= 1.098611 yards). This value is as accurate as any that can be deduced from
existing data.

Tbe ffram Weislis, by Prof. W. H. Miller's determination,! 15.43234874

Sains. An examination made at the International Bureau of Weights and
easures in 1884 makes it 15.43236639 grains. The leeal value in the United
States is 15.432 grains.

• 1 centimeter =» r^ meter = 0.3937 inch. 1 milliliter {^^ liter) or cubis centi-
meter =3 0.061 + cubic inches,
t Anpendix No. 22 to report of 1876, page 6.
X Philosophical Transactions, 1866, pp. &3y ets.

218 rOEBIGN COINS.

Approximate Talses of Foreign Coins* in U. S. Honey.
The references 0, ^, ^ and *) are to foot-notes on next page.

From Circular of U. S. Treasury Department, Bureau of the Mint, Jan. 1, 1887;
from " Question Mon6taire," by H. Costes, Paris, 1884; and from our 10th edition.

Argentine Repub.— Peso = 100 Centavos, 96.5 ots.** Argentino = 5 Pesos, $4.82.

Austria.— Florin = 100 Kieutzer,47.7 cts.,2 3o.9 cts.s Ducat, $2.29. Maria Theresa
Thaler, or Levantin, 1780, $1.00.2 Rix Thaler, 97 cts.* Souverain, $3.57.*

Belgium.i— Franc = 100 centimes, 17.9 ct8.,« 19.3 ots.*

Bolivia— Boliviano = 100 Centavos, 96.5 cts.,* 72.7 cts.« Once, $14.95. Dollar,
96 cts *

Brazil.— Mil reis = 1000 Reis, 50.2 cts.,* 54.6 cts.3

Canada. — English and U. S. coins. Also Pound, $4.*

Central America.*— Doubloon, $14.50 tu $15.65. Reale, average S^ cts. See
Honduras.

Ceylon.— Rupee, same as India.

Chili.— Peso = 10 Dineros or Decimos = 100 Centavos, 96.5 cts.,* 91.2 ct«.» Con-
dor = 2 Doubloons = 5 Escudos = 10 Pesos. Dollar, 93 cts.*

Cuba.— Peso, 93.2 cts.* Doubloon, $5.02.

Denmark.— Crown = 100 Ore, 26.7 ct8.,« 26.8 cts.a Ducat, $1.81.* Skilling, % ct*

Ecuador.— Sucre, 72.7 cts.» Doubloon, $3.86. Condor, $9.66. Dollar, 93 cts.*
Eleale 9 cts *

Egypt.— Pound = 100 Piastres :« 4000 Paras, $494,3.*

Finland.— Markka = 100 Penni, 19.1 cts.* 10 Markkaa, $1.93.

France.1— Franc =100 Ceniimes, 17.9 ct8.,« 19.3 cts.8 Napoleon, $3.84.* Livre,
18.5 cts.* Sous, 1 ct.*

Germany.— Mark = 100 Pfennigs, 21.4 cts.,2 23.8 cts.* Augustus (Saxony), $3.98.*
Carolin (Bavaria), $4.93.* Crown (Baden, bf,varia, N. GermanyX $1.06.*
Ducat (Hamburg, Hanover), $2.28.* Florin (Prussia, Hanover), 66 eta.*
Groschen, 2.4 cts.* Kreutzer (Prussia), .7 ct. Maximilian (Bavaria). $3.30.*
Rix Thaler (Hamburg, Hanover), $1.10* (Baden, Brunswick), $1.00* (Prussia,
N. Germany, Bremen, Saxouy, Hanover), 69 cts.*

Great Britain. — Pound Sterling or Sovereign (£) = 20 Shillings = 240 Pence,
$4.86.65.* Guinea = 21 Shillings Crown = 6 Shillings. ShilUng (*), 22.4
cts.,s 24.3 cts. (^ pound sterling). Penny (d), 2 cts.

Greece.!— Drachma = 100 Lepta, 17 cts.,« 19.3 cts.*

Hayti.— Gourde of 100 cents, 96.5 cts.s*

Honduras.— Dollar or Piastre of 100 cents, $1.01. See Central America.

India.— Rupee = 16 Annas, 45.9 cts.,^ 34.6 cts.* Mohur = 16 Rupees, $7.10. Star
Pagoda (Madras), $1.81.*

Italy, etc.i— Lira = 100 Centesimi, 17.9 cts.,2 i9.3cts.* Carlin (Sardinia), $8.21.*
Crown (Sicily), 96 ctfi.* Livre (Sardinia), 18,6 cts.* (Tuscany, Venice), 16
sts.* Ounce (Sicily), $2.50.* Paolo (Rome), 10 cts.* Pistola (Borne), $3.37.*
Scudo* (Piedmont), $1.36 (Genoa), $1.28 (Rome), $1.00 (Naples, Sicily), 95
cts. (Sardinia), 92 cts. Teston (Rome). 30 cts.* Zecchino (Rome), ^.27.*

Japan.— Yen = 100 Sen rgold), 99.7 cts.* (silver), $1.04^, 78.4 cts.*

Liberia.— Dollar, $1.00.* *

Mexico.— Dollar. Peso, or Piastre = 100 Centavos (gold), 98.3 cts. (silver), $1.05,«
79 cts.* Once or Doubloon = 16 Pesos, $15.74.

Netherlands.— Florin of TOO cents, 40.5 cts.," 40.2 cts.« Ducatoon, $1.32.* Guilder,
40 cts.* Rix Dollar, $1.05.* Stiver, 2 ctfl.*

New Granada.— Doubloon, $15.34.*

Norway.— Crown = 100 Ore = 30 Skillings, 26.7 ct8.,« 26.8 cts.«

Parascuay .—Piastre = 8 Reals, 90 cts.

Persia.— Thoman = 6 Sachib-Kerans = 10 Banabats = 25 Abassis — 100 Scahia,
$2.29.

Peru.— Sor= 10 Dineros = 100 Centavos, 96.5 cts.,a 72.7 cts.* Dollar, 93 eta.*

Portugal.— Milreis = 10 Testoons = 1000 Reis, $1.08.* Crown = 10 Milreis.
Moidore, $6.50.*

Russia.- Rouble = 2 Poltinniks = 4 Tchetvertaks = 6 Abassis = 10 Griviniks =
20 Pietaks = 100 Kopecks, 77 cts.,« 58.2 cts.* Imperial =-« 10 Roubles, $7.72.
Ducat = 3 Roubles, $2.39.

Sandwich Islands.- Dollar, $1.00.*

Sicily.— See Italy.

Spain.— Peseta or Pistareen = 100 Centimes, 17.9 cts.,* 19.3 cts.* Doubloon (new)
= 10 Escudos = 100 Reals, $5.02. Duro = 2 Escudos,* $1.00.2 Doubloon (old),
$15.65.* Pistole = 2 Crowns, $3.90.* Piastre, $1.04.* Reale Plate, 10 cta.^
Beale vellon, 6 cts.*

1, 2, 3, 4. See foot-notes, next page.

FOBEIGN COINS.

219

(Foreign Coins QnUinMd. Small flsnreft Oi *» 'i *) ^^^ ^ M^ noUs.)

Sweden.— Crown = 100 Ore, 25.7 ct8.,« 26.8 cta.» Ducat, $2.20.* Rix Dollar, $1.05.«

Switzerland.!— Franc = 100 Centimes, 17.9 et8.,2 19.3 ct8.«

Tripoli.— Mahbub = 20 Piastres, 65.6 ct8.»

Tunis.— Piastre = 16 Karobs, 12 cts.2 10 Piastres, f 1 .16.6.

Turkey.— Piastre = 40 Paras, 4.4 cts.' Zecchin, J1.40.*

United States of Colombia.— Peso = 10 Dineros or Decimos = 100 Centaros, 96.5

cts.,« 72.7 ct8.3 Condor = 10 Pesos, $9.65. Dollar, 93 5 cts.*
Uruguay.— Peso = 100 Centavos or Centesimos (goldl, $1.03 (silver^ 96.5 cts.s
Venezuela.— Bolivar — 2 Decimos, 17.9 cta.,2 19.3 cts.* Venezolano = 5 Bolivars.

Standard Blameiers and Welgrbte of United States

Coins.

Valae.

Diam«ier.

Wetgbt.

€k>ld, 10 per cent, alloy :

Double Eagle

Eagle

TTfLlfFagle . .

1

20
10
'5
2.50

1.00
0.50
0.25
0.10

0.05
0.01

Inches.

1.350
1.060
0.848
0.700

1.500
1.205
0.955
0.705

0.835
0.750

Millimeters.

34.29
26.92
21.54
17.78

38.10
30.61
24.26
17.91

21.20
19.09

Grains.

516.00

258.00

129.00

64.50

412.60

192.90

96.45

38.58

77.16
48.00

Grams.

33.436

16.718

8.359

Quarter "kagle

Silver, 10 per cent alloy :

Standard Dollar

TTalf Dnllfif . .

4.180

26.729
12,50

Quarter Dollar

Dime

JHlnor

Five Cents, 75^^ copper, 25^«
nickel . . .•

6.25
2.50

5.00

One Cent, 95^^ copper, 5^ tin
and zinc

3.11

Perfectly pure sold is worth $1 per 28.22 grs = $20.67183 per troy oe =*
$18.84151 per avoir oz. Bttandard (U. 8. coin) is worth $18.60465 per troy oz =
$16.95736 per avoir oz. It consists of 9 parts by weight of pure gold, to 1 part
alloy. Its value is that of the pure gold only ; the cost of the alloy and of the
ooini^ being borne by Government. A cable f€»ot of pure cold irelgphs
about 1204 avoir lbs ; and is worth $362963. A cubic ineh weighs about 11.148
avoir oz ; and is worth $210.04.

Pure gold is called fine, or 24 earat gold ; and when alloyed, the alloy is sup-
posed to be divided into 24 parts by weight, and according as 10, 15, or 20, 4&c, of
these parts are pun gold, the alloy is said to be 10, 16, or 20, Ac, carat.

The averaipe fineness of California natlTe void, by some thou-
sands of assays at the U. S. Mint in Philada., is 88.5 parts gold, 11.5 silver. Some
from Georgia, 99 per cent. gold.

•Pure sllTer fluctuates in value : thus, during 1878-1879 it ranged between
$1.05 and $1.18 per troy oz., or $.957 and $1,076 per avoir, oz. A cubic inch weiglfs
about 5.528 troy, or 6.065 avoir, ounces.

1 France, Belgium, Italy, Switzerland, and Greece form the Latin Union.
Their coins are alike in diameter, weight, and fi^ieness.

t __ 19.3 times the value of a single coin in francs as given by Costes.

» Par of exchange, or equivalent value in terms of U. S. gold dollar.— Treasury
Giicalar.

« Erom our 10th edition.

220 WEIGHTS AND MEASURES.

Troy Weifrbt. U. S. and British.

24 grains 1 pennyweight, dwt.

20 pennyweights 1 ounce = 480 grains.

12 ounces 1 pound = 240awtB. = 5760 grains.

Troy welcht is nsed for grold and silver.

A carat of the jewellers, for precious stones is, in the U. S. = 3.2 grs. ; in
London, 3.17 grs. ; in Paris, 3.18 grains., divided into 4 jewellers' grs. In troy,
apothecaries' and avoirdupois, tbe grain is tbe same.

Apotbecaries' Weiffbt. U. 8. and British.

20 grains 1 scruple.

3 scruples 1 dram = 60 grains.

8 drams 1 ounce = 24 scruples = 480 grains.

12 ounces 1 pound = 96 drams = 288 scruples = 5760 grains.

In troy and apothecaries' weights, the grain, ounce and pound are the same.

Avoirdupois or €oniniereial Weiffbt. U. 8. and British. .

27.34875 grains - 1 dram.

16 drams 1 ounce = 437V grains.

16 ounces 1 pound = 256 drams = 7000 grains.

28 pounds 1 quarter = 448 ounces.

4 quarters ~ 1 hundredweight = 112 fl)8.

20 hundredweights 1 ton = 80 quarters = 2240 fts.

A stone «> 14 pounds. A quintal = 100 pounds avoir.

Tbe standard of tbe avoirdupois pound, which is the one in
common commercial use, is the weight of 27.7015 cub ins of pure distilled water.
at its maximum density at about 39°.2 Fahr, in latitude of London, at the level
of the sea ; barometer at 30 ins. But this involves an error of about 1 part in
1362, for the IS) of water = 27.68122 cub ins.

A troy lb = .82286 avoir ft. An avoir ft = 1.21528 troy ft, or apoth.

A troy OS. = 1.09714 avoir, oz. An avoir, oz. = .911458 troy oz., or apotb.

IiOn§: Measure. U. 8. and British.

12 inches 1 foot = .3047978 metre.

3 feet 1 yard = 36 ins = .9143919 metre.

5^ vards 1 rod, pole, or perch =» 16U feet = 198 ins.

40 ro^s 1 furlong = 220 yards -= 660 feet.

Sfurlongs 1 statute, or land mile = 320 rods = 1760 y^ =.6280 ft « 63360 iiM.

3 miles 1 league = 24 fUrlongs = 960 rods = 5280 yds = 15840 it.

A point =y, inch. A line = 6 points =*t^ inch. ^ palm = 3 ins. A
banS = 4ins. Aspan = 9ins. A fatbom = 6 feet. A cable's lenKtb

= 120 fathoms = 720 feet. A Gnnter's surveying cbain is 66 feet, or 4
rods long. It has 100 links, 7.92 inches long. 80 Gunter's chains = 1 mile.
A nautical mile, geoffrapbical mile, sea mile, or knot, is

variously defined as being = the length of

metres feet statute miles

1 min of loniritude at the equator = 1856.345 6087.16 1.15287

1 « latitude « " = 1842.787 6045.95 1.14507

1 ^^ lauiuu ^^ ^ 1861.655 6107.85 1.15670

1 '« «* atlat46° = 1862.181 6076.76 1.15090

1 "a great circle Qf a true') (value adopted .by U. S. Coa»t
mhere whose surface area is V -=< and Geodetic Survey
fqutl To that of the earth j ll853.248 6080.27 1.15157
British Admiralty bnot = 1853.169 6080.00 1.15152
The above lengths of minutes, in metres and feet, are those published by the U. S.
CoMt and Geodetic Survey in Appendix No 12, Report for 1881, and are calculated
from Clarke's spheroid, which is now the standard of that Survey.

At the equator, 1° of lat =-- 68.70 land miles; at lat 20° = 68.78 ; at 40° =
69.00 ; at 60° - 69.23 ; at 80° = 69.39 ; at 90° = 69.41.

WBiaHT8 AKD MEASURES.

221

I^en^tlis of a Dflgr— 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.

0 1

W.16

14

67.12

28

61.11

42

61.47

66

88.76

70

28.72

a 1

ie.i2

16

66.50

80

69.94

44

49.88

68

86.74

72

21.43

4 1

M.N

18

65.80

S3

58.70

46

48.13

60

84.67

74

19.12

6 (

B6.76

20

66.02

34

67.39

48

46.88

62

82.56

76

16.78

8

B&tt

22

64.16

36

56.01

50

44.54

64

30.40

78

14.42

10

118.12

24

63.21

88

64.56

63

43.67

66

28.21

80.

12.05

13

17.66

96

62.90

40

53.06

54

40.74

68

26.98

82

9.66

InelieB redaeed to Deeimals of a

Foot.

Ao errors.

Ina. ]

root.

las.

Foot.

IDI.

Foot.

Ins.

Foot.

Ins.

Foet:

Itti.

Foot.

•

.0000

%

.1867

4

.8383

6

.5000

S

.6667

10

.8833

1-SS

.0026

.1693

.3359

.5026

.6693

.8859

1.16

.0062

.1719

.8886

.6052

.6719

.8886

8-n

.0078

.1746

.8411

.5078

.6746

.8411

Ji .

.0104

H

.•771

H

.9488

H

.5104

H

41771

H

.8438

OUO

.1797

a Jig 4

mOVfn

.6130

.6797

.8464

S-16 .

0166

.1828

.3480

.5156

41823

.8490

f-tt

0182

.1849

.8516

.6182

41848

.8616

Ji :

0208

H

.1876

H

.3542

H

.6208

H

.6875

H

.8643

0284

.1901

.3568

.5284

.8801

.8568

fr-16

0280

.ion

.3594

.6200

.6927

.8694

11-S9

0286

.1953

.3620

.6286

.6953

.8620

H

0313

H

.1979

H

.3646

H

.5313

H

.6879

H

•oDvO

ust

0339

.2006

.8672

.5339

.7006

.8672

7«1«

086&

.2031

.3698

.6866

.7031

.8688

U^

0381

.2067

.3724

.5391

.7057

.8724

.^

0417

H

.2083

H

.3750

H

.6417

H

.7083

H

.8750

17-SS

0443

.2109

.8776

.5443

.7109

.8776

9-M

0469

.9186

.8802

.5469

.7135

.8802

IMS

0485

.2161

.8828

.5495

.7161

.8828

nji :

0621

H

.2188

H

.3854

H

.5521

H

.7188

H

.8854

0647

.2214

.8880

.5647

.7214

.8880

ii.i«

0573

.2340

.8906

.5573

.7240

.8806

ss-ss

0680

.2966

.3932

.6599

.7266

.8692

H

0626

H

.2392

H

.8958

H

.5625

h

.7292

H

.8958

Sft^

6661

.2318

.8964

.5651

.7818

.8964

lft.lC

oon

^2844

.4010

.5677

.7344

.9010

S7-»

0703 .

.2370

.4036

.5703

.7370

J8006

y •

0729

%

.2396

X

.4063

X

.6729

H

.7396

}i

.9063

f^

0765

.2432

.4069

.6755

.7422

.9089

mi .

0781

•9vfto

.4115

.6781

.7448

.9115

• Sl-SS

0807

.2474

.4141

.6807

.7474

.9141

1

06SS

S

.2509

0

.4167

y

4i688

9

.7500

11

.9167

1«

0869

.2626

.4193

.6859

.7526

.9193

1-lC

0885

.3563

.4219

.7562

.9219

8-S2

0911

.2678

.4245

.6911

.7578

.9246

H

0888

H

.2004

H

.4271

H

.5038

H

.7604

H

.9271

5-St

096A

.3660

.4297

.5964

.7680

.9297

S-I6

0800

.3866

.4323

.6990

.7656

.9823

7-8i

1016

.3683

.4.')49

.6016

.7682

.9349

3< •

1042

H

■S&

H

.4876

3i

.6043

H

.7708

H

.9375

9-Si

1068

.4401

.8068

.7784

.9401

6-16

1684

.2768

.4427

.6094

.7760

.9427

11-32

1198

.2786

.4453

.6120

.7786

.9468

K

1148

H

.2811

H

.4479

H

.6146

H

.7813

H

.9479

lS-3t2

1172

.2889

.4505

.6172

.7889

.9506

7-16

1198

.2666

.4531

.6198

.7865

.9531

«

16-32

1224

.«9l

.4567

.6234

.7881

.9557

^

1260

H

S&

H

.4583

H

.6250

H

.7917

H

.9583

17-.%

1276

.4809

.6276

.7948

.9609

9-16

UOS

.2M9

.4635

.6302

.7969

.9636

19-32

1828

:SSi

.4661

.6828

.7995

.9661

2i.l^i :

1864

H

H

.4688

H

.6354

H

.8021

H

.9688

1380

.lOiV

.4714

.6380

.8047

.9714

11-16

1406

.8978

.4740

.6406

.8073

» .9740

SS.S2

108

J089

.4766

.6432

.8099

.9766

9i

1468

H

.8136

h

.4792

H

.6456

H

.8125

h

.9792

25-S3

1484

.8161

.4818

.6484

.8151

.9618

13-16

1610

.8177

.4844

.6510

.8177

OtlAA
•von

27-S2

1686

.8908

.4870

.65.<{6

.8203
.8229

.9870

H

1668

H

.8228

H

.4896

K

.6bea

X

H

.9896

n.n .

1689

41256

.4922

.6589

.8255

.9922

16-16

1816

. .8281

.4948

.6615

.8281

.9948

n« •

1641

.8807

.4974

.6641

.8307

.9974

WEIGHTS AND MEABUBBS.

— —"■H-Ij

» iq ill = 10a» aq tOl.

rodi = W40 iq Ida = UMt K Ml-

Cnblp. or Solid M^amare.

A CBbt* a

M Dik THd, or i.Ma» ■•knlg^ (I.
HI iu^>llln. •> ««HHn,

n. A tim i.iw> ai^ ci iennijtiu

A cnbl« luch Is midaI to

l.nuta snlllllni; e.r.ie3S«e3 arellLLnir a

A cubic yard la emnMl l4

1 aphere I toot In diameter, tiontnlna

A sphere 1 Inek In diameter, eonlnlna

WEZGHTS AUTD HBASimiiB.

22a

cylinder 1 foot In diameter,

.02909 oub yard.
.7854 cub foot.
I35T. 1712 cub inches.

.63112 U. S. di7 bushels.
2.5245 U. S. dry pecks.
a0.1958 U. S. dry quarts.
. 40.3916 U. S. dry pints.
5.8752 U. S. liaaid gallons.
28.5008 U. S. liquid quarts.

A eylinder 1 ineli in diameter, and

.005454 cub foot.
9.4248 cub inches.

.2805 U. 8. dry pint.

.3264 liquid pint.
1.3056 U. S. gill.

and 1 f<N>t bisrta, coui^Jiins

47.0016 U. S. liquid piuta.
188.0064 U. b. liquid gills.
4.8947 Brit imp gallons.
19.5788 Brit imp quarts.
39.1575 Brit imp pint*.
156.6302 Brit imp gills.
222.S95 decilitres.
22.2395 litres.
2.22395 decalitres.
.222895 hectolitre.

1 foot liiji^li, contains

.2719 Brit imp pint.
1.0677 Brit imp gill.
15.4441 centilitres.
1.54441 decilitres.

.164441 litres.

I«iqald JHeasnre. u, g. only.

The iMMda of this measure in the U. S. is the old Brit wine gallon of 231 oub ins; or 8.3S888 Ibr
aToir of pure water, at its max dennity of about 39^.2 Fabr ; the barom at 30 ins. A cylinder 7 in»
iiam, and 6 ins high, contains 230.904 cob ins, or almost precisely a gallon ; as does also a oube of
t.lS68 ina on an edge. Also a gallon = .13368 of a cub ft ; and a cub ft contains 7.48052 galls ; nearly
1H gall*-. TUs bastfl howerer InTolres ab err«r of about 1 part in 1363, for the water adtn-

63 gallons 1 hogshead.

2 hogsheads 1 pipe, or butt.

2 pipes. 1 tun.

In the U. S. and Great Brit. 1 barrel of wine or brandy = 31i^ galls ; in Pennsylvania, a half
barrel, 16 galls; a double barrel, 64 galls; m puncheon, 84 galls; a tierce, 42 galls. A liquid
Beasore barrel of 81^ galls contains 4.211 cub ft = a oube of 1.615 ft on an edge ; or 3.38v U. S. struck
bosbals. A sill = 7.21875 oub ins. The followlns cyliinders contain some o.' these measure*
very approximately.

ally weighs 8.3450(tti tbi

cub ins.

4glUa Ipint =28.875.

2 pints 1 qnart = 57.750 = 8 gills.

4 qxaaU 1 gallon = 231 . =8 pints— 32 gills

DIam. Height,

enb ins. Ins. Ins.

Omj.21875) IH 3

><pint 2« 3«

Pint 3« 3

quart S^ 6

Diam.
Ins.

Gallon 7 .

2 gallons 7 .

8 gallons 14 .

10 gallons 11 .

Height.

Ins.

6

. 12

. 12

. 15

Apotbecaries* or Wine Measure.

1 Gallon

mnt...

1 Fluid ounce . .
1 Fluid drachm.
IMmim

Symbol.

Pints.

Floid
.ounces.

FJoid

draohms.

Minims.

Coble

inches.

Cong*

m

8

1

• ■ • •

• • • •

• • • •

128
16

1

• • • •

• • • •

1024
128

8
1

• • •

61440
7680

480
60

1

231
28.875

1.8047
0.2256
0.0088

Weight of water4

Pounds, av. Grains.

8.345

1.043

Ounces, av.

1.043

68415
7301.9

456.4
57.05
0.96

To redoce U. H» liquid measnres to Brit ones of the same denomina*

tlon, divide by 1.30032; or near enough for common use, by 1.2; or to reduce Brit to U. S. multiply
by 1.2.

Dry Measure.

U. S. only.

Tlie basis of tliis is the old British Winchester struck bushel of 2150.42 cub

las; or 77.627418 pounds avoir of pure water at its max density. Its dimensions by law are 18^ ins
iaaer diam ; 19>t id> outer diam; and 8 ins deep ; and when heaped, the cone is not to be less than 6
ins Ugh ; which makes a heaped bushel equal to 134 struck ones ; or to 1.55556 cub ft.

Bdge of a cube of
equal capacity.

2 pints 1 qoart, =67.2006 cub ins = 1.16365 liquidiit 4.066 ins.

4 quarts 1 gallon. = 8 pints, = 268.8026 cub ins, :^ 1.16:i65 liq gal 6.454 "

2 gallons 1 peek, = 16 pints, = 8 quarts, = 537.6050 cub ins 8.131 "

4 pe<d(s 1 stmok bushel, = 64 pinls, = 32 quarts, = 8 gals, = 2150.4200 cub ins. 12,908 "

* Abbreviation of Latin, Congius.
t Abbreviation of Latin, Ootarios.

} At its maximum density, 62.426 pounds per eubio foot, correspouding to a temperature of 4°
Ceotigrade = S9.2P Fahrenheit.

224

WMGH1B AKD MBA8X7BBS.

A 9trnck bnshel =» 1.24445 cub a. A cub ft * .80356 of a struck bushel.
Xhe dry flour barrel = 8.75 cub ft; =8 struck bushels. The dry barrel la

not, howe%'er, n legMliied measure; and no great attention is given to its capacity; consequently,
barrels rar^ cunsiderablT. A barrel of Qour conuins by law, liW Its. In ordering by tbe barrel, the
amount of its contents sboald be specifled in pouods or galls.

To reduce IJ. S. dry measures to Brit imp ones of the same name, di?

by 1.031516 ; and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposes use 1.033.

Brltlsb Imperial Measure, botb liquid and dry.

This system is established throughout Great Britain, to the exclusion of tbe old ones. Its basis is
the imperial gallon of '277.274 cub ins, or 10 lbs avoir of pure water at the temp of 62^ Fahr, when

the barom is at 30 Ids. This basis Involves an error of about 1 part im

18S6, for 10 lbs of the watar =:only 277.128 cab ina.

Aroir Ihe.
of water.

Oob. las.

Cab. ft.

Edge of a cube «f

equal capaeity.

Inches.

Acllla 1 pint

1.25

8.50

6.
10.

80. -1

80. I Dry
820. { meaa.

84.6688

e».8l85

188.687

877.874

554.648

9818.188

8878.768

in45.686

8.8605

Ipinta 1 quart

S quarts 1 pottle

8 Dottles I Kallon

4.1079

6.1756

6.6908

S fftllODB 1 p6C!C ••••••••••• ■••

8. 2157

4 Dooki 1 buhel.a.a ••••••••••

1.8R87

6.1847

10.2694

1*041?

4 basbelsl coomb

8 coombs 1 quarter 1

6i0.

TiM) imp gall = .16046 cub ft; *Dd 1 Ottb ft =<.9B918 galls.

Measure.

Symbol.

Pints.

Fluid
ounces.

Fluid
drachms.

Minims.

Oubic
iochM.

Weight of watar4

Pounds, AT.

Graimt.

1 Gallon

1 piDt

Of

fl. OS.

fl.dr.
mill*

8

1

• • • •

• • • •

160
90

1

• •••

• • ■ •

1280
160

8

1

• • • •

78800
9600

480

60

1

877.274
86.669

1.783
0.217
0.0086

10
1.85
Ounces, ar.

• •••

70068
•750

487.5
54J875
0.9114

1 Fluid ounce . . .
1 Fluid drachm..
1 Minim

The weight of water aflbrds an easy way to find the cubic contents of a tressel.

To' obtain the slae of commerelal measai^ea by means Qf tlio
* welg^bt of water.

At the common temperature of fh>m 70*^ to 75° Fah, a cub foot of ftesh water weighs wrr appnud>
mately 6214 \bi avoir. A cubic half foot, (6 ius on each edge,) 7.78125 0>a. A cub quarter foo^ (8 ins
on each edge.) .97266 n>. A cab yard, 1680.75 lbs; or .75034ton. ▲ cub half yd, (18 ins on each «das,)
210.094 lbs ; or .0938 ton. A cub inch, .036024 0) ; or .576384 ounce ; or 9.2222 drams ; or 252.170 grama.
An Inch square, and one foot long, .432292 Bk. Also lib = 27.76908 cab ins, or a cube of 8.096 ins on IB
edge. An onnce, 1.785 «ub ins ; a ton, 85.964 cab ft, all near enoof h for common me.

Original.

Uquld Measures. i^^\^^«^-

of Water.

V. S.Gill 26005»

U. 8. Pint 1.0409

U. S. Quart 2.0804

U.S. Gallon 8 lbs 5l 01 8.8916

U. S. Wme Barrel, 31 H Gail 969.1810

Dry Measures.

U. S. Pint 1.2104

U. 8. Quart 2.4208

17. S. Gallon 9.6834

V. 8. Peck 19.3668

U. a. Bushel, struck 77.4670

' * Or 4 ounces ; 2 drams ; 15.6625 grs.

I«lqal€l and I>ry. Um AT*ir.

^ ot Water.

British Imp Gill S1914*

*' Pint 1.94858

" «• Quart 9.49715

•• •' Gallon 9.9886

" •• Peek..M. 19.9779

" Bushel 79.9088

* 4.9949 ; or rery nearly 5 onnoas.

Metric Measnires.

Centilitre .03196t

pMilltre siMt

Litre J.1981

Decalitre, or Centlatere 91.9606

Btere (eubio meter) 9198.0786

t Or 5.6271 drams; or 153.866 gra.
{ 3.5169 onnoes.

* Abbreviation of Latin, Congius.
t Abbreviation of Latin, Ootarius.
t At the standard lemperatore, 929 Fahrenheit a

about 16.r> Oentlf rada.

WEIGHTS, AND UEABUBB0.

225

Metrle Measures of I^eni^^b.
By U. 8. and Brltfsli StaaiUrd.

Ins.

Ft.

Yds.

Miles.

Millimetre*

.089370

.89370428

8.9370428

89.370428

393.70428

Road
measures.

.008281

.082809

.8280869

3.280869

32.80869

328.0869

3280.869

82808.69

CeTltim«tre+t--,T,---r r ^,,r,r-,„r „f

•

]|[)ACini6tTA

.1093628
1.093628
10.93623
109.3623
1093.628
10936.23

Metret

Dnftiunetrft ")

Hectometre

.0621875

Eflometre

.6218760

Kyriametre j

6.213750

• N«arl7 the ^ part of ao inoh. t Full K inob.

} Yerj nearly 8 ft, 3H ioB. wbioh is too long hj onlj 1 part in 8616.

Hetrlc Square Measure-
By U. S. m4 British Slradard.

8q Millimetre

8q Centimetre

Sq Decimetre

Sq Metre, or Centlare.,
Sq Decametre, or Are.

D«care (not nsed)

Hectare

8q Kilometre

8q Myriametre

Sq. Ins.

.001550
.155003
1S.500B
1550.03
155008

.3861090 so miles.

38.61090 "

Sq. Feet.

.00001076
.00107641
.10764101
10.764101
1076.4101
10764.101
107641.01
10764101

Sq.Yd8,

.0000012

.0001196

.0119601

1.19601

119.6011

1196.011

11960.11

1196011.

Acres.

.000247
.024711
.247110
2.47110
247.110
24711.0

Metric Cubic or Solid Measure.

Aaevrdlns to V. 8. Standard.

Only thoM marked '« Biit" are Britiah.

Mill1]itr«,oroab
Centimetre....

Centmtre

Decilitre

Litis, or cubic
Dscimetre....

Decidltre, or
Coitiatere....

Hectolitre, or
Decistere

Kflolitre, or
Cubic Metre,
or Stere

[friolitie, or
Decastere

Cub Ins.

.0610254

.610254
6.10264

61.0254

610.254
Cub Ft.

.858156

8.53156

86.3156
863.156

riiiaoid.
(Dry.
J Liquid.
(Dry.
J Liquid.
(Dry.

.0084537 gill.
.0070428 Brit gill.
.0018162 dry pint

.084537 ffUl.
.070428 Brit gill.
.018162 dry pint.

.84537 gill = .21184 pint.

.70428 Brit gill = .17607 Brit pint.

.18162 dry pint.

{

Liquid,
Dry.

2.1134 pints.

fUpi
.11351 peck = .9081 dry qt « 1.8162 dry pt

1.05671 quart » 2.1134 pii
.88036 Brit quart = 1.7607 Brit

)ints.

(Liquid.
(Dry.

(Dry.
I Liquid.
(Dry.

2.64179 U. S. Uquid gal.

2.20000 Brit gaL

.283783 bush ^ 1.1851 peck « 9.061 dry qts.

26.4179 U. S. Uquld gal.
22.0090 Brit gal.
2.83783 bush.

264.179 U. S. liquid gal.
220.090 Brit gal.
28.3783 bush.

Onb yds, 1.8080.

Liquid. 2641.79 U. S. Uquid gal.
283.783 busb.

r Liquj
iDry.

}

I Cub yds,

18.060.

15

226

WXI6H1S Ain> KBAMUMBB*

Metric Welflrhta* redoeed to eonnnon Commercial or AtoIc
Welfffitt of 1 poand = 16 ounces, or 7000 yralns.

MiUigramme..
GentigrEunme.
Decigramme ..
Gramme

Decagramme

Hectogramme

Kilogramme

Mynogramme

Quintal*

Tonneau; Millier; or Tonne.

Grains.

.015432

.15482

1.6482

15.432

Pounds aT.

.022046
.22046
2.2046
22.046
220.46
2204.6

The graniiM is the YtaaHa of Tr«neh wdgtatt r u>d !■ the welf ht of a cab eendmetre of ^*«^^
Vater at its max deniity, at lea level, la lat of Parle ; barom 29.922 ins.

k

Frencb Measures of tlie *' Systeme Usuel.**

This iTstem wae In nse from about 1812 to 1840, when It was forbidden by law to nse eren its naoMB.
This was done in order to expedite the general nse of the tables which we have before glTen. But ss
the Systema Usnel appears In books pnbUshed daring the above interral, we add a taUa of sobw oC its
valnes.

Measures of liOiiflrtli*

Ugnensml, orliae

Pouee vsael, or inch, = 12 Ugnes.
Pled nsnel, or foot, =12 peaces ..

JLnne nsael, or elL

Toise asnel,=6pieds

Yards.

.8M&4

i.si2se

2.18727

PecC.

.09118
1.09862
8.9S706
6.M181

.09118

i.oasa

U.lStM

47.346

78.T4in

Weights, VsueL

Qrala nsnel...
GrosnsoeL...
Once nsnel...
Marensnd...
Lirre nsnel, I
1,5

or pound,

^75 grains.
60.297 '•

1.10268 arotr os.
.66129 avoir lb.

1.10268 avoir n>.

Onbio, or Solid. TTsueL

= 1.7606 British pis*.
S.TSU British sate.

1811, or before the '*8jsteme nend," the Old System, " Systeme Anolen," was in

Frencli Measures of tbe '* Systeme Anden.**

LlneaL

Point anclen, .0148 Ins. •.....•••....., ...........

Ligne anoien, .06881ns

Pouoeanden. 1.06677 ins =.0888 ft

Pled anoien, 12.76^2 ins = 1.06677 ft

Anne anoien, 46.8989 lns=8.90782ft=l.S0261 yds

Toise anoien. = 6.3946 ft= 2.1816 yds

Leagne= 2282 toises= 2.7687 miles

Sqna

Sq. ins.
.00789
1.1359

Sq.ft.

1.1859
40.8908

Sq. yds.

4.6484

Onbio.

0. ins.

.0007

1.2106

C.ft.

1.2106
261.482

G.yda.

•.68a

There is, however, much oonfosion about these old measures. Dliferent measnfas had the same
same in diibreBt provinces.

^^Ml 1 I I I.. - ^ I l'

• The m99tr4¥foU qniatal is 100 avelrdapois p«aui4s.

WEIGHTS AND MEASURES. 227

Biuwlan.

Foot; same as U. 8. or British foot. Sacblne = 7 feet. Temi * 50C
sachine » 3600 feet ai 116^ yards » .6629 mile. Pood » 86.114 lbs avoirdapoisi

Spanlsb.

Tlie eastellano of Spain and New Granada, for weighing gold, is varlouslf
estimated, from 71.07 to 71.04 grains. At 71.0S5 grains, (the mean between th«
two,) an avoirdapois, or common commercial oaoce contains 6.1572 castellano;
and a lb aToirdupois contains 98.51ff. Also a troy ounce =s 6.7553 casteliano ; and
a troy lb » 81.064 castellano. Three U.S. gold dollars weigh about 1.1 castellano.

Tlio Spanisli nuirlL, or mareo^ for precious metals, itf South America,
may be taken in practice, as .5065 of a lb aroirdupois. In Spain, .5076 lb. In
other parts of Europe, it has a great number of values : most of them, however,
being oetween JH and .54 of a pound avoirdupois. The .6065 of a lb =3 8545^

Sains ; and J5076 9) «■ 8553.2 grains. 1 marco = 60 castellanos = 400 tomine =»
90 S^nish jjroM-grains.

The arroba has various vslues in difl^rent parts of Spain. That of Cas-
tile, or Madrid, is 25.4025 lbs avoirdupois; tlie tonolada of Castile =- 2082.2
fts avoirdupois ; tlie quintal = 101.61 lbs avoirdupois ; the libra » 1.0161
fta avoirdupois; tbe eantara of wine, Ac, of Castile a 4^268 U. S. gallons;
that of Havana a 4.1 gallons.

"nie wara of Castile =3 82.8748 inches, or almost precisely 82j^ inches; or 2
feet 8Ji inches. Tbe iianeyada of land since 1801 » 1.5871 acres = 69134.08
sqaare feet. Tbe ftmeffa of corn, Ac « 1.69914 U. & struck bushels. In
California, tbe vara by law »» 88.872 U. S. inehee ; and tbe leipui - 6001
varaa; or 2.6888 U. SL miles.

fit iill^lfii

I III -1 Slj,
Ppini.1T!

mm ^i'

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i I !

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253

Si

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WEIOHTB AKD HEAHCBRS.

TABI.E or ACRES

■XaiJIBED pme mU*, a

tor dimrent wldtka.

ijk"

jiT

^.

^

k!

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^fSi'

E

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8.82

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82

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1.62

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88

0.7

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88

4.81

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0.8

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M

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4.81

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88

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11.

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80

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87

8.13

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88

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T»bl« vr rntde" P«r mile, and per 100 fket newiaiwd hop|>
sontallr, aad evrreapondlnr te dUferrat auBlea ot iaelb

Ab7 bmw dl>t is — sloping dist >

» alOBlnsdtot 19-hordlat i

" vertltelKbt IS'hardJgt >

or = sloping dlBl>

A gnde of n fKt rlH per 100 f«et li uwiillf ci

WeiUUXS AND HE&SCREa.
H PBKT PHR 100 FT, HOROOHTAIh

The trutlou of mfnnteg us eiren onlj la 34 feet In 100.

A eUnonutcr gisduaud by Uie 3d column, ind numbuwd by the flnt on*,
will gin U Hgbt tb* ilopH In feel per 11X1 Uei. So (inn. Origiunl.

ltJ

«"£

i

Sc

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ly

STc'

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1

1

ML

Si
si

«. HIE.

is!
11

ii

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SI

II

is

1

II
1

1

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)iit,^S,of Ui.

4dlx T;iprcz<fniif«^^pn>portioaHl luS; but tbje et^gpegt
grad^p surmounted by traction onlT^ even on olecttlo

ft horijoiiial ' B """"*■' '" ft TanicaJ • " A" '""

^ Is the cotangent of the augle, a, with the horiiontil, or ths Ungenl of the
int-le (9g°-a) wtth the >eiilcaJ. Tbus stated, a dope of 2 to 1 means t, slope of 2

OBASBS.

257

Table of nrrades per mile; or per 100 feet meaaared liorl«
■ontally.

Grade

Grade

Grade

Grade

Grade

Grade

Grade

Grade

in ft.

in ft.

in ft.

in ft.

in ft.

in ft.

in ft.

in ft.

per mile.

per 100 ft.

per mile.

per 100 ft.

per mile.

per 100 ft.

per mile

per 100 ft.

1

.01894

39

.73S64

77

1.46833

115

2.17803

2

.03788

40

.76758

78

1.47727

116

2.19607

S

.05682

41

.77652

79

1.49621

117

2.21601

4

.07576

42

.79546

80

1.51615

118

2.28486

5

.09470

43

.81439

81

1.53409

119

2,25379

6

-11364

44

.83333

82

1.55803

120

2.27278

7

.13258

46

.85227

88

1.67197

1-21

2.20167

8

.15152

46

.87121

84

1.59091

122

2.31061

9

.17045

47

.89016

86

1.60985

123

2.32966

10

.18939

48

.90900

86

1.62879

124

2.34848

LI

.20833

49

.92803

87

1.64773

126

2.36742

12

.22727

60

.94697

88

1.66666

126

2.38686

18

.24621

61

.96591

89

1.68661

127

2.40680

14

.26515

52

.98485

90

1.70466

128

2.42424

16

J28409

63

1.00379

91

1.72848

129

2.44818

16

.80803

64

1.02278

92

1.74212

130

2.46212

17

.32197

65

1.04167

93

1.76186

131

2.48106

18

.31001

66

1.06061

94

1.78080

132

2.50000

10

.35985

57

1.07956

95

1.79924

133

2.51894

20

.87879

68

1.09848

96

1.81818

134

2.53788

21

^773

69

1.11742

97

1.83712

135

2.56682

22

.41667

60

1.13636

96

1.85606

136

2.57676

28

.43561

61

1.15530

90

1.87500

137

2.59470

24

.45455

62

1.17424

100

1.89391

138

2.61364

25

.47348

63

1.19318

101

1.91288

139

2.63258

26 •

.49242

64

1.21212

102

1.93182

140

.'T.65152

27

.51186

66

1.23106

103

1.95076

141

2.67046

28

.53030

66

1.25000

104

1.96970

142

2.68039

29

M924

67

1.26894

106

1.98864

143

2.70833

80

.56818

68

1.28788

106

2.00768

144

2.72727

81

.58712

69

1.30682

19Z

2.02662

145

2.74621

82

.60606

70

1.32576

108

2.04646

146

2.76516

83

.62500

71

1.34470

109

2.06439

147

2.78409

84

.64804

72

1.36364

110

2.08333

148

2.80308

86

.66288

73

1.38258

111

2.10227

149

2.82197

36

.68182

74

1.40152

112

2.12121

150

2.84091

87

.70076

75

1.42045
1.43939

113

2.14016

151

2.85986

88

.7mo

76

114

2.15909

152

2.87870

If the grade per mile should consist of feet and tenffuj add to tbe grade per 100
iMt in the foregoing table, that corresponding to the number of tenths taken firom
the tabl« below ; thus, for a grade of 48.7 feet per mile, we have .81439 -f- .01826 «
.82766 feet per 100 feet.

Ft. per Mile.

Per 100 Feet.

Ft. per Mile.

Per 100 Feet.

Ft. per Mile.

Per 100 Feet.

.06

.00094

.4

.00768

.7

.01328

.1

.00189

.46

.00852

.75

.01420

.16

.00283

.6

.00947

.8

.01516

J

.00379

.65

,01041

.86

.01609

?fi

.00473

.6

.01136

.9

.01706

4

.00668

.66

.01230

.95

.0179i

.36

.00662

258

WEIGHTS AND MEASUBE8.

TABUE OF HEADS OF WATEB COBBESPONDIHO TO

OIYEN PBESSVBES.

Water at maximum density, 62.425 lbs. per cubic foot ^ 1 gram per cubit
centimeter ; corresponding to a temperature of i° Centigrade = ^.2^ Fahrenheit.

Head in feet — 2.306768 X pressure in lbs. per square inch.
*• ** ^ 0.0160192 X pressure in lbs. per square foot.

Heads corresponding to pressures not given in the table can be found by theae
formulc. or taken from the table by simple proportion.

Premare.

Head.

Preaanre*

Head.

Preaanre.

Head.

lbs. pei
■q. in.

' lbs. per
sq. ft

Feet.

lbs. pel
sq. in.

> lbs. per
sq. It

Feet

Ibe. per
•q. in.

lbs. per
sq. ft.

Feet

1

144

2.3068

61

7344

117.646

101

14644

282.984

2

288

• 4.6135

62

7488

119.952

102

14688

235.290

8

432

6.9203

68

7682

122.259

108

14832

287JS97

4

676

9.2271

64

7776

124.565

104

14976

289.904

5

720

11.6338

65

7920

126.872

106

16120

242.211

6

864

13.8406

66

8064

129.179

106

16264

244.617

7

1008

16.1474

67

8208

181.486

107

16408

246.824

8

1162

18.4541

68

8352

133.793

108

16652

249.181

9

3296

20.7609

69

8496

186.099

109

16696

261.488

10

1440

23.0677

60

8640

188.406

110

15840

268.744

11

1684

25.3744

61

8784

140.718

111

16984

256.061

12

1728

27.6812

62

8928

143.020

112

16128

268.868

13

1872

29.9880

68

9072

145.326

113

16272

260.666

14

2016

82.2948

64

9216

147.633

114

16416

262.972

16

2160

84.6016

65

9360

149.940

116

16560

266.278

16

2304

86.9083

66

9504

162.247

116

16704

267Jm

17

2448

39.2151

67

9648

164.568

117

16848

269.892

18

2692

41.5218

68

9792

156.860

118

16992

272.199

19

2736

43.8286

69

9936

159.167

119

17186

274jW

20

2880

46.1354

70

> 10080

161.474

120

17280

276J12

21

8024

48.4421

71

10224

163.781

121

17424

279.119

22

8168

60.7489

72

10368

166.087

122

17568

281.426

23

8312

68.0367

78

10512

168.394

128

17712

28S.7«2

24

8456

55.3624

74

10656

170.701

124

17856

286.088

26

8600

67.6692

76

10800

173.008

125

18000

288.84«

26

3744

69.9760

76

10944

175.814

126

18144

290.698

27

3888

62.2827

77

11088

177.621

127

18288

292.960

28

4032

64.5895

78

11232

179.928

128

18432

295.266

29

4176

66.8963

79

11376

182.235

129

18576

297J$7S

80

4320

69.2030

80

11520

184.541

130

18720

299.880

81

4464

71.5098

81

11664

186.848

181

18864

802.187

82

4608

73.8166

82

11808

189.166

132

19008

804.498

83

4752

76.1233

88

11952

191.462

138

19162

806.800

84

4896

78.4301

84

12096

193.769

184

19296

809.107

85

5040

80.7369

86

12240

196.075

186

19440

811.414

86

6184

83.0436

86

12384

198.382

186

19684

818.720

87

5328

85.3504

87

12528

200.689

187

19728

816.027

88

6472

87.6572

88

12672

202.996

188

19872

8184184

39

6616

89.9640

89

12816

205.302

189

20016

820J641

40

6760

92.2707

90

12960

207.609

140

20160

822.946

41

6904

94.5775

91

13104

209.916

141

20804

826.264

42

6048

96.8843

92

13248

212.223

142

20448

827.961

48

6192

99.191U

93

13392

214.529

143

20592

829.668

44

6836

101.4978

94

13536

216.836

144

20736

882.175

46

6480

103.8046

96

13680

219.143

145

20880

884.461

46

6624

106.1113

96

13824

221.450

146

21024

886.766

47

6768

108.4181

97

18968

223.756

147

21168

48

6912

110.7249

98

14112

226.063

148

21812

641.402

49

7056

113.0:U6

99

14266

228.870

149

21466

846.706

60

7200

115.3384

100

14400

280.677

160

21600

M6.016

I

WEIGHTS AKD MEAStTBES.

259

TABUB OF PRESSURES COBRESPOMDINQ TO OITEH

HEADS OF WATER.

Water at maximum density, 62.425 lbs. per cubic foot » 1 gram per cubio
•tntlmeter ; eorrespondiug to a temperature of 4° Centigrade — Z9:J9 Fahrenheit.

Pressure in lbs. per square inch = 0.433507 X head in feet.
Pressure in lbs. per square foot = 62.425 X head in feet.

Pressures corresponding to heads not given in the table can be found by these
formulK, or taken from the table by simple proportion.

Head.

Pressure.

Head.

Inches.

Pressure.

Inches.

lbs. per sq. in.

lbs. per sq. ft.

lbs. per sq. in.

lbs. per sq. fL

0.086126
0.072251
0.108377
0.144502
0.180628
0.216753

5.202083
10.4U4167
15.606250
20.808333
26.010417
31.212500

7

8

9

10

11

12

0.252879
0.289005
0.825130
0.861256
0.897381
0.488507

86.414583
41.616667
46.818750
52.020833
57.222917
62.425000

Prevnife.

1
2
3

4
5
6
7
8
9

tb
11

12
18

14

10
16
17
18
19
20
21

24
2S
26

rf

28
29
SO
31
88

0.4885
0.8670
1.3005
1.7340
2.1675
2.6010
ZJ0S45
3.4681
3.9016
4.3801
4.7686
5.2021
8.6806
6.0691
6JM)26
6.9361
7.3696
7.8031
a2366
8.6701
9.1036
9.0372
9.9707
10.4042
10.8377
11.2712
11.7047
12.1382
12.6717
18.0002
1&48S7
18.8722
14.8087
14.7392
16.1727
16.8008

Pressure.

Ibe. per
sq. in.

62.425

88

124.850

80

187.275

40

249.700

41

312.125

42

374.500

48

486.975

44

499.400

40

561.825

46

624.250

47

686.675

48

749.100

49

811.625

60

873.950

01

986.375

02

99SJBO0

08

1061.225

64

1123J650

05

1166i)76

56

1248.000

57

1310.925

58

1373.350

09

1435.775

60

1498.900

61

1560.626

62

1628.050

63

1685475

64

1747J0O

65

1810|25

66

1872.750

67

1935475

68

1997 JOO

69

2060105

70

2122J0O
2184i70

71
72

28O8J20

78

74

16.4733
16.9068
17.3403
17.7738
18.2073
18.6408
19.0743
19.0078
19.9413
20.8748
20.8088
21.24;8
21.6758
22.1089
22.5424
22.9759
23.4094
23.8429
24.2764
24.7099
25.1434
25.5769
26.0104
26.4439
26.8774
27.3109
27.7444
28.1780
28.6115
29.04.50
29.4785
29.9120
30.3455
30.7790
31.2125
31.6460
32.0795

Ibe. per
sq. ft.

2372.150
2434.575
2497.000
2559.425
2621.850
2684.275
2746.700
2809.125
2871.550
2933.970
2996.400
3058.82.5
3121.250
3183.675
3246.100
3308.525
3870.960
8438.378
3495.800
3558.225
3620.650
3683.075
3745.500
3807.925
3870.350
3932.77.')
3995.200
4057.625
4120.060
4182.475
4244.900
4307.825
4369.750
4432.175
4494.600
4557.025
4619.400

Head.

Feet.

Pressure.

76

76

77

78

79

80

81

82

83

84

80

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

Ibe. per
sq. in.

82.0130
32.9460
83.8800
83.8180
34.2471
84.6806
35.1141
35.5476
35.9811
86.4146
86.8481
37.2816
37.7151
38.1486
38.5821
39.0156
39.4491
39.8826
40.3162
40.7497
41.1832
41.6167
42.0502
42.4837
42.9172
43.3507
43.7842
44.2177
44.6512
45.0847
45.5182
45.9517
46.3852
46.8188
47.2523
47.6858
48.1193

lbs. per
sq. ft.

4681.870

4744.300

4806.720

4869.150

49.^1.575

4994.000

5056.4^

5118.850

5181.275

5243.700

5306.125

5368.550

5430.975

5493.400

5555.825

5618.250

5680.675

5743.100

5805.525

5867.950

5930.375

5992.800

6055.225

6117.650

6180.075

6242.500

6304.925

6367.350

6429.775

6492.200

6554.625

6617.050

6679.475

6741.900

6804.325

6866.750

6929.170

260

WEIGHTS AND MEASURES.

TAMIiE

OF PBESSVBES (€iMitinaed).

Pressure. |

Presirare.

Presiare.

Head.
Feet.

Hemd.

Feet.

Head.

Feet

lbs. per

lbs. per

lbs. per

lbs. per

lbs. per

lbs.psr
sq. ft

sq. in.

aq. ft.

sq. in.

sq.ft.

sq. in.

112

48.5528

6991.600

144

62.4260

8989.200

176

76.2972

10986.800

lis

48.9868

7054.025

145

62.8686

9051.626

177

76.7807

11049.226

114

49.4198

7116.450

146

63.2920

9114.060

178

77.1642

11111.660

115

49.8533

7178.875

147

63.7266

9176.476

179

77.5978

11174.076

116

50.2868

7241.300

148

64.1690

9238.900

180

78.0313

11236.600

117

50.7203

7303.725

149

64.5926

9301.825

181

78.4648

11298.926

118

51.1538

7366.160

160

65.0260

9363.750

182

78.8988

11861.360

119

51.5878

7428.576

161

65.4596

9426.176

183

79.3318

11423.776

.120

52.0208

7491.000

162

65.8931

9488.600

184

79.7658

11486.200

121

52.4543

7653.425

163

66.3266

9651.026

186

80.1988

11648.626

122

52.8879

7615.860

164

66.7601

9613.460

186

80.6828

11611.060

123

53.3214

7678.275

165

67.1936 -

9675.876

187

81.0668

11678.475

124

53.7549

7740.700

166

67.6271

9738.900

188

81.4998

11736.900

126

54.1884

7803.125

157

68.0606

9800.726

189

81.9328

11798.826

126

54.6219

7865.530

168

68.4941

9863.150

190

82.8668

11860.760

127

55.0554

7927.975

169

68.9276

9925.675

191

82.7998

11923.176

128

55.4889

7990.400

160

69.3611

9988.000

192

83.2338

11986.600

129

55.9224

8052.825

161

69.7946

10050.425

193

83.6669

12048.025

180

56.3569

8115.260

162

70.2281

10112.850

194

84.1004

12110.460

131

66.7894

8177.675

163

70.6616

10175.276

196

84.6889

12172.876

132

67.2229

8240.100

164

71.0951

10237.700

196

84.9674

12236JI00

183

57.6664

8302.625

165

71.5287

10300.125

197

85.4009

12297.726

184

58.0899

8364.950

166

71.9622

10362.550

198

85.8344

12860.150

183

58.5234

8427.876

167

72.3957

10424.975

199

86.2679

12422JJ75

186

58.9570

8489.800

168

72.8292

10487.400

200

86.7014

12485.000

187

59.3905

8552.226

169

73.2627

10549.825

201

87.1349

12647.426

188

59.8240

8614.650

170

73.6962

10612.250

202

87.6684

12609.860

139

60.2575

8677.075

171

74.1297

10674.675

203

88.0019

12672.276

140

60.6910

8739.500

172

74.5632

10737.100

204

88.4364

12734.700

141

61.1245

8801.925

173

74.9967

10799.525

205

88.8689

12797.126

142

61.5580

8864.350

174

76.4302

10861.950

206

89.3024

12869.650

143

61.9915

8926.775

176

75.8687

10924.375

207

89.7359

12921.976

Table sbowlnar the total pressure against a Tertleal plane

one foot wide, extending froip the surface of the water to tJie depth named in
the first column.

Water at its maximum density, 62.425 lbs per cubic foot =» 1 gram p«r cubic
centimeter, correBpondins to a temperature of 4° Cent. = 39.2° Fahr.

Total pressure in pounds = 31.2125 X square of depth in feet.

Depth.

Total
pressnre.

Depth.

Total
presrare.

Depth.

Totol
prewire.

Depth.

Total
pt-essare

Feet

Pounds.

Feet

Pounds.

Feet

Pounds.

Feet

Pounds.

1

31.21

17

9020

33

38990

49

74941

2

124.85

18

10113

34

36082

50

78081

3

280.9

19

11268

35

38235

51

8118«

4

499.4

20

12485

86

40461

62

84899

6

780.3

21

13765

37

42730

63

87676

6

1124

22

15107

38

45071

64

9101C

7

1529

23

16511

39

47474

65

94418

8

1998

24

17978

40

49940

60

112866

9

2528

25

19508

41

62468 .

65

181878

10

3121

26

21100

42

55069

70

162941

11

3777

27

22754

43

57712

76

176570

12

4495

28

24471

44

60427

80

199760

13

5275

29

26260

45

63205

86

225610

14

6118

30

28091

46

66046

90

2S2821

16

7023

31

29995

47

68948

96

28169S

16

7990

32

31962

48

71914

100

8121S8

WEIGHTS AND MEASUSES.

261

TABIiE OF 1»ISCHAB«1» Ilf CUBIC F£ET PCR SECOSTB

coBBESPonrBiire to eiysjir DiscuABOfis in v. s.

eAI.I.ONS P£R 24 HOVBS.

n. S. gallon

Discharge in cubic feet per second

231 cubic inches.

1.54723 X discharge in miUiwu of U. S. gal-
lons per 24 hours.

Millions

Millions

Millions

Millions

ofU. a

Cubic feet

of U.S.

Cubic feet

of U. 8.

Cubic feet

of U. S.

Cubic feet

gals, per

per second.

gals, per

per second.

gals, per

per second.

gals, per

per second.

24hrs.

24hrB.

24hr8.

24hr8.

.010

.0164728

18

20.1140

43

66.6808

72

111.400

.020

.0809446

14

21.6612

44

68.0781

73

112.948

.080

.0464169

16

28.2084

46

69.6258

74

114.496

.040

.0618891

16

24.75P7

46

71.1726

76

116.042

J080

.0778614

17

26.8029

47

72.7197

76

117.689

.060

.0928837

18

27.8601

48

74J»70

77

119.137

.070

.108806

19

29.3978

49

76.8142

78

120.684

.080

.128778

20

80.9446

60

77.8614

79

122.281

.000

.189261

21

82.4918

61

78.9087

80

123.778

.100

.164728

22

84.0390

52

80.4569

81

126.326

.200

.309446

28

36.6868

63

82.0081

82

126.873

.800

.464169

24

87.1886

64

83.6508

83

128.420

.400

.618891

26

38.6807

56

86.0976

84

129.967

.600

.778614

26

40.2279

66

86.6448

85

131.614

.600

.938887

27

41.7752

67

88.1920

86

133.062

.700

1.08806

28

43.8224

68

89.7398

87

134.609

.800

1.28778

29

44.8696

69 •

91.2866

88

136.156

.900

1.89261

80

46.4169

60

92.8337

89

137.703

1

1.64728

81

47.9641

61

94.3809

90

139.251

2

8.09446

82

49.6118

62

96.9282

91

140.798

3

4.64169

88

61.0586

63

97.4764

92

142.345

4

6.18891

84

62.6068

64

99.0226

93

143.892

5

7.78614

&9

64.1530

66

100.670

94

145.489

6

9.28887

86

e».7002

66

102.117

95

146.987

7

10.8806

87

67.2476

67

103.664

96

148.584

8

12.8778

38

68.7947

68

105.212

97

150.081

9

13.9261

89

60.8419

69

106.759

98

151.628

!•

16.4728

40

61.8891

70

108.306

99

153.176

n

17.0196

41

68.4364

71

109 J68

100

154.728

12

18.6667

42

64.9836

262

WEIGHTS AND MBASURBS*

TABIiE OF BISCHAROlMi IN CUBIC FEfiT PEB SBOOUD
CORRESPONDING TO OITEN BISCHABOES IN IM-
PERIAIi GAIiliONS PER 24 HOURS.

Imperial gallon «> 277.274 cubic inches.

Discharge in cubic feet per second = 1.85717 X discharge in Imperial gallons per

24 hours.

Millions

MilUons

Millions

Millions

of Imp.

Cubic feet

of Imp.

Cubic feet

of Imp.

Cubic feet

of Imp.

Cubic feet

gals, per

per second.

gals, per

per second.

gals, per

per second.

gals, per

per second.

24hrs.

24hr8.

24hr8.

24hr8.

.010

.0185717

13

24.1432

43

79.8583

72

133.7162

.020

.0871434

14

26.0004

44

81.7155
83.5727

73

135.5734

.030

.0557151

15

27.8576

45

74

187.4306

.040

.0742868

16

29.7147

46

86.429A

76

139.2878

.050

.0928585

17

31.5719

47

87.287^1

76

141.1449

.000

.111430

18

33.4291

48

89.1442

77

143.0021

.070

.130002

19

35.2862

49

91.0013

78

144.8593

.080

.148574

20

37.1434

50

92.8585

79

146.7164

.090

.167145

21

39.0006

51

94.7157

80

148.6736

.100

.185717

22

40.8577

52

96.5728

81

160.4308

.200

.371434

23

42.7149

53

98.4300

82

162.11879

.900

.557151

24

44.5721

54

100.2872

88

164.1451

.400

.742868

25

46.429$

55

102.1444

84

156.0028

.500

.928585

26

48.2864

56

104.0015

86

167.8595

.600

1.11430

27

50.1436

67

105.8587

86

169.7166

.700

1.30002

28

52.0008

58

107.7159

87

161.6738

.800

1.48574

29

53.8579

69

109.5730

88

168.4310

.900

1.67145

80

55.7151

60

111.4302

89

166.2881

1

1.85717

31

57.5728

61

113.2S74

90

167.1453

2

3.71434

32

59.4294

62

115.144$

91

169.0025

3

5.57151

33

61.2866

68

117.0017

92

170.8696

4

7.42868

34

63.1438

64

118.8589

98

172.7168

5

9.28585

35

65.0010

66

120.7160

94

174.6740

6

11.1430

36

66.8581

68

122.5732

96

176.4S12

7

13.0002

37

68.7153

67

124.4304

96

178.2883

8

14.8574

38

70.5725

68

126.287$

97

180.1465

9

16.7145

39

72.4296

69

128.1447

98

182.0027

10

18.5717

40

74.2868

70

130.0019

99

183.8698

11

20.4289

41

76.1440

71

131.8591

100

186.7170

12

22.2860

. 42

78.0011

WEIGHTS AND MEASURES.

263

TABIiE OF DISCHAB«ES IN OAIil^OMS PER 84 HOUIIA
COBKESPONDINO TO OITEST DISCHARGES IN CUBIC
FEET PER SECOND.

U. S. gallon = 231 cubic inches. Imperial gallon = 277.274 cubic inchea-
Diaoharge in U. S. gallons per 24 hours = 646317 X discharge in cubic feet

per second.
Discharge in Imperial gallons per 24 hours » 538454 X discharge in cubic fe«i

per second.

Onb. ft.

Millions of

Millions of

Cub. ft.

Millions of

Millions of

U. S. gHllons

Imperial gallons

per sec.

U. S. gallons

Imperial gallons

per 24 hours.

per 24 hoars.

per 24 hours.

per 24 hours.

1

0.646317

0.538454

53

34.254795

,28.5880U

2

1.292634

1.0769O7

54

34.901112

29.076488

8

1.938951

1.615361

55

85.547428

29.614951

4

2.685268

2.158815

56

36.193745

30.153405

i

3.281584

2.692266

CT

36.840062

30.691859

6

3.877901

3.230722

58

37.486379

81.230312

7

4.524218

8.769176

59

38.132696

31.768766

8

5.170535

4.307629

60

38.779013

32.307220

9

5.816852

4.846088

61

39.425330

32.845678

10

6.463169

5.384537

62

40.071647

33.384127

11

7.109486

5.922990

68

40.717963

33.922581

12

7.755808

6.461444

64

41.364280

34.461034

18

8.402119

6.999898

65

42.010597

34.999488

14

9.0484S6

7.538351

66

42.656914

85.537942

15

9.694753

8.076805

67

43.303231

36.076395

16

10.341070

8.615259

68

43.949548

36.614849

17

10.987387

9.153712

69

44.595865

37.153303

IB

11.633704

9.692166

70

45.242182

37.691756

19

12.280021.

10.230620

71

45.888498

38.230210

20

12.926338

10.769073

72

46.534815

38.768664

21

13.572654

11.307527

78

47.181132

89.307117

22

14.218971

11.845981

74

47.827449

39.845571

28

14.865288

12.384434

75

48.473766

40.384025

24

15.511605

12.922888

76

49.120083

40.922478

28

16.157922

13.461342

77

49.766400

41.460932

.28

16.804289

13.999795

78

50.412717

41.999385

27

17.450556

14.538249

79

51.059034

42.537838

28

18.0968(73

15.076702

80

51.705350

43.076293

.29

18.743190

15.615156

81

52.351667

43.614746

80

19.889506

16.158610

82

52.997984

44.153200

81

20.085828

16.692063

83

63.644301

44 691654

82

20.682140

17.280517

84

54.290618

45.230107

88

21.328457

17.768971

85

54.936935

45.768561

84

21.974774

18.307424

86

55.583252

46.307015

85

22.621091

18.845878

87

56.229569

46.845468

86

23.267408

19.384332

88

66.875885

47..'W3922

87

23.913725

19.922785

89

67.522202

47.922376

88

24.560041

20.461239

90

58.168519

48.460829

89

25.206a'W

20.999693

91

68.814836

48.999283

40

25.852675

21.588146

92

59.461153

49.537737

41

26.498992

22.076600

93

60.107470

50.076190

42

27.145309

22.615054

94

60.753787

50.614644

48

27.791626

23.158507

95

61.400104

51.153098

44

28.487943

23.691961

96

62.046420

51.691561

46

29.084260

24.280415

97

62.692737

52.230006

46

29.730576

24.768868

98

63.389054

52.768459

47

30.376893

25.307322

99

63.985371

53.306912

48

81.028210

25.845776

100

64.631688

53.845366

49

81.669627

26.384229

101

65.278005

54.383820

50

32.315844

26.922683

102

65.924322

54.922273

61

32.962161

27.461187

103

66.570639

55.4WJ727

62

83.608476

27.999590

104

67.216956

55.999181

264

WEIGHTS AKD MEASURES.

TABI4E OF BISCHABOES (Continned).

Cub ft.

Millions of

Minions of

Cub. ft.

MilUonsof

Millions of

per sec.

U. S. galloDB

Imperial gallons

per sec.

U. S. gallons

Imperial gallons

per 24 hours.

per 24 hours.

per 24 hours.

per 24 hours.

i05

67.863272

66.687684

167

107.934919

89.921761

106

68.509589

67.076088

168

108.581236

90.460215

107

69.155906

57.614542

169

109.227553

90.998669

208

69.802223

58.162995

170

109.873870

91.537122

109

70.448540

68.691449

171

110.520186

92.075576

110

71.094867

69.229903

172

111.166503

92.614030

111

71.741174

69.768356

173

111.812820

93.152488

112

72.387491

60.306810

174

112.459137

93.690937

118

73.033807

60.845264

175

113.105454

94.229891

lU

73.680124

61.383717

176

113.761771

94.767844

115

74.326441

61.922171

177

114.898088

95.806298

116

74.972768

62.460625

178

115.044406

96.844761

117

75.619075

62.999078

179

115.690722

96.388206

118

76.265392

63.537532

180

116.337038

96.921669

119

76.911709

64.075986

181

116.983355

97.460112

120

77i»8026

64.614439

182

117.629672

97.998666

121

78.204342

66.152893

188

118.275989

98.537020

122

78.850659

66.691347

184

118.922306

99.075478

123

79.496976

66.229800

185

119.568623

99.618927

124

80.143293

66.768254

186

120.214940

100.152881

125

80.789610

67.306708

187

120.861257

100.690684

126

81.435927

67.845161

188

121.507578

101.229288

127

82.082244

68.383615

189

122.153890

101.767742

128

82.728561

68.922068

190

122.800207

102.806196

129

83.874878

69.460522

191

123.446524

102.84464»

180

84.021194

69.998976

192

124.092841

108.388108

131

84.667511

70.537429

193

124.739158

103.921666

132

85.313828

71.075883

194

125.»85475

104.460010

183

85.960145

71.614337

195

126.081792

106.098464

184

86.606462

72.162790

196

126.678108

106336917

186

87.262779

72.691244

197

127.324425

106.076S71

186

87.899096

73.229698

198

127.970742

106.618825

187

88.545413

73.768151

199

128.617059

107.162278

188

89.191729

74.306605

200

129.268376

107.690782

189

89338046

74.845059

201

129.909698

106.229186

140

90.484363

76.383612

202

130.566010

108.767689

141

91.130680

76.921966

203

131.202327

109.306098

142

91.776997

76.460420

204

131.848644 •

109344647

148

92.423314

76.998873

205

, 132.494960

110.388000

144

93.069631

77.637327

206

133.141277

110.921464

146

93.715948

78.075781

207

133.787594

111.45990S

146

94.362264

78.614234

208

134.433911

111.998861

147

95.008581

79.152688

209

135.080228

112.536815

148

96.664898

79.691142

210

135.726545

113.075269

149

96.301215

80.229596

211

136.872862

118.618722

150

96.947532

80.768049

212

137.019179

114.152176

151

97.593849

81.306503

213

137.665495

114.690680

152'

98.240166

81.844956

214

138.311812

116.229088

163

98.886483

82.383410

215

138.958129

116.767887

154

99.532800

82.921864

216

139.604446

116.806891

155

100.179116

83.460317

217

140.25U768

116344444

156

100.825433

83.998771

218

140.897080

117.882898

157

101.471750

84.537225

219

141.643397

117321882

158

102.118067

85.075678

220

142.189714

118.468806

159

102.764384

85.614132

221

142.836030

118.998SB»

160

108.410701

86.152586

222

143.482347

119386n8

161

104.057018

86.691039

228

144.128664

120.07616ft

162

104.703335

87.229498

224

144.774981

120.618620

163

105.349651

87.767947

225

146.421298

121.163074

164

ia5.995968

88.306400

226

146.067615

121.600B87

165

106.642285

88.844854

227

146.713982

122.228881

166

107.288602

89.883308

228

147.860249

122.78704

TIME.

265

TABIiE OF I»lS€HAReES (Contlnae^i).

Oab. ft

MillioDs of

Millions uf

Oub tt

MilUons of

Millious of

per sec.

U. S. gallons

Imperial gallons

per sec.

U. S. gallons

Imperial gallou

per 24 hours.

per 24 hours.

per 24 hours.

per 24 hours.

229

148.006566

123.306888

240

155.116061

129.228878

230

148.652882

123.844342

241

155.762368

129.767332

231

149.299199

124.382795

242

156.408685

180.305786

232

149.945516

124.921249

243

157.065002

130.844239

233

150.591833

125.459703

244

167.701819

131.382693

234

151.238150

125.998156

245

158.847636

131.921147

235

151.884467

126.536610

246

158.993962

132.459600

236

152.680784

127.075064

247

169.640269

132.998054

237

163.177101

127.613517

248

160.286586

133.636608

238

163.828417

128.151971

249

160.932903

134.074961

239

164.469734

128.690426

250

161.579220

134.613416

TIME.

60 seconds,*! marked s, =■

60 minutes,! *' m, =

24 hours, " h, =

7 days, " d, =

Arc Time
1° = 4 minutes
r s= 4 seconds
V = 0.066... second

1 minute
1 hour =
1 day =
1 week =

3600 seconds

1440 minutes = 86400 seconds

168 hours = 10U80 minutes

Time Arc

24 hours =360°
Ihour = 15°
1 minute = 0° 15'
1 second =* 0° 0' 15"

Bletbods of reekonins time. Astronomers distinguish between mean
solar time, true or api)arent solar time, and sidereal time.

At a standard meridian (see page 267) mean solar time is the same at
ordinary clock time. At any point not on a standard meridian, standard time
is the local mean solar time of the meridian adopted as standard for such point ;
and local time is = time at a standard meridian phu correction for longitude
from that meridian if the place is east of the meridian, and vice versa. For the
amount of such correction, see second table above. A true or apparent
aolnr day is the interval of time between two successive culminations of
the sun, «.«., between two successive transits or passages of the sun across the
meridian of the same point ou the earth ; but, since these intervals are unequal,
they do not correspond with the uniform movement of clock time. A fictitious
or imaginary sun, called the "mean sun," is therefore supposed to move along
the equator in such a way that the interval between its culminations is con-
stant. This interval is called a day, or mean solar day, and is the average of the
lengths of all the apparent solar days in a vear. Apparent and mean time
agree at four points in the year, viz., about the middle of April and of June,
September 1 and December 24. The sun is sometimes behind and sometimes
in advance of the mean sun, and is called " slow " or " fast " accordingly. The
sun is " slow " in winter, the maximum being about February 11, when it passes
any standard meridian, or "souths" (making of^Mrent noon), about 14m, 28s,
after noon by a correct clock. The sun is " fast," or in advance of the clock, in
MJty and in the £all, with a maximum, about l^ovember 2, of about 16m, 20s.

The difference between apparent and mean time is called the equation of
time. It can be obtained from the Nautical Almanac, or, approximately, by
taking the mean between the times of sunrise and sunset, as given in ordinary
almanacs.

As solar time is measured by the apparent daily motion of the sun, so sidereai
time is measured by that of the fixed stars, or, more strictly speaking, by the
motion of the vernal equinox which is the point where the sun crosses the
equator in the spring.

* The second was formerly divided into 60 equal parts called thirds (marked
'") ; but it is now divided decimally.

f The old and confusing practice of designating minutes, seconds and thirds
of time (see footnote *) as % " and ''', is no longer in vogue. Days, hours, min-
utes and seconds are now designated by d, h, m, and s, respectively, thus : 2d,
20h, 48ni, 65.43 s.j and the symbols ' and " designate minutes and seconds of are.

266 <TIMB.

A sidereal dAy" is the interval of time between two tueeeisiye paaaages of
the vernal equinox (or. practically, of auy star) past the meridian of a ^ven
point on the earth. It is, practically, the time required for one complete revo-
lution of the earth on its axi£, relatively to the stars.

The length of the sideral day is 23 h, 56 m, 4.U9 s, of mean solar time, or S m,
56.91 A of mean solar time less than the mean solar day of 24 hours. In other
words, a star will, on any night, appear to set 3 m, 55.91 s earlier by a correct
clock than it did on the preceding night. Hence, substantially, the number of
sidereal days in a year is greater by 1 than the number of solar days.

The sidereal day, like the solar day, is divided into 24 hours. These hours
are. of oourse, shorter than those of tne solar day in the same proportion as the
sidereal day is shorter than the solar day. They are counted from 0 to 24, com-
mencing with sidereal itoon, or the instant when the vernal equinox passes the
ujmer meridian.

Tlie etwil day (» 24 hours of clock or mean solar time) commences at mid-
night ; and the astronomical solar day at noon on the civil day of the
same date. Thus, on a standard meridian, Thursday, May 9, 2 a. m . civil time,
is Wednesdav, May 8, 14 h, astronomical time; but Thursday, May 9, 2 p. M.,
eivil time, is Thursday, May 9, 2 h, astronomical time.

Tbe cItII month is the ordinary and arbitrary month of the calendar,
varying in length from 28 to 31 mean solar days.

A sidereal montb is the time required for the moon to perform an entire
revolution with reference to the stars. Its mean length, in mean solar time, is
about 27 d, 7 h, 43 m, 12 s.

A lunation, or synodic month is the time from new moon to new
moon. Its mean length is about 29 d, 12 h, 44 m, 8 s.

The tropical or natural year is the time during which the earth
describes the circuit from either equinox to the same again. Its mean length,
in mean solar time, is now about 365 d, 5 h, 48 m, 49 s.

The sidereal year is the time during which the earth describes its orbit
with reference to the stars. Its mean length, in mean solar time, is about 365
d, 6 h, 9 m, 10 s.

The elwll year is that arbitrary or conventional and variable division of
time comprised between the 1st of January and tbe 31st of the following Decem-
ber, both inclusive. It contains ordinarily 365 mean solar days of 24 hours, bat
each yenr whose number is divisible by 4 contains 366 days, and is called a leap
year, except that those years whose numbers end in 00 and are not multipMB
of 400 are not leap years.

To regulate a watch hy the stars. The author, after having rega-
' lated his chronometer for a year by this method onlv,diffiereid but a few seconds
from the actual time as deduced from careful solar observations. Select a
window, facing west if possible, and commanding a view of a roof-crest or oth^
fixed horizontal line, preferably about 40^ above the horizon, in order to avoid
disturbance due to refraction, and distant say 50 feet or more. Note the
time when any bright fixed star (not a planet) passes the range formed between
the roof, etc., and any fixed horizontal line about the window frame, as a pin
fixed in <>it her Jamb. The sight in the window, and the watch, must be illumi-
nated. The star will pass the range 3 m. 55.91 s. earlier on each suooeeding
evening. Those stars which are nearest the equator appear to move the fastest,
and are therefore best suited to the purpose. If the first observation of a given
star lie made as late as midnight, that saron star will answer for about three
months, until at last it will begin to pass the range in daylight. Before this
happens, transfer the time to another star which sets later. By thus tabidating,
throughout the year, about half a dozen stars which follow each other at
nearly equal intervals of time, we may provide a standard by means of which
correct clock time may be ascertained on any clear night. Experinfenting in
this way with two of the best chronometers, the author found that tWr
rates varied, at times, as much as from three to eight seconds per day.

An average man takes two steps (one right, one left) per Bcca»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-
psssd to take place, lies half>way between them. But it would, of course, not be
practiesble for the railroads to use an imaginary line in passing from one time
standard to another. The changes are made at prominent stations forming the ter-
mini of two or more lines; or, as in the case of the long Pacific roads, at the ends
(tf divisions. As far as practicable, points at which changes uf time had previously
basn made, were selected as the changing points under the new system. Detroit,
Wch., Pittoburgh, Pa., Wheeling and Parkersburg, W. Va., and Augusta, Ga., al-
though not situated upon the same meridian, are points of change between «a$tem
and central standard times. A train arriving at Pittsburgh from the east at noon,
and leaving for ths west 10 minutes after its arrival, leaves (by the figures shown
npon its time-table, and by the watches of its train hands) not at 10 minutes afker
ISjbat at 10 minntss alter 11.

The necessity for making the changes of time at principal points, instead of on a
true meridian line, necessitates also some "overlapping** of the times, or of their
eolors on the map. Thus, most of the roads between Buffalo and Detroit, on the
north side of Lake Brie, run Irf ** eastern," or **red,** time; while those on the $ouih
side of the Ijske, between Buffalo and Toledo, immediately opposite to and directly
south of them, run by ** central ** or " blue ** time.

If the chauMs of time were made at ths meridians midway between the standard
ones, it woula not be necessary for any town to change its time more than 30 min-
utes. As it is, somewhat greater changes had to be made at a few points. Thus,
standard time at Detroit is 32 minutes ahead, and at Savannah 86 minutes back, of
mean locaf time.

In most cases the necessary change was made upon the railways by simply setting
docks and watches ahead or back the necessary number of minutes, and without
making any change in time-tables.

Raliux, and a few adjacent cities, use the time of the 60th meridian, that being
the nearest one to them ; but the railroadM in the same district have adopted the
T6th meridian, or eastern, time; so that, for railroad purposes, intercolonial time
has never come into force.

In 1878 there were 71 time standards in use on the railroads of the United States

and Canada. At the time of the adoption of the present system this number had

■been reduced, by consolidation of roads, Ac, to hS, By its adoption, the number be-

tame 5, or, practicslly, 4, owing to the adoption of eastern time by the intercolonial

roads; as aJrcHsdy explained.

268

DIAIA

DIALLING.

To malKe a borlxontal San^dlal,

Draw a line a h ; and at right angles to it, draw 66. From any convenient point, bb c,
in a fr, draw the perp c o. Make the angle cao equal to the lat of the place ; aJfo
the angle e o « equal to the same ; Join o e. Bfake e n equal to o e; and from n as a
center, with the rad e n, describe a quadraat e «; and div it into 6 equal parts. Draw c
y, parallel to 6, 6; and
firom n, through the 5 ^ DIAL ^

points on the quadrant, ^

draw lines n t^n t, ^c,
terminating in ey. From
a draw lines a 6, a 4, Ac,
passing through t, i, Ac.
From any convenient
point, as c, describe an
arc r nt A, as a kind of fin-
ish or border to half the
dial. All the lines may
now be effaced, except
the hour lines a 6, a 6,
a 4, Ac, to a 12, or a A;
unless*, as is generally
the case, the dial is to
be divided to quarters
of an hour at least. In
this case each of the
divisions on the quad-
rant « «, must be subdivided into 4 equal parts; and lines drawn from n, thioaf^
the points of subdivision, terminating in ty. The quarter-hour lines must be drawn
from a, as were the hour Unes. Subdivisions of 6 min may be made in the same
way ; but these, as well as single min, may usually be laid off around the border, by
eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi-
nary dial. To draw the other half of the Fig, make a d equal to the intended thick-
ness of the gnomon, or style, of the dial ; and draw d 12, parallel, and equal to a 12 ; and
draw the arc x^ to, precisely similar to the arc rmh. Between x and to, on the arc ng «0,
space off divisions equal to those on the arc r7nh\ and number them for the hoan,
as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be
triangular; its thickness must throughout be equal to a<2 or &«o; its base murt
cover the space adhv)\ its point will be at ad; and its perp height Av, over A.«^
must be such that lines vd^uii, drawn from its top, down to a and d, will make the
angles u a A, « d io, each equal to the lat of the place. Its thickness, if of metal, may
conveniently be fh>m ^ to ^ inch ; or if of stone, an inch or two, or more, aooording
to the siie of the dial. Usually, for neatness of appearance, the back A u « to of the
style is hollowed inward. The opper edges, ua, v d, which cast the shadows, moat
be sharp and straight. The dial must be fixed in place hor, or perfectly level ; ah
and dw must be placed truly north and south ; ad being south, and A«o north. Th»
dial givee only sun or solar time ; but clock time can be found by means of the ** fiurt'
or slow of the sun," as given by all almanacs. If by the almanac the tun is 6 miB.
Ac, fast, the dial will be the same ; and the clock or watoh, to be correct, must be f
Bin slower than it ; and vice versa.

To make a Vertical Snn-Dlal.

Proceed as directed above, except that the angles eao and eo« on the drawing,
and the angle t«a A or v dir of the style, must lie equal to the oo-latitnde (» dif-
ference between the latitude and 90^) of the place, and the hours must be num-
bered the opposite wav from those in the above flgare ; i e, from A to y number
12, 11, 10, 9, 8, 7 ; and from to tog number 12. 1, 2, 8, 4, 6. The dial plate muat be
placed vertically, in the position shown in the figure, (kcing ezacuy south, and
with a A and dw vertical.

BOABD HBABCBE.

BOABD HEASTTBE.

■ fMlowlnv t»Me. Tha u.

BOARD HEASURB.
niMe at Bo»r« Mcaanre— (ConUaud.)

£i

--

„..

"Kr-tf

d.M Id a

"loriilil.)

f

35

P

IM

THIOUt

MM SS

IKOIUS.

P

i

i

1

1

J

.«™

if

1
1

1

,i

S
1

1
1

^

.HSU

i
i

1

1
i

1

t

I

1

■a
1

1

1

s

i

1
1

i

i

nuK.

1

1

1

)
J

3

1

I."

„

1

1

BOASO HEABURB.

T»M« of Btmrd Heu

DTC—

(OonUn

Md.)

1

a|-

sx

M(

TK

OKK>

Ba DT

nraa

■S.

'H

IH

1^

5
i
i

i

i

i
s
J

s

It"

i!

i

1

1

i

is

1

i

if

i

1

1

if

IS

sir
;.s

1

1

i

is
Is

is

!:|

i

1

1
J
1

i

•-

i

BOARD UEASUBE.
Table of Board Mcaaar* — (Contlnutd.}

BOARD MEASURE.

273

Table of Board Biewiare~(Continued.)

si

H
H
1.

H

2.
H
H
H

8.

H
H
H
4.

H

6.

7.

H

H
8.

H
H
H

9.

H

10.

H

IS.

IS.

H
u.

16.
It.

17.

18.
10.

ao.

21.
».
IS.

M.

of Board Meuure oontaiBOd In on« raaning tfnA of Softotlinga
of dilftrent dimenaiODS. < Original.)

THIOKNEStt IK ZKOHIBS.

10

lOii

io«

lOH

Ft Rd.M.

FtBd.M.

ptBa.M:

FtBd.lC

.1083

.2136

.2186

JfM

.4167

.4271

.4375

.4479

.6250

.6406

.6363

.671*

.8333

.8642

.8750

.8956

1.042

1.068

1.094

1.120

1.250

1.821

1.318

1.344

1.458

1.495

1.631

1.568

1.667

1.708

1.750

1.793

1.875

1.922

1.969

2.016

2.0A3

2.135

2.188

2.240

2.292

2.349

2.406

2.464

2.500

2.563

2.625

2.688

2.708

2.776

2.844

2.911

2.917

2.990

3.063

3.135

3.126

3.208

3.281

3.359

.S.333

3.417

3.600

3.583

3.542

3.630

8.719

3.807

3.750

3.844

3.938

4.031

3.958

4.057

4.m

4.255

4.167

4.271

4.479

4.375

4.484

4.594

4.703

4.5a3

4.608

4.813

4.927

4.792

4.911

5.061

6.161

5.000

5.126

5.250

6.376

5.208

5.339

5.469

6.599

5.417

5.562

6.688

6.898

5.625

6.766

6.906

6.047

5.833

5.979

6.126

6.271

&04S

6.193

•M4

6.486

«.2S0

<.406

•.56B

6.719

6.468

6690

«TB1

6.943

6.667

6.833

T.00O

7.167

6.875

tMl

T.219

7.391

7.083

T.960

7.438

7.616

7.292

1JIT4,

7.056

T.838

7.500

7J88

7.676

8.068

7.708

7.901

8.004

8.286

7.917

8.115

8.313

8.510

8.125

8.828

8.631

8.734

8.3S3

8.542

8.760

8.968

8.643

8.766

8.960

9.182

8.760

8.969

9.188

9.406

8.958

9.182

9.406

9.630

9.167

9.396

9.626

9.864

9.876

9w699

9.844

10 08

9.583

9.823

10.06

10.30

9.792

10.04

10.28

10.63

10.00

10.26

10.50

10.75

10.42

10.68

10.94

11.20

10.83

11.10

11.88

11.66

11.26

11.68

11.81

12.09

11.67

11.96

12.26

12.64

12.06

12.39

12.69

12.99

12.50

12.81

13.13

13.44

12.92

1324

18.66

13.89

18.83

13.67

14.00

14.33

13.76

14.09

14.44

14.78

14.17
14.58

14.62

14.88

15.23

14.95

15.81

15.77

16.00

15.88

16.76

16.13

15.88

16.23

16.63

17.02

16.67

17.08

17.60

17.92

17.60

17.94

18.38

18.81

18.33

18.79

19.26

19.71

19.17

19.06

90.13

90.60

WJOO

99.60

21.00

21.60

1

11
rtBd-M.

.9893
.4683
.6875
.9167
1.146
1.376
1.604
1.833
3.063
2.292
2.621
2.750
2.979
8.308
.3.438
3.667
8.896
4.136
4.354
4.583
4.813
5.042
5.271
5.500
5.729
6J68
6.188
6.417
6.646
6.876
7.104
7.333
7.563
7.792
8.021
8.250
8.479
8.709
8.939
9.167
9.396
9.626
9.854

10.06

10.81

10.64

10.77

11.00

11.46

11.92

12.38

12.83

13.29

13.76

14.21

14.67

16.13

16.68

16.04

16.50

17.42

18.33

19.25

20.17

21.08

22.00

llji

rt.Bi.lL

.23U

.4688
7031

.9376
1.172
1.406
1.641
1.875
2.109
2.344
2.578
2.813
3.017
3.281
3.516
8.730
3.984
4.219
4.453
4.688
4.922
5.156
6.391
5.625
5.869
6.094
6.328
6.363
6.797
7.081
7.366
7.500
7.734
7.969
8.303
8.438
8.672
S.906
9.141
9.376
9.600

]0j08
10.31
10.66
10.78
11.02
11.26
11.72
12.19
12.66
13.13
13.59
14.06
14.63
15.00
16.47
13.94
16.41
16.88
17.81
18.75
19.69
20.63
21.56
32.50

UH

FtBd.lC

.9306
.4792
.7188
.9688
1.198
1.438
1.677
1.917
2.156
2.396
2.636
2.876
3.113
3354
.3.594
3.833
4.073
4.313
4.552
4.791
6.031
5.270
6.510
6.750
5.990
6.229
6..469
6.708
6.948
7.188
7.427
7.667
7.906
8.146
8.386
8.625

IIH

9.104
9.3a
9.583
9.823
10.06
10.30
10.54
10.78
11.02
11.36
11.60
11.98
12.46
12.94
13.42
13.90
14.38
14.85
15.33
13.81
16.29
18.77
17.26
18.21
19.17
20.13
21.08
22.04
23.00

FtBd.M.

.2448
.4896
.7344

1.224
1.469
1.714
1.958
2.203
2.448
2.693
4.938
3.182
3.427
3.67S
8^17
4.161
4.406
4.651
4.896
5.141
5.385

,5.680
5.875
6.120
6.366
6.609
6.854
7.090
7.344
7.589
7.833
8.078
8-32B
8.566
8.813
9.057
9.302
9.547
9.793

10.04

10.28

10.53

10.77

11.02

11.36

11.61

11.75

12.24

12.73

13.22

13.71

14.20

14.69

15.18

15.67

16.16

16.65

17.14

17.63

1840

10.58

20.56

21.54

32.52

23.60

12

FCBd.M.

.8600
.5000
.7500
1.000
1.250
1.500
1.730
2.000
2.250
2.600
2.750
8.000
8.250
3.600
8.750
4.000
4.250
4.500
4.730
5.000
6.250
5.500
6.750
6.000
6.250
6.500
6.750
7.000
7.250
7.500
7.750
8.000
8.250
8.500
8.750
9.000
9.250
9.500
9.750

10.00

10.26

10.50

10.76

11.00

11.25

11.50

11.75

12.00

12.50

13.00

13.50

14.00

14.50

15.00

15.50

16.00

16.50

17.00

17.50

18.00

19.00

20.00

21.00

22.00

23.00

94.00

*>2
♦"9

$

1.

H
H

2.

If

H

3.

14
H
H

4.

H
H
H

3.

H
H

6.

H

H

t

• .

H
H
H

8.

H

9.

H
11.

12.

H

13.

H

14.

>i
15.

H
16.

H

17.

H

18.
19.
20.
21.
32
IS.
24

18

274

IiAITD SUKYBZIHa.

LAND SURVEYING.

In surveyliie • tnet of gimiml, the sites which eoMpose its outline are deri»
nated by nuraben in the order in which they ocoor. Thst end of each side which
first presents itself in theooarseof the surrey, may be called its near end ; and the
other its /or end. The oamber of each side is plaoed at its far end. Thus, in Figr. 1,
the sarTey being supposed to comroeDce at the corner 6, and to follow the direc-
tion of the arrows, toe irst side is <>, 1 : and its number is placed at its far end at 1 ;
and so of the rest. Let NS be a meridian line, that is, a north and south line;
and EW an east and west line. Than in any side which runs northwaidly;

Flff.1.

whether northeast, as side 2; or north westL as sides 8 and 1; or doe north; the
distance in a due north direction between its near end and its far end, is called
its lunihing; thus, a 1 is the northing of side 1; Ibthe northing of side 2 ; 4e
of idde 5. In like manner, if any side runs in a southwardly direction, whether
southeastwardly, as side 8; or south westwardly, as sides 4 and 6; or due south ;
the corresponding distance in a due south direction between its near end and its
far end, is called its southing; thus, d3 is the southing of side 8; 80 of side 4;
/6 of side 6. Both northinss and southings are included in the general term
jD^erence of Latitude of a side ; or, more commonly but erroneously, its kUiiude,
The distance due east, or due west, between the near and the far end of any side,
is in like manner called the Muting^ or westing^ of that side, as the case nuy be;
thus, 6 a is the westing of side 1; 6/ of side 6; e6 of side 5; e4 of side 4; and
6 2 is the easting of side 2 ; 2 d of side 8. Both eastings and westing are included
in the general term Dqaarture of a side; implying that the side d^xxrU so far
from a north or south direction. We may say that a side norths, wests, sontheasta^
Ac. We shall call the northings, southings, Ac. the Ks, Ss, £b, and Ws ; the lati-
tudes, lats; and the departures, d^.

Perfect accuracy is unattainable in any operation inyolyinff the measur»^
meuts of angles and distances.* That work is accurate enough, which cannot
be made more so without an expenditure more than commensurate with the
object to be gained. There is no great difficulty in confining the uncertainty
within about one-half per cent, of the content, and this probably never pre-
▼ents a transfer in farm transactions. But errors always become apparent when
we come to work out the field notes; and since the map or plot of the surrby, and
the calculations for ascertaining the content, should be consistent within them-
selres, we do what is usually called eorreding the errors, but what in fact is simply
humoring them, in, no matter how scientific the nrocess may appear. We distrib-
ute them all around the survey. Two methods are used for this purpose, both
based upon precisely the same principle * one by means of drawing; the other,
more exact but much more trouolesome. by calculation. The graphic method, in
the hands of a correct draftsman, is sufficiently exact for all ordinarv purposes.
Add all the sides in feet together; and divide the sum by their number, for the
average length. IMvide this average by 8 ; the quotient will be the proper scale
in feet per inch. In other words, take about 8 ins. to represent an average side.
We shall take it for granted that an engineer does not consider it accurate work to

• A 100 ft. ehalii may Tary Its length 5 feet per mile, between winter and sammer. bj m«rc
ehange of temperature; and this alone will make a differenoe of about 1 acre in 6X1. The turn-
dent aboald praetiao ploitlng from perfeetUr accurate dau : aa tnoL tSa ejuunpto la table. ^ 181, ot

LAXD BUBYETINQ. 275

■Mwatv hto MiglM t9 the nearwi qoarter of a degree, wtaieh 1« tbe atnal prMtiM amonf land'torrey
tn. Tbey OMi, Df idmbi of tbe engineer's tmntlt, now in aniTonal ose on our pobllo works, be readfq^
■eMMOd within a minute or two ; and being thus nocb more accurate than the oompass oonrsee,
(wtaiob eanoot be read off so eloselr, and which are moreover subject to many lonroes of error,) th«f
serve to correct the Utter in the oflloe. The noting of the coarses, however, should not be confined t«
the nearest quarters of a degree, btit should be read as closely as tbe observer oan guess at the minutes.
The back courses also should be taken at every comer, as an additionid cheek, and for tbe deteetioa
ef local attraction. It la
well in taking the oom-
pass bearings, to adopt
as a rule, always to point
the north of tbe compass*
box toward tbe ohJeet
whose bearing is to be
taken, and to read off
from tbe north end of the
needle. A person who
uses indUEerentiy th» M
and tbe S of the box, and
of the needle, will be very
liable to make mistakee.
n ie beet to measure the
least angle (shown by
dotted arcs, Tig 2.) at the
sther it

; whether it be
exterior, ae that at oomer
ft; or interior, as all the
others; because it is al-
ways less than 180° ; so ^ , . •,. ^
that there is less danger >; .' Fig. 8.
ef reading it off ineor- '"
reetly, than if Itezeeeded
180P; tiUdBf It for grant.

ed that the transit InstmnMDt Is graduated fhnn the same lero to 180° each way ; If it is gradnatai
fkvm sevo to 180° tfte preeaatlon is useless. When the small angle is exterior, subtract it from SIMP
for the interior one.

Snppoelng the fleld work to be finished, and that we require a plot from which the oontenta may
be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be
measnred br scale, and thtir areas calculated one by one,} a protraction of it may be made at once
from tbe field notes, either by uslQg tbe angles, or by first oorrtictiag the bearinga by means of the
angles, and then nsing them. The last is tbe best, because in the first tbe protractor must be moved
to each angle ; whereas In tbe last it will remain sUtionary while all the bearings are being pricked
off. Kverj movement of it Inoreasea the liability to errors. The manner of oorreotlng the bewrings
Is explained on tbe next page.

In either case the protracted plot will oertainly not eloee precisely ; not only in oonsequence of errors in
tbe field work, but also in the protracting itself. Thus the last side. No 6, Fig S, Instead of closing in at
eomer 6, will end somewhere else, say, for instance, at (; the diet 1 6 being the etoting orror, which,
however, as represented in Pig 3, is more than ten times as great, proportionally to the siie of the
snrrey. as would be allowable in praetice. Now to hnmor-ln this error, rule through every oomer
a short line parallel to ( d; and. in all eases, in the direetion from t (wherever it mav be) to tbm
Btartlag point 6. Add all the sidoB together ; and measure ( fi by the scale of the plot. Then befl)i>
BiBg at oomer 1, at the fsr end of side 1, say« as the

Sum of aU . Total dosing . • oiii^ i • Error

the sides • error «d •• ""^* • Ibrsidel.

Lay eff tbia error fh>m 1 to a. Then at comer 3, say, as the

Sum of all . Total olosing . , Sum of • Error

the sldea • error 16 • • sides 1 and S • for side 1

Which error lay off from 2 to 6 ; and so at each of the comers; always using, a« the third term, the
sum of Uie sides between the starting point and the ^ven ooAier. Finally, Join the points a, b, e,
li, e, 6 ; and the plot la finished.

The oerreotiec has evidently changed the length of every side ; lengthening some and shortening
others. U has also changed the angles. Tbe new lengths and angles may with tolerable accuracy
be fonnd by means of the scale and protractor ; and be marked on the plot Instead of the old ones.

tnm those to be fbond in books on survering. This Is the only way In which be oan learn what la
Mt by aecorate work. His semlolrealar protractor should be about 9 to 12 Ins in diam. and gradn-
I to 10 min. His straight edge and triangle should be of metal: we prefer (vorman silver, which
I not rast as steel does ; and they should be made with teniptUou* aeeuraey by a skilfUl lustra-

jt-naker. A very fine needle, with a sealing-wax beiul, should be used for pricking off disU and

aaglcs; it mnst be held vertically ; and the eye of tbe draftsman most be directly over it. The lead
peaeU should be hard (Paber's No. 4 is good for protracting), and must be kept to a sharp point by
rabMiv on a fine file, after nsing a knife for removing the wood. Tbe scale should be at least as long
aa the longest side of tbe plot, and should be made at the edge of a strip of tbe same paper as the plot
Is drawn on. This will obviate to a considerable extent, errors arising from contracUon and expao-
ilea. Unfortunately, a sheet of paper does not contract and expand in the same proportion length*
•Iss and eroaswlae, thus preventing the paper scale n-om being a perfect corrective. In plots of com-
1MB farm survi^s, iko, however, the errors rh>m this source may be neglected. For such plott as mav
m pretraoted. divided, and computed within a time too short to admit of appreciable change, theordi-
iarf seales of wood, ivory, or metal may be used ; but satisfHctory accuracy oannot be obtained with
Asm on plots requiring several days, if tbe air be meanwhile alternately moist and dry, or subject to
ssnsldarable variations in temperature. What is called parehmont paper is worae in this respect thaa
fsed ordinary drawing-paper.
With tba ArMoliic preoaatii«8 wa maj work tnm a drawing^ with as mnoh aoenra^ as is iwnaQf
~i in tli« Md WW*.

276

LAND BUBYETINa.

When U)« plot taM nuny sldM. tula Mlonlating the error for eaob eC tfieai _

4aoe, In a weU'performea aurrey and protraoUon, the entire error will be but a verj unall qoanti^,
jjA abould not exoeed about -r^jr P^>^ of the periphery,) it may uanallj be divided among the sidee by
merely placing about ^, ^, and H of it at oomera aboat ^ yi, and H way around the plot ; and at

Intermediate cornera propor-
tion It by eye. Or caloulatioB
may be avoided Mtlrely bt
drawing a line a 6 of a length

Sual to the united lengtha
all the aidea ; dividing it
Into diatanoea a, 1 ; 1, 3 ; Sm. equal to the reapeotive aidea. Make b e equal to the entire oloaing error ;
join a e ; and ilraw 1 , 1' ; 2. 2' , 4o, which will give the error at each oorner.

When the plot ia thus completed, it may be divided by One pencil llnea into trianglea, whoaa
baaea and heights may be measured by the aoale, in order to compute the oontenta. With care In
both the anrrey and the drawing, the error ahould not exeeatf about -r-Itt V^ ot the true area. At
leaat two distinot aeta of trianglea abould be drawn and computed, as a guard against miatakea ; and If
the two aeta dlflbr in calculated oontenta more than about -^^ part, they have not been aa carefully

frepared aa they abould have been. The doaing error due to imperfect fleld- work, may be accurately
Mloulated, aa we shall ahow, and laid down on the paper before beginning the plot ; thua furnishing
• perfect teat of the accuracy of the protraction work, which, if correctly done, will not cloae at the
point of beginning, but at the point which indicates the error. But this calculation of the error, by
a little additional trouble, furniahea data alao for dividing it by calculation among the diff aides;
besides the means of drawing the plot co)-r«c(Zy at once, without the use of a protractor ; thna en»>
bling uB to make the aubaequent meaaurementa and oomputationa of the triangles with more oar-
tainty.

We shall now describe thia proceaa, but would recommend that even when it la employed, and
aapeeially in complicated surveys, a rough plot should first be made and oorreoted, by the first of the
two mechanical methods already alluded to. It will prove to be of great service in using the method
by oalonlation, inaamuoh aa it fumisbes an eye check to vexations mistaken which are otherwise apt
to occur: for, although the principles involved are extremely simple, and easily remembered when
once understood, yet the oonUnual changes in the directions of the sides will, without great ears,
•auae na to uae Na inatead of Sa; Bs instead of Wa, Ac.

We auppose, then, that such a rough plot has been prepared, and that the angles, bearings, and
diatancea, aa taken ft'om the field book, are figured upon it in leadptneU.

Add together the interior angles formed at all the cornera : call their sum a. Unit the number o*
aidea by 1909 ; from the prod aubtract 360" : if the remainder la equal to the aum a, it ia a proof that
the anglea have been correctly meaanred.* This, however, will rarely if ever ooeur ; there wHl
always be aome discrepancy ; but if the field work has been performed with moderate eare, tliis wUl
not cxcMd about two mln for each angle. In this case div it <n tqttal part* among all the anglea,
adding or aubtracting, as the caae may be, unleaa It amounta to leaa than a min to each angle, when
it may be entirely disregarded in common farm surveys. The corrected angles may then be marked
0n the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be aa
•hown in Fig S.

Next, by meana of these
oorreoted angles, oorreet ths
bearings alao. thua. Fig t ;
Select some aide (the longv
the better) trom. the two enda
of which the bearing and ths
reverse bearing agreed ; thns
showing that that bearinc
was probably not infloenesd
by local attraction. Let ilds
t be the one so selected ; ••»
sume iM bearing, N 76° ST I,
as taken on the ground, to be
correct; through either end
of it, as at its far end S, draw
the short meridian line ; par-
allel to which draw others
through every ooraer. Now,
having the bearing of side S,
M nP 8*i' B, and reqnirfaig
that of side S, it is pltfn that
the reverse bearing fromoor>
ner 8 is 8 75° S2' W ; and
that therefore the angle 1. %,
m, is 76° 32'. Therefore, if we
take IfP 38' trom the entire
oorreoted angle 1, 8, S, or lUP
67', the rem 68° 86' wiU bn
the angle m 83 ; consequently
the bearing of aideS mstaC be
8 MO 86' E. For finding the bearing of aide 4, we now hare the angle 88 a of the reveraebearing af
•Ide S, alao equal to 6»o 26' : and if we add this to the entire corrected angle 234. or tofito 88*. we havs
theangleaS4 = «8O23'+e»°S3' = 1380 67'; which taken f^m 180°. leaveo the angle 684= il^S';

FI9.8.

• BecaoM in evenr atralght*llned figure the sum of all its Interior 1
light angles as the figure has sides, minus 4 right angles, or 300°.

iglos Is eqnal to twlea a«

LANS SUBYEYINa.

277

Mrtftal obMrrstion Is BMestaiy to B«e how tbe aereral angles are to be employed at eaeh oanmt,
Biilea are sometimes given for this purpose, but unless frequently used, they are soon forgotten.
The plot ueehanioally prepared obviates the necessity for such rules, inasmuch as the principle of
proceeding thereby beoomes merely a matter of sight, and tends greatly to prarent error from asing
the wrong bearings ; while the protractor will at onoe detect any serions mistakes as to the angles,
and thus prevent their being carried farther along. After having obtained all the corrected bearings,
Utev may be figured on the plot instead of those taken in the field. Thej will, however, require a
slUi farther oorreetion after a while, since they will be affected by the adjustment of the closing error.
We now prooeed to ealoalate the closing error <6 of Fig t, which is done on th« principle that in a
aorreet survey the northings will be equal to the southings, and the eantings to the westings. Pre*
pare a tabia of 7 columns, as below, and in the first S cols place the numbers of the sides, and their '^or.
rsotedooarsee; also the diets or lengths of the Mdes, as meanured on the mugh plot, ifsnchaonQ
has been prepared ; bnt if not, then as measured on the ground. Let them be as follows :

Side.

Bearing.

Dist. Ft.

Latitudes.

Departures.

N.

8.

£.

W.

1
3
8

4-
6

•

N10O40'W
N 750 82' X
8 69° 25' X
8 41° 3' W
N 790 40' W
8 53030'W

1060

1202

1110

850

802

706

1015.5
300.3

143.9

800.2
<U1.

419.3

11fl3.9
1039.2

804.

658.2

789.
566,7

1459.7
1450.6

1460.5

Error In
Lat.

2203.1

Error in
Dep.

2217.9
2203.1

9.2

14.8

Kow. bj means of tne Table of Sines, etc., And the N, 8, R, W, of the several sides, and place
them in the oorrespAoding four columns. Thus, for side 1, which is 1(M0 feet long, with bearing
N 1|0 40' W ; cos ItP 4(K &s 0.9580 ; sin 16P 40' = 0.2868.

Hare N s 1000 x 0.9580 s 1015.5; and W s 1060 X 0.2^ = 304. Prooeed
tbvs with all. Add vp the foor eols ; find the dllT between the N and S ools ; and also between
the B and W ones. In this instance we find that the Ns are 0.2 feet greater than the Ss ; and that
the Wa are 14.8 ft greater than the Is ; in other words, there is a eleslntf error which wonld cause a
mrrtct protraotion of oar first three eels, to terminate 9.2 feet too far north of Um starting point : and
14.8 feet too ter west of it. 80 that by placing this error npon the paper before beginning to protraet,
We should bare a ten ftnr the aoenraoy of the protracting work ; bnt, aa before remarked, a little more
IrenUe will now enable us to div the error proportionally amonc all the Ms, Ss, Sa, and Ws, and thereby
give aa data for drswing the plot correctly at once, without using a protractor at all.

To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaeea
are farther apart : and that the addings-np ef the old N, S, B, W oolunns are omitted. The additioai
here aotloed are made subseqaently.

The saw table is on (ha nasi pafs.

Bkm AKX. Tbe l>earinir And ibe reverse bearing from the two ends
of a line will not read preciHt'ly the same argle; and the differauce varies with the
latitode and with the length of the line, but not in the same proportion with either.
It is, however, generally too small to be detected by the needle, bein^p, according ^o
Gummare, only three quarters of a minute in a liue one mile long in lat 40°. In
higher lata it is more, and in lower ones less. It is caused by the fact that meridians
or north and soath lines are not truly parallel to each other; but would if extended
■eet at the poles.

Heaee tbe only bearing (bat can be run in a straigbt line,

eilh ttrlet aocnraey, is a true N and 8 one ; except on the very equator, where alone a due E and w

one will also be straight. But a true curved E and W line may be found

■lywhere with suffioient accuracy for the survevor's purposes thus. Having first by means of the N
ttMrmtUt or otherwise got a true N and 8 bearing at the starting point, lay off from it 90*, for a true
land W DMtring at that point. This B and W bearing will be tangent to the true E and W curve.
Baa this tangent carefully : and at intervals (say at the end of each mile) lay off ftrom it (towards
the N If in N lat, or vice versa) an ofltet whose length in /Ml is equal to the proper one from the
Wlowinff (able, multiplied by the sotiare of the distanee in mtlM from the star«iug point. These
•bets will mark points in the tme K and W curve.

10°

lao

SOO

liatitade IT or H.

250 80° 960 409 46°

500

550

003

«•

OAieUi in ft one mile ft*oni startinfr point.

4M .118 .179 .34S .311 .885 .467 .559 .667 .795 .952 1.15 1.43

te, any offiiet in ft = .6666 X Total Dist in miles> X Nat Tane of Lat.

A rtiainb line is any one that crosses a meridian obliquely, that is, ia
■•flher d«S ir ttitf 8, nor E and W.

278

LAND SURVEYING.

Side.

Bearing.

Dist. Ft.

Latitudes.

Departures.

N.

S.

K.

W.

1

N 16° 40^ W
N 75° 32' E
S 69° 25' E

a 410 3' w
N790 40^ W
S 53° 30' W

*

1060

1202

1110

850

802
705

1015.6

1.7

3O4.0

2.7

1013.8...

... 301.3

2

300.3
1.9

390.2
1.8

1163.9
3.1

3

298.4

143.9
1.3

... 1167.0

1039.2
2.9

4

392 ...

641.0
1.3

... 1042.1

558.2
2.2

5

642.3...

419.3
1.1

.. 556.0

789.0
2.1

6

142.6...

... 786.9

666.7
1.8

420.4...

664.9

5729

Sum of

Sides.

1454.8
Cor*d Na.

1464.7
Cor'd Ss.

2209.1
Cor'd Es.

2209.1
Cor'd Ws.

Kow we have alrewlj foaAd by the old Uble that the Ns and th« W« are too long; oonaoquent^
fhey must be shortened ; while the Be, and E«, maet be lengthened ; all in the following proportieBa:
▲•the

Sum of all . Any given .. Total err of . Err oflat, erdep,
the Eidee * side * * lat or dep • of giren elde.

Thng, oommencing with the lat of side 1, we hare, as

Sum of all the aides. . Sldel. .. Total lat err. . Lat err of side L.
6729 • 1060 • • 9.2 • l.t

Now as the lat of side 1 is north, It mnst be shortened ; henee tt keooma«'=:10IS.5-~l.T3dCtaj^ as
Bgured oat in the new table. Again we hare for the departinv of side 1,

Snm of all the aides. . Sldel. .. Total dep err. . Dep err of aide 1.
6729 • 1060 • • 14.8 • 2.7

Vow as the dep of side 1 is west, it most be shortaned; faenes it beaoiMB9M— S.T=^m;S, «a figvraa
out in the new table.

Prooeedlng thus with eaeh
side, we obtain all the corrected
lats and deps as shown in the
new table : where thej are oon-
nected wfth their reepeotlT*
sides by dotted lines; but la
praotioe it is better to oross oal
the original ones when the oal"
onlatlon is finished and proved.
If we now add upthe 4 eols of
oorrected N, S, S, W,w« And *^%t
the Ns =: the Ss ; and tha S8=
the Ws; thus proving (hat the
work is right. There la. It la

5fi \ / true, a dlsorepanoy of .1 of a ft

I \- ^^j^ — y betweentbeNs, andtheSs; bat

tbis is owing to oar oarryiBg
out the oemotions to only oaa
deoimat plaoet and la too small
to be regarded. Diaerepmnofaa
of 8 «r 4 t^thi of a foot wtn
sometimes ooear f^m this
cause; but may ha n^lootad.
The oorrsolod late and dioM
mast ovUaatty ehaiifa tha
bearing aad dlstanoa or a
bnt wttheut knowing either of these, we eaa aew plot the survey by means of the

FUr.4.

LAND SUBTEYIMQ.

ir.iM.

i. iM.

-"'"."■

-',-■"■

1

su

no.«

ino.o

i

|g«;^5^?;^|,s-£ Stt-J'A i.

^

«,d^.

•W.i,^

KJKE

USi

i

§

280

LAND BURTEYING.

•r the •orragr.* The oomoted northings and southian we have already found ; ae alio the eaatinfi
and wesUngi. The middle diata are fouDd by meau of the latter, by employing their holvM ; adkUng
hair eaatinge, and lubtraeting half wectinga. Thne it ia evident that the middle dist 2' of aide a, is

Snal to hair the easting of side S. To this add the other half easting of side 2, and a half easang
side S ; and the sum is plainly equal to the middle dist 8' of side 8. To this add the other half
easting of Ride 3, and subtract a half westing of side 4. for the middle dist 4' of side 4. From this
subtract the other half westing of side 4, and a half westing of side 6, for the middle dist 6' of side
6i and se on. The actual calAulation mi^ be made thus :

Half easting of side 3 =

2

= fi8lS.5 E £= mid dUt of side 1
S8S.6 I

Half easting of side 8 =

IMll 1167.0 E
— = 521.0 E

1688.0 E = mid dist of side t.
621.0 E

■Of

556

ting of ride 4 = —

2

2209.0 E
= 278.0 W

19S1.0 E = mid dist Of aide 4»

278.0 W

786.t 166S.0 E

Balf vesting of side 6= = 8W.5W

2

1259.5 E = mld«iator«ide6.
88S.5W

Half westing of side 6 =

564.9

866.0 E
282.4 W

688.6 EsmMdlstefiUett.
282.4 W

Balf veeting of side 1 =

801.8

801.2 E
lfi0.6W

160.6 Est mid dist of side 1.

The work always proves Itself by the last two results being equal.

Next make a table like the following, in the first 4 ools of whioh plaoe the numbers of th« sldaa,
the middle dists. the northings, and loathings. Mult each middle dist by its corresponding northing
or southing, and place the products in their proper col. Add up each col ; subtract the least flrom the

Side.

1
2
8

4
6
6

Middle dist.

150.6
583.5

1688

1931

1259.5
583.6

Northing.

1013.8
298.4

142.6

Southing.

392
642.3

420.4

North prod.

152678
174116

179605

506390

Sonth prod.

661606
1240281

245345

2147322
506399

43560)1640923(37.67 Aont.

• Proof. To lllnatrate the principle npon whioh this
mle is based, let a 6, be, and c a. Fig 6, represent in
order the 8 sides of the triangular plot of a survey, with
a meridian line <l^ drawn through the extreme west cor*
ner, a. Let lines o d and ef be drawn from eaeh oomer,
perp to the meridian line ; also from the middle of eaeh
side draw lines w e, m n, « o, also perp to meridian ; and
representing the middle dlsts of the sides. Then sinoe
the sides are regarded in the order a 6, 5 e, e a, it is
plain that a d represents the northing of the side a b ;
fa the northing of ea; and d/ the southing of 6e.
Aow if we mult the nothing ad ot the side ab, by its
mid dist ew, the prod Is the area of the triangle abd.
In like manner the northing fa of the side ea, mult by
its mid dist « o, gives the area of the triangle a ef. Again,
the $otUhing dfot the side b e, mult by lu mlddistmn,
gives the area of the entire flg dhefd. If ftom this
area we subtract the areas of the two triangles at tf,
and aef, the rem is evidently the area of the plot •6«.
^ith any other plot, however oomi^lflated.

Fi|r.&

IJLND SURVKTINQ.

281

■natMt. Th« ran will be tbe area of the rarvey in aq ft ; which, div by 4S6M, (the namber af aq ft

la an aore,) will be tbe area in aor^a ; in this iusiauoe, 37.67 ac.

It now remaina enly to oaloalate the eorreeted beariugs and lengptha of the sides of the sorrey, all

of which are neceaaarUy changed by the adoption of tbe eorreeted lau and deps. To And the bearing

of any aide, dir lu departure (K or W) by Ita 1m (N or S) ; in the table of nat tang, find (he qnot ;

HOI 3 W
the angle opporite It Is (he reqd angle of bearing. Thus, for the oourae of aide 1, we hare >-— ' — —

=: .3972=rnat tang ; oppoaite which in the table is the reqd angle, l(P 8S' ; the bearing, therefore. Is
K 1«» M' W.

Again : fer the dial or length ef any aide, from the table of nat cosines take the cos opposite to
tbe angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the oos. Tons
for tlie diet of side 1, we find opposite 16° S3', the coa .9686. And

Lat. Cos.
1013.8 -i- .9686 » 1067.6 the reqd disk

Tte MlaiwiBc table oontaias all the cMreotifOiis ef the foregoing snnr^y ; eonaeqaeatly, if the bear.

Side.

Bearing.

Dist.7t.

1

S
8
4
6
6

N 16® 33' W
N 760 Sy E
S e«0 23'K
S40O63' W
N 78«> 44' W
8 63® 21' W

1057.6

12M.0

1118.3

849.6

800.1

704.3

.*.

tags anA dlsts are correctly plotted, they will close perfictly. The yeang asatatant Is adTised ta
prafBtiae doing thla, as well as dtviding the plot Into triangles, and oempottng the content. In this
manner be will soon learn what degree of care is neoeseary to insiue aocarats resalis.

The following hlsta may often be ef serrloe.
1st. ATold taking bearings and
Aisle along a eirenitoas bound- a

atyUnelikeate, Fig7;bQtma •. ......................_.._=' » .«>

the etralght line a c ; and al - . -r*

right anglea to It, measure ofT
sets to tbe crooked line. 94.
iTisblng to surrey a straight
flna fMm a to e, bat being ana"
ble to direct the instrument
precisely toward e, on account
ef iBierreainv woods, or ether
ebattMlea; first nm atrialUnab
as • «». as nearly in the proper

direotlon aa can be guessed at. .

Measure m e. and say, as a m is to in e, so ts 100 ft to T Lay off a o equal to 100 ft, and o • equal
to r ; and run the final line a s e. Or. if m 0 is quite small, calculate offsets like o s for erery 100 ft
alnc a », and thus avoid the aeeesslty for running a second line. Sd. When e is Tisible from a, but
dia uitervenlng ground dllBcnIt to measure along, on account of marshes, Ice, extend the side y a
to good ground at t : then, making the angle ytd equal to y a o, run the line t n to that point d at
wlaiA the ma^ ndel» found by trial to be equal to the angle atd. It will rarely be necessary to
mmkm asore than one trial for this point d; for, suppose it to be made at x, see where it strikes a e at
<; aioaeaw 4 e, and eontinoe ftxmi x, making a <( =< c 4th. In case of a very irregular piece of
laad. or a lake, Fig 8, surround it by straight lines. Surrey these, and at right angles to them,
■MMaro ofbets to the crooked boundary, ftth. SurTeyiBg a straight line from w toward y, Fig ft

m

Ffff.ft.

« d
Flff.lO.

n

FI9.0.

s

o. Is net To iMMs It, lay off aright aagletptw; measure any <«; make It* OS
I v; make «» v < =90°; make « < = ( i»; make •<y = 90°. Then is ti = uv; and
ly la in the straight line. Or, with less trouble, at g make I g a=aOPt measure any g a; make
#«s3=d0O; and«s = |r0: make a«< = 60O. Then is y • = 9 a or ••; and < s, continued toward
r. Is la the etralght Hue. fth. Being between two ol^eets, m and n. and wishing to place myself ia
laagi with them, I lay a straight rod s b on the ground, and point it to one ef the objects m ; then
to the end e, I And that It does not point to the otaT ofejeet. By suoeessire trials, I find tbe
e # te vhleh H polats to both otjects, and eoaseq. wtly is ia range with them.

282 CHAINING.

CHAINUrO.

Chains. EDgineers have abandoned the Gunter's chain of 6& ft, divided
into 100 links of 7.92 ins each. They now use a chain of 100 ft^ with 100 links
of 1 ft each, and calculate areas In sq ft, the number of which, divided by
43,560, reduces to acres and decimals, instead of to acres, roods, and perches,
Giinter's chain is used on U. S. Government land surveys.

Chains are commonly made of iron or steel wire. Each link is bent &i each
of its ends, to form an eye, by which it is connected with the adjacent linki,
either directly, as in the Grumman patent chain, or, more commonly, by from
1 to 3 small wire links. The wear of tnese links is a fruitAil source of inaccuracy,
inasmuch as even a very slight wear of each link considerably increases the
length of the chain. Hence, chains should be compared with some standard,
sucn as a target rod, every few days while in use. For transportation, the
lengths are folded on each other, making a compact and sheaf-like bundle.

Tapes. With improved facilities for the manufacture of steel tape, the chain
is going out of use. The tape, being much lighter, requires much less pull, and,
as there are no links to wear, its length is much more nearly constant than that
of the chain. It is replacing, to some extent, the base-measuring rod for
accurate geodetic work. Steel tapes are made in continuous lengths up to 600,
600, and even 1000 ft, but those of 100 ft are the most commonly used. Very
long tapes are liable to breakage in handling. Even the shorter lengths, unless
handled carefully^ are apt to kink and breaC Breaks are difficult to mend, and
the repaired joint is seldom satisfactory ; whereas a kink in a wire chain seldom
involves more than a temporary change of length. Being run over by a car or
wagon will often kink steel tapes very badly, if it does not break them.* How*
ever, the lightness, neatness, and reliability of the tape ofiG^et these disadvan*
tages, which, indeed, the surveyor soon learns to overcome.

Tapes for general field work are usually narrow (from 0.10 to 0.25 in) and
thick (from 0.018 to 0.025 in),t and are graduated by means of small brass
and copper rivets, spaced, in general, 6 ft apart, 1 ft apart in the 10 ft at eac^
end, and 0.1 ft apart in the ft at each end. They are usually mounted on reels.

Tapes for city work are wider (from 0.25 to 0.5 in) and thinner (from 0.007 to
0.010 in)t and are graduated (usually to 0.01 ft) throughout their length by
means of lines and numerals etched on the steel.

Pins are ordinarily of wire, pointed at the lower end, and bent to a ring at
the upper end. They can be forced into almost any ground that is not exceed-
ingly stony. A steel ring, like a large key rin^ is often used for carrying the
pins. Each pin should have a strip of bright red flannel tied to its top, in order
that it may be readily found, among the grass, etc., by the rear chainman.

Corrections for Hofs and tStretcll. The following diagram ^ (seep.
283) gives the correction for a steel tape weighing 0.75 fi> per 100 ft.t

*The Nichols Engineering & Contracting Ck>., Chicago, guarantees that its
tapes will not be injured by beins run over by wagons.

fThe sizes of tapes, as made by different manufacturers, vary greatly. In
applying the corrections, therefore, the width and thickness of the tape to be
used should be carefully measured, and its weight per ft computed.

X Deduced from diagrams constructed by Mr. J. O. Clarke, Proceedings Engi-
)ers' Club of Philadelphia, April, 1901, Vol. XVIII, No. 2. from the formuU :

Stretch, in feet

neers'

PS

EA

where

P = pull on tape, in fl>s.

S = span of tape, in feet.

E = modulus of elasticity for steel = 27,600,000 flt>s per sq in.

A = area of cross-section of tape weighing 0.76 B> per 100 ft.

= 0.0022 square ins,

and from the equation of the parabola, according to which

W> S*
shortening by sag, in feet = ^

where W = weight of tape, in pounds per foot.

Except for very light pulls, this last formula gives practically the same reaalts
as the equation of the catenary, which is absolutely correct, but much more
cumbersome.

, an StHi Tape Wallihing f,

TbuA, a tupBj of uj teiigthf weiohlug 1 lb

iDj-giTenooiiKtioD,m pull oti-^j=lHy.

J, - r le OOfTactlan on tbaata-adard tape, weighing 0.70

CoDveTselT : cItct a pull Qf 10 bs on a SO ft ipan of a tape wdfthlnd; D.fl Tb per
lOOrt; requiredtheaorrectian. Ta produaelbeBameemirln tbeUpe welgbtng

0.7S lb per 100 ft "onlii require
the diagram at 1Z.G Bn on tb
Tble ia thfl proper AorrAcliQi
li^itar tape vllh 10 *■ pull.

bB of ■Undn'd ferrglh at M^»^r. For' ordinarr eteel tape, Uie t
MJnperature it about (10000085 ft pCT ft per degree " '

lU of y = 10 X j;^ - 12.0 lbs. Beftrrl
'or » » ft span, we flna comictinn " -

□ ight, a

Wben measuring oter slopliiL , .

tapesbnuld beheld as Dearlf boriioutal as possible, trsnsferrlug the poaitloD of
Ibe raised end to the ground bj means of a plumb line. Where the ground Is
■teep, It b^xiines necessary to use a short length of tape, as the down-hul ebain-

psraliel with theslope, and the disUncecarrecledGr the (ullowlDg form

284

LOCATION OF THE MERIDIAN.

IiO€ATIOIir OF THE HERIDIAHT.

By means of clrcampolar stars.

(1) Seen from a point O (Figs. 1 and 2) on the earth, a circumpolar star e
(•tar near the pole P) ap(>ears to describe daily* and counterclockwise a
small circle, euwl, about the pole. The angle P O e, P O u, etc., subtended
by the radius P e, P u, etc., of this circle, or the apparent distance of the
star from the pole, is called its polar distance. The polar distances of
stars vary sligntly from year to year. See Table 3. They vary slightly also
during each year. In the case of Polaris this latter yariation amounts to
about 50 seconds of arc.

(3) The altitude of the pole is the angle N O P of the pole's elevation
above the horizon N E S W, and Is = the latitude of the point of obser-

FiG. 1.

Pig. 2.

ration. Decl Inatlon = angular distance north or south from the celestial
equator. Thus, declination of pole = 90°. Declination of any star = 90°— its
polar distance.

(3) Let Z e H be an arc of a vertical circlet passing through a circumpolar
star, e, and let H be the point where this arc meets the horizon N E S W.
Then the angle N Z H at the zenith Z, or N O H at the point O of observa-
tion, between the plane N Z O of the meridian and the plane H Z O of th©
star's vertical circle (or the arc N H), is called the azlmutlkt of the star.
If this angle N O H be laid off from O H, on the ground, the line O N will be
in the plane of the meridian N Z S, or will be a nortb-and-sontii
llne.||

(4) When a star is on the meridian Z N of the observer, above or below
the pole P, as at u or ^, it is said to be at its upper or lower culmina-
tion, respectively. Its azimuth is then = 0, tne line O H coinciding with
the meridian line O N.

(5) When the star has reached its greatest distance east or west ftom the
pole, as at e or w, it is said to be at its eastern or western eloni^A-
tlon.{

« In 23 h. 56.1 m.

t A great circle is that section of the surface of a sphere which is formed
by a plane passing through the center of the sphere. A vertical circle is a
great circle passing through the zenith Z.

I Astronomers usually reckon azimuth from the south point around
through the west, north, and east points, to south again ; but for our pur-
pose it is evidently much more convenient to reckon it f^om the north
point, and either to the east or to the west, as the case may be.

II The point N, on the horizon; is called the north point, and must not
be confounded with the north pole P.

g As seen ttova. the equator, a star, at either elongation, is, like the pole
Itself, on the horizon ; and the two lines Pe,Tw, joining it with the pole,
* — I a single straight line perpendicular to the meridian, and lying in the

LOCATION OF THE MERIDIAN.

285

(6) The boar anffle of any star, at any given mconent, is the time
which has elapsed since it was in upper culmination.'"

(7) Evidently the azimuth of a star is continually changing. In cir-
cumpolar stars it varies from OP to maximum (at elongation) and back to
(P twice daily, as the star appears to revolve about the pole ; but when the
star is near either elongation the change in azimuth takes place so slowly
that, for some minutes, it is scarcely perceptible, the star appearing to
travel vertically.

(8) Given the polar distance of a star and the latitude of the point of
observation, the aaimutli of the star, at eloiiirAtlon, may be found
by the formula.f

Sine of azimuth of star =

sine of polar distance of star

cosine of latitude of point of observation

or see (11) and Table 3.

(9) The following circumpolar stars are of service in connection
obeervations for determining the meridian. See Fig. 3.

Constellation Letter

Ursa minor (Little bear) a (alpha)

Ursa major (Great bear) € (epsilon)

( " " i <(zeta)

with

Cassiopeia

S (delta)

Called

Polaris
Alioth
Mizar
Deltas

Jfora»r^.^tet:»

July

Fig. 8.

(10) Polaris^ or the nortb star, is fortunately placed for the determi-
nation of the meridian, its polar distance being only about 1%^. See Table
3. Fig. 3 shows the circumpolar stars as the}r appear about midnight in
July ; inverted, as in January ; with the left side uppermost, as in April ;
ana, with the right side uppermost, as in October. R

horizon. The azimuth of the star is then == its polar distance. But in
other latitudes Pc and Pit; form acute angles with the meridian, as shown,
and these angles decrease, and the azimuth of the star at elongation in-
creases, as the latitude increases.

* In lat. 40° N., the hour angle, ZPc = ZP«>, of Polaris, at elongation, is
= 5 h. 55 m. of solar time. Caation. It will be noticed that, except for
an observer at the equator, the elongations do not occur at 90° from the
meridian.

t In the spherical triangle Z P «, we have :

sin e Z P ^ sinPe

sin Z e P ^ Bin P Z

But, since Z « P = 90°, sin Z « P = 1. Also, sin P Z = cos (90° — P Z), and
< Z P — azimuth of e.

sin Pe _ sin polar distance P O e

cos latitude

Hence, sin azimuth of e . ^ „

sm F Z

1 « Cassiopeia is here called Delta, for brevity.

I Polaris is easily fonnd by means of the two well-known stars

called the *^ pointers '' in " the dipper," Fig. 3, which forms the binder

286 LOCATION OF THE MERIDIAN.

(11) Table 3 ffives the polar distances of Polaris and their log sines for
January 1 in each third year from 1900 to 1990 inclnsive, the log cosines
of each fifth deeree of latitude from '2/iP to 50°, and the corresponding
azimuths of Polaris at elongation. Intermediate values may be taken by
interpolation.*

(12) By olMervatlon of Polaris at elonntlon. This method
has the convenience, that at and near elongation the star appears to travel
vertically for some minutes, its azimuth, during that time, remaining
practically constant : but during certain parts of tne year (see Table 1;, the
elongations of Polaris take place in daylight; so that this method cannot
then be used. | See (18), (19), (22). Nor can it be used at any time in places
south of about 4° N. lat., because there Polaris is not visible.

(18) The approximate times of elongation of Polaris for certain dates,
in 1900, are given in Table 1, with instructions for finding the times for
other dates. Or, watch Polaris in connection with any of those stars which
are nearly in line with it and the pole, as Delta, Mizar, and Alioth. See
Fig. 3. The time of elongation is approximated, with sufficient clofleneas
for the determination of the azimuth, by the cessation of apparent hori-
zontal motion duriftg the observation.

(14) From fifteen to thirty minutes before the time of elongation, have
the transit, see (21). set up and carefully centered over a stake previously
driven and marked with a center point. The transit must be in adjust-
ment, especially in regard to the second adjustment, p. 294, or that or the
horizontal axis, by which the line of collimation is made to describe a ver-
tical plane when the transit is leveled and the telescope is swung upwMrd
or downward.

(15) Means must be provided for illuminating the cross-hairs of the tran-
sit. X I'h^ T^^y ^ done by means of a bull's
eye, or a dark lantern, so neld as not to throw
its light into the eye of the observer ; or, better,
by means of a piece of tin plate, cut and per-
forated as in Fig. 4, bent at an angle of 45^, as
in Fig. 5, and painted white on the surface
next to the telescope. The ring, formed by
bending the long sirip, is placed around the
object end of the telescope. A li^ht, screened
from the view of the observer^ is then held,
at one side of the instrument, in sucb a way Fig. 4. . Fig. 6.
that its rays, falling upon the oblique and

whitened surface of the tin plate, are reflected directly into the telescope.

(16) Bring the vertical hair to cut Polaris, and, bv means of the tangent
screw, follow the star as it appears to move, to the right if approaching eoM.-
em elongation, and mce versa, keeping the hair upon the star, as nearly as
may be. As elongation is approached, the star will appear to move more
and more slowly. When it appears to travel vertically along the hair, it
has practically reached elongation, and the vertical plane of the transit,
vriih the vertical hair cutting the star, is in the plane of the star's vertical circle.
Depress the telescope, and fix a point in the line of sight, preferably 300
feet or more distant from the transit.f Immediately reverse the transit,
(swinging it horizontally through an arc of 19XP), sight to the star again.

^

portion of the " great bear " (Ursa major), a line drawn through these two
stars passing near Polaris. .\s the stars in the handle of the dipper form
the tail of the great bear, as shown on celestial maps, so Polaris and the
stars near it form the tail of the little bear (Ursa minor.) Polaris is also
nearly midway and in line between Delta and Mizar. Polaris forms, with
three other and less brilliant stars, a quite symmetrical cross, with Polaris
at the end of the right arm. In Fig. 3 this cross is inverted. Its height is
about 5°, or == the distance between the pointers.

* Part of a table computed by the Surveying Class of 1882-8, School of
Engineering, Vanderbilt University, Nashville, Tenn., and published by
Prof Clin H. Landreth.

t The stake must be illuminated. This may be done bv throwina' light
upon that side of the stake which faces the transit, or, better, by holding a
sheet of white paper behind the stake, with a lantern behind the paper. In
the latter case, the cross-hairs of the transit, as well as the stake, and the
knife-blade or pencil-point with which the assistant marks it, show out
dark against the illuminated surface of the paper.

\ See Note, page 290.

LOCATION OF THE MERIDIAN. 287

•gain depress, and» if the line of sight then coincides perfectly -with the
mark first set, both are in the plane of the star's vertical circle. If not,
note where the line of sight does strike, and make a third mark, midway
between the two. The line of sight, when directed to this third mark, is in
the required plane, from which the azimuth, found as in (8), has yet to be
laid off to the meridian, to the l^ from. eaMem elongation, and vice vena,

(17) To avoid driving the distant stake and marking it during the night,
a fixed target at any convenient point may be used, and the horizontal
angle formed between the line ox sight to the star and that to the target
merely noted, for use in ascertaining and laying off the azimuth of the
tarvet.

(19) By otMervation of Polaris at cnlmtnaiioii. Owing to
its greater difficulty, this method will generally be used only when that
by elongation is impracticable. It consists in watching Polaris in connec-
tion with another circumpolarstar (such asMizar *or Delta) until Polaris is
seen in the same vertical ]^ane with such star, and then waiting a short and
known time T, as follow8,t until Polaris reaches calminatlon, where-
upon Polaris is stehted and the line of collimation is in the meridian. At
their upper culniinations, Mizar and Delta are too near the zenith to be
conveniently observed at latitudes north of about 25° and BOP respectively.
At their lower culminations they are too near the horizon to be used to
advantage at places much below about 88° of N. latitude. In general.
Delta is conveniently obeexved at lower culmination ttom. February to
August, and Mizar during the rest crf^kie year.

Mizar Delta

T= T =

In 1900 2.6 mins 8.4 mins

In 1910 6.5 mins 7.2 mins

Mean annual increase, 1900-1910 . 0.39 min 0.38 min

(19) "By obsenration of Polaris at any point In Its path*

Table 1 gives the mean solar times of upper culmination of Polaris on the
1st of each month in 1900, and directions ibr ascertaining the times on other
dates ; and Table 2 gives the azimuths of Polaris corresponding to different
values of its hour angle in civil or mean solar time, for different latitudes
fh)m 30° to 50°, and for the years 1901 and 1906. For hour angles and lati-
tudes intermediate of those in the table, the azimuths may be taken by
interpolation. See Caution and formula, p. 290.

(SO) The local time} of observation must be accurately known, and the
time of the preceding upper culmination (as obtained from Table 1) dedu<!ted
from it. The difference is the hour angle. If the hour angle, thus found,
is 11 h. 58 m. or less, the star is west of the meridian. If it is greater than
11 h. 58 m., the star is east of the meridian. In that case deduct the hour
angle from 28 h. 56 m. and enter the table with the remaiTuier as the hour
an^le. See Fig. 1.

(»1) Where great accuracy is not required, Polaris may be observed by
means of a plumb-line and sight. A brick, stone, or other heavy object
will answer perfectly as a plumb-bob. It should hang in a pail of water.
A compass sight, or any other device with an accurately straight slit about
1/16 inch wide, may be used. The sight must remain always perfectly verti-
cal, but must'be adiustable horizontally for a few feet east and west. The
plumb-line and sight should be at least 15 feet apart, and so placed that the
star and plumb-line can be seen together through the sight, throughout the
observation. The plumb-line must be illuminated. It is well to arrange
all these matters on an evening preceding that of the observation. When
the star reaches elongation, the sight must be fastened in range with the
plumb-line and the star. From the line thus obtained, lay off the azimuth ;
to the toest for ea^em elongation, and vice versa.

{fSS9) Bjr any star at eqnal altitudes. This method, applicable
to south as well as to north latitudes, consists in observing a star when it
is at any two equal altitudes, £. and W. of the meridian, thus locating, on the
horizon, two points of equal and opposite aziQiuth. The meridian will
be midway between the two points.

• Mizar will be recognized by the small star Alcor, close to it.

t Deduced from values calculated in astronomical time (p. 266) by the
U. S. Ckiast and Oeodetic Survey.

X Ijocal time agrees with standard time (p. 267) on the standard
meridians only. For other points add to standard time 4 minutes for each
degree of longitude east of a standard meridian, and trice versa.

288

LOCATION OF THE MERIDIAN.

(as) By e^aal sliadows from the sun. Piir. 6 ADDroximAtP
At the solstices (about June 21 and December 21) the path a b c <J traveraed
before and after noon, by the end of «*«'<'"■ tniveraea

the solar shadow O o, etc., of a verti-
cal object O, or by the shadow of a
knot tied in a plumb-line suspended
over O, will intersect a circular arc
a N d, described about O, at equal dis-
tances, am^ md, from the meridian
O N. The observations should be
made within two hours before and
after noon. At the vernal equinox
(March 21) the line thus located will
then be west, and at the autumnal
equinox (Sept. 21) east, of the merid-
ian, by less than 7.}4 minutes of arc. For intermediate dates the error is
nearly proportional to the time elapsed. It is well to draw several arcs
of different radii, O a, O 6, etc., note two points where the path of the shadow
intersects each arc, and take the mean of all the results. A small piece of
tin plate, with a hole pierced through it, may be placed with the hole
vertically over O ; and the bright spot, formed by the light shining through
the hole, used in place of the end of the shadow.

Table 1.

^^^S'V^^*'?"^** **'^" times of elongratlon and calmlnatlon

muilJh hTlScX)" ■^•' ^ong. 90° W. from Greenwich, on the first of each

The times given in this table are mean solar or local times.
fn^ti^^^^o iS^Y.^"" 5^22i^.^i,.TJi»l^.^iL«^ i« bold-faee.

In lattude 25^, W. elongations occur later and E. earlier K« , . ,
latitude 50°, W. " " earlier and E. later f ^^ nearly 2 mins.

le correction fc%r Inno-iti-iHA amr\tt-n*a *ex ana■m.^^^■^ » «..• *. jfj.i ,

In

'TK^ —-w > y. cttiijcrtiiiuji. later) * -f -•"**«'•

For other days of the month, deduct 8.94 min. for each succeedinp fl*v

In general, the times are a little later each vear In iSith^^S?! i^^v ^:
b}A minutes later, but in 1905, only about 3 mlnm^s latefthan^iJT^iJ? *S2!}*
discrepancy is due to the occurrence of leS^yeaMni^'^ ^ ^^' ^^^

Inasmuch as this table serves chiefly to out the obsPrvlV «« «r.,««^ ^
he- should be at his post from 15 to S m?nmk in advance^S^^^ ""S^^
the gradual increase in the times is of little conseauence Thi^oUl'^®*;
the star at.elongation is determined by observS ^ position ot

At culmination, where the change in azimuth is most ranid a»i o-,^. <«

At elongation,

an error in time of

20 minutes
10 minutes

5 minutes

1 minute

will make an error in azimuth of

less than 90 seconds
less than 6 "

less than 2 "

about 0.06 second

, „, ~' — aooui 0.06 second

Jan. 1.
12.31 A.

July 1.
12.51 A.

W.
M.

E.
M.

Jan. 1.
6.38 P.

July 1.
«.44 P.

U.
M.

L.

Elongratlons. (E, eastern : W, western.) 1900.

AVp.V IfSSkli. .^SSk^. Km«: \Zl
.»P.^«. i%-k^. .?J^kV rj.J:S: V^l

Cnlmlnatlons. (U, upper ; L, lower.) 1900.

E
Mi

w!

M.

Feb. 1. L.
4.38 A. M.

U.
M.

Aug. 1
4.45 A.

}^fn'}'h >P^- 1- ^- May 1. L.

2.47 A. M. 12.45 A. M. 10.48 P. M.

Sept. 1. U. Oct. 1. U. Nov. 1. U.

2.43 A.M. 12.46 A.M. 10.40 P.M.

Jane 1.
S.«8P.

Dec. 1.
S42P.

^

LOCATION OP THE MERIDIAM.

l«.»5°4e°4IS'' HW

0 410 43 0 47I 0 51
0 440 47 0 5ll 0 Sa

0 m'o 58

11

SI i

1

wu" uiiuillr ba 'S

290

LOCATION OF THE MERIDIAN.

Table a.

POLARIS. POLAR DISTANCES, AND AZIMUTH AT ELONGATION.

Azimuth at Elongation, in Latitude

u

Polar
Dist. of
Polaris

Log sin
poldist.

1

S0<>

JWO

BOO

85°

40<>

400

50^

O / ft

o /

o t

O f

o /

O f

o /

o /

1900

1 18 33

8.38027

1 18.8

1 21.1

1 24.9

1 29.8

1 36.1

1 44.1

1 64.4

1908

1 12 37

8.32 472

1 17.3

1 20.1

1 28.8

1 28.7

1 34.8

1 42.7

1 58.0

1906

1 11 41

8.31 910

1 16.3

1 19.1

1 22.8

1 27.6

1 33.6

1 41.4

1 51JS

1909

1 10 45

8.31 341

1 15.3

1 18.1

1 21.7

1 26.4

1 32.3

1 40.1

1 60.1

1912

1 9 49

8.80 765

] 14.3

1 17.0

1 20.6

1 25.2

1 31.1

1 88.7

1 48.6

1915

1 8 53

8.30 181

1 13.3

1 16.0

1 19.6

1 24.1

1 29.9

1 37.5

1 47.2

1918

1 7 58

8.29594

1 12.3

1 15.0

1 18.6

1 28.0

1 28.7

1 36.1

1 46.7

1921

17 2

8.28 999

1 11.4

1 14.0

1 17.4

1 21.9

1 27.6

1 34.8

1 44.8

1924

16 7

8.28 401

1 10.4

1 13.0

1 16.3

1 20.7

1 26.8

1 33.5

1 42.9

1927

1 6 12 8.27 794

1 9.4

1 11.9

1 16.3

1 19.6

1 25.1

1 82.2

1 41.4

1980

1 4 16 8.27 169

1 8.4

1 10.9

1 14.2

1 18.5

1 28.9

1 30.9

1 40.0

Log 008

Ut

9.97 299

9.95 728

9.98 753

9.91 337

9.88426

9.84949

9.80807

. Owing to changes in the position of Polaris during the year, the positions
given in the table may at times be in error by as much as a minute. The
error is greater in the nigher latitudes.

Having the north polar distance,/), of a star, and the latitude, L, of the
point of observation, we have, declination of star = 6 = 90° — p ; and ^e
aslmutb, a, of the star, corresponding to any hour angle, a, may be
found by the following formulas :

TanM = ^ = -^. Then Tan a = <^ " ' ^° * .
cos h cos h cos (L— M)

The declinations, fi, of Polaris are given in the U. S. Ephemeris or Nautical
Almanac. From these the polar distances may be obtained more accurately
than from our Table 3.

Caution. When it is desired to determine the meridian within one
minute of arc, it is well to use more than one method and compare the
results. For example, observe Polaris both E. and W. of the meridian, aitd
a star at equal altitudes south of the zenith.

NoTK. — Lf Polaris be found during twilight, iu the morning or evening, obsei--
▼atinns of it luuy be made without artificial illumiaation of the cross-haira.
For times of elongation, see Table 1,

CouTertiion of Arc Into Time, and vice versa.

Arc Tike
1° = A minutes
1' = 4 seconds
l» =1 0.066... second

Time Abc

24 hours =860°
Ihour = 150
1 minute = OP 16'
1 second « (PVl^

1

TBE ENGINEBB's TRANSIT. 291

THE ENGINEER'S TRANSIT.

292 TtTB EHQINEE&B TRAITBET.

Thb dtMIlB of the transit, like thme of the IstgI, are dllTerCDtlT trmtgei hf
diff nukem, and to mlt pirtkuUr purpoAU. We deocribe it In iti modern Ibrm,
SB uude by Heller ud Brightly, arPhlladiL without the lone bBbU«-tBke
F F, Fig 1, onder the telescope, and the BrrndBstcd an p, It la theli plklB
teBMalt. With tb«e sHiendage*, or nther vltta a, eradusted cirde ia fite* of
the Bra It becomea Tirtiullr s COBipl«te Ttaeodiillle.

B D D, Fig 1, Is the tripo<l>ke«d. The Krew-tbRwds at v loeelfe the sciew
of a wooden trlpod-head-cover vhen the inetniment Ia out of use. S B A la Qu
l«w«r panulel |»l«t«. After the traaiit has been set tstt dmtIt oier the
center ofa sl^e, the mlilftlns-plat«, <f d e c, enables lu, bf illabClr lonealng
the I«TelllBK-BCrem K, to shlA the upper paiU boriiontallT a (rifle, and
■haa bring the plumb-bob eiactlj OTer tbe center -with leaa trouUe than bf the
elder method of puiblog one or tiro of the legs furibei Into the giouod. or apread-
InE tbem more or leaa. Tbe acreirfl, E, are taea tightened, thereby puablDg up>
ward the upper BBiwllel pl«M n « ni z i, and vitb It tbe balT-bkll t, ibni
pr^alng o c llghtl; up afalnat the under lida at 8. Ths plomb-UnB paana

throngb the yert hols in 6- Scraw-eaja, / g, protect the leTalUnMcrewi ttom.
dual, ia The feet, i. of tbe icrewa, work In looea aocketa^^, made flat at bottom.
to^presene S from being Indented, The paita thui far dTeKribed are guaiBUw
left atUiched lo [he legs at all Uraea. Flj? 1 show, the method of attachmmt.

To set (he upper puM up*a «m panllal l>l«te|k Plaoe tbe
lowerendor UU Id 1 1, holding tbe Instrument so that the thrw bloekaonaawe
(of which the one ahown at Fis morable) mar ^oter the three oorreapondtiiK

THE engineer's TRANSIT. 293

rMeeses in a, thus allowing a to bear fully on m, upon which the upper pute
then rest. (The inner end of the spring-catch, I, in the meantime enters agroov6
around U, Just below a, and prevents the upper parts from falling off, if the in*
strument is now carried over the shoulder.) Kevoive the upper parts horizontaUj
a trifle, in either direction, until thev are stopped by the striking of a small lug
on a against one of the blocks F. Tne recesses in a are now clear of the blocks.
Tighten g, thereby pushing inward the movable block F, which clamps the
bevelled flange a between it and the two flxed blocks on m m, and confines the
spindle U to the fixed parallel plates. It remains so clamped while the instrument
is being used.

To remoTe the upper parts ft^m theparallel plates. Loosen
g, bring the recesses in a opposite the blocks F. Hold back I, and lift the upper
parts, which are then held together by the broad head of the screw inserted into
the foot of the spindle w.

T T is the oater reTOlTlng: spindle, cast in one with the support*
Ing^plate Z 2^, to which is fastenea the s^radnated limb 0 O. The limb
extends beyond the compass-box, and thus admits of larger graduations than
would otherwise be obtainable, to wis the Inner revolving^ spindle. At
its top it has a broad flange, to which is fastened the vernier plate P. To the
latter are fastened the corapass-box C, the two bubble-tubes M M, the standards
y Y, supporting the telescope, &c. Each bubble-tube is supported and adjusted
by four capstan-head nuts, two at each end. The bent strip, curving over the
tnbe, protects the glass from accidental blows in swinging the telescope.

<k»iatrol of motions of ir>*»dnAl«d limb O O and wernler
plate P. — ^The tangent-screw 6 and a spiral spring (not shown) opposite to it
are fixed to the graduated limb 00, and hold between them a projection y from
the loose collar t, which is thus confined to the limb and made to travel with it.
The clamp-screw H passes through the collar t and presses against the small lug
shown at its inner end. When H is tightened, this lug is pressed against the
fixed spindle U U, to which the graduated limb is thus made fast. A slow mo-
tion may, however, still be given to the limb by means of the tangent-screw G.

The motion of the vernier plate P over the graduated limb O 0 is simUarly
governed by the tangent-screw 6 and its spiral spring (not shown), fixed to the
ternier plate P, and the clamp-screw e, which passes tnrough the collar z, and

{>re88es against the small lug shown at its inner end. In Heller and Brightly's
nstraments, the screw b is provided with means for taking up its ** wear," or
"lost-motion."

There are two verniers. One is shown at ja. Fig 1. Both may be read, and
their mean taken, when great accuracy is required. Ivory reflectors, c, facilitate
their reading. Before the instrument is moved from one place to another, the
eompaas-needle, ib. Fig 2, should always be pressed up against the glass cover
of the compass-box by means of the upright miUed-head screw seen on the ver^
nier-plate m Fig 1, Just to the right of the nearest standard. The pivot^point is
thus protected from injury.

R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope
in any required position. It is oest to let the eye-end. 1C, of the telescope revolve
dowHiffard, as otherwise the shade on O, if in use, may fall off. The tangent-screw,
il. moves a vert arm attached to R, and is thus used for slightly changing the
elevation of the telescope. In the arm is a slit like that seen in the vernier-arm
L Bt mesns of the screw D. the movable vernier-arm Y may be clamped at
tDT desired point on the vertical limb g. When (P of the vernier is placed at
9(Pon the arc ^, and the index of the opposite arm is placed over a small notch
on the horizontal brace (not seen in our figs) of the standards, the two slits will
be opposite each other, and may be used for laying off offsets, oc, at right-angles
to the line of sight.

One end, R, of the telescope axis rests in a movable box, under which is a screw.
By means of the screw, the box may be raised or lowered, and the axis thus ad-
justed for very slight derangements of the standards. For E, B, O, and A, see
iaulf p 306. a is a dust-guard for the object-slide.

StaaiA Kalrs. Immediately behind the capstan-screw, p. Fig 1, is seen a
nnaller one. This and a similar one on the opposite side of the telescope, work
in a ring inside the telescope, and hold the ring in position. Across the ring are
itretched two additional horizontal hairs, called stadia hairs, placed at such a
distance apart, vertically, that they will subtend say 10 divisions of a graduated rod
placed 100 ft from the instrument, 15 divisions at 150 ft, Ac. They are thus used for
asttsuring hor and sloping distances.

Tbe lonff babble-tube« F F, Fig 1, enables us to use the transit as a level.
•Ithoof h it Is not so well adaotsd as the latter to this purpose.

294 THE engineer's transit.

To aAinmt a plain Transit*

When either a lerel or a transit is purchased, it is a good precaution (but one
which the writer has never seen alluded to) to first screw the oltject-glass firmly home
to its place ; and then make a short continuous scratch upon the ringt>f the glass, and
upon its slide ; so as to be able to see at any time when at work, that the glass is
always in the same position with regard to the slide. For if, after all the adjustments
are completed, the position of the glass should become clumged, (as it is apt to be if
unscrewed, and afterward not screwed up to the same precise spot,) the acyustments
may thereby become materially deranged ; especially if the object-glass is eccentric,
or not truly ground, which is often the case. Such scratches should be prepared by
the maker. In making adjustments, as well as when using a transit or lerel, be
careful that the eye-glass and object-glass are so drawn out that there shall be ne
parallax. The eye-glass must first be drawn out so as to obtain perfect distinctness
of the cross-hairs ; it must not be disturbed afterward; but the object-glass must
be moved for different distances.

First, to ascertain tliat tlie bnbble-tnbes, M Bf • are placed
parallel to the vernier-plate, and that therefore when both bubbles are in
the centers of their tubes the axis qf the inst is vert. By means of the four levelling-
screws, K, bring both bubbles to the centers of their tubes in one position of the
inst ; then turn the upper parts of the inst half-way round. If the Dubbles do not
remain in the center, correct half the error by means of the two capstan-nuta
rr; and the other half by the levelling-screws K. Repeat the trial until both
bubbles remain in the center while the inst is being turned entirely around on
its spindle.

Second, to see that the standards have snfTered no deranire-
ment ; that is, that they are of equal height and perpendicular to the vernier-
plate, as they always are when they leave the makers hands. Level the inst
perfectly ; then direct the intersection of the hairs to some point of a high object
(as the top of a steeple) near by ; clamp the inst by means of screws H and e,
and lower the telescope until the intersection strikes some point of a low object.
(If there is none sucn drive a stake or chain-pin, Ac, in the line.) Then un-
clamp either H or e, and turn the upper parts of the inst half-way round ; fix the
intersection again upon the high point ; clamp ; lower the telescope to the low
point. If the intersection still strixes the low point, the standards are in order.
If not, correct one-ltalf of the difference by means of the adjusting-block and
screw at the end, R, of the telescope axis. Fig. 1, and repeat the trial de novo,
resetting the stake or chain-pin at each trial. If the inst has no adjusting-block
for the axis, it should be returned to the maker for correction of any derange-
ment of the standards.

A transit may be used for running ^raight lines^ even if the standards become
slightly bent, by the process described at the end of the fourth adjustment.

Third, to see that the cross-hairs are traly vert and hor
^rhen the inst is level. When the telescope inverts, the cross-hairs are
nearer the eye-end than when it shows objects erect. The maker takes care to place
the cross-hairs at right-angles to each other in their ring, or diaphragm ; and gene-
rally he so places the ring in the telescope, that when levelled, they shaJl be reii
and hor. sometimes, however, this is neglected ; or the ring may by accident be-
come turned a little. To be certain that one hair is vert, (in which case the other
must, by construction, be hor,) after having adjusted the bubble-tubes, level the in«
strnment carefully, and take sight with the telescope at a plumb-line, or other yert

straight edge. If the vert hair coincides with this object,
it is, sofar^ in adjustment ; but if not, then loosen sKghtlv
only two adjacent screws of the four, pp i t. Fig 1 ; and
with a knife, key, or other small Instrument, tap verj
gently against the screw-heads, so as to turn the rin^ »
little in the telescope; persevering until the hair be*
comes truly vertical. When this icr done, tighten the
screws. In the absence of a plumb-line, or vert stsulgfat
edge, sight the cross-hair at a Tery small distinol
point; and see if the hair still cuts that point, when
the telescope is raised or lowered by revolring it on
its axis.
The mode of performing the foregoing will be readily
understood ft'om this Fig, which represents a section across the top part of the tele>
acope, and at the cross-hairs. The hair-ring, or diaphragm, a; vert hair, v; tele*
scope tube, g ; ring outside of telescope tube, d; & is one of the four capstMi-headed
screws which hold the hair-ring, a, in its place, and also serve to a^jnst it. The
lower ends of these screws work In the thickness of the hair-ring; so that when
they are loosened somewhat, they do not lose their hold on the ring. Small

THE EKOIKEES'S TIUXSIT.

295

mO

washers, c, are placed under the heads h of the screws. A space ^ y is left around
each screw where it passes through the telescope tube, to allow the screws aud ring
together to be moved a little sideways when the screws b are slightly loosened.

Fourth, to see tliat the wertical hair is In the line of colU-
matlon. Flant the tripod firmly upon the ground, as at a. Level the inst ;
clamp it; and direct the vert hair by means of tangent-screw O ffigs. 1 and 2)
upon some convenient object h\ or if there is none such, drive a thin stake, or a
ennin-pin. Then revolving the telescope vert on its Hxis, ^

observe some object, as c, where the vert hair now strikes ; ^ a ^^^

or if there is none, place a second pin. Uoclamp the instru- « «^^

ment by the clamp-screw H ; and turn the whole upper • "

part of it around until the ven hair again strikes b. JPig, 4,
Clamp again ; and again revolve the telescope vert on its
axis. If the vert h»ir now strikes e, as it did before, it shows that c is really
at 0 ; and that 6, a, e, are in IM^ same straight line ; and therefore this adjustment
is in order. If not, observe where it does strike, say at m, (the dist a m being
taken equal to a c,) and place a pin there also. Measure m c ; and place a pin
at v, in tne line m c, making m v <— one-fourth of m c. Also put a pin at 0, half-
way between m and c, or in range with a and b. By means of the two hor
screws that move the ring carrying the cross-hairs, adjust the vert hair until it
euts V. Now repeat the fntire operation ; and persevere until the telescope, after
being directed to b, shall stVike the same object 0, both Hmes, when revolved on
its axis. See whether the movement of the ring in this 4th adjustment has dis-
turbed the verticality of the hair. If it has, repeat the 3d adjustment. Then re-
peat the 4th, if necessary ; and so on until both adijustments are found to be right
at the same time. Thus a straight line mav be run, even if the hairs are out of
adjustment ; but with somewhat more trouble. For at each station, as at a, two
back-sights, and two fore-sights, a c and a m, may be taken, as when making the
adjustment ; and the point 0, half-way between c and m, will be in the straight line.
The inst may then be moved to 0, and the two back-sights be taken to a ; and so on.

Angles measured by the transit, whether vert or hor, will evidently not be
tifected by the hairs being out of a4justment, provided either that the vert
liair is truly Tert. or that we use the inler^oHon of the hairs when measuring.

The foreproiniT ^^^^ All the a^instments needed, unless the tran-
sit is reqnlrea for levelUi^, in which case the following one muse be attended to :

To adjust the lontr bnbble*t1Ihe« F F, Fie. l, we first place the line
of sight of the telescope hor, and then make the bubble-tube hor, so that the
two are parallel. Drive two pegs, a and b Fig. 5, with their tops at precisely
the same level (see Bem. p. 296) and at least about 100 ft. apart ; 800 or more
will be better. Plant the inst Armly, in range with them, as at c, making^ c
an aliquot part of a b, and as short as will permit focusing on a rod at 6. The
inst need not be leveled. Suppose the line of sight to cut e and d. Take the
readings b e and a d. Their diff is be — ad=^an — ad=*dn\ and ah-.ac:
dnids'i s being the height of the target at a when the readings (a «, b 0) on the

two stakes are equal. as==ad-\-ds^ad-\ r — ' If the reading on a

taceeeds that on b (as when the line of sight is vfg) the diff of readings is = a ^ —

bf=sag — ai^gi\ smd as = a g — g s=aaff — ^ — j- — • Sight to «, bring the

babble to the cen of its tube by means of the two small nuts n n at one end of the
tube. Fig. 1, and assume that the telescope and tube are parallel.* The zeros of

* Thla B«0eeM s mnmll «iTor due to the oarralnre of the earth ; fDr a hor line at v ia v h, tao*
flaatiml to tlM earved (or " tofwl") torfiaoe of still water at «, whereae » • Is tangential to water aarf
at a point midwaj between a and h. Henoe if the telesoope at « points to a li will not be parallel te
the level bobbto-tnbe. To allow for this, and for the reftvotloa bj the air, wUeh diminUhM the
error, rsiae the tarfet on • to a point h above a. h* — .0000000205 x square of a 0 in (I ; bat when
• e is S30 ft, Jk a is only aboni one tenth of an inoh and barely oovers the apparent thlekness of Um
-bnlrlatkn '

296!

THE ENaiNEER'S TRANSIT.

the vert circle, and of its vernier, may now be aAjiitted, if they require it, by
loosening the vernier screws and then moving the vernier until the two coin-
cide. ^ , . - ,

Rem. If no level is at hand for levelling the two pegs o and &, it may be done
by the transit itself, thus : Carefully level the two short bubbles, by means of the
levelling-screws K. Drive a peg m, from 100 to 300 feet from the instrument o.
Then placing a target-rod on m, clamp the target tight at whatever height, as sv,

the hor hair happens to cut it ; it being of no im-

l^ L portance whether the telescope is level or not;

TV (J) although it might as well be as nearly so as can

\ X conveniently be guessed at. Clamp the telescope

g^ — JJ. in its position by the clamp-ring K, Fig. 1. Re-

^ volve the inst a considerable way round; say
iJifiT. 0. nearly or quite half way. Place another peg n,

atprecUdy the same diet from the instrument that m is; and continue to drive it un-
til the hor hair cuts the target placed on it, and still kept clamped to the rod, at the
same height as when it was un m. When this is done, the tops of the two pegs are
on a level with each other, and are ready to be used as before directed.

When a transit is intended to be used for surveying farms, Ac, or for retracing
lines of old surveys, it is very useful to set the compass so as to allow for the ** va-
riation" during the interval between the two surveys. For this purpose a
'' TArlatton- vernier " is added to such transiCB ; and also to the oompaos.
When the graduations of a transit are figured, or numbered, so as to read both

H) 0 10

ways from aero, thus, i n ii 1 1 1 h i 1 1 1 1 1 1 1 1 1 1 1 1 I m the vernier also is mada

double ; that is, it also is graduated and numbered from its sero both ways. In thia
case, if the angle is measured from zero toward the right hand, the reading must be
made from the right hand half of the vernier ; and vice versa. If the figuring la
single, or only in one direction, from zero to 360^, then only the single vernier la
necessary, as the angles are then measured only in the direction that the figuring
counts. ICngineers differ in their preferences for various manners of figuring the
graduations. The writer prefers from zero each way to 180^, with two double ver-
niers.

To replace cross-hairs in a IcTel, or transit. Take out tiie tube
from the eye end of the telescope. Looking in, notice which side of the oroM-
hair diaphragm is turned toward the eye end. Then loosen the four screws which
hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new
hairs with beeswax, varnish, glue, or gum-arabic water, Ac. This requires care.
Then, to return the diaphi-agm to its place, press firmly into one of the screw-holes
on the circumf of the diaphragm itself, the end of a piece, of stick, long enough to
reach easily into the telescope as far as to where the diaphragm l^Iongs. By this
stick, as a handle, insert the diaphragm edgewise to its place in me telescope, and hold
it there until two cpposUe screws are put in place and screwed. Then draw the stick
out of the hole in the diaphragm ; and with it turn the diaphragm until the same
side presents itself toward the eye end as before ; then put in the other two screws.

The so-called cross- hairs are actually spider-web, so fine as to be barely visible to
the naked eye. Holler A Brightly use very fine platina wire, which is much better.
Human hair is entirely too coarse.

To replace a spirit-level, or bnbble^lass. Detach the level from
the instmment; draw off its sliding ends; push out the broken glass vial, and the
cement which held it ; insert the new one, with the proper side up (the upper side
is always marked with a file by the maker); wrapping some paper around its ends,
if it fits loosely. Finally, put a little putty, or melted beeswax over the ends of the
vial, to secure it against moving in its tube.

In purchasing instruments, especially when they are to be used far from a maker,
it is advisable to provide extras of such parts as may be easily broken or lost ; such
as glass compass-covers, and needles; atjQusting pins; level vials; magniflen, Ao,

Theodolite adjustments are performed like those of the level and transit.

let. That of the cross-hairs; the same as in the level.

2d. The long bubble-tube of the telescope ; also as in the level.

8d. Th^ two short bubble-tubes ; as in tne transit.

4th The vernier of the vert limb ; as in the transit with a vert circle.

5th. To see that the vert hair travels vertically ; as in the fourth adjustment
of the transit. In some theodolites, no adjustment is provided for this ; but in
Isrm onaa it is provided for by screws under the feet of the standards.

Somttimw • second telescofKi is added ; it Is p^iic«d belov the hor limb, and to

THE BOX OB POCKET SEXTANT.

297

called a toate?ur. It has its own clamp, and tangent-screw. Its use is to ascertain
whether the sero of that limb has moved during the measurement of hor angles.
When, previously to beginning the measurement, the zero and upper telescope are
directed to^inund the first object, point the lower telescope to any small distant
object, and then clamp it. During the subsequent measurement, look through i^
from time to time, to be sure that it still strikes that object ; thus proving that nt
slipping has occurred.

THE BOX OR POCKET SEXTANT.

Ths portability of the pocket sextant, and the fact that It reads to single minutes,
render it at times very useful to the engineer. By it, angles can be measured while
in a boat, or on horseback ; and in many situations which preclude the use of a
transit. It is useful for obtaining latitudes, by aid of an artificial horizon. When
closed, it resembles a cylindrictu brass box, about 3 inches in diameter, and 1)^
inches deep. This box is in two parts ;
by unscrewing which, then inverting
one i>art,,and then screwing them to-
gether again, the lower part becomes a
handle for holding the instrument.
Looking down upon its top when thus
arranged, we see, as in this figure, a
movable arm I C, called the index,
which turns on a center at C, and car-
ries the vernier Y at its other end. Q
6 is the graduated arc or limb. It
actually subtends about 13P, but is di-
vided into about 146^. Its zero is at
one end. Its graduations are not shown
in the Fig.

Attached to the index is a small mov-
able lens, (not shown in the figure,)
likewise revolving around C, for read-
ing the flue divisions of the limb. When
measuring an angle, the index is moved
by turning the milled-head P of a
pinion, which works in a rack placed within the box. The eye is applied to a eir*
cnlar hole at the side of the box, near A. A small telescope, about 3 inches long,
; accompanies the instrument; but may generally be dispensed with. When so, the
eye-hole at A should be partially closed by a slide which has a very small eye-hole
in it ; and which is moved by the pin A, moving in the curved slot. Another slide,
at the nde of the box, carries a.dark glass for covering the eye-faole when observing
the ran. When the telescope is used, it is fastened on by the milled-head screw T.
The top part shown in our figure, can be separated from the cylindrical part, by
removing 3 or 4 small screws around its edge ; and the interior can then be exam-
ined, and cleaned if necessary. Like nautical, and other sextants, this one bm
two principal glasses, both of them mirrors. One, the Index-fplass, is attached
to the underside of the index, at C; its upper" edge being indicated by the
two dotted lines. The other, the Moriaon-KliMiS) (because, when meas-
uring the vert angles of celestial bodies, it is directed toward the horizon,) is also
within the box; the position of its upper edge being shown by the dotted lines at
R. The horizon-glass is silvered only half-way down ; so that one of the observed
objects may.be seen directly through its lower half, while the image of the other
object is seen in the upper half, reflected from the index-glass. That the instrument
may be in a4justment, ready for use, these two glasses must be at right angles to the
plane of the instrument ; that is, to the under side of the top of the box, to which they
are attached; and must also be parallel to each other, when the zeros of .the vernier
and of the Umb coincide. The index-glass is already permanently fixed by the
ma^T, and requires no other a4ju8tment. But the horizon-glass has two adjust-
ments, which are made by a key like that of a watch, and having a milled-head K.
It is screwed into the top of the box, so as to be always at hand for use. When
needad, it is unscrewed. This key fits upon two small square-heads, (like that for

298

THE COMPASS.

winding a watch;) one of which is ihown at S; while the other is near it, but on the
SIDE of the box. These squares are the heads of two small screws. Jf the
horlEon glass H should, aa in this sketch, (where it is shown endwise,) not be at
right angles to the top U HJ of the box, it is brought right by turning the square-
bead S of the screw S T ; and if, after being so far rectified, it still is not parallel to

thn index-glass when the zeros coincide, it is moved
a little backward or forward by the mjuare head
at the side.

To adjust a box sextant, bring the two
aeros to coincide precisely ; then look through the
eye-hole, and the lower or unsilvered part of the
horixon-glasB, at some distant object. If the instru-
ment is in adjustment, the object thus seen directly,
will coincide precisely with its reflected image,
seen at the same time, at the same spot. But if it
is not in ac^ustment, the two will appear separated
either hor or vert, or both, thus, * • ; in which case
apply the key E to the square-head S ; and by turning it slightly in whichever direc-
tion may be necessary, still looking at the otjject and its ima^e^ bring the two into a hor
position, or on a level with each other, thus, * •. Then apply the key to the square-
head in the side of the box; and by turning it slightly, bring the two to coincide
perfectly. The instrument is then ai^justed.

In some instruments, the hor glass has a hinge at v, to allow it play while being
adjusted by the single screw S T ; but others dispense with this hinge, and use two
screws like S on top of the box, in addition to the one in the side.

If a sextant is used for measuring vert angles by means of an artificial
boriEon, the actual altitude wilF be but one-half of that read off on the
limb ; because we then read at once both the actual and the reflected angle. The
great objection to the sextant for engineering purposes, is that it does not measure
angles horizontally, as the transit dues ; unless when the observer, and the two ob>

jects happen to be in the same hor plane.
Thus an observer with a sextant at A, if
measuring the angle subtended by the
mountain-peaks B and C, must hold the
graduated plane of the sextant in the
plane of A B C ; and must actually meas-
^^ ,' ,-' I ^ ; ure the angle BAG; whereas what he

g^k^*^':'- ' wants is the hor angle nAm. This is

^f""" -'Wl greater than BAG, because the dista An

A and A m, are shorter than A B and A G.

The transit gives the hor angle n A m, be-
<iau8e its graduated plane is first fixed hor by the levelling-screws ; and the subse-

Suent measurement of the angle is not affected by his directing merely the line of
[ght upward, to any extent, in order to fix it upon B and G. For more on this sub-
ject ; and for a method of partially obviating this objection to the sextant, see the
note to Example 2, Case 4, of " Trigonometry."

Tbe nautical sextant, used on ships, is constructed on the same principle
as the box sextant ; and its adjustments are very similar. In it, also, the index-
glass is permanently fixed by the maker ; and the horizon-glass has the two adjust-
ments of the box sextant. It also has its dark glasses for looking at the sun ; and
a small eight-hole, td be used when the telescope is dispensed with.

•-•-

THE COMPASS.

To adjust a Compass*

The first adjustment is that of the bubbles. Plant firmly ; and level th«

Instrument, in any position ; that is, bring the bubbles to the centers of their tubes.
Then turn the instrument half-way round. If tlie bubbles then remain at the cen-
ters, they are in adjustment;. but if not, correct one-half the diff" in each bubble,
by means of the adjusting-screws of the tubes. Level the instrument again ; tun
it half roimd ; and if the bubbles still do not remain at the center, the atiUusting-
■crews must be again moved a little, so as to rectify half the remaining diff. Gener*

THE COMPASS. 299

ally several trials must be thus made, until the bubbles will remain at the oente
while the compass is being turned entirely around.

Seeond adjustmeiit* Level the compass, and then see that the needle it
hor; and if not, make it so by means of the small piece of wire which is wrapped
around it ; sliding the wire toward the high end. A needle thus horizontally ad-
justed at one place, will not remain so if removed fietr north or south from that place.
If carried to tiie north, the north end will dip down ; and if to the south, the soutii
end will do so. The sliding wire is intended to counteract this.

Tliird a^Jnatment. This is always fixed right at first by the maker; that
is, the sights, or slits for sighting through, are placed at right angles to the compass
plate ; so that when the latter is levelled by the bubbles, the sights
are vert. To test whether they are so, hang up a plumb-line ; and
having levelled the compass, take sight at the line, and see if the
slits coincide with it. If one or both slits should prove to be
out of plumb, as shown to an exaggerated extent in this sketch.
It should be unscrewed from the compass, and a portion of its foot
on the high side be filed or ground off, as per the dotted line ; or
as a temporary expedient, a small wedge may be placed under the
low side, so as to raise it.

Foortb BdJaBtmeilt, to straighten the needle, if it should become bent.
The compass being levelled, and the needle hor, and loose on its pivot, see whether
its two ends continue to point to exactly opposite graduations, (that is, graduations
18€P apart ;) while the compass is turned completely around. If it does, the needle
is straight ; and its pin is in the center of the graduated circle ; bat if it does not,
then one or both of these require adjusting. First level the compass. Then turn It
until some graduation (say 90^) comes precisely to the north end of the needle. If
the south end does not then point precisely to the opposite 90° division, lift off the
needle, and bend the pivot-point until it does ; remembering that every time said
point is bent, the compass must be turned a hairsbreadth so as to keep the north end
of the needle at its 90^ mark. Then turn the compass half-way round, or until the
opposite 90° mark comes precisely to the north end of the needle. Make a fine pen*
<^ mark where the touth end of the needle now points. Then take off the needle,
and bend it until its south end points ha^f^ay between its 90° mark and. the pencil
mark, while its north end is kept at 90° by moving the eompass round a hairsbreadth.
Tlie needle will then be straight, and must not be altered in making the following,
adjostment, although it will not yet cut opposite degrees.

Flfih a4ius^i»eiit, of the pivot-pin. After being certain that the needle is
straight, turn the compass around until a part is arri ved at where the two ends of the
needle happen to cut opposite degrees. Then turn the compass quarter way around,
or through 90°. If the needle then cuts opposite degrees, the pivot-point is already
in adjustment ; but if the needle does not so cut, bend the pivot-point until it does.
Bapeat, if necessary, until the needle cuts opposite degrees while being turned entirely
•round.

Oare and nicety of observation are necessary in making these adjustments properly ;
because the entire enor to be rectified is, in itself^ a minute quantity; and the novice

it must be held parallel to the graduated circle. Otherwise annoying errors of
several minutes will be made in a single observation ; and the accumulation of two
or three such errors, arising from a cause unknown to him, may compel him to
abandon the ac^ustments in despair. This su^estion applies also to the reaiding of
angles taken by the transit, Ac ; although the errors are not then likely to be so
great as in the case of the compass. In purchasing a magnifier for a compass, see
that DO part of it, as hinges, or rivets, are made of iron ; for such would change the
direction of the needle.

If the sight-slits of a compass are not fixed by the maker in line with the two
opposite zeros, the engineer cannot remedy the defect. This can be ascertained by
passing a piece of fine thread through the slits, and observing whether it stands
precisely over the zeros.

THE COMPAfiS.

THE COUFABB.

I!
II'

|3|3i||| III |3

111

til

1^

> ts

f i
i i

II
if!

I L,

; Hi

bSj

302 OOlffTOtm LIKEB.

United StatflB, by Henry GtuuMtCi In 17th Annual Beport ef tf. 8. Geological
Survey, 1896-^

Electrietty, either atmospheric, or excited by rubbing the glass ooy«r of
the compass box, sometimes gives trouble. It may be removed by touching the
glass with the moist tongue or finger.

DEMAOHETIZATIOV.

The needle, if of sqft metal, Bometimeo loses part of its magnetism, and consequently
does not work well. It may be restored by simply drawing the north pole of a
common magnet (either straight or horseshoe) about a dozen times, from the center
to the end of the south half of the needle ; and the south pole, in the same way, along
the north half; pressing the magnet gently upon the needle. After each stroke,
remove the magnet several inches from the needle, while bringing it back to the
center for making another stroke. Each half of the needle in turn, while being thus
operated on, should be held flat upon a smooth hard surface. Sluggish action of the
needle is, however, more generally produced by the dulling or other iujury of the
point of the pivot. RemagnetiEing will throw the needle out of balance ; which must
be counteracted by the sliding wire.

In order to prevent mistakes by readlnn^ sometimes from one end,
and sometimes from the other end of the needle, it is best to always point the N of
the compass-box toward the object whose bearing is to be taken ; and to read off
from the north end of the needle. This is also more accurate.

OONTOUB LINES.

A OOHTOUB um is a curved hor one, every point in which represents the same level ;
thus each of the contour lines SSc, 91c, 94c, itc. Fig 1, indicates that every point in
the ground through which it is traced is at the same level ; and that that level or
height is everywhere 88, 91, or 94 ft above a certain other level or height called
datum ; to which all others are referred.

Frequently the level of the starting point of a survey is taken as being 0, or zero,
or datum ; and if we are sure of meeting with no points lower than it, this answers
every purpose. But if there is a probability of many lower points, it is better to
assume the starting point to be so far above a certain supposed datum, that none of
these lower points shall become minus quantities, or bdow said supposed datum or
zero. The only object in this is to avoid the liability to error which arises when
some of the levels are -|-» or plus ; and some — ^ or minus. Hence we may assume
the level of the starting point to be 10, 100, 1000, Ac, ft above datum, according to
circumstances.

The vert dists between each two contour lines are supposed to be equal ; and in
railroad surveys through well-known districts, where the engineer knows that his
actual line of survey will not require to be much changed, the dist may be 1 or 2 ft
only ; and the lines need not be laid down for widths greater than 100 or 200 ft on
each side of his center-stakes. But in regions of which the topography is compara-'
tively unknown ; and where consequently unexpected obstacles may occur which
require the line to be materially changed for a considerable dist back, the observa-
tions should extend to greater widths ; and for expedition the vertical dists apart
may be increased to 3, 5, or even 10 ft, depending on the character of the country,
Ac. AlsOj when a survey is made for a topographical map of a State, or of a county,
vert dists of 5 or 10 ft will generally suffice.

Let the line A B, Fig 1, starting from 0, represent three stations (S 1, S 2, 8 3,) of
the center line of a railroad survey ; and let the numbers 100, 108, 101, 104, along
that line denote the heights at the stakes above datum, as determined by levelling.
Then the use of the contour lines is to show in the offlcH what would be the effect
of changing the surveyed center line A B, by mrving any part of it to the right oi

CONTOUB JUNES.

303

Iflft hand.* Thug, if it should be moved 100 ft to the left, the starting point 0 wonl^
be on ground about 6 ft higher than at present ; inasmuch as its leyel would then
be about 106 ft above dktum, instead of 100. Station 1 would be about 7 ft higher,
or 110 ft instead of 103. Station 2 would be about 7 ft higher, or 108 ft instead of
101. If the line b<« thrown to the right, it will plainly be on lower ground.

The field obeervat^'ons for contour lines are sometimes made with the spirit-level;
but more frequently oy a slope-man. with a straight 12-ft graduated rod, and a slope
instniment, or clinometer. At each station he lays his rod upon the ground, as

FIg.l.

•
nearbr a^ right angles to the center line A B as he can Judge by eye ; and placing
the slope instrument upon it, he takes the angle of the slope of the ground to the
nearest ^ of a degree. He also observes how far beyond the rod the slope continuee
the same ; and with the rod he measures the dist. Then laying down the rod at that

Kint also, he takes the next slope, and measures its length ; and so on as far as may
Judged necessary. His notes are entered in Ids field-book as shown in Fig 2 ; the
angles of the slopes being written above the lines, and their lengths below ; and
should be accompanied by such remarks as the locality suggests ; such as woods,
rocks, maryih. sand, field, garden, across small run, ftc, Ac.

* la thni aiing the word* right and left wc an lUppoMd to have our baeki turned to the ■tartiog
point of the survey. In a river, the rliplit bniik or shore is that which
IS on the right band as we descend it, that is, in speaking of its right or left

huk. ve are lODpoMd to hare oar backs turned toward! Ita head, or origin ; and bo with a surrey

804-

CONTOUR LINES,

I-

91

''m^i'

64- 70

It is not abeolately necessary to represent the slopes roughly in the fleld-book, aa
in Fig 2; for by usin^ the sign + to signify "up;" — "4own;" and = "'leTel,*'
the slopes may be vrnt-
ten in a straight line,
as in Fig 2^.

The notes naving been
taken, the preparation
of the contour lines by
means of them, is of
course office-work ; and
is usually done at the
same time as the draw-
ing of the map, &c. The.
field observations at each
station are then sepa-
rately drawn by protrac-
tor and scale, as shown
in Fig 3 for the starting
point O. The scale should not be less than about -^ inch to a ft, if anything Iik«
accuracy is aimed at. Suppose that at said station the slopes to the right, taken in
their order, are, as in Fig 2, U°, 4°, and '26P ; and those to the left, 20°, lO^, and IQP ;
and their lengths as in the same Fig. Draw a hor line h o. Fig 3 ; and consider the
center of it to be the station-stake. From this point as a center, lay off these angles
with a protractor, as shown on tho arcs in Fig 3. Then beginning say on the right
hand, with a parallel ruler draw the first dist a c, at its proper slope of 16^ ; and of
its proper length, 45 ft, by scale. Then the same with c y and yt.Do the same with
those on the left hand. We then have a cross-sectitm of the ground at 8ta 0. Then
on the map, as in Fig 1, draw a line as m n, or A 10, at right angles to the line of road,
and passing through tha station-stake. On this line lay down nie Jior dists a d, d «, s «,
ae^eg^gk^ marking them with a small star, as is done and lettered in Fig 1, at 8ta O.

When extreme accuracy is pretended to, these hor dists must be found by measure
on Fig 3 ; but as a general rule it will be near enough, when the slopes do not ex-
ceed 10°, to assume them to be the same as the sloping diets measured in the field.
Next ascertain how high each of the points cy tint is above datum. Thus, measure
by scale the vert dist ae. Suppose it is found to be 5 ft ; or in other words, that e
is 5 ft below stationHBtake 0. Then since the level at stake 0 is 100 ft above datum,
that at c must be 6 ft less, or 100 — 6 = 95 ft above datum ; which may be marked in
light lead-pencU figures on the map, as at d, Fig 1. N6xt for the point y, suppose
we find « 2/ to be 11 ft, or y to be 11 ft below stake 0 ; then its heiglit above datum
must be 100 — 11 =s 89 ; which also write in pencil, as at s. Proceed in the same
way with t. Next going to the left hand of the station-stake, we find « I to be say
2 ft ; but Z is above the level of the station-stake, therefore its height above datum is

Biff. 8.

100 4- 2 » 102 ft, as figured at e on the map. Let ng be 5 ft; then is n, 100 -f- 0 ^
105 ft above datum, as marked at a ; and so on at eacn station. When this has been
done at several stations, we may draw in the contour lines of that portion by hand
thus: Suppose they are to represent vert heights of 3 ft. Beginning at Station O
(of which the height above datum is 100 ft) to lay down a contour line 103 ft abova
datum, we see at once that the height of 103 ft must be at ^, or at ^ the dist from «
to g. Make a light lead-pencil dot at t ; and then go to the next fetation 1. Here
we see that the height of 103 ft coincides with the station-stake itself; place a dot
there, and go to Sta 2. The ^evel at this stake is 101 ; therefore the contour for lOP

CONTOUli LDcaa.

305

ft mtut evidently be 2 ft higher, or at <, ^ of the dl^t fh>m Sta 2 to +104 ; theretiort
make a dot at i. Then go to Sta 3. Here the leTel being 104 aboye datum, the con-
tour of 103 must be at y, or i of the diet from Sta 3 to +99 ; put a dot at y. Finally
draw by hand a curving line through ^ SI, i, and y ; and the contour line of 103 ft
ii done. All the others are prepared in the same way, one by one. The level of each
must be figured upon it at short intervals along the map, as at 103 c, 106 c, Ac

Or, instead of first placing the + points on the map,l;o denote the slope dists actu-
ally measured upon the ground, we may at once, and with lees trouble, find and show
those only which represent the points ty S 1, t, y, Ac, of the contours themselves.
Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec-
tion c < of the ground has been prepared as before ; and that we want the hor dista
from the stake, to contour linea for 94, 97, 100 ft, Ac, 3 ft apart vert.

Draw a vert line t; 2, through the station-stake, and on it by scale mark levels
of 94, 97, 100, dba ft. This is readily done, inasmuch as we have the level 104 of
the stake already given. Through these levels draw the hor lines a. b, m, n, <&c.
to the ground-slopes. Then these lines, measured by the scale, plainly give the
requirea dists.

When the ground is very irregular transversely, the cross-sections must be
taken in the field nearer together than 100 ft. The preparation of contour lines
will be greatly facilitated by the use of paper ruled into small squares of not less
than about ^ inch to a side, for drawing the cross-sections upon.

When the ground is very steep, it is usual to shade such portions of the map to
represent hill-side. The closer together the contours come, the steeper of course
is the ground between them ; and the shading should be proportionally darker
at such portions. But for working maps it is best to omit the shading.

In surveys of wide districts, the transit instrument with a graduated vertical
circle or arc, g, p. 291, ia used for measuring the angles of slope, instead of
the common slope-instrument.

In many cases, notes similar to the following will serve the purpose of contour
lines on railroad surveys.

BUCO..
61..

es..

6S.

... — S.1B. +S. IL.
... + 2.2B. — 1.8L.
... = 1. E. + 4. 1 L.

Wblek meaai tbat at ttotlon 40, the slope of tbe groand on the right, m nearly as he can Jadge by
0jm, •r by hi* band-lerel, is aboat S ft downward, for 1 ehain, or 100 ft ; and on the left, about 2 ft
apward In 1 ehnln. At 61, 2 ft ap, in Zehatns t* the right; and 1 ft down in S chains to the left.
A% tS, l«y«l for 1 ohaln to tbe right; and ascending 4 ft in 2 chains to the left. At 6S, the same as at
n, Ai aoBie spots it will be well to add a sketch of a orons-seotion, like Fig 2 ; only, instead of the
■agies, use ft of rise or fall, to indicate the slopes, as J udged bj eye, or by a haod-level. By this
■ethod, the resolt at every station will be somewhat in error; bat these small errors will balance
•aeh other m» nearly that the total may be regarded as sufBeiently correot for all the parpoees of a
pnUmioMxy eettmate of the oost of a rood. When the final stakes for guiding the workmen are
pioflod* the slopes should be sorefliUy taken, in order to ooloalato the qnontity of ezeavation aooa-
ratoly for payment.

20

THE LEVEL.

Qui ptDS 1 J which coDHoe the semlclrculsr clipi 1 1, aud iheu oprnlg; the clip*.
The pins should be tied l« the Ys, by pieces ot string, to preveut Ihofr being iom.
(be ilide of the oMwt-gbai O, is nio>ed burkward ur rorw&rd by a rauk niid plnian,
bf meBDB or the mlHeS hewl A. The slide of the lyt-gkia £. la moved Id [be same
WB* br the milled head e. A cTlindrlual lube ef brass, oallal s lAmfe, is usua]]*
hirnMied with «*eh kTel. It la Intended to bo slid on to the objeci-cnd O of the
teleicape, to prerent Ibe ^are of the sun upua the objecl-glass, when the nun ia
low. At Biaui outer rlDseiiclrcUag the telescope, and carrrlOE 4 small cspstan-
beadea wrewa; tmof wfilcb.pp, are at lop and boiiom; while the other two,
of whkh I la ODe, areittba^ea, and M right inelei top p. laslde a[ this outer
ring la another, loaldo of the telescope, atid wblc^b bas stretched acrosa it two

«, when cairjlQ^

.._ _... th'm ™r'!J'^

be Juii^ed'bf ejar^ls euablea the lereller lo'^see^ tbaTt/o i(^m»n hulda b"^

ia desired, as la KHuetimei the case, when itsking out work, ^t may ba obtained (^
IA* tiutrumrnt ij in perfiet a^ailmtnl, and UvUai) by tighllng at a plmnb-llne. or
olhor »ert oltjoct, and then turning tbe tideetopo a little in Its tiw aa to bring the

Uw teleKopa and Y, to aare that tron'cle In fiiiure. Heller & Brightly, howaTw.

The small holes around the beadaofthe 4 small capstaii-screwsti,l,JustnrerfedU^
are for admitting the end of a small steel pin, or lerer, fbr tumlogtbem. If flnt

will be lDworM^ and Iba liorltnntal hair with it. But un loiAing through the tal» I

THE LEVBU 307

■cope th«7 will appear to be mSsed. If first the lower one be looeened, and the npper
one tightened, the hor hair will be Mctnally raised, but apparently lowered. This is
because the glasses iu the eye-piece B reTerse the apparent position of objects intid€
cf the telescope ; which effect is obTiated, as regcurds exterior ol^Jects, by means of
the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a
hair appears to strike too high, it must be raised still higher ; if it appears to be
already too &r to the right or left, it must be actually movcKl still more in the same
direction.

This remark, however, does not apply to teleacopn which make objects appear
iUTerted.

There is no danger of li^urlng the hairs by these motions, inasmuch as the four
screws act against the ring only, and do not come in contact with the hairs them-
t^lves.

Under the telescope is the bubble-tube D D. One end of this tube can be raised or
lowered slightly by means of the two capstan-headed nuts n n, one of which must
be looeened before the other is tightened. On top of the bubblo'tube are scratches
for showing when the bubble is central in the tube, frequently these scratches, or
marks, are made on a strip of brass placed above the tube, as in our fig. There are
several of them, to allow for the lengthening or shortening of the bubble by changes
of temperatuie. At the other end of the bubble-tube are two smidl capstan-screws,
placed on opposite sides horizontally. The circular head of one of them is shown
near L By means of these two screws, that end of the tube can be slightly moved
hor, or to right or left. Under the bul>ble-tube is the bak Y F ; at one end of which,
as at y, are two large capstan-nuts to w, which operate upon a stout interior screw
which forms a prolongation of the Y. The holes in these nuts are lai^r than the
others, as they require a larger lever for turning them. If the lower nut is loosened
and the upper one tightened, the Y above is raised ; and that end of the telescope
becomes farther removed from the bar; and vice versa. Some makers place a similar
screw and nuts under both Ys ; while others dispense with the nuts entirely, and
substitute beneath one end of the bar a large circular milled head, to be turned by
the fingers. This, however, is exposed to accidental alteration, which should be
avo&ded.

When the portions above m are put upon m. and fastened bv the screw Y, all
the upper part may be swung round hor, in either direction, oy loosening the
elamp-serew H ; or such motion may be prevented by tightening thatecrew.
It frequently happens, after the telescope has been sighted very nearly upon an
object, and then clamped by H, that we wish to bring the cross-hairs to coincide
more precisely with the object than we can readily do by turning the telescope by
kand: and in this case we uee the tanfrent-ticrew 5, by means of which a
Bliffht but steady motion may be given after the instrument is clamped. For
fuller remarks on the clamp and tangent-screws, see '* Transit."

The parallel plates m and S are operated bv four levelllnipHierews ;
three or which are seen in the figure, at K K. The screws work in sockets B;
which, aa weU as the screws, extend above the upper plate. When the instrument
is placed on the ground for levelling, it is well to set it so that the lower parallel
plate S shall be as nearly horizontal as can be roughly judged by eye ; in order
to avoid much turning of the levelling screws K ^ in making the upper plate
m hor. The lower plate S, and the brass oarts below it, are together called the
tripod-taead ; and, in connection with three wooden legs Q Q Q, constitute
the tripod. In the figure are seen the heads of wing-nuts J which confine the
legs to the tripod-head. Under the center of the tripod-head should always be
placed a small ring, from which a plumb-bob may be suspended. This is not
needed in ordinary levelling, but becomes useful when rangmg center-stakes, &c.

To adjast a Irevel.

This is a qnite simple operation, but requires a little patience. Be careful to avoid
thraininff any of the screws. The large Y nuts ie w sometimes require some force to
ttoH them ; but it should be applied by pressure, and not by blows. Before begin-
rJDg to su^nst, attend to the o^ect-glass, as directed in the first sentence under ^^ To
•i^nst a plain transit.**

Three at^nstments are necessary ; and rrnist be made in the following order:

First, that of tlie cross-bairs ; to secure that their intersection shall
toatinue to strike the same point of a distant object, while the telescope is being
tnnu'd round a complete revolution in its Ys. This is called ac^usting the line
sf eolllmation, or sometimes, the line of sight; but it is not strictly the line
of (tight until all the adjustments are finished; for until then, the line of coUimation
vni not serve for taking levelling sights. If eross-liairs brealK* see p 296.

Second* Miat of Uie bnbble-tnbe D D, to place it parallel to the Une

308 TBB LBYBL.

0f coUimatlon. preTiomly •4|asted; so that when the bahble stands at the centra o(
ItD tube, indicating that it is lerel, we know that onr sight through the telescope is
hor. To replace broken bubble tabe, see p 296.

Tbird, tbat of tbe Ts, by which the telescope and bubble-tube a^re supported;
flo that the bubble-tube, and line of sight, shall be perp to the yert axis of the instru-
ment; so as to remain hor while the telescope is pointed to objects in diff directions,
as when taking back and fore sights.

To make tbe first adjastmenty or that of the cross-hairs, plant the
tripod ^r/n2y upon the ground. In this adjustment it is not necessary to lerel the
instrument. Open the clips of the Ys ; unclamp ; draw out the eye-glass E, until
the cross-hairs ieure aeen perfectty cUar ; sight the telescope toward some clear dis-
tant point of an object ; or still better, toward some straight line, whether yert or
not. More the object-glass 0, by means of the milled head A, so that the object shsJI
be clearly seen, wltbout parallax, that is, without any apparent dancing
about of the cross-hairs, if the eye is moved a little up or down or sideways. To
secure this, the object-glass alone is moved to suit different distances ; the eye-glass
is not to be changed after it lb once properly fixed upon the cross-hairs. The neglect
of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the
tangent-screw d, bring either one of the cross-hairs to coincide x>reciM/y with the
object. Then gently, and without jarring, revolve the telescope naif-way round in
its Ys. When this is done, if the hair still coincides precisely with the object, it is
in adjustment ; and we proceed to try tbe other hair. But if it does not coincide,
then by means of the i screws p, t, move the ring which carries the hairs, so as to
rectify, as nearly as can be judged hy eye, only one-fuUf of the error; remembering
that the ring must be moved in the direction opposite to what appean to be the
right one ; unless the telescope is an inverting one. Then turn the telescope back
again to its former position : and again by the tangent-screw bring the cross-hair to
coincide with the object. Then again turn the telescope half-way round as before.
The hair will now be found to be more nearly in its right place, but, in all probabil-
ity, not precisely so ; inasmuch as it is difficult to estimate one-half the error accu-
rately by eye. Therefore a little more alteration of the ring must be made ; and it
may be necessary to repeat the operation several times, before the adjustment is
perfect. Afterward treat the other hair in precisely the same manner. When both
are adjusted, their intersection will strike the same precise spot while the telescope
is being turned entirely round in its Ys. This must be tried before the aci^ustment
can be pronounced perfect; because at times the adjustment of the second hair,
slightly deranges that of the first one ; especially if both were much out in the b»
ginning.

To make the second adjustment, or to place the bubble-tube paralW
to the line of collimation. This consists of two dis>
tinct adjustments, one vert, and one hor. The first
of these is effected by means of the two nuts n n on
the vert screw at one end of the tube ; and the second
by tbe two hor screws at the other end,^, of the tube.
Looking at the bubble-tube endwise, from t in tbe
foregoing Fig, its two hor adjusting-screws 1 1 are
seen as in this sketch. The larger capstan-headed
nut helov), has nothing to do with the adjustments ;
it merely hold^ the end of the tube in its place.

. To make the vert adjustment of the bubble-tube, by means of the two nuts nn. Place
the telescope over a diagonal pair of the levelling-ecreWH K. K ; and clamp it there.
Open the clips of the Ys; and by means of the levelliug-screws bring the bubble to
the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and
put it back again in its reversed position. This being done, if tbe bubble still remains
at the center of its tube, this adjustment is in order ; but if it moves toward one end,
that end is too high, and must be lowered ; or else the other end must be rftised.
First, correct htdf the error by means of the levelling-screws K K, and then the re-
maining half by means of the two small capstan->headed nuts n». To roiM the end
n, first loosen the upper nut and then tighten the lower one ; to do which, turn each
nut so tiiat the near side moves toward your right. To louwr it, first loosen the lowei
nut, then tighten ttie upper one, moving the lutar side of each nut toward your ^fU
Having thus brought the bubble to the middle again, again lift the telescope out of
its Ys ; turn it end for end, and replace it. The bubble will now settle nearer the
center than it did before, but will probably require still further adjustment. If so,
correct haif the remaining error by the levelling-screws, and half by the nuts, as be*
fhre; and so continue to repeat tbe operation until the bubble remidns at the cental
in both positions. For another method, see '* To adjust the long bubble-tube,** p 2ML
Horizontal adjustment of bubble-tube ; to see that its axis is in the same plans
with nhat of the telescope, as it usually is in new instruments. It is not eesily d»

TEE LEVEL. 309

ranged, except by blows. Have the bubble-tube, as xxearly as may be, directly under
the telescope, or over the center of the bar T F. Bring the telescope over two of the
leTellingHScrews K K ; clamp it there ; center the bubble with said screws ; turn the
telescope in its Ts, say about ^ inch, bringing the bubble-tube out from over the
center of the bar, first on one side, then on the other. If the bubble stays centered
irhile so swung out, this adjustment is correct. It it,runs towajrd opposite ends of its
tabe when swung out on opposite sides of the center, move the end t of the tube by
the two horizontal screws 1 1 until the bubble stays centered when the tube is swung
out on either side. If the bubble runs toward the same end of its tube on both sidesy
tiie tube is not truly cylindrical, but slightly conical,* so that if the telescope is
tamed in its Ts the bubble will leave the center, even when the horizontal a^just-
ment is correct. It is known to be correct, in such tubes, if the bubble runs the Kune
diikmce from the center when swung out the same distance on each side.

Having made the horizontal adjustment, turn the telescope back in its Ys until the
bubble-tube is over the bar. Bepeat the vertUxU adjustment (p 308), which may have
become deranged in making this horizontal one. Persevere until both adjustments
are found to be correct at the same time.

To mabe tibe tliird adjustment, or to a4just the heights of th« Ts, m
■s to make the line of coUimation parallel to the bar V F, or perp to the vert axis
of the instrument. The other adjustments being made, fasten down the clips of the
Ts. Make the instrument nearly level by means of all four of the levelling-screws
K. Place the telescope over two of the levelling-screws which stand diagonally;
and leave it there undamped. Then bring the bubble to the center of its tube, by
the two levelling-screws. Swing the upper part of the instrument half-way around,
BO that the telescope shall again stand over the same two screws; but end for end.
This done, if the bubble leaves the center, bring it half-way back by the large cap-
stan nuts to, 10 ; and the other half by the two levelling-screws. Remember that to
raise the T, and the end of the bubble over «o, io, the lower tv must be loosened ; and
the upper one tightened ; and vice versa. Now place the telescope over the pttier
diagonal pair of levelling-screws; and repeat the whole operation with them, ilav-
Ing completed it, again try with the first pair; and so keep on until the bubble re-
mains at the center of its tube, in every position of the telescope.

Correct levelling may be performed even if all the foregoing adjustments are
out of order; provided each fore-sight he taken at preeiidy the tame distance from
the instrument as the back-sight is. But a good leveller will keep his instrument always
in acyustment; and will test the ac^ustments at least once a day when at work. As
much, however, depends upon the rodman, or target-man, as upon the leveller. A rod-
man who is careless about holding the rod vert, or about reading the sights correctly,
ibould he discharged without mercy.

The levelling-screws in many instruments become very hard to turn if dirty. Clean
with water and a tooth-brush. Use no oil on field instruments.

Forma for level note-books. When the distance is short, so as not to
fsqnire two sets of books, the following is perhaps as good as any.

I 8^olI.'S£tU^".,.| »»• |l*"l.|«»«««.| Cut. I «IL I

Bat on pnblic works generally the original field-books have only the first five cols.
After the grades have been determined by means of the profile drawn from these,
the re«nlta are placed in another book, which has only the first col and the last four.
In both cages, the right-hand page is reserved for memoranda. The writer considers
it best, both witii the level and with the transit, to consider the term " Station " to
apply to the whole dist between two consecutive stakes; and that its number shall
be that vrrftten on the last stake. Thus, with the transit, Station 6 means the dist
fin>m stake 5 to stake d; that it has a bearing or ocnirse of so and so; and its length
is so and mo. And with the level, Station 6 also means the dist from stake 5 to stake
6; the back-sight for that dist being taken at stake 5, and the Ibre-sight on stake
6; and thait the level, grade, cut, or fill is that at stake 6. The starting-point of the
nwej, wbether a stake, or any thing else, we call and mark simply 0.

• This defect can be remedied only by removing the tube and inserting a correctly-
ihaped one, and this is best done by an instrument-maker ; but correct work can
be done in qpite of it, Ihus: Make all the acyustments as nearly correct as possible.
Level the instrument. By turning the telescope in its Ts, make the vertical hair
coincide with a plumb-line or other vertical line, and make a short continuous knife-
Kiatch on the collar nearest the object-glass, and on the adjoining T. Lift the tele-
Kope ont of its Ts, turn it end for end, replace it in its Ts ; again bring the upright
hair vertical, and make on the other T a scratch coinciding with that on the collar.
Then, in levelling or in a4justing, always see that the scratch on the collar coincides
Mitt thai on the ac^oining T when the bubble-tube is under the telescope.

THB HAKD-IiBVKL
TOE BASD-LETEI.

ffll. M arpuTged bj Prof€«ir Locke, of (;indlDll»U,l«

SLmpljhuKIl

.IR it in DM hind, u

idlookinethroBgh
'nd!^TinVni^1

BDd 0 Ihe oW«t .

ebotlomof-hlA

™,ghtl,.top

KO.'^mmrftottl^

.»1dopcniog..nd

for sijurtizig

tie «irs, (an be 1

loFhed hKkwird a

ir poahod fom

•»rdby»™al1>pri

p1»c«J at so 1

iiglBo''HS=,«M^

■h the f.>rfB.B.

nlioned DpBniogB,

lil'y* Jrf fa^ a^^

f. M shown I.)

' tho^nlle dotl^'

linMCMdK; Mid

nWUi of™h8 tnbe rTi. Throup?^'

(b« wire shull Btaoir no piinllu ; bat ■pp**' tHd; BCBinM the dIi)boI irEui Ibe <J«
la allghll; moTod Dp or don. At «ch and oT t)i» tube B O la ■ dmdu pl«oB of

To adlaat tbe bond-level, lint fli
&U fbet Id 100 J'ards ^art, 'nitB beiDg done« ]
level marlij. ud take atght a the oUisr. If, then, tbe wire does not appau-

aleht a the oUisr. If, then, tbe wire does not appau- to be
■ illghtly huckwiLrt or forward, M the

hand-level tWelf, eieii If i[ la onUrely od( o( odjoil- ^ "^f

nhlecL u d. an that tha wire aoneui to cnl the eenter

ro''"iVhVr°CMermito"ri^rk"f^"h»l1--w'ay"to\wee"c°i^^^^ Then (> und in will be Che
two iBTel mirkB reijuirod. With o»re, these adjualnimti, when once msdo, will
remain in ordet for ream. The Intlrumenl gsnenllyhas aBmall ring r, for hanging

eiplorlng a roule. The heigh 1 of « bar* iiil I can be found bybeelnning st the ftiot.
and ijgtiling aheed at anj little chance objei^t which the onm-wlre ma; Btrlka, ■• a
pebble, cnlg, Ic; then going fonrard, ataud at Ibat object, and fix Che win m

a height eqnal 10 thac of the eye, lay bK^ feet, or whateTar it may be, WheUier
going DP or down It, If the bill la coTered with grau, bnihea, te, a target rod moR
be need for the fore-aighw ; and the tonstant height of the eye may be reganlsdH

IiBTXU.

311

To adiast a bailder*s plnmb-
leTei, todi stand it npon any two sup-
borta «» and it, and mark where the plumb-
line cuts at o. Then reverse It, placing the
foot t upon n, and d upon m, and mark where
the line now cute at e. Half-way between o
and e make the permuient mark. Whenerer
the line cuts this, the fiaet t and d are on a
level.

To adjast a slope-lnstrament, or clinometer. As usually made,
the bubble-tube is attached to the movable bar by a screw near each end^ and the
head of one of the screws conceals a small slot in the bar, which allows a slight vert
motion to the scr^w when loose, and with it to that end of the tube. Therefore, in
order to adjust the bubble, this screw is first loosened a little, and then moved up
«r down a trifle, as may be reqd. It is then tightened again.

312 ZJSVBLLING BY THB BABOMETEB.

liETEIililire BT THE BAROIHETER.

1. Many drcnmstancM combine to render the results of this kind of WTellino^ no*
reliable where great accuracy is required. This fact was most concluslyely proved
by the observations made by Captain T. J. Cram, of the U. 8. Coast Stirvey. See
Beport of U. 3. C. S., toI. for 1864. It is difficult to read oiT from an aneroid (the
kind of barom generally employed for engineering purposes) to within from two to
five or six ft, depending on its size. The moisture or dryness of the air aflTects the
results; also winds, the ricinity of mountains, and the daily atmospheric tides,
which cause incesHant and irregular fluctuations in the barom. A barom hanging
quietly in a room will often vary -^jf of an inch within a few hours, corresponding
to a diff of elev.ition of nearly 100 ft. No formula can posiiibly be deyised that shall
•mbrace these sources of error. The variations dependent upon temperature, latir
tnde, Ac, are in some measure provided for; so that with very ddicate instruments, •
skilful observbr may measure the diff of altitude of two points dose together, such
as the bottom and top of a steeple, with a tolerable confidence that he is within two
or three feet of the truth. But if as short an interval as even a few hours elapses
between his two observations, such changes may occur in the condition of the atmo-
sphere that he may make the top of the steeple to be lower than its bottom ; or at
least, cannot feel by any means certain that he is not ten or twenty ft in error; and
this may occur without any perceptible change in the atmosphere. Whenever prac-
ticable, therefore, there should be a person at each station, to observe at both points
at the same time. Single observations at points many miles apart, and made on dif-
ferent days, and in different states of the atmosphere, are of little value. In such
cases the mean of many observations* extending over several days, weeks, or months,
and made when the air is apparently undisturbeid, will give tolerable approximAtionB
to the truth. In the tropics the rang^ of the atmospheric pres is much leas than
in other regions, seldom exceeding ^ inch at any one spot; also more regular in
time, and, therefore, less productive oferror. Still, the barometer, especially eitiier
the aneroid, or Bourdon^s metallic, may be rendered highly useftil to the civil engi-
neer, in cases where great accuracy is not demanded. By hurrying from point tO
point, and especially by repeating, he can form a Judgment as to which of two sum-
mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying
party, may materially lessen their labors, especially in a rough country, by select-
ing the general route for them in advance. The accounts of the agreement within
a few inches, in the measurements of high mountains, by diff observers, at diff
periods ; and those of ascertaining accurately the grades of a railroad, by means of
an aneroid, while riding in a car, will be believed by those only who are ignonmt
of the subject. Such results can happen only by chance.

When possible, the observations at different places should be taken at the same
time of day, as some check upon the effects of the daily atmospheric tides ; and In
very important cases, a memorandum should be made of the year, month, day, and
hour, as well as of the state of the weather, direction of the wind, latitude of the
place, Ac, to be referred to an expert, if necessary.

The effecto of latitade are not included in any of our formulas. When
reqd they may be found in the table page 814. Several other corrections must be
made when great accuracy is aimed at ; Dut they require extensive tables.

In rapid railroad exploring, however, such refinements may be neglected, Inas-
much as no approach to such accuracy is to be expected ; but on the contrary, errors
01 from 1 to 10 or more feet in 100 of he^ht, wul frequently occur.

As a very roa§rli avera^r® ^^ iQ^y assume that the barometer falls -J^
inch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But
in fact its rate of tall decreases continually as we rise ; so that at one mile high it
fiEdls ^ inch for about 106 ft rise. Table 2 shows the true rate.

JLEVSLLING BY THE BABOM£T£B.

813

To «aeert«in tlie dUT of lieiirbt belweew two points.

Jlcn^E 1. Take readings of the barom and therm (Fah) in tlie siiade at both
stations. Add together the two readings of the barom, and div their sum bj 2, for
their mean ; which call b. Do the same with the two readings of the thermom,*and
call the mean t. Subtract the least reading of the barom from the greatest ; and call
the diff d. Then mult together this diff d; the number from the next Tablt: No. 1,
opposite ( ; and the constant number 30. Div the prod by b. Or

Height Diff (d) of ^ Tabular number opposite v, n«„„*..„* on
in feet "^ barom ^ mean (f) of thermom X constant du.

mean (b) of barom.

ExAMPLi. Beading of the barom at lower station, 26.64 ins ; and at the upper
sta 20.82 ins. Thermom at lowest sta, 70^; at upper sta, 4^. What is the diff io
height of the two stations? Here,

Sarom, 26.64 Therm, 70^

" 20.82 *• iOP

— — Also^ — —

2)47.46 2)110

23.78 mean of bar, or b. 669 mean of

therm, or t.
The tabular number opposite 66°, is 917.2.

Bar. Bar.
Again, 26.64 — 20.82 = 5.82, diff of bar ; or d. Hence,

d. Tab No. Con.
Height _ 5.82 X 917 Si X 30 _ 160143.12 ^^^ ^ ^^^,^
in feet 23.73 (or 6) "*" 23.73

Then oorrect for latitude, if more aooaracy is reqd, by rule on next page.

mie screw at tlie baekof an aneroid Is for adjusting the index by a stand-
ard barom. After this has been done it must by no means be meddled with. In
some instruments specially made to order with that intention, this screw may bo
used also for turning the index back, after having risen to an elevation so great that
the index has reached the extreme limit of the graduated arc. After thus turning
it back, the indications of the index at greater heights must be added to that at-
tained when it was turned back.

TABIiB 1. For Rale 1.

Mean

Mmd

Mean

Mean

•

of

No.

of

No.

of

No.

of

No.

Ther.

Ther.

Ther.

Ther.

oo

801.1

80°

864.4

60O

927.7

90O

991.0

1

803.2

31

866.6

61

929.8

91

993.1

3

805.3

32

868.6

62

981.9

92

995.2

S

807.4

38

870.7

63

934.0

98

997.3

4

809.6

84

872.8

64

936.1

94

999.4

6

811.7

86

874.9

66

938.2

95

1001.6

. «

818.8

36

817.0

66

940.3

96

1003.7

7

815.9

87

879.2

67

942.4

97

1005.8

8

818.0

38

881.8

68

944.6

98

1007.9

9

820.1

80

883.4

69

946.7

96

1010.0

10

822.2

40

886.4

70

948.8

100

1012.1

11

824.3

41

887.6

71

950.9

101

1014.2

12

826.4

42

869.6

72

953.0

102

1016.3

13

828.5

48

891.7

73

955.1

103

1018.4

li

880.6

44

893.8

74

967.2

104

1020.5

16

833.8

46

896.0

76

969.3

105

1022.7

16

834.9

46

898.1

76

961.4

106

10i4.8

17

887.0

47

900.2

n

968.6

107

1026.9

18

889.1

48

902.3

78

965.6

108

1029.0

1»

8«1.3

49

904.6

79

967.7

lOB

1031.1

20

84SJI

60

906.6

80

860.9

110

ia'M.2

21

8A5.4

61

908.7

81

972.0

111

1035.3

23

847.6

63

910.8

82

974.1

112

1037.4

28

848.6

63

913.0

83

976.2

118

1039.5

3i

861.8

64

916.1

84

978.3

lU

1041.6

25

853.9

66

917.2

86

980.4

116

1043.8

96

8G6.0

66

919.3

86

982.6

116

1045.9

27

868.1

67

921.4

87

964.7

117

1048.0

28

800.2

68

923.6

88

966.8

118

1050.1

»

863.8

69

925.6

89

988.9

119

1052.2

314

LEVSLLINO BT THE BAROMETEB.

RuLi 2. BelTlUe's short approx rale is the one beit adapted to rapid
Aeld use, namely, add together the two readings of the barom only. Also find the
diir between said two readings; then, as tbe sam of the two readlnffs
is to tbelr dlff, so Is 55000 feet to the reqd altitude.

<3orreetion for latitude is usually omitted where great accuracy is not
required. To apply it, first find the altitude by the rule, as before. Then divide it
by the number in the following table opposite the latitude of the place. (If the two
places are in different latitudes, use their mean.) Add the quotient to the altitude
if the latitude is leea than 45°. Subtract it if the Utitude is more than 45°. No cor-
rection required for latitude 45°.

Table of corrections

for latitude.

Lat.

Lat.

Lat.

Lat.

Lat.

Lat.

0°

S52

14°

890

280

630

420

8867

640

1140

680

490

a

S54

16

416

80

706

44

10101

66

941

70

460

4

856

18

486

82 .

804

46

00

68

804

72

486

6

860

ao

460

U

941

46

10101

60

705

74

416

8

867

22

490

86

1140

48

8867

62

680

76

990

10

8T5

M

527

88

1468

60

9028

64

572

78

886

IS

886

26

672 40

9038 1 63

1458

66

527

80

876

lieTCllins by Barometer; or bjr the bollini^ point.

Rule 3. The following table. No. 2, enables us to measure heights either by means
of boiling water, or by the barom. The third column shows the approximate alti-
tude above sea-level corresponding to diif heights, or readings of the barom ; and to
the diif degrees of Fahrenheit's thermom,at which water boils in the open air. Thus
when the barom, under undisturbed conditions of the atmosphere, stands at 24.08
inches, or when pure rain or distilled water boils at the t«mp of 201° Fah ; the place
is about 5764 ft above the level of the sea, as shown by the table. It is therefore
rery easy to find the diffoi altitude of two places. Thus : take out from table No 2,
the altitudes opposite to the two boiling temperatures ; or to the two barom readings.
Subtract the one opposite the lower reading, from that opposite tbe upper reading.
The rem will be the reqd height, as a rough approximation. To correct this, add
together the two therm readings ; and div the sum by 2, for their mean. From teble
for temperature, p 816, take out the number opposite this mean. Mult the ap-
proximate height just found, by this tabular number. Then correct for lat if reqd.

Ex. The same as preceding ; namely, barom at lower sta, 26.64 ; and at npper ata,
20.82. Thermom at lower sta, 70° Fnh ; and at the npper one, 40°. What is the diff
of height of the two stations ?

Alt.

Here the tabular altitudes are, for 20.82 9579

and for 26.64 3115

To correct this, we have

70° + 40° 110°

6464 ft, approx height. .
65° mean ; and in table p 816, opp to

55°, we find 1.048. Therefore 6464 X 1.048 = 6774 ft, the reqd height.
This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft In each 100 ft.

At 70° Fah, pure water will boil at 1° less of temp, for an average of about 660 ft
of elevation above sea-level, up to a height of U a mile. At the height of 1 mile, V*
of boiling temp will correspond to about 560 ft of elevation. In table p315 the
mean of the temps at the two stations is assumed to be 32° Fah ; at which no correc-
tion for temp is necessary in using the table ; hence the tabular number opposite
32°, in table p 316, is 1.

This diff produced in the temp of the hailing pointy by change of elevation, most
not be confounded with that of the atmotpherej due to the same cause. The air be-
comes cooler as we ascend above sea-level, at the rate (very roughly) of about 1^ Fah
for every 200 ft near sea-level, to 350 ft at the height of 1 mile.

The followingr table, "So. 2, (so tar as it relates to the barom^ was da^
dncnd by the wnter from the standard worU on the barom 'by Lieut.-Ool. R. S. Wil-
liamson, U. S. army."*

• FablUbed by penaiMton of OoTernmeni In 1868 by Vao Koetraod. N. T-

lAVELLINQ BT THE BABOKBTEB, ETC. 315

TABI.E 9.
I.«ivellliifc by Bfkrometer ; or by the bnllliift p»liil.

imed templn theebide 32° Full. JI pot S2°, mult harnni sk us per TBbIe,p

316

SOUND.

Corre«il«iis f«r temperatare; to be used in eonnecUon wltb
Bule 3, wlien irreater aecuracy is necessary. Also in con-
nection witli TaMe 2 wlien tlie temp is not 33°.

Mean

•

Mean

Mean

Mean

*

temp

Malt

temp

Mult

temp

Mnlt

temp

Mult

in the

by

in the

by

In the

by in the

by

shade.

shade.

»

ihade.

shade.

Zero.

.933

28°

.992

5«o

1.050

84°

1.108

20

.937

30

.996

68

1.064

86

1.112

4

.942

32

1.000

60

1.058

88

1.117

6

.946

34

1.004

62

1.062

90

1.121

8

.960

36

1.008

64

1.066

92

1.126

10

.954

38

1.012

66

1.071

94

1.129

12

.958

40

1.016

68

1.076

96

1.133

14

.962

42

1.020

70

1.079

98

1.138

16

.967

44

1.024

72

1.083

100

1.142

18

.971

46

1.028

74

1.087

102

1.146

20

.976

48

1.032

76

1.091

104

1.150

22

.979

60

1.036

78

1.096

1U6

1.154

U

.983

62

1.041

80

1.100

108

1.168

»

.987

64

1.046

82

1.104

110

1.163

SOUND.

u

— 20°

M

1040

«

— 10°

u

1060

it

0

u

1060

it

10°

«

1070

U

20°

u

1080

M

• 32°

u

1092

«

40°

u

1100

M

50°

u

1110

t(

60°

it

1120

H

70°

M

1130

U

80°

U

1140

(«

90°

U

1160

K

100°

t(

1160

M

110°

<(

1170

«

120°

a

1180

((

((

t:

u

u

it

It
It

it

t(

M

U

M

tt

It
tt

tt

U

tt

tt
4(

<t

tt

it
it

tt
tt

U
tt

u

M

(«

«

tt

«. 1

tt

6.08

.. 1

u

5.03

■B 1

tt

4.98

*■ 1

«

4.93

^ 1

((

4.8S

IBS X

u

4.83

■> I

«

4.80

^ 1

«

4.78

^ 1

H

4.73

m^ 1

U

4.68

m= 1

It

4.63

*B 1

l(

4.69

■B X

u

4.65

IM \

tt

4.61

— 1

tt

4.47

«

(«

-reloeitjr at sound in quiet open air, haa been experimentally deter>
mined to be very approximately 1090 feet per second, when the temperature is at
freezing point, or 32° Fahienheit. For every degree Fahrenheit uf increase of
temperature, the velocity increases by from V^ foot to 1^ feet per second, according
to different authorities. Taking the iucreasu at 1 foot per second for each degree
(which agrjBes closely with theoretical calculations), we have

at ^ 30° Fahr 1030 feet per sec '^ 0.1951 mile per sec — 1 mile in 6.13 seconds.

— 0.1970

— 0.1989

— 0.2008
» 0.2027

— 0.2045

— 0.2068

— 0.2083

— 0.2102

— 0.2121

— 0.2140

— 0.2169

— 0.2178

— 0.2197

— 0.2216

— 0.2236

If the air is calm, fog or rain does not appreciably affect the retult ; but wisds do.
Very loud sounds appear to travel somewhat faster than low ones. The watchword
of sentinels has been heard across still water, on a calm night, 10^ miles ; and a
cannon 20 miles. Separate sounds, at intsrvals of ^ of a second, cannot be distin-
guished, but appear to be connected. The distances at whieh a speaker can be
understood, in front, on one side, and behiud him, are about ab 4, S, and 1.

Dr. Charles M. Cresson informs the writer tliat, by repeated trials, he found that
in a Philadelphia gas main 20 inches diameter and 16000 feet long, laid and covered
in the earth, but empty of gas, and having one horizontal bend of 90^, and of 40 fast
reuUus, the sound of a pistol-shot travelled 16000 feet in precisely 16 seconds, or 1000
feet per second. The arrival of the sound was barely audible ; but was rendered
very apparent to the eye by its blowing off a diaphragm of tissue-paper placed over
the end of the main.

Turo bosits anchored some distance apart may serve as a base line for
triangulating objects along the coast; the distance between them being first found
by firing guns on board one of them.

In ivater tliie velocity is about 4708 feet pef second, or about 4 times that
in air. In iwroodsy it is from 10 to 16 times ; and in metalSf fh>m 4 to 10 times
greater than in air, according to some authorities.

w

t«

g!^;.'.:

Eaeb 13^ M IS" of bekt prodncaln wr*t Ir^ i

°* ^ I"" '° "" '^t^' Id Iki no «!• niv U(>. n IM Iha I«(Ilu i

, „ ezpsnalon of HtVD* wUlfmi*

TinM mcRlnS points «re qnlM lUiccrtatM. W« ^n (be miu of

•atanwIborMH. iMoWlii tUiMlJi »(aiMjtor Bnool^tioiit l«l»,»t«|Mlro» "HlilHriH

■aiBnUHT wUh Uu H« olulsiuil • «'n n» will diiim lu l>ii|lk [au? of in Iniik.

THEBUOIfETEBS.

T« «liBnc« derreea of Fitlirenbelt 1« Ike eorrMipOBdIns de-
■re«a •rc«ntl)?«de) l&kBiir>)irBidliiK32°liivn-ihitnih« ilit|aoiis: mnlt
— lD°IMiit Agali,— 190F>b = r— II— a])>cCi-t = — •&XC'i-?=— l^Oaal. ~ ~

To cli»iiKe P>h taMBOi uks & Fati rudlug 32^ Iswrthu ihs (Ina

• ti^-eoBi.o. linln,— IPlVlisI— ls-ijfx'*+»='--«SXt-H'=— *1°B*M.
ToeliAnce £«ntto l'nb|niiill ihe Oni nwllDg b> 0; dirlds by !

eIlAn» O

5l*i>^'=r^^?"°

>. Tkkaft

i^taii.—tfOmn=i--xx'-T6>+ai=—*''r^.

shannlMml to Fohri-

niiu:Tir'i«i™=("X9*«)+»t

,^'-4i«

'^^oekanreBtenMCentiiDnltbyH; div by 4. Thna: -fB°R«u — + 8°
TABI>E1> FHtarenhelteomiMredwItliCeiitlBr^deaiidK^a*

THERHOUETERS. Zl^

TABLE 3. CcntlrnMie eom|»a>«d with rahrcnbclt a>«

C F. K. C. F. R. C. F. I K. C F. R.

TABLE S. Ktaaoiar coiapitrfld wltb FabMiBbelt luul

tlTradc.

K. F. C. B. F. C. R. F. C. R. F. C.

M III.M KnOO 4< 4119 tl.iS I* TS Jt.n —1 1.16 — I1.1S

II M!ffi K.a M ullS SI.M It 00 wiot — < O^UI — IiIm

ig 303.01 (MM 11 lU.ie M.it IS lb e.11 — s —1.75 — i8.7e

Ts via.Ti 0.11 H 9ij» u.m 11 u 1.U — s -4.in _».og

TS IK.r, •l!«i •> MM i'lUI II M s!lM - 8 -H 60 -h'm

II I "" — SLIl (U n.oi> uiloo ID M LU — » — isigo ~ia!oi>

a Bi.n 91 ot!eo iiiw i ti.ui l.oo ~h ~ie'^ -si'm

a i nils n mIn leM i km i.m - la ~-9i.oo — SSM

« 17.50 II 01,11 a.ib I U.1S ].^ —a -u.a -x.u,

11 iZn MM 10 J7!oo »!oo -10 g'so '\-ii.sb -« -».a> -ooiot

320 Aia

AIR-ATMOSPHERE.

The atmospliere is known to extend to at least 4S miles

abore the earth. It is a mixture of about 79 measures of nitrogen gas and 21
of oxygen gas ; or about 77 nitrogen, 23 oxygen, br weight. It generallr con-
tains, however, a trace of water, and of carbonic acid and carbu retted hydrogen
gaaes, and still less ammonia.

Density of air. Under *' normal ** or " standard " conditions (sea level,
lat 45^, barometer 760 mm => 29.922 ins, temperature O^C^ZTP F) dry air
weljirhs 1.292673 kilograms per cubic meter * = 2.17888 fi>s avoir per cubic yard.
For other lats and elevations —

Density, in kg per cu m, =i 1.292673 X j^^^A ^ ^^ —0.002837 oos 2 lat) •

where B = earth's mean radius =» 6,366,198 meters ; A >« eleTation above aea
level, in meters. For other temperatures, see below.

Under normal conditions, but with 0.04 parts carbonic acid (0 O,) in 100 parts
of air, density = 1.293052 kg per cu m.f » 2.17952 fi» avoir per cu yd.^

The atmospherie pressure, at any given place, may yarr 2 inches or
more from day to day. 'rhe averagr® pressure, at sea level ^ varies from
about 745 to 770 millimeters of mercury according to the latitude and locality.
760 millimeters * is generally accepted as the mean atmospheric pressure, and
called an atmosphere. The '* metrie atmosphere,** taken arbitrarily
at 1 kilogram per square centimeter, is in general use in Continental Europe.
The pressure diminishes as the altitude increases.f Therefore, a pump in a high
region will not lift water to as great a height as in a low one. The pressure of
air, like that of water, is, at any given point, equal in all directions.

It is often stated that the temperature of the atmosphere lowers at
the rate of 1<^ Fah for each 300 feet of ascent above the earth's snrfhees
but this is liable to many exceptions, and varies much with local causes. Actual
observation in balloons seems to show that, up to the first 1000 feet, 1^ in aboat
200 feet is nearer the truth ; at 2000 feet, 1° in 250 feet ; at 4000 feet, 1° in 300 feet;
and, at a mile, 1° in 350 feet.

In breathingr, a grown person at rest requires from 0.25^ to 0.35 of a cubic
foot of air per minute : which, when breathed, vitiates from 8.5 to 5 cubic feet.
When walking, or hard at work, he breathes and vitiates two or three times as
much. About 5 cubic feet of fresh air per person per minute are required for the
perfect ventilation of rooms in winter; 8 in summer. Hospitals M to 80.

Beneath the ipeneral level of the surface of the earth, in temperate
regions, a tolerably uniform temperature of about 50° to 60^ Fah exists at
the depth of about 50 to 60 feet ; and inereases about 1° for each additional 50 to
60 feet ; all subject, however, to considerable deviations owing to many local
causes. In the Rose Bridge Colliery, England, at the depth of 2424 feet, the
temperature of the coal is 93.5° Fah ; and at the bottom of a boring 4169 feet
deep, near Berlin, the temperature is 119°.

The air is a werjr slow eondnetor of heat; hence hollow walls
serre to retain the heat in dwellings ; besides keeping them dry. It mahea
into a waeunm near sea level with a velocity of about 1157 feet per second ;
or 13.8 miles per minute ; or about as fast as sound ordinarily travels through
quiet air. See Sound. ^

Iiike all other elastie fluids, air expands eoually witik
e^ual increases of temperature. Every increase of o° Fah, expands
the bulk of any of them slightly more than 1 per cent of that which it has at 0^
Fah ; or 500° about doubles its bulk at xero. The bulk of anv of them diminishes
inversely in proportion to the total pressure to which it is subjected.

This holds good with air at least up to pressures of about 750 fte per sqnare
inch, or 50 times its natural pressure ; the air in this case occupying one-flxtietii
of its natural bulk. In like manner the bulk will increase as the total preasuiv
is diminished. Substances which follow these laws, are said to be perCeetiy

* H. V. Regnault, M6moires de 1* Acaddmie Royale des Sciences de Plnstitiit de
France, Tome XXI, 1847. Translation in abstract. Journal Of Franklin Insti-
tute, Phila., June, 1848.

fTravaux et M6moire8 du Bureau International.desPoidset Mesnres, Tomel

£age A 54. Smithsonian Meteorological Tables, 1898, publiabed In Smithsooian
[iscellaneous Collections, Vol. XXXV, 1897.
I See Conversion Tables.
f See Leveling by the Barometer.

WIND.

321

1

elAstle. Under apressure of about 6^ tons persqiiaie Indi, air would become
as dense as wa^er. Since the air at the surface of the earth is pressed 14^ !ba per
square inch by the.atmosphere above it, and since this is equal to the we^ht of a
oolumn of water 1 inch square and 34 feet high, it follows that at the depths of
84, 68, 102 feet, &4i, below water, air will be compressed into ^, 3^, 3^ Ac,
01 its bulk at the surface.

In a divliiK-bell, men, after some experience, can readily work for seyeral
hours at a depth of 51 feet, or under a pressure of 2^ atmospheres ; or 37^ ftis

Kir square inch. But at 90 feet deep, or under 3.64 atmospheres, or nearly 55
8 per square inch, they can work for but about an hour, without serious suffer^
ing from paralvsis. or even danger of death. Still, at the St Louis bridge, work
was done at a deptn of 1103>^ feet ; pressure 63.7 9>8 per square inch.

The dew point is that temp (varying) at which the air deposits its vapor.

Tlie gnreatest beat of tlie air in the sun probably never exeeeds
145° Fah J nor the greatest cold — 74P at u ight. About 130° above, and 40° below
zero, are the extremes in the U. S. east of the Mississippi ; and 65^ below in the
N. W.; all at common ground level. It is stated, however, that —81° has been
observed in N. E. Siberia: and +10lo Fah in the shade in Paris; and +153° in
the sun at Greenwich Observatory, both in July, 1881. It has frequently ex-
'beaded -i-l(XP Fah in the shade in Philadelphia during recent years.

WIND.

The relation between the weloeity of wind, and its preas*
lire against an obstacle placed either at right angles to its course, or inclined
to it, has not been well determined ; and still less so its pressure against curved
surfaces. The pressure against a laige surface is probably proportionally greater
than gainst a small one. It is generally supposed to vary nearly as the squares
of the velocities; and when the obstacle is at right angles to its direction, the

Sressure in lbs per square foot of exposed surface is considered to be equal to
lie square jof the velocity in miles per hour, divided by 200. On this basis,
which is probably quite aefective, the following table, as given by Smeaton, is
prepared.

YeiL in MUes

Vei. m Ft.

Frea. in Lbs.

Remarks.

per Hour.

per Sec.

per Sq. Ft.

1

1.467

.005

Hardlj perceptible. ^..^^
PleMsnt. ^C~J>g

s

2.933

.020

8

4.400

.045

^

4 .

5.867

.OBO

^

5
10

7.38
14.67

.125
.5

zJo/rt

12H

18.S3

.781

Fresh breexe. O

lb

n.

1.125

20

S9.33

8.

^ . Th« prei acainit

25

86.67

3.125

Brlakwind. « iiemioylindrioal

so

44.

4.5

Strong wind. sarfftoe ac&nom

40

S6.67

8.

High wind. ig about half that

60

73.88

12.5

Storm. against the flat

60

88.

18.

Violent storm. gnrf abnni.

SO

117.3

32.

Hurricane.

100

146.7

60.

Violent hunieane, uprooting large trees.

TreddTOld reeommends to allow 40 lbs per sq ft of roof for the

pras of wind against it ; but aa roob are oonstruoted with a slope, and oonsequentty do not receive
<ke ftill foree or the wind, this is plainly too much.* Moreover, only one>half of a roof is usually ex-
I, even thas partially, to the wind. Probably the force in suoh cases varies approximately as the
of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per
prodmsed a pre* of 14 lbs per sq ft againut an object perp to it: and one of 70 miles, per hour,
(the Mvterect gale on reoord at that city.) 43 lbs per sq foot. These would make the ores per sq ft,
More nearly equal to the ■qoAre of the vel iq miles per hour, dlv by 100 ; or nearly twice as great as
glvea in Smaaton's table, we should ourselves give the preference to the Liverpool observations. A
very violent gale in Scotland, registered by an excellent anemometer, or wind-gauge, 45 lbs per sq
ft. It la stated that aa high as 55 lbs has been observed at Glasgow. High winds often l^ roots.

The gaoge at Oirard Coliese, Fhilada, broke onder a strain of 43 lbs per sq ft ; a tornado passing
St the moment, within a mils.
By inrersion of SoMaton's rule, if the force in Iba per sq ft, be mult by 200, the sq rt of the prod
Igive the vel in milec per hoar. Smeaton's rule is used by the U. S. Signal Service.

«i/c

• The writer thinks 8 lbs per sq foot of mrdinarn doubte-aloping roofi, or 10 lbs for •Ked-rooft, suffl ■
«imt allowanee for prea of wind.

21

322

RAIN AND SNOW.

RAIN AND SNOW.

The annaal preelpitatlon * at any giyen place varies greatly from
year to year, the ratio between maximum and minimum being frequently greater
than 2 : 1. Beware of averai^es. In estimating ^oo^«, take the maximum
falls, and in estimating water supply, the mtnimttm, not only per annum, but for
short periods. In estimating water supply, make deductions for evaporatios
and leakage.

Maxima and minima deduced fh>m observations covering only 4 or 5 years are
apt to be misleading. Data covering even 10 or more years may just miss includ-
ing a very severe flood or drought. Becords of from 15 to 20 years may usually
be accepted as sufficient.

Table 1. Averafre Preelpltatlon * In tbe United States, in ins.
(Frmn Bulletin C of U. S. Department of Agriculture, compiled to end of 1891.)

Steto. Spr.

Alabama 14.9

Ariisona 1.3

Arkansas 14.8

California. 6.2

Colorado 42

Connecticut 11.1

Delaware 10.2

Dist. Columbia.11.0

Florida 10.2

Georgia 12.4

Idaho 4.4

Illinois 10.2

Indiana 11.0

Indian T'y 10.6

Iowa 8.3

Kansas 8.9

Kentucky 12.4

Louisiana 13.7

Maine 11.1

Maryland 11.4

Massachusetts. ..11.6

Michigan 7.9

Minnesota 6.5

Mississippi 14.9

Missouri 10.0

8am. Aat. Win. Atxn'l

13.8

10.0

149

53.6

43

2.2

3.1

10.9

12.5

11.0

12.8

50.6

0.3

3.5

11.9

21.9

5.5

2.8

2.3

148

12.5

11.7

11.5

46.8

11.0

10.0

9.6

40.8

12.4

9.4

9.0

41.8

21.4

14.2

9.1

549

15.6

10.7

12.7

51.4

2.1

3.6

7.0

17.1

11.2

9.0

7.7

38.1

11.7

9.7

10.3

42.7

11.0

8.9

6.7

36.2

12.4

8.1

41

32.9

11.9

6.7

3.5

31.0

12.5

9.7

11.8

46.4

15.0

10.8

144

53.9

10.5

12.3

11.1

45.0

12.4

10.7

9.5

440

11.4

11.9

11.7

46.6

9.7

9.2

7.0 83.8

10.8

5.8

8.1

26.2

12.6

10.1

15.4

53.0

12.4

9.1

6.5

38.0

SUte.

Spr. Sum. Aat. Win. Annl

Montana 4.2

Nebraska S.9

Nevada 2.3

N. Hampshire. 9.8

New Jersey 11.7

New Mexico..... 1.4

New York 8.5

N. Carolina 12.9

N. Dakota 46

Ohio 10.0

Oregon 9.8

Pennsylvania...l0.3
Rhode Island. ..11.9

S. Carolina 9.8

S. Dakota 7.2

Tennessee 18.6

Texas 8.1

Utah 3.4

Vermont 9.2

Virginia 10.9

Washington 8.6

W. Virginia 10.9

Wisconsin 7.8

Wyoming 4.8

United States... 9.2

49

2.6

2.8

140

10.9

49

2.2

26.9

0.8

1.3

3.2

7.6

12.2

11.4

10.7

44.1

13.3

11.2

11.1

47.8

5.8

8.5

2.0

12.7

10.4

9.7

7.9

86.5

16.6

12.0

12.2

68.7

8.0

2.8

1.7

17.1

11.9

9.0

9.1

40.0

2.7

10.5

21.0

440

12.7

10.0

».6

42.6

10.7

11.7

12.4

46.7

16.2

9.7

9.7

46.4

9.7

8.5

2.5

22.9

12.5

10.2

145

60.7

8.6

7.6

6.0

80.3

1.5

2.2

8.5

lao

12.2

11.4

9.8

42.1

12.5

9.5

9.7

42.6

3.9

10.5

16.8

89.8

12.9

9.0

10.0

42.8

11.6

7.8

6.2

82.6

8.5

2.2

1.6

11.0

10.3

8.3

8.6

80.8

At Philadelphia, in 1869, during which occurred the greatest drought known
there for at least 50 years, 43.21 inches fell ; August 13, 1873, 7.3 inches in 1 day ;
August, 1867, 15.8 inches in 1 month ; July, 1842, 6 inches in 2 hours ; 9 inches
per month not more than 7 or 8 times in 25 years. From 1825 to 1893, greatest
in one year, 61 inches, in 1867 \ least, 30 inches, in 1826 and 1880. At Norristown,
Pennsylvania, in 1865, the writer ^aw evidence that at least 9 inches fell within

5 hours. At Genoa, Italy, on one occasion, 32 inches fell in 24 hours ; at Geneva,
Switzerland, 6 inches in 3 hours ; at Marseilles, France, 13 inches in 14 hoars;
in Chicago, Sept., 1878, .97 inch in 7 minutes.

Near iJondon, Eng^land, the mean total fall for many years is 28 inches.
On one occasion, 6 inches fell in 1% hours! In the mountain districts of the
English lakes, the fall is enormous: reaching in some years to 180 or 240 inches;
or from 15 to 20 feet ! while, in tne adjacent neighborhood, it is but 40 to 00
inches. At Liverpool, the average is 34 inches ; at Ckiinburgh, 30 : Glasgow, 22;
Ireland, 36; Madras, 47; Calcutta, 60; maximum for 16 years, 82; Delhi, Si;
Gibraltar, 80 ; Adelaide, Australia, 23 ; West Indies, 36 to 96 ; Rome, 89. On the
Khassya hills north of Calcutta, 500 inches, or 41 feet 8 inches, have Allien in the

6 rainy months I In other mountainous districts of India, annual falls of 10 to
20 feet are common.

A moderate steady rain , continuing 24 hours, will yield a depth of about an indu

As a seneral rule, more rain fhlls in warm tban in 99MA

€SonntrIes; and more in elevated regions than in low ones. Local pecuUaxw

* Precipitation includes snow, hail, and sleet, melted,
estimated at 10 inches snow » 1 inch rain.

Unmelted snow ia

BAIV AND SNOW.

323

KieB, howerer, sometimeB reyerae this : and also oanse great differences in the
amounts in places quite near each other ; as in the English lake districts Just
alluded to. It is sometimes difficult to account for these variations. In some
lagoons in New Granada, South. America, the writer has known three or four
heavy raiiio to occur weekly for some months, during which not a drop fell on
hills about 1000 feet high, within ten miles' distance, and within full sight. At
another locality, almost a dead-level plain, fully three-quarters of the rains that
fell for two years, at a spot two miles from his residence, occurred in the morn-
ing ; while those which fell about three miles from it, in an opposite direction,
were in the afternoon.

Tlie relation between precipitation and stream»0ow is greatly
ai^cted by the existence of forests or crops, by the slope and character of ground
on the water-shed, especially as to rate of absorption, by the season of the year,
the frost in the ground, etc. The stream-flow may ordinarily be taken as vary*
ing between 0.2 and 0.8 of the rainfall. Streams in limestone regions frequently
loee a very large proportion of their flow through subterranean caverns.

Aasnminff a fall of 2 feet in 1 year (=3 76,379 cubic feet per square mile per
day), that half the rainfall is available for water supply, and that a per capita
consumption of 4 cubic feet (^t 30 gallons) per day is sufficient, one square mile
will supply 19,095 persons ; or a square of 88.26 feet on a side will supply one
person.

Ineb of rain amonnto to 3630 enble fiBet; or 27156 U. SL

EkUonB ; or 101.3 tone per acre ; or to 2323200 cubic feet ; or 17378743 U. S. gal^
ns ; or 64821 tons per squ&re mile at 62^^ fts per cubic foot. • ^
The most destructive rains are usualhr those which fall upon snow, nnder
which the ground is frozen, so as not to absorb water.

Table 2. Kaxlmnm intensify of rainlMl for periods of 5, 10, and

60 minutes at Weather Bureau stations equipped with self-registering

gauges, compiled from all available records to the end of 1896.

(From Balletin D of U. S. Department of Agriculture.)

Stations.

Rate per hour for—

Stations.

Rate per hour for—

6min.

lOmins.

60 mins.

6min.

10 mins.

60 mins.

Bismarck.

Ins.

9.00
8.40
8.16
7.80
7.80
7.50
7.44
7.20
7.20
6.72
6.60
6.60
6.60

Inches.

6.00
6.00
4.86
4.20
6.60
6.10
7.08
6.00
4.92
4.98
6.00
3.90
4.80

Inches.

2.00
1.30
2.18
1.25
2.40
1.78
2.20
2.15
1.60
1.68
2.21
L60
1.86

Chicago

Ins.

6.60
6.48
6.00
6.00
5.76
6.64
6.46
5.40
6.40
4.80
4.56
8.60
3.60

Inches.

6.92
6.58
4.80
4.20
6.46
3.66
5.46
4.80
4.02
3.84
4.20
3.30
240

Inches.
1 60

St. Paul

Galveston...

Omaha

2 55

Kew Orleans

1.65

Milwaukee

Dodge City

Norfolk

1.84
1 55

Washington

Jacksonville

Cleveland

'Atlanta.

1.12
1.50

Detroit.

Key West

Philadelphia...

St Louis...

Cincinnati

Denver...

2.26
1.60
2.25

New York aty>
Boston

Savannah

1.70
1 18

Indianapolis......

Memphis..

Duluth—

1.35

The welirbt of firesbly flallen snow, as measured by the author,
varies from aoont 5 to 12 lbs per cubic foot ; apparently depending chiefly upon
the degree of humidity of the air through whicn it had passed. On one occasion,
when minsled snow and hail had fallen to the depth of 6 inches, he found its
weight to Be 81 fbs per cubic foot. It was very dry and incoherent. A cubic foot
of heavy snow mav, by a gentle sprinkling of water, be converted into abont
half a cubic foot of slush, weighing 20 9>s.; which will not slide or mn oflf
from a shingled roof sloping 30^, if the weather is cold. A cubic block of snow
•atorated with water until it weighed 45 Tba per cubic foot, Just slid on a rough
board inclined at 45''; on a smoothly planed one at 30^ ; and on slate at 18° : all
ipproximate. A prism of snow, saturated to 62 lbs per cubic foot, one inch
square, and 4 inches high, bore a weif^t of 7 fi»s ; which at first compressed
it abont one-quarter part of its length. European engineers consider 6 n>s per
square foot of roof to oe snffielent allowance for the weight of snow;

324 RAIN AND 6NOW.

and 8 lbs for the pressare of wind ; total. 14 lbs. The writer thinks that in the
U. S. the allowance for snow should not be taken at leu than 12 fi>8 ; or the total
for snow and wind, at 20 Bm. There is no danger that snow on a roof will
become saturated to the extent Just alluded to ; because a rain that would supply
the necessary q^uantity of water would also by its violence wash away the snow ;
but we entertain no doubt whatever that the united pressures from snow and
wind, in our Northern States, do actually at times reach, and even surpass.
20 fbs per square foot of root The limit of

perpetnal snow at the equator is at the height of about 16000 feet, or say
3 miles above sea-level; in lat 45° north or south, It is libout half that neight;
while near the poles it is about at searleveL

Rain Oaoi^es. Plain cylindrical vessels are ill adapted to service as rain
gauges ; because moderate rains, even though sufficient to yield a large run-off
from a moderate area, are not of sufficient depth to be satisfactorily measured
unless the depth be exaggerated. The inaccuracy of measurement, always con-
siderable, is too great relatively to the depth.

In its simplest and most usual form, the gauge (see Fig.) consists essentially
of a funnel. A, which receives the rain and leads it into a measuring
tube, B, of smaller cross-section. The funnel should have a verticci

and fairly sharp edge, and, in order to minimize the loss through xA/
evaporation, it should fit closely over the tube, and its lower end ^

diould be of small diameter. '

The depth of water in the tube is ascertained by inserting, to the
bottom of the tube, a measuring stick of some unpolished wood
which will readily show to what aepth it has been wet. The stick
may be permanently graduated, or it may be compared with an ordi-
nary scale at each observation. The tube is usually of such diameter
that the area of its cross-section, minus that of the btick, is one-tenth
of the area of the funnel month. The depth of rainiaU is then one-
tenth of the depth as measured by the stick.

B

DiiCENsiONS OP Standard U. S. Wbathbb Bubbau Bain Gauge. Ins.

A. Beceiver or funnel. Diameter 8

B. Measuring tube. Height 20 ins. " 2.53
C C. Overflow attachment and snow gauge. " 9

Such gauges, with the tubes carefully made from seamless drawn brass tubing,
«08t about $5.00 each ; but an intelligent and careftil tinsmith, given the dimen-
sions accurately, can construct, of galvanized iron, for about ^.00 a gauge that
will answer every purpose of the engineer.

Tbe exposure has a very marked effect upon the results obtained. The
funnel should be elevated about 3 ft, in order to prevent rain from splashing back
into it from the ground or roof. If on a roof, the latter shoald be nat, and pref-
erably 50 ft wide or wider, and the gause should be placed as far as possible
from tbe edges ■ else the air currents, produced by the wind striking the side of
the building, will carry some of the rain over the gauge. No objects much higher
than the gauge should be near it, as they produce variable air currents which
•may seriously affect its indications.

An overflow tank, G, should be provided, for cases of overfilling the tube.

Water, freezing in the gauge, may burst it, or force the bottom off, or at least
<ao deform the gauge as to destroy its accuracy.

To measure snow, the funnel is removed, and the snow is collected in
the overflow attachment or other cylindrical vessel deep enough to prevent the
snow from being blown out, and the cross-sectional area of which is accurately
known. The snow is then melted, either by allowing it to stand in a warm
place, or, with less loss through evaporation, by adding an accurately known
quantity of luke-warm water. In the latter case, the volume of the added water
must of course be deducted from tbe measurement.

Rainfall equivalent of snow. Ten inches of snow are usually taken
as equivalent to 1 in of rain ; but, according to various authorities, the equiva-
lent may vary between 2>^ and 34; i. e., between 25 and 1.84 &». per cubic foot.

Self-reeordinir g^ngr^s, of which several forms are on the market, are

Jiuite expensive, and, even when purchased from regular makers, seldom per-
ectly reliable. Gauges using a small tipping bucket register inaccurately la
heavy rains ; those using a float are limitea as to the total depth which they c
xegister ; while those which weigh,tbe rain, if exposed, are aflbcted by wind.

BAIir AND 81I0V.

Bulletia Cot U.S. DeparUuent of AftlcaUura, IBM.)

•F0riinlaiu)^Abbmmm,UablIa,tiioata0.2Mniih,34.S mauuthBton!4,3per
pt. of the dnjA ombrsc^d ivltfain tb* 30 yean, ram fall to a depth of from a

tFMaiOiitobarU7Bcail;. t Fnm Juaarr 1S14 oeOj. iFiomliUy ISTi odI;.

326 WATEB.

WATER.

Pure water, as boiled and distilled, Is eomposed of the tiro gases, hydro-
gen and oxygen ; in the proportions of 2 measures hydrogen to 1 of oxygen ;
or 1 weight of hydrogen to 8 of oxvgen. Ordinarily, however, it contains sev-
erid foreign ingredients, as carbonic and other acids ; and soluble mineral, or
organic substances. When it contains mirch lime,- it is said to be h€a^; and will
not make a good lather with soap. Tbe air in its ordinary state conlwiiis
about 4 grains of water per cubic foot.

The average pressure of tlie air at sea level, will balamee a
colamn of water 34 feet high ; or about 30 inches of mercury. At its boil-
ing point of 212° Fah, its bulk is about one twenty-third greater than at IQP.

Its welg^lit per cubic foot is taken at 62^ fi>fl,or 1000 ounces avoir; but 62}^
lbs would be nearer the truth, as per table beh>w. It is about 816 times hearier
than air, when both are at the temperature of 62°; and the barometer at 80
inches. With barometer at 30 inches the weight of perfectlv pure water is as
follows. At about 39*^ it has its maximum density of 62.425 ros per cubic foot.

Temp, Fah. Lbs per Cub Ft.

929 62.417

40° 62.423

50° 62.409

60° 62^7

Temp, Fah. Lbs per Cub Ft.

70° 62.302

80® - 62.218

90°- 62.119

212°- « 69.7

Weifflil; of sea ivater 64.00 to 64.27 B>s per cubic foot, or say 1.6 to 1.9 9>8

per cubic foot more than fresh water. See also p 328.

Water has its maxlmnm density when its temperature is a littler above
89° Fah ; or about 7^ above the freezing point. By best authorities 39.2°. From
about 39° it expands either by cold, or by heat. When the temperature of 320
reduces it to ice, its weight is but about 57.2 lbs. per cubic foot ; and its specific
gravity about .9176, according to the investigations of L. Dufour. Hence, as
ice, it has expanded one- twelfth of its original bulk as water; and the sadcleii
expansive force exerted at the moment of freezing, is sufficiently great to
split iron water-pipes; being probably not less than 30000 lbs per square inch.
Instances have occurred of its splitting cast tubular posts of iron bridges, and
of ordinary buildings, when full of rain water Arom exposure. It also loosens
and throws down masses of rock, through the Joints or which rain or spring
water has found its way. Retaining- walls also are sometimes overthrown, or
at least bulged, by the freezing of water which has settled between their backs
and the earth filling which they sustain ; and walls which are not founded at a
sufficient depth, are often lifted upward by the same process.

It is said that in a irlass tube ^ Incli in diameter, water will not
freeze until the temperature is reduced to 23°; and in tubes of less than^
inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in
rapid running streams. Ancbor lee, sometimes found at depths as great as
26 feet, consists of an aggregation of small crystals or needles of ice frosen s*
the surface of rapid open water ; and probably carried below by the fbroe of ths
stream. It does not form under frozen water.

Since ice floats in waters and a floatinff body displaces a weight of the
liquid equal to its own weight, it follows that a cubic foot of floating ice weighing
57.2 lbs, must displace 57.2 fSs of water. But 67.2 lbs of water, one foot square, is 11
inches deep: therefore, floating ice of a cubical or paralleloplpedal shape, will
have \^ of its volume under water; and only ^ above: and a square foot of ice
of any thickness, will require a weiffht equal to ^ of its own weight to sink it
to the surface of the water. In practice, however, this must be regarded merely
as a close approxima}iion, since the weight of ice is somewhat iSfocted by en-
closed air-bubbles.

Pure water is usually assumed to boll at 212° Fah In the open air, at the
level of the sea ; the barometer being at SO inches ; and at about 1^ less for every
620 feet above sea level, for heights within 1 mile. In fsct, its boiling point
Varies like its freezing point, with its purity, the density of the air, the material
4>f the vessel, dbc. In a metallio vessel, it may boil at 210°; and in a glass one,
at from 212° to 220°; and it is stated that if all air be previously extracted, it
requires 275°.

It evaporates at all temperatures; dissolves more substances than any
other agent : and has a greater capacity for heat than any other known substanosi

It is eomjpressfHl at the rate of about one-21740th. (or about ^^ of an
inch in 18^ feet,) by each atmosphere or pressure of 16 lbs per square Inclk
When the pressure is removed, it* »>\»uHniXj restores its orisinal boUk

J

WATER. .327

Effeet on metals. The lime contained in many waters, forms deposits In
metallic water-pipes^ and in channels of earthenware, or of masonry ; especially
if the current oe slow. Some other substances do the same ; obstructing the
flow of the water to such an extent, that it is always expedient to use pipes of
diameters larger than would otherwise be necessary. The lime also forms very
hard inemstatioiis at tbe bottoms of boilers^ very much impair-
ing their efficiency ; and rendering them more liable to burst. Such water is
unfit for locomotives. We have seen it stated that the Southwestern B R Ck>,
England, prevent this lime deposit, along their limestone sections, by dissolving
1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms
similar deposits in boilers; as uso does mud, and other impurities.

Water, either when very pure, as rain water; or when it contains carbonic
acid, (as most water does,) produees carbonate of lead in lead
pip^ ; and as this is an active poison, such pipes should not be used for such
waters. Tinned lead pipes may be substituted for them. If, however, sulphate
of lime also be present, as is very frequently the case, this effect is not always
produced; and several other substances usually found in spring and river
water, also diminish it to a greater or less degree. Fresh uraier corrodes
vrronslit Iron more rapidly tban cast; but the reverse appears to
be the case with sea water; although it also affects wrought iron very
quickly ; so that thick flakes may be detached from it with case. The corrosion
of iron or steel by sea water increases with the carbon. Cast-iron cannons
from a vessel which had been sunk in the fresh water of the Delaware River
for more than 40 years, were perfectly free from rust. Gen. Pasley, who had
examined the metals found in the ships Royal George, and Edgar, the first of
which had remained sunk in the sea for 62 years, and the last for 133 years,
"stated that the cast iron had generally become quite soft; and in some cases
resembled plumbago. Some of the shot when exposed to the air became hot;
and burst into many pieces. The wrought iron was not so much injured,
except when in eantaet vkth copper, or brcus gun^metal. Neither of these last was
much affected, except when in contact with iron. Some of the wrought iron
was reworked by a blacksmith, and pronounced superior to modern iron." **Mr.
Cottam stated that some of the guns had been carefully removed in their soft
state, to the Tower of London : and in time (within 4 years) returned their orig^
inal hardneu. Brass cannons rrom the Mary Rose, which had been sunk in the
sea for 292 years, were considerably honevcombed in spots only ; (perhaps where
iron had been in contact with them.) The old cannons, of wrought-iron bars
hooped together, were corroded about }^ inch deep; but had probskoly been pro-
tected bv mud. The cast-iron shot became redhot on exposure to the air; and
fell to pieces like dry clay I"

** Unprotected parts of cast-iron sluice-valves, on the sea gates of the Cale-
donian canal, were converted into a soft plumbaginous substance, to a depth
of % of an inch, within 4 years; but where they had been coated with common
Swedish tar, they were entirely uninjured. This softening effect on cast iron
appears to be as rapid even when the water is but slightly orackish ; and that
only at intervals, it also takes place on cast iron imbedded in salt earth. Some
water pipes thus laid near the Liverpool docks, at the expiration of 20 years
were soft enough to be cut by a knife ; while the same kind, on higher ground
beyond the influence of the sea water, were as good as new at tne end of 60 years."

Observation has, however, shown that the rapidity of this action
depends ntncn on the quality of the Iron ; that which is dark-
colored, and contains much carbon mechanically combined with it, corrodes
most rapidly : while hard white, or light-gray castings remain secure for a long
time. Some cast-iron sea-piles of this character, showed no deterioration in 40
years.

Contact wltli brass or copper is said to induce a galvanic action
which greatly hastens decay in either fresh or salt water. Some muskets were
recovered from a wreck which had been submerged in sea water for 70 years
near New York. The brass parts were in perfect condition ; but the iron parts
had entirely disappeared. Galwanlstng: (coating with zinc) acts as a pre*
serrative to the iron, but at the expense of the sine, which soon disappears.
The iren then corrodes. If iron be well heated, and then coated with toot
coal-tar, it will resist the action of either salt or freshwater for many years.
It is very important that the tar be perfectly purified. Sucji a coat«

ing, or one of paint, will not prevent barnacles and other shells from
attaching themselves to the iron. Asphaltum, if pure, answers as well aa
4M>a]->tar.

Copper and bronse are very little affected by sea water.

Ko galvanic action has. been detected where bnun leroles are inserted intt
the water-pipea in Philadelphia.

328 TIDES.

Tbe most prejudicial exposure for Iron, as well as for wood, is
that to alternate wet and dry. At some dangerous spots In Long Island Sound,
it has heen the practice to drive round bars of rolled iron about 4 inches diam-
eter, for supporting signals. These wear away most rapidly between high and
low water; at the rate of about an inch in depth in 20 years ; in which time the
4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under
frenh water especially, or under ground, a thin coating of coal-pitch vamishi
carefully applied, will protect iron, such as water-pipes, Ac, for a long time.
See page 655. *The sulphuric acid contained in the water from coal minei
corrodes iron pipes rapidly. In tbe ft'esli water of canals, iron boata
have continued In service from 20 to 40 years. Wood remains sound for
centuries under either fresh or salt water, if not exposed to be worn away by
the action of currents : or to be destroyea by marine insects.

fitea urater welgrns from 64 to 64.27 ft>s per cubic foot, or say from 1.6 to
1.9 ft)s per cubic foot more than fresh water, varying with the locality, and not
appreciably with the depth. Theexcess, over the weight of fresh water, is chiefly
common salt. At 64 lbs per cubic foot, 35 cubic feet weigh 2240 fi>s. Sea water
freezes at about 27° Fahr. The ice is fresh ; but (especially at low tempera-
tures) brine may be entrapped In the ice.

A teaspoonful of powdered alum, well stirred into a bucket of dirty w^ater,
will generally purify it sufficiently within a few hours to be drinkable. If «
hole 3 or 4 feet deep be dug in the sand of the sea-shore, the infiltrating watei
will usually be sumciently fresh for washing with soap; or even for drinking.
It is also stated that water may be preserved sweet for many years by placing
in the containing vessel 1 ounce of black oxide of manganese for each gallon
of water.

It is said that water kept in zinc tanks ; or flowing through iron
tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc
formed thereby; and it is recommended to coat zinc surfaces with asphalt
varnish to prevent this. Yet, in the city of Hartford, Conn, service pipes of
iron, galvanized inside and out, were adopted in 1855, at the recommendation
of the water commissioners ; and have been in use ever since. They are like-
wise used in Philadelphia and other cities to a considerable extent. In many
hotels and other builaings in Boston, the *' Seamless Drawn Brass Tube" of the
American Tube Works at Boston, has for many years been in use for service

Eipe ; and has given great satisfaction. It is stated that the softest water may
e kept in brass vessels for years without any deleterious result.

Tlie action of lead upon some waters (even pure ones) is highlr poison-
ous. The subject, however, is a complicated one. An injurious ingredient may
be attended by another which neutralizes its action. Organic matter, whether
vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm-
less.

Ice may be so impure that its water is dangerous to drink.

Tke popular notion tbat hot water freezes more qniclLljr
than cold, with air at the same temperature, is erroneous.

TIDES. .

The tides are those well-known rises and falls of the surface of the sea
and of some rivers, caused by the attraction of the sun and moon. There are
two rises, floods, or high tides ; and two falls, ebbs, or low tides, every 24 hoars
and 50 minutes (a lunar day) ; making the average of S hours 12^^ minutes
between high and low water. These intervals are, however, subject to
fpreat variations; as are also the heights of the tides; and this not only
at different places, but at the same place. These irregularities are owing to the
shape of the coast line, the depth of water, winds, ana other causes. ImuMy at
new and full moon, or rather a day or two after, (or twice in each lunar month,
at intervals of two weeks,) the tiaes rise higher, and fall lower than at other
times; and these are called spring tides. Also, one or two days after the
moon is iu her quarters^ twice in a lunar month, they both rise and fall less than
at other times ; and are then called neap tides. From neap to spring they
rise and fall more daily ; and vice versa. The time of hifrii water at any
place, is generally two or three hours after the moon has passed over either
the upper or lower meridian ; and is called the establishment of that
place; because, when this time is established, the time of high water on any
other day may be found from it in most cases. The total height of spring tides
is generally from 1}^ to 2 times as great as that of neaps. The great ii<t*I
wave is merely an undulation, unattended by any current, or progressive motion
of the particles of water. Each successive hijgh tide occurs STOUt 24 mlnatei
later than the preceding one ; anil so with the Um tides-

EVAPOBATIOK AND LEAKAGE. 329

EVAPOEATION, FUTBATION, AND LEAKAGE.

Tbe amount of evaporation from surfaces of water exposed to

tlM natural effect* of the open air, is of cooree greater in aammer than in winter ; althoagh It is quite
perceptible in even, the coldest weather. It is greater in ahalloir water than in deep, inasmuch aa th*
bottom also beoomes heated by the sun. It is greater in running, than in standing water ; on much
the same principle that it is greater daring winds than calms. It is probable that the average dailj
loss from ^ reservoir of moderate depth, m>m evaporation alone, throughout the 3 warmer months

of the year, (June, Jniy, Aagust,) rarely exceeds about -^ inch, in any part of the United States. Or

JL inch daring the 9 colder months ; except in the Soathem States. These two averages would give

adaily one of .16 inch ; or a total annual loss of $6 ins, or 4 ft 7 ins. It probably is S.5 to 4 ft.

By some trials by the writer. In the tropics, ponds of pure water

8 ft deep, in a stiff retentive day, and ftally exposed to a very hot san all day, lost during the dry sea-
son, preoijiely 2 ins in 16 days ; or H ^^oh per day ; while the evaporation from a glass tumbler was
V inch per day. The air in that region is highly charged with moisture ; and the dews are heavy.
Every day during the trial the thermometer reached ftt>m 115° to 126° in the sun.

The total annual evaporation in several parts of England and Scotland is stated to average fhmi 22
to 38 ins ; at Paris, 84; Boston, Mass, 32 ; many places in the U. 8.,' SO to 36 ins. This last would give

a dailj average of -aA^ ineh for the whole year. Such statements, ho.wever, are of very little value,

nnless accompanied by memoranda of the circumstances of the case ; such as the depth, exposure,
sixe and nature of the vessel, pond. Ac, which contains the water, Ac. Sometimes the total annua)
evaporation from a district of country exceeds the rain fall ; and vice versa.

On canals, reservoirs, Ac, it is usual to combine the lofis bj eyaporation*

with that by filtration. The last is that which soaks into the earth ; and of which some portion
passes entirely through the banks, (when in embankt;) and if in very small quantity, may be dried
up by the son and air as fast as it reaches the outside ; so as not to exhibit itself as water ; but if is
greater quantity, it becomes apparent, as leakage.

E. H. Gill, € E, stat^ the average evaporation and filtra-
tion on tlie Sandy and Beaver canal, Oliio, (38 ft wide at ^ater snr-

Cmo; 26 ft at bottom ; and 4 ft deep.) to be but IS cub ft per mile per minute, in a dry secuon. Here
the exposed water surf in one mile is 200640 sq ft; and in order, with this surf, to lose 13 cub ft per

mln, or 18720 cub ft per day of 34 hours, the quantity lost must be innjWV ~ '^'^^ f^> — ^H loch fa
depth per day. Moreover, one mile of the canal contains 675840 cab ft ; therefore, the number of days
teqd for the combined evaporation and filtration to amount to as mach as all the water in the canal, is

^-I^ ^J^ = 36 days. Observations in warm weather on. a 22'mile reaeh of the Chenango canal, N
18720
York, (40; 28 ; and 4 ft,) gave 9SH cub ft per mile per min ; or 6 times as much aa in the preceding
ease. This rate would empty the canal in about 8 days. Besides this there was an excessive leakage
at the gates of a look, (of only bH ft lift,) of 479 cub ft per min, 22 cub ft per mile per min ; and at
aqnedneta, and waste-weirs, others amounting to 19 cub ft per mileper min. The leakage at other
locks with lifts of 8 ft, or. less, did not excMa about 350 cub ft per min, at each. On other canals, it
has been found to be fhom 60, to 500 ft per min. On the Chesapeake and Ohio canal, (where 60, 82,
and 6 ft.) Mr. Fisk, C E, estimated the loss by evap and filtration in 2 weeks of warm weather, to be

Moai 10 all the water in the canal. Professor Baublue assumes 2 Ins per
day, for leafcaffe of canal bed, and evaporation, on Eni^llsb

canals* i. B. Jervls, 0 B, estimated the loss trom evap, filtration, and leakage through lock'
gates, on the original Erie canal, (40, 28, and 4 ft.) at 100 eub ft per mile per min; or 144000 cub It
per day. The water surf in a mile Is 211200 sq ft ; therefore, the daily loss would be equal to a dsjpth of

<Hi tbe Belaware division of tbe Pennsylvania canals, when

the sapply is temporarily shut off f^m any long reach, tbe water falls from 4 to 8 ins per day. The
filtration will of course be muoh greater on embankta, than in eota. In some of our canals, the depth
at high embankta beoomes quite considerable ; the earth, from motives of economy, not being filled in
level under the bottom of the canal ; but merely left to form its own natural slopes. At one spot at
least, on tbe Ches and Ohio canal, where one side Is a natural face of vertical rock, this depth is 46
ft. Sooh depths increase the leakage very greatly ; especially when, as is frequently the case, the em-
baakta are not paddled; and the practice Is not to be commended, for other reasons also.

Tbe total averaire loss from reservoli^ of moderate deptbs.

In ease tbe earthen dams be constmeted with proper oare, and well settled bv time, will not exoeed
ahont f^om ^ to 1 inch per day ; Imt in new ones, it will usually be oonsiderabiy greater.

Tbe loss flrom dltcbes, or cbannels of small area, is much

greater than that from navigable canals ; so that long canal feeders usually deliver but a small pre*
psrtion of the water which enters them at their heads.

330 FORGE IN RIGID BODIES.

MECHANICS. FORCE IN BIQID BODIES.

In the following pages we endeavor to make clear a few elementary prinoiples
of Mechanics. The opening articles are devoted chiefly to the subject of matter m
motion; for, while an acquaintance with this is perhaps not absolutely required in
obtaining a loorking Itnowledge of those principles of Statics which enter so largely
into the computations of the civil engineer, yet it must be an Important aid to their
intelligent appreciation.

Art. 1 (a). Meolianlcs may b« dellned as that branch of science which
treats of the effects of force upon matter.

This broad definition of the word *' Mechanics" includes hvdroetatics, hydraulics,
pneumatics, etc., if not also electricity, optics, acoustics, and indeed all branches of
physics ; but we f^hall here confine ourselves chiefly to the consideration of the action
of extraneous forces upon bodies supposed to be rigid, or incapable of change of shape.

S) Mechanics is divided into two branches, namely :
Inematlos $ or the study of the moliona of bodies, without reference to the
causei of motion ; and
Dynamlesy or the study of force and its efiiects.
The latter is sob-divided into

Kinetics; which treats of the relations between force and motion; and
Statics t which considers those special, but very numerous, cases, where etpui
and opporite forces counteract each other and thus destroy each other's motions.

Art. $8 (a). Matter, or substaitoey may be defined as whatever occupies spao^
as metal, stone, wood, water, air, steam, gas, etc.

(b) A iMKly is any portion of matter which is either more or less completely
separated in fact from all other matter, or which we take into consideration by itself
and as if it were so separated. Thus, a stone is a body, whethsr it be falling thronngh
the air or lying detached upon the ground, or built up into a wall. Alao^ the wall is
a body ; or, if we wish, we may consider any portion of the wall, as any particulsr
cubic foot or inch in it, as a body. The earth and the other planets are bodies, and
their smallest atoms are bodies.

A train of cars may be regarded as a body; as may also each car, each wheel or
axle or other part of the car, each passenger, etc., etc

Similarly, the ocean is a body, or we may take as a body any portion of it at plsss-
nre, such as a cubic foot, a certain bay, a drop, etc.

(c) But in what follows we shall (as already stated) consider chiefly rigid bodies:
i. «., bodies which undergo no change in shape^ such as by being crushed or str^chea
or pulled apart, or penetrated by another body. AH actual bodies are of course more
or less subject to some such changes of shape ; t. «., no body i* in fact absolutely
rigid; but we may properly, for convenience, suppose such bodies to exist, because
many bodies are so nearly rigid that under ordinary circumstances they undergo
little or no change of shape, and because such change as does occur may be con-
sidered under the distinct head of Strength of Materials.

(d) But while bodies are thns to be regarded as incapable of change at form, it Is
squally important that we regard them as smeeplihle to change of p^ititm as wholm.
Thus, they may be upset or turned around horizontally or in any other direction, or
moved along in any straight or curved line, with or without turning around a point
within themselves. In short they are capable of moHon, as wholes.

FORCE IN RIGID BODIES. 331

A.ictm 3 (a). Motion of a body is change of its poeitton fn relation to another
body or to some real or imaginary point, which (for conyenieiice) we regard as fixed,
or at rest. Thns, while a stone &11b from a roof to the ground, its position, relatively
to the roof, is constantly changing, as is also that relatively to the ground and that
relatively to any given point in the wall ; and we say that the stone is in motion relor
tively to either of tkote bodies, or to any point in them. But if two stones, A and B,
flail from the roof at the same instant and reach the jironnd at the same (subsequent)
instant, we say that although each moves, relatively to roof and ground, yet they
have no fi^otum rebxtivdy to each other; or, they are at rest relatively 1o each other;
for their position in regard to each other does not change ; i. e., in whatever direction
and at whatever distance stone A may be from stone B at the time of starting, it
remains in that same direction, and at that same distance from B during the whole
time of the fall. Similarly, the roof, the wall and the ground are at rest relatively
to each other, yet they are in motion relatively to a falling stone. They are also in
motion relatively to the sun, owing to the earth's daily rotation about its axis, and
iti annual movement around the sun.

(b) If a train-man walks toward the rear along the top of a freight train Just as
flwt as the train moves forward, he is in motion relatively to the train; but, as a
whole, be is at real relatively to ImUdingSf etc. near by ; for a spectator, standing at
a little distance from the track, sees him continually opposite the same part of such
building, etc. If the man on the train now stops walking, he comes to rest relatively
to the irotn, but at the same time comes into motion relatively to the surrounding
bnHdinffSt etc., for the spectator sees him begin to move along with the train.

(c) Since we know of no absolutely fixed point in space, we cannot say, of any
body, what its absoltUe motion is. Consequently, we do not know of such a thing as
absolute re«^ and are si^e in saying that all bodies are in motion.

Art* 4 (u). The ▼eloetty of a moving body is its rate of motion. A body (as a
railroad train) is said to move with uniform -velocttFy or constant velooit^y
when the distancee moved over in equal times are equal to each other^ no matter how
tmall those times may be taken.

(b) The -velocity la cxprcsacd by stating the dittance passed over during some
giv0n feme, or which tBovid be passed over during that time if the uniform motion
continued so long Thus, if a railroad train, moving with constant velocity, passes
over 10 miles in half an hour, we may say that its velocity, during that time, is
(». «., that it moves at (he rate of) 20 miles per hour, or 105,600 feet per hour, or 1780
feet per minute, or 2Si^ feet per second. Or, we may, if desirable, say that it moves
at the rate of 10 miles in half an hour, or 8R feet in three seconds, etc. ; but it is
generally more convenient to Htate the distance passed over in a unit of time, as in
one day. one hour, one second, etc.

(c) I^ of two trains, A and B. moving ^ith constant velocity,

A moves 10 miles in half an hour,

B moves 10 miles in quarter of an hour,

then the veloeitieB are,

A, SX) miles per hour,

B, 40 miles per hour.

In other words, the velocity of a body (which may be defined as the distance passed
over in a given time) is inversely as the time required to pafis over a given distance.

(d) By nnlt velocity is meant that velocity whieh, by common consent, is taken
as equal to unity or one. Where English measures are used, the unit velocity gen-
erally adopted in the study of Mechanics is 1 foot per second.

(e) When we say that a body has a velocity of 20 miles per hour, or 10 feet per
second, etc.. we do not imply that it will necessarily travel 20 miles, or 10 feet, etc. ;
for it may nc^ have snfBcient time for tbat. We mean merely that it is traveling at
the rate of 20 miles per hour, or 10 feet per second, etc. ; so that if it coniimied to move
at that same rate for an hour, or a second, etc., it would travel 20 miles, or 10 feet. etc.

(t) When velocity inereaget. it is said to be accelerated. When it decreases.
It is said to be retarded. If the acceleration or retardation is in exact proportion
to the time ; that is, when during any and every equal interval of time, the same degree
of change takes place, it is uniformly accelerated, or retarded. When otherwise, the
words vcuriahle and variaMy are used.

(s) A body may have, at the same time, tivro qr more Independent veloel-
requlring to be considered. For instance, a ball fired vertically upward from a

J

332 FOBOE IK RIGID BODIES.

Sn, and then falling again to the earth, has, daring the whole time of its rise and
1, (iBt) the tmiform vptoard Telocity with which it leaves the muzzle, and (2nd) the
continually acceUrated dovmward Telocity given to it by gravity, which acts upon it
daring the whole time. Its remUant (or apparent) velocity at any moment is the
d^ertnoe between these two.

Thus, immediately after learlng the gun, the downward velocity given by
gravity is very small, and the resultant velocity is therrfore npwanl and Teiy
nearly equal to the whole upward velocity due to the powder. But after awhila
the downward velocity (by constantly increasing) beoomes equal to the upward
velocity ; i. «., their difference, or the resultant velocity, becomes nothing ; the ball
at that instant stands still ; but its downward velocity continues to increase, and
immediately becomes a little greater than the upward velocity ; then greater and
greater, until the ball strikes the ground. At that instant its resultant velocity is

rthe downward Telocity which it would ) , ( the uniform upward
*» •€ have acquired by falling dwring the V — < velocity given by the
(, vahoU tivM of its rite and faU. ) ( powder.

We have here neglected the resistance of the air, which of course retards botb
flie ascent and the descent of the ball.

(li) As a further illustration, regard a b n c as a raft drifting in the direction
ca ox nh. A man on the rait walks with uniform velocity from comer n ta
corner c while the raft drifts (with a uniform velocity a
little greater than that of the man) through the distance n b. /^a\

Therefore, when the man reaches corner c, that comer has v'H^vVs^

moved to the point which, when he started, was occupied by xTff-"-^^

a. The man's resultant motion, relatively to the bed of the / ; /

river or to a point on shore, has therefore been » a. His / j. /

motion at right angles to n a, due to his walking, is t c, but ^<" — -fi /
that due to the drifting of the raft is o 6. These two are ***--.i'''*

equal and opposite. Hence his resultant motion <U right il

angles to n a is nothing ; he does not move from the line n a.
His walking moves him through a distance equal to n i, in the direction n a;
and the drifting through a distance equal to t a, and the sum of these two is n a.

(i) All the motions which we see given to bodies are but €hang«a in their unknown
absolute motions. For convenience, we may conflne our attention to some one or
more of these changes, neglecting others.

Thus, in the case of the ball fired upward from a gun (see. (9) above) we may
neglect Its uniform upward motion and consider only its constantly accelerated
downward motion under the action of gravity ; or, as is more usual, we may oonaldar
only the retuUard or appatrmi motion, which is first upward and then downvrard. In
both cases we neglect the motions of the ball caused by the several motions of
the earth in spaed.

Art. 5 (•)• Forcoy <be «Miiu« of change of motion. Suppose •
perfectly smooth ball resting upon a perfectly hard, frictionless and level surfiMS^
and suppose the resistance of the air to be removed. In erder to merely move the
ball horizontally (i. e., to set it in motion — ^to change its state of motion) some /orc«
must act upon it. Or, if such a ball were already in motion, we could not retard
or hasten it, or turn it from its path without exerting force upon it. For, as stated
in Neiirton's flrat \wk-v¥ of motion, ''•-rerjr body continues In its
•tstte of rest or of motion in a straight line, except in so far as it may be com-
pelled by impressed forces to change that state." On the other hand, if a force act*
upon a body, the motion of the body must undergo change.

(b) Force Is an action betifreen t-wo bodies, fending eitber i»
separate them or to bring them closer togeU&er. For Instance, when
a stone falls to the ground, we explain the Csct by saying that a force (the attnction
of gravitation) tends to draw the earth and the stone together.

Magnetic and electric attraction, and the cohesive force between the particles of a
body, are other instances of ottmcttee force.

(c) Force applied by contsMst. In practice we apply force to a body (B)
by causing contact between it and another body (A) which has a tendency to motwti
toward B. A repulsive force is tljus called into action between the two bodies (io
■omo way which we cannot understand), and this force pushes B forward (or in the

FOBCE IN RIGID BODIEBL 333

direction of A's tendency to move) and pushes A backward, thus diminishii^ its for-
ward tendency *

If, for instance, a stone be laid npon the ground, it tends to moTe downward, bat
does not do so, because a repulsive force pushes it and the earth apart Just as hard as
the force of gravity tends to draw them together.

Similarly, when we attempt to lift a moderate weight with our hand, we do so by
giving the hand a tendency to move upward. If the hand slips from the weight
this tendency moves the hand rapidly upward before our will force can dieck it.
But otherwise, the repulsive force, generated by contact between the hand (tending
upward) and the weight, moves the latter upward in spite of the force of gravity,
and pushes the hand downward, depriving it of much of the upward velocity which
it would otherwise have. It is perhaps chiefly fh>m the eftortf of Vhich we are
conscious in such cases, that we derive our notions of <Yorce."

When a moving billiard ball. A, strikes another one, B, at rest, the tendency
of A to continue moving forward is resisted by a repulsive force acting between it
and B. This force pushes B forward, and A backward, retarding its former velocity.
As explained in Art. 23 (a), ' the repulsive force does not exist in either body

ontil the two meet

(d) The repulsive force thus generated by contact between two bodies, continues to
act only so long as they remain in contact, and only so long as they tend (from *
■ome extraneous cause) to come closer together. But it is genenJly or always
accompanied by an additional repulsive force, due to the compreuion of the particles
of the bodies and their tendency to return to their original positions. This eUutic
repulsive force may continue to act after the tendency to compression has ceased.

(e) Force acts either sui a P^I or sui a puatai. Thus, when a weight
Is susiMnded by a hook at the end of a rope, gravity jmU« the weight downward, the
weifrht ptuhn the hook, and the hook puUi the rope, each of these actions being
accompanied, of course, by its corresponding and opposite "reaction.** When two
bodies collide, each pushps the other, generally for a very short time.

(1) EjqiiaUtjr of actloni aad reaction. A force always exerts itself equally
upon the two bodies between which it acts. Thus, the force (or attraction) of
gravitation, acting between the earth and a stone, draws the earth upward just as
hard as it draws the stone downward ; and the repulsive force, acting between a
table and a stone resting upon it, pushes the table and the earth downward just as
bard as it pushes the stone upward.' This is the fact expressed by Ne'vrton's
tl&lrd lainr ot motloiiy that **to every action there is always an equal and
contrary reaction.*' For measures of force, see Arts. 11, 12, 13.

If a cannonbidl in its flight cuts a leaf from a tree, we say that the leaf has reacted
against the batt with precisely the same force with which the ball acted against the
leaf. That degree of force was sufficient to cut off a leaf, but not to arrest the ball.
A ship of war, in running against a canoe, or the fist of a pugilist strikint; his
opponent in the foce, receives as violent a blow as it gives ; but the same blow that
will upset or sink a canoe, will not opprecto&Iy affect the motion of a ship, and the
blow which may seriously damage a nose, mouth, or eyes, may have no such effect
upon hard knuckles.

The resistance which an abutment opposes to the pressure of an arch ; or a retain-
ing^wall to the pressure of the earth behind it, is no greater than those pressures
themselves ; but the abutment and the wall are, for the sake of safety, made capable
of sustaining much greater pressures, in case accidental circumstances should pro-
duce such.

(§p) In most practical cases 'we liaT-e to consider only one of the two bodies
between which a force acts. Hence, for convenience, we commonly speak as if the
force were divided into two equal and opposite forces, one for each of the two bodies,
and confine our attention to one of the bodies and the force acting upon it, neglect-
ing the other. Thus we may speak of the force of steam in an engine as acting
upon the pitton, and neglect its equal and opposite pressure against the head of
the founder.

(h) That point of a body to which, theoretically, a force is applied, is called the
pplnt ot application. In practice we cannot apply force to a point, according
to the seientlflo meaning of that word ; but have to apply it distributed over an ap-
preciable area (sometimes very large) of the surface of the body.

* We ordinarily express all this by saying simply that A pushes B forward, and this
is sufficiently exact for practical purpoees ; but it is well to recognize that it iH merely
a convenient expression and does not fully state the facts, and that every force neees-
aarUff consists of two equal^and opposite pulls or pushes exerted between two bodiai.

334 FORCE IN RIGID BODIBS.

For the present we shall aasume that the line of action of the force passes
through toe center of gravitf of the body and forms a right angle with the sur-
face at the point of application.

Art* 7 (a). Acoeleratlon. When an unresisted force, acting upon a body,
sets it in motion (i. «., gi^es it Telocity) in the direction of the force, this velocity
increases as the force continues to act; each equal interval of time (if the force
remains constant) bringing its own equal increase of velocity.

Thus, if a stone bu let full, the furce of gravity gives to it, in the first in-
conceivably short interval of time, a small velocity downward. In the next equal
interval of time, it adds a second equal velocity, and so on, so that at the end of
the second interval the velocity of the stone is twice as great, at the end of the
third interval three times as great, as at the end of the first one, and so on. We
may divide the time into as small equal intervals as we please. In each such
interval the constant* force of gravity gives to the stone an equal increase of
velocity.

Such increase of velocity is called accelerstion-f When a body is thrown verticaUy
uptoardy the downward acceleration of gravity appears as a retardation of the upward
motion. When a force thus acu offaitut the motion under consideration, its acceleivp
tion is called negatim.

Art* 8 (a). Tbe rate of aujoeleratloiii is the acceleration which takes
place in a given Hmsj as one second.

rb) The unit rate of acceleration is that which adds unit of velocity in a
unit of time ; or, where Bnglish measures are used, one foot per second, per geeond,

(o) For a given rate of acceleratioo, the total accelerations are of course propor-
tional to the HmsM during which the velocity increases at that r&te.

Art. 0 (tt), Iia^rs of acceleration* Suppose two blocks of iron, one fwhich
we will call A^ twice as large as the other (a), placed each upon a perfectly fricnonless
and horizontal plane, so that in moving them horizontally we are opposed by no force
tending to hold them still. Now apply to each block,

through a spring balance, a pull such as will keep the pointer of each balance always
at the same mark, as, for instance, constantly at 2 in both balances. We thus have
equal forces acting upon unequ^ masses.^ Here the rate of acceleration of a Is
double that of A ; for nrlien the forces are equal tbe rates of auseelera-
ration are Inversely as tike masses*

In other words, in one second (or in any other given time) the small block of iron,
a, will acquire twice the increase of velocity that A (twice as lai^e) vdll acquire ; so
that if both blocks start at the same time from a state of rest, the smaller one, a, will
have, at the end of any given time, twice the velooitff of A, which has twice its mass.

(b) Again, let the two masses, A and a, be equal, but let the foree exerted upon a
be twice that exerted upon A. Then the rate of acceleration of a will (as before) be
twice that of A ; for, 'vrl&en tl&e masses are cqnatly tbe rates of aoeelera*
ration are alrectljr as the forces*

(e) We thus arrive at the principle that, in any case, the rate of acceleration
Is dlreotljr proportlonsil to the force and Inwerselw proportional
to the n&ass*

* We here speak of the force of gravity, exerted in a given place, as constant,
because it is so for all practical purposes. Strictly speaking, it increases a very little
as the stone approaches the earth.

t Since the rtUe of acceleration is generally of frreater conseq-qence. in Meohanies,
than the total acceleration, or the "acceleration" proper, srienttfic writers (for the
sake of brevity) use the term "acceleration" to denote that rate, and the term
"total acceleration" to denote the total increase or decrease of velocity occnrrinK
during any given time. Thus, the rate of acceleration of gravity (about 32.2 ft. per
second per second) is called, simply, the "acceleration of gravity.'* As we shall not
have to use either expression very frequently, we shall, generally, to avoid misappre'
hension, give to each idea its full name ; thus, <* total acceleration " for the whoU
change of velocUy in a given case, and " rate of acceleration " for the rate of that
change.

t The mass of a body Is the quantity of matter that it contains.

FOBCB IN BIOID BODIES. 335

*

(d) Htticat if we make the two forces propmrtloiial to the two maases, tbe rases
of aoceleratioQ will be equal ; or, t»r m fpiwea vmtm of acosleimtloii^ tbe
forces most be dlrectljr as the masses.

(e) Hence, also, a greater force is required to Impart a g^Ten Teloci^ to a girea
body in a short time than to impart the same Telocity in a longer time. For instance,
the forward coupling links of a long train of cam wonld snap instantly nnder a pull
safBoIent to give to the train in two seconds a Telocity of twenty miles per hour, sup-

Ciing a suflBoiently powerftil looomotiTe to exist In many such cases, therefore) we
Te to be contented with a slow, instead of a rapid acceleration.
A string may safely sustain a^ weight of one pound suspended from our hand. If
we wish to impart a great upward Telocity to the weight in a very sJiort timey we eTi-
dently can do so only by exerting upon it a great force; in other words, by Jerking
the sMng Tiolently upward. But if the string has not tensile strength sufficient to
transmit this force from our hand to the weight, it will break. We might safely
giTe to the weight the desired Telocity by applying a le$» Jbre^ during a longer time.

{t) When a stone falls, the fi>rce pulling the earth upward is (as remarked aboye)
equal to that which pulls the stone downward, but the tncun of tne earth is so Tastly
greater than that of the stone that its motion is totally imperctptible to us, and
would still be so. eTen if it were not counteracted by motions in other directions
in other parts of the earth. Hence we are pracHcaUy^ though not abtolutelif, right
when we say that the earth remains at rest while the stone fiJls.

(§;) Bat in the case of the two billiard balls (Art 5e. p. 388), we can dearly see

the result of die action of the force upon each of the two bodies; for tbe second

ball, B» which was at rest, now moTes forward, while the forward Telocity of tbs

lint OB», A, is dimiidshed or destroyed, its backward mention thus appearing as a

. ntenlaMBa of ita forward motion. And, (since the same force acts upon both balls)

mass . mass . rate of acceleration . rate of negatlTe acceleration
ofA'ofB'* ofB ofA

or (siaee the ibrce acts Ibr tbe saate time upon both balls)

miuMy mass forward Telodty . loss of forward Telocity
oTA * ofB *' OfB OfA

' Ok) "RgwAng. A man oaamot 1^ a weight of 20 tons; but if it be placed upon
prv^r friction rollers^ he can move it horisontally, as we sea in some drawbridges,
tumtablea, Ac. ; and if friction and the resistance of the air could be entirely remoTed,
he could BOTC it by a ringle breath ; and it would continue to uoto forerer after the
foiee of the brecrth had ceased to act upon it. It would, howsTer, moTe Tory slowly,
because the force of tbe single breath would hsTe to diffuse itself among 20 tons of
matter. He can more it, if it be placed in a suitable Teasel in water, or if snqiended
from a long rope. A powerful locomotlTe that may moTe 2000 tons, cannot lift 10 tons

Terticaltar.

If we imagine two bodies, each as large and heaTy as the earth, to be precisely
balanced in a pair of scales without friction, a single grain of sand added to either
icale'paa, would giTe motion to both bodies. '

Art. lO (a). The constant force of gravity is a uniformly accelerating force
when it acts upon a body falling freely ; for it then Increases the Telocity at uie uni-
form rate of .322 of a foot per second during every hundredth part of a second, or 32,2
feet per second in eTery second. Also when it acts upon a body moving down an in-
clined plane; although in this case the increase is not so rapid, becatise it is caused
l^ only a part of the graTity, while another pert preeses the body to the plane, and a
third part OTercomes the friction. It is a uniformly retarding force, upon a body
thrown Tertically upward; for no matter what may be the Telocity of the body
when projected upward, it will be diminished .322 of a foot per second in each
hundredth part of a second during its rise, or 82.2 feet per second during each
entire second. At least, such would be the case were it not for the varying resistance
of the air at difforent Telocities. It is a uniformly straining force when it causes a
body at rest, to press ux)on another body ; or to pull upon a strinfi; by which it is
suspended. The foregoing expressions, like those of momentum, strain, push, pull,
lift, work, &c., do not indicate different hinde of force ; but merely different kinds of
^eets producM by the one grand principle, force.

(b) The aboTe 82.2 feet per second is called the aeceleratton otgrm.'vltr f and
by scientiile writers is conTcntlonally denoted by a small g % or, more correctly qieak-

336 FOBOB IN BIGTD BODIEa.

tag, ifnce the aoc«l«ratloii li not precisely the eame at ftll parti of Che Mrtb, g
denoteethe aooeleratloii ptf aeoond, whateTer it may be, at Any particular idaoe.

. Art. 11 (a). Ralatton b«tw«eit force and nuuM* The mass of a body
is the quantity of matter which It contains. Ons cubic foot of water has ttotei
AS great a mass as /la^ a cubic foot of water, but a lesi mass than one cubic
foot of iron. Thus, the n'Mof a body is a measure of mass between bodies
of the tame material, but not between bodies of different materials.

(b) When bodies are allowed to fall freely in a racuum at a given place,
4hey are found to acquire equal velocities in any eiven time, of whatever
different materials they may be Qomposed. From ibis we know (Art. 9 (dV,
p. 335). that the forcee moving them downward, viz. : their respective tMighu
at that place, must be proportional to their maeaee.

Thus, in any given placet the weight of a bodv is a perfect measure of its mam.
But the weight of a given bodv changes when the body is moved from one level
above the sea to another, or from one latitude to another; while the mass of
the body of course remains t/ie same in all places. ThuSga piece of iron which
weighs a pound at the level of the sea, will weigh leee than a pound by a spring
balance, upon the top of a mountain close by. because the attraction between
the earth and a eiven mass diminishes when tne latter recedes from the earth's
center. Or if tne piece of iron weighs one pound near the North or South
Pole, it will, for the same reason, weigh leet toan a pound by a spring balaliioe
if weighed nearer to the equator and at the same level above the sea.

The difference in the weight of a body in different localities is so slight as
io be of no account in questions of ordinary practical Mechanics ;• bat
scientific exactness requires a measure of mass which will give the same
expression for the quantity of matter in a given body, wherever it may
be; and, since weighing Is a verv convenient way of arriving at the quantity
of matter in a body, it is desirable that we should still be able to express tiie
mass in terms of the weight. Now, when a given body is carried to a hieher
level, or to a lower latitude, its loss of weight is simply a decrease in the Jores
with which gravity draws it downward, and this same decrease also causes
a decrease of the velocitu which the body acquires in falling during any
given time. The change m velocity, by Art. 0 (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 sxessa
1 in 300; or. under ordinary circumstances, 1 in 1000.

FOBOE IN BiaiD BODIES. 337

By method A, therefore, the unit of masn is g times (or about 33 times) the
mass of the standard piece of metal called a pound; i. e., a body containing
one such unit of mass wei^s g lbs. or about 32 lbs.; or, tijr method A,

the weight of any given body ^ ^ y the mass of the body,
in lt«. — Sf A jjj oaitg Qf mass.

file moss of a body, in units of mass - l^.^ ^^^g^^ ^^ ^^^ ^Q^y> ^^ POP°<^

g
For instance:

in a body weighing the mass is about

y^ pound ^ unit of mass

1 •*

2 «

82 " i

64 « 2 *• ••

It has been suggested to call this unit of mass a *' Matt/*

(•) In naetlMPd By the moM of the standard pound piece of platinum is taken
as toe unit of nuum and is called a pound} and the force which will give
to it in one second a velocity of one foot per second is taken as the unit of force.
This small unit of force is called a ponndal* In order that it may in one
second give to the mass of one pouna a velocity of only one foot per second, it

must (by Art f b), be -i. f or about Jjj of the weight of said pound mass.

Hence, "by m«tl&od By

the ma8$ of any given body, in jxrunds - the weight of the body in poundaU

and

the weiifht of a body, in potmdale — gr X the maea of the body in pound§.

Forinstar'jce:

in a body weighing the mass of the body is about

^ pouitdal — > JL pound JL pound

82 " — 1 " 1 ••

64 " — 2 « 2 "

tty VoT coi&'renlenoey we sometimes disregard the scientific require-
ment that the unit of force must be that which will give unit rate of accele-
ration to nnit mass, and take a pound of matter as our unit of maes^ and a
pound weight as our unit of force. Our unit of force will then in one second
give ft velocity of g (or about 32.2 feet per second) to our unit of mass. In
Sicties^ we are not concerned with the masses of bodies, but only with the
fijrees acting upon them, including their weights.

Art* 13 (a). Impnlse. By taking, as the unit of force, that force which, in
one second, will give to unit mass a velocity of one foot per second, we have
(by Art. 9, p. 334), in any case of unbalanced /orc« acting upon a mass during a
iSvon timei

Velocity - force X time ^^

mass

Force - volocity X mass ^gj

time
Mass - .force X time ^3^

velocity
Time - niass Xvelocity ^^

force

Force X time — mass X velocity. . . . (5> *

25

338 FORCE IN RIQID BODIES.

Tb the prodnoft^ force X time, in equation (5), writers now give the name
Impolee^ which was formerly given to eoUUion (now called liiip««t}* See
Art 24 (a). The term impuUe, as now used^ conveys merely the idea

of force acting through a certain length of time. Equation (5) tells us that an
impulse (the product of a force by the time of its action) is numerically equal
to the momentum* which it produces. Eqilation (2) tells us that any force is
numerically equal to the momentum which it can produce in one second. In
other words, the monftentmn of a body moving with a given veloci^ is
numerically equal to the force which in one second can produce or destroy
that velocity in that body; or, a force is numerically equal to the rate pw
second at which it can produce momentum. Thus, forces are proportional to
the momentums which they can produce in a given time; or, in a given time^
equal forces produce equal momentums. Therefore a force must always give
equal and opposite momentums to the two bodies between which it acts.

Art* 13 (»)• Tlk* luiiial -wajr of meaanrln^ a fbrce is by ascertaining
the amount of some other force which it can counteract. Thus we may meas-
ure the weight of a body by hanging it to a spring balance. The scale of the
balance then indicates the amount of tension m the spring: and we know thai
the weight of the body is equal to the tension, because the weight just pre
vents the tension firom drawing the hook upward.

Thus, fbremm are conveniently expressed In -vrelfpltt|i9 as in pounds,
tons, &c., and they are generally so measured in Statics, and in our following
articles.

(b) A fbroe mav' be aonstant or Tarlable* When a stone rests upon
the ground, the pull of gravity upon it (i. e., its weight) remains constanlh
neither increasing nor decreasing. But when a stone is thrown upward its
weight decreases very slightly as it recedes from the earth, and again increases
as it approaches it during its fall. In this case, the force of gravity, acting
upon the stone, decreases or increases eteadUy, But a force may change
euddenlyf or irregtUariyf or may be intermittent $ as when a series of uneqiul
blows are struck by a nammer. In what follows we shall have to do only with
forces supposed to be conatarU,

Art. 14b («)• "DonuUjr* The deneitiea of materials are proportional to the
mauee contained in a given volume, as a cubio inch j or inoerselff as the volume
required to contain a given mass. Or^ since the weights at a given place are
proportional to the masses, the densities are proportional to the weights per
unit of volume (or ** specific gravities **) of the materials. Thus, a body weigh*
inff 100 lbs. per cubio foot is twice as dense as one weighing only 60 Iba. per
cuoic foot ait the same place.

Art. 15 (a). Inertia. The inability of matter to set itself in motion^ or ta
change the rato or direction of its motion, is called its inertia, or inertneeaL
\Blien we say that a certain body has twice the inertia (inertness) of a smaller
one, we mean that twice the /or<:0 is required to give it an equal ratoof acoete*
ration ; and that, since all force (Art. 5f)t acts equally in both direo*

tions, we experience twice as great a reaction (or so-called ** resistance*^ from
the larger body as from the smaller one. The ** inertia** of a body is therefora
a measure of the/ore« required to produce in it a given rato of acceleration; Ob
which is the same thing, it is a measure of the mass of the body. We mi^
therefore consider ** inertia'* and **mass** as identical.

(b) What is called the ** resistance of inertia** of a body, ia simply
reaction, (i s., one of . the two equal and opposite actions) of whatever
force we apply to the body. Hence, its amount depends not only upon tiia
mass of the Dody, but also upon the rato of acceleration which we choose to

*The momentam of a body (sometimes called its ** quantity of motion")
is equal to the product obtained by multiplying its ma»9 by its velocity. If «•
adopt the pound as the unit of mass, as in '* method B/* Art. 11 («), tha

proauct, voeight in pownds X velocity, is numerically either exactly or neurly
the same as the product, m(M8 in pounds X velocity, depending upon whether
or not the body is in that latitude and at that level where a ma8$ of one poufltdl
is said to weigh one pound. But the product, weight in poundats X velocity; It
exactly a times (about 82.2 times) the product, mass in pounds X velocity; afeo^
k^ ** menod A,** iMi^M in |K>uncte X velocity — y X «MM in ** matte '* X

FORCE IN RIGID BODIES. 339

giTe to it. Therefore we cannot tell, from the mass or weight of a bodj alone,
what its " reeistance of inertia " in any given case will be.

Art. 16 (a). Forees In opposite directions. When two equal and
opposite forces act upon a body at the same time, and in the same straight line,
we say that they destroy each other's tendencies to more the body, and it remains
at rest. If two unequal forces thus act in opposition, the smaller force and an
equal portion of the greater one are said to counteract each other in the same
way, but the remainder of the greater force, acting as an unbalanced or unresisted
force, moves the body in its own direction, as it would do if it were the only
force acting upon it.

Thus, when we move bodies, in practice, we encounter not only the " resist-
ance of inertia" (i. e., we not only have to exert force in order to move inert
matter), but we are also opposed by other /otom, acting against us, as friction,
the resistance of the air. and, often, all or a part of the wHght of the body. By
'' resistances," in the following, we mean such resisting /oroM, and do not include
in the term the " resistance of xHertia,"

(b) If separated, the two bodies, A and B, of 8 &m and 2 lbs respectively, would

fidl with equal accelerations = g ; each unit, — , of mass being acted upon by its

own weight, W, Bat. connected as they are, A will

move downward, and B upward, with an acceler- T^2A

ation =» only f ; for now an unbalanced force of

5

only 8 — 2 = lfb must give acceleration to a mass

(J'*^0

T*2j4

ation =» only | ; for now an unbalanced force of

5

,8 + 2 6 « " 2 T-2.4

of = -. But, to give to a mass, B, of -, an

aoeelof |, requires a force of -. I »^]b=»a4 lb. 3|liji EjI^

This, plus 2 lbs (required to balance the weight of Al

B) is the tension, 2.4 lbs existing throughout the

cord. Exerted at A, this tension balances 2.4 of

the 3 lbs weight of A. The remainder (8 — 2.4 = 0.6 !b) of the weight, acting

downward upon the mass, -, of A, gives to it the required acceleration of ^;

, - force .- 8 0.6 g .. g

for here = 0.6 -«- - = -r-2 = 0.2 g = f .

mass g 8 * 6 ^

Or we may regard the total tension, 2.4 lbs, in the cord at A, as acting upon A

O

and giving to it a negative or upward acceleration of 2.4 -t- - = 0.8 g, which,

g

dedacted from g (the acceleration which A would otherwise have) leaves

Acceleration = g — 0.8 g = 0.2 g = |.

Let W = weight of A
w ss weight of B
F a> net force available for acceleration » W — w

mr -4- w
M =3 combined mass of both bodies = — — —

g

m => mass of B » -

g
a => acceleration
T a tension in cord.

Then: a = ^ = (W - w) ^ :5^^ = «-^£i:^
M ^ g W + w

m . . ^ , ^ g(W — W) / W — W\

T = w + ma = w +— a==wH ^-vi?— ; = w 1 1 + ^5,— — |.

g 'gW + w \Wh-w/

(e) An ** Ati¥00«l*s Machine'* consists essentially of a pulley, a flexible
cord passing over the pulley, two equal weights (one suspended at each end of
the cord), and a third weigfnt, generally much lighter than either of the other
two. The two equal weights balance each other by means of the pulley and
cord. The third weight is laid upon one of the other two weights. The force
of gravity, acting upon the third weight, then sets the masses of the three
weights in motion at a small but constantly increasing velocity. In order to do
this it mast also overcome the friction of the pulley and cord, and the rigidity

340 FORCE IN RIGID BODIES.

of the latter ; bat, as these are made as slight as possible, they are, fbr ooo-
venience, neglected. The machine is used for Illustrating the acceleration given
to inert matter by unbalanced force, and forma an excel^nt example of the two
distinct duties which a moving force generally has to perform, vis: (1st) the
balancing of resistance, and (2nd) acceleration.

(d) In the case of a lo«oiiiotlire. drawfngr a train on a leTel, fHc-
tioa and the resistance of the air are the only resistances to be balanced ; for Uie
weight of the train here opposes no resistance. Unless the force of the steam is
more than sufficient to balance the resistances, it cannot mote the train. If it
exceeds the resistances, the excess, however slight, gives motion to the inert
matter of the train. If, at any moment while the train is moving, the force of
the steam becomes jtut equal to the resUtcmces (whether by an increase of the
latter or by diminishing the force) the train will move on at a uniform velocity
equal to that which it had at the moment when the force and resistance were
equalized ; and, if these could always be kept equal, it would so move on forever.

But so lone as the excess of steam pressure over the resistances continues to act,
the velocity Is increased at each instant ; for during eaeh such instant liie excess
of force gives a small velocity in addition to that already existing.

On a level railroad, let
P »- the total tractive force of the locomotive = say 13 tons <•
W sa weight of locomotive = 50 tons
w sa weight of train = 336 tons

R *= resistance of locomotive (including internal fHction, etc.) «> 8 tona
r a resistance of train =■ 1 ton
F » net force available for acceleration — P — R — r-s9 toms

M « mass of engine and train =■ — — ,. _ -* 12

* g 82.2

- ^ , w 8S6 ^^ , .
m «■ mass of train = - — —- = 10.44

g 32.2

a = acceleration

T = tension on draw-bar.

F 9
Then : Acceleration at a » ^ — r^ = 0.75 ft per second per second.

The tension T on the draw-bar « resistance of train + force causing accel-
eration a, orT=r + ma — 1 + 10.44 X 0.76 = 1 + 7.83 = 8.83 tons.
This tension, T, pulling backward against the locomotive, causes there a

T a aa «r

retardatim, or negative acceleration, of masa of tocomoUve = -go- = »» «

per sec per sec, and thus reduces, by that amount, the acceleration which the

(P ■■'' r) s 10 X 8S.8
locomotive would otherwise have, and which would be — ^ — ka ■■ ,. —

oO 60

~ 6.44. This, less 6.69, » 0.75 ft per sec per sec — acceleration of train.

(e) If the tractive force of a locomotive exceeds the resistances, due to friction,
grades, and air, the velocity will be accelerated ; but it then heoomeB more dilB-
cult to maintain the excess of force, for the pistons must travel fast«r through
the cylinders, and the boiler can no longer supply steam fast enough to maintain
the original cylinder pressure Besides, some of the resistances increase with
increase of velocity. We thus reach a speed at which the engine, alUiough
exerting its utmost force, can do no more than balance the resistances. T^e
train then moves with a uniform velocity equal to that which it had when thia
condition was reached.

When it becomes necessary to stop at a station some distance ahead, steam It
shut off, so that the steam force of the engine shall no longer counterbalance or
destroy the resisting forces; and the number of the resistances themselves is in-
creased by adding to them the friction of the brakes. The reBistanoea, thus
incneased, are now the only forces acting npon the train, and their acoeleration
is negative, or a retardation. Hence, the train moves more and more slowly, and
must eventually stop.

(f) Caution. When two opposite forces are in equilibrium, an addition to
one of the forces does not always form an unbalanced force ; for in many cases
the other force increases eguallyy up to a certain point. For Instance, when we
attempt to lift a weight, W, its downward resistance^ R, remains constantly Just
equal to our upward pull, P, however P may vary, until P exceeds W. Thas, R
can never exceed W, but may be much less than It. Indeed, when we atop pull-
ing, R ceases, although W (the attraction between the eartn and the weight) of

FORCE IN RIGID BODIES. 841

eoarse remnins unchanged throaghout. Such Tariation of resisting force, to meet
varying demands, occurs in all those innumerable cases where structures sustain
varying loads within their ultimate strength.

Art. 17 (a). Work. Force, when it moves a body,* is said to do " work "
upon it. The whole work done by the force in moving the body through any dis-
tance is measured by multiplying the force by thedutance; or: Work = Force
X distance. If the force is taken in pounds, and the distance in feet, the product
([or the work done) will be in foot-pounde ; if the force is in tons and the distance
in inches, the product will be in inch-tons ; and so on.f

Thus, if a force of moves a body through we have work =

1 pound 10,000 feet 10,000 foot-pounds

100 pounds 100 '* 10,000 "

10,000 " 1 foot 10,000 "

or, in any case, if the fiprce be F pounds, the whole work done by it in moving a
body through s feet, is F « foot-pounds.

(I») The foot-pound, the foot-ton, the inch-pound, the inch-ton, etc., etc., are
called unlto oi wwrfc.f

For practical purposes, in this country, forces are most frequently stated in
pounds, and the distances (through which they act) in feet. Hence tbe ordi-
nary anii of work, is the foot-pound. The metric nnit of work
is the klloflrram-meter, i e. l Kilogram raised 1 meter = 2.2046 pounds
raiaed 3.2800 feet, = 7.23S1 foot-pounds. 1 foot-pound = 0.13825 kilogram-meter.

(«) In most cases, a portion at least of the work done by a force is ex-
pended in owereomlnv reflistiunees. Thus, when a locomotive begins
to move a train, a portion of its force works against, and balances, the resist-
anoM of friction or of an up-grade, while the remainder, acting as unbalanced
toroe upon the inert mass of the train, increases its velocity.

An upward pull of exactly one pound will not raise a one pound weight, but
will merely biuanoe the downward force of gravity. If we increase the upward
pail from one pound (=» 16 ounces) to 17 ounces, the ounce so added, being
unbalanced foroe, will give motion to the mass, and will acceleirate its upward
velocity as long as it continues to act. If we now reduce the upward pull to 1
pound, thus miking it just equal to the downward pull of gravity, the body will
move on upward with a uniform velocity : but if we reduce the upward force to

15 ounces (= || pound), then there will be anjunbalanced dovmward force of 1
ounce acting upon the body, and this downward force will generate in the body
a downward or negative acceleration or retardation, and will destroy the upward
velocity in the same time aa the upward excess of 1 ounce required to produce it.

Daring any time, while the 17 ounces upward ** force" were acting against the

16 ounces downward " resistance," the product of total upward force X distance
mast be gre<Uer than that of resistance X distance. The excess is the work done
in accelerating the velocity, by virtue of which the body has acquired kinetic
energy or capacity for doing work in coming to rest.

On the other hand, while the npward velocity was being retarded, the product
of total upward force X dist was less than that of resistance X dist, the difference
being the work done by the kinetic energy against the resistance of gravity.

In practice, the term " work" is usually restricted to that j9or<ion of the work
which a force performs in balancing the resistances which act against it ; in other
words, to the work done by so much of the force as is equal to the resistance.

With this restriction, we have work ^ force X dist, = resistance X dist.

Thus, if the resistance be a friction of 4 ft>s., overcome at every point along a
distance of 8 feet; or if it be a weight of 4 S>s., lifted 3 feet high, then the work
done amounts to 4 X 8 » 12 foot-9>8, provided the initial and the final velocities
are equal.

(d) In cases wbere tbe weloeity Is nnlform, as in a steadily running
macbine, tbe force is necessarily equal to the resistance ; and where the velocities
at the beginning and end of any work are equal (as where the machine starts
from rest and conies to rest again) the mean force is equal to the mean resistance.
In such cases, therefore, the two products, mean force X distance, and mean
resistance X distance, are equal, and we have, as before,

Work =^ force X dist = resistance X dist.

♦ A man who Is standing still is not considered to be working, any more than
is a post or a rope when sustaining a heavy load ; although he may be support-
ing an oppressive burden, or holding a car-brake with all his strength ; for his
force moves nothing in either case.

t These products must not be confounded with momerUs, — force X leverage.

342 FOKCE IN RIGID BODIES.

(f ) In calculating the work done by machinery, etc., allowance must be made for
this expenditure of a portion of the work in overcoming resistances. Thus, in pump-
ing water, part of the applied force is required to balance the friction of the different
parts of the pump; so that a steam or water "power,** exerting a force of 1(H) &8.,
and moving 6 feet per second, cannot raise 100 fi>8. of water to a height of 6 feet
per second. Therefore machines, so far from gaining power ^ according to the popular
idea, actually lose it in one sense of the word. In Uarting a piece of machinery, the
forces employed have (1st) to balance, react a^rainst, or destroy the resisting force
of friction and the cohesive forces of the material which is to be operated on ; and
(2d) to give motion to the unresisting matter of the machine and of the material
operated on, after the resisting forces which had acted upon them have thus been
rendered ineffective. But after the desired velocity has been established, the forces
have merely to bcUance the resistances in order that the velocity may continue uniform.

(g) That portion of the work of a machine, etc., which is expended against fric-
tion is sometimes called <* lost -work " or ** prejudicial ^rorky" M'hile only
that portion is called " useful -vrork " which renders visible and tangible service
in the shape of output, etc. Thus, in pumping water, the work done in overcoming
the friction of the inimp and of the water is said to be lost or prejudicial, while the
useful work would be represented by the product, weight of water deliverwl X height
to which it is lifted.

The distinction, although artiflcial| and somewhat arbitrary, is often a very con-
venient one ; but the work is of course not actually ** lost," and still less is it ** pre-
judicial ;" for the water could not be delirered without first overcoming the resist-
ances. A merchant might as well call that portion of bis money lost which he
expends for clerk-hire, etc.

(it) For a given force and distance^ tlie i^ork done is independent of the

time $ for the product, force X distance, then remains the same, whatever the time
may be. But the distance through which a given force will work at a given velocity
is of course proportional to the time during which it is allowed to work. Thus, in
order to lift 50 pounds 100 feet, a man must do the same work, (= 6000 foot-pounds)
whether he do it in one hour or in ten ; but, if he exerts constantly the scrnie foroey
he will lift 50 &>s. ten times as high in ten hours as in one, and thus will do ten times
the work. Thus, for a given force, the vrork is proportional to the tinte*

Art. 18 (a), Poorer. The quantity of any work may evidently be considered
without regard to the time required to perform it ; but we often require to know the
rate at which work can be done ; that is, how much can be done within a certain
time.

The rate at which a machine, etc. can work is called its -power. Thus, in selecting
a steam-engine, it is important to know how much it can do per minute, hour, or dag.
We therefore stipulate that it shall be of so many horse-powers; which means nothing
more than that it shall be capable of overcoming resisting forces at the rate of so
many times 33,000 foot-pounds per minute when running at a uniform velocity, i. e.,
when force X distance = resistance X distance.

(b) The liorse-poiver, 33,000 foot-pounds per minute, or 550 foot-pounds per
second, is the unit of ponrer, or of rate of ivork, commonly used in connec-
tion with engines. The metric horse-poorer, called "force dt
cheval," " cheval-vapeur," or (German) " Pferdekraft," is 75 kilogram-meters pel
second = 542.48 ft-ibs. per sec. = 32,549 ft.-ft>s. per minute = 0.9863 horse-power. 1
horse-power = 1.0138 " force de cheval." In theoretical Mechanics the foot-ponud
per second is used in English measure ; and the lUlo§;ram-meter per ceo-
ond in metric measure,

1 foot-pound per second =» 0.13826 kilogram-meter per second.
1 kilogram-meter per second = 7.2331 foot-pounds per second.

(c) Up to the time when the velocity becomes uniform, the po-wer, or rate 9t
vrork, of the train, in Art. 16 (d), is variable, being gradually axelerated.
For in each second it overcomes its resistances (and moves its point of application)
through a greater distance than during the preceding second. Also, after the steam is
shut off, the rate of work is variable, being gradually retarded. When the force of
the steam just balances the resistances, the rate of work is uniform.

(d) Po-«rer = force X velocity. Since the rate of work is equal to the work

done in a given Hrne, as so m&xxy foot-pounds per second, we may find it by dividing the

work in foot-pounds done during any given time by the number of seconds in tkst

time. Thus

_ ^ * , force in pounds X distance In feet

Power =■ rate of work = \. , ; •

time in seconds

FOBGE IN RIGIB BODIES.

343

Bat this is eqaivalent to

- . * . J V ^ distance in feet

Power -rate of work -force in pounds X time in seconds

— -orce in lbs. X velocity in feet per second.

Or if we treat only of the work of that force which overcomes resUtancea: or i«
eawes where the velocity is either uniform throughout or the same at the
beginning and end of the work;

Power rate of work _ resistance, w velocity,

in ft-lbs. per sec " in ft-lbs. per sec in lbs. ^ in ft per sec.

Thus if the resistance is 3300 lbs. and is overcome thrpugh a distance of 10
feet in every minute; or if the resistance is 33 lbs. and is overcome through
?di8tonce of 1000 f4et per minute, the rate. of the work i^J^J^^^J'^
the same, namely, 33,000 foot-pounds per mmute, or one horso-power; Sat

lbs. vel. lbs. vel. .

8300 X 10 — 33 X 1000 — 33,000 foot-pounds per mmute.

M The same "power" which will overcome a given resistance through*
riven distance, in agiven time, will also overcome any other resistance through
Wiy other distance, in that same time, provided the «:<»w**^°®**°^.*^^®.5S!?
when multiplied together give the same amount as m the first case. Thus.
the power that will lift 60 pounds through 10 feet in asecond, will m a second
Hft 600 pounds, 1 foot; or 25 pounds. 20 feet; or 6000 pounds ^ oi a foot.
El practice, the adjustment of the speed to suit different resistances, is usually
effected by the medium of cog-wheels, belts^or lever.. By "^eans of
these the engine, watei>wheel, horse, or other motive power, exerting a given
force and ruhning at a given velocity, may be made to overcome small resist*
ances rapidly, or great ones slowly, as desired.

Art. 19 (a). The 'vrork 'vrhldi a bodjr ean do hy -rlrtiie ot its
motion j or (which is the same thing) the 'vrorh reonircd to brins
the body to rest. Kinetic energy* -vim -viTa^ or "living ttorce.'*

As already remarked, a force equal to the weight of any body, at any place,
will, in one second, give to the mass or matter of the body a velocity — g, or
(on the earth's surface) about 32.2 feet per second. Or if a body be thrown
\Lpward with a velocity — ■ g, its weight will stop it in one second.

Since, in the latter case, the velocity at the beeinning and at the end of the
■econd are, respectively,— g feet per second, ana — 0, the mean velocity of the

iody is -£- feet per second. Therefore, during the second it will rise _^ feeC^

2 2

or about 16 feet. In other words, the work which any body can do, by virtue
of being thrown vertically upward with an initial velocity (velocity at the
gtart) otg feet per second, is equal to the product of its weight multiplied of

-J- feet Or,

work in foot-pounds — weight X -^

Ifotioe that in this ease (since the initial velocity v Is equal to jy), JL. — 1.

^ 9

Smppose now that the same body be thrown upward with double the former

velocity; i. e., with an initisd velocity equal to.2 g (or about 64 feet per seconds
dince gravity requires (Art 8 c), two seconds to impart or destroy this

velocity, the body will now move upward during two seconds, or twice as long
a Urns as before. But its mean velocity now is p. or twice as great as before.
Therefore, moving for double the time and with double the velocity, it will
teavel /our times as far, overcoming the same resistance as before (viz. : its
own weight) through /our times the distance.

Thus, by making its initial velocity v — 2 p, {. «., by doubling its -L-. making

g
it — 2, we have enabled the body to do four times the work which it could

io when its — !L was 1; so that the work in the second case is equal to the

9

344 FOBOE IN RIGID BODIE&

product of that in the first case multiplied by the 8quar$ of -2L( Qg^

- weight X -2- X ^

— weight X —

And it is plain that this would be ithe case for any other velocity. Now the
total amount of the work which the body can do, is independent of the
amount of the resistance against which it is done; for if we increase the
resistance we diminish the distance in the same proportion, so that their
product, or the amount of work, remains the same. The above formula^
therefore, applies to all cases ; i. 6., the total amoiuit ot 'vrorfc, in fo^
pounds, whicn any body will do, f^ainst any resistance, by virtue of its motioii
Alone, in coming to rest, is

Work - weight of moving body, in lbs. X square of its velocity in ft per sec^d

f/
— weight of moving body, in lbs. X fall in ft required to give the velocity

_ weight of moving body, in lbs, y square of its velocity in ft per second
g 2

In these equations, the weight is that which the body has in any given plaoe^
and g is the acceleration of gravity at that same place.

(b) Since the weight of a body j^ j^^ ^^^ ^^^^ 1^^ ^ 336), the last formula
becomes, by "method A,^* Art. 11 (d).

mass of moving body w square of its velocity in ft per second
in foot^ot^mb " in "matU^' '^ 2

and by "method B," Art. U (e),

mass of moving body v> square of its velocity in ft per eeobad

infoo^poundato" in potmdij ^ 2

(c) In the above equations the left hand side represents the work (or resis-
tance overcome through a. distaiice) in any given case, while the right hand
side represents the Unetlo energy of the body, by which it is enabled to do
that work. Some writers call this energy "via ▼!▼»,»» or " living force" a
name formerly given (for convenience) to a quantity just double the energy,
or — mass X velocity*.

(d) As an illustration of the foregoing, take a train weighing 1,120,008
pounds, and moving at the rato of 22 feet per second. The kinetic energy
ef such a train is

energy - weight X I5!2^; or.

1,120,000 lbs. X — — 8,400,000 ft.-lbs.
64.4

That is, if steam be shut off, the train will perform a work of 8,400^000 fL-lba.
in coming to rest. Thus, if the sum of all the resistances (of friction, air,
grades, curves, ete.) remained constantly — 6000 lbs.,* the train would travel

8,400,000 ft.-lb8. _ lesott,
5000 lbs.

(e) We thus see that the total quantity of work which a body can do by virtua
of its motion alone, and without assistance ft-om extraneous forces, is in pi^
portion to the weight of the body and to the square of its velocity when it
begins to do the work. For example, suppose that a train, at the momaDft
when steam is shut off, has a velocity of 10 miles an hour and that the kinetio
energy, which that velocity gives it, will by itself carry the tram against th»

•In practice, this would not be the case.

9OB0B IK RIGID BODIES. 345

CMistances of Che road, etc^ for it distance of ons quarter of a mile before it
stops. Then, if steam be shut off while the train is moTing at 5, 20, 30 or 40
miles per hour (t. e^ with ^^ 2, 8 or 4 times 10 miles per hour) the train will
tiavel JL, 1, 2 ^ or 4 miles (or ^ 4, 9 or 10 times ^ mile) before coming to

rest*

Bat the rate of work done is proportional simply to the resistance and the
ntoeity (Art* IBd, p. 842). Therefore, the locomotive whose steam is shat oft
at 20, 80 or 40 miles per honr, will require, for running its 4. 9 or 16 quarters
tf a mile, but 2, 3 or 4 times as many seconds ae it required at 10 miles per hour.
The same principle applies to all cases of acceleration or of retardation.f
For instance, in the case of a falling body, the distance through which it
mnst fall in order to acquire any giren velocity is as the square of that
Telocity, but the time required is simply as the velocity. Also, if a body is
ttirown Terticanlly upward with any given velocity, the height to which it will
rise bvh the time gravitv destroys that velocity, will be as the square of the
Yelooity,but the time wiU be simply as the velocify.

Art. SO (a). The momentnin of a moving body (or the product of its
mass by its velocity) is the rate, in foot-pounds per second, at which it works
against a resisting force equal to its own weighty as in the case of a body thrown
vertically upward. At the instant when it comes to rest, its momentum, or rate
of work, is of course = nothing. Therefore its mean rate of work, or mean
momentum, is one-half of that which it has at the moment of startiug.

Thus, suppose such a body to weigh 5 lbs. Then, whatever its velocity may
be, 6 pounds is the resisting force, against which it must work while coming
to resL Let the initial velocity be 96 feet per second. Then its

momentum ■• mass X velocity «— 6 X 96 — 480 foot-pounds per second?

Mid, while ooming to rest, its

•Moa momentum -» mass X T .^r^ ■« 240 foot-pounds per second.

Now, in falling, the weight of the body (5 lbs.), would ^ve it a velocity of 96
foet per second in about three seconds. Consequently, in rising, it will destroy im

lelooity in the tame time. In other words, the time — ,. velocity ^ velocity

•^ acceleration g

M £| 1. 3. Three seconds, therefore, is the time during which it can work.

How, if the mean rate of work in foot-pounds per seeond (at which a body
ean work against a resistance) be multiplied by the time during which it can
ooBtinue so to work, the product must be the total work done. Or, in this case^

work mean rate of work v^ time, oji* v <» ion *r^* »wvn»^.

to IWbe, - in flrlbs. per sec. X or No. of sees. - 240 X 3 - 720 footrpounds.

-weight X 12}2£ife X ^^l^^ifc
2 g

.weight X y^'?^ ,asinAjt.l9(o),-6 X ^ - 720 ft.ponnda

(b) We may notice also that since, in the case of a falling body, or of one

ihixywn upward, . ^"^^ is the time during which it must fall in order to

0
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— irucUblc. Energy, expended In wortt, to
not destroyed. It is either transterred to other bodies, or eue stored ap in the
body itself; or part may be ithua transferred, and the re^t thus stored. Bnt^
althoagh ener^ cannot be destroyed, it may be rendered useless to us. Thn^
amoTing train, in coming to rest on » level track, transfers its kinetic enei
into other kinetio energy: namely, the useless heat due tofidctioo at the r
brakesand Journals ; and this heat, although none of itiadeatrayed, is disai]
Jed the earth and air so as to be practicallyoeyond our recovery.

Alt. sa (a). Potential •nergy* or possible energy, may be defined as
•toted-np energy. We lift a one-pound body one-foot oy expending upon it
one foot-pound of energy. But this foot-pound is stored up in the **sy8tem **
(composed of the earth and the body) as an addition to its stock of potential
energy. For, while the stone falls through one foot, the system wilt acquire
a kinetic energy of one foot-pound, and will part with one foot-pound of its
potential energy. •

(b) The potentiai energy of a ''system*' of bodies (such as the earth and a
weight raised above it, or the atoms of a mass of powder, or those of
a bent spring) depends upon the relative poaitiona of those bodies, and
upon their tendencies to change those positions. The kinetie energy of a
system (such as the earth and a moving train of cars) depends upon the tnaM«6
m its bodies and upon their motion relatively to each other.

Familiar instances of potential energy are— the weight or spring of a clock
When fully or partly wound up, and whether moving or not; the pent-up water
In a reservoir; the steam pressure in a boiler; and the explosive energy of
powder. We have mechanical energy in the case of the weight or springs or
water; heat energy in the case of the steam, and obemica! energy in that
df the powder.

(o) In many oases we ma3r conveniently estimate the total potential enei^
of a systenu Thus (neglecting the resistance of Uie air) the explosive energy
of a pound of powder is » the weight of any given cannon ball X the height
to which the force of that powder could throw it. •» the weight of the ball X
(the square of the initial velocity given to it by the explosion) -i- 20. But in
other cases we care to find only a certain definite portion of the total potential
energy. Thus, the toM potential energy of a olock-weight* would not be
exhausted until the weight reached the center of the earth: but we generally
deal only with that portion which was stored In it by winding-up. and which
tt will give out again as kinetio energy in running down. This portion is -• th^
weight X the height which it has to run down -• the weight X (the square of
the velocity which it would acquire in fallin^/V>oe{y through that height) -i- 2if.

(d) There are many cases of energy in which we may hesitate as to whether
the term "kinetic" or "potential** Is the more appropriate. Thus, the pres-
sure of steam in a boiler is believed to be due to tne violent motion of the
particles of steam, which bombard the inner surface of the boiler-shell; so
that, from this point of view, we should call the energy of steam kinetie. But,
on tne other hand, the shell itself remains stationary; and, until the steam is
permitted to escape from the boiler, there fs no outward evidence of energy
in the shape of work. The energy remains stored up in the boiler ready kt
nse. From this point of view, we may call th e energy of steam potential energy.

(e) It seems reasonable to suppose that further knowledge as to the nature
of other forms of energy, apparently potential (as is that of steam), might
reveal the fact that all energy is ultimatiely kinetio.

Art. 23 (a). There is much confusion of ideas in regard to those
actions to which, in Mechanics, we give the names, *' force," *• enerfry«'*
** power," etc. This arises from i he fact that in every-day language these
terms are used indiscriminately to express the sime ideas.

Thus, we commonly speak of the " force " of a cannon-ball flying through the
air, meaning, however, the repulsive force which would be exerted between the
ball and a building, etc. with which it might come into contact. This force
would tend to move a part of the building along in the direction of the flight
of the ball, and would move the ball backward ; (i. e., would retard Its forward
motion). But this great repulsive "force" does not exist until the ball strikes
the building. Indeed, we cannot even tell, from the velocity and weight of the
ball, what tne amount of the force will be, for this depends upon the strength,
etc., of the building. If the building is of glass, the foroe mav be so slight as
scarcely to retard the motion of the ball perceptibly, while,'if the building is an

* For convenience we may thus speak of the energy of a mdem of bodies (the
earth and the clock-weight) as resiaing in only one of the bodies.

FORCE IN RIOIB BODIES. 347

earth embankment, the force will be much greater, and may retard the motion
oX the ball so rapidly as to entirely stop it before it has gone a foot farther.

The moving ball has great (kinetic) energy; but the only force that it exerts
during its flij^ht is the comparatively very slight one required to push aside the
particles of air.

The energy of the ball, and therefore the total work which it can do, are inde^
pendent of the nature or the obstruction which it meets ; but since the work is
the product of the resistance oifered and the distance throu^^h which it can be
overcome, the distance must be inversely as the resistance offered ; or (which is
the same thing) inversely as the force required of, and exerted by, the ball in
balancing that resistance.

Since work, in ft.-lb8. => force, in &>s., X distance traversed, in feet, we have

force in lbs. = work, in ft.-lbs. _ rate of work,

' distance traversed, in feet in ft.-lbs. per fool.

Art. S4 (a). An impact, blow, stroke or collision takes place when a
moving body encounters another body. The peculiarity of such cases is that
the time of adion of the repulsive force due to the collision Is so short that een-
erally it is impossible to measure it, and we therefore cannot calculate the force
ttovsx the momentum produced by it in either of the two bodies : but since both
bodies undergo a great change of velocity (i.e., a great acceleration) during this
Short time, we know that the repulsive force acting between them must be very
great.

We shall consider only cases of direet Impact, or impact where the centers
of gravity of the two bodies approach each other in one straieht line, and where
the nature of the surfaces of contact is such that the repulsive

force caused by the impact also acts through those centers and in their line of
approach.

(b) This forcCj acting equally upon the two bodies (Art. fi/), for the
same length of time (namely, tne time during which they are in contact), neces-
sarily produces equal and opposite changes in their momentums (Art. 12, p. 888).
Hence, the total momentum (or product, mass X velocity) of the ttoo bodies is
always the same after impact as it was before.

(c) But the relative behavior of the two bodies, after collision, depends upon
their elasticity. If they could be perfectly inelastic, their velocities, after im-
pact, would be equal. In other words, they would move on together. If they
could be perfectly elastic, they would separate from each other, after collision,
with the same velocity with which they approached each other before collision.

(d) Between these two extremes, neither of which is ever perfectly realized in

Enictice, there are all possible degrees of elasticity , with corresponding differences
1 the behavior of the bodies. The subject, especially that of indirect impact, is
a very complex one, but seldom comes up in practical civil engineering.

(e) " In some careful experiments made at Portsmouth dock-yard, England, a
man of medium strength, and striking with a maul weighing 18 lbs., the handle
•f which was 44 inches long, barely started a bolt about '% of an inch at each
blow ; and it required a quiet pressure of 107 tons to press the bolt down the
same quantity ; but a smsQl additional weight pressed it completely home."

348

GRA.VITY — ^PALLING BODI£S.

«RATITT. FAIililire BOBIK8.

Bodies flAlllngr Tertleally. A body, falling freely in racuo
from a state of rest, acquires, by the end of tbe first second, a Telocity of about
32.2 feet per second ; and, in each succeeding second, an cuidition of velocity, or
aoceleratiod, of about 82.2 feet per second. In other worda^ tbe Telocity receivM in
each second an acceleration of about '62.2 feet per second, or is accelerated at tbe
raU of about 32.2 feet per second, per B^cond. This rate ie generally called (fbr
brerity, see foot-Bote,t p. 334), simply the sM)oeleratloia of gravity (bat see *
below), and is denoted by |p« It increases ftx>m about 82.1 f«et per second, par
second, at the equator, to about 32.5 at the poles. In the latitude of London it if
82.19. These are its values at sea-level ; but at a height of 6 miles above that level
it is diminished by only about 1 part in iOO. For most practical purpoeee it may be
taken at 32.2.

Caution. Owlnar to tbe resistance of the air none of the follow-
ing rules give perfectly accurate results in practice, especially at great vela.
The greater the specific gravity of the body the better will oe the rMnlt. The air
ffeelets botn rislnir and fklllnir bodies.

If a body be tbrown vertically upwards with a given vel, it will
rise to the same height from wiiich it must have fallen in order to acquire said
vel; and its vel will be retarded in each second 32.2 It per lec* Its average ascend'
ing velocity will be half of that with which it startled ; as in all other cases of
uniformly retarded vel. In falling it will acquire the same vel that it started
up with, and in the same time. See above Caution.

Acceleration acquired*

in a given time = ff X time

in a given fall from rest = \^ 2 g X fall.

in a given fall from rest ) __ twice the fall

and given time j *~ time

Time required

- , , x» acceleration

for a giyen acceleration >- —

9

for a given fall flrom rest

fall

fall

3^ final velocity
fall

for a given fall from rest i _^

or otherwise / ™ mean vel ~~ J^ (initial vel + final vel)

FaU

In a given time (starting from rest) — time X H ^^"^ ^^ ^ timeS X ^iff
in . giren time (.t«:ttagi _ inltl.1 t«1 + ftn.l r«|

from rest or otherwise) J 2

reqd for a given acceleration "i __ acceleration^

(starting from rest) ) 2g

during any one given second (counting from rest)

■» ^ X (number of the second (Ist, 2d, Ac) — \\

during any equal consecutive times (starting from rest) « 1, 3, 5, 7, 9, Ae.

wfti^e f ^^*- 2d. 3d. 4th. 6th. 6th. 7th. 8th. 9th. 10th.
' seconds

Velocity; ft per sec.
Dist fallen since end
of preceding sec ; ft.

Total diet fallen; ft.

32.2
16.1

64.4
48.3

96.6
80.5

16.1 1 64.4 144.9

128.8
112.7
257.6

161.0
144.9
402.6

193.2
177.1
679.6

225.4
209.3
788.9

267.6

241.6

1090.4

289.8

278.7

1904.1

822.0

805.9

1610.0

* By " acceleration,** in thi» article, we mean the total aooelerstion ; C «., tbe whole
change of velocity occarring in the givwi tins or fUl. For the raft oC *rflHtwrtn>
we use simnly the letter g.

DESCENT ON INCLINED PLANES.

349

I^escent on Inclined plirnes. When a body, U. is placed
upon an inclined plane, AC, its whole weight W is not employed m giviug it
▼elocity (as in the case of bodies falling vertically)
but a portion, P, of it (= W X cosine of o = W X
cosine of a*) is expended in perpendicular pressure
against the plane; while only S, (= W X sine of o
— W X sine of a*) acts upon U in a direction parallel
to the surface AC of the plane, and tends to slide it
down that surf.

The acceleration, generated in a given body in a
given time, is proportional to the force acting upon
the body in the direction of the acceleration

Hence If we make W to represent bv scale
tbe ttccfeleration g (say 32.2 ft per l*c) which gray
would give to U in a sec if falling freely, then S will
give, by the same scale, the acceleration in ft per
sec which the actual sliding force 8 would give to U in one sec if there were
no friction between U and the plane. We have therefore

theontio^ acceleration down the plane = gr x siae of a.

Therefore we have only to substitute "^. sin a" in place of "flr;" and the
</omn^ distance or "slide" AC in place of the corresponding vertical distance
or " fall " A £ in the equations, in order to obtain the acceleratioos etc as

follows :

on an inclined plane witbont friction.

In the foUowingr* tbe slides A € are in feet, tbe times in
seconds, and tbe velocities and accelerations in feet per
second.t

Accelerationfof sliding velocity

i« - -j««« n^^ "^^rt accel acquired in falling) w i^ _
in a given time = ^^^^ ^^^j^^ the same time / X sin a

B g. sin a X time

in agiven slide, as AC,> slide
from rest i 14 ti°>e

f vert accel acquired in falling)
=< freely thro the corresponding >•■
{ verthtAE J

» y' 2 ^. sin a X slide

V^7^'^

for a given sliding acceleration

Time required
sliding acceleration

ff, sin a

for a given slide, as A C, lirom _^ slide

wst "" y^ final sliding veloc

« /__8lid
iity "" V H flf. si

slide
sin a

time reqd to fall freely thro the correspond-
ing verthtAE

sin a

slide

slide

for a given slide, from > ^ ^

rest or otherwise J "* mean sliding vel "" H (initial + final sliding vels)

Cosine a

Sine a

horizontal stretch, as E C»

base EC of any length, aa A C ^ l/AC» — Al?

length A C ^ that length A C

height A E _ fall, A E. in any given length, A C ^ T/AC2 — te<>

length AC"" "^

that length

AC

* Because o and a are equal.
tHr acceleration,*! flW» cartielet we mean the total acceleration, t. «., the whok
eha&flle in telodty occurring in the given time or slide, for ttie rate of acceleration
ire nse tiaiolT the letter a.

350 GRAVITY — PENDULUMS.

Slide, u A C

in a glyen time, starting from rest = time X }4 final sliding Tel

= time *X}iff. sin a.

in a given time, s<«rting from rest ., ,,

or otherwise — ""*® X mean sliding Tel

- time X H (initial + final, sliding rels)

required for a ffiren sliding accel- „ sliding acceleration*
oration (starting from rest) *" 2 p. sin a

But in praetlce the sUdlmr on the plane ts always on-

£!?;^ ^X ™««»- To Inclnde the emJt of Metionrwe hJ^

only to substitute

sin a - (cos a. ooeff fric)] " in place of « g. sin a " in the abore eqoatlona.
Lse

Friction = Perpendicnlar pressure P X coefficient of friction

= weight W X cosine a X coefficient of friction
and

retardation of firletlon '^gX cosine a X coefficient of friction.

Besnitant slidinir acceleration

« theoretical sliding accel (due to the sliding force, S) — retardation of fHo
=- iff. sin a) — (g. cosine a. coeff fric)

= ffX fsin a — (cosine a. coelTfrlc) j

If the retardation of friction (•= y. cos a X coeff fVic) is not leu than the total
•r theoretical accel ("^. sin a") the body cannot slide down the plane.

"PX
Because

• ^

PENDULUMS.

Tex numbers of ribrations which diff pendulums will make in any ^Ten place la
a giren time, are inversely as the square roots of their lengths : thus, if one of them
Is 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; but
its number of vibrations will be but ^ /^i or i^ as great. The times in which diff
pendulums will make a yibration, are directly as the sq rts of their lengths. Thna,
if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, S, or 4 times aa
great ; and so also will be the time occupied in one of its vibrations.

The length of a pendulum vibrating seconds at the level of the sea, in a Taonmn.
in the lat of London (51^ North) is 39.1393 ins ; and in the lat of N. York (409^

North) 39.1013 ins. At the equator about ^ inch shorter ; and at the poles, about -ffg
Inch longer. Approximately enough for experiments which occupy but a few sec,
we may at any place call the length of a seconds pendulum in the open air, 89 ins ;
half sec, fl^^ ins ; and may assume that long and short ribrations of the same pen-
dulum are made in the same time ; which they actually are, very nearly. For mea»-
nring depths, or dists by sound, a sufficiently good sec pendulum may be* made of a
pebble (a small piece of metal is better) and a piecfi of thread, suspended fh>m a
common pin. The length of 39 ins should be measured from the centre of the pebble.

PXBJSTDULUMS, ETC. 351

In Btartliig tlie Tibratlons, the pebble, or boby must not be thrown into motion, but
meroly lei drop^ after extending the string at the proper height..

To find the lenKrtb of a pendalam read to make a given number of
vibrations in a min, divide 375 by said reqd number. The square of the quot will bo
the length in ins, near enough for such temporary purposes as the foregoing. Thus,
for a pendulum to make 100 vibrations per min, we have |^^ =» 3.75 ; and the square
of 3.75 = 14.06 ins, the reqd length.

To find (lie namber of ▼ibrationti per min for a pendulum of
given length, in ins, take the sq rt of said length, and div 375 by said sq rt. Thus,

for a pendulum 14.06 ins long, the sq rt is 3.75 ; and z-=i » 100, the reqd number.

Rkk. 1. By practising before the sec pendulum of a dock, or one prepared as Just
stated, a person will soon learn to ooant 5 in a sec, for a few sec in succession ; and will
thus be able to divide a sec into 5 equal parts ; and this may at times be oseftil for
▼ery rough estimating when he has no pendalam.

Oentre of Oscillation and Pereusslon*

Bsv. 2. When a pendulum, or any other suspended body, is vibrating or oscillating
backward and forward, it is plain that those particles of it which are far front the
point of suspension move faster than those which are near it. But there is always
a certain point in the body, such that if all the particles were concentrated at it, so
that all should move with the same actual vel, neither the number of oscillations,
nor their angular vel, would be changed. This point is called the center of oKiUa-
Hon. It is not the same as the cen of grav, and is always farther than it fh)m the
point of suspension. It is also the cerUre of percussion of the suspended vibrating
body. The dist of this point fh>m the point of snap is found thus : Suppose the body
to be divided into many (the more the better) small parts ; the smaller the better.
Find the wc^gnt of each part. Also find the cen of grav of each part ; also the dist
firom each such con of gray to the point of susp. Square each of these diets, and
mult each square by the wt of the corresponding small part of the body. Add the
products together, and call their sum p. Next mult the weight of the entire body
by the dist of its cen of grav from the point of susp. Gall the prod p. Divide p hyg*
Thinp is the moment of inertia of the body, and if divided by the wt of the
body the sq rt of the quotient will be the Radius of Gyration.

Angrnlar Telocity.

When a body revolves around any axis, the parts which are farther from that
axis move faster than those nearer to it. Therefore we cannot assign a stated
linear velocity in feet per second, or miles per hour etc, that shall apply to every
patriot it. But every part of the body revolves around an entire circle, or
through an angle of 860P, in the same time. Hence, all the part« have the same
▼elocity in deare^i per second, or in revolutions per seoond. This is called the
angular velocity. Scientific writers measure it by the length of the arc de-
scribed by any point In the body in a given time, as a second, the length of the
arc being measured by the number of times the length of Us ottn radius la con-
tained in it. When so measured,

Angular velocity __ liaear velocity (in feet etc) per sec
in radU per second - length of radius (in feet etc)

Here, as before, the angular velocity is the same for all the points in the body,
because the velocities of the several points are directly as their radii or dis-
tances from the axis of revolution.

In each revolution, each point describes the circumference of the circle in
which ft revolves =» 2 v r (ir = 3.1416 etc ; r = radius of said circle). 0>nse-
qaently, if the body makes n revolutions per second, the length of the arc de-
scribed by each point in one second is 2irrn; and the angular velocity of the
body, or linear velocity of any point measured in its own radii, is .

2irr»

2 w » «= say 6.2832 X revs per second = say .1047 X revs pe» f^inute.

Moment of Inertia.

Sappose a body revolving around an axis, as a grindstone; or oscillating, like
apeodnlum. Suppose that the distance from the axis of revolution (which, in
the pendulum, Is the point of suspension) to each individual particle of the
body, has been measured; and that the square of each such distance has been
multiplied by the weight of that particle to which said distance was measured.

352

MOMENT OF INERTIA.

The sum of all these products is the moment of inertia of the bodf . Or

Moment
of Inertia

-{

the sum,
lor all the particles

}•'

r weight square of dist

-< of X of particle from
(.particle axis of revolution

or, I = 'S,<fiw.
Scientific writers frequently use the mass of each particle ;

ie,

its weight

instead of its weight, in calcnlatiug

acceleration (g) of gravity, or about 32.2
the moiueiit of inertia.

Ill practice we may suppose the body to be divided into portions measuring
a cubic inch (or some other small size) each : and use these insteaO of the theo-
retical infioitely small particles. The smaller these portions are taken, the
more nearly correct will be the result.

When the moment of inertia of a mere surface is wanted (instead of that of a
body), we suppose the surface to be divided into a numl)er of small areaSf and
use them instead of the weights of the small portions of the body.

weight of body, Muare of

Table of Radii of Clyratlon.

Body

Any body or
fig^nre

Solid cylin-
der

ditto

ditto, infinitely

short (circular

surface)

Hollow cyl*
inder

ditto, infinitely
thin

ditto, of any
thickness

ditto, infinitely
thin

ditto, infinitely
thin and infinitely
short (circumfer-
ence of a circle)

Solid spbere

Rewolwinff
around

any given axis

Its longitudinal
axis

adiam, midway
between Its enas

a diameter

its longitudinal
axis

ditto

a diam midway
between its ends

ditto

a diameter
a diameter

V

Badlas of Gyratioii

'moment of inertia around the given axis
weight of body, or area of surface

radius of cylinder X 'Xj-k-
* radius of cylinder X aboat .7071

V

'length' radiu8> of cylinder
12 "*" 4

V

radius of oylinder

inner rad» + outer radi
2

nidlUB of eylfnder

V

dinner rad' + outer rad* ^ length^
4 ■*" 12

V

radius^ of cylinder length*

12

radias of cylinder X
■at radius of cylinder X

about .7071

V

radius* of sphere
2.5

= radfus of sphere X V'Ti"

= rndins of sphere X about .68246

BADU OP GYRATION.

353

Table of Radii of Oyration,— CoimiruBD.

Hollow
•plioro of any

thickness

ditto, thin

ditto, inflnitelT

thin (spherical

surface)

8traiirl>t line,

ab

Solid eone

Circular
plate, of rect-
angular cross seo-
tion

Circular
ring^, of rectan-
$oIftr cross section

Square, rect-
angrle and
otlier snr*

RevolviniT
aroand

a diameter
ditto

ditto

any point, £, In its
length

either end, a or 6

Its center, e
its axis

S«e Solid cylin-
der

See Hollow cylin-
der

Badins of Oyration

V

2 (outer rad* — inner rad»)

5 (outer rad* — inner rad')
approz (outer rad + inner rad) x ^065

radius of sphere
■■ radios of sphere X al>but .8166

Sab

length aft X "\-^
— length abX about JB776

•- length abX about .2887

radius of base of cone X '\/~S'
M radius of base of cone x .5477

For the thidcnest of plate or ring,
measured perpendicularly to the plane
of the circumference, take the length of
the cylinder.

For Uasl radius of gyration, or that around the longe$t aafs,
see p 496 and 497.

2a

364 CENTRIFUGAL FORCE.

GEBTTRIFVOAI. FORCE.

When a body a, Fig. 1 , moves in a circular path abd^ it tends, at each point, as
a or 6^ to move in a tangent at or bif to the circle at that point. But at each
point, as a, etc., in the path, it is <ie;fiected from the tangent by a force acting
toward the center, c, of tlie circle. This force may be the tension of a string, ca,
or the attraction between a planet at e and its moon a^ or the inward pressure
of the rails, ah,OB & curve, etc., etc. Like all force, it is an action between two
bodies, tending either to separate them or to draw them closer together, and act-
ing equally upon both. (See Art. 5 (6), p. 882). In the case of the string, itpnlU
the body a, Un/xird the center, r, and the nail or hand, etc., at c, toward the body
at a or 6, etc. ; i. e.^from the center. In the case of a oar on a curve it pushes the
car toward the center, and the rails from the center. The pull or push on the
revolving body toward the center is-called the eentripetttl forc«; while the
pull or push tending to move the defecting body from the center is called the
cenArira^al force. These two *^ forces," being merely the two ** sides " ^as
it were) of the same stress, are necessarily equal and opposite, and can onlv exist
toffether. The moment the stress or tension exceeds ihe strength (or inherent
conesive force) of the string, etc., the latter breaks. The centripetal and centrif-
ugal forces therefore instantly cease ; and the body, no longer disturbed by a
deflecting force, moves on, at a uniform velocity,* in a tangent, at or M', etc., to
its circular path*; «. «., at right angles to the direction whloh the centrifugal force
had at the moment it ceased.

4

(a). A singrle revoliringr body, a, Fig. 1. Let
= the centrifugal or centripetal force, in pounds.

= the weight of the body a, in pounds,
= the radius ca of the path of the center of arc
V = the uniform velocity of the body a in ftt circular path dbd,'iu feet per

= the radius ca of the path of the center ofgraviiy of the body a, in feet.

second,

ft — the number of revolutions per minute,
^ a the acceleration of gravity = say 32.2 feet per second per second,

900 ^ = about 28980.
«■ = circumference -^ diameter » say 3.1416. ir* « about 9.869ft.

Then, for the centriAigal force, /:

If we have the velocity v in feet per second : / = W ^ t • • • (1)

If we have the number n of revolutions per minute : / = W ' t • • • (2)

9U0 g

/» about .0008406 WB»* 2 ... (8).

* Neglecting friction, gravity, the resistance of the air, etc.

t For let a/. Fig. 1, represent the amount and direction of the velocity • of the body
at a in feet per second. Then at the end of one second the body will have reached
the point b (the arc ab being made = a/), and the amount and direction of its
velocity at b will then be represented by the line bt' = a< in lengtli, but differing; in
direction. Drawing cu and cm' at the center, equal and parallel respectively t<i at
and bt'y we find that the change in the direction of the motion (».«., the acceleration
towaid the center) during the second is represented by the arc mm' ; and, since angle
aeb = angle ttcu', we have the proportion, radius H or m : ab or at :: cu or at: arc
Mt\ In other words, the acceleration tnt' in one second, or rate of acceleration, is ^

aC V*

■o '^ ^* ^"^> ^^^ ^^® f<°^^ causing that acceleration, we have

/ = mass of body X rate of acceleration =» mass of body X v "■ ^ ^S~'
JByformula(l),/ = W . But»=— — — :andv*

^g 60 * 3600 900 '

It X „, »r*R'n« _.ir«Rn»

$ Formula (3) is obtained from (2) by substituting the values 9.8696 and 2898U for
v" and 900 g respectively.

CENTRIFUGAL POftCE.

355

(b) Wbecls and dines. Suppose the rim of a wheel to be cut into verj short
dices, as shown (much exaggerated) at a, Fig. 2. Then for each slice, as a, by

formula (1): /= weight W of slice X ^ ;* and if each slice were connected

ti m o

with tb« eenter by a separate string, the mini of the titresses in all the strings
(taeglectlhg friction between adjacent slices) would be:

F — sum of centrifugal forces of all the slices f = weight of riin X

'Big'

(4).

But the stress with which we ure usually concerned in such cases (viz. : the
tension f n tbe rim Itself in the direction of a taiugent to its own cir-
cumference) is much Uss than the theoretical quantity F obtained from formula

(4), being in fact only T^j^n ^^ ^^* ^^^ suppose first that the same thin rim is

cut only at two opposite points m and n, Fig. 3, and that its two halves are held
together only by toe string S.

* If the rim is very thin in proportion to itB diameter mn^ we may take the center

•f gravity of each slice as bein^ io a circle mn midway between the inner and outer

M * A*. _» ^ Ai..^-.. inner radius 4- outer radius - - - . ^,

edges of the rim, so thAt K = ^ . In a rim of appreciable

thidnMSS, this is not the case, because each slice is a little thicker at its outer than at
its inner end. See Fig. 6. Hence its center of grHvity is a little outside of the curved
line AMI, Fig. 2.

t In a perfectly balanced rim (». «., a rim whose center of gravity coincides with its
eenter of rotation, as in Fif;. 3) the centrifugal forces of tbe particles on one side of c
counterbalance those on the opposite side. Here, too, K = 0. Hence, as a lehole^
nich a rim hss lu> centrifugal force ; i. «., no tendency to leave the center in any one
Abrection by rirtue of its rotation. But if the two centers do not coincide (Fig. 4),
then the rim is a single revolving body, and its centrifugal force is : / = weight

•f entire rim X ^~ ; where B is the distance between the two centers, and v the

&g
velocity of the center of grarity a. The force / acts in the line Joining the two
wnters.

356 CENTRIFUGAL FORCE.

Then : •

F
semi-circumference mzn : diameter tnn : '• 'tt ' pull on the string S ;

80 that

pull on half weight ^ i^ ^ _2 weight t>* F F

■trtny S "^ of rim '^ R^ ^ ir "" of rim '^ R ^ir"" ir"" 3.1416 ' ' * ^ ''

and if the rim is now made complete bv joining the ends at m and n, and if the
string S is removed, then the pull on the string by formula (5) will be equallv
iLivided between m and n. Hence each cross-section, as m orn, of the rim, will
sustain a tensile stress equal to half the pull on the string; or

«^».«^« *- «««. il ?- weight of rim Xt>' .-.

tension m rim »- - ^28^ ^ 6.2882 Ry ' <®>'

The centripetal force,/, Fig. 2, holding any part o of the rim to its circular path,
is the resultant of the two equal tensions at the ends of that part.
For the streu per square inch of cross-section of rim, we have :

_ tension in rim

~ area A of cross-eection of rim, in square inches

F _ weight of rim X v* ,-.

""6.2832A~ 6.2882 A R^ ^ '*

We shall arrive at the same result if we reflect that the pull in the string S
or the sum of the two tensions at m and n, is equal to the centrifugal force /of
either half of the rim, revolving, as a sinsle body, about the center e. Find the
center of gravity G of the half rim, and then, in formula (1), use the velocity of
that point, and the radius cG instead of velocity at g and radius cz reepectively ;
thus:

«.,ii *« »*^«« — / ^ centrifugal force_ „^i„ut «<• u„i/ ^„ >• (velocity at G)« .
pull in string == / = ^^ haff-rim = '^^'^^^ ^f half-rim X ^^^ ;

and half of this Is the tension in each cross-section of the rim.t

If the rim were Infinitely thin, cG, Fig. 3, would be 0.6366 ex.
If Its thickness must be taken Into consideration, and If it is of rectangular
crofls-section, find the centers of gravity g and jK, Fig. 6, of the whole semiolrcular
segment cz and of the small segment c6 respectively (eg *» 0.4244 oe, and eg' =
0.4244 eb. Then

. , area of entire segment cz

g'^ = gg'X area~of halTrim *

For rims of other than rectangular cross-secticHi, use formulae (4), (5) and r6).

In a disc, sncli as a irrlndstone, the tension In each full cross-section
mn. Fig. 7, is equal to the centrifugal force / of ha^ the disc. Let W » weight
of half disc. The distance cG from the center c to the center of gravity G of
the half disc, Is cG = 0.4244 cz ; and the

* In Fig. 2, let the centrifugal force of any slice, o, be represented by the diagonal,
/, of a rectangle, whose sides. H and Y, are respectively parallel and perpendicular
to the given diameter mn. Tnen H and V represent the components of / in those
two directions. The equal and opposite horizontal components H, of o and of th<*
corresponding slice o', being parallel to mn, have no tendency to pull the rim ^^art at
m or n. Hence, the pull on a string S, Fig. 3, perpendicular to mn, is the sum of the
components Y of all the slices. For each very thin slice. Fig. 6 (greatly exaggerated)
we have (since angle A = angle A') :

Length I . its horizontal . . centrifugal force , its vertical
of slice • projection, p ' ' /, of slice * component V.

Hence, for the entire half-rim mn^ Fig. 3 (made up of such slices), we have:

«rh.lf.rim • prelection «, ' ' sT^or^. llfiim' ' f*^ J,,^'. •>'

which is identical with the proportion at top of page.

t The rim* of revolving wheels are usually made strong enough to resist the tension
due to the centrifugal force, without aid from the apohe$^ which thus have merely to
support the weight of the wheel. But if the rim breaks, the centrifugal forces of its
fragments come entirely upon the spokes; and, since the breakage is always irregn-
lar, some of the spokes will always receive more than their share.

CENTRIFUGAL FORCE.

357

rad. cGXfl' 0.4244 czx^

(8).

(»).

= W

= w

0.4244 (vel. at g)«

czXg
0.4244 g« n» cz

900^

The stress per square inch in any full section mn is

tension in mn

unit stress =>

area of cross-section in square inches
0.4244 (velocity at g)'

= W

-W

diam. mm, ins. X thickness, ins. XczXff

0.4244 ir« w« cz

diam. mn, ins. X thickness, ins. X 900 ^

. .(10)1
. . (11).

Fig. 5

n m c n
Fig. 7

f= the centripetal force, in pounds, acting uvon a single revolving body, a,
Figs. 1, 2, 4 and 5, or upon the halt-rim or half-disc, Figs. 8, 6 and 7
= the centrifugal force exerted by such body.'

TP = the sura of the centrifugal forces f, of all the particles of a rim, Fig. 3.

W = the weight of the body, in pounds.

R = the radius c<iy Figs. 1, 4 aud 5, of the path of the center of gravity of the
body.

V = the uniform velocity of the body in its circular path, in feet per second.

n = tlie number of revolutions per minute.

g = the acceleration of gravity = say 32.2 feet per second. 900 g = about
28980.

oircumfereuce

w =

= say 3.1416. ir* = about 9.8696.

diameter

In m rolling wlieel, each point in the rim, during the moment when it
touches the ground, is stationary tpith respect to the earth; but each particle has
the same velocity abont the center as if the latter were stationary, and hence the
•entrifugal force has no effect upon the weight.

368 8T4TICS.

STATICS.

FORCES.

!• Statics Defined. The science of 3tatics, or of equilibrium of forces;
takes account of those very numerous cases where the forces under con-
sideration are in equilibrium, or balanced. It embraces, therefore, all cases
of bodies which are said to be "at rest."*

2. In the problems usually presented in civil engineering, a certain
given force, or certain given forces, applied to a stationary* body (as a bridge
or building) tend to produce motion, either in the structure as a whole or in
one or more of its members; and it is required to find and to apply another
force or other forces which will balance the tendency to motion, and thus
permit the structure and its members to remain at rest. See If 33, below.

3. Equilibrium* Suppose a body to be acted upon by certain forces.
Then those forces are said to be in equilibrium, when, as a whole, they pro-
duce no change in the body's state of rest or of motion, either as regards its
motion as a whole along any particular line (motion of translation), or as
regards its rotation about any point, either within or without the body.
In such cases the body also is said to be in equilibrium. See % 84, below.

4* A body may be in equilibrium as regards the forces imder consideration,
even -though not in equilibrium as regards other forces. Thus, a. stone, held
between the thumb and finger, is in equilibrium as regards their two equal
pressures, even though it may be lifted upward by the excess of the muscular
force of the arm over the attraction between the earth and the stone. Simi-
larly, on a level railroad, a car is in equilibrium as regards gravity and the
upward resistance of the rails, although the horizontal pull of the locomotive
may exceed the resistance to traction.

5. molecular Action. Any force, applied to a body, is in fact made
up of a system of forces, often parallel or nearly so, applied to the several
particles of the body. Thus, the attraction exerted by the earth upon a
grain of sand or upon the moon is, strictly speaking, a cluster of nearly par-
allel forces exerted upon the several particles of those bodies ; but, for con-
venience, and so far only as concerns their tendency to move the body as a
whole, we conceive of such forces as replaced by a single force, equal to
their sum and acting in one line. In thus considering the forces, we as-
^me that the bodies are absolutely rigid, so that each of them acts as a
angle " material particle" or " material point."

6. Transmission of Force. The upward pressure of the ground, upon
a stone resting upon it, acts directly only upon those particles which are
nearest to the ground. These, in turn, exert a (practically) equal upward
force upon those immediately above them, and so on; and the i<^rce is thus
transmitted throughout the stone.

7. Rigid Bodies. In treating of bodies as rigid, we assume that the
intermolecular forces hold the several particles absolutely in their original
relative positions.

It is not the material that resists being broken, but the forces which hold its
particles in their places. Thus, a cake of ice may sustain a great pressure;
but its particles yield readily when its cohesive forces are destroyed by a
melting temperature.

8. Force Units. The force units generally used in statics are those of
weight, as the pound and the kilogram. See Conversion Tables, p. 235.

In statics we have no occasion to consider the masses of bodies (except

* Strictly speaking, absolute rest is scarcely conceivable, since all bodies
are actually in motion (see Art. 3, p. 331). so that unbalanced forces produce
merely changes in the states of motion oi bodies. Yet, for a body to be at
rest, relative to other bodies, is a very common condition, and, in practical
statics, we usually regard the body under consideration as being at rest
relatively to the earth or to some other large body, so that the oaange of
state of motion, due to the action of unbalanced force upon it, consists in a
change from relative rest to relative motion. See % 33, below.

FORCES. 359

in so far as these determine their weights, or the force of gravity exerted upon
them), bodies being regarded merely as the media upon and through which
the forces under consideration are exerted. Hence we require, in statics,
no units of mass; and, as the bodies are regarded as being "at rest," no upits
of time, velocity, acceleration, momentum, or energy.

0. Forces, how Petermlned. A force is fully determined when we
know (1) its amount (as in pounds, or in some other weight unit), (2) its
direction, (3) its sense (see % 10), and (4) its position or its point of applica-
tion.

10. When a force is represented by a line, the length of the line
mav be made to represent by scale the amount of the force, and its direction
and position may often be made to indicate those of the force, while the sense
of the force may be shown by arrows or letters affixed to the lines, or by the
signs, + and — .

Thus, the directuma of the forces represented by lines a and 6, Fig. 1, are
vertical, and those of e and d are horizontal. The sense of a is upward, of b
downward, of c right-handed, of d left-handed. Thus, a and b are of like
direction, but of opposite sense; and so with c and d. In treating of vertical
or horieontal forces, we usually call upward or right-handed forces posi-
tive, and downward or left-handed forces nef^rative, as indicated by the
signs^ 4- and — , in Fig. 1.^ When a force is designated by two letters, at-
tached to the line representing it, one at each end of the line, the sense of the
force may be indicated by the order in which the letters are taken. Thus, in
Fig. 1, having regard to the directions of the arrows, we have forces, ef, ha*
k Cand n m,

11. Hfine of Action, etc. The point (see ^ 6) at which a force P, Fig. 2,
is supposed to be applied, as a, is called its point -of application. But
the force is transmitted, by the particles, throughout the body (see ^ 6), and

:t:i

k I *n n

— y ■< —

g

ri». 1.

the e€fect of the force, as regards the body as a whole, is not changed if it
be re^rded as acting at any other ix>int, as 6, in its line of action. We
may therefore regard any point in that line as a point of application of the
force. For instance, the tendency to move the stone, Fig. 2, as a whole, will
not bo changed if, instead of pushing it, at a, we apply a puU (in the same
direction and in the same sense) at b; and if a weight, P, be laid upon the
top of the hook, at b. Fig. 3, it will have the same tendency, to move the
hook as a whole, as it has when suspended from the hook as in the Fig.

A force cannot actually be applied to a body at a point outside of the sub-
stance of the body, as between the upper and lower portions of the hook in
Fig. 3, yet this portion also of the line a 6 is a part of the line of action of the
force. The vertical force, exerted by the weight, P, is transmitted to b by
means c^ bending moments in the bent portion of the hook.

12. Stress. (See Art. 1, Strength of Materials, p. 454.) Opposing
forces, applied to a body by contact (see Art. 5 c, p. 332), cause stress, or the
exertion of intermolecular force, within it, or between its particles, tending
to pull them apart (tension) or to press them closer together (compression).
The stress, due to two equal opposing forces, is equal to one of them.

Tension and Compression. Ties, Struts, etc. If the action of
the forces tends to pull farther apart the particles of the body upon which
they act, the stress is called a tension or pull, or a tensile stress. If it
tends to press them claser together, the stress is called a pressure, com-
tvession or push, or a compressive stress. A long slender piece sustaining
tension is called a tie. One sustaining compression is called a strut or
|X)8t. One capable of sustaining either tension or compression is called a
tie-strut or strut-tie.

360 STATICS.

MOMENTS.

13. Moments. If, from any point, o, or </, Fig. 4, a line, o c or o' «, be
drawn normally to the line of action, n m, of a force. Pi, whether the point, o
or o\ be within or outside of the body upon which the force, Pi, is acting, said
line, ocot </ «, is called the arm or leverage of the force about such point;
and if the amount of the force, in lbs., eto., be multiplied by the length of the
arm, in ft., etc., then the product, in ft.-lbs., etc., is called the moment of
the force about that point.* The moment represents the total tendency of
the force to produce rotation about the given point. A force has evidently
no moment about any point in its line of action.

14. Sense of Moments. Since the moment of Pi about o. Fig. 4,
tends to cause rotation (about that point) in the direction of the motion of
the hands of a clock, as we look at the clock and at the figure, or from left to
right, as indicated by the arrow on the circle around o, it is called a clock-
wise or right-hand moment ; but the moment of the same force about </
tends to produce rotation from right to left. Hence it is called a counter-
clockwise or left-hand moment, as is also that of P« about o. Right-
hand or clockwise moments are conventionally considered as positive,
or +t and left-hand or counter-clockwise moments as negativet or — ;

15. The pl£ine of a moment is that plane in which lie both the line
of action and the arm of the force.

16. The resultant or combined tendencv of two or more moments in
the same plane is equal to the algebraic sum ox the several moments. Thus,
Fig. 4, if the forces, Pi, P2, and Pa, are respectively 6, 5, and 3 lbs., and if
the arms, oc, oy, and o 0, of their moments about o are respectively 7* 6, and
8 ft., we have

Pi . 0 c — Pi .0 y 4- Ps . o «
-6X7—6X6 + 3X3
- 42 — 30 + 9 =21 ft.-lbs.

^Zy'm.

I K — n — •

©i^^o

5*— IF— ^
k- — f ^

FIgr. S. Figr. 6.

17. If the algebraic sum of the moments is zero, they are in equilibrium
and tend to cause no rotation of the body about the given point.

Thus, in Fig. 6, where W is the weight, and G the center of ^pavity of the

body, and R the upward reaction of the left support, a, taking moments

about the right support, b, we have R / — W a; — zero ; or R i — W «. Hence,

W X
having W, x and Z, to find R, we have R — - .- .

Similarly, in Fig. 6, where W — weight of beam alone, and g^ the center of
gravity of W, is at the center of the span /, so that the leverage b g of the

weight of the beam about h, is -■ - -, we take moments about &, thus:

R Z i- O o — W- - — Mm — N n — zero; or

Mm + Nn + W — — Oo

R- 2 .

I

'*'Note that a very small force may have a great moment about a point,
while a much greater force, passing nearer to the same point, may have a
smaller moment about it ; or, passing through the jwint, no moment at all.

MOMENTS.

861

In Fig. 7, where W is the weight of the beam itself, and w its leverage, tak-
ing moments about b, we have

+ RZ + O0 — Nn — Ww-|-Mm = 0;

Wi£> + Nn — Mm — Oo

Hence,

Reaction at a

R

I

In any case, if W be the combined weight and G the common center of
gravity, of the beam and its several loads, and x the horizontal distance of
that center from the right support, h\ and if I be the span, R the reaction of
the left support, a, and R' that of the right support, 6, we have

R -

Wx

I W

Ifx-|-, Ria-^ -R'.

I

and R' - W — R.

Flff. 7.

Note that the moments^ of two or more forces, about a given point,
may be in equilibrium, while the forces themselves are not in equilibrium.
See 1 84, below.

18. Center of Moments* So far as concerns equilibrium of moments,
it is immaterial what point is selected as a center of moments ; but it is gen-
erally convenient to take the .center of moments in the line of action of
one (or more, if there be concurrent forces, see ^ 19) of the unknown forces,
for we thus eliminate that force or those forces from the equation.

CLASSIFICATION OF FORCES.

19. Classification of Forces,
and Parallel Forces.

Concurrent, Colin ear, Coplanar.

Forces are called concurrent when their lines of

Figr* s.

Figr. 9.

action meet at one point, as a, b, c, d, e and /, or / and g. Fig. 8 ; non-concur-
rent when they do not so meet, as c and g; colinear when their lines of action
coincide, as a and b. or c and d; non-colinear when they do not coincide, as
b and /; coplanar when their lines of action lie in one plane,* as a, b, c, d and
c, or b, f and (7, etc. ; non-coplanar, as c and g, or 6, / and d, when they do not
he in one plane; parallel wnen their lines of action are parallel, as 0 and g\
non-parallel when those lines are not parallel, as b and /.

*Acting wpon a plane, as in Fig. 9, must not be confounded with acting in
that plane, as in Figs. 70, etc.

862

STATICS.

Any two parallel forces must be coplanar. Three or more parallel forces
may or may not be coplanar. Any two concurrent forces must be coplanar.
Three or more concurrent forces mav or may not be coplanar. Any two
ooplanar forces must be either parallel or concurrent.

COMPOSITION AND BESOLUTION OF FORCES.

SO. Kesultant. A single force, which can produce, upon a body con«
sidered as a whole, the same effect as two or more given forces combined, is
called the resultant of those forces. Thus, in Fig. 10 (b), a downward pres-
sure, G, ■= to + W, is the resultant of the downward pressures w and W;
and, in Fig. 11 (6), a downward pressure, =■ W — tr, is the resultant of the
downward pressure W and the upward pull w of the leit-hand string.*

31. Component.- Any two or more forces which, together, produce,
upon a body considered as a whole, the same effect as one given force, are
called the components of that force, which thus' becomes their resultant.
Thus, in Fig. 10 (6), w and W are the components of the total force, G, =
«; + W. In Fig. 1 1 (6), + W ( = 5) and m) ( - — 3) are the components of G.*

22. If we take into account the resultant of any given forces, those forces
(components) themselves must of course be left out of account, as regards
their action upon the body as a whole; although we may still have to con-
eider their effect upon its particles. Vice versa, if the forces (components)
are considered, their resultant must be neglected.

Fflff. 10.

6

(C)

3

S^

»

FI9. 11.

23. Anti-resultant. The anti-resultant of one or more forces is a sinsle
force which, acting upon any body or system of bodies considered as a wh(ue,
produces an effect eoual, but opposite, to that of their resultant. In other
words, the anti-resultant is the force reouired to hold the given force or
forces in equilibrium. Thus, in Fig. 10 (o), the upward reaction, G, of the
sround, is the anti-resultant of the two downward forces, w and W ; and the
downward resultant, W 4- to, of W and to, is the anti-resultant of G. In
Fig. 11 (6), G (upward) is the anti-resultant of W (downward) and to (acting
upward through the left-hand string). Similarly, this upward pull of tff is
the anti-resultant of W and G.

24. In any system of balanced forces (forces in equilibrium), any one of
the forces is the anti-resultant of all the rest ; and any two or more of them
have, for their resultaht, the anti-resultant of all the rest. In such a system,
the resultant (and the anti-resultant) of all the (balanced) forces is zero.

25. Anti-component. The anti-components of a given force, or of a
given system of forces, are any two or more forces whose resultant is the anti-
resultant of the given force or of the given system of forces.

26. Composition and Resolution of Forces. The operation of
finding the resultant of any given system of forces is called the composition of
forces; while that of finding any desired components of a given force is called
the resolution of the force.

♦ For convenience, we here reverse the convention of H 10.

COLINEAR FORCES.

363

Colinear Forces.

27* Let the vertical line, w. Fig. 10 (6), represent, by any oonyenient
scale, the weight of the upper stone in Fi^. 10 (a), and W that of the lower
stone. Then, w + W, ■". G, ~ the combined length of the two lines, gives,
by the same scale, the combined weight of the two stones, and a verticu line
G, coincident with them, equal to tneir sum, and pointing upward, would
represent their anti-resultuit, or the reaotioii of the ground.

(a)

0

\

\

V

/

/

a

\

\

A

/

/

6 .

W^

w*

lb)

(C)

z=io<

JUve to to to to to to
Tauat» t9 t9 t» t9 t» ^

r^)

^ J

::i

B'-'Sei

1>«

>t»

94

\ J

Fi«. 13.

!88. Similarly, if, at each panel point of the lower chord in the bridgo
truss in Fig. 12 (a), we have 2 tons dead load (weight of bridge and floor,
etc.*) axul 10 tons live load (train, vehicles, cattle, passengers, etc.), the com*
bined length of the two lines in Fig. 12 (b), L - 10, and D - 2, gives the tota*.
panel load of 12 tons.

29. In Fig. 11 the prenure, 5 lbs., of W upon the ground, is diminished by
the 3 lbs. upward pull of the cord, transmitted from the smaller weight i9,
leaving 2 lbs. upwara pressure to be exerted bv the ground in order to main-
tun equilibrium. The upward reaction, R, of the pulley is — w + W — G
-■8 + 6 — 2 -* 6. This is represented graphically in Fig. 11 (c).

30. In the truss shown in Fig. 12 (a), the total dead and live load is — 6
X 12—72 tons, and half this total load, or 36 tons, rests upon each abut-
ment. Hence, to preserve equilibrium, each abutment must exert an up-
ward reaction of 36 tons; but, in order to ascertain how much of these 36
tons is iranamiUed through the end-pott, a e, we must deduct from it the 12
tons which we assume to be originally concentrated, as dead and live load*
at the panel jpoint a; for this portion is evidently not transmitted through
a e. Accordingly, in Fig. 12 (c), we draw R upward, and equal by scale to
36 tons: and, from its upper end, draw p downward and — 12 tons. The
remainder of R, — R — p -• 36 ^ 12 — 24 tons, is then the pressure trans-
mitted through a e.

31* Golinear forces are called similar when they are of like sense, and
opposite when of opposite sense. The tame distinction applies to result*
ants.

b

h

a

■^— f

o c

Figr. 13.

d

•

33* For equilibrium, under the action of colinear forces, itia,
ci oo^irse, necessary that toe sum of the forces acting in one sense be equal to
the sum of those acting in the oppomte sense, or, in other words, that the
algebraic sum of ail the forces be zero. Thus, in Fig. 13, if the forces are in
equilibrium, the sum, b a ■{• a o, ot the two right-handed forces must be
equal to the sum, ed + dc + co, of the three left-handed forces. Or, con-
sidering the right-handed forces, b a and a o, as positive, and the left-handed
forces, e dj d c and c o, as negative, as in ^ 10, we have, as the condition of
equilibrium of colinear forces :

ba •{- ao — oc—'cd — de — O.

*The dead load is, of course, never actually concentrated upon one chord,
as here indicated ; but It is often assumed, for convenience, that it is so
concentrated.

364 STATICS.

In other words, the algebraic sum of all the forces must be zero; or, more
briefly,

2 forces — 0,

where the Greek letter S (sigma), or sign of summation, is to be read "The
sum of — ."

33* Two equal and opposite forces, acting upon a body, are com-
monly said to keep it at rest ; but, strictly speaking, they merely prevent each
other from moving the body, and thus permit it to remain at rest, so far as
they are concerned ; for they cannot keep it at rest against the action of any
third force, however slight and in whatever direction it may act; and the
body itself has no tendency to move.

34. Unequal Opposite Forces. If two opposite forces, acting upon
a body, are unequal, the smaller one, and an equal portion of the greater
one, act against each other, producing no effect Uf^n the body as a whole;
while the remainder, the resultant, moves the body in its own direction.

Concurrent Coplanar Forces. The Force Parallelogram.

35. Composition. Let the two lines, ao,bo, in any of the diagrams of
Fig. 14, represent, in magnitude, direction and sense, concurrent forces
whose lines of action meet at the point o. Then, in the parallelogram, acbo,
formed upon the lines a o^ b o, the resultant of those two forces is repre-
sented, in magnitude and in direction, by that diagonal, R, which passes
through the point, o, ci concurrence. The parallelogram, a c & o, is called a
force parallelogram.

a' (a)

o

"^^^^V*

ligr. 14.

36. Resolution. Conversely, to find the components of a given force,
o c, Fig. 14, when it is resolved in any two ^ven directions, o a, o 6, draw the
lines, o a\ o b\ in those directions and of mdefinite length, and upon these
lines, with the diagonal R » o c, construct the force parallelogram a ch o»
The sides, o a, ob, of the parallelogram then represent the required compo-
nents in amotmt and in direction. ^

37. Caution. The two forces, a o and b o. Fig. 14, may act either toward
or from the point o; or, in other words^ they may act either as pulls or as
pushes ; but the lines representing them m the parallelogram, and meeting at
the point, o, must be drawn, either both as pushes or both as pulls; and the
resultant, R, as represented by the diagonal of the pandlelogram, will be a
pull or a push, according as the two forces are represented as pulls or as
pushes.

38. Thus, in Fig. 15 (^a), the inclined end-post of the truss pushes obliouely
downward toward o, with a force represented by a' o, while the lower chord
pulls away from o, toward the ri^ht, with a force represented by o V, If,
now, we were to construct, in Fig. 15 (a), the parallelogram o a' cf V^ we
should obtain the diagonal o cf or c' o, which does not represent the true re-
sultant. In fact, as one of the two forces acts toward, and the other from,
the point, o, we could not tell (even if R' were the direction of the resultant)
in which sense its arrow should point.

We must first either suppose the push, a' o, in the end-post, toward o,^ to be
carried on beyond o, so as to act as a pull, o a. Fig. 16 (o) (of course, in the
same direction and sense as before), thus treating both forces as pulls; or

FOBCE PABALIiELOOKAH.

366

rise we must similarly suppose the pull, o V, in the chord, to be transformed
into the push, 6 o, of Fig. 15 (c), thus treating both forces as pushes. In
either case we obtain the true resultant, R ( » a' 5', Fig. 15a), which, in this
ease, represents the vertical downward pressure of the end of the truss upon
the abutment.

FtfT- IS-

Caution. The tensile force, exerted at the end of a flexible tie, neces-
sarily acts in the line of the tie; but, in general, the pressure, exerted at
the end of a strut, acts in the line of the axis of the strut only when all
the forces producing it are applied at the other end of the strut. Thus,
in Fig. 15 id), the components, R and H, of the weight, W, do not coin-
cide with the axis of the beam which supports the Toad; but in Fig. 15
(e), where the weight acts at the intersection of the two struts, its com-
ponents, R and H, do coincide with the axes of the struts. See idso Figs.
143 and 145 (b).

39. Demonstration. The rational demonstration of the principle of
the force parallelogram is given in treatises on Mechanics. (See Bioliog-
raphy.) It may be established experimentally as indicated in Fig. 16,
where c o represents by scale the pull shown by the spring balance C, while
o a and o h represent those shown by A and B respectively.

40.- Equations for Components and Resultant. Given the
amounts of the forces, a and c, or of the resultant, R, and the angles formed
between them. Fig. 17 (a), we have'*':

♦ See dotted lines, Fig. 17 (a), noting that c* ^ c; c. sin (x + i/) -» R. sin
X, and a. sin (x + y) >« K. sin y.

366

STATICS.

rt = c

sin (x 4- y)

R^

Bin X
sin X

— a

sin (x + y)
sin y

„ = R »>» »

sin (x + y)*

sin (X -i- 1/)* , . ^,

If the angle between the two forces is 90**, Fig. 17 (b), these formulas be<
me:

come:

cos y cos X
c — R cos y; a =» R cos x.

FiiT- 17.

41. Position and Sense of Resultant. Figs. 18. If the lines
representing the components be drawn in accordance with Iff 37 and 38,
and if a straight line, m n or m' n', be drawn through the point, o. of concur-
rence, in such a way that both forces are on one side of that line, then the line
representing the resultant will be found upon the same side of that line with
the components, and between them ; and it will act toward the line, m n or
tn* n'r ii the components act toward it, and vice versa. The resultant is
necessarily in the same plane with its two components.

tm

/^^MiosS^

nC^T^^^'

^r

-s/'

Fiir. 18.

Fis. 19.

42* If one of the components is colinear with the force, it is the force itself,
and the other component is zero. In other words, a force cannot be resolved
into two non-colinear components, one of which is^n the line of action of the
force. Thus the rope, o e. Fig. 19, may receive assistance from tu}o ad-
ditional ropes, pulling in the directions a c, and c b; for the resultant of their
pulls may coincide with o c; but, so long as o c remains vertical, no aingU
force, as c a or c b, can relieve it, imless acting in its own direction c o.

43. In Fig. 20, the load, P, placed at C, ia suspended entirely by the verti-
cal member B C, and exerts directly no pull along the horizontal member,
C £. Neither does a puU in the latter exert any eneot upon the force acting
in B C, so long as B C remains vertical. But the tension in B C, acting
at B, does exert a thrust o a along B D, although that member is at right
angles to B C; for B G meets there also the inclined member A B; and
the tension o d \3 thus resolved into o a and o 6, along B D and B A
respectively. The horizontal thrust, o a, in B D, is really the anti-resultant
of the horizontal comp>onent, db, of the oblique thrust in the end-poet B A,
at its head, B, which thrust is — the pull in A £, due to P.

FORCE TRIAKGLE.

367

44. In Fig. 21, the tension, o e, in the inclined tie, D G, is resolved, at D,
into o a and o b, acting at right angles to each other along D F and D £ re-
spectively.

45. A resultant may be either greater or less than either one of its two
oblique components, but it is always less than their sum. If the components
are equal, and if the angle between them » 120^, the resultant is eaueii to one
of them. Therefore the same weight which would break a single vertical
rope or post, would break two such ropes or posts, each inclined 60° to the
vertical.

Fly. 91.

The Force Triangle.

46. The Force Triangle. Inasmuch as the two triangles, into which a
paralldogram is divided by its diagonal, are similar and equal. It is suffi-
cient to cu^w either one of these triangles, aoc or h oc. Figs. 14, 16, 18, in-
stead of the entire parallelogram.

47. If three concurrent coplanar forces are in equilibrium, the lines repH
resenting them form a triangle; and the arrows, indicating their senses,
foUow each other around the triangle. Thus, in Fi^. 22 (a), we have, acting
at o and balancing each other there, three forces: vu., (1) the vertical down-
ward force o c of the weight, acting as a pull through the rope o c, (2) the
horizontal thrust a o through the oeam a o, and (3) the upward inclined
thrust 6 0 of the strut o b, all acting in the senses (o c,ao,b o) in which the
letters are taken, and as indicated by the arrows.

48. Each of the forces in Fig. 22 (&) and (c) is th6 anti-resultant of the
other two in the same triangle ; and, if its sense be reversed, it becomes their
resultant. Thus, o c, Fisj. 22 (b), is the anti-resultant, and c o the resultant,
ofea and a o; and o c. Fig. 22 (c), is the anti-resultant, and c o the resultant
of e & and bo,cb being parallel to a o. Fig (b), and representing the thruflt
exerted by the horizontal beam against the joint o, Fig. (a).*

ib) (c) id) (e)

^^ •

c e^t,

Flff. 33.

*Fig. 22 (tO and (e), representing the same two forces, a o, b o, of Fig.
22 (a), show the erroneous resultant (a b) obtained if the lines are drawn
with their arrows pointing both toward or both from the meeting-point of the
lines. See ^1f 37, 38. A comparison of any force parallelogram, as that
in Fig. 18, with either of the two force triangles composing it, will show
that this, while apparently contradicting Ht 37 and 38, is merely another
statement of the same fact. The apparent contradiction is due to the
fact that, in the force triangle, the lines representing the forces do not
meet at the point, o, of concurrence of the forces.'

368

STATICS.

40. Converselsr, if the three sides of a trian^e be taken as representing,
in direction and in amount, three concurrent forces whose senses are such
that arrows, representing them and affixed to their respective sides in the
triangle, follow each other around it, then those forces are in equilibrium.

50. The three forcest Fig. 23, are proportional, respectively, to the
sines of their opposite angles. Thus:

Force a : force b : force e
— Sin A : sin B : sin C. Fly. 2S.

51* Example. In Fig. 24, the half arch and its spandrel, acting as a
nngle rigid bodv, are assumed to be held in equilibrium by their combined
weight, W, the horizontal pressure h at the crown, and the reaction R of the
skewback, which is assumed to act through the center of the skewback. In
the force triangle c « t, e «, acting through the center of gravity of the half
arch and spandrel, represents the known weight W, and 8 t ia drawn hori-
sontal, or parallel to h . From c, where h, produced, meets the line of ac-
tion of W, draw c t through the center of the skewback. Then • t and e I
give us the amounts of h and R respectively.

Fig. 24.

Figr- 9Xi,

52. Example. Let Fig. 25 represent a roof truss, resting upon its abut-
ments and carrying three loads, as shown by the arrows. Draw a R ver-
tically, to represent the proportion of the loads carried by the left abut-
ment, a, or, which is the same thing, the vertical upward reaction of that
abutment. Then, drawing R c, parallel to the chord member, a <2, to inter-
sect a 6 in c, we have, for the stresses in a e and a d, due to the three loads:

Stress in a « s a e
" od = Re

It

%4 ^'^bA

63. While any two or more given forces, as o 6 and h c. Fig. 26 (a) (arrows
reversed), or o b' and b' c,oroa and a c, or o a' and o' c, can nave but one re-
sultant 0 c; a sinffle force, as o c. may be resolved into two or more concur-
rent components in any desired directions. In other words, there is an
infinite number of possible systems of concurrent forces which have o c for
their resultant.

SECTANOULAB COMPONENTS.

869

Bectangular Gomponenti.

54. ResoluteSt or Rectangular Components. A very common case
of resolution of forces is that where a force, as the pressure, c n, of the post,
fig. 27, is to be resolved into components at rieht angles to each other, as are
the vertical and horizontal components c t and tn in Fig. 27 (a). Two such
components, taken together, are called the resolutes or rectangular compo-
nents of ibjb force. The joint, o d, in Fig. 27 (a), is properly placed at right
angles to e n; but the joint c ib. Fig. 27 (5), provides also against accidental
changes in the direction of c n. In Fig. 27 (6), the surfaces, c i and i b, are
preferably {proportioned as the components, c i and t ih Fig. 27 (a), respec-
tively, by simUarity of triangles, ctb, ctn^

Tig. 27.

Fl«r. 28.

55* Example. In bridge and roof trusses it is often required to find the
vertical and horizontal resolutes of the stress in an inclined member, or to
find the stress brought ui>on an inclined member by a given vertical or hori-
zontal stress applied at one of its ends, in conjunction with another stress
(whose amount may or may not be given) at right angles to it.

Thus, in Fig. 28, the tension C p in the diagonal C d is resolved into a com-
pression e p along the upper chord member CD* and a compression C e in the
• post Cc.*^ Addmg to C c the load at c, and representing their sum by / c, we
nave tension f g in chord member e d, and tension c g in the diagonal B c.
Making B A = c g,-we have i A, compression in B C, and B j, compression in
the end-post or batter poet B A. But the load at b also sends to B, through
the hip vertical B 6, a load (tension) equal to itself. Representing this by
B ;fc, we have ( A; as its component along the chord member B O, and B I as its
oom]M)nent along the end-post B A. Now, making A *» = the sum of B;
and B /, we find the vertical resolute A » = so much of the vertical reaction
of the abutment as is due to the three loads only, and the horizontal resolute
mn '^ the corresponding stress in the chord member, A c.

'\

>a

Flff. 80.

56. Example. Inclined Plane. Again, in Fig. 29, let it be required
to find the two resolutes of P (the weight of the ball) respectively parallel and
perpendicular to the inclined plane. The former is the tendency of the ball
to move down the plane, and is called the tangential component. The

^'Ilie stress^ thus found is not necessarily the total stress in the member.
The compression in C c (neglecting its own weight and that of the top chord)
fe due entirely to the tension C p in C <i, acting at its top, and hence C e rep-
tmenta the total compression in C c; but e p ia only a portion of the com-
pression sustained by C D ; for B C also contributes its share toward this.

24

btl«rut

ofthebBUaCUDBt

h« plui«, aod ia (wlled the noimal

compon

enl.

Herew

to draw

the triftogl

of for.

OEOC-Pto

JdiiMtkma.

S£

Iho weight

of the b

•11. Uld »

undo

s^/F-

ely the do

mid wid the taaeeatia

87. If

the ineli

ed plane g

m, Fig

29, to be fri«t

o»le«, and if

the body

018 to be prevented t

rom sliding

down

'o're^J^^'g."-

umotafone

■.ppliedm&direotiDTi

parsJIel

to the plwi
the pluu

e. that

thua,

n Fig. 30, B

ihe stoi

be friotionlees,

mhave a e

™ agamsl

SS. Table, of a

tordiffenot

T>Ft. H«.

i: i'
1: t

Id Il.t

\>»«

Dl-K

• Or 0 * c. It both triangles are drawn, we have the foro
trhs line a « (or c a) is called the prolecllon of o c ui

BTBE8S CX>HFONENT8.

371

59. Equations. In Fig. 29.

o a »- P . cos e o a
a c ■" P . sin e o a

and, since the angle eoa between the vertical o e and the normal component
o a is equal to the angle A of inclination between the plane g m and the hori-
sontal a n, we have :

Normal component, o a » P . cos A.
Tangential component, a c » P . sin A.

60. When a force is resolved into rectangular components, as in Figs. 29
and 30, each of these components represents the total effort or tendency which
that force alone ean exert in that direction.

FI9. 81.

Thus, in Fig. 31, the utmost force which the weight o e alone can exert
perpendicularly againat tke plane is that represented by the component o a.
iVue, if, in order to prevent the bo<ly from sliding down the plane, we apply
a force in some other direction, such as the horisontal one, h o, instead of the
tangential one h o, and find the components of o c in the directions h o and o a,
weuiall find the normal component o d greater than before; but the increase
a d is due entirely to the normal component, h fr, of the horisontal force h o.
Thus, the only effect upon the body o, and upon the plane, of substituting
h o for b o, is to add the normal component, h 2>, of the former, to that (o a)
doe.

Stress Components.

61. Stress Components. In Fig. 32, let a o and & o be any two forces,
and c o their resultant. From a and 6 draw a a' and h 6' at right angles to
the diagonal o c of the force parallelogram a o b c^ and construct the sub->
parallelograms (rectangles), oa' a a" and oVh If'. Each of the original com-
ponents, o a, o- h, is thus resolved into two sub-components, perpendicular to
each other, one of which is perpendicular also to the resultant, o c, while the
other coincides with o c in position and in sense. Now, perpendiculars, let
fall from the opposite angles of a paralldogram upon its diagonal, are equaL

.// (a)

.//

,, (6)

^

/

4

< /

f<

"-,

/

6 -X

Flff. 32.

Bence the two colinear forces, o a'\ and o 6", acting upon the body at o, are
equal and opposite (although the lines, a' a and h' 6, representing them, are
not opposite). Hence also they are in equilibrium, and their only effect
upon the body is a stress of compression in Fig. 32 (a), and of tension
in Fig. 32 (6). They may therefore be called the stress components. The
other two sub-components (o a' of o a, and o 5' of o h) combine to form the
resultant o e, which is equal to their sum, and which tends to move the body
0 in its own dir&ction.

372

STATICS.

62. The two great forces, o a, ob, in Fig. 33 (6) have the same reeultant,
oc, = o c', as the two small forces, o a' o b\ in Fig. 33 (a), although their
stress components, a" a, = V b, are much greater.

63. It often happens that one of the components is itself normal to the
resultant. Thus, in Fig. 22, where o c is vertical, its component, o a, is hori-
zontal, and the perpendicular, let fall from a upon o c, represents its hori-
zontal anti-component, a o. Here the horizontal and the inclined beam
sustain equal horizontal pressures; but the vertical pressure, o c, "^ the
weight, W, is borne entirely by the inclined beam.

Flip. 33.

Flip. 34.

64. When, as in Fig. 34, the resultant, o c, forms, with one of the original
components, o a and o b, an angle, aoc, greater than 90^, the perpendicularB,
a a', b 6', from a and &, must be let fall upon the line of the resiiltant produced.
Here, however, as before, the two equal and opposite sub-components, o a"
and o b"j are in equilibrium at o, while the other two sub-components, o b* and
o a', go to make up the resultant o c; which, however (since o 6' and o a* here
act in oppottte senses) is equal to their difference, and not to their sum, as in
Fig. 32.

Fig. 34 shows that a dowrvward force, o e, may be so resolved that one of its
components is an upward force, o a, greater than the original downward force,
and that the pressure, o 6, has a component, o b* or V &, parallel to o c, and
greater than o c itself; for b" b — o 6' '^ o c -\- cV.

Applied and Imparted Forces.

65. Applied and Imparted Forces* In Fig. 29, the ball is free to
roll down the inclined plane. Hence, although the entire weight P of the
ball is applied to the body g mn, only the normal component o a is imparted
to it or exerts any pressure upon it, and this pressure is in tlie direction o a.

But in Fig. 30, the body g mn ceeeives and resists not only the normal
component o a, but also (by means of the stop «) the tanaential component
o b; and the entire force P, or o c, is thus imparted to the body g mn, pres»-
ing it in the direction o c.

Comiposition and Kesolution of Concurrent Forces by Means

of Co-ordinates.

66. In Fig. 35 (a) let the three coplanar forces E, F and G act through
the point x. Draw two lines, H H, and V V, Fig. 35 (b), crossing each
other at right angles, as at o.* These lines are called rectangular co-ordin-
ates. From o, draw lines E o, F o, G o, parallel to E a:, F x, (jrx, Fig. 35 (o),
and equal respectively to the forces E, F, and G by any convenient scale. Re-
solve each of these forces, Fig. 35 (6), into two components, parallel to H H
and V V respectively. Thus, E o is resolved into t o and n o, F o into u o
and e o, G o into i o and m o. Then, summing up the resolutes, we have:

Sum of horizontal resolutes = u o — io — to — — so, and

Sum of vertical resolutes ==

no + e o — m,o

— ao,
ao;

*It is only for convenience that the co-ordinates are usually drawn (as in
Fig. 35) at right angles. They may be drawn at any other angle (see Fig.
36) ; but. in any case, the forces must of course be resolved into components

EaraUel to the co-ordinaUa, whatever the directions of those co-ordinatee may
e.

COMPOSITION AND RESOLUTION.

373

and — 9 0 and a o are the resolutes of the resultant, R, of the three forces, E,
F and G.

67. When a system of (concurrent) forces is in equilibrium, the algebraic*
sum of the components of all the forces, along either of the two co-ordinates,
is zero. Thus, in Fig. 35 (6) or 36, if the sense of R be such that it shall act as
the anti-resultant of the other three forces E, F and G, its component, o « or
o a, along either co-ordinate, will be found to balance those of the other
forces along the same co-ordinate.

Flff. 35.

Henoe we have the very important proposition that : When a system of
ooncurrent coplanar forces is in equilibrium, the algebraic sums of their com-
ponents, in any two directions, are each equal to zero.

Fig:. S6.

68. Conversely, in a system of concurrent forces, if the algebraic sums of
the components in any two directions are each jequal to zero, the forces are
in equilibrium.

If the sum of the components in one of anv two directions is not equal to
zero, the forces cannot be in equilibrium. Thus, in Fig. 35 (6) or 36 (b), the
sum of the components, along either one (as VV) of the two co-ordinates,
may be zero; and yet, if the sum of those along the other co-ordinate is
not zero, their resultant, or algebraic sum, will move the body, on which
they act, in the direction of that resultant.

♦The components being taken as + or — , according to the sense of each.

374

STATICS.

69. With Tertical and horizontal co-ordinates, the condition of
equilibrium* becomes:

. The sum of the horizontal resolutes must be equal to zero ;

The sum of the vertical resolutes must be equal to sero;

or, more briefly:

2 horizontal resolutes ■- 0

2 vertical resolutes ■» 0

Conversely, if these conditions are fulfilled, the forces are in equilibrium.

Tig. 37.

Tig. 3S.

Flip. 39.

70. Resultant of More than Two Coplanar Forces. Where it
is required to find the resultant of more than two concurrent and coplanar
forces, as in Fig. 37, we may first find the resultant Ri of any two of them,
as of P] and Ps; then the resultant, R^, of Ri and a third force, as Pa; and so
on, until we finally obtain the resultant R of all the forces. This resultant is
evidently concurrent and coplanar with the given forces.

71. It is quite immaterial in what order the forces are taken.

Thus, we may, as in Fig. 38, first combine Pi and Ps; then their resultant Ri
with Ps, obtaining R2; and, finally, R^ with P4, obtaining R;or, as in Fig. 39,
we may first combine any two of the forces, as Pi and Ps, obtaining their
resultant Ri ; then proceed to any other two forces, as Ps and P4, and obtain
their resultant R^; and finally combine the two resultants, Ri and R^, ob-
taining the resultant R.

The Force Polygon.

73. The Force Polygon. Comparing Figs. 37 and 38 with Figs. 40
and 41, respectively, we see that we may arrive at the same resultant R by
simply drawing, as in Fig. 41, lines representing the several forces in any
order, but following each other according to their senses. It will be noticed
that this is merely an abbreviation of the process of drawing the several force
parallelograms.

73. Resultant and Anti-resultant. The line, — R, required to com-
plete the polygon, represents the an<i-resultant of the other forces if its sense
IS such that it follows them around the polygon, as in Fig. 40. If its sense is
opposed to theirs, as in Fig. 41, it is their reavUant, R.

74. In other words, if any number of concurrent forces, as Pj, Pj, Pj, P*
and R, Figs. 37 and 38, f are in equilibrium, the lines representing them, if
drawn in any order, but so that tneir senses follow each other, will form a
closed F>olygon, as in Fig. 40 (or in Fig. 41 if the sense of R be reversed).

75. Conversely, if the lines representing any system of concurrent
coplanar forces, when drawn with their senses following each other, form a
closed polygon, as in Fig. 40, those forces are in equilibrium.

*With non-concurrent forces, another condition must be satisfied. See ^ 83.

tR is here regarded as tending upward, so as to form the anft-resultant of
the other forces.

FORCE POLYGON.

375

It will be noticed that the force triangle, and the straight line representing
a system of colinear forces, Figs. 10 and 11. Hlf 20, etc., or a system of
parallel forces, Figs. 55, etc., tf 111, etc., are merely special cases of the
force polygon.

76. In a force polygon. Fig. 42, any one of the forces is the anti-resultant
of all the rest. Any two or more of the forces balance all the rest ; or, their
resultant is the anti-resultant of all the rest.

If a line a c or 6 d, Fig. 42, be drawn, connecting any two comers of a force

rig. 40.

Fiff. 49.

polygon, that line represents the resultant, or the anti-resultant (according i
its arrow is drawn) of all the forces on either side of it. Thus :

a c is the resultant of Pi Ps and the anti-resultant of P3 P4 Pg
<5 a " " " Ps P4 Ps * " '* Pi Fa

6 rf " " " Ps Ps " " " P4 P6 Pi

d b " •* " P4 Pft Pi " " " Pi P3

77* Knowing the directions of all the forces of a system, as Pi P5,

Fig. 42, and the am&unta of all but two of them, as Ps and P3, we may find the
amounts of those two by first drawing the others, P4, Ps and Pi, as in the
figure. Then two lines b c and c d, drawn in the directions of the other two
and dosing the polygon, will necessarily give their amounts.

Tig. 48.

Tig. 44.

78. If any two points, as o and c. Fig. 43. be taken, then the force or forces
represented by any line or system of lines joining those two points will be
equivalent to o c. Thus :oe''oabc'^ode^onpc''ohkmc =
on mc '^ o fc " o gc, etc., etc.

Similarly, in Fig. 42, the force polygon abe deais equivalent to the force
polygon ab fdea, and to the force triangle, abca, eacn being = zero.

•
Non-concuirent Coplanar Forces.

79. Non-concurrent Coplanar Forces. Fig. 44. The process of
finding the resultant of three or more coplanar but non-concurrent forces is
the same as if they were concurrent. Thus, let Pi, Ps and Ps represent three
sueh forces.* We may first find the resultant Ri of any two of them, as Ps

*Apy two coplanar non-parallel forces, as P; and P2, or P^ and Ps are
necessarily concurrent (see % 19); but there is no single pomt in which
the three forces meet.

376

STATICS.

and P3; and then, by combining Ri with the remaining force Pi, we find the
resultant R of the three forces. Here the line R represents the resultant, not
only in amount and in direction, but also in position. That ls, the line of
action of the resultant coincides with R.

80. The resultant R is the same, in amount and in direction, as if the
forces were concurrent, and its position is the same as it would have been if
their point of concurrence were m the line of R. If there are more than three
forces, we proceed in the same waj'.

81. Conversely, the resultant R, or any other force, may be resolved
into a system of any number of concurrent or nonconcurrent coplanar forces,
in any direction^, at pleasure. Thus, we may first resolve R into Pi and Ri;
then either of these into two other forces, as Ri into P2 and P3, and so on.

83. If a system of non-concurrent coplanar forces is in equilibrium, the
forces will still be in equilibrium if they are so placed as to be concurrent;
provided, of course, that their directions, senses and amounts remain un-
changed ; but it does not follow that a system of forces, whicl> is in equilib-
rium when concurrent, will remain in equilibrium when so placed as to be
non-concurrent.

Thus, the five forces, Pt Pr„ Fig. 45 (a), may be so placed, as in Fig.

45 (6), that the resultant a c, of Pi and Pa, does not coincide with the re-
sultant c a of P3, P4 and Ps. but is panUlel to it. These two resultants -then
form a couple. (See tlf 155, etc.)

Fig. 45.

83. Third Condition of Equilibrium. Hence,
equilibrium for concurrent forces, stated in \ 69,

the oondHiona of

2
2

vertical
horizontal

components ■■ 0
components -= 0

do not suffice for non-concurrent forces, and a third condition must be added,
viz. : —

2 moments « 0;

t. e., the moments of the forces, taken about any point, must be in equilib-
rium.

A system of forces in equilibrium has no resultant ; hence it has no moment
about any point. In other words, the moments of the forces, as well as the
forces themselves, are in equilibrium.

84. The resultant of a system of unbalanced non-concuireiit
forces, acting upon a body, may be either

(1) a single force, acting through the center of gravity of the body; or

(2) a couple; t. e., two equal and parallel forces of opposite sense (see
m 155, etc.) ; or

(3) either (a) a single force, acting through the center of gravity of the
body, and a couple ; or (b) a single forca acting elsewhera th»r throu^k ( he
center of gravity of the body.

^ In Case (3), the two alternative resultants are interchangeable; t. e.. a
single force, acting elsewhere than through the center of gravity of the body,
may always be replaced by an equivalent combination consisting of an eqijuu

CORD POLYGON.

377

parallel force*, acting through the center of gravity of the body, and a couple,
and vice versa. See HI 161', etc.

The resultant gives to the body, in Case (1), motion of translation in a
straight line, without rotation; in Case (2), rotation without translation;
and m Case (3), both translation and rotation. See foot-note (*), t !•

85. The force polygon, ^ 72, Figs. 40, etc., and the method by co-
ordinates. H 66, Fig. 35, therefore, give us only the amount, direction and
sense of the resultant of non-ooncurrent forces, and not its position. To find
the position of the resultant of non-concurrent forces, we may have recourse
to a figure, like Fig. 44. where the forces are represented ib their actual posi-
tions, or to the cord polygon, H1[ 86, etc., Fig. 46.

The Cord Polygon.

86. In the force triangle any two of the three lines may be regarded as
representing, by their directions, the positions of two members (two struts
or two ties, or one strut and one tie) of indefinite length, resisting the third
force ; while their lengths give the amounts of the forces which those mem-
bers must exert in oTaer to maintain equilibrium.

FlfT* 20 (repeatefl).

87. Thus, in Fig. 26 (6), are shown four different systems, of two mem-
bers each, inclined respectively like the forces c h and b o in Fig. 26 (a) and
balancing the third force o c. The stresses in these two members are given
by the lengths of the lines e b and b o in Fig. 26 (a).

Tlie members acting as struts are represented, in Fig. 20 (b), as abutting
against flat surfaces, while those acting as ties are represented as attached
to hooks, against which they pull.

In Fig. 26 (c) and (d) are indicated systems of members, inclined like the
forces c a' and a' o, ca and a o, respectively, of Fig. 26 (a), by which the third
force o c might be supported.

88. In the force polygon abed ea. Fig. 46 (6), representing the four
forces, Pi, Ps, P3, Pj, of Fig. 46 (a), if we select, at pleasiue, any point o
(called the pole) and draw from it a series of straight lines oa,ob, etc. (called
mys), radiatinff to the ends, a, b, c, etc., of the lines Pi, Ps, etc., representing
the forces, we snatl form a series of force triangles, aobthoc, etc.

Thus, in the triangle d b o we have the force Pi, or a b, balanced by the two
forces o a and b o; m the triangle b c o, the force P2, or b c, balanced by the
two forces o b and c o; and so on.

89. The Cord Polygon. If, now, in Fig. 46 (a), we draw the lines a
and b, parallel respectively to the rays o a and o b of Fig. 46 (b) and meeting
in the une representing the force Pi, they will represent the positions of two
tension members of indefinite length, which will balance the force Pi by ex-
erting forces represented, in amount as well as in direction, by the rayS 0 a
and b o, Fi^. 46 (b). Again, taking pol? o'. Fig. 46 (b), instead of o, we have
a' and b*. Fig. 46 (aO, parallel respectively to the rays, o' a and 0' b, and rep-
resenting a pair of struts performing the same duty.

90. Similarly, the lines b and e. Fig. 46 (a), parallel respectively to rays o h
and o c, represent two tension members, which, with stresses equal respec-
tively to o b and c 0, Fig. 46 (b), balance the force Pg.

378

STATICS.

01. We thus obtain, finally, a system of five tension members, ab e de.
Fig. 46 (a), which, if properly fastened at the ends a and e respectively, will^
by exerting forces represented respectively by the rays, o a, ob, oc, etc.. Fig.
46 (6), balance the four given forces Pi, P{, Ps and P4.

92. The figure abode. Fig. 46 (a), is called a cord polygon, funicular
polygon, or equilibrium polygon.

03. Resultant, Anti-resultant. Amount and Direction. In the

force polygon, Fig. 46 (6) or (d), the line e a, joining the end of the last force-
line d e with the beginning of the first one a b, represents the anti-resultant of
the given system of four forces, and a e their resultant. Evidently, there-
fore, the rays^ a o and o e, which represent two components of a e, represent
also, in direction and in amount, two forces which would balance e a, or which
would be equivalent to the given system of (four) forces.

Flffs. 46 (a), (a') and (fr).

04. Position of Resultant. Hence, in the cord |>olygon. Fig. 46 (a)^
the intersection, i, of the cords a and e, parallel respectiv^y to the rays o a
and e o, is a point in the line of action of the resultant R; and. if we imasine
a i and e i to be rigid rods, and apply, at t, a force, — R, equal and parallel to
a e, but of opposite sense, that force will be the anti-resultant of the (four)
given forces, and we shall have a frame-work be di of cords and rods, kept in
equilibrium by the action of the five forces, Pi, Pg, Pg, P4 and — R.

06. By choosing other positions of the pole, as o\ Fig. 46 (fi), or by differ*
ently arranging the given forces, as in Fig. 46 (c), we merely change the
shape of the cord polygon, and (in some cases) reverse the sense of the
stresses in the members. Thus, in Fig. 46 (a), all the stresses are tensions, or
pulls : while in Fig. 46 (c) a, b, d and e are tensions or pulls, and c is a com-
pression or push.

06. In constructing the cord polygon, Fig. 46 (a), (aO. (c), and (e), car*
must be taken to draw the cords m their proper places ; and for this it is neo-
essary to remember, simply, that the two rays pertaining to any particular
force line in the force polygon. Fig. 46 (6), represent those members which,
in the cord polygon. Fig. 46 (a), take the components of that force.

CORD POLYGON.

379

Thus, o a and h o» Fig. 46 (6), pertain to the force Pi ; o b and e o to the
force Pj. Hence, in Fig. 46 (a") or {c) we draw a and h (parallel respectively
to o a and 6 o) meeting in the line of action of Pi : h and c (parallel respect-
ively to o 6 and c o) meeting in the line of action of Ps, etc., etc.

97. Each ray in the force polygon. Fig. 46 (6), including the outside ones,
is thus seen to pertain to two force?, and each force has two rays. The two
oords,^ parallel respectively to the two rays of any force, must be drawn to
meet in the line of^aetion of that force; and each cord must join the lines of
action of the two forces to which its parallel ray pertains. The lines, a, &, c.
etc., in the cord polygon. Fig. 46 (a) and (c), give merely the incLinatwM oi
members which, as there arran^d, would sustain the given forces. ' The
lengths of these lines have nothing to do with the amounts of the stretaes.
These are given by the lengths of the corresponding raya in the force polygon,
Fig. 46 (6).

Flffs. 46 (o), {d) and («).

08« If the anti-resultant force, — R, is not applied, the cords a and e may
be supposed fastened to firm supports, against which they exert stresses rep-
resented, in amount and in direction, by the rays a o and o e respectively.
But the resistances of those two supports are plainly equal and opposite to
those stresses, or equal to o a and e o respectively. Hence, their resultant is
the anti-resultant, — R, of the foiu> origmal forces.

99* If, Fig. 46 («), the two end members a and e were attached merely to
two ties, V and V, parallel to the anti-resultant, — R, they would evidently
draw the ends of those ties inward toward each other. To prevent this, let
the strut k be inserted, making it of such length that the ties V and V may
remain parallel to — R, and draw o k, Fig. 46 (6), parallel to k. Then a k
and k e give the stresses in V and V respectively.

ipO. If the anti-resultant, — R, found by means of the force pply^n, be
applied in a line passing through the intersection of the outer (initial and
final) members in the cord polvgon, all the forces, includinff of course the
aati-resuliant, will be in equilibrium. In other words, coplanar forces are
in equilibrium if they may be so drawn as to form a dosed force polygon, and
if a closed cord polygon may be drawn between them. But if the anti-re-
soltant be applied elsewhere, we shall have a couple, composed of the anti-
rwnltant, — K, and the resultant R of the forces.

380

STATICS.

Concurrent Xon-eoplanar Forces. .

101. Any two of the concurrent forces, as o o and o c. Fig. 47 (a) or (6), are
necessarily coplanar. Find their resultant, o r, which must be coplanar with
them and witn a third force o h. Then the resultant, R, of o r and o 6 is the
resultant of the three forces. If there are other forces, proceed in the same
way.

102. No three non-coplanar forces, whether concurrent or not, can be in
equilibrium.

103. Force Parallelopiped. The resultant of any three concurrent
non-4ioplanar forces, o a, o\ o c. Figs. 47, will be represented by the diagonal
a R, of a parallelopiped, of which three converging edges represent the three
forces.

104. Methods by Models, (a) For three forces. , Construct a
box, Kg. 47 (a) or (6), with three conver^nt edges representing the three-
forces in position and amount. Then a stryig o R, joining the proper corners,
will represent the resultant.

Fig. 47.

Or, let ao,ho, c o. Fig. 48 (o), be three forces, meeting at o, "DnM on
pasteboard the three forces a o, b o, e o, as in Fig. 48 (6), with their actual
angles aob, boc, coa, and find the resultant wooi the middle pair, b o and
c o. Cut out neatly the whole figure, a o a c w b a. Make deep knife-
scratches along o 6, o c, so that the two outer triangles may be more readily
turned at angles to the middle one. Turn them until the two edges o ci^oa
meet, and then paste a piece of thin paper along the meeting joint to keep

\ \ /

w

(«)

(ft)

Fig:. 48.

(«)

them in place. Stand the model upon its side o & tp c as a base, and we aball
have the slipper shape a ob w. Fig. 48 ic)\o w being the sole, and aob the
hollow foot. In the model, the force a o and the resultant to o of the other
two forces, are now in their actual relative positions. To find their resultant,
cut out a separate piece of pasteboard, R a o to, with R a and R w parallel
respectively to w o and a o. Draw upon each side of it the diagonal R o.
Paste this piece inside the model, with its lower edge tt; o on the line to o. Fig.
48 (6), and its edge a o in the comer a o. This done, R o represents the re«
sultant oia o,b o, c o, Fig. 48 (a), in its actual position relative to them.

105. (b) For four forces, aaa o,bo,co,d o, in Fig. 49. Draw them as in
Fig. 40 (a), with their angles aob, boc, etc. Draw also the resultants « o, of
c o and b o; and wo,oico and d o. Then out out the entire figiire, as before,
and paste together the two edges a o, a o. Hold the model in such a way
that two of its jylanea (as a o 6 and boc) form the same angle with each other

NON-COPLAXAR FOBCES.

881

as do the two corresponding; planes between the forces. Then we have the
two resultants vo^wo, Fig. 49 (6), in their ctctiMl relative poeitiona. Cut out a
separate piece of pasteboard R v o w, Fig. 40 (&), draw the diagonal R o on
each side of it, and paste it inside the model, with o v and o to on the oorre*
sponding lixteB of the model. Then R o will represent the resultant of the
four forces, ao^bo,cOtdo, in its actiial position relative to them.

The model may be made ol wood, the triangles aobth oc, etc., being cut
out separately, the joining edges bevelled, and then glued to«^ther.

(«)

FUr. 49.

(*)

Non-concurrent Non-coplanar Forces.

lOG. Non-concurrent Non-coplanar Forces. Fig. 50 (a). (For par-
allel non-coplanar forces, see ^'^ 110, etc.) Resolve each force mto two rec-
tangular components, one normal to an assumed plane, the other coin-
ciding with the plane.* Find the resultant of the (coplanar) components
coinciding with the plane, by methods already given, and that of the normal
(parallel) components, by 1ft 110, etc. If these two resultants are coplanar,
tney are also concurrent, and their resultant (which is the resultant of the
system) is readily found.

107. If not, let V, Fig. 50 (6), be the resultant normal to the plane, and H
the resultant lying in tl^e plane. By If 162, substitute, for H, the eqtial and
parallel force H', meeting V at O, and the couple H . O a, and find the result-
ant, R', of V and H'. The system of forces is thus reduced to the single force
R' and the couple H . O a. For Couples, see If 155.

108. Moments of Non-coplanar Forces. Th<% action of the weight
W of the wall. Fig. 51 (a), and of the non-coplanar forces Pi and Pe, may be
represented as in Fig. 51 (&), where the axle a* cf represents the edge a c
about which the wall tends to turn, while the bars or levers represent the
leverages of the forces. So far as regards the overturning stability of the
wall, regarded as a rigid body and as capable of turning only about the edge
a e, it is immaterial whether an extraneous force, as Pi, is applied at p or at
g; but it is plainly not immaterial as regards a tendency to swing the wall
around horizontally, or to fracture it; or as regards pressures (and conse-
quent friction) between the axle a' <f and its bearings. For equilibrium. Pi vik

■- Pc A + W. — . Here a torsional or twisting stress is exerted in the axle.

*Wires, stuck in a board representing the plane, will facilitate this.

382

STATICS.

and the presBures of its ends in the bearings are more or less modified ; bui,
so far as merely the equilibrium of the moments is oonoerned, we may sup-
pose all of the forces and their moments to be shifted into one and the same
plane, as in Fig. 51 (c).

109* In oases like that represented in Fig. 51, it is usual, for convenience,
to restrict ourselves to a supposed vertical alice, «, 1 foot thick, and to the
forces acting upon such slice ; supposing the weight of the slice to be concen-
trated at its center of cavity, and the extraneous forces to be applied in the
same vertical plane with gravity. In eflfect, we are then dealing with a
slice indefinitely thin, but luiving the weight of the 1-ft. slice.

Flff. 51.

PARALLEL FORCES.

110. The resultant of any number of parallel forces, whether
they are in the same plane or not, and whether in the same direction or not,
is parallel to them and — their algebraic sum.

Coplanar Parallel Forces.

111. The resultant of any number of coplanar parallel forces

is in the same plane with them, whether the forces are of the same or
of opposite sense; and the leverages, or arms, of such forces, and of their
resultant, about any given point in the same plane, are in one straight line.
Thus, in Fig. 56 (a), where the five forces, a, b, c, d and e are in one plane, their
resultant, R, is in that same plane; and tne levera^ of the forces, and
of R, about any point, as 6 or v, in the same plane, are in the straight line R v.

Fig. 02.

113. The resultant, R, or anti-resultant, Q, Fig. 52, of two parallel
forces, a and b, intersects any straight line, u v, joining the directions of
the two forces. Hence, if three parallel forces are in equilibrium, they ara
m the same plane. In Fig. 62 (a), the two forces, o and 6, are of like
sense. R is then between a and b, and R = 6 -H a. In Fig. 52 (6), a and
b are of opposite sense. R is then not between a and 6, and R — fr — a.

PABALL£1< FORCES.

383

113* To find the position of the resultant, draw and measure any straight
line* u V, joininjs the lines of action of the forces. It is immaterieil whether
u « is perpendicular to said directions, or not. The line representing the
resultant cuts u v, and its position is found thus:

M i — tt « X -p- ; and v i = u v X -^.

FliT* 93.

*-i

114. This may be conveniently done by making u v equal, by any conve-
nient scale, to the sum of the forces, as in Fig. 53, where uv^ 42. Then
make u i equal, by the same scale, to the force at v,oxvi equal to the force at
u. Then a line, R, Fig. 52 (a), drawn through t parallel to a and h, gives the
position and direction of their resultant ; and its amount is equal to the sum
of a and h; or R =- a + 6. In other words, if a force, Q, parallel to a and 6,
and equal to their sum, but of opposite sense, be applied to the body any-
where in a line passing through i, it will balance a and 6, or will be their anti-
resultant.

\~ —

.-.^1^

/^

I

x^l

y

Ftgr. 55.

(ft)

115. The position of the resultant, so found, satisfies the condition of
equilibrium of moments : thus, h.vi — a.ui « zero.

If the two forces are equal, their resultant R is evidently midway between
them.

116. In the common steelyard, Fig. 54, the two forces a and &, of
Fig. 52 (a), are represented by the two weights, a == 3 pounds at i«, and h =•
1 pound at v, with leverages ui and vi respectively, as 2 : 6, or as 1 : 3.

384

STATICS.

It will be noticed that in Fig. 56 (a) the resultant, R, owing to the posi-*
tions and amounts of the several forces, falls outside of the system of given
forces.

117. Figs. 65 to 58 illustrate the application of the cord polygon (^^ 86
to 100) to coplanar parallel forces. Here the force polygon is necessarily a
straight line.

Jt

a Jft \d _^__y

(«)

Tig, 56.

118. Resolution. Let Fig. 57 (a) represent a beam bearine a single
concentrated load* a, elsewhere than at its center; and let it be required
to find the pressure on each of the two supports, w and x.

FIgr. 57.

(6)

Draw X a. Fig. 57 <fe), to represent the load a by scale, and rays X O, a O,
to any point O not in the line X a. In Fig. (a), from any point, t. in the
vertical through the point, a, where the load is applied, draw t • and t r,
parallel respectively to O X and O a. Join r «, and in Fig. (h) draw O to par-
allel to r 8. Then the two segments, w a and X w,^ of X a, give by scale the
pressures upon the two supFK>rts, w and x respectivelv. ,The greater pres-
sure will of course be upon the support nearest to the load; but we may
be guided also by remembering that the segment X w, adjoining the radiv
line O X in Fig. (6) represents the pressure on that supi>ort, x, Pig. (a),
which pertains to the line i 8 parallel to O X; and vice versa.

119. Fig. 58 represents a case where there are several loads on the
beam. Here the intersection, i, of the lines h a and k r, Fig. (a), drawn
parallel respectively to O X ana c O, Fig. (6) shows the^ i>osition of the
resultant of the three loads. Here, as in Fig. 57. we join r «, ftc. (a),

PARALLEL FORCES.

385

and draw O w, Fig. (b), parallel to r ». Then X to, Fig. (5), gives the
pressure upon x^ and w c that upon w.

(a) Fl|r. SS.

Non-coplanar Parallel Forces.

(&)

120. Non-coplanar Parallel Forces. Fig. 59 (a). Between the
lines of action of any two of the forces, as a and b, draw any straight line, u v,
and make

u i = u V X 1 r ; or v i — uv X

a ^ b *

a + 6 *

Through i draw R', parallel to a and ft, and equal to their sum. Then
is R' the resultant of a and 6. Then, from any pomt, t, in the line of action
of R', draw i z to any point, z, in the line of action of c, and make

c R'

ik — i z X p^ ; or « A; = t z X —3757 • Through k «draw R parallel to

a, ft and c, and equal by scale to their sum. Then is R the resultant of the
three forces, a, b and c. If there are other forces, proceed in the same way
with them.

c-tf

(«)

Fiir« 99.

(6)

131. In Fig. 59 (a) we have shown the forces, a and c, acting upon surfaces
raised above the general plane, merely in order to illustrate the fact that it is
not at all necessary that the forces be supposed to act upon or against a plane
surface.

122. Although Fig. 59 (o) illustrates the method of finding the resultant
of non-coplanar parallel forces, yet it plainly does not give the actual relative
positions of the forces and their resultant * because it is necessarily drawn in a
jcind of perspective, and therefore all the parts cannot be measured by a
scale. The true relative positions may of course be represented in plan, as
by the five stars, a, ft, c, % and k, Fig. 59 (6), corresponding to the points where

26

386

STATICS.

the forces and resultants intersect some one chosen plane. But it is now
impossible to represent the forces themselves by lines. They must there-
fore be stated in figures, as is here done. It is then easy to find the positions
of the resultants, as before.

1^. If there are also forces acting: In the opposite dfrection, as

d and e. Fig. 59 (a), find their resultant separately. We thus obtain, finally,
two resultants of opposite sense. These resultants may be equal or unequal,
and colinear or non-colinear. If they are non-colinear, see ^ 84, and Couples,
nil 155, etc.

134:. Metliod by projections. Fig. 60. First find the projections.
a\ h* and cf of the forces, a, h and c, upon any plane, bs x y, parallel to
them; and then their projections, a", 6", and c'', upon a second plane, x r,
parallel to them and normal to the first. Find the position, R', of the re-
sultant of a^ h' and c^, in plane x y, and that R'', of a'^ 6" and c", in plane
X V. Now, as the lines, a', b\ c', and o", b", c", are projections of the forces,
a, b and c, so R', R'', are projections of the resultant, R, of the forces. The
position of R is therefore at the intersection of two planes, R R' and R R'',
perpendicular \o the planes, x y and x v, and standing upon the projections
K' and R", of the resultant, R. R = o + 6 + c.

CENTER OF GRAVITY.

195. If a body. Fig. 1,* or a system of bodies. Fig. 2, be held successively
in different positions, (a), b), etc., the resultant of the parallel forces of grav-
ity, acting upon its particles and indicated by the arrows m the figures, will
occupy different positions, relatively to the figure of the body or system.
That point, where all these positions, or lines of gravity, meet, is called the
center of gravity of the body or system. Thus, if a homogeneous cylinder
be stood vertically upon either end, the line of gravity will coincide with
the axis of the cylinder; but if the cylinder be then laid upon its side, the
line of gravity will intersect the axis at right angles ana will bisect it.
Hence, in the cylinder, the center of gravity is at the center of the axis.

1|26. About the center of gravity the moments of all the forces of mivity
are in equilibrium, in whatever pK)sition the body or system may be. Hence,
the body, or system, if suspended by this point, and acted upon by gravity
alone, will balance itself; t. e., if at rest it will remain at rest;^ or, if set ib
motion revolving about its center of ^avity, and then left to itself, it wilt
continue to revolve about that center indefinitely and with uniform anfculaf
velocity. Or, if suspended freely from any point, it will oscillate until the
center of gravity comes to rest vertically under such point.

* Figs. 1 to 45, relating to Center of Gravity, are numbered independently

of the rest of the series oi figures relating to Statics.

CENTER OF GRAVITY.

387

127* In some bodies, such as the cube, or other parallelopiped, the sphere,
etc., the center of gravity is also the center of the v>eigfU of the body; but
very frequently this is not the case. Thus, in a body a b. Fig. 2, with its
center of gravity at G, there is more weight on the side a G, than on the side
G6.

Tig. 1,

Stablet Unstable, and Indifferent Equlllbrluni.

128* A body is said to be in stable equilibrium when, as in the pendulum,
it is so suspended that, if swung a little to either side, it tends to oscillate
until it comes to rest again, with its center of gravity vertically under the
point of su8f>en8ion.

129» It is said to be in unstable equilibrium when, as in the case of an
efx, stood upon its point, it is so supported that, if swung a little to either
side, and left to itself, it swings farther out from the vertical and eventually
falls.

130. It is said to be in indifferent equilibrium when, as in the case of a
grindstone, supported by its horizontal axis, or of a sphere resting upon a
horizontal table, it is so suspended or supported that, if made to rotate about
ita center of gravity and then left to itself, it will continue in that state ol rest
or of angpilar motion in which it is left.

(a) (b)

Wig. s.

General Rules,

131. The following general rules (I) to (6), form the basis of the special
rules, (7) to (39).

In speaking of the center of gravity of one or more bodies, we shall assume,
for simplicity, that they are homogeneous (i. e., of uniform density through-
out) and of the same density with each other. The center of gravity is then
the same as the center of volume, and we may use the volumes of the bodies
(as in cubic feet, etc.) in the rules, instead of their weigfda (as in pounds, etc.).

In applying these general rules to surfaces, use the area^ of the surfaces,
and' in applymg them to lines, use the lengths of the lines, in place of the
weights or volumes of the bodies.

In all of the rules and figures, pp. 388 to 398, G represents the center of
gravity, except where otherwise stated.

388

FORCE IN RIGID BODIES.

(1). Anjr turo Itodies, Fig. 3. Havine found the center of gravity, g^ ^^
of each body, by means of the rules given oelow: then O is in the line joining
yand^S' and

weight of flf^

sum of weights of g and g^

weight of g

sum of weights of g and g^

G

iris.3

(5S). An^ number ot bodies^ as «, & and e, Fig. 4, whether their centen
of gravity are in the same plane or not.

First, by means of rule (l> find the center of gravitv, gr, of any two of th«
bodies, as a and 6. Then the center of gravity, Gp of the three bodies, a, b
and c, 18 in the line ^p' joining g with the center of gravity, g^ of e; and

gG=^ gg' X

weight of e

sum of weights of a, & and e *

</ 6 — -^ w sum of weights of o and b ,
sum of weights of a, 6 and c
and so on, if there are other bodies.

(3)* In many cases, a slnffle eomplex bodjr may be supposed to be diylded
into parts whose several centers of gravity can be readily fonnd. Then the
center of gravity of the whole may be found by the foregoing and following
rules. Thus, in Fig. 6, we may find separately the centers of gravity of the

two parallelopipeds and of the cylinder between them (each in the center of
its respective portion of the whole solid) ; and in Fig. 6 the centers of gravity
of the square prism and the square pyramid (the latter by rule (36),
and then, knowing in either case the weightis of the several parts, find their
common center oi gravity as directed in rules (1) and (2).

CICNTEB OF GRAVITY. 889

(4^, Anjr l&olloiw- iMdy, or body Gontaining one or more openings. Fig.T
fmd the oommon*center of gravity, g'^ of the openings by role (1) or (2X anc,

the center ot gmyit^, p, of the entire figure, as though it had no opeatega
Then G is in the line £ry'» extended, and

aQ ^ Off V snm of volumes of openings

Tolome of entire body — yolumes of openings

t^-nifx

volume of entire body

volume of entire body — volumes of openings

Bbmabk. For convenience, we have shown the several centers of gravity,
9t ?/ ^9 upon the awrface of the figure. In the real solid (supposed to be <»
uniform tniokness) they would of course be in the middle oi its thickness-
and immediately under the positions shown in the figure.

(•). In any line, figure or body, or iaany system of lines, figuresorbodie8,any

Slane passing through the center of gravitv is called a " puuse «f §;r»vlt]r ^
>r said line, etc., or system of lines, ete. The intersection of two such planes
of gravity is called a '< line ot ^pcmrvU^** The center of gravity is (Ist) the
intersection of two lines of gravity; (2nd) the intersection of three planes
of gravity, or (3rd) the intersection of a plane of gravity with a line, of gravity
not lying m sMd plane.

If a figure or body has an axis or plane of swn&aaetrjr {i. a., a Mne or plane
dividing it into two equal and similar portions) said axis or plane ia a line of
place of gravity. If a figure or body has a central point, said point is the
center of gravity.

In Fig. 1, the string represents a line of gravity; and any piano with

which the string coincides Is a plane of gravity. Thus Qt may often be con-
veniently found, especially in the case of a flat body, by allowing it to hang
freely from a string attached alternately at different comers of it, or by bal-
ancing it in two or more positions over a fanife-edge, etc., and finding G in
either case by the intersecnon of the lines or planes of gravity thus found.

(6). Tlie irraplilc metliod of finding the resultant of parallel forces
may often be advantageously used for finding the center of gravity of a com-
pound body or figure, or of a system of bodies or figures, when the centers of
gravity of the several parts are xnown.

Thus, in Fig. 8, let a^ 6 and e represent three figures or bodies whose centers
of gravity are in one plane. Draw vertical lines through said centers, and
construct the polygon of forces, xa 6e, Fig. 9. making the lin's xa^ahy etc.,
proportional to the weights of a, 6 and e; and from any convenient point O
draw radial lines Ox, Oa, etc. In Fig. 8, draw m ti^mrunp, and p A;, parallel
IMpectively to O a;, O a, O 6, 0 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 aemi-eirelef*

eQ — radius a 0 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 « 0

U M

M

a

u

« —.40
« —.46
— .60

ft

u
**

M

— .645ao

— .641 « 3

— .6»7 « 0

(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 sides, 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 tbotriaasleagti).

(IBs). Or, Hg. 19, Dnd Bnt (hi
angles, d A d and abd, into vhlch
nala, i a. lata m n. Then find

Then « is the In

peilnm Is' d

OENTBB OP GRAVITY.

393

(16). Tntpcowiffi onlv-f Fig. 2a 6 is in the tine 0/ joining the centera,
eand/, of the two parallel sides, ab and cd. To find its position in said line,
prolong either parallel side, as a 6, in either direction, say toward i; and make
(t equal to the opposite side, cd. Then prolong s€ud opptosite side, e d, in the
q>posite direction, making ah'^ah. Join hi. Then G is the intersection oi
Aiande/. Or

fa - */ w 2«fe -H cd .

or oO

en 2a6 4- cd
8 ^ a6 -)- ed

c n Of

B'ifif. so

~'2aai#«.A>

(1T)« RegnlMP polygon. G is the center of the flgme.

(17 a). Irreguimr polyson. If the polygon be divided into any two
pcnrtions, as by any diagonal, G must be in the line (of graTiify) Joining ^e
centers of grayity of those two portions. If we again divide the whole polygon
faito two oVher parts by another diagonal, and join the centers of gravity of
ihoee two parts, G is the intersection of the two lines of gravity.

(17&). Or we may divide the polygon into triangles, find the center of
mvi^ of each triangle, by Boles (14)» etc., and then find G by general Bole
&X(2)or(6).

(M)

S^ig. 31

I. CJIvonlar Motory aobe, Vig, 21. (Center of circle at e).

-0-. 2 •^f»«^« vr eh<»^g^ radius* X chotjl

3 arc ao 6 8 X Area

For length of arc, see p. 14L

(18a)* If the seotor is a Mxteaty

cG — radios X— — Tadios X 0.6801

IT

(18&)* If the aeoior is a qiio<lraiit» Fig. 122,

— -i radios X ^^--^ — radios X Oi600t.

3 V

cG

em

3 'V

— as G — — radius X — •

a «•

(ISe). If the sector is a seiiil-eircle»

cG — — radios X —

« IT

— radios X 0^4211

14
«• (approximately) radios X -^ •

70BCE IN BIQID BODIES.

12 X are* of segment
<19a). It thesegmeutisaHHii-sliolat

(Q "J radiiu X -^ - radlns X «.

— OVfinHdinataW Mdins X ^ •

^Mt). Orelold, Pig. 31. (Vertex at V).

F)l). Pusl»lii,ohe, FIg.26. a
la Ihe hexe; ax »nd 01, ordinatei
and thf height or a:i1% bx. m al

er of gtaTlty &c Q*, mid

•lll[iH In at tEe»iiter <:rthallEa

O f B the omter of gnvlty of theqDartere1ll[we,0i>a.

Ctr—jOPX — -•UaUoe — (approximtteM^oB-
«CI*-0'G-l«ii X -^ il.4244eii-(i4>pcosliiuiteIy}-^aih

CENTER OF GRAVITY.

396

(93)* Amy pliu&e flfl[iur«« Draw the figure to scale on stout card-board.
Out it out and balance it in two or more positions over the edge of a table or
on a knife-ed^e; and mark on it the several positions of the supporting edge.
Where these mtersect is the center of gravity. Considerable care is of course
necessary to obtain very close results by this method. Before balancing the
card, its upper edges should be marked off into small equal spaces. Otherwise
it will be difficult to locate the positions of the supporting edge. The papc4r
on which the figure is prepared must of course be so stiff that the figure will
not bend when balanced on the knife-edge. See Bule (6).

B. Sarlinces of Solids.*

(94). Curved surfiAce* of spliere or ■]^«rold(ellip«old). G is the center
of the figure.

(il5). Curved sur&ce* of any anborloal soiie» as a splkcrlttal amgnunU
hiemlaplfccre, etc., Figs. 27. O i»ihe center of the axis or height, a o.f

In the liemtsplMre, o O ^ Vi radius-f

Fiff.ar

(96). Bight or oblique piiam, whose «nd8 are either regular figures
or parallelograms (this includes the «alM and other jMtimlleloplp«ds)| and
rigtit or oblique cyllmder (circular or elliptic). Surface* (either including
both or excluding both of the two parallel ends). G is the center of the axis,
or line Joining the centers of the two parallel ends.

(JiT). Curved surflftce *t of right oone, Fig. 28 (circular or elliptic), or slanting
flurfaoes't of right regular pyramid. Fig. 29. O is in the axis oa (the line
joining the apex and the center of the base); and

o G — 3^ o a.

In an oblique cone or pjrramid, the perpendicular distance of G* firom tha
base is one-third of the perpendicular height, as in the right cone and pyramid;
bmt doM not lie in the axis.

^ ^*?1' JP>'«<»*«>»« with top and base parallel, Figs. 80 and 31. Carved sur-
»co*t of frustum of right cone (circular or elliptic); or slanting surfaces *t of
frnstnm of right regular pyramid. G is in the axis o a (the line joining the
centers of the two parallel ends) ; and ^

oG — — oa X

circumference of o + 2 circumference of a.
circumference of o + circumference of a.

• We treat now of the turfaees of solids, not of their contents or volumes or
weights. For these, see Rules (29), etc.

t If the top or boae is to be included, see Rules (1) and (2),

FOBC& IN RIQID BODIES.

In the eomlc flnutam. Fie. 30, we mt^ uie the rada at the two ends
In ihe trmMtmm of • renjiir pTnuntd, Fig. 31, any tidt of each ei
icnnd de) ianeadaf the clrcuinfereDoes.

ITiB.SO

the rolloirlni Talea for oenter of e^vlty of Bollda, the eolld ia lappoaed
I lionioaetuicut! i. e_ of uniform denalty throughout; bo that the oenlor of
Ity is the center of magnitude or of volume,

t). S^un and *|Acnld (eUlpasldJ. O ia the center of the body.
){. BioBlapbere, Fig. 32. (Oenter of ephere at c). Height « T — ndia*
O la in the axis, cT, sad

(81). KpkerKAl

Center of tplUTa at e. Canter
■ h. eialnUewusmT;uid
3 (aradiUB c ft of ipttf* — height h)'
"" ■" 4 -^ aradiuaebof apAfl-a — belghtA
__ height. * a (radtUB wh of hate'fl + (height, >y
9 -^ 3 (radiua m b of baH}> + (height, A)>

_ helghl^ h i X radluB eh of tphtn — be^ht, h
~ i 3 X rBdiuBeb of ipAva — height,* '

It bua)> + 3 (i&dlua ( e of top)* + (height o 0> '
irregular, right or ohlique (including the •

(34). Frlnn.reguUr or irregular, right or ohllque (including the aaba
and other puiallelaplpMli). and ejrlludwr, circular or elliptic, etc., ragolar
or irregular, right ot oblique. Ols the center of the axle Joining the centeia
of gravity of the Im ends.

CEHTEB OF OBAVITT.

(Ma). A flat body, Hich u aa
IhortcjliDder or prism. See (34)

(30). DngDla of B cjUnder, circu
af tbe ellipse caiacides witb Uie oblique ooUiiig
Figs. S« and 3T.

ij be treated ai a Teir

>lltl)tli!(piMVidedODeDtth«ueg
itUog pUoe); rigbt oi aULqiu.

) alia Ooining the oantenof gisTltyorthe ends), and XR •
a psrsLlaL to the axis, In the pUoe, ABC D, passing through tha

1. 1, 4.U . -_j 1 TnioBt])ointBOBBdDof (JieobUqoe

"le plana A B O D, ii foond Urnac

ffl«|. Cone, Figs, «_^Biid 41,
mil.

9«|. Cone, Figs, 40 and 41, circular. _ _

iptio, etc., Hghl or obltquB; or ny™. ! T

la, regular or iiTBguW, nBht or A ^^

j,Ti«""""'""-*°""" Mm. .^m

(37). rmrtnm of a, cone. Pigs. 42 and 43. slrnular or elllptie, right
abiiqae; or of a pTramid, reffnlar or Irregular, right or oblique ; proridedl
tvaends ABand CDnre piLralleL

1 ; Mid let A ba

FOBCB IN BiaiD BODIS.

a is Id the ula O Z, vhlall foinB ">" ceuleis ot n&Tll7 O and S ot i
•Bdaj Mid iti distance Aom ths bus, A B, tMOdarnf oimg Ik* axil, is

!■, t£ts bwomes

B, right or oMIqoB, wtth p

■here R ftni
(38). Pigs. 44 Mid U.

* W + Hr + ri'

the ndil of the Ivge imd anuU ends of t>

etc., right orobllquei or ot a pyTunld, regufsr or Iriegulor, right i
whether tha ends in puillBl or Dot By rule (38) And the center
N of the (nttrs pTramld (or cone, as the case msr be) A B T, ot
frustum forma the lower partj and the eealar of in^iitv n nf H
pyrunld or cone D C T (~ entire pynirald or ooae, t
nod the vclamt ot eachi thns.

frnBtum). Alto

ZriB,44

Tolame of pyramid or ocas

Volume of

the frustum ■■ t , ._ _

ABCD or cone, A BT oue.DCT

Then the center of gravity O of the frustum ABCD is In tlia<

' Tolumeof sTUBller pyramid or cone, POT

(39). FmrBbololdL O la la tt

CENTER OF PRES8UBE. 399

UNE OF PRESSURE.
CENTER OF FORCE OR OF PRESSITRE.

Pos^lon of Resultant.

133. In fl 133 to 154 we discuss the position of the resultant, or line of
pressure, of a system of parallel forces acting against a surface. For the
changes in that position within a structure, due to the action of non-parallel
forces, see Arches, Dams, etc., ^^251, etc.

134. In a system of parallel forces, acting against a surface, the line of
pressure, or pressure line, is the position of the resultant of the forces; and-
the center of force cr center of pressure is the point where the pressure line
meets that surface against which the forces act.

135. If the lengths of the lines which represent the forces be taken as rep-
resenting weights, to scale, the:, the position cf the pressure line is the line of
gravity (see (5), ^ 131) corresponding to those weights.

136. Thus, in Fig. 55 (a), f 117, if the three forces, o, h and c, be taken
as weights, represented to scale by the arrows, a, b and c, respectively, then
the resultant R of the three forces occupies the position of the line of gravity
of the three weights. '

137. Again, in a mass of sand. Fig. 61,* with an irregular surface, we may

Fiff. 61.

suppose the mass to consist of innumerable vertical columns of sand, of
different heights, and exerting pressures proportional to those heights. Here*
also, the pressure line is the vertical line of gravity of the mass, and the cen-
ter of pressure against the base of the containing box is the point where said
pressure line meets that base.

138. Although we are usually concerned with forces acting against atir-
jiaces, so that «he lines representmg the forces form a solid and not merely a
surface, yet, iu a majority of the cases which occur in civil engineering, we
may, for convenience, regard the forces as concentrated in a single plane,
and therefore as acting against a mere line.

130. Thus, in the case of an arch, pressing against its skewback, the pres-
sure is ordinarily distributed over all or a considerable part of the bearing
Burfaoe of the skewback; but we may, for convenience, regard it as concen-
trated in a single plane, midway between, and parallel to, the two faces of
the arch.

140. Similarly, in the case of the water pressure against the back of a dam
Cor against a small strip of the back, extending from the water surface to the
bottom, or to any other depth), the water, of course, presses upon the entire
surface of such strip; but we may, for convenience, regard the pressure as
concentrated in a vertical plane normal to the back of the dam and meeting
it in the vertical axis of the assumed strip.

141. We have just seen (11[ 138'to 140) that, when a system of parallel
pressures acts agaeunst a surface, they may often be assumed to act, in oue
plane, against a single line — viz., the intersection of that plane with the sur-
face. It also frequently happens that such forces are so distributed along
that line that the hues representing the forces are either of equal length or <^
lengths increasing uniformly from one end of the line to the other.

^Following Fig. 60, of Parallel Forces, ^ 124. Figs. 1 tc 45, illustrating
Center of Gravity, are numbered independently of the rest of the series of
figures relating to Statics.

400

STATICS.

142. Thus, in the case of water restins upon a horisontal surface, Fig. 62,
the pressure is uniformly distributed, and the diagram. Fig. (b), representina
the pressures, is a rectangle bounded by a horizontal line, and its center oi
gravity, G, is at the center of the figure. Hence, the center of pressure, c, i>
at the center of the line a 6, or I. m

Here the unit pressure, p, is uniform, and R — p 2.

xa)

FIs* 68.

Fiff. es.

143. But when the water presses horizontally against a vertical or in*
clined surface, a 6, Fig. 63, the unit pressure increases uniformly from zero,
at the water surface, &, to a max at the bottom, a ; and the hor pressures
are represented, in Fie. (&), by the ordinates of the triangle V a' d. Since
the resultant passes through the center of gravity, G, of the triangle, the
center of pressure, c, is at such a depth that c a « i a &, and c^ a' « i h.
See Rule (14 c) under Center of Gravity .

Here the mean horizontal unit pressure, p, is half the maximum horizontal
pressure at a, and the total horizontal pressure is — p A.

Figr. 64.

144. Again, if we consider onlv the water pressures against a certain part,
a h. Fig. 64, of the depth of the back of a dam. the diagram. Fig. (6), repre-
senting the horizontal unit pressures, becomes a trapezoid, composed of a
parall^ogram 1/ /, and a triangle 2/ a' d, with their centers of gravity at g
and g* respectively; and the center of pressure, c, on a b, is opposite thur
common center of gravity (center of gravity of trapezoid), G. If h be the
vertical depth of the portion considered, then

h ^ 2 h'e±afj

o' c »= — X -, 7-7- •

3 6' 6 + a' /

See Rule (16) under Center of Gravity. See also Center of Pressure^
under Hydrostatics.

Distribution of Pressmre.

145. Conversely, if two surfaces, as those of a masonry ioint, are in suoh
contact that the pressure is, or may be regarded as, regularly distributed,
and if the position of the resultant is known, the rectilinear figure, represent-
ing the distribution of pressure, may be drawn by means of the priiMuplM
just stated.

DISTRIBUTION OP PBESSURE.

401

146. In Figs. 65 to 68 inclusive, let

0 «> the center of the joint a h between the two surfaces;
R =■ the total pressure = resultant of all the pressures;

c =* point of application of resultant, R ;

1 ^ ab = the length of the joint;

X *^ o c = the distance of the center of pressure from the center of the
joint;

y "^ "o — X = a c >= distance of center of pressure from nearest end of

joint;

p « the mean imit pressure =» -: ;

pa » the maximum unit pressure ;
ph *" the minimum unit pressure.

^1[ 147 to 154 apply equally whether the surface is horizontal, vertical or
inclined, and whether the forces are normal or inclined to it.

147* If X is not greater than — , or, in any case, if the joint is capable of

o ^

muitaining tension, as well as compression, w4 have:

6 X
Maximum unit pressure = p» *=• p (1 -I — j-) ;

6 X
Minimum unit pressure =* Pb = P (1 r-).

If X exceeds -^^ and if the joint is incapable of resisting tension, see 1f^
161, 152, 154.

Figr. 65.

Tig. 66.

14S. Demonstration. In Fi^. 06, where the parallelogram a' d repre-
sents the total pressure R aa it would be if uniformly distributed along l^ we
see that the moment of K, about o, which changes the parallelogram a' d mto
the trapezoid a' 6' n m, is equivafent to a couple (see Couples, If 165, etb.)
composed of two forces — viz., a pressure, / (not shown) distributed over 0 a
and represented by the shaded triangle on the left, and a tension, — /, or
dimintition of pressure, distributed over o h and represented by the triangle
on the right.^ The forces, / and — /, act through the centers of gravity of
these two triangles respectively; and the distance of each of these centers
of gravity from the center, o, of the joint, measured parallel to the joint,

2 I
is » -^ . '^. Hence ihie distance between the two centers of gravity, meas-

26

402

STATIGB.

21
ured parallel to the joint, is => -^. Let x be the eooentrioity, c o, of R,

measured alonff the joint, and let Ak and Ao (not shown) be the lever arms
of R and of the couple, respectively, about the center, o, of the joint.
Then, since R is parallel to / and — /, Ak to Ac, and x to 2, we have:

A.:Ac=x:Ai.

21

If R is normal to the joint, we have : Ax ^ x; and Ao =

3*

Tig. 65 (repeated).

moment of R

Now
Hence,

/

arm of couple

FflflT* <^tt (repeated).

R.Ab
Ao *

f 21 I ' 2 ^ 2 '

The mean additional pressure on o a (or mean tension on o 6) is — -^

and the corresponding maximum additional pressure is

- „ f 4 . 4 3x 6x

Now p» =p + /„ = p + p-p-p(14. ^)

J 1 6 X ,- 6 X.
and pb - p — fm-P — p-j- -p(l J-)'

149. If, as in Fig. 65, the center of pressure, c, is at the center, o, of the
surface, we have x ^ o c '^ zero, and the pressure, R, is uniformiy distrib-
uted over the surface.

150. " The Middle Third." If, as in Fig. 67, x - |^,— *. «., if the re-
sultant, R, of all the forces, meets the surface at the edge of the middle third
of that surface, then p» — 2 p ; and Pb "" 0. See 1[1 143 and 148.

151. When, as in Fig. 68 (a), x exceeds -^, — i. «., when the center of pres-

o

sure, c, falls beyond the middle third of the surface of pressure, a portion,

8 b, of the surface, is in tension, the maximum tension, pb. Fig. 68 (6),

6 X 6 Xv

being = p (1 ^) as above; maximum pressure = p (1 + -j-). and total

pressure on a « — ' • »■ R plus the tension in sh; but if, as usually

DISTRIBUTION OF FBE8BCRB.

luLppena m moAonry, the eurfocefl .
prBMure,R, isairnplyoant— •— •-

oftpable of niataining toisioo. Fig. 6S (r). If ft —

1S4. The inflneooe diagrama. Flp, 6fl (aes ^% 33B. etc.. and Tmeses,
Y1 70< etO-), show the changes in the maximum and minimum unit pn»-
■ur^ ft and tH- as the center of preHeijre» c. recedes from the center, 0, of
the joint. The dlaeramH are constructed for a mean unit preaauie, p, of i.
If the surfaces of the joint are capable of sustaining tension, every part of
the joint always Bustaina either preeeure or teaaioDi and (eee dotted hnesd

404

8TATIG8.

i'

Fi^ 69) the maximam unit pressure, p»
proportionally with x; becoming — 4 p — -

4R

+ — *)

see f 146, inereases

I

of the joint, and -j- —

when c reaches the end, a.

The maximum tension, Pb, is then — 29 —

/ ~ 2*
But if the surfaces are incapable of sustaininiT tension (see solid lines. Fig.

60), the increase of p^ is proportional to x only so long aax < -^ ; — t. e., so

6

long as the resultant of all the pressures falls within the middle third of the
base a b. ^ When that limit is exceeded, the maximum unit pressure, p^,
begins to increase more rapidly than does the distance, x, of c, from the
benter,, o, of the joint, the diagram becominxj^ a rectangular hyperbola; so
thirty uF the resultant could be actually applied at the very edge of the
joint, the unit pressure there would become infinite.

9 e

center ofJiHn^to center ofpres»ttr§

COUPLES.

165. Couples. Two equal parallel forces, p and q, or p' and g'. Fig. 70,*
of opposite sense, are called a couple. A couple has no tendency to move
the body t as a whole in any straight line. In other words, the two fotoes,
forming a couple, can have no resultant. Their only tendency is to make the
body revolve about its center of gravity, G, and in the plane of the eoHple
— i. e., the plane in which the two forces lie. A body with a fixed axis can
revolve only in a plane normal to that axis. The actual plane of rotation of
a free body depends upon the distribution of mass in the body, and is not
necessarily the plane ox the couple.

* Figs. 70 to 75 are supposed to be seen in perspective, and the forces art
supposed to act in the planes shown.

t See foot-note (♦), H 1.

0OUPLE8.

405

156. The moment of a couple is equal to the product of one of the
two forces, porq, mto the perpendioular distanoe, d, between the two forces.
Or, in our ngures, ^

moment of couple •= p . d '^ q . d.

157. Graphic Bepresentatlon of Couples. A couple, M or N-, Fig:.
70, is indicated, in amount, in direction and in sense, by a line, L or L'.
normal to the plane of the couple, so placed that, looking along it toward
that plane, the couple appears positive or right-handed, and of such length
as to repissent, by scale, the moment of the couple. In Fig. 70. the two
couples M and N aT<e of opposite sense. Hence the lines L and L', repre-
senting them, project in opposite directions from their respective planes.

Wig. 70.

Fiff.Tl.

158* Composition of Couples. If the lines, L and L', Fig. 71, repre-
sent two couples, in accordance with ^ 157, then the line R, completing the
triangle, will^ in the same way, represent their resiiltant or anti-resultant.
As drawn, with its arrow foUowing those of the other two sides, it represents
their arUt-resultant. For their resultant, tiie arrow on H, and that indicating
the direction of rotation, must be reversed.

150. Eouality of Coupler* Two couples, M and N, in the same plane.
Fig. 72 or Fig. 73, or in paralld planes, Fig. 70, are equal if their moments
are equtil, whether or not the forces of one of the couples be equal or parallel
to those of the other. In Fig. 73, the two couples* M and N, are of like sense ;
in Figs. 70 and 72, of opposite sense.

Fly. 72.

Flff. 78.

160. Since a couple has no resultant (% 155), it can have no anti-resultant;
i. e., no single force can balance a couple and thus preserve equilibrium.
(But see U 168.) To do this requires an equal and opposite couple. Thus,
in Fig. 72 the couple M is balanced by the equal and opposite couple N. If,
as in Fig. 72, the two couples are in the same plane, and if we find first the
resultant of either pair of non-parallel forces, as p and p', and then those of
the other pair, ^ and 9^, we shall find these resultants equal and opposite,
maintaining eqmlibrium.

161« Any couple* as M, Fig. 73, may be replaced by any other
equal couple, N, in the same plane or in a parallel plane, and of like sense.

162. If» to a force, P, Fig. 74 (a), we add a couple, M. Fig. 74 (b), in
the eahie plane with the force, we may replace the couple, M, by an equal
and like couple, N, Fig. (c), composed of the forces, — P and P', each — P.
placing ■ — P opposite P, as shown. Then P and — P counteract each other, ana
we have left only P', equal and parallel to P; and, since Pd » M, we have

406

STATICB.

d"

M

In other words, the efiFect of the addition of the couple, M, Fig. (ft);

to the force, P, is simply to shift the line of action of P, parallel with itself,
through the distance, d. If the couple M is left-handed, as in the figure, P
will be shifted to the right (looking in its own direction), and vice versa.

163. CoiiTersely« the force, P', Fig. (c), is equivalent to the combination
of force P and couple M, Fig. (jb).

164. Again, having only the force P', Figr. (c), if we apply, at a distance,
d, from P', the two opposite forces, P and — -P, each equid and parallel to P',
we shall thus substitute, for P', the equal and parallel force, P, and a couple
= Pd =. M.

165. Hence, also, the combination of the force P and the couple M, Fig.
(6), is equivalent to the combination of the force P and the couple N, Fig.
(c).

166. If the moment of the couple, M, Fig. (b), or N, Fig. (c), be equal and

opposite to the moment of the force P about the center of gravity, G, of a

■ M
body, we have d *= p ■■ distance from P to G. In other words, the eflfeot of

such a couple is to shift the force, P, parallel with itself, to a line passing
through the center of gravity, G.

167. Hence, the efiFect of a force, P, Fig. (a), applied to a body at a dis-
tance, d, from its center of gravity, G, is equivalent to the combined effect oC
an equal and parallel force, P', Fig. (c), applied at the center of gravity, and
a couple (as M, Fig. h) >=» P<2, and ox like sense, applied to any part of the
body m a plane parallel to P and P'.

168. It will be seen that, although (t 160) no single force can balance a
couple and establish equilibrium, yet, if a force, P, be so applied that its
moment, Pd, about the center of ^avity, G, of the body, is equal and oppo-
site to the momenit of the couple, it will counteract the tendency to rotatioOt
due to the couple, and substitute for it a motion of translation only.

FliT. 75.

169. Thus, in Fig. 75, where the force, p, acts through the center of
gravity, G, of the body, let a force, — q, equal and opposite to a, be applied
m the same line with it. Then rotation will be prevented, and the boour will
move * under the action of p ( — the resultant of the three forces), which acts
through the center of gravity, G, of the body. The rotation will similarly
be prevented if a force 7efi« than q be applied farpier from G than 9 ia; or if a
force greater than ^ be applied nearer G than g is ; provided always that the
moment of said third force, about G, be equal and opposite to that of the
couple p q. But in the first case the resultant of the three forces (being always
equal to the third force) will be less, and in the second case greater, than p.

170. If, to a couple, be added a third force, colinear with one of the forces
of the couple, we have the case of two unequal parallel forces of opposito
sense. See t 112, under Parallel Forces.

♦ See foot-note (*), ^ 1.

FRICTION. 407

FRICTION.*

171. When one rough body rests upon another, the projections and de-
pressions, forming the roughnesses of their surfaces of contact, Interlock
to a greater or less extent ; and, in order to slide one over the other, we must
expend a portion of the sliding force, either in separating the bodies (as by lift-
ing the upper one) sufficiently to clear the projections, or in breaking off some
of the projections and clearing the others.

17!S« Even the most highly polished flat surface, as x y, Fi^. 76, is not (as it
appears to the eye) a ptone, but is, in fact, a more or less jagged surface, as
would appear under a sufficiently powerful microscope; so that the force, a 6,
instead oi forming the apparent angle, a b x^ with one smooth surface, x y, of
application, really becomes a series of parallel forces, as c, d and e, which form
other angles with a number of surfaces, m m, nn, etc., of application, inclined
(often in different directions) to the general surface, x y, as shown. Among
these surfaces may be some, as m m, at right angles to the applied force ; and,
the force c will be imparted to them in its original direction, although applied
cblimiely to the apparent surface, x y. In the case of the two forces, d and «,
appfied to the surfaces, n n and « «, if the sliding tendencies along the two
surfaces are equal and act in opjHMttion to each other, the cornbined resistance
of the two surfaces, n n and e «, is directly opposite to the forces, as would be
that of a single surface at right angles to those forces.

Flff. 76.

173. It is of course entirely out of the question to ascertain the exact
resistance of each such microscopic projection in any given case. Instead of
this, we find by experiment the combined resistance which all of the projec-
tions, in a given case, offer to the sliding force, and give to this resistance the
name of friction.

174. Friction always tends to prevent relative motion of the txoo bodies
betaken which it acta; i. e., motion of one of the bodies relatively to the other.
In doing so, however, it tends equally to catiae relative motion f between
each of those two and a third, or outsit body. Thus, the fric between a belt
and the pulley driven by it tends to prevent slipping between them; but thus
tends to make the belt slip on the driving pulley, and sets the driven pulioy
and its shaft in motion relatively to the bearing in which the shaft revolves.
This motion is resisted by the fric between journal and bearing \ and this fric,
in turn, tends equally to make the bearing revolve with the journal, and to
make the belt slip on the driven pulley.

175. The fric between two bodies at rest relatively to each other is called
static triction» orfric of rest. Th^t between two bodies in relative motion
is called Icinetic friction or fric of motion.

176* The ultimate or maximum static fric between two bodies,
as U and L, Fig. 77 (or the greatest fric resistance which they are capable
of opposing to any sliding force wheiji at rest), is equal to a force (as that of

* " Friction" (meaning rubbing) is a misnomer in so far as it implies that
rubbing must take place in order to produce the resistance. For we meet
this resistance, not only during rubbing, but also before motion (or nibbing)
takes place. *' Resistance of roughness " would better express its nature.

t See foot-note (♦), If 1.

408

0TATIGB.

the wt F) which is just upon the point of making U begin to slide upon L.*.
Thus frio, like other forces, may be expressed in weigfUa^ as in lbs.

177. A resistce cannot exceed the force which it resists.! Therefore if F
is less than the ult static fric between U and L, the ffictxonal resiatce actttaUy
exerted by them is also less. When F is =- the ult fric (and U is therefore on
the point of sliding) the actual resistce is ■> the ult stat fric. If F exceeds
the ult Stat fric, the excess gives motion to U.

178* If, when a body is in motion, all extraneous forces and resistces am
removed or kept in equilib, it moves at a uniform vel. Hence, if the force, F,
Fig. 77, is just — the ultimate kinetic fric between U and L, their vel is uni-
form. If F exceeds this, the excess acceleratea the vel. If the ult kinetic
fric exceeds F, the excess retard* the vel. Thus the actual frictional
resistce exerted by two bodies in relative motion is "> their ult kinetic
trie " that force (as F) which can just maintain their relative vel uniform.

179. Hence, if the hor surf S upon which L rests, could be made perfectly
frictionless, the i>res of L against the lug m (which would then always be' ■*
the actual fric resistce between U and L) would also be ■■ their utt fric so long
as U continued in motion over L, and might therefore be greater or less than
or »■ F; but when U was at rest the pres against m would be ■- F, and less
than (or at most just -= ) the ult fric.

Coefficient of Friction.

180. Since no surface can be made absolutely smooth, some separation d
the two bodies must in all cases take place in order to clear such projections
as exist. Hence the fric is siways more or less a£fected by the amount of the
perp pres which tends to keep them together.

181. The ratio of the ult fric, in a given case, to the perp pres, is oalled
the coefficient of friction for that case. Or,

Coefficient of friction -a

ultimate friction

and

perpendicular pressure
Ultimate friction •» perp pres X coeffof fric.

Thus, if a force F, Fig. 77, of 10 lbs, just balances the ult frio between U
and L, and if the wt of tJ (the perp pres in this case since the surf between U

and L is hor) is 60 tbs, then the ooefF of fric between U and L is « SoTS

- 0.2.

Tig. T7.

FtflT. 78.

182. The coeff is usually expressed decimally, or by a common

fraction ; but sometimes, as in the case of railroad cars and engines, in fbb
(of fric) per ton (of perp pres). Or by the "angle of frio" in degs and mins.

* We here neglect the frio of the string and pulley, and assume that oU the
force of the wt F is transmitted by the string to U.

t If a resisting force exceeds the foroe resiatedt the excess is not reeiatee,
but motive force.

ANGLE OF FBICTION.

409

183. Ansrle of Friction. la Fig. 78* lefe W -> the tveight of the body,
P = its pressure normal to the plane, and S «» the component tending to
slide (he body down the plane.

When the angle a is such that the body is just on the point of sliding down
the plane, it is called the angle of friction, or angle of repose. The mction
F and the sliding force S are then equaL

S "R" A

But p "* p "■ B ** coefficient of friction — tan a. Hence F = P tan a
*- W ooein « . tan a,

184. Frlctional Stability. Let R, Fig. 79, be the resultant of all the
forces pressing a body against a plane, and Jf a normal to the plane. If the
angle i between R and N exceeds the angle of friction (a. Fig. 78) between the
two surfaces in contact, the body will slide on the plane, but not otherwise.
If t does not exceed the angle of fxietion, the entire resultant R will be im-
parted to the plane and in its own direction, and not merely its normal com-
ponent V, as would be the ease if the surfaces were friotionless.

Figr. 79.

FI9. so.

185. To find the coeff of klnetle trie, allow one of the bodies, U,
Fig. 80, to slide down an inclined plane A G formed of the other one and hav-
ing any convenient known steepness ACE greater than the angle of frio (%
183). Note the vert dist A £ through which U descends in sliding any dist
asACfAE — ACX sine of A C E) ; also its actual sliding vel in ft per sec
on leacning C. Calculate the vert dist A D through which it would nave to
descend along the plane (from A to B) to acquire that vel if there toere no frie.

(

AD

velocity^ in ft per sec

0-

twice the accel g of grav '
Find P E (» A £ — A D), and the hor dist £ C corresponding to A C
(EC - ACX cosineACE - T^AC«— AeO. Then

Coeff of the average frie in sliding from A to C -•

DE
EC

because, if we let A E represent the total sliding force expended (in accelera-
tion and in overcoming the frio), then A V> represents the portion of A E ex-
pended on vel, and D £ that expended on frie, and, since C £ represents the
perpendicular pressiu*e (if 183),

DE _ friction^
iE C prep pres

— coeff.

186. Or, find sine and tangent of A C E ; and the dist A C ( » tlme^ in
les X i 0 * X sine of A C E) through which U would slide in a given time

ii there toere no frie. Measure the dist A B through which it actuaUy elidea in

tiuittime; andfindBC >« AC — AB. Then

coeff of the average ) . t^/^« ^ A/-n^v,BC

- . . ,. J. - A i n f " tan DCE — tan ACE"'
f nc in sliding from A to B j

because

* g " about 32.2 ft per second per second.

AC

410 FBicnoN.

(lit) AO:AB:BO :> ABsADtDI

«^dSie toTlSf^Udiif fJroe '' ^^ ~^ ^^^^^^^ '• «^« «««*>*«» ret«d»tl«

sliding force employed the friction, or the sliding
: : the total sliding force : in giring the actual : force required to balance

velocity the fiiction.

And, if A £ is «- the total sliding force, then S C is — the perpendicular pressure,

and Tk IS

tL^ = the coefficient of friction =^ tangent of D 0 E.

BO

(2nd) Owing to the similarity of the twa triangles, A B D and A G E, we hare

AG : BG :: A£ : D fl :: ^:£-E ::tangentAGE : tangentDGB.

B fl fit G

187. In 1831 to 1834, Oen'l Artlinr Horln* experimented with
pressures not exceeding about 30 lbs per sq in ; and arrived at the following
couclusions in regard to sliding fric where the perp pres is considerably less
than would be necessary to abrade the surfs appreciably. These were for a long
time generally regarded as constituting the tliree ftmdamentol la^rsox
ttie.

1st. The ult fric between two bodies is proportional to the total perp force
which presses them together; 1 6,the eoeflTis Independent of tne perp

pres and of its intensity (pres per unit of *wf). Hence

2d. For any given total perp pres, tlie eo^T Is independent of the
area of snrf in eontaet.

If upon a hor support we lay a brick, measuring 8X4X2 ins, first upon its
long edge (8X2 ins) and then upon its side (8X4 ins), we double the area of
contact, while the total pres (tlie wt of the brick) remains the same, and thus r»-
duee the pres per aqvn^aj une-balf. Consequently (the coeff remaining practically
the same) we have only half thefricpersq m. But we have twice as many sq ins
of contact, and therefore the same toial fric.

But if we can increase or diminish the area of contact without affedimg the pres
per sq in, tlie totaJ pres will of course vary oi the area, and the total fric will vary
m the same proportion, for the coefT remains the same. Thus, if we place two
similar sheets of paper between the leaves of a book (taking care not to place
both sheets between the same two leaves) and then squeeze the book in a letter-
copying press, it will require about twice as much force to pull out both sheets
as to pull out only one of them.

3d. Although the coeff of «to/tc fric between two bodies is often much greater
than their coeff of kinetic fric ; yet the eoelf of kinetie flrie is Inde-
pendent of the wel.

This applies also (approx) to the fric, and hencetothetrorA;(in>bo<-pounds etc)
of overcoming fric through a given dist; for then the work ( — resistce X dist) is
independent of the vel. But in a given Hme, the dist (and consequently the
work also) of course varies as the vel.

188. (a) Some kinds of surfaces appear to Interlock their projections
much more perfectly when at rest relatively to each other, than when in even
very slow motion ; and in some cases the degree of interlocking seems to in-
crease with time of contact. Hence there is often a great diff in amount between
fric of rest and fric of motion. Thus, Gen'l Morin found that with oak upon
oak, fibres of the two pieces at right angles, the resistce to sliding while still at
rest, and after being for " some time in contact," was about one eighth greater
than when the pieces had a relative vel of from 1 to 6 ft per sec.

(b) But experience shows that even very slight jarring suffices to remove this
diff; and since all structures, even the heaviest, are subject to occasional jarring^
(as a bridge, or a neighboring building, or even a hill, during the passage of a
train ; or a large factory by the motion of its machinery ; or in numberless cases,
hv'the action of the wind) it is expedient, in construction, not to rely on fric for
stability any further than the coeff for movina fric will justify. When it is to bs
regarded as a resistce, which we must provide force for overcoming, it should be
taken at considerably more than our tabular statement.

• See hla "Fundamental Ideas of HeohaniM", tranilatad br Joe. Bennett; D. Appltoa Jt GSb
New York. 1860. » -.-r • *.^

FRICTION.

411

Table of moTintr tMetlon, of perfectly smootb, eleAn, and
^bry, plane snrfoees, chiefly from Morln.

MaterUlB Experimented with.

«

tl

«

O«kono«k; all the llben parallel to the motion <

moTing fibres at right angles to the others ; and to the motion. . .

all the fibres at right angles to the motion

moving fibres on end ; resting fibres parallel to the motion

cast iron , fibres at right angles to motion

Blm on oak, fibres all parallel to moUon

Oakonelm, •• " "

Kim on oak, moTing fibres at right angles to the others, and to motion

Ash on oak, fibres Hi parallel to motion

Vironoak. " " " «

Beeohonoak " " " "

Wrooght iron on oak, fibres parallel to motion

Wrenght iron on elm, " " " ••

Wronght iron on east iron, fibres parallel to motion

" " on wronght iron, fibres all parallel to motion

Wreoghtiron on brass

fTrongb t iron on soft limestone, well dressed ] '

*• *• " bard " " *'

•' " or Steel on hard marble, sawed. By the writor.V.V.V.V.about! !
*' " " " *' smoothlj planed, and nibbed mahogaaj-, fibres par-
allel to motion

" " " •• " smoothly planed wh pine !!*.".!!*.!!!".*'.!

Oaet iron on oak, fibree parallel to motion

i« II «< ^im •< «i •< II

" *' *• oast iron *.*.*''.'.'.'.*.*.**.'.'.'.*.!!*.!!I*!!!*.l!!!!!***

M It II 5rn„

Steel on east iron

8teelon steel. By the writer

Steel on brass

Steel on polished glass. By the writer about..

" qnite smooth, hot not polished; on perfeotly dry planed wh pine, fibres

parallel to motion about . .

** quite smooth, but net polished; on perftotly dry planed and smoothed

mahogany, fibres parallel to motion about..

TeUow oopper on oust iron

" •• onoak

ferass on oast iron.

" on wronght iron, fibres parallel to motion

" on brass

" on perfectly dry planed wh pine, fibres parallel to motion about. .

** " " " and smoothed mahogany, fibres parallel to mo-
tion .•..••.••...•.....about..

Polished marble on polislied marble. By the writer Arerage.

" '* on oommon brick "

Oemmon briek on common brick «

Soft limestone well dreesed, on the same

Common brick, on well-dressed soft limestone

•• •• •• " •' hard "

Oak aflross the grain, onsoft limestone, well dressed..*...! !1!'..'! ill. i!!!!!*.!!!
II 11 «• >• II bard " " "

Hard limestone on hard limestone, both " •' !!!!!!!!!!!!!!!!!!!!!!!!!!

•4 •• <• gof( II II It II

Soft " " hard " •• «• " !!*!!!!!!!!!!!!!!!!!!!!!*.!

Wood on metal, generally, .2 to .62 mean..

Wood, ««nr smooth, on the same, generally, .25 to .5 « ..

Wood, " '* onmeul, •♦ .2 to .62 •' ..

Metal on meul, very smooth, dry '• .15 to .22 " ..

Masonry and brickwork, dry " .6 to.T " ..

" •• '• with wet mortar about..

slightly damp mortar .

II
ii

n
II

II

fl

II

M U II

" ondry clay

" " moist"

Marble, sawed ; on the same ; both dry. By the writer ••• . . .

both damp " ••••...,

on perfBctlT dry planed wh pine. " . .•

on damp planed wh pine " .......

potislMd, on perfeotly dry planed wh pine '*

VThlte pine, perfeotly dry ; planed; en the same; all the fibres parallel to

motion.. about..

" ** damp, planed ; on the same •« ..

•4
«l

it
II
II
jH
II
II
II
II

Coeffof
Frio; or
Propor-
tion of
Frio to the
Pres.

.48
.32
.34
.19
.37
.43
.25
.45
.40
.86
.36
.62
.25
.19
.14
.17
.49
.24
.80
.17

.18
.16
.49
.20
.15
.15
.20
.14
.15
.11

.16

.18
.19
.62
.22
.16
.20
.19

.24

.16

.44

.64

.64

.65

.60

.88

.38

.38

.67

.65

.41

.38

.41

.18

.65

.47

.74

.51

.33

.4

.55

.45

.6

.6

Angle of
Fric

Dec. Min.

& 38

17 46

18 47
10 46

20 19
28 17
14 8
24 16

21 49

19 48
19 48
31 47
14 8
10 46

7

9

26

58

39

6

18 80

16 42

9 89

10 12
9 6

26 6

11 19
8 32
8 32

11 19

7 69

8 82
6 IT

9 6

10 li

10 «

81 48

12 25
'9 6

11 19
10 46

13 80
9 6

28 46

82 88

82 88
S3 2
31 00
20 48
20 48
20 48

83 60
33 2
22 18

20 48
22 18
10 12
83 2
» 30
36 SO

27 00
18 15

21 49

28 49
24 14
81 00

14 36

21 48

31 00

* But after a few trials the surfaces become so much smoother as to reduce the angles as much as
from 2<> to 6<' ; the sliding blocks weighing about 30 Tts « -4x

■b gnatar 'irlMJaiii ot pres M)d or
'" "'"" " -"--'-' ^haD^» Id the cop*^
elawa in f i

«vjn In the danipneBaof the air, will often ciuae much gre»ter ohBOMB of ;
ilie Itmlu of alirnsion, we may genetaliy laka Morln'i rula» u luffleientij

" E
TtvlllbeBeeu thititic

Tt vll] be Been that at low vpIb the co^dpcrpued when tbc prn pemqln waa
aliDHt Imperceptiblr Inereaned ; but this dlff diuppearcd u tbe vel Increaaed.
At •«]■ fiwn 4 ID iW Ina per kc, the coeff gsoerally deoreiHd aa tbe tei in-
ereiuied ; rapidly at flrtt.but more alow ly a» the Tel becaniB greater. This asRea
wllh other recent eipls. But at verj low .els (.08 lo S ins per »ec) Prof. Kimball
IbuDd the ooeff [llpo E) (7«7-saii~o ttry rapHUy urMlheaH.

are go Blight thai they would otherwise be KamstT peroeHlbla. Lm JelleaM
azpti would haie failed to show Uiem at aU.

IBl. <■) In ins Onpt. DoBgliu ClKltan>iidMv.<l«ornWMt*
Inriioiiav, Jr.* made carpful a<p«HiD0nia In BDaland to aaHrtaln the «Aal4l
Mctlon In cannectlDn with nllwur lmkea.t The IhcllDii and pruHBra ■«•

vpcDdoDtDt eat PTfi. Batlr hi(h piHwMiiw IM
«aki 4*vn Uw'pgglMtloM whil* Uw lowir — mrt

Ur 1 (|T« WBI pm, tut, »t MkKlU >k> tmnnltf

irvx

PRICTION,

tifel-ttnd wcndBD wIimIi, «U looh» tn «...»<:„„ >,.

H abon In Fte. 4.

mis poinU bi lint* A, B Uld C sboir the STenge bra , ..

Ing fric beiwcCD ih« tnmi of ■ nUing wheel md ibe ftra*»UiKt.
S»e*4 arOi

range bnutf coeOk. or co«lb of lU

Une A elunn bnke saefi obulned Immed'T alter ■ppliaMlaa ef bnke

" B ■• " nspc.

" C " " 15 " " '

■• D show* raU ooeffl. or ooeffi of iHdlng frlc betireep the treid of ■ dU-
4HOT"ijHdd>ii^" wheel Ibeld rut brtbebrBtelund the mif.

(b) From Itnai A B end C II appeHri thsi the brshe coeff obtHlned it ■
ElTan jeagth of time ifter the appUcitloa of the brake wu Keaertllj creator
at l*wlluu> M hlcbTeU. But where the rel wtii malDtelnefunirorm
U>< brake e*eff dloilBlihed aa block aiMl wb««d remained
lonnr In contact. Tbua, llnee A sod B show chat at S^^ miles per hour
(he tiralie gaeirwa>.lS4 when itae brake wu fint applied (ddIhI a), but fell !•■
JIM In S wa (i). Liae A (immed'T after application) ihowi a bleber brake coefT
LlNttfiuttimaai than llneS(5 sees artcr applkstlon) Bhone al ST^ ullea

The dlralniitlan of tliara<J coeffwltb leoEtti of time of appllcalloii of brakes

crease of coefT) becomes - the" adbnloD" or static frlc belween the rail and
the tire of the n>)llng irheel, the tel of rotattan rapldW falls below [bat di>e to
tbHrslar thecar; if, (he wheel begins to ''■kid" or slide along tbs
nit ; and Id from .79 to Slecs the rotation of the wheel ceaeee entirely.

(d) Ttae rail coeN*. line D. la cenerallT mncta lem tban tbc
krake coeir, Unei A, fi Bad C. The preson the rail ( — the wl on (-'-—'>
«>■ about GOOa DM per an In, orfreail; in axoMi of (be limit of abrailon
at the brake was about lOD In per h in. A few eipta were made wKli
BlockibaTlngbiitloftb. «~ "< - i~r . _ .

•anHH anr markM ehanee Id "the
(C) Tbcralleaeir.TlnsD.U

brake eoelf J uat before iklddlag. With steel tlrrs obVon rails at hieta reli It wu
•ainewbit greater than on iteel ralli, but tbli dlffdllappeared u the rel diiala>

4 per eq in. A few expta wen

UBl area of contact, and therefi .
er u iDunder atiien total prei. Thej failed to sliow conclualiely that tbis
Miaed any marked chanee '~ "~~ —-"

(f) IiOcomoMwca

■Ilthednfe»:i>,tlie

■odlffttaataslml

The pree per tq in thin (traatlr eiceeds nut only that upon which the tablea are
b>Kd. bnt also the limit of abruioa. Baidei, an; point in ttae (r**d, during

■odlfftliat a similarity Id their coeO^ could hardiT be expected. Ttae great wt.
wn from 2 to e orereD T tuna, on a drlrer. Is concentrated on aaurf Iwliere Ifae
wheel tonches the rail] ahout 2 ins long X about ) Inch wido, or — saj 1 »q in

414

FRICTION.

tfae instant when it is acting as the fulcrum for the steam pres in the cyl, ia
stationary upon the rail. Its fric (miscalled " adheetou ") is therefore Uatie.

Capt. Galton found that tlie C€»«flr of *' adbeslon *' was independent of
the Tel, and depended onlv on the character of the surftt in contact. With »
four-wheeled car having about 5000 lbs load on each wheel, it was generally over
.20 on dry rails ; in some cases .25 or eyen higher. On wet or greasy rails, with-
out sand, it fell as low aa .15 in one case, but averaged about .18. Witli sancl
on wet rails it was over .20. Sand applied to dr^ rails before starting gave .35
and even over .40 at the start, and an average of about .28 during motion ; but
eaad applied to dry rails while the car was in motion was apt to be blown away
by the movement of the car and wheels.

(s) Owing to the constancy of the coeff of '* adhesion " under given conditions
of tire and rail, the brake fric necessary to "skid" the wheels in any case was
also practically constant for all vels. But at high vels, owing to the lower brake
coeffi a higher brake pre* was reqd to produce this fixed amount of brake frit
The skidding also reqd a longer time than at low speeds.

192. If the pres is sufficient to produce abrasion (indeed, while it is
much less) the fric often varies greatly, but no jorecise law has yet been discov-
ered for estimating it. Rennie gives the following table of coeflb of fi*ie
of dirf surfaces, under pressures ipraaually increased up to
tbe limits of abrasion. It will be noticed that in tbis table tbe
coeff i^eneralljr increases with the intenHty of the pres :

Coeflb of friction of dry surfaces, under pressures grad-
ually increased up to tbe limits of abrasion. (By G. Rennie, C E.)

Prei. in Lb*.

W roaght Iron

Wroaght Iron

Steel

Bna

per

on

on

on

on

Sqaare Ineh.

Wroaghtlron.

Gut Iron.

Cut Iron.

OMt Iron.

32.6

.140

.174

.166

.167

186

.350

.275

.800

.396

334

.871 .

.293

.388

.319

886

.813

.833

.847

.315

448

.876

.365

.854

.206

660

.409

.367

.858

.388

673

.376

.408

.888

709

.434

.884

784

.883

831

.378

193. (a) RoUinir friction, or that between the circumf of a roll-
ing body and tne surf upon which it rolls, is somewhat similar to that of a
{dnion rolling upon a rack. In disengaging the interlocking projections, or in
ifting the wheel over an obstacle o. Figs 5 and 6, the motive force F, instead of
dragging one over the other, as in Vic 76, p. 407, acts at the end of a bent lever
F B W Figs 0 and 6, the other end W of which acts in a direction perp to the
contact surf; and in practical cases of rolling fric proper the leverage B W of
the resisting wt of the wheel and its load is very much lees, in proportion to
that (FB) of the force F, than in our exaggerated figs. Hence the force F reqd
to roU a wheel etc is usually very much less than would be necessary to MlidtvL

(b) There are nsuallv two ways of applyiuf^ tbe force in overoom-

Ing lolling fric : Ist (Fig 5) at the axit of the rolling body ; as the force of a

horse is applied at the axle of a
wagon-wheel ; or that of a man at the
axle of a wheel-barrow : 2d (Fig 6) at
the ciremr\f; as when workmen pusli
along a heavy timber laid on tm> of
two or more rollers ; or as the en<M of
an iron bridge-truss plav backward
and forward t>y contraction and ex-
pansion, on top of metallic rollers or
balls (p725). in Fig 6 we have, in ad-
dition to the rolling fric of the dr-
cumf of the wheel on its sapport« the
sliding fric of the axle in its bearing.
In Fig 6 we have only rolling tno.

iTig.e

but at both top and bottom of the wheel.

(c) When the obstacles o are very small, as in the case of cart-wheels on
smooth hard roads, or of car-wheels on iron or steel rails, the leverage

FBICnON.

415

PTE) of F becomes, praotioally, In Fig 0 the radkut and in Fig 6 the diam, of the
wheel; while that (RW) of the resistce is Tery small. Hence, neglecting axle
fric in Fig5, the force F reqd to overcome rolling fric in such cases is directly
u the wt W of and on the wheel, and Inyersely as the diam of the wheeL

The few expts that hare been made npon the coeffs of rolling fric, apart firom
axle firic, are too incomplete to serre as a basis for practical rules.

(d) The fric (or ^^ adhesion") between wheel and rail, which enables A
locomotiye to moye itself and train, or which tends to make a car-wheel rerolTe
notwithstanding the pros of the brake, is a resistce to the sliding of the wheel
on the rail ; ana is therefore not rolling but Hiding fric ; ttaiie when the wheels
either stand still or roll perfectly on the rails ; and kinetie when they dip or
"akid".

1S>4* Tlie frletlon of liquids moving in contact with solid bodies
is independent of tlie pressure, because the "lifting'' of the particles
of the fluid over the projectioos on the surf of the solid body, is aided by the
pres of the surrounding particles of the liquid, which tend to occupy the places
of those lifted. Hence we have, for liquids, no coeff of fric corresponding with
that {=» resistce -t- pres) of solids. The resistce is bslieved to be directly as the
area of surf of contact. Recent researches indicate that Besistce = a coeff X
area of surf X vel» in which both n and the coeff depend upon the vel and
opon the character of the surf: and that at low vols n => 1, but that at a certain
"critical ** vel (which varies with the circumstances) n suddenly becomes a> 2,
owing to the breaking up of the stream into marked counter currents or eddies.
The resistance of fluid trie arises principally from the counter currents thna set
in motion, and which must be brought into compliance with the direction of
the force which is urging the stream forward.

IMS. Table of eoeilleients of mowinir flriction of smooili
plane snrCsces, wben kept perfeetly Inbrleated. (Morin.)

SabatanoM.

Oftkonoak. flbrw parallel to motion....

» <• !• flbrM perpendicular to motion

" on elm, ilbrei parallel to motion

" on east iron, flbrei parallel to motion

*' on wToaght iron, " '* "

Beech on oak, ibroa «< " «' w

Ilmonoak, " " " "

'*onelm, " " " ••

•• eaatlron, " " " "

▼rongbt Iron on oak, flbree parallel, greased and wet, .256.

" " '* *• flbree parallel to motion

« • «* onelm, *' •* " "

" « onoaetlron, " " ",

** «• on wroDgfat iron, " ** "

" *• on braH, fibres " " "

Oaet ifon on oak, fibres parallel to motion

.4 <• «• M 4« " «. .4 greased and wet, .318

'* *• onelm, " " " "

" ** on east iron, with water, .814

" ** onbrass •

Coppar onoak, fibres parallel to motion

Twow ooppor on east iron

Brass on oast iron

' * on wroagh t iron

" onbrass....

Sled on oast iron

" on wrought iron

" onbrass •

Tninod oxhide on oast iron, greased and very wet, .365

" " onbrass

** " on oak, with water, .29

Dry

Olive

Soap.

OU.

.164

oe • •

• • • •

• o • •

.186

o* ••

• • ■ •

ooae

• « • •

O O 0 0

.187

• • • •

.138

• • • •

• • • 0

.214

• • • •

• • • •

.055

• • ■ •

.066

• • • •

.070

■ • • •

.078

.189

• • • •

.076

• • • •

.061

.197

.064

• « • •

.078

« • • •
■ • • •

■ • • •

.066

■ • * •

.077

• • • •

.072

• • • •

.058

• • • •

.079

• • ■ •

• • • •

• • • ■

.053

• • • •

.133

• • • •

.191

Tal.
low.

.075
.063
.073
.080
.006
.066
.070

.006

.066
.078
.103
.062
.103

.078
.077
.100
.103

Lard.

.072
.066

.081

.067
.072
.066

.060

.076
.076
.061
.076

.076

.076
.075

.068

.105 .061
.098 ' .076
.066
.159
.241

.601
.06ft

.089

.067

Tike laitnelkliky ftictton of the wooden fdgate Princeton was fimnd by •
«oinmittee of the Franklin Institute in 1844, to average about .067 or one-flft«enth
«f the pressure during the first .76 of a second and .022 or one forty-fifUi for tike
hext 4 seconds of her motion. The slope of the ways was 1 in 13, or 4 degrees 24
hiinntes. They were heavily coated with tallow. Pressure on them =- 16.84 lbs.
per f quare inch, or 22S0 B>8. per square foot In the first .75 of a second the vessel
•Hd 2.5 inchest in the next 4 seconds 16 feet 6.5 inches; total for 4.76 seconds 16.7§

416

FR1CTI0(N.

196. The frl«tloa «f InbrtMiied surfllwes Tarlefl greatty vitli
tbe character of the slirfs and with that of the labrioant and the manner of lit
application. If the lubricant is of poor qualitj, and scantilj and unevenly ap'
plied under great preo, ft may wear away in places and leave portions of the dr?
surfs in contact. The conditions then approximaM to those of unlubrioated
surfaces. But If the best lubricants for the purpose are used, and supplied rea-
ularly and in proper quantity, so as to keep the surfs always perfectly separated,
the ease beootnes praotically one of liquid friction, and the resistce is very
small. Between these two extremes there is a wide range of variations (see
table, 1L.197 id)\ the coeff being affected by the smallest change in the condi-
tions. Where any degree of accuracy is reqd, we would refer the reader to the
experimental results given in Prof. Thurston's very exhaustive work,* devoted
exclusively to this intricate subject-.

197. (a) Expts by Mr. Arthur M. Wellington upon the pc%e Of
labrleateo Jenraalsf gave a gradual and continuous inereitse or coeff af.
the vel of revolution diminished from 18 ft per sec ( s»= a car speed of 12 mflet
per hour) to a stop. This increase was very slight at high veh, but much more
rapid at low ones ; as in Figs 3 and 4. At vels from 2 to 18 ft per sec the coeir
was much less under high pressures than under low ones; but at starting there
was little diff in this respect. The coeff increased rapidly as the tempera*
tare rose from 100<=> to 12XP and 160^ Fahr.

Thurston, also experimentina with InbrieateA Jo«nimls,t

at starting, the coeff increaseq with increase of pares, as it did also

(b) Prof,
found that

when in motion, if the pres greatly exceeded the max (say £iuO to 600 fi)s per so
in) allowable in machinery. He also found that at high vels the coeff incrwsM
very slowly (instead of continuing to decrease) as the vel increased.

(e) Prof. Thurston gives the following «MlPOX fennvlae fei* JowihA

f^i^tion at ordinary temperatures, presaures and spdeds, with Jotiriial and
bearing in good crtndinon ahd weU lubneated:

CoeflT for fttarttnff » (.015 to .02) X ^pres in 9>s per s<| in.

Coeff when the shaft / ^^ 4^ «o\ w 1^ vel in ft per min

is revolvlniT -(•<» ^ '^) X .s..--, ^ ft. _,, ^ .„

l/^ pres in ids per sq id«

At pressnrea of about 200 l>s per sq In :

Temp*r.tare ofmtatm^m _ ,, ^ ,^,,1 in ft pe, ^<»

Cantlon. Tbe leweracgr^, with which Journal frlc resists mo^MB, la-
creases witb tbe diam of the JoumaL *

(d) Tbe following figures, selected from a table of experimental results given
by Prof. Thurston, merely show tbe extent to wbicb tbe coeff of
Journal fSrle Is affected by pres, wet and temperature; and

hence the risk incurred In rigidly applying general rules to such casea. la
these expts the character of Journal and bearing, the lubricant and its method
of application, remained the same throughout, where these vary, still further,
and much greater, variations in the coeff may occur.

Steel Journal In bronse bearing, lubricated wfltb stamdwd

sperm oil.

m ■■ II.. ■ 1,11.1 ^

1^

9J

130O
909

Speed of rewblatlon

80 feet per minute I TOO feet per minuie Isoo ft per mifJlSOO ftperanin

200 100 I 4
lbs per sq in

Coeff

.0160

.0056

Coeff

.0044

.0081

Coeff

.126

.094

200 100 4
lbs per sq in

C^eff Coeff
.0087 I .0019
.0040 1 .0019

Coeff

.0630

.0680

200 100
lbs per sq In

Coeff

.0058

.0076

Coeff

.0087

.0061

200 I ido
lbs per sq ia

Coeff

.0065

.0100

Coeff

.0075

.0160

• Friction and I^ont Work In MMchlnerr iind Mill Work. John Wltoy * Boni. New York.
t Tr«i.« Amer Soo of Thil Knar*. New York. Oeo. 1884.
t J(tnrn^l of the hVtinklli) 1if<tltiii'. Jhoff* ItfM.

i

FRICTION.

417

(•) Wliere tlie force to applied first on one side of the Jour-
nal and tlien on tlie opposite side, aa in crank pinsi the fric is less
than where the resultant pres is always upon one side, as in fly-wheel shafts;
because in the former case the oil has time to spread itself alternately upon both
aides of the journal.

(f) Friction rollers. If a journal J, in-
stead of revolving on ordinary bearings, be sup-
ported on friction rollers R, R, the force required
to make J revolve will be reduced in nearly the
same proportion that the diam of the axle o or
o of the rollers, is less than the dlam of the
rollers themselves.

Mr. Wellington experimented with a patent
hearing on this principle, invented by Mr. A.
Higley. Diam of rollers BB, 8 ins; of their
axles 0 0 If ins ; of the journal c, 3| ins. Here,
theoretically,

r' c . .. I 1 /•! i>oi*i...iv> diam of axles 00 Iflns

fric of patent journal — fric of 34 m journal X r- ? — n s^ — -^: —

*^ -^ s ' diam of rollers B B 8 ms

or as 1 to 4.6. Under a load of 279 lbs per sq in, Mr. Wellington found it about
as 1 to 4 when starting from rest; and about as 1 to 2 at a car speed of 10 miles
per hour.

198. (a) Resistance of railroad rolllne stock. This con-
siats of rolling fric between the treads of the wheels and the rails (the treads
also sometimes slide on the rails, as in going around curves) ; of sliding fric be-
tween the journals and their bearings, and between the wheel flanges and the
rail heads; of the resistce of the air; and of oscillations and concussions, which
consume motive power by their lateral and vert motions, and also increase the
wheel and journal fries.

Its amount depends greatly upon the condition of the road-bed and rails (as
to ballast, alignment, surf, spaces at the joints, dryness etc); upon that of the
rolling stock (as towt carried, kind of springs used, kind and quantity of lubri-
cant, condition and dimensions of wheels and axles etc) ; upon grades and curv-
ature; upon the direction and force of the wind; and upon many minor con-
siderations. Experiments give very conflicting results.

(*) During the summer of 1878, Mr. Wellington experimented with
loaded and empty box and flat freight car% passenger and sleeping cars, and at
speeds varying from 0 to 35 miles per hour. The cars were started roiling (by
giav) down a nearly uniform grade of .7 foot per 100 feet, or 36.6 feet per mile,
and 6400 ft long. Their resistces were calculated as in f 185. "The rails

were of iron, 60 lbs per vd, and the track was well ballasted and in good line and
surf, but not strictly first class." The following approx figures are deduced
from Mr. Wellington's expts upon cars fitted with ordinary journals:*

Car Resistance In pounds per ton (2240 lbs) of weight of
train, on straight and level track in good oondition.

Speed of

Empty pars

Loaded can

train in

mOesper

hour

Axle,

tire and

flange

Oscilla.

tion

and

con-

cuss'n

Air

Total

Axle,

tire and

flange

Oscilla-
tion
and
con-

cuBs'n

.Air

•
rD>tal

0
10
20
SO

14
6
6
6

0

.6

2.7

6.3

0

.4
1.3
2.7

14

7

10
14

18
4
4
4

0
.6

2.

4.7

0

.4
1.
2.3

18
6
7

11

(c) With the Higley patent anti-fric roller journal, the resistce to «to7fin^ was
bat about 4 lbs per ton. '

(d) About midway in the track experimented upon, was a cnrre of !<' de-
lection angle (6780 ft rad) 8000 ft long, with its outer rail elevated 3 to 4 ins

— ' . _ . ^_ _ _

• Tranaaottons, American Society of Civil Enelneers, Feb 1879.

27

418

FRICTION.

above the inner one. The rise of the outer rail was began on the tangent, about
000 ft before reaching the curre. In the first 600 ft of the curve the reeistce was
greater than that encountered just before reaching the curve, bv from .6 to 2.1
(average i.l) lbs per ton. In the last 600 ft of the curve this excess bad diminished
to from .2 to .9 (average .6) lbs per ton. Owing to the continuance of the down

Srade on the curve, the vel increased as the train traversed the curve ; but it
oes not clearly^appear whether the decrease in curve resistce was due to the
increase in vel, or to the fact that the oscillations caused by entering the curre
graduidly ceased as the train went on.

(e) Mr. P. H. Dudley, experimenting with his *' djrnair>*»plK ''* ob-
tained results from which the following are deduced:

Vraln Beslatanee in pounds per ton (2240 lbs) of weigrlit otf

train, inelndintr fri^ades.

Description of train

Lf>aded
cars

79
87
25

Empty
cars

S
0
2

Weight tons
(2240 lbs)

036
688

458

Trip

Toledo to Cleve-
land. 95 miles

Cleveland to Erie
95Ji miles

Erie to Buffalo.
88 miles

Average
speed.

Miles per
hour

20
20
20

Average
resist-

8JM
7.67
8.89

**With the long and heavy trains of the L. S. & M. S. Rv, of 600 to 650 tons, it

nl less fuel with the same engine to run trains at 18 to 20 miles per hour than
id at 10 to 12 miles per hour", owing to the fact that at the nigher speeds
steam was used expansively to a greater extent, and hence more economically.

190. The work, in ft-lbs, reqd to owereome fric through
any dist, is = the fric in lbs X the dist in ft. In order that a body.started slid-
ing or rolling freely on a hor plane and then left to itself, mrij do this work ; is,
may slide or roll through the given dist, its kinetic energy ( = its wt in lbs X its
vel< in ft per sec ■*- 2^) must ^ thefir«t«named prod, (inversely, tlie dist
in ft through which such a body wm slide or roil on a hor plane, is

its kinetic energy in ft-lhs, at start
fric in lbs

wt of body in lbs X Initial vel* in ft per sec initial vel* in ft per see
"" wt of body in lbs X coeff of fric X 2Vt "" coeffof fric X 2yt

The time reqd, in sees, is '^

dist in ft

dist in ft, so found
mean vel, in ft per sec \ initial vel in ft per

Suppose two similar locomotives, A and B. each drawing a train on a levri
straight track ; A at 10 miles, and B at 20 miles, per hour. The total resistce of
each eng and train (whicli, for convenience, we suppose to be independent of
vel) is 1000 lbs. Hence the force, or total steam pres in the two cyls reqd te
balance the fric and thus maintain the vel, is the same in each eng.' In travel-
ing ten miles this force does the same amount of work (lOOO lbs X 10 miles
H 10000 pound-miles) in each eng, and with the same expenditure of steam in
each; although B must supply steam to its cyls twice tu jaat as A, in order to
WMintain in them the same pres. It? one hour the force in A does 10000 lb-miles
as before, but that in B does (1000 lbs X 20 miles = ) 20000 lb-miles, and with
twice A*s expenditure of steam.

But in fact the resistce of a given train is much greater at higher vela. See
table, f 198 (6) And even if we still assumed the resistce to be the same at
both vols, B must exert more force than A in order to acquire a vel of 20 miles
per hour while A is acquiring 10 miles per hour.

* An iHBt for meuaring the atrain on th« draw>bar of a looomotf t«, or Uit AVM whieh Um latMl
•xerti Qp«n the train,
t ^H aeeeleratlon of gruyitj k say 83.2 ; Syas tuj 64.4.

LEVERS.

419

200. Natural Slope. When granular materials, as sand, earth, grain,
etc., are deposited loosely, as when they are shoveled from a cutting or
dumped from a cart, the angle, formed between a level plane and the sloping
surface of the pile of material, is called the natural slope. This angle de-
pends upon the friction and adhesion between the separate particles of the
material, and often varies, in one and the same material, from time to time,
with changes in weather conditions, etc., especially with dampness.

Fl«. 85.

201. Any force, p, Fig. 85, actins^ upon a body, B, will suffice to move the
body (see foot-note (*), If 1), provided it exceeds the sum, S. of all resist-
ances, including friction between B and the surface upon which B rests, or if
it forms, with any other force or forces, P, a resultant, R, greater than S.

If, before the application of p, the body is already in uniform motion, P is
"■ S; and any force, p, however small, will suffice to change the direction of
motion. This accounts for the ease with which a revolving shaft may be slid
longitudinally^ in its bearings, and for the fact that a cork may be more
easuy drawn if we first give it a twisting motion in the neck of the bottle.

202. Classes of LeTers. Figs. 86. Levers are classc according to
the relative positions of ' ' power, " * " weight " * and fulcrum iS follows :

K3

fr

"l

wO

(Ot)

)w

-l-

iTlL

'5)(W

B

w

w

Jt

I

?r

(C)

Class • 1 . Fulcrum R between power w and weight W ;
" 2. Weight W between power w and fulcrum R ;
" 3. Power W between weight w and fulcrum R.

In class 2, the leverage of the power is necessarily greater than that of the
weight. In class 3, vice versa.

203. In Fig. 86, taking the moments of the forces about any point at
pleasure, as o, we have, for equilibrium :

Fig. (a), W . ^w — R 2r + «? . iw = 0;
Fig. (6), W . Zw — R Zr - «» . iw = 0;
Fig. (c), W ../w — R /r + u? . Zw •= 0.

* When levers are used for lifting weights or for overcoming other resist-
ances, the force applied is called the "power," and the resistance to be over*-
eozne is called the 'weight."

420

STATICS.

!804. Compound leverst Fig. 87, may be used where there is not room
for the arms of a single lever of sufficient length. In a compound lever,
neglecting friction,

\ weight product of lengths of power arms ^ 8 X 10 X 2 _ 160 __ ^q

power product of lengths of weight arms 2X1X4

8

The three levers of Fig. 87, taken separately and beginning at the power
end, give:

weight 1 .^. 10 _ ,0.1 _ 1.

power "2 *' 1 "' 4 2*

and 4 X 10 X ^ =■ 20. as before.

ir-»o

i

■10

"T

-&-

I

Fig. 87.

Fly. 88.

^^05. Toothed or Cos Gearing. Wheels and Pinions. Fig. 88.
These are a series of continuous compound levers. The power is usually ap-
plied to a crank, c, and the weight is attached to a drum, d. The larger
wheel, w, on a given shaft, is called the wheel ; the smaller one, p, the pinion.

Let c = the radius of the crank, d = that of the drum, m » the product
of the radii of the {pinions, and n = the product of the radii of the wheels.
Then, neglecting friction,

weight c . n

power

m .d

Instead of the several radii, we may of course use the corresponding diam-
eters or circumferences; and, as the teeth are necessarily of eaual pitch"
(length, measured along the circumference), the number of teetn on a wheel
or pmion is usually taken instead of the radius.

When the ratio, — ~ — , is great, the s3rstem is said to be of hiarh sear.
power

When that ratio is small, we have low gear.

Compound levers and gearing are used for converting low into high veloc-
ity, as well as for lifting great weights by means of small powers. When
used for increasing the velocity, the positions of power and of weight are the
reverse of those shown in Figs. 87 and 88.

306. Whenever the power and the weight balance each other, either
in a single lever, or in a connected system of levers or leverages, of any
kind whatever, then if we suppose them to be put into motion about the
fulcrum, their respective velocities will be in the same proportion or ratio
as their leverages; that is, if the leverage of the power is 2, 5, or 50 times as
great as that of the weight, the power will move 2, 5, or 50 times as fast
as the weight. Therefore, by observing these velocities,, we may determine
the ratio of the leverages. The weight and the power are to each other,
therefore, inversely as their velocities, as well as inversely as their leverages.

S07. No mechanical advantage is gained by merely increasing the lentfth
of a lever, as by curving it, as at abo. Fig. 4, H 13, or by giving it an in-
clination to the line of action of the power, P, as at o m. o u or o n.

LEVERAGE.

421

208. Thus, in Fig. 89, representing a bent lever, a f b, the length of the
lever, or of any of its members, as f b, must not be confounded with the arm
or leverage of the force acting upon the lever. These may or may not be
equal. Thus, the member / o is much longer than the member / a; yet, if
the arms, / a and / c, of the forces or weights are equal, the weights n and m
must also be equal in order to insure equilibrium.

Tig. 80.

209. If the weight m be removed, a force c, or a, or y, or d, with leverage
= c', «', y^, d\ respectively, may be applied at any point, as &, to balance the
moment of n. In any case this force must be such that

force X its leverage -• n.a f.
Hence,

n . a f

force —

leverage of force'

210* Hence also the force required is leaet when, as at y, it is perpendicular
to the length of the member /b; for the leverage (which evidently cannot ex-
ceed f b) is then greatest. The force required increases as it deviates in
either direction from the line & y (perpendicular to / 6) and approaches more
nearly to the direction of / 6 itself; for its leverage then constantly decreases.
No force, however great, could balance the moment of n about /, if applied
in the direction / &, or b /; for such a force would have no moment- about /.

211. Similarly, in Fig. 90, the moment, about a, of a load W, placed at 6,
is — W. a c or the same as if it were placed at c, and not » W.a o.

Fiff. 00.

Flgr. 91.

212. In Figs. 91, also, the moments W. o « and W. o' e', of the equal
weights, W and W, are equal. But if forces, p and p', be applied in direc-
tions perpendicular to the longer beam, o t, the leverage o t oi p becomes
about 6 times that (<K tO of p^. Hence a force, p, applied at t, has about the
same bending moment as a parallel force *» 6 p, applied at e'.

422

STATICS.

STABILITY.

213. Stability. Figs. 92. If the resultant, R, Fig. (a), of the force P
and weight W, falls beyond the base, as shown, then the overturning moment
of P, Fig. 92 (6), about the toe n, will exceed the moment of stability of the
weif^ht W about the same point, and the body will overturn about n. If not,
it will stand. •

314. Assuming stability against overturning, the bod^ will slide if the
horizontal component, k, of R, Fig. 92 (a), exceeds the f notional and other
resistances.

215. In practice, the toe, n, or the ground beneath it, might yield if the
stone revolved upon it, or if R fell near n (see«1[1[ 145, etc.); but this is a
question of strength of materials. Cement, clamps, etc., between the base
and the ground, would add a third force, and thus change the problem.

(a)

(6)

Fig:. 92.

h-«r-*t^«a

Fls. 98.

Tig. 94.

216. Owing to the greater leverage, l^,. Fig. 93, of W about a, the moment
of stability is much greater about a than about b.

217. In Fig. 94, let G — 2 lbs.: g =• 1 lb.; leverages = 3, 4 and 6 ft., as
shown. Then the moment of stability of the rectangular body, G, against
a horizontal force, P, is = 3G = 3X2 = 6 ft.-lbs. ; and the moment of the
lower triangular body, g,ia = 4g = 4Xl= 4 ft.-lbs.; so that, although the
larger body weighs twice as much as the smaller one, yet its moment of
stability is only 1.5 times as great.

218. Work of Overturning. In Figs. 95 (a) and (6), let the shaded
portion of each figure be of lead, and the remainder of wood, and let the
center of gravity of the entire body, in each case, be at G. Then, since the
weight, W, is the same in both cases, as is also its leverage of stability, about

o, = ^, the moment of stability, « — &.W, is the same in both cases, as is

also the force, P, required to balance that moment when applied at a given
elevation, e. As overturning proceeds, the weight, W, remaining un-
changed, the leverage and moment of stability, and the overturning moment
required, decrease, becoming » 0 when the bodies reach the positions

STABILITY. 423

shown by the dotted lines. If the elevation, e, remains constant, the force,
P, required for overturning, decreases in the same proportion as the lever'
age, etc.

219. But in order that the bodies may be overturned by the force of grav-
i]^ alone, they must be brought into the positions shown by the dotted unes.
Tnis requires that the weights of the bodies be lifted through a height >- the
distance, h, through whicn their centers of gravity, G, are raised:. Hence

work of overturning — Vf.h.

•

Since h is greater in Fig. (6), the work of overturning is greater in that case.

In civil engineering we are generally concerned with the amount of the
force which will begin overturning, rather than with the amount of work
required to complete the overthrow.

2!30. Stability against overturning is of course affected, and may be in-
cr^tsed, by forces other than the weight of the body itself. Thus, the
stability of a bridge pier is ordinarily increased by the weight of the bridge
itself if this be brought upon the pier symmetrically. Otherwise the weight
of the bridge may either increase or diminish the stability of the pier, accord-
ing to circumstances.

221* The coefficient of stability, in any given case, is the ratio of the
moment of stability to the overturning moment. Or,

^ ic • I. e ^ uM-x moment of stability

Coefficient of stability =■ ; 7.

overturning moment

222. Let the weight, W. of the stone in Fig. 96 be 10 lbs., G its center of
gravity, and og = 2 feet. Then the moment of the weight about o, or the
moment of stability about o, is 10 X 2 » 20 ft.-Ibs. ; and, if o n » 5 feet, a

Of)

force P *■ ~ = 4' lbs., will just hold in equilibrium the moment of the
5

weight, so that, except at the corner, o, no pressure will be exerted upon the

base o tn^ although the stone remains in contact with the base. If tne force

P exceeds 4 lbs., the stone will.begin to turn about o. If P is less than 4 lbs.,

the stone will exert a pressure upon the base o m.

Let the stone be supported at o and at m only. The leverage of the sup-

Krting force R, at m, is » the length o m of the base, » I. Let P — 1 and
se o m •-* 4.5 ft. Then, for equilibrium,

W .a g — P. on — R.om =» 0;
or, 20 ft.-lbs. — 1 X 6 - 4.5 X R;

or, R "" — v-g — " 3.33 .... lbs.

In other words, a vertical upward force, R, of 3.33 .. . lbs., at m, will
maintain equilibrium.

(a) Ob (P) b
FI9. 96. Fiff. 97.

223. In Fig. 97 (b), let g be the center of gravity of the load W and the

W eg
table, combined. Fig. 97 (a). Then, upward reaction of 6 = — r^—. Those

of a and e may be similarly found.

424

STATICS*

2Z4» In Fijp;. 08, let h be the horisontal force exerted at the crown by the
left-hand half of the arch, agamst the half -arch shown, and e its leverage
about o. Let W be the weight of the half-arch with its spaiidrel, acting as a
single, rigid body, and I its leverage about o. Then, for equilibrium, we nave

h ,e — W.Z; or A —

W .1

Fiff. 9S.

Fi^r. 09.

Stability on Inclined Planes. .

Z2S. StabUity on Inclined Planes. Fig. 99. Here, as in t 213, if the re-
sultant, R, of the force P and weight W, falls beyond the base, — i. e., if the
overturning moment exceeds the moment of stability, — the body will over-
turn. If not, it will stand.

The force, P, in any given direction, required to prevent overturning, is
= the anti-resultant, A, of weight W and reaction R; and reaction R =»
anti-resultant of force P and weight W.

2/S6. Neglecting friction, as in Fig. 99 (a), R will be normal to the plane.
Taking friction into account, Fig. 99 (b), R may form, with a normal, N, to
the plane, an angle, a, not exceeding the angle of friction between the body
and the plane. R may be either uphill or downhill from N.

227. In Fig. 100, the body B has less stability against overturning about
its toe, a, than has the similar body. A, when the force, n, tends to upset it
downhill; but a greater stability than A against overturning about c under
the action of a force tending to upset it uphill.

228. The body C, which would upset if upon a horiiontal base, would be
stable against overturning if placed upon an mclined plane, as at D. Assum-
ing ao '^ tCfA given upward vertical force would have the same overturning

FliT* 100.

moment, whether applied at a or at c. But a given horizontal force, applied
at any given height, as at g, has a greater leverage, g o, when pushing down-
hill than when pushing uphill. In the latter case its leverage is only g t.

229. Structures built upon slopes are liable to slide. This may be ob-
viated by cutting the slope into horizontal steps, as at d y. Fig. E; but the*
vertical faces of such steps break the bond of the masonry ; and, moreover,
the joints being more numerous, and the mortar therefore in greater quan-
tity, on the deeper side, 8 d, than on the shallower uphill side, e y, the struc-
ture is liable to unequal settlement, the downhill side settling most and tend-
ing to split away from the uphill portion, as might be the ease with a founda-

THB COBD.

426

tion firm in some parts and compressible in others. ^ Hence, when circum-
stances permit, it is preferable to level off the foundation, as at d v; or, if the
structure has to withstand downhillward pressures, v should be lower than
dt and the courses of masonry laid with a corresponding inclination.

THE COBD.

330. The Cord. Figs. 101 (a) and (b) and 102 (a) and (b). In Ift 230
to 239 we deai with cords supposed to be perfectly flexible, inextensible,
frictionless, weightless and infimtely thin.

P^Qr

FliT. 101.

231* Let P be the external force applied to the cord at the knot or pin, o^
and let R be the resultant of the stresses, 8i and «2, or o a and o b, in the two
seenxents, o m and o n, of the cord. Then, for equilibrium, R must be equsJ
to and colinear with P.

IS3!3. Knowing the amount of P ( => R), the tensions «i and «8 may be
found by means of ^ 36: and, vice versa, given <i and «s, we may find R
( - P) by 1 35. Or see 1 40.

2S3d. If, as in Figs. 101 (a) and (&), the force P be applied to the cord, at o,
by means of a fixed knot, incapable of sliding along the cord, so that the seg-
ments, o m and o n, of the cord, are of fixed lengths, and the an^le, x + v,
between them, of fixed magnitude, then the force may be applied in any
direction, as P or P', passing between the two segments of the gokI; and the
components, «i and «8, will be equal only when K (P produced) forms equal
angles, X and y, with the two segments of the cord. If the direction of the
force, as P", coincides with either segment, as o n, of the cord, that segment
transmits the entire force, P", and the other segment none.

(*) /

Qr

Fl«r. 102.

234. But if, as in Figs. 102 and 103, the force P be applied to the cord by
means of a frictionless ring, slip-knot, pin or pulley, etc., then, for equilib-
rium, the two stresses, 8\ and so, must be equal, as must also the two angles,
X and y; and, if we suppose the direction of the force P to be changed, as to
P', the pin and the cord will readjust themselves, as indicated by the dotted
lines in Fig. 103. until the pin finally comes to rest at that point, o\ where
the angles, x' and j/, are equal, and also the stresses, «i' and 82' .

426

STATICS.

235. Even though the pin or pulley be rigidly fixed to some external ob-
ject, as at o. Fig. 104, yet, if there is no friction at its axle, or between it
and the cord, the components, «i and «i, will still be equal, and their resultant,
R, will bisect the angle, x + 2/1 oetween them. In other words, the angles*
X and v^ will be equal.

FliT. 104.

ri». 105. ^

236. When the pin is movable. Figs. 102 and 103, to find the position, o.
Fig. 105, which it will assume. From the end, n, of one of the segments, o n,
of the cord, draw n v parallel to P. From the end, m, of the other segment,
with radius = mo -\- on^ » length of cord, describe an arc, cutting nv in d.
Bisect ndxne. Draw e o normiu to n v, intersecting m d in o. Then o is the
required p>oint.

237. Whether o be a fixed knot or a movable pin or pulley, it is always in
the circumference of an ellipse whose foci are at the ends, m and n, of the
cord.

Fig:. 106.

238. From the foregoing it follows that, if o, Fig. 106, be a fixed knot, and
if the other pins or pmleys, etc., are frictionless, the stress a o, or «!, will be
transmitted uniformly throughout the left segment of the oord, from o to its
end at m; and b o, or as, throughout the right segment, from o to n.

FTTNICULAK MACHINE.

427

239. Caution. Note that, in Fie. 107 (6), the stresses in all the cords are
twice as great as the stresses in the corresponding cords in Fig. 107 (a),
although each Fig. shows a load «* 4 suspended from the pulley. Thus, if
the weight be that of a man, hanging by the rope, and if the roi>e, in Fig. (a),
be just sufficiently strong to hold, it will break if he gives one end of the rope
to another man to hold, or makes it fast, as in Fig. (6).

\8

(b)

rig, 107.

The Funicular Machine.

340. When the angles, x and y. Figs. 101, etc., are very great, a very small
force, P, will balance a very great stress, «i or «a, in the cord. When x = y
— 90*^, we have cos x = cos y — Q, and »i = «a •= infinity, however small P
may oe. If a line, m n, joining the ends of the cord, is horisontal or inclined,
the weight of the cord itself acts as a force P. Hence

"There is no force, however great, can stretch a cord, however fine, into
a horizontal line that shall be absolutely straight. ''

^1. The funicular machine takes advantage of the fact that, when the
total aujp^le, x + y, between the two segments of the cord, approaches 180°,
a small force, P, may balance great stresses, si and 83. Thus, in Fig. 108, let
W represent a heavy boat (seen in plan) whicn is to be hauled ashore. One
end of a rope being made fast to the bow of the boat, the rope is passed
around one smooth post, n, to another, m, around which it is given one or
more whole turns ; and a man stands at the end, e, to take in the slack ; while
others, taking hold of the rope between m and n, pull it, in the direction of P,
into a position mon. If the two angles, x and 1/, are equal, the component
in the segment o n exceeds P, so long as the an^le x exceeds 60*^, and a
puU, equal to this component (except in so far as it is reduced by the rigidity
of the rope and by its friction against the poet n), is exerted upon the boat at
W, drawmg it a shorty distance up the beach. The rope is then straightened
again, from m to n, by taking in the slack at e» and the operation is repeated
as often as may be necessary.

f9»

•10*

Flff. 108.

\W

Tig. 109.

The Toggle Joint.

242. The toggle joint, Fig. 109, is simply an inversion of the funicular
machine with a fixed knot, the force P and the components, «i and 80, being
pushes or compressions, instead of pulls or tensions.^ The joint being unable
to move along the arms, the force P may be applied in any direction at pleas-
ure, but it is usually exerted in a direction forming approximately equal
angles with the two arms.

428 STATICS.

The PnUey.
243. Fiiig. 110 shoir the relatioDa of BtnaKB and wsights in i
rHneements of fixed and movable pulbys. Thus, in (a), 1 lb
1 lb.. In (b) 2 lbs., in (c) Bnd in id} 4 lbs. In eacb case, if tb«
weights be set in motion, their velooilieB are inversely as their mi
1 20fl.

BhsQKiQg direction of Btress. for the

equar; but |a the compound pulley, FiKs ,' '0 (b). (>;). (lO

ttSe "weight"). moviDK slowly, at another part. 1.1m™. t-a luu-muun
pulley is used for the purpO!« of overeomine ffreat lesiatanoea slowly, by
means of small forces, moviog rapidly.

345. To set such a system in ^ot^' (t. e..,to raiM the "weight") re-
will continue indefinitely if the "power "is made sufQcieatly p«ater than tha
"weight" to balance the resistances of friction, etc.

The Loaded Cord or Chalo.

246. InFias. Ill the principle of the cord polygon, HH 88, etc., ia applied

to Che case of a flexible cord or chain, sustaimiig four loads, p, . . . pt,

by the vertical line, 0-4, Fig. lll(o); the honionlalpulX H. by 0-c; the
amount and direction of the inclined pull. R. at the upper end of the oord. by
4-f. and the teosioDB in tbe segments, 1-2, 2-3 and 3-4, by the rays, l-c, 2-e

. 247. The horizontal tension, H ( - the horiiontal somponeDt of the tan-
betwsea that segment and the pulley, m. l^us,

LOADED CORD.

429

(0-2; Fig. a) of the texudon, €-2^ in segmeiit 2-3, is >- pi + Pi; that in aes-
ment 3-4 is 0-3 = Pi i- Ps + Pa. ©tc.

Z4S* If all the loads (includineW) be inoreased in the same proportion, as
indicated by the dotted lines in Fig. Ill (a), or diminished in the same pro-
p>ortion, the new triangles, c' 4' 0, etc.. Fig. (a), will be similar to the old,
and the profile of the cord, Fig. (b), will remain unchanged, although the
stresses in its segments will of course be increased or diminished in the same
proportion.

249* In Fig. Ill we make the weight, W, which is necessarily equal to
the horizontal* pull, H (see The Cord, IfK 230, etc.)f equal also to the

Fig:. 111.

» ^8 P,

Flic. 112.

W

Tig. 118.

sum of the loads, pi . . . pt. When this is the case, the cord segment, a-A,
next to the sup|K)rt, a, and the corresponding line, c-4. Fig. (o), will be in-
clined 45° to the vertical.

250. But if, while the loads, Pi . . . P4, remain unchanged, we raise the

fulley t», so as to keep H horizontal,* we shall obtain a flatter curve, as in
ig. 112; and, for equilibrium, H (■» W) must be made greater than the
sum of Pi . . . P4. On the other hand, if we place the pulley, m, lower
than in Fig. Ill (still keeping H horizontal), we obtain a deeper curve, as in

Fig. 113; and H ( — W) must be made leae than the sum of pi

P4.

* In Figs. Ill, 112 and 113 we suppose the weight, W, and the position of
the pulley, m. to be so adjusted, relatively to the support, a, that the pull, H,
■hall remain horizontal.

ASCHE8, DAHB, ETC. THBU8T AND SEBISTANCE UITES.

Tbe Arch.

oi the commonLy mcc«pted theory
s, see 11 256 to 260. and Stone

\be stonea of a maaoory arch. Fig.

Z5». lntbeca8eofaQanili.F«.114.as3uniiag*tlu
H, at tbe crown, m. and tbe reaction, R. of the skewba

,„ „ d of aliewbacii. reapectiTely,

their amounts, and tbe direction of the reaction, R. in» be found by means
of the Foroe Triangle, 151, or by Momenta. 1224, (See T 25T.) Wethen
Buppoae tbe balf-arab and ita apandrel to be divided, by vertical phuieo,* Fig.

114 (b). into a number of segmenta, as shown; and, finding the weight and
the center of gravity of each such segment (see HH 257 and 206), we treat
these segments as ve treated the loads, p, . . . Pj, of Figs. Ill to 113, 1»T-
iag them off from 0 to 6, Fig, 114 (a), and laying off 0-c horiiontal and - H,

" Thel^X™ q" m^d «*/ , *^
thrust line, orlineof reaultan
Figs, 111 (6), etc,'
2S3. The

254. When the planes, by which the arch is supposed to be divided into
aiohes, the thrust and 'resistanee lilies. Fig. I'lS, prai^tieajly coincide; but if

™ThS",Tn 'pfg, l'rs*(wh?re the thl
line dotted). notieioK where result
ioiut B. etc,, it will be seen that th
oint C. where they begin to diverf

• See Practical Cod»

ARCHIPg.

431

ZS5* In f 252 we aasumed that the arch and its spandrel are divided into
vertical segments, incapable (except in the arch ring) of exerting other than
vertical pressures. The theoretical resistance line, thus obtained, may,
especially^ in deep arches, pass from the thickness of the arch ring in places;
so that, if no other forces were acting, the arch would open at such places;
on the intrados when the resistance Tine cuts the extraaos, and vice versa;

FiiT. 115.

•

but such opening is usually prevented by other forces, such as the horizontal
or inclined pressures of the spandrels. The actual resistance line is thus
confined within the thickness of the arch ring. In general, the actual resist-
ance line, Fig. 116, approaches the extrados at the crown, and the intrados
at the haunches, so that the arch tends to sink at the crown (opening there
on the intrados), and to rise at the haunches (opening there on the extrados),
as shown.

FliT* 11«-

356. In order to avoid any tendency of the joints to open at either side,
the arch should be so designed that the actual resistance line shall every-
where be within the middle third (see Ifl 145, etc.) of the depth of the arch
Xing.

257* In general, the design of an arch is reached by a series of approxima-
tions. Thus, a form of arch and spandrel must be assumed in advance, in
order to find their common center of gravity for the purpose of determining
the horisontal thrust, H, and the skewback reaction, R, as in f 252 ; and, if
it is afterward found necessary to modify the form first assumed, in order to
satisfy the reqtiirements of ^ 256, or for other reasons, we may have to re«
compute H and R, again modifying the design, and so on.

432

STATICS.

Practical Considerations.

IS58. While the theoretical thrust and resistance lines, based upon tho
foregoing assumptions, are easilv found, much uncertainty exists as to the
positions of the actual thrust and resistance lines in a masonry arch.

559. In the first place, we do not know through what points in the crown
and skewback, respectively, the resultants, H and R, pass.

560. Again, we have assumed that the loads on the arch, like those on the
cord. Figs. Ill to 113, are incapable of acting otherwise than vertically;
whereas the spandrel walls and filling, which form a large portion of the load
on a masonry arch, may offer resistances acting in other directions. If the
loading were a liquid, like water, its pressures upon the arch rin^ would be
radial, like those of the particles of steam, in a boiler, upon the boiler tubes;
and this condition is probably more or less closeljr approximated in the case
of a loading of clean dry sand; and, less closely, in the case of earth filling.
Hence, although the determination of the theoretical thrust and resistance
lines in an arcn is facilitated by the assumption that the arch is correctly
represented by the inverted eord, the distinction between the two cases must
be borne in mind when drawing practial conclusions from the lines so found.

261. Thus, in many cases, the theoretical thrust and resistance lines out
the intrados or the extrados in places, thus passing entirely out of the arch
ring; so that this would inevitably fall (see 1 255), were it not for horizontal
or inclined resistances exerted by the upper parts of the abutments through
the spandrel walls and filling.

263. Hence, in order to determine the actual resistance line, we should
not only have to know through what points, in crown and in skewback
respectively, the resultants, H and R, pass, but we should also have to ascer-
tain and take into account the possible horizontal and inclined resistances of
the spandrel waUs and filling. But, as this is ordinarily impracticable, we
content ourselves either with determining the theoretical thrust and resist-
ance lines, as directed above, and then estimating, as well as may be, the
resistances of the spandrels, or with reasoning by analogy from the behavior
of actual structures. See Stone Bridges.

263. If the inverted cord correctly represented the actual thrust line in
a masonry arch, the arch stones, in elliptic or in deep segmental arches,
would have to be made inordinately deep, in order that the resistance line
should nowhere leave the middle third of their depth (see %% 145, etc.);
and it might therefore appear rational to make the profile of the arch corre-
spond approximately with the thrust line,- which usually approaches a para-
bola. ^ But, owing to the spandrel resistances, the actual thrust line, even in
semicircular arches, probably seldom greatly oversteps the middle third.

264. With a wall or a deep continuous filling, over an arch, if the arch were
to settle, or were to be removed, the. wall and the filling above it would form
an arch, as indicated by the broken lines in Fig. 117; and only that portion

Flff. 117.

below this arch would fall out. Hence, only this portion can properly be
regarded as pressing upon the arch.

265. Neglecting the strength of the mortar, the inclination of each joint
between two arch stones must of course be such that the angle, between
the thrust, at any joint, and a normal to that joint, shall be less than the
angle of friction. See t If 183, 184.

IUM8.

483

It is oltan the oaae thai th<l gpandrels or tha fiiMkndrel filling ai« of

less specific gravity tlian the aroh ring. In such casesp in ordeac to facilitate
the finding of the lines of eiavity of the segments, we may, before dividing
the haif-arch and its spandrels into, vertical segments (% 252) , consider the
lighter structure of the spandrels as being reduced to an equivalent depth
of material haviiig equal specific gravity with the arch. Ine areas of the
i^everal s^^ments, as seen In profile, and as thus reduced, may then be taken
as repj^esenting their weights. Thus, in Fig. 118, where ttt represents the

riff. lis.

top of the spandrels, the curved line e 0 e represents the top of a filling of
equal weight per foot run with the spandrels, but of equal specific gravity
with the arch ring. When, as in Fig. 1 19, the spandrels consist of a series of
transverse arches, we may assume that the main arah carries a series of loads
conoentrated at the piers of these transiTerse arches.

riff. no.

-The lltasonry Dam.

2S1» A daih must be secure aigainst sliding, on its base or on any plane
within the body of the dam, against overturning, and against crushing of the
material at any point and consequent opening of a seam at either face of the
dam.

268. The dam will be secure against sliding if the resultant of all the pres-
sures, upon any. surf aceib /orms, with a normal to that surface, an angle
lees than the angle of friction of the surface. See f If 183, etc. In
practice, the base of the dam is let well down into the rock foundation, as
indicated in Fi£.- 122 (a), and continuity of joints is avoided by making all
the stones break joints. The angle of friction thus becomes, in effect, 90°,
and sliding cannot occur without shearing the stones themselves.

969. If the material is sufficiently strong to resist crushing, under the
maximum unit stresses brought upon it, and if the resultant of all the foraes
aotlng upon any section falls within the body of the dam, the dam will be
«eeure against overturning. But see % 270.

tS70« For a given total pressure upon any section, the maximum unit
pressure in the section' would be least when the resultant out the middle
point of the seotioA. See Center of Pressure^ tH 13^ etc. It is generally
impraotioable to secure this; but the dam must be 00 designed that, imder
the maximum unit pressure, the given material shall not be taxed beyond
its safe crushing strength. If this is done, and if, under all conditions, the
center of pressure is kept within the middle third (see t 150) of each hori-
mootaX section throughout the dam, there will be no tendency to open on
either face of the dam.

2ft

434

STATICS.

«1, Let Fig. laOrepresentftBlonBblMlc, resting upon a solid foundation
and intended to BuatoiD the prsBBure, p, of quiet wateroDoneeide. Thioush
the c«nter of erBvity, a, of the block, drawo'N vertioally, tu represent the
weiriit, W, of tlie block. Thea the point. 1. where o' N meets the founddlion,
IB the i»n(*r of preaaure for the block alooe, i. e.. wliBn the water is removed.

273. Let A be the depth of water back of the block, and let the block be
one foot in length, measured normally to the paper. Then the amount, in
l»uada,of the water pressure.againet the vertical back. ob, 13^=62.5 AXH&
Bod its center of preaaure iaat s depth, d'=%h, below the water surface.

213. Combining p with W (1 35) we obtain R as their resultant, ai "
as the cenUr of preBfliue upon the foundation when thf ■-■--■-■ -

14. Let

(. 121 n

>□ the block is
:h blocks superposed. Let

" " pi — " " " " " " 1. 2, and 3 combined,

indiiig r, and fj, r, and «i, r, and «. etc., for joints 1-2, 2-3, 3-4,
fore, we liave the points/,, r,. rs. etc., in the resistance line for fuU

275. In Kb. 122 (a) t
down-streamlimits, resp

276. Whiletheory wouldre<iuirethi
u rndiclw"* "* " "' '

u shown by dotted

g bloeka l ani]

ippose that block 2 had ai

ular, aa shown by dotted Ime e a (Fie, 122 (c), ah

_ enlarged) ; but this makes the center of pressure, r'. for the full dam. Ml
be^ondthe middle third of the narrow base, a b. We therefore try the traps-

. of pressure, rj, although further down-stream than before, falls' within tli»

sr base. The remaini

DAio. 436

918. Graphic Method. Suppose the croas-Bection to be divided, by
horiiontal sections, into nuniBTOUBblookB. 1, 2, a 4, alo., of fl depth approii-
inately - tho top width of the dam. In Fig. 122 (fc), draw 0-1. 1-2. 2-3. 3-4,
tie., vertically, to tepresenl, by Hcale, tGe weights of the aevaral blocke,

the water preasureH against aaid bloclia respeit'iyely; and draw I'-l, Z'-2.

premureB upon jomlo 1-2, S-3, 3-4. etc., sad upon the 'base, rfiapEctively.
Thus, 2'-2 represente the resultant of the water preaeuie (see Uydrostatics)
upon blocks 1 and 2, and the combined weight of those blocks. From the

Oi ri, tp, ri. etc.. parailel respectively to I'-l. 2'-Z, etc., Fip. 122 (6),. to the

re thee

ia filled

.-_.,--- „-_Ji for » dam already completed oi

ir'S"l''tnd' l^'irViL.'wTor'^'' " *■ *■ ■'"'^' ^^ ^^- ""

id 2'-2, for the second block, ai

n1277.

irse begin at the top. and lay o(
) for the Jirst block: then line

279. In order that the resietauce lii
be brought well within the middle thin
adopt a eomewbat unwieldy cross-sect.,-,. ,,--. ^-
daager involved in the amallest opening on the upi
(He H 281), it is well here to err on theB^eside.

2SO. Aa tL_ - .,

third, in each ot the lower ji
The centers ot pressure, i, . .
ibe middle third, on the up-stri
To obviate this, the up-stream

I further dona, the ai

L, may then fall beyond
d at joints E-6 and 6-7.
von » ourved profile, a*

436 STATICS.

Practical Consideratloiu*

381. The assumption of ideal conditions is pftrticulariy danger-
ous in the case of masonry' dams. Thus, any compression of the material
at the down-stream face may open seams on tne upHrtream face; and "water,
entering these seams, will exert a wedge-like action, shifting tne resistance
line further downnstream, thus still further increasing the tendency to crush-
ing on the dowuHstream face and to opening on the up-stream face. Again,
if any relatively smooth joints have been left, the water, thus penetrating
into or under the dam, increases the tendency to slide, not onl^ by diminish-
ing the efifective weight of the upper portions, but also by acting as a lubri-
cant upon the seam where it penetrates.

It has been suggested that failures of dams may have been occasioned,
in part at least, by vacuum, formed in front of the down-stream face, by the
action of the sheet of water falling In front of that face.

• .

282. Theoretically, the deflections of arches, dams and other structures
composed of blocks, may be found by means oi the formulas in %^ 162-167
of Trusses; but, owing to uucertalnty as to the values of the moduli of
elasticity, E, of building stones and of mortar, and to the relative inaccu-
racy of finish in masonry work, the formulas are of but little practical
value in such cases.

THE SCBEW.

283. The screw is a spiral inclined plane. The force (or "power") de-
scribes a spiral, at the end of a lever arm, while the res&tanee (or "weight")
moves along the axis of the screw. During the time in which the force
makes one revolution, the resistance traverses the "pitch," or distanpe be-
tween the centers of two adjacent threads.

284. Hence, if P ■• power, to =- weight, rf •■ pitch, I -f lever arm, v —
rectilinear velocity of Weight, and V = linear (cumular) velocity of pbvrer,
are have, theoretically:*

t^ ^ V ^ 2 irl

P V ~ d '

* Neglecting friction, which, however, very greatly modifies th« result.

EQUILIBRXtm 09 BEAMS.

487

FORCES ACTINO UPON BEAMS AND TBUSSES.

Conditions of EqiiiUlirinm,

285. In beams and trusses, for equilibriuctf, it is necessary and eroffieient
that the resisting forces, exerted by the material of the structure, and the
moments of those forces, shall balfmce the external ot destructive forces and
their moments. We here discuss chiefly the destructive forces. For the
resisting forces^ see Stresses, under Trusses^ and Beams or Transverse
Strength, under Strength df Materials. ^

286. The destructivetorora Site CtT'the-ldads upon the structure, includ-
ing its own weight^' live'' or moving Ictads, wimd, etc., and (2) the reactions
of the supports. We shall he^ discuss the action of vertical loads only, in-
cluding (a) the dead load, or t^ weight of fli0 s^cture itself, together with
the roadway, etc., and (b) the live, moving or-extraneous load of vehicles,
trains, persons, etc. The action of ixArixeUtel loads (wind, centrifugal force,
etc.) is governed by similar laws, and is discussed under Stresses, in Trusses.

287* Let Fig. 123 (a) represent a cantilever, resting upon a support, h,
and iMaring m Umd, >W« at its cuter tenii cc TbsroaatUeyee is nreventedfrom
tumiog about b* by %he tensioxv T« of a horiaontal cdiain, and by the oompres*
sion, C, in ft 40riso9tal stiut«* JDfeitlMtitng 4he- weight of the cantilever

IHf!. 123*

itself, ihe cantilever Is acted upon by four external forces, forming two
couples; one couple consisting of two vertical forces — viz., the load, W, and
the reaction, R', of the supjport ; the other couple consisting of two horizontal
forces— •^ris., ihe tension, T, near th» top. and |bhe eompression, G, near the
bottom. Were it not for tho resetion, R', of the support, b. the load, W,
would pull tile cantileiver dow|iwaid, as indicated in Fit. 123 (&).

288. In Fig. 123 (4^ w»kave:

Aleeb'raic sum of vertical forces '» E' — W^—'O;

" " horizoirtai '* - T — O •* Ot

" '' moments, about any point, as o.

II

W.w — R'.r + T.< + C.c « 0.

* In Pigs. 123 to 127, inclusive, and Figs, 132 and 133, showing cantilevers,
beams and part beams, acted upon by loads, by reactions, by pulls of chains
and by pushes of Struts, the arrows denote forces acting upon the cantilever
or beam or upon iU augments, and not forces acting upon the load, the sup-
ports, or the connecting chains or struts. Thus, the tension in a chain tends
to draw together the two bodies which it connects. Hence, in these cases, the
corresponding arrows point toward each other. On the other hand, the com-
pression in a strut tends to separate the two bodies between which it acts.
Hence its two arrows point away from each other.

438

STATICS.

280. If, as in Fig. 124, the horizontal forces are exerted at the end farthest
from the support, and at the same distance apart as before, their amounts
and senses must remain respectively the same as before; but we now have
compression, C, at the top, and tension, T, below. Or, if Fi«. 123 be m-
verted, R' acts as the load, and W as the upward reaction; and we have, as
in Fig. 124, compression, C, at top, and tension, T, below. Thus, Fig. 124
is practically Fig. 123 inverted.

290. The condition described in J 289, Fig. 124, represents also the condi-
tion in each segment, A, B, of a beam, Fi^ 125 (a) and (6) or Figs. 126(a)
and (6), supported at both ends and beanng a concentrated load, W + w;
Fig. 126, or W + w, Fig. 126.

(b) ll^ O^

Fiff. 125.

201. Suppose the beam. Fig. 125 (a) or Fig. 126 (a), to be divided
into two cantilevers, or part beams, as in Fig. 125 (b) or Fig. 126 (b) ; each
part sustaining, at its end, a part of the original load. (See If 292.) The
stresses in the strut and chain, Figs. (&), take the place of stresses in the ma-
terial (situated in the dotted line) of the truss or beam, Figs, (a) . In a truss,
these forces are exerted by the chords: in a beam, by the particles or fibers
throughout the section.

202. If, as in Fig. 125 (a), the load is at the center of the span, the spans,
X and y, of the cantilevers, Fig. 125 (6), are equal, as are also the loads,
W = W, carried by them. But if, as in Fig. 126 {a), the load. W + w, on
the beam, is not at the center of the span, the partial loads, W and to, sup-
posed to be supported at the ends of the two cantilevers, or part beams, re-
spectively, Fig. 126 (b), are unequal, and inversely proportional to their
leverages about their respective supports. Hence, the moments of the two
opposite couples are equal. The reaction of each support is equal to the
weight carried by the cantilever resting upon it.

EDTD BEACTIONS.

439

End Reactions.

293. In a cantilever, Fig. 127, there is but one vertical support; the reac-
tion, R', of that support, is >« the sum of all the loads, including the weight
of the cantilever itself; and the reaction due to each partial load is — such
partial load. Thus, if B — weight of cantilever,

R' - W + w + B.

294. The reaction, R', must not be confounded with other vertical forces.
Thus, a cantilever is often supported as in Fig. 128 (a). The couple, com-
posed of two horizontal forces, T and C, Fig. 127, is then replaced by a couple
oompoeed of two vertical forces, V and V, Fig. 128 (6) ; one of which, V, co-
incides with the reaction, R'. Here, R' + V', acting upward, is the anti-
reeultant of W, w, B and V , acting downward.

295. In a beam, Fi^. 129, the sum of the two end reactions is — the sum
of all the loads, including the weight of the beam itself.

296* The reaction, R, of the left support, a, Fig. 129, due to the load,
W, alone, is R -".W . -y (see If 17), and the reaction, R', of the right

support, 6, is — W — R

R-R'=:-

If the load is central.

I

I

-g.and

W±g. 129.

FiflT. 130*

297* Graphically, Fig. 156, suppose a concentrated load, W ^not shown).
to be placed on the beam at any i>oint, as c. Draw a' a* and b' b't vertical
and each — W. Join a" &'; also join a' 6^ and draw g h vertically through
</. Then the ordinate, e' g, to the upper line, a" b\ and the ordinate, </ h,
to the lower line, a' 6', give the left and the right end reactions, R and R',
respectively.

298. Where, as in Fig. 130, there are two or more loads (in which the
weight of the beam may or may not be included), the reactions due to each
load may be separately obtained, the sums of these reactions giving the total
reactions; pr, the common center of gravity, G, of all the loads may first be
found (see ff 125, etc.), and then the reactions found as for a single load,
W, Fig. 129; the combined Weight of the loads, whose center of gravity is at
G, being supposed oonoentrated there.

4^ 9TATIC8.

299. In a beam. Fig. 131, undar aload; W,, ultfformly distributed over any
part of the spaa* let G be the center of gravity of the load, and let x «ndv be
the segments of the span, 4 to the left and right of Q respectively. Then,

neglecting the weight of the beam. R - W -^-; and R' — W — R - W 4.

Fly. 181.

^ »

800. If the load is uniformly distributed over the entire span, its center
of gravity is at the center of the span, and we have:

f-^-i, andE-E'-f.

Momente and She^nb

801 • In order to determine what internal stresses are required, at ansr
point in the span, to maintain equilibrium, we may suppose the cantilever
or beam to be cut in two by a section, e e. Fig. 132 or Fig. 133, at such poixitj .
and inquire what forces must be applied, in the section, in order to mam*
tain equilibrium and hold in position the two ei^Ements, £ and F, into whicH
the section, cCf divides the span. Fig. 132, or that part of the span between
the load and a support. Fig. 133. The forces, so ascertidned, are evidently
equivalent to those actually exerted, for the sfune purpose, tgr the material
of the beam itself.

802, In Figs. 182 and 133, moments of loads and of rfpictions,or-ea;ter»
not or bending moments, are indicated by arrows below the cantilever
and beam respectively; while the resisting moments of the internal forcea
are indicated by arrows within the body of the cantilever or beam respeo-*
tively.

303. In the cantilevert Fig. 132, the load, w, — 4 lbs., distant 6 ft.

from the section, e c, produces there a left-hand or negative moment ^f 6 tr -^

6 X 4 — 24 ft. -lbs. Hence, for equilibriuq[i, the horjuontal strut and

chain, at ec, must exert a right-hand or positive resisting moment of 24

f t.-tbs. : and, being 2 ft. apart, tney must exert n tension, T, and compressioiw

24
G, of -TT — 12 lbs. each. At the support, moment of load «>0ii;"«0X4«

36 ft.ilbs.? and T - C - ~ - 18 lbs.

304. But, considering only the forces thus far discussed, we shoidd find th*
right segment, F, acted upon, at ce, by a left-hand covpip* — <i X T«
c(XC">2X12«' 24ft.-ibs.; and, at the support, by a right-hand couple^
- d X T' - d X C - 2 X 18 - 36 ft.-lbs. In other words, there would
be an unbalanced excess of right-hand moment, •*• 86^— 24 -• 8 R* •i*' 8.X
4 » 12 ft.-tbs., acting upon F. F also receives, at the eupport, the upwaiv
reaction, R', — 4 lbs., of that support. Similarly, the couple, d X T ••
d X C, at tf c, exerts, upon the left segment, £, an apparently unbalanced
right-hand moment of2X 12»6i0-"6X4m24 f£.<*lbs., and £ raoeivsflb
from the load, to, a downward pull •■ 4 lbs.

305. For equilibrium, therefore, the verti^ chain at e e must eatart a
tension — S — W'-R' — 4 lbs., pulling F downward, and E upward. Ilia
downward tension, ^—S, acting on F at e c, forms, with the reaction. R\ <tf
the support, a left-hand couple *"3R'»«3X4""12 ft.-lbs., balancsag
the excess of right-hand moment acting upon F; while the upward tension,
+ 3, acting on E! at e c. forms, with the wei^t, w, a he^hana ooupAe, ** 6 «
* 6 X 4 ■- 24 ft.-lbs., balancing the excess of rightrhand mom. actios on 1^

MOMENTS JLND SHEARS.

441

306. Similarly, if we 8aploo«e' tdie csAiiWfet eut 'through by a section at
aay othfar poii^t, we< shall find that as vertical foroe^ =». g » ti; «= K^ aeting
upward upon the lett segment and downward upon the right segment*
is required io order to maintain equilibrium and to transmit the load, w^
to th^ support^ fio that the two segments may act unitedly as a single
cantilever. This force, S, is called a shear- >3ee %%■ 32d, etc* Without it,
section E would fall, as in Fig. 123 (6).

307* In the beam. Fig. 133. the total load is 16 lbs.; and, its distances,
3 ft. and 9 ft., from the left and froiQ the right support respectively, being
as 1 to 3, the €Qid I»ft^i^9|»^ (f^.^ 2fii3, ietoO Are us 3 to 1 j or R = 16 X t
= 12 lbs. ; R' = 16 X i = 4 lbs. We therefore regard the beam as be-
ing cut by a section at the load (as well aa at c c), and the total load of
16 .lbs. as divided into twp portions; one, W =f R = 12 lbs., attached to the
end segment, M ; and ^HiA o^her, «; = R' = 4 lbs,, wapported b^ the mid-
die a^ment, ^E. Here, as hi Fig. 132, segmentiiE and F .together form
a cfmUIevei:, WH, Tong, lofided with a weighty w, oIASm^ at ita end; but.

Fiir* 19^*

m\[{\mtp;=4

A

r-Szfod

w^' : :

c««

T^$9i ^*S^ tT^8

., M,P^

Wwl^:

Fiif. 138.

in Fiff. 13^ the^hortaontal MSiSting forces, T' and C, by which the entire
ouitilever (E -f F) is upheld, are exerted, not at the support, as in Fig. 132,
but at the end farthest from the support.

308. .We haW, therefore, in Fig. 133 ^— at e c.
Bending moment, positive,

Actjs|?onE«»2T^— 6w = 2X 18 — 6 X 4 = 36 — 24 - 12 ft.-lbs.
Actin^f on F *= 3 R' — 3 X 4 = 12 ft.-lbs.

Itesistmg moment, negative, -.2T-2C-2X6-12 ft.-lbs. Hence,
T - C -^ 6 ft>s.; and shear, S - «? - R' - 4 lbs.

309. In Fig»432 cbt i» Fig. 133, considering the segment extending from
ttio load to either Support (in Fig. 132 there is but one such segment), it will
beseen that, at the free end of any such segment, the horizontal stresses are
eero, and that the^ Increase unif<wmly to a maximum at the other end of the
■e^ent. Thrai, bi Pig. 132, they increase uniformly from 0, at the loaded
or free end, to 18 lbs., at the support; while, in Fig. 133, they increase uni-
formly from 0, at each support, or free end, to 18 ros., at the load.

UomentB In CantlleTcn.
810. Id ■ cantilever. Fie. 134. each Load exem

itself Bud tho support, a tnomcnt — ila weijjht X .—

ils mnMr ot atavity from such point; and the total momt

iflthBfluni of the momcnu - -' -

nefllectiDC the weiRht of the
a^utb. in(

'• d.

" A. or any point beyond A.

I i i '

, , ,^ Fi i

Tiff. 1S4.

FIc. ISO.

311. In a <»ntilarer. Fig. 136, the mAiimum hrenM of anf load. W,
is evidently itfl distance, i, from the support, b. Hence, the mari mum bend-
ing mom^Dt a{ any load upon a i»ntilever i« at the support, and is — W J.
From this maximum, the moment diminishes uniformly to uro, at the lewd.
See 1 309.

.312. Dran b" m. Fig. 135 (b). to represent the maiimum moment by scale,
and join m W, llien, (or any ptunt, c,

313. In a cantilever. Fis- 13tt (a), with tiro or more concentrated loads.

b' m. Fig. ISe ib). ■
b' W, Fig. 136 lb). ■

Then, for both loads, W and v, otvleeting the weight of the beam,
at d. moment — moment of W alone. — ordinate at d';
— Bum ci two otdinates, c* nandc'n', ate*.

■■■■II

ilil Y

1 h

■> .h

t

*'"^\

\

(6)

Fl«. 18«. Fit. m.

314. Id a cantilever, Fis, 13T (a), under a load. W, uniformly distributed
over a length. I. beginning at the supnort. b. the maumum moment, at the
support, 6. 18 — W . -i .

MOMENTS.

443

In Fig. 137 (5), make 6' m "^ said max moment, and draw a semi-^rabola
m k\ with apex at A;'. Then,' at any other section, c, the moment is repre-

sented by the ordinate, c^, of said parabola, and is — w . -^, where to = the

weight of that portion of W beyond c, and x -■ the length of that portion.
At kf or at any point beyond k, moment « 0.

315. In Fig. 138, neglecting the weight of the cantilever itself, let W repre-
sent the weight of the whole load, and w, that of the shaded portion, concen-
trated at their respective centers of gravity, G and g. Then,

about h, moment -■ W . x;

c, " - W.y;

d, " — to . v;

«4
44

44

kf or any point beyond k, moment ■- 0.

k ! ! I

I i
I I
■ I

-I— •-

H

•«

6

■Ih-^

I i

■w-

-i

Fl«. 138.

Moments in Beams.

318* In a beam. Fig. 129, the upward reaction of each abutment exerts,
about any point, a moment <« reaction X distance of support from such
point ; but any load, between such point and the support, exerts a contrary
moment — load X distance of load from such point. Thus,

. about c, moment =» R' . z "■ R (Z — *) — W (i/ — z).

At each support, the moment is 0.

317. In a beam. Fig. 129, carrying a single concentrated load, W, the
moment, R'.«, at any point, c, is = R'« = W-y . « = R (Z — z) — W (y — z)

=» W . y (Z — «) — W (y — z). At a point, as e, not under the load, the

moment, R'c, is evidently less than the moment, R'.f/, about the point, o,
under the load. In other words, the maximum moment is at the point, a,
under the load.

W

J^

I

r^«'

-y >

Fly. 129 (repeated).

Fiff. 139.

318. From the point, </, Fig. 130 (b), corresponding to the point, o, Fig.
(a), where the load is applied, erect an ordinate, (/ m^ equal by scale to the
(maximum) moment, =■ R' . y ■- R . x, at that point. Join a'm 6'. Then
the ordinate to a'tn^ or to mV, at any point, c', d', e', etc., represents by
scale the moment at the coiTesix>nding point, c, d, e, etc., in the span.

444

BTA'CXC&

819. When the load, W« is at the center of thespAn, 2» Fig. 140, eaeh end

W

reaction is <-> -^ • Hence, the moment, a', at any point, «, distant y from a

support, as 6, is

W

moment

2 •

y-

At tfa^ center of the span (u 0., at the pojat tmdesr the central, load, W)

wehave^

W I W il
maximum moment, M, = _ . -., = / -.

Z Z 4

Fig^. 139 (repeated).

Flff. 140.

In order that the maximum moment (at 0, Pig. 139) due to an ecoen-
tric load, W« may be ec(iial to the maximum moment («t eenteir of spfuir 0
due to a given center load. C, we must have

W

I

W = C.^;

or

w = c4--=c(2;

4 xy — t-

320. Whsen there are two or more concentrated loads* c d, «, Fig.- 1^1*
treat each load as in Fig. 139, making each short ordinate, m, m', m* repre-
sent the maximum moment of its single load, c, d or «, aloiM. Make tne long
ordinates, M, M' and M'' — the sums of the separate moments, as measured
at t'y at d\ and at e', respectively. Then the ordinate, to a' If M' M"' \/\ at
any point, represents the total moment at that point, due to the several
loads combined.

'(^ Q^ 9^

Fi«..141.

Flgr. 142,

3)31. In a beam, Fig. 142, under a uniform load, W, covering the span, i^
the maximum moment is at the center of the span, and ia

moment -= R. 2— 2*4 " 2-2 ~ 2-4 " 2 ''4

8*

MOK927TS. .hMS

«

M»J(e o'Mr by so^e^ » the maximum momeat: and dmw the pamhola,
m' M h\ with vertex at M. Then the moment, at any seotioa, as t, is nptt-
sented by the corresponding ordinate, «', to that parabola.

Let w and « = the portions of W to the left and right of a, respec-
tively. Thei^ moment at»»*»-qpy* — -5-35* '^ ^ moment due to whole

* *'i -

load, W, concentrated at 8.*

At either support, moment — 0.

In Fie. 131, at a point, c, under the center of gravity of a load, W, uni-
formly distributed over a portion, t, of the span, neglecting the weight of
the beam,

moment « Rjc "" o" • X " •^•* ¥~ "" ^ -^ — ~^~*

822. Let W — the total load, whether concentrated or uniform, and let
I => the span. Then the maximum moment, M, is as 'given beldw:

Cantilever. Load, W, at end. M at support lil = W Z;

" " " uAllorm. * " " " M»=^;

W I
Supported beam.f ** ** at center. " at center M =

^i'.

•* uniform. " " " M = -V^;

o

Fixed beam.t " " at cwiteh: " " " or support M = ^;

" " uniform. " "support M - ^.

823. In the inolined beam, Fig. 143, the inclined distances may be used,
instead of the horieontal distances, in finding the reactions. Thus,

reaction R' — W . -f- - W . I-.

But. in finding ^moments of vertical forces, we must of course use the
UKurisoBtaU not. toe inelined^ distances; Thus^ at e, moment R'c; not RV.

Fig:. 143.

♦ Moment at« =• R'y — t;|^--^ I/ — -gl/- — g — V "^ ^V'
_. X Vf w W — w V

With W concentrated at «,

moment ata—W-ry — wy «W-ya;s»»«.

t Beam supported at each end, but not fixed.
X Beam fixed at each end.

446

STATICS.

334. In curved beams, the same principles apphr as in straight beams.
Thus, Fig. 144, at », moment — W . Z. Again, in Fig. 145 (a), reaction R'

I

\ and at a, moment °= R' . {/. Or, as in Fig. 145 (6), from o, where

the load is applied, draw o a and o b, to the two supports respectively, and,
by means of the force parallelogram, find the components, p and 9, of W.
Then, at a,

moment = p . n.

^

o

W

////////////////A

Flff. 144.

FiflT. 145.

Shear.

325. In the beam, a &, Fig. 146 (a), consider the segments, a c and c &, to
the left and to the right respectively of the plane n n. Besides the horizontal
forces acting across the plane n n, we have seen {% 305) that we reciuire also,
for equilibrium, a vertical force, •" the left end reaction. R, acting down-
ward upon the left segment, a c, and forming a couple with R; and, at the
same time, acting upward on the right segment, c 0. being = the load, W,
minus the right end reaction, R'. This force is called the shear, S, in the
section n n. It may be regarded as the transmission of the vertical forces
from loads to supports or vice versa.

326. The two segments, a e and e 6, thus tend to slide vertically past each
other, the right segment, e &, tending downward, owing to the preponder-
ance of the load, W, over the right end reaction, R'; and this tendency is
resisted by the shear, S, which is » the left end reaction, R. The same ten-
dency exists uniformly between W and a, and is resisted throughout by a
shear — S = R.

327. Between the load, W. and the right support, 6, also, a uniform shear
exists; but here the shear, S% is = the right end reaction, R', — R~- W;
and, whereas the shear, S, to the Uft of the load was rnrAf-handed or e^odbwiM
(the portion to the Hght of any section, n n, receiving the downvxvrd toTot\
and is called positive, or +, the shear on the right of the load is 2e/l-handed or
counterclockvnee (the portion to the left of any section receiving the doum*
toard force), and is called negative, or — .

328. The shears, S and S', to the left and to the right of the load, W, are
represented by the diagrams in Fig. 146 (b) ; that, S, on the left of the load
being drawn above the sero line, a &', to indicate a poaiiive shear, and vice
versa.

329. Comparing Figs. 146, 147 and 148, notice that, between the left sup-
port, a, and tne load, W, Fig. 146, we have positive shears, S ■" 90, Fig. 1^,
and a =• 15, Fig. 147; so that, in Fig. 148, where both loads, W and w^ are
placed upon the same beam, we have, between a and W, a total positive
shear ofS+« = 90-fl5=- 105. Between the right support, 6, and the
load, w. Fig. 147, we have negative shears, S' -= — 30, Fig. 146, and t* ■-
— 45, Fig. 147 ; so that, in Fig. 148, between b and w, we have a total negative
shear — S' + «' =» — 30 — 45 = — 75. But, between the points of appli-
cation of W and of w, we have S' = — 30, Fig. 146, and « - + 16, Fig. 147;
leaving, between W and w. Fig. 148, s + S' - 15 — 30 - — 15. If the
total right end reaction, R' + r', exceeds w, as we here suppose, the shear, at
.any point between the two loads, W and w. Fig. 148, is negative, as indi-
cated ; and vice versa.

SHEAR.

447

330. In any section, the shear is "- the reaction at either end, minus any
loads between that end and the given section.

331. If, as in Fig. 149, the right end reaction, R' + r', is » the load, v>,
then the left end reaction, R + r, is » the load, W ; and there is no shear at
any point between the two loads. In other words, if the beam be cut by a
section at any point between W and w, horizontal forces alone will pre-
serve equilibrium, no vertical forces being required, since the two segments
have no tendency to slide vertically past each other.

332. A similar condition exists in any section where the sign of the shear
changes from + to — or vice versa. Thus, if the beam be cut by a section
immediately under W, Fig. 146 or 148, or under to, Fig. 147, horiaontal forces
equivalent to the fiber stresses in the beam, will suffice to preserve equilib-
rium, without a vertical force, or shear; there being no tendency of the two
segments to slide past each other. Also, when, as in Fig. 149, under W
and under 10, the shear changes, in amount, from any value, on one side of
a section, to 0, on the other side, the shear in the section itself is « 0.

(«)

U^S6

(b)

Fl«. 140.

Q^-

60

ir^lS

'4S

-90

Flff. 147.

190

'Q Q-

eo

"SO

TiK. 148.

Fly. 149.

333« But in the section under ir, Fig. 148, where the shear changes in
amount, although not changing sign from + to — or vice versa, there is a
shear » the lesa of the two shears on the opposite sides of the section, for
this is the amount of the shear transferred through the section, or is the
tendency of either segment to slide past the other.

334* With any number of loads, if that portion of the total load to the
left of any section be called X, and that portion to the right of the same sec-
tion be called Y, it will be found that the shear in the section is equal to the
dififerenoe between that part of X which goes to the right support, b, and that
part of Y which goes to the left support, a.

335. With a load, W, Fig. 150 (a), imiformly distributed over the entire

W
span, the maximum shear, = R = R' = -^, is at each support, o and 6.

The minimum shear, = 0, is at the center, c, of the span, which is also the
point of iT»ft^^'""T» bending moment, see ^ 321 and Fig. 142. At any point.

448

BTATWR.

d, the shear is given by the eoxrespondinff ordinate, d', £'ic- 150 0>)-> See
Relation between Moment and Shear, %% 359, etc.

336. With a load, W, luulonzkly distributed over any part, y^ of the span.
Fig. 151 (a), find the end reaetioiia, R and R', as in % 209. Then

between a and rf, shear — S •• R; '

e and b, " - S' - R';
at e, " - 0.

R R'

X ■■ dc

V'^'* ^

y — X — C« — V

'W

w

i!i:iiiiiiJiiiii'iii:!nii:i!iii!iiii!''!'i!:ii!i—

«

...^

MB—

6 d

' Wra^>w--« — >i:Bp

Tig. 150.

FI9. 151.

IS

8

Haj»|t — 4; — >p
! I I

Flgr. 152.

Wi^. 158.

337* When the loaded portion, y, of the span, begins at one of the sup-

ports, 6, Fig. 152 (a), then since R -» W

-W-U

X = dc

R

W

I

y

21

^ 2 T* ^® liave

2/

%L,^M1

•W" W "22 2r

338. When a concentrated load, W, Fig. 153, is added to a load uniformly
distributed over the entire span, or over a part of it, each load produces the
same shears as if it alone were upon the span. Those due to W are repre-
sented in Fig. 153 (6), while those due to the uniform load are represented
in Fig. 153 (c). The resultant shear, due to both loads combined, is repre-
sented in Fig. 153 (d). Note that, between v and r, the addition of W, with
its positive shear, reducea the negative shear due to the uniform load, and
that, between r and z, the addition of W reverses the negative shear; also
tliat it shifts the zero point from t to r.

For Continuous Beams, see Beams, under Strength of Materials.

lyFLCSKOE DIAOBAUa

339. The end

maEaents, Bheus and st

IDS. due to a oiTea load, and eoiuequentlr (ha

iDSuence DtiiK''"'! for Homenta.

in Fig. 154 (b). a'af is the momeat iuBueDoe dutgiam lor the

H eilicle coiioeiitrBt«d load. W.f

^ 3*l^Inri£.lM,leti
ft aupport.

beTuUblediatanoeoftheload, W,t
my position of W, 'the left end^esc-
tlon, n, i» — W . 4 ; and the moment of that feaetion about c, — R . ii.
-W, j.j. TheriihteudraaetioiiiaR'-W^^.&nd its moment, about
« is R' (1 — 1,) _ W^l-'« — V).

SokoicaaWisbetwesnbandctbemomeiitatciB - R . ir -W.j.u.

343. Sine* W, v and I are eooBtatit, the momeot, at c, while W ia between

b and 0, ifl ptoportional to the variable distance, x, of the load from b. It

„ ly, fromO. TrfieoWiBatt, toJUmBximuiDvaluo,

1 317. Hanoe, if the ordinate, e* M, bo made
■ M.thBntb. - ■ •

thenfoi

H, when W . .
equal, by uale.

poeiljon, rf. flor/. of W, between cand b. aaivtia by the eorreapoodiiiK ordi^
nate, d'. ^ or f. to the line 6' M. Similarlv, the momenta, at e, for any
positions of W between e and a, are given by the ordinatfls to the line a' M.

momemlB of a load. W. - u
Each ordinate must then bf
in the comflponding unit , in

:ruot the m
(Iton, Ipoi

thousand kilo^a.
eeponding load, i

Henoe, ordinate.

s of the diaciam, a-W/, is - -^ M - -^

L as the load, in thia diMuasion, ocinipiea diflerent p.

450

STATICS.

346* If a load, ■■ 1, be distributed over a length, « 1, at e. Fig. 155 (a),
the resulting moment, at c, may be represented by the area of the rectangle
standing on c\ Fig. (6), the height of said rectangle being the ordinate, c' li,
and its length ■= 1. Similarly, the moment, at c, due to a uniformly dis-
tributed load, « /, of 1 per unit length. Fig. (a), may be represented by the
sum of the areas of the rectangles between e' and /', Fig. (b) ; and, if we sup-

{>ose the load, e /, Fi^. (a), of 1 per unit length, to be divided into a very
arge number of very narrow vertical strips, the resulting moment, at c, may
be taken as represented by the area of the shaded trapezoid over e' K Fig.
155 (b) . The moment, at c, due to a load of p <lbs., tons, etc.) per unit length,
and occupying the same length, e /, is » p X area of trapezoid over ef f. Fig.
155 (6).

347* Hence, the maximum moment, at e, due to a uniform load of p Cbs^
tons, etc.) per unit of length, occurs when that load covers the entire span.

This maximum moment is ■■ p X area a'Mfr', — p -^ — • See H 345.

Flgr* 1S5.

FliT* 18««

Influence Diagram for Shear.

348. Under a single concentrated load, the shear, at any point between
the load and either support, is »■ the reaction of that support. See ^^ 326
and 327.

349. In the shear influence diagram, Fig. 156, as in the moment influence
diagram, Fig. 154, let I be the span ; x the variable distance of tiie load, yi^
from the right support, 6, and y the constant distance, ac, oi a given point,
c, from the left suppo^, a. Then, for any position of W, the left end reac-
tion, H, or the shear, S, at any point between the load and the left support, is

«> W . y ; and ihe right reaction, R', or the shear, S\ at any point between

I y

the load and the right support, is =- W . — j — .

350. The influence line for shear, like that for monaents, t 344, is usually
eonstructed for a load = unity, so that S =» R ■- -7-; and S' — R' —

I

I

Each ordinate of the shear diagram must then be multiplied by W*

in order to obtain the required e^ear.

351. Since W ( -» 1) and I are constant, R and S vary directly (and R' and
S' inversely) with x. Thus, when W ( = 1) is at b, we have a; — 0 ; S - R
= 0, and S' = R' = W = 1. When W ( = 1) is at a, we have x - /; S - R
-= 1, and S' « R' = 0. Draw a' a" and 6' 6* each - W ( - 1), and jdn
a" b' and a' b". Then, with W at c, the (positive) shear, S, at each point, as
/, between c and a, is given by the ordinate, c' g, to the line a" b' ; while the
ordinate, c' K to the line, a' 6^ gives the (negative) ^ear at each point, as «,
between c and b.

352. Similarly, with W at e, the ordinate, e' t, gives the (positive) shear at
each point, as c, between e and a; while ^ p gives the (negative) shear at
each point between e and b.

INFLUENCE SUGRAHB.

461

■e total chugs

353. It will be noticed that, u tha load, W, paa
other of any point, u c, the shear st that poiat is re'
in shear being - he- + e" e - hg - the toad, W.

3M. With a load, W ( - 1), at », the shear at c is - 0. See 1 3S1. As
the load advaaoea from b toward a, the positive shear. S — R — -j-i at c iit-
ereaaes in pniportion to the ordinata to the line I/ g, becoming — c* ir ••
"T", when W ii just to the right t^ e. With W just to the left of <; we have;
■Motive shear ate-S" — R' — e"* — y. But as W proceeds from c to o,
this negative shear, at c, deenAoee in proportion to the ordinatea to the tins
■ -* "—ooniing O when W reaoheg a. Tbos, a'kgb' is the eh — '-"

diagram for^e pi^Dt. e. SiniiUu'l)'. a'ptb' is tJia sliear ii
tor the point, e, etc.

3AA. IF a series of nenib UDlfonn and equldiBtaat oo:
SDoh aa the wheel loads of a locomotive and train, cameupo

__ ., ntrated loads.

,_ ,— ., id train, come upon the span, at the

- -. b, and advance toward a, the shear at c evidently increaaes until the
first load reaches c. It is then euddenly diminished, by an amount = tha
" ' — -* - ■■ •'■ •' o diminish, as each

iinilarmlji diitrAuM load, of ui

first load, aa
wheel pasMHi
SOS.

3M. With
Mini"-- "

of the
wbent

/,the^

857. Similariy, the sheaiB at I
gram, afptV.

358. Tig. 167* showstheinfluenoe diagrajil,ad

aeef

ir unit length, moving

'ered by the load, portiona

_.-_ . - . _ taken as neptLve, Thus,

n the head of the load naohes e. the (positive) shear at e is given by the
of the triangls, Vi^t. With head of load at e^ the shear at c nscbes ita
imum, and is given by the area of triangle, V e B. With head of load at

--^ ' Vc'if — areaf c-^n.

nax by the areas of portions of tb* dia-

nv. 107.

pofait 6, tor a given UBiformly distributed
ooming upon tbr - '

pcdntO, passing aoross it, and leaving it at point 8;

. ^^ „ Jiaenuii,0« 15. tor the left support. 8r and that.

w Die. tor (be rj^t support. 0.

For the action of internal resisting (orcee in beams and trasses, ee*
Traiwvene Strength, under Stiength (rf UaterUls, and Stresso, under

462

STATICS.

Relation between Moment and Shear.

359. The shear, at any point in the span, is simply the rate at which
the bending moment is changing: at that point.

360. Thus, in Fig. 158, the moment, M, Fig. (b), at the support, 6, due to
the concentrated load, W. of 6 Ibe., ia-Wi-6X4-24 ft.-tbs.; but,
between the support and the load, the moment is decreasing at the uniform
rate of 6 ft.-lbs. for each foot of x, or 6 ft.-lbs. per foot — 6 lbs. ; and this 6 lbs.
is the uniform s?iear, V, Fig. (c), throughout the beam. Hence the shear-
diagram. Fig. (c), is a horizontal line ; i. e., its ordinates are of e^ual length.

361* Again, in Fig. 159, the shear diagram ordinates between a'' and o^\
Fig. (c), are positive; showing the (algebraic) increase of the bending mom^it,
M, Fig. (6), as we proceed from the left support, a, toward the center, o, of
the span ; while the negative shear diagram ordinates, between </' and 1/\
show the (algebraic) decrease of the bending moment as we proceed from
the center, o, to the right support, h. At the center, o, the rate of change of
bending moment is zero, as is also the vertical shear.

^(C)

Fig. 1S8.

Fig. 159.

362« Both in Figs. 158 and 159, the bending moment, M, b constantly
changing; but in Fig. 158 its rate of change (—6 ft.-lbs. per ft. of span) is
constant. Hence, the moment dia^tim b a straight inclined line, and the
s?iear diagram b a horizontal line ; whereas in Fig. 159 the rate of change of
bending moment b constantly varying, being — 12 ft.-tbs. per foot of span
(shear — 12 lbs.) at the support, and diminidiing to lero at the center, o, of
the span. Hence, in Fig. 159, the moment diagram. Fig. (b), b no longer
straight, but curved; and the shear diagram, Fig. (c), is no longer horiaonta],
but inclined.

363. But, in Fig. 159 (c), the shear, V, or the rate of change of the bending
moment (although no longer corwton/, as it was in Fig. 158 (c)), nevertheless
diminishes uniformly, as we proceed from a toward &. Thus, at the point, 1,
Fig. 159, midway between a and o, the bending moment b changing at the
rate of 6 ft.-lbs. per foot, or half as fast as at a. Hence, the shear diagram,
although no longer a horizontal line, is still a atraight line; and the uniform
decrease ( » 6 ft.-lbs. per foot per foot) in the rate of change of the bending
moment, or the uniform decrease ( ■" 6 lbs. per foot) in shear, is indioated by
the horizontal diagram in Fig. 159 (d).

^ 364. In either Fig., let a straight line be drawn, tan^ntial to the moment
dbgram (&), at anypoint, c', and forming, with the horisontal sero line, a* 6',
an angle, A. (In Fig. 158, this line coincides with the moment diagram.)
Then the tangent of A b given by the shear diagram ordinate, e^, ooire*
spending to the point, c; or, for any point, V — tan A.

UOMWrt - Alffl> BHEASL 468

965. In Plff. 158, where this angle. A, Fig. (6), is eorutani, the shear ordi*
nates, Fi^. (e), are of constant length. In other words* the shear diagram.
Fig. 158, IS a Aorinm/aZ line.

366* Since the shear diagram ordinates represent forces (as in tbs., etc.)
and the abscissas represent distances (as in ft., etc.) the product of the dis*
tanoe between any two shear ordinates, multiplied by the mean of those ordi-
nates, is an area representing a mom«n< (in ft. -lbs., etc.). Tl^s moment is —
the difference between the two moments represented by the corresponding
ordinates in the moment diagram.

367* Thus, in Fig. 158 (&), the increase in (negative) bending moment,
between points 1 and 3, is»18 — 6»12 ft.-'lbs.; and, in Fig. 158 (c), the
moment represented by the (shaded) area, between the same two pomts, is
" 2 ft. X o lbs. » 12 ft.-tbs. In Fig. 159, the (algebraic) increase in bend-
ing moment. Fig. (6), between the left support, o, and the center, o, of the
qpan, is ->• 8 + 4 — 12 ft.-lbe.; and the moment, represented by the shear
diagram area (triangle) between the sanw two points, Fig. (c>, is

368. Again, in Fig. 159, at any two points equally distant from the center,
9, of the span, the moments are equal; or difference of momenta — lero:
and. since shcAr ordinates below the zero line, a" h". Fig. (c), are considered
as negative, the algebraic sum of the two corresponding shear triangles. Fig.
ie), is also ■■ sero.

Similarly, areas in fig. 159 (<0 correspond to differences of ordinates in
Fig. 150 (c).

464

BTREMGTH OF HATERtAU.

STEENGTH OF MATERIALS.

OEBTERAI. PRIBrCIPI.ES.

Art. 1 (a) Stress or Strain* occurs when force acts upon a body in sach
a way that its particles tend to moye (at the same time) with different velocities
or in different directions ; to do which they must either separate from each other
or come closer together. This occurs, for instance, when a body is so placed
as to oppose the relative motion of two other bodies ; as when a block is placed
between a weight and a horizontal table. In this case, each of the two oodles
(the weight and the table) imparts a force to the opposing body (the block) ; and
the stress is the opposition of these equal forces. The tendency of the jMrticles
of the block to separate or to come closer together calls into action the innerent
forces of its material, and these act between the particles and tend to keep
them in their original relatiye positions.

(b) Compression and Tension. If two opposite forces are simul-
taneously imparted to a body in the same straight line, the stress is either com-
pressive (when the forces act toward each other) or tensile (when they SLciJrom
each other).

Oompressive stress tends to push the particles closer together. Tensile stress
tends to pull them farther apart.f

(c) If two imparted forces, as a o, & o, meet at an angrle, as at o ;

then two equal ana opposite components, a" o and b" o, will cause compressive
or tensile stress in the body, while the other two, a' o and 6' o, unite to form the

i»"-4''

resultant, c o, which, unless balanced by other forces, moves the body in its own
direction, and, in doing so, produces another stress among the particles of th«
body. See (a), above.

(d) If tiie two forces are parallel, forming a " couple,** as in a punch
and die, the stress is a sbear (tending to slide some of the particles over the
others, see p. 499), and is accompanied also by a transverse stress (causing a
tensile stress in some of the particles and a compressive stress in others) as in the
case of a beam. The transverse stress is proportional to th^ distance between the
two forces (i. e., to the arm or leverage of the couple), so that, when they are very
close together, as in a pair of shears, the transverse stress is very small and u
neglected, and the shearing stress alone is considered.

(e) If two contrary couples, in different planes, act upon a
body, the stress is called torsion or twisting. See p. 499. Thus, torsion takes
place in a brake axle when we try to turn it while its lower end is held fast by
the brake chain.

(f) But the ultimate tendency of any of these forms of stress is either to
separate certain particles or to drive them closer together, as in cases (tensile or
compressive stress) where the two forces are in one lincf • We shall, therefore,
in these introductory articles, consider only this simple and ftmdamental form
of stress, assumine that it is caused by the action of two opposite imparted
forces, acting in tlie same straight line so that they are entirely employed in
causing the stress.

(gr) A stress may be stated in any unit of weight, as in pounds, and
is equal to one of the two opposite forces.

* For another use of the word "strain,** see Art. 2 (<f).

t Indeed, even in cases of compressive stress, it is only by the separaiion of the
particles that the structure of tne body and its inherent forces can be destroyed.

STRENGTH OF MATBRIALB. 455

Art* Si (it) Steetdft and Ri&|itarA. It appean firom ex|>eriment thai the
inherent cohesive forces called into action by the first application of any stress
are always less than that stress, hoWerer small it may be. In other woras ; any
stress, however slight, is believed to produce some derangement of the particles.
But the inherent loroeB inertaae with this derangement (up to a oertain point) -
and thus, in many cases, they become equal to the strera and so prevent further
derangement, when the stress exceeds the greatest inherent force which the
body can exert, the particles separate to such an extent that the inherent forces
cease to act. The body is then said to be broken, or ruptured.

(b) Dlflfevemt mAteiiaU belut'r* jrmiry cllffereiitljr when under stress.
Brittle ones seem to resist almost perfectly up to a certain point, allowing no
perceptible deran^ment of the particles ; and then yield suddenly and entirely.
In ductile materials, on the contrary, considerable derangement takes place
before the inherent resisting forces finally yield.

(«) The nltlnuito limaa of a body n that which is just sufficient to break
it or crush it, or, in short, to destroy it« stmoture so that it can no longer resist
In other words, a stress just less than the ultimate is the greatest stress to which
the body can be subjected.

Cstifttiona In brittle materials, such as brick, stone, cement, glass, cast-iron,
etc., especially when subjected to tension^ the point of rupture is dearly marked,
and hence the ultimate strength mav in such cases be stated with precision.
But with ductile or malleable materials, such as copper, lead and wrought iron,
especially when under eompresaiony it is often difficult or impossible to state the
ultimate strength definitely. For instance, a cube of lead may be gradually
crushed into a thin flat sheet without rupture. In other words, there is
practically no load which can break it by crushing. In such cases, we may
arbitrarily assume some given amount of distortion as marking the point of
ultimate stress. Thus, by the " ultimate " load of a rolled iron beam
we mean "that one which so cripples the beam that it continues to yield
indefinitely without increase of load." Such assumptions, however, necessarily
give rise to some ambiguity, and care should therefore always be taken to define
or to ascertain clearly m what sense the term " ultimate stress " is employed.

The ultlnaato atrenf^ of a material (or, more briefly, its atreng^) is
the greatest inherent force which its particles can exert in opposition to a
stress. In other words, it is that inherent resistance which is just e^ual to the
ultimate stress. Hence strength, like stress, is stated in units of weight, and
we may use the terms " ultimate strength " and " ultimate stress " indifferently,
as denoting practically the same thing.

{&) For want of a convenient and appropriate name for the ehang^e of
ihape caused by stress ; modern writers have, rather unfortunately, given to
it the name of strain,* which, in ordinary language, is used to signinr itrest^
■8 above defined. We prefer to use the word ** streteh '* i >r change of shape,
in inches, etc (regarding ocmipretfion as negative "stretch"), and "strain*' or
** stress'' for the action of the two opposing forces, in pounds, etc.

(e) By the <*Iemf^*> of a body, we mean its dimension measured in the
Une of the strei8 ; and, by « are««'' the area of the resisting oross section at
light angles to that line. Thus, if a slab of iron, 2 inches thick and 10 inches
square, be laid flat npon a smooth and horizontal surface, and if a load be placed
upon it so as to be uniformly distributed over its upper flat surface, the " length "
is 2 inches, and the " area," 10 x 10 « 100 square inche6.t

(f) Units SMlopted. Unless otherwise stated, we shall understand the
stress in any case to be given in pounds, the stretch and the length in inches,
sad the area in square Inches.

•The word ** strain" is not thus defined, even as a scientific term, in either
Webster's or Worcester's dictionary.

f Undet stresses approaching the ultimate stress, the area of the cross section
generally increases under compression, and diminishes under tension, to diflTer-
•nt extents in diArent materials ; but we are here concerned only with cases
within the limit of elasticity, (Art. 4 a. p. 458) and in such cases the change of
area is generally very slight and may be neglected.

456 BTRENOTH OF MATERIALS.

Art. 3 (a). If th« total atreaB (in lbs., etc) upon a body be divided by the

area of tbe retistiog surface (in square inches, etc.) the quotient, ?"''^^ , ia the

area

mean atreM per unit of area, or (as it is sometimes called) the intensity oi
the stress. Or,

StreM per unit of area =- total stress

area

Thu2, if a bar of iron, 2 inches wide by 1 inch thick, (having therefore 2 square
inches of area of cross section) and 10 feet long, be subjectmi to a total tensile
stress of 20,000 lbs. in the direction of its (10 ft.) length, we have

mean stress ) total stress 20,000 lbs. mnnn ik- «o, .««--.» 4««k

i>Ar unit nf arAft f "" ™ JT-^ T" — l^j^OO ibs pcrsquaro inch.

per unit 01 area) ^^^^ 2 square in.

Caution. Strictly speaking, the stress on a surface is seldom distribated
uniformly over it. Thus, in the case of the bar just referred to, if the stress is
applied by means of grips, clamping the sides and edges of the bar, the stress
per square inch near those sides and edges is probably greater than that near
the center of the bar, because the stress is not perfectly and uniformly trans-
mitted from the outer to the inner fibres. And, in cases of compression, the
load, instead of being uniformly distributed over the surfkce, as it appears to be,
is often in fact supported by a few projecting portions of it. In practice, theee
considerations are often of the greatest importance, but in studying the
principles of resistance, we may, for convenience, temporarily neglect tnem, and
assume the stresses to be uniformly distributed over their respective areas.

(b) If the total stretch of a body (in inches, etc.), under any given stress, b«
divided by the original length of the body (in the same measure), the quotient
is the etretoli per unit of lenfftb. Or,

Stretoh per unit of lencth total stretch

original length

Thus : if the foregoing bar. 10 feet (or 120 inches) long, is found to streteh
.04 inch, under its load of 20,000 lbs. total, or 10,000 lbs. per square inch, we hav«

stretch per unit of length _ total stretch .04 ftAAM <««u «^« i^^u

unSir said lo«l "^ - originri length " IJO— -"^ ^^ "*' *^

(c) The Modnliia of BUastielty. In materials which undergo a per-
ceptible stretch before rupture, it has been found by experiment that up to a
certain degree of stress, called the limit of elasticity (Art. 4 a, p. 458), the n^io^

Jotaljstress ^ ^ ^^^ given body, remains very nearly constant. In other
total stretch

words, within the limit mentioned, equal additions of stress cause practioally
equal additional stretches.

In order to compare bodies of diffbrent dimensions, we state the same faet by

paying that, within the elastic limit the quotient, stress per unit of area

stretch per unit of length
remains practically constant.

This quotient, as found by experiment with any given material, is oalled the
Modiilaa of Slaatlcit^ of that material, and is usually denoted by the
capital letter B. It is of course expressed in the same unit as the atreaa p«f
unit of area, as, for instance, in pounds per 'square inch; but it is usually sti^
simply in pouiuft, the words " per square inch *' being understood.

(d) It will be noticed that the greater the stress required to produce a fflven
stretch in a body, the greater is its modulus of elasticity. Hence the moaolui
B is a measure of the resistance which the body can make asainst a chanse in
shape. This resistance we call the *< elaetidtjr >' of the body, althouffn in
every-day language (and, indeed, often in a scientific sense also) we apply th*
term " elasticity " rather to the ability of a body to sustain considerable dutof"
tion without losing its power of returning to its original shape.

1

MBrararra of hatshi^ia. 457

area
uid sinot

ttnUik por uoii of length >» .■■;■■■, — r-= -«- «

onginal length

1P8 may tnA the modalns of alastlcity of any matariel, from •zperimflnl npoa
any apecinien of it| thuas

BCodiavaar elMtleitr - total etieee X original length .... 0)

Ftom this we haxe the following equations:

Total atreasy in lbs. modulus of total stretch .. area
i«qaired for a given — elasticity in lbs X in inches '^insquateina
total streteh, in inches per square inch original length in inohea

• • •

Streaa per «nlt of area^ modulus of stretoh oar

In Ibo. per square inoh, required » elasticity in lbs. X miitof lanffth * *
for a given stretch, in inehes per square inoh w«0iaii«u

Total stretcl&y in total stress ^ original length, in inehea
Inches, under any stress- in lbs. ^ modulus of elssticity, ^ area, in • • • C9

in lbs. per square inch ^ sq. ins

oilsinal length w^ stress, in lbs, per square inch ^t

"" Ininchea ^ modulus of dasticity in lbs. per sq. inch' ' '

■tNteH per imtt of leaKtl& - stress, in lbs, per square inch „.

modulus of elasticity, in lbs per sq. in.

The modulus of elasticity is used chiefly in connection with the stiflhesa d
beams. In any beam, supported at both ends and loaded at the center :

^Mtleltr*^ — (tolba- **" 8pairoffKSam,kribs.) ^ ^"in incSS" ... (7)
Iba. per so. inoh aa \y deflection, v^^ moment of inertia

^- **X ininches ^

^* m ^ (W + ^w)<» ... (8)

If the beam ia rasftMi^iilar, this beoomea

I «r i ^<^ 4. ^ webrht of dear \ ^ eube of spea
ffSk — \fa> itM. ^ apan ofbeam, in lbs/ ^ in inches
Wm ner square inch a v deflection, ^ breadth, v^ cube of depth
'^ ^ «X ininchea ^ inches ^ ininchea

OoResponding fbrmnlss for modulus of elasticity In beams otherwise sup*
ported and loaded, may be readily deduced from those for deflection

{t) If equal additions of stress could produce equal additional stretches In a
body to an indefinite extent, both within and beyond the elastic Umit, then a
stress equal to the Modulus of Elasticity would daubU the length of a bar when
applied to it in tenHon, or would aAortea it to gm> when applied in ccmmrtmkm
In other woide, if equation (5),

total atretch. mm <»^^sl length ^ stress per square Inch
•**^ Ininches ^ modulus of elasticity *

held jraod &eifond the tiastio limit, as it does (approximately) within that UmIL
and u we oonld make the stress per square inch mimI to the moduluafli
elasticity, we aheuld have total stretoh ■■ original length.

(»)

458 flTBUrOTH OF MATBBIAUL

For ezamplei a ono-inch sqaaie bar of wrought iron will, within the limit ol

elasticity, stretch or shorten, on an average, about Tiioo of its length nadei
each additional load of 2240 lbs. If it could continue to stretch or shorten
indefinitely at this rate, it is evident that 12000 times 2240 lbs., or 26 880 000 Iba^
(which is about the average modulus of elasticity for su^ mrs) oould either
■tretch the bar to double its length or reduce it to zero.

If equal additional stresses applied to a bar could Indefinittiy piodaee
■tretches, each bearing a constant proportion to tA« inereated length of the har^ II
in tension; or to the dimvniehed length, if in eommretaion; then the same load
which would double the length of the bar if applied In tension, would rediio9
it to hcUfiU length, if applied in compression.

(S) ^^ S^^® below a taMe at aveimge Modnli ot Klaatlciiyy in round
numbers, for a few materials ; remarking, by way of caution, that, even in the
case of auctile materials, the stretches produced by stresses within the elawtie
limit are so small, and (owing to difforences in the character of the material) so
irregular, that a satisfactory average can be arrived at only by comparing many
experiments *, while, in the case of materials^ such as stone, brick, etc., where
almost no perceptible stretch takes place before rupture, it is scarcely worth
while to ^ve any vidues as representing the actual moduli* Thus^ eighteen
eimeriments upon a single brand of neat cement for the St. Louis bridge, indi*
cated a Modulus varying from 800 000 to 6 980 000 (!) pounds per square inch
In tmeion, and from 500 000 to 1 600 000 in compression,

(h) Owing to the fact that the stretches within the elastic limit are 8eldo■l^
if ever, exacUy proportional to the stresses, but only approximately so, th#
modulus of elastioity, as found by experiment for a given material, will
ftnerally vary somewhat with the stress at which the stretcn is taken.

Art. 4 (a) The stress beyond which the stretches in any body increase peiw
eeptibly faster than the stresses, is called the limit ot elasticity of that
body. Owing to the irr^ularity in the behavior of difibrent specimens of the
■ame materiiu, and to the extreme smaHness of the distortions caused in numt
materials by moderate loads, and because we often cannot decide just when the
•tretch begins to increase faster than the load, the elastic limit is seldom, if
%ver, determinable with exactness and certainty.* But by means of a Isffge
number of experiments upon a given material we may obtain useAil avenge
or minimum values for it, and should in all oases of practice keep the stressee
well within such values ; since, if the elastic limit be exceeded (through rnl^
ealculation. or through subsequent increase in the Stress or decrease in tht
atrength or the material) the structure rapidly fails. The table^ below, giw
approximate avenge elastic limits for a few materials.

(h) Brittle materials, such as stones, oements, bricks, ete., ean eeaiesty be nld

10 nave an elastic limit; or, if they have, it is almost impossible to delemiiBt
it; since rupture, in such bodies, takes place before any stretch ean be aati»
ihctorily measured. Thus, in tne 18 specimens of one brand of oemen^
referred to in Art. 3^ above, the experiments indicated an elastic limit varyiBg
between 16 and 104 (I) pounds per square inch in tetution, and from 424 to IMl
in eomprsssion,

(e) Experiments show that a small permament **9Ut*' (stretch) fWObeHf
takes place in all cases of stress even under very moderate loads; but oidlnariM
it first becomes noticeable at about the time when the elastic limit is ezeeeded.
Kany writers define the elastic limit as that stress at which the first marked
pannanent set appears.

(d) The elaatic ratio of a material is the quotient, elastic limit ^ ]|
^ ' H -I ammnte strangS

11 naually e9q>res8ed as a decimal i^aetion.

* The U. S. Board appointed to test Iron, Steel, Ac., found a variation of nearly
4000 lbs. per square inch in the elastic limit of bars of one make at rolled Iroi^
px'epared with great csre and having very uniform tensile stnngth ; and, ia
another very carefiiUy made iron, a difference of over 80 per cent between twa
bars of the same sise. Beport, 1881, Vol. 1, p. 81.

8TSBHGKFH OF MATBBIAI&

469

(e) Flartitc McMlall and Elastle Itimito. Approximate aTertgecf
JB =* elastic modnlns, in inillions of pounds per square inch ;
f = stretch or compression, in a length of 10 feet, under a load of
1000 pounds per square inch.
= (10 X 12 X 1,000) -i- (1.000,000 E) ;
Sg = stress at elastic limit, in thousands of pounds per square inch.

Metals.

Iron, cast

" " ordioarily

" wrought*

Steel, structural*

Brass, cast

•* wire

Copper, cast

" wire

liead ,

Tin, cast

Bronzes

Stones, etc.t

Masonry t

Wood J

E

10 to 80
12 to 15
27 to 81
" to ♦'
8 to 10

12 to 16
10 to 14
10 to 14

0.8 to 1.0
6 to 7

13 to 15
4 to 8

0.5 to 2
1.5 to 2

0.012 to 0.004

0.010 to 0.008

0.004

0.015 to 0.012
0.010 to 0.007
0.012 to 0.009
0.012 to 0.009
0.150 to 0.120
0.020 to 0.017
0.009 to 0.008
0.030 to 0.015
0.240 to 0.060
0.080 to 0.060

4to 8
6 to 7

20 to 40

34 to 38
5to 7

14 to 18

6to 7

8 to 12

Ito 1.2

1.4 to 1.6

14 to 15
1 to 2
Art. 4 (b)
5 to 7

(/) Yield point. Commercial, Relative or Apparent Elas-
tic l^lmlt. In testing specimens of iron and steel, it isconimonlj found that,
at a stress slightly exceeding the true elastic limit (Art. 4 a), the stretch begins
to increase without further Increase of load. This point is usually called **the
yield point," or ** the elastic limit" in commercial testing. The French Com-
mission on Methods of Testine the Materials of Construction called it the
" apparent elastic limit." The late Prof. J. B. Johnson (" The Materials of Con-
struction," New York, John Wiley A Sons, 1906, p. 19) applied the term, " rela-
tire or apparent elastic limit" to that point on the stress diagram at which the
rate of deiormation is 50 -pet cent, greater than at points below the true elastic
limit.

*In rolled iron and steel, the elastic modulus is remarkably constant for all
grades. In wrought iron, the elastic limit depends chiefly upon the degree of
redaction of eross section in rolling; the smaller sizes having the higher elastic
limit. In steel, this eflisct is less marked.

f See Art. 8 (g) and (h), and Art. 4 (a) and (b).

J In wood, "the extreme fiber stress at the true elastic limit (Art. 4 a) of a
b^m is practically Identical with the oompressive stress endwise of the material,"
Uble, p. 958. See discussion by a T. Neely, in '* Timber Physics," 1889 to 1898,
by Fifibert Roth, House Document No. 181, 55th Congress, 3d Session, Wasb^
ington, 1899, p. 374.

460 srauBRoxH of xatsbxaul

Art. B (m\ Brnw/ntmrnp* If a name givni to Um work (as in inoli-poaiidbi)
which must be done in order to produce a certain edeteli in a given body.
This work is equal to

resilience _ said stretch v. mean stress in pounds employed in producing
tn inch-pounds "" in inches ^ the stretch.

The totcU resilience is the work done in causing rupture. The eUuUe resilience
(firequently called, simply, the retiUertee) is that done In causing the greatest
Itretch possible loithin the elastic limiL

(b) Suddenly applied loads. Place a weight of 4 lbs. in a spring
baiaDoe, but let it be upheld by a string fastened to a firm support in such a
way that the scale of the balance shall show only 1 lb. By now catting this
string with a pair of scissors, we auddenly apply 4 — 1 as* 8 lbs. ; and the weight
will descend rapidly, until, for an instant, the scale shows about 1 + twice
8 «■ 7 lbs. In other words, the load of 3 lbs. applied suddenly (but without jar
or shock) has produced nearly iwiee the stretch tnat it could produce if added
grain by grain, as in the shape of sand.

For, when the load is first applied, the inherent forces, as noticed in Art. 2 {a\
are insufficient to counteract its stress. Hence the loaa begins to stretch the
spring. The work thus done is equal to the product, suddenly applied weight
of 8 Ids. X the stretch of the spring ; and it has been expendea (except a small
portion required to counteract friction) in bringing the resisting forces into
action, thus storing in the spring potential energy nearly

sufficient to do the same work ; i. «., to lift the weight (8 lbs.) to the point (1 lb.
on the scale) from which it started. But a portion of this energy has to work
sgainst friction and the resistance of the air. Therefore the weight does not
nse quite to its original height.

The shortening of the spring nearly to its originsl lenath has now reduced
its inherent forces almost to aero ; and the weight again nils, but not so flur as
before. It thus vibrates through a less and less distance each time, and finallT
comes to rest at a point (4 lbs. on the scale) midway between its highest ana
lowest positions (1 lb. and 7 lbs.) Thus, within the Umit of elasticity, a losUI
applied anddenljr (though without shock) prodnees tenaporaaiilj^ a
•tretoli nearly eqnal to tiwlee tliat iwluoik It could prodne* If
applied gradwalljr | i. e^ twice that which H can maiiintain after it comts
to rest.

Remark. If the load Is added in small instalments, each applied suddenly,
then each instalment produces a small temporary stretch and ailerwaid matn«
tains a stretch half as great. Thus, under the last small instalment, the bar
ttretohes temporarily to a length greater than that which the total load can
maintain, by an amount equal to half the small temporary stretch pradaeed hf
the sudden application of the last small instalment.

(•> The Bfodnlne of ESlastle Reallleaoe (often called, alnnily, tte
Modulus of Resilience) of, a material, is the work done upon one saMc Msa qf II
by a gradually applied load equal to the elastic limit. Or,

HT/wiiiiiia stretch in inches mean stress

^rSSautm^ - P^ *»»«* <>/ ^'^^*^ X ii^1h9. per square Uuk
or resilience ^^ ^^le elastic Umit causing that stretch

B^ as Is nsually done; we assume this mean stress to be >^ the elastle Umt^
then, by formula (6)

Modalus _ elsstic limit ^ ^ ^^,^ H^l
or fesilienoe modulus of elasticity ^ ^ ^^

■« 14 aquare of elastic limit
modulus of elasticity

elastle resllienoe of any piece is then
PMHwDoe •» modulus of reaiUenoe X ▼olome of piece in onblo tochifc

STRENGTH OF MATBBIAUL 461

The modulus of reBiUence of a material ia a measure of its oapaolty tor reeiBt-
Ing shocks or blows.

Elastic Ratio. The elastic ratio of a material is the ratio between its
elastic limit (Art. 4 a, . and its ultimate strength (Art. 2 c.

Thus, if the ultimate tensile strength of a steel bar be 70,000 poudds per
square inch, and its elastic limit in tension 39,900 pounds per square inch, its
(Blastio ratio is

89,900

70,000

= 0.67.

Inasmuch as it is now generally conceded that the permissible working load
of a material should be determined bj its elastic limit rather than by its ulti*
mate strength, it follows that, other things being equal, a high elastic ratio is
in general a desirable qualification ; but, on the other hand, it is possible, by
modifying the process of manufacture, to obtain material of high elastic ratio,
but deficient in " body " or in resilience— t. e.. in capacity to resist the effdct of
blows or shocks, or of sudden application or nuctuation of stress.

In the manufacture of steel it is found that the elastic ratio is increased by
increasing the reduction of area in hammering or rolling, and that the rate of
increase of elastic ratio with reduction of area increases rapidly as the reduc-
tion becomes very great. This is indicated by the following experiments by
Kirkaldy on steel plates :*

Plates 1 inch thick, mean elastic ratio 0.63
" H " " " " 0J5&

*• U " " " " 0.64

" % " " " « 0.61

* Annual Report of the Secretary of the Navy, Washington, 1885, Vol. L p.
499; and Merchant Shipping Experiments on Steel, Parliamentary Papar, Gl
2897, London, 1881.

462

STRENGTH OF MATERIALS.

Art. 6. A seetlon wMtmy be weakened by ineremmiMBg its
width. On pp 400, etc., we considered the case where the width o"^ .he base
if fixed and where the point of application of the resultant of the forces actmg
upon it is shitted to diirerent positions along the base. We will now notice the
case where the resultant is applied at a constant distance ftrom one end of the
base, but where the base is of yaryine width, so that this constant distance may
be equal to, or greater or less than, the half width of the base.

Let Fig. £ represent a side view of a bar of uniform thicknees = 1,* but (m

Fig.B.

Scale of Unit Vremmrtt fbr FIff. B.

h i k h i k ^

1.761
1.60

(S

1.26

1.00

0.76

0.60

•40

0JB6

0
-0.86

-0.601

\

\

\

\

\

F

^'

P.

\

1
I

*^^^^^m

tr —

N

/"

^^»

*1

-t

!!««

i:

^

^

/

s^

/

•

\
\
\

V,

^

.^

/
/

\

^^

. J

Urn

I_5«i

t^

■e«

\
\

V

j

^-v

"-^,

Infl#^

prei

•an

?:•!

VPP

l»A

1

•

1

■*-.-

'-■■III

m^Z

LM

0.86 0.60 0.76 LOO 1.86 1.60 8.00 8.60 8.00

Widths (ab»a)

shown) of varying width, and subjected to pressures, the resultant P of whieh
is =: 1,* and passes through the center of that section ab whose width is 1.* f

* We adopt the value 1 for the pressure P, the width ab, and the thick ne^^
merely in order to facilitate the explanation. It is not essential to the applica-
tion of the principle.

t We here suppose ourselves dealing with a perfectly rigid and homogeneous
material. In practice, these values would be more or less modified by yieMUnc
of the particles under stress, by unevennesses in the surfaces of the sapposeS
cross sections, etc. Nevertheless, the general principles here laid down hold
good.

STRENGTH OF HATEKIALS.

463

The pressure per unit of area of croas section (or "unit stress ") in the section

P P
ab is then "7-77^; = — = P=1* and may be assumed to be uniformly dis-

trlbuted over it.

But at other sections of the bar the resultant is nearer to one edge than to the
other, and the unit stress can then no longer be assumed to l>e uniformly dis-
tributed over the cross section, but, as explained on i>p. 400 to 408 is a max-
imum at the edge nearest to the resultant, and diminishes gradually and uni-
formly to a minimum at the farther edge.

This iis indicated by the shaded triangles, etc, fn Fi^g. E and: by the curves in
Fig. F, which show, for the several sections in Fig. £, the mean unit stress*
and the unit stresses at the upper and lower edges respectively, calculated by the
rules on pp. 401-404.

These stresses are also given in the following table :

p
Vnlt Stresses in Fig. £ ; the unit stress -- in section a b being taken as l.f

flArtinn

Width.

Stress per unit of area of cross section.

Mean.

At lower edge m e.

At upper edge nf.

ef

cd

ab

mnf

4.00
8.00
2.50
2.00
1.50
1.25
1.00
0.75
0.60
0.25

0.25

&

0.50

1.00

4.00

0.8125
1.00
1.12
1.25

\>k

1.00
0

— 32I

— 0.8125 1

= '^2

— 0.25

0

0.32

1.00

2%

8.00

40.00

It is imnortant to vi<»tiee that for a given force P, and for widths leas
than Sab, the strongest section of this bar is not the toidat one^ but that (a b) at
wbJeh the resultant P passes through the center of the section. In other words.
a bar may be weal&ened by additions to its cross section if

those additions are such as to cause the resultant of the pressures to pass else-
where than through the center of any cross section. This fact is entirely inde-
pendent of the ufeif^fu of the added portion.

Among the sections wider than ab, the weakest is that (ed) whose width
is — IJiab, At that section the lowfer edge vie has its maximum unit stress

cd

be-

J sa= 1/^X — t] while at d in the upper edge there is no pressure. Beyond

tne upper edge n/ is in tension X and the unit pressure along m« decreases,

p
coming again = — at ef, where the width ef iB = Sab, and decreasing still

further with further increase in width.

* In the case discussed on pp. 400 to 403, the mean pressure un = — ,

uv

remained constant so long as the entire surface u v was called into play. Here,

on the contrary, the area of the section varies. Hence the mean unit pressure

▼ariee also, and inversely as the area.

t See foot-note *, p. 462.

X In the present discussion, as well as in that on pp. 400 to 403, we have
assumed cases of eontpresxitm for illustration, but the principle involved applies
equally to cases where the force applied is tensile. In such cases, however, the
terms " pressure " and " tension " are of course reversed.

^ The unit stresses at the ed^es in section i k are too great to be shown con-
Teniently in either figure ; while those in section m n (as the table shows) far

JO

exceed the limits of the figures. The pressure at k would be — =: cc (infinity)
were it not for the tensions in the lower part of the section.

464

STRENGTH OF MATERIALS.

When the width becomes less than that at a6, sb BXgk^ etc., the vpper edge of
the bar comes nearer to the resultant than the lower edge, and hence receives
the maximum pressure.

When the width = Vafr, as at gh, the distance of the resultant from the
upper edge is ^ the width of the section. The pressure at the lower edge is

PI P

then = 0 ; the mean pressure in j^ A is — r X r^ = 1/^ X — r, and the pressure at

a 0 0.76 o, 0

p
the upper edge is twice the mean pressure in g A, or 2^ X —r*

When the width becomes less than ^ a 6, as at < A and mn^ the pressure at the
lower edge m « becomes negative or tensile.* Thus, when, as at i jfc, the width
is -» V^aft, and the resultant passes through the upper edge, the unit pressure

P P

at that edge is = 8 x ^ , while the lower edge sustains a terwion of 4 X — ; and.
" ah ah

as the section is further reduced, these stresses are still further and very rapidly

increased.

The condition of those sections (such as m n) where the line of the resultant

passes outside the section, is similar to that of the section m n of a bent hook

sustaining a load, as in Fig. G.

Mi'iff-n

Ftir*«-

Fly.H.

Messrs. William Sellers h Co., of Philadelphia, had occasion to test a number
of cast-iron beams, each having a large circular opening, as in the annexed
figure. These beams broke, not at the smallest section directly under the center
of the opening, but a little to one side, where the section was deeper, as indicated
in Fig. H .

8TRKNQTH OF KATBBIAI^ 466

Fstl^rne of BIal;erlaUu In the fbllowing artioles on StreDgth of Mate-
rials, the ultimate or breaking load i8 that which will, during its first application^
rupture the given piece within a short time. But Wohler's and Spangenberg's
ezperiuents show that a piece may be ruptured by reMated applica*
tiotts of a load much Uu than this ; and that the oftener tne load is appued the
less it needs to be in order to produce rupture. ThvB, wrought iron which re*
quired a tension of 53000 lbs per sq inch to break it in 800 applications, broke
with 35000 lbs per s() inch applied about 10 million times ; the stress, after each
ap^ication, returning to zero in both cases.

The d^ between the maximum and minimum tension in a piece subjected to
tension only, or between the max and min compression in a piece subjected to
comp only ; or the sum of the max tension and max comp in a piece subjected
alternately to tension and comp ; is called the raiig« of stress in the pieee.
Stresses alternating between 0 and any point within the elastic limit may be
repeated many million times without producing rupture.*

For a given number of applications, the load required for rapture is least when
the range of stress is greatest. If the stress is alternately comp and tension,
rupture takes place more readily than if it is always comp or always tension.
That is, it takes place with a less ran^ of stress applied a given number of
times, or with a less number of applications of a given range of stress. For a
given ran^e of stress and given number of applications, the most unfavorable
condition is where the tension and eomp are equal.

The above facts are now generiAy taken Into consideration in designing
members of important structures subject to moving loads. For instance, Mr.
Jcfl. M. W^ilson, G. R, Mem. Inst. C. E. (London Eng.K Mem. Am. Soc. C. E., uses
the following formulae for determining the ** permissible stress*' in iron bridges,
in Iba per sq inch ; in order to provide th« proper area of cross section for each
member.

For pieoea subject to one kind (^f stress only (all comp or all tension)

. / min stress in the piece \
' \ max stress in the piece /

For a piece subject aUsmately to oomp smd tension^ find the max comp and the
max tension in the piece. Call the lesser ot these two maxima*' max lesser",
•and the other or greater one, *' max greater '*. Then

max lesser \
2 max greater/

For a piece whose max eomp and max tension are efrnU^ thia becomes

«t(l-

..„t(l-4-)-

u
2

The above a Is the permissible tensile stress in lbs per sq inch on any mem-
ber ; but the permisslole compressive stress Is found by " Gordon's formula" for
pillars, p 496, using a (found as above) as the numerator, Instead of/. For a In
the divisor or denominator of Gordon's formula (which must not be confounded
with the a of the foreffoing formulae) Mr. Wilson uses for wrought iron :

when both ends are fixed »,^^..., 86000

when one end is fixed and one hinged ..» 24000

when both ends are hinged 18000

Experiments show that materials may fail under a longr continued
atross of much less intensity than that produced by the ult or bkg load.

• This does not always hold in cases where the elastic limit has been artificially raised
bv process of mannfkcture.etc. Oft- repeated alternations between tension and compres-
sion below such a limit reduce it to toe natural one. A slight flaw may cause rupture
under comparatively few applications of a range of stress but little greater, or even less,
than the elastic limit. Rest between stresses increases the resisting power of a piece.
In many oases, stresses a little beyond the elastic limit, even if oft- repeated, raise that
limit and the strength, but render the piece brittle and thup more liable to rupture from
shocks ; and a little further increase or stress rapidly lessens, or may entirely destroy,
the elasticity. A teruiU stress above the elastic limit greatly lowers, or may even destroy,
the eompreanoe elasticity, and vice versa. If a tensile stress, by stretching a piece, reduces
its resisting area, it may thus reduce its toUU strength, even though the strength per
«9 in has increased. Mr. B. Baker finds that hard steel fhtignes much faster under re-
peated loads than soft steel or iron.

t a = 0500 lbs per nq Incli for rolled iron in comprestiion

■s 7000 tt« " ** " tension (plates or shapes).

■■TSOOIb* ** for doable rolled iron in tension ninks or rods).

30

466 8TBENGTH OF MATERXAUB.

TBAKSYEBSE STRENGTH.

1. In Statics, 1[f 285, etc.. we disoum the action of external or destnio-
tive forces upon cantilevers, beams and trusses. We here discuss the reao-
tion of the internal or resisting forces (stresses) iu solid cantUeven and
beams, in order to determine their loads. See also Trusses.

2» Unless otherwise stated or apparent, we assume that the stresses in all
pwrts of the cantilever or beam are within the elastic limit.

Conditions of Equilibrium.

3* For eouilibrium, the internal forces, and their moments, must balance
the external forces and their moments. In other words, if the cantilever or
beam be supposed cut by a section at any point, we must have

2 vertical forces = 0
S horisontal forces — 0
2 moments = 0

Or:

(1) Algebraic sum of the internal vertical atreaaea *- algebraic sum of the
external vertical farces on either side of the section;

(2) Sum of horisontal tenaHe stresses -■ sum of horizontal compreaaive
streeees; and

(3) Algebraic sum of the moments of the internal atreaaea ■■ algebraic
sum of moments of external foreea on either side of the section.

4. Oantilevers and beams of uniform eross-section have usually a super-
abundance of strength against shearing, and fail (if at all) near the
middle, where the bending moment is greatest. Henoe the discussion of
their resistance turns principally upon equilibrium of momenta. For their
resistance to vertical shear, see Statics, it 325, etc., and p. 490. See also
Horisontal Shear, ^^ 51 to 53, below.

FiflT. 1.

5^ For equilibrium, therefore, the resisting moment, R ("■ the sum of the
resisting moments, r, of all the particles in any cross-section of the canti-
lever or beam, Fig. 1 or 2), must be equal to the bending moment, M, or alge-
braic sum of the moments of all the external forces on either side of tad
section.

FlflT. 2.

Reactions of Fibers.

6. In a truaa or framed beam (see Trusses) the resistance of each of its
two chords is regarded as acting in a line passing through the centers of grav-
ity of the cross-sections of the chord; but, in a solid cantilever, Fig. 1, or
beam, Fig. 2, the total resisting moment is the sum of the separate resisting
moments of the several fibers throughout the cross-section.

Neutral Surface* Neutral Axis.

7. When a cantilever (or beam) bends, the fibers in the upper (or lower)
part of each cross-section are extended, while those in the lower (or upper)

TKANBVBBSB STBEKGTH.

467

iMut are compressed (see Figs. 1 and 2); the extension and compression
being greatest at tlie top and bottom of the section, and thence decreasing
uniformlsr inward toward a surface, n n. Figs, (a), near the center of the
cross-flection. la this surface, which is called the neutral 8urface» the
fibers are neither extended nor compressed. The line, o o. Figs, (b), formed
by the intersection of the neutral surface with any crossHsection ot the canti-
lever or beam, is called the neutral axis of that section.

8. In order that the algebraic sum of all the horizontal stresses in the
crooo section may be sero, as required for equilibrium, the neutral axis must
pasB through the center of gravity of the section. Hence, the neutral sm--
lace passes through the centers of gravity of all the cross-sections.

9. The neutral axismav be found by balancing the section (put out of
cardboard) over a knife-edge. Or see Center of Gravity, under Statics, H
125, etc. Every section has an indefinite number of neutral axes, all passing
through its center of gravity in as many dififerent directions. The axis re-
qiiireo, in any given case, is that one which is normal to the plane of the
bending moment under consideration.

In^the following discussion, we assume that the neutral axis of the sec-
tion is normal to the line of action (usually vertical) of the load, as it gen-
erally is. For other cases, as, for instance, the case of roof purlins, see
"The Determination of Unit Stresses in the General Case of Flexmre/' by
Prof. L. J. Johnson, Boston Soc. of Civil Engineers, in Jour. Ass'n of
Ihig'ng Soos., vol. xxvni, No. 5, May, 1902.

Beslstlng Moment. Unit Stress.

10* It is 'assumed that the extension or compression of each fiber, and
therefore the resisting force actually exerted b^ it, is proportional to its
vertical distance, t, above or below the neutral axis.

o^

Fiff. 8.

In Fig. 3, let
T -

the distance from the neutral axis, o o, to the fiber farthest from
that axis, either above or below the axis ;

S -■ the unit stress in said farthest fiber;

t — the distance from the neutral axis to any given fiber;

a » the unit stress in said given fiber;

a — the area of said given fiber;

F -" the total stress in said given fiber;

r -" the resisting moment of said given fiber about the neutral axis;
M "" bending moment at the cross-section under consideration;
R — the resisting moment of the entire cross-section ;

— S r *■ the sum of the resisting moments of all the fibers ;

I » the moment of inertia of the cross-section. See ^1[ 14, etc.;

— 2 1^ a >• the sum, for all the fibers, of fi a;

I R
X — the section modulus, -• ts ■■ s- . See Iflf 25, etc.

Tlien the imit stress, in any given fiber, is » « — S fp ; its total stress;

F, is — a« — Sa^; and its resisting moment, r, is — F ^ — S a -=. Hence^
the resisting moment, R, of the entire section, is

R- M - 2r- 2Sa^ - ^2<«a= ^.I.

Henoe, alsoi

fl MT

Since,

1«
«

468 STRENGTH OF MATESIAD3.

T S •

-7-t it follows that ^ *" "7 * *^*^

R = M= ^. I = y.I.

The reeisting moment, R, is ■■ S X, and the moment of inertia, I, « T X.
When beams are tested to destruction, the value attained by S is called
the Modulus of Rupture.

11. It will be noticed that the strengths of similar beams of any shape,
and those of rectangular beams, whether similar or not, are directly propor-
tional to the product, width X square of depth. See 1 63.

12* When the stress, S, upon the extreme fibers, is « the elastic limit of
the material, failure is imminent. The permissible unit stress is usually
taken as not more than half the elastic limit, and the safe load is that under
which S does not exceed the permissible unit stress.

13* The same quantity of material that composes a solid beam. Fig. 2,
\70uld present greater resistance to bending or breaking if it were cut in two
lengthwise along the neutral surface, n n, and converted into top and bot-
tom chords of a truss; because, first, the leverage with which the resLstanoe
acts is thus greatly increased ; and, second, the depths of the chords are so
small, compared with their distances from the neutral axis, that their fibers
may be assumed to act unitedly and equally. Hence, practically, (M the
fibers in the upper chord must be crushed, or cM those in the lower pulled
apart, at the same instant, before the truss can give way; whereas, in the solid
beam, the extreme upper or lower fibers yield first; then those next to them,
and so on, one after the other.

Moment of Inertia.

14. Unlike the moment of a force, which is the product of a force and a
distance, the moment of inertia, being the sum of the products of areas of
fibers by the squares of their distances from ^ the neutral axis, is a purely
geometrical quantity. Thus, the moment of inertia of a ^iven section de-
pends solely upon the dimensions and shape of that section, and is inde-
pendent of the material and the span of the beam and of the manner in
which it is supported or loaded.

Unit of Moment of Inertia* The moment of inertia of a figure

being the product of an area by the square of a distance, its unit ia the

fourth power of a unit of length. Thus, in a rectangle 3 ins. wide and 4

, ^ m d^ 3X64 192 ,_ , . J X. . u ,« . ui

ins. deep, I = r^- = — ^ — = -r^ = 16 biquadratic mohes =« 16 inch*.

1X6*
In a rectangle 1 inch wide and 6 ins. deep, I = — ^^ — ■■ 18 inch*.

15. Comparing similar sections of any shape, their moments of inertia
are proportional to the product, breadth X cube of depth. Compare ^11.

16. The following illustrated table, pp. 469-471. gives, for several
figures of frequent occurrence,

(1) I =- the moment of inertia =- It^ a;

(2) T — the distance from the neutral axis to the farthest fiber;
<3) X — the section modulus — >r — — sr- ■- -«" " a'*

(4) A — the area of the croas-section.

17. In sections where the distance from the neutral^ axis to the lower-
most fiber, and the corresponding section modulus, differ from those (T
and X) pertaining to the uppermost fiber, those corresponding to the
lowermost fiber are distinguished as T' and X' respectively.

18. In each figure the neutral axis is indicated by a horiaontal line
crossing the section.

HOMENTS OF INEBTIA.

469

Moments of Iitertta* etc

I

Moment
of
inertia.

Distance from
neutral axis
to farthest
fiber

T

Section
modulus.

A
Area of
Section.

kB*i

4

fa

BD«
18

2

BD«

BD

kB«i

2

-^n

B (D»-*»)

12

D
2

GD

B(D-d)

U~S-J

8

"*■

T

r

Bi

12

B^

a

B*
12.

B*-ft*
13

B*-6*
12

B

V5

B
2

B8
6

BD»
SG

V2

:i^B3

12
> 0.U8 B

B*- ft*
OB

3

I'-T"

12 ' B

B*-6*
— 0.118 — = —

X —

BD'
U

x'-bd:

12

B =

B

B*-6*

B^-b''

BD
2

8TBEK6TH OF UAIERIALS.

MOMBNTB OF INERTIA.

471

4S A

I

.A
+

+■

g

^1^

n
t

«

t

9

91

A|<

ftl<

M

•o
I

w

n

A

<N

M

H

l_

+

I

n

I

+

I

H

•o

n

•o
I

n

I

I

I

T L e J

t^M

C9

00

472

STRENGTH OF MATERIALS.

19. The moment of inertia of any figure, about its neutral axis, is the sum
of the moments of inertia of its severalparts, about that same axis.

290* Let I — the moment of inertia of the entire figuoe about its neutral

axis, o o;

% «■ the moment of inertia of any part, about the neutral axis,
o o, of the entire figure;

m — the moment of inertia of that part, about its own neutral
axis;

a — the area of that part;

t « the distance of its center of gravity from the neutral azia^
o Ot of the entire figure.

Then I - S i; and i " m-^af,

21. Thus, in Fig. ^

and I — ti + »8.

i^^l^+hdt^\

*-6-»

TZ..

B—

Tig. 4.

Fly. 5.

22. Hence, in anv hollow section, aa in the hollow rectangle. Fig. 5, let
I' — the moment of inertia of the whole figure (including both the shaded
and the unshaded rectangles), i *- that of the missing or unshaded reotanglflk
and I » that of the shaded portion; all referred to the neutral axia, oo,<»
the shaded portion. Then I — I' — i.

-o

Flff. 6.

23. In the case of an irregular section, as Fig. 6, let the section be clfvided
Into numerous strips, parallel to the neutral axis and narrow enou|^ to be
considered as lectangular; and proceed as id lH 19 to 21.

TRANSVEBSE STBENGTH* 473

{S4. The narrower the strips are taken, the lees m beooi.ies. If the strips
be taken so narrow (relatively to the depth of the section) tba^ m may be
neglected, then I — 2 ^ a, as in ^ 10. The strips need not be of unuorm
width.

The Section Modulus.

25* Definition. If the resisting moment, R »- i- . S ^ a, be divided

by the unit stress, S, in the extreme fibers, the quotioit, X *- -^ "" ^

o 1

— Tp, 18 called the Section Modulus. This, like the moment of inertia. If If

14, etc., is a purely aeometrical quantity, depending solely upon the dimen-
sions and shape of the section, and belag independent of the material, of the
span, and of the manner of loading.

136. Having the section modulus, X, we have only to multiply it by the
unit stress, S, in the extreme fibers, in order to obtain the resistmg momenl^
R; or E -SX.

37* Multiplying the section modulus. X, bv the dlstanoe, T, fiom the
neutral axis to the farthest fibers, we obtain the moment of inertia, I; or,
I-TX. «*. . .

2S» The section modulus is usually pven in tables Df rolled beams, chan-
nels and shapes. See tables >f Oaxnegie Beams, etc.

Loading. Strength.

29. The following illustrated table gives (1) the max moment, M,
corresponding to a given load, W; and (2) the load, W,* oorreeponding to a
given unit stress, S, for different conditions of support and of loading. In
this table,

M » maximum bending moment ;
R » M = resisting moment of cross section ;

W — the total extraneous load * un the beam, whether concentrated
at one point (as shown) or uniformly distributed over the span;
I ■" the span;
8 -■ the unit stress, in the fibers farthest from the neutral axis, du» to

the extraneous load, W; *
T ■« the distance from the neutral axis to the farthest fibers;
I ■■ the moment of inertia.

In rectangular beams,

b *- breadth;
d "- depth;

I -" moment of Inertia —

n —

12

S6d«*

Of the two diagrams under each loading, the first represents the mo-
ments, and the second the shears, in the several parts of '^he span.

30. If S — the permissible unit fiber stress, then, in the foreg ing formulas,
W » the permissible extraneous load.*

81* It will be noticed that the strengths of similar beams ar- pr portional

heP
to their values of -j- ; i. e., the strengths of beams of similar crrjfls-sections

are directly proportional to their breadths and to the squares of their depths,
and inversely proportional to their spans.

* The beam is here supposed to be without weight. See %^ 42, etc.

474

STRENGTH OF MATERIALS.

For symbols* see

opposite page.

K-

ir

Q.

I
I

W3

9

mHQQi

-nrnlllli^^

""rns^

Haximnm bendtac

At
Support.

M-WZ

At M-^

Support. %

At
Oentes.

H-

W*

At
Oenter.

At

Oeater

and at

Support.

IjOAd

Geaersl

w-x

-s

'il

W— « R

— 8S'Yj

R

Ml

Support. '^ 18

— 8S

TT

w-ia
-us*

I

Tl

In

re«t»iiKMl*r

beams

S_ bd*
^■"l* ' Ti

6 t

W

8

_8.
8

It
bd*
1

8

6d»

b^'

** 8 Ti

48
8

W-8

-tB

6d'

TI

bd*

9

1^

8

T

T

TRANSYEBSE 8TRENOTH. 475

Symbols In Table Opposite.

M — maximum bending moment ;
R -» M "■ resiBting moment o£ cross section ;
S «- unit stress, due to W, in the extreme fibers ;
T "- distance from the neutral axis to the extreme fibers ;
I » moment of inertia ; M — maximum bending moment ;

W -» load ; I ■■ span.

In rectangular ) b «- breadth ; ^ _ W I , w _ « o ^ <^
beams. / d - depth ; ^ iT5« ' w - n s -p-.

Beam 1 Inch Square, 1 Foot Span*

3!3* In a beam, 1 inch square and 1 foot (12 inches) span, supported at
both ends, we have, for the extraneous center load : *

33* For any other rectangular beam, let L « the span in feet. Then the
extraneous center load,* W, required to produce the same unit stress, &, in
the extreme fibers, is

W - w V-.

34. Thus, for yellow pine, let S -• the permissible unit stress ■« y the

3240
elastio limit ^ - _ — 1620 tbs, per sq. in. Then, for a beam 1 Inch square,

1 foot span, supported at both ends and loaded at center, the permissible
load,* W', IS

^ l2 "18 18 -«^"*'

and. for a joist, 3 X 12 ins., 20 ft. span; the permissible extraneous * oeoter
load is

W - W'^ - 90 X ^-^y^ - lW41be.:

and the permissible extraneous uniform load is = 2 W » 2 X 1044 = 3888
tbs.

86. If the load, W. the span, L, and the coefficient, W, are giren, we

have b cP = ^, . Thus, in the case of the yellow pine .joist, mentioned

in f 34, of 20 ft span, with a uniform extraneous * load, where 2 W =>

W T. 1044 v 9fl

2 X 1944 = 3888 !bs., we have b (P => ^^ = gn ^ ^^'

30. Then, if either & or d is given, the other is easily found. If not,
assign, to either of them, an arbitrary value. Thus, if 6 = 6, we have cP =

432

^ = 72; and «/ »= 1/ 72 = say 8*. With 6 = 3, d» = 144, and d = 12.

37* )^th the slide rule, in the foregoing example, place the runner at 432;
and, assuming & -■ 6, place 1 for 10) on the shde, opposite 6 on the rule.
Then, in the scale of square roots, on the slide, opposite 432 on the rule, will
be found 838 and 265. The former of these represents the desired root, and
we take 8.5 as a sufficient approximation.

38. If the relation, S »• 18 W, held beyond the elastic limit, and if W "^
the center brectking load, in tbs., on a beiun of any materiaL 1 inch square,
1 ft. span, supported at each end ; then, for any othj^r beam of the saMe mate-
rial, and of breadth b ins., depth d ins., and span L ft., the center breaking

load would be W - W ^.

39* Notwithstanding the defective basis of this method, as applied to
loads beyond the elastic limit, its simplicity renders it very convenient, and
it is much in use. See the following table of values of W^ and example, % 40.

* The beam is here jupposed to be without weight. For the weight of the
beam itself, see HIF 42, etc.

476

STRENGTH OP MATERIALS.

Center Breakingr Loads, W» In Ponnds, for Beams 1 Inch
Square* 1 Foot Span, Supported at Each End.

WOODS.

Ag\ Bnglish

" Amer White (Author).

" Bwamp

« Black

Arbor VitOj Amer.
Balsam^ Canada....

Beech^

Amer

it

p.

I

s

2?

Birch, Amer Black 2,9

" Amer Yellow g-g

Cedar, Bermuda prS

** Ouadaloupe g p^

** Amer White, ) ff^

or Arbor Vitae J o "5

C9ie»tnttt 3 S

Mm, Amer White o 2:

" Rock, Canada. a®

Henifock f*g*

Hickory, Amer.. . ^ g*

" Bitter nut |.|

Iron Wootl, Canada ~ ^

Locust ^d

Lignum Vitae.... * %

Larch »' S

Mahogany %

Mangrove, yrh\tA ,_,

" Black »

Jfaple, Black ..«. r^

" Soft Z

Oak, English

" Amer White (by Author).

" ** Red, Black, Basket...

" Live

Pinf^ Amer White. ..(by Author)

" " Ycfllow '^ "

« " Pitch " « .

" Georgia

Poplar.

Poon

Spruat (by Aathor).

" Black

Sycamore

Tamarack...

Teak....^

Walnut

WUUyw

MXTALS.

Brasi

Iron, cast, IdOO to 2700.. ..average

commonpig..

castings fk>om pig

employed in our ta-
bles

for castings 2}^ or 8

ins thick

Mnn, wroughi, 1900 to 2d00.....aT
Wrought iron does not break ;

M
M

((
M

t4

w

650
650
400

eoo

250
850

850

550
850
400
600

250

450

650
800
500
800
800
600
700
650
400
750
650
550
750
750
550
600
850
600
450
600
550
850
550
700
450
650
500
400
750
650
860

860

2100
2000
2800

2026

1800 1
2260

but at tAioui the average of 2250
lbs its elas limit is reacned.
S^el, hammered or rolled; elas
destroyed by 3000 to 7000..
Under heavy loads hard steel
snaps like cast iron, and soft
steel bends like wrought iron.

8T0NE8, ETC.

Blue stone flagging, Hudson River

Bricky common, 10 to SO.^^verage

" good Amer pressed, 30 to

50 ....average

OaeM Stone

W
6000

Concrete, see article .on Con-
crete.

Orardte, 60 to 150 average

" Qnincy

Glass, Millville, N. Jersey, thick

flooring ....(by Author).

Mortar, of lime alone, 60 d^TS old

** 1 measure of slacked lime

in powder, 1 sand

** 1 measure of slacked lime

in powder, 2 sand

Marble, Italian, White (Author)
* Manchester, Vt, " "

East Dorset, Vt, « «

Lee, Blass, •* <*

Montg'y Co, Pa, Gray **
" " Clouded^ ••
RntlaQd,Tt,6raj ^

« Glenn'8FaIl8,N.Y3Uusk «
*< Baltimore, jkd, white,

coarse.. ....»•.•.. **

Oolites, 20 to 60

Sandstones, 20 to 70 average

** Bed of Connecticut and

New Jersey

Slate, laid on its bed, 200 to 450, av

n

(I
(t

««

126
20

40
26

100

100

170
10

8

7
110

06
111

85
108
142

TO
166

lot

86
45

46
826

TB^NBVEBSE STBENQTH. 477

4S^ Bxamplek In th« yellow pine joist of ^% 34 and 3A, 3 X 12 ins.,
20 ft. 8pan» we have, from the table, W -■ eay 500 Ibe. Henee

Center breaking load ♦ W - W ^ - ^^ ^^^ ^^ - 10,800 tbs., or

about 5.6 times the permisBible load, found, by means of the permissible unit
■tress, ia ^di.

Dimensions.
41. SinotW-nS-^ - W'-^,andW'=.^,weh»v«»

Breadth - * - ^-g^ - ^^7^:

"WT

Dept. ..-VS-V^'

W w extraneous load * raqiuied;

W «* CKtianeous load on beam 1 inch square, 1 ft. spaa)

n «• coefficient from last column of table •• ^^ a;

8 ■■ unit fiber stress;
i "* span in inehes;
L » span in feet.

Weight of Beam Considered as Load*

42. For rimplieity we hate hitherto rMsided our cantilevers and befims
as having no weight of their own; and. in beams of the moderate dimensions
usually employed in buildii^ their own weight, v^ is so small, in comparison
with their loMS, W, that it may often be aafely^neglected; but in larger
beams it must generally be ti^n into account. The loads, found as above,
with S ■• greatest permissible unit stress, must then be regarded as including
not only the extraneous load, W, but also the wei^t, w, of the beam itself,
for a length ■■ span.

43. If the beam is prismatic, — ». e.. of uniform cross-section, — its weight,
fir, acts as a uniformly distributed load, and we have, for the extraneous load.
W, in the case of a concentrated center load on a bwrnf or of a concentrated
load at the end of the span, 2, in cantilevers,

W - whole load—- ~;

In the case of a uniform load,

W — whole load — is.

44. In finding the breadth or the depth of a rectangular beam,
required to carry a ffiven load with a given span and given unit stress, we
may provide for the weight by successive approximations. Thus.

45. To find the breadth, 6. required for a beam of given depth, d.
Neglecting the weight, w, of the eeam, find the first approximate breadth, b.
by the formulas in If 41. for the extraneous load, W. Next, calculate the
weight, to, of a beam with width, 5, treat said weight as a uniform load ; and,
by the same formulasLfind the additional breadth. 6', required to carry this
additional load, 10. Then & + 6' » a second approximate breadth. If nec-
essary, find the weight, ti''. of a beam of breadth, 6', and. from this, a second
additional breadth. 5", required to carry it. Then 5 + 6' + 6" = a third
approximate breadth, and so on.

48. To find the depth, d, required for a beam of given breadth, h;
find a first approximate depth, d, by the formula, 1 41, for the extraneous
load, W. Find the weight, w, of a beam of that depth; and again apply the

__ __ to

formula, using (in place ofW)W + ti>ifWiBa uniform load, or W + ^ if W

is a concentrated load. The depth, d'. so found, is a second approximation.
We'may again appl^ the formula, as before, using the weight, w\ of beam of
depth a'; or, more simply, increase the breadth, as in H 45.

*The beam is here supposed to be without weight. See t1[ 42. etc.

478 STRENGTH OF MATERIALS.

47* In practice, beams of rectangular section are almost alwajm of timbers
and such beams are eoonomioally obtainable onlv in oertain oommercial
sixes. Hence, the second approximation will usually be all that is required.

Strengths and Welshts of Sin&ilar Beams of Different Dimen*
slons* Comparison between Models and Actual Structures.

48. In any given beam, let W| — the load causing any given unit stress*

8. Then, Wi — j (for n, see table, p. 474) ; and, in any similar beam.

of a times the breadth, depth and span, the corresponding load, W >■
n S aba* <fl jj^^^^ ^^ ^^^^ ^^ ^^^^ ^^^^^ -^^ - o*; or W - a» Wii

but the ratio of their weighta is — — ■ . . , — a'; ortp — a*tPi.

40. In other words, oomi>aring one beam with another, of a times its
breadth, depth and span, their atrengtha are as the aguarea of their respeetnra
dimensions; but their weighta are as the ctiies of those dimensions.

fiO. Hence, if a model of a beam will just break under a uniform load
Gncludin^ its own weight, to) -> 2, 3 or 4, etc., times its own weight, then a
beam of sunilar crossHsection. but of 2, 3 or 4, etc., times its breadth, depth
and span, will just break under its own weight alone.

Horliontal Shear*

51. When (Figs. 7 and 8) deflection occurs in a eantUever or beam oom-
posed of separate horisontal layers, like a pile of loose boards, the several
Winers slide upon each other; but, if they are firmlv joined together, or other-
wise prevented from sliding, they exert, upon each other, a horisontal sheai^
ing foroe. In any section, this force diminishes uniformly from a maxi-
mum, at the neutral surface, n n, to sero, at the top and bottom of the section.

n'

Fiir. ▼• Wim* s.

52. In any section of a rectangular beam, the maximum horisontal
shear, per unit of neutral surface, is

" 2bd'
where V — the vertical shear in the section, and 6 and d — the breadth and
the depth of the section, respectively.

In words, the unit horisontal shear, at any point, is directly pro-
portional to the vertical shear at that point. Hence, the horisontal
shear diagram is similar, in character, to the vertical shear diagram;
but is opposite in sense, i>o8itive vertical shear corresponding to nega*
tive horisontal shear.

53* If the horisontal shear is resisted by a fastening applied at only
one point, said fastening must be made sufficiently strone to resist the
aum of all the horizontaf shears between such point and tnat where tlM
•hear is ■■ 0.

54. In Fig. 9, diagrams (h) and (c) show respectively the moments
and the vertical shears due to concentrated and distributed loads on a
beam as shown; and, in Fig. (d), each ordinate represents the force
which must be applied, at the corresponding point, m order to resist
the sum of all the horisontal shears between that point and the point
of sero shear. Ordinates above a sero line indicate positive moments or
shears, and vice versa. In poaiiive moments, the segment to the Uft ef a

HORIZOMTAIi BHEAB.

479

section tends to turn dockwUe. In poniive shears, the left-ha.nd segment
tends to slide upward or the upper segment to slide toward the right.
Between a and c, between e and d, between g and A, and between h and
b, all the diagrams are straight lines. Figs; (6) and (d) being inclined,
and Fig. (e) horisontal. At e and at h. Figs. (&) and id) change their
inclination, and Fig. (c) shifts its position. Between d and /,*and
between / and g (i. «., under the distributed load), Fiss. (b) and (cO are
parabolic curves, ana Fig. (c) shows inclined straight lines. At /, Figs.
(2>) and (d) change curvature, and Fig. (c) shifts its position. At e, the
point of maximum moment. Figs, (c) ana (d) change signs. See Relation
between Moment and Shear, Statics, Iff 359 to 368.

ntalShearB

Fly. 9.

55* Inasmuch as the horisontal sheas is a resistance to bending, its n^ect,
in the common theory of beams, as heretofore explained, is in general on the
side of safety. But, in beams composed of horizontal layers, means must
be provided for its transmission from one layer to the next.

I

l®l

L — .J

ir_j

f — i
I®.

« — i

■i

II

■^

ii
ii
•I

II
ii

-^

t

.11.

T

II

ii

w

Tig. 10.

56* Thus, deep wooden beams. Fig. 10, are frequently built up of two or
more timbers, one above the other. In order to prevent deflection, due to
the sliding of these timbers upon each other, blocks are inserted between
them at intervals, as shown, or the adjacent sides of the timbers are so
notched as to interlock. In either case, the timbers are tightly bound
together. The blocks or notches then serve to transmit the horizontal shear
from one timber to the other. In Fig. 10 the blocks are more numerous near
the ends of the span, as required by the diagram of horizontal shear. Fig. 9 (cQ.

I m T

BTKENGTH OF MATERIAIA

i!!|* * ^

I>£FL£CTIONS. 481

BeflectloBS.

57. Tli« opposite table gives the defleetlons within the elastic limit,
q{ €Lnj prvfmatichea.m (beam of uniform cross section throughout) under dif*
ferent arransrements of support and of load; also (in the last
column) the extraneom load wblcn will produce a grlTen defleetion,

Without assistance from the weight of the beam itself. All the formulae are based
upon the assumption that the increase of deflection is proportional to increase
of load.

The letters signify as follows :

d ■■ deflection of beam, in inches (see Figs).
W =s weight of extraneoaa load, in jpounds.
w =i *' " clear span of beam, in pounds.
I = clear span of beam, in inches (see Figs).

X| =9 modulus of elasticity of the material of the beam, in lbs per sq Inch.
I => moment of inertia of the cross section of the beam, in inchos.

"Wtcm the piiDclitlM embodied in the apposite table, we find tkai in beeai* of
■imilar cross section aud of the same matetial, and within liie elastifi limit, the toe^
and deflections (neglecting the weight of the beam itself are as follows :

With the same

The deflections under a giren extraneous load are

*< and breadth
^ ** « depth
Weadfli'' *"

in^wrsely as the breadths and as the cubes of the depths

« " breadths
directly ** cubes of the spans

With the same

The extraneous loads for a given deflection are

span
** and breadth
" " depth

breadth " "

directly as the breadths and as the cubes of the depths

t( U U (( it

" " breadths
inversely ** cubes of the spans

Defleetton In Terms of Extreme Fiber Stress. In table, p. 474,

the load W •■ iS j^; where > =- a coeflicient, as below; S = unit stress

in extreme fibers; I — moment of inertia; T = distance from neutral axis
to extreme fiber, and I = span. From the table opposite, we have:

W ■■ m — j^ — ; where m -» a coefficient, as below; d -• defleetion, and

T <2 El I h P S

E ■■ modulus of elasticity. Hence, k S sk-» = »» — tj— J and d — — . ^g =

PS, m

" WT-c ' "^^^'^ ^ ' *•

In a cantilever, loaded at end, !»■■ 3; ib"- l;o** 3.

uniformly, to - 8; t - 2; c - 4.

In abeam, supported, and loaded at center, m «■ 48; j^ » 4; c » 12.

uniformly, m- te.S; ft- 8; c - 9.8.

•• •• fixed, " " at center, m - 192; ft - 12; c - 16.

'f •• *" " " uniformly, m - 384; * - 12; c - ?2.
31

482 . STRENGTH OP MATERIALS.

Elastie liimlt.

58. Under moderate loads, the deflections are practically proportional to
tb» load. When they begin to increase perceptibly tester than the toady the latter kt
■aid to have reached the elastic limit, or limit of elasticity. It Is generally at thia
point that the " periiian«mt set " first becomes noticeable ; L e., after remoTsl
of tke load, the beam fails to return to its original unstrained condition, and remains
more or less bent. The deflections then also begin to increase irr0gularltf ; and to
continue indefinitely without farther increase of load. In short, the beam is in
danger. Hence, the actual load must never exceed the elastic limit ; and should no4
exceed from one-third to two-thirds of it, according to circumstances.

TIm limit ot elastioitjr olT at b«sun of any particular form, or material, !■
determined bjr experiment with a similar beam, as in the case of constuita
for breaking loads, Ac. Thus, load a beam at the center, by the carefol gradnal
addition of amall eqniU loads; carefully note down the deflection that takes plaee
within some minutea (the more the better) alter each load haa been applied; in order
to ascertain when the deflections begin to increase more rapidly than the loads) Cor
when this takes place, the load for elastic limit has been reached.*

It is not the deflections of the whcie beam that are to be noted, but fhoee of ito
clear span only. Seyeral beams should be tried, in order to get an average constaDt*
for eren in rolled iron beams of the same pattern, and same Iron, there is a yerj
appreciable difference of strengths and deflections.

Than, to get the constant, using theeoCoZ load applied daring the equal deilectioiia^
inohiding half the weight of the beam itself,

Gonirtant Ibr elsurtic limit - Span in feet X Total load in Iba.

Breadth in inches X Square of depth in inehea

TMe oonstau&t* for urooden bestma^ may be had, near enough for oommon
practice, by taking one third of the breaking constants in the table, page 476.

Said constant, thus calculated. Is the elastic limit of a beam of the gfren tHrngm
and material, 1 inch broad, 1 Inch deep, and of 1 foot span, supported at both ends
and loaded at the center. To obtain from It the elastic limit ^ any other beam of
the same design X and the same material, similarly supported and loaded, but of oihsr
dimensions,

AlsuitlG _ constant X ^'^^'"^*** ^ inches X sqqare of depth in inches ^
limit " span in feet

If the beam is

supported at both ends and loaded at center,
" u «. it it .. uniformly,

fixed t " " " " " at center,
« K <c «i u u uniformly,

" " one end *' '• at other end,

" *' u u u ,1 uniformly,

• Of course, in practice, It Is frequently difficult to ascertain with precision, when,
or under what load, the deflections actually do begin to incr««8e more rapidly than tihs
successiYS loads. For although by theory thfi deflections are practically equal Ibr
equal loads, until the elastic Umit is reached, yet in fad they are siilb||ect to
more or less irregularity ; for no material composing a beam is perfectly nnlfbrm
throughout in texture and strength. Hence, instead of regular increase of defleo-
tion, we shall have an alternation of larger and smaller ones. iTherefore, Nome Judg*
ment is required to determine the flnal point ; in doing which, it Is better, in case
of deubt, to lean to the side of eafety. It is assumed always that the load Is not
subject to Jars or Tibrations. lliese would increase the deflections.

f A beam Is said to be " fixed ** at either end when the tangent to the longitudinal
axis of the defiectod beam at that end remains always horizontal.

tThe tkapes of the two beams need not be similar. For instaaee^fhe constant
deduced from experimento upon any rectangular beam Is applicable to any ottier
rectangular beam, whether square or oblong.

DEFLEOTIONS.

483

The Elastic Curve.

59. When a cantilever. Fig. 1, or a beam, Fis.
any manner, bends, under the action of any load,
forms a curve, such that, at any section.

M " MA'

2, supported or fixed in
the neutral surface, n n«

where

M

I

the radius of curvature,, at the section ;
the bending moment, at the section ;
the moment of inertia of the section;

E = the elasticity coefficient of the material, — -j-

S
k

S » any unit stress within the elastic limit ;

k »■ the unit "stretch" (elongation or compression) produced by S in
the given material.

m

(CI) (b)

Flg^. 1 (repeated).

(a) ib)

FiiT* 2 (repeated).

The Deflection Coefficient.

00. Definition. The deflection coefficient, for any given material, is the
deflection, in inches, of a beam, of that material, 1 inch square and of 1 foot
span, supported at each end, and carrying, at its center, an extraneous load
of 1 lb. — i w\ where v/ =» weight of clear span of beam alone, in lbs.

61. Let y =« the deflection coefficient for any given material. Then, in
any rectangular beam, of the same material, with center load or uniform
load, let

h — the breadth, in inches;
d •= the depth, in inches ;
L = the span, in feet;

to = the weight, in lbs., of the clear span of the beam itself;
W -= center load -f- i to;
= i (uniform load + w) .
Then, in the given beam.

Deflection = Y = y

Iioad

W

b . d»

L«

Breadth* == b

W.

y

Depth *

d =

L8.y.
<i«.Y*

^\6Y

62. The deflection coefficient, y, for any given material, is obtained by
experiment, thus: At the center of any rectangular beam, of the given mate*
rial, placed horizontally upon two supports, at any convenient and known
distance apart, place any load that is within the elastic limit, and measure
the resulting deflection, Y. Let W =» the extraneous center load + I- to,
where to «= the weight, in lbs., of the clear span of the beam itself. Then
the deflection coefficient is

V ^'^
^ "" W . L3'

where b and d » the breadth and depth, in inches, and L » the span, in

feet, of the experimental beam.

♦ In calculating the breadth or the depth, if it is necessary to provide for
the weight of the beam itself, we first let W = the extraneous load only,
and then proceed by successive approximations, as in 1[1[ 45 and 46, remem-
bering, however, that in the case of deflections, 5-8 of the weight of each
additional section is to be taken as equivalent center load, and not 1-2 as in
the case of strengths.

484

8TRENQTH OF MATERIALS.

63. The ratio between anjf two homologoua lines, in any two similar
figures, is constant. Hence, in determinins or using coefficients, whether
for strength or for deflection, by comparing beams of similar sections but of
different sizes, we may use any two homologous lines in place of the two
breadths, or in place of the two depths, or the same line may be taken in
place of both breadth and depth. Thus, in Figs. 11,

h-8 ^-«

(«)

Fier. 11.

Hence,

Also,

B
b

D
" d

b<P

384

48

-8 -

2» =

B»D

1728

ifi .

^ =-

R8R

1000
125

10.000

8 - 2»;

= 16-2*.

.0.00032

bsd 108 *- - r3r 626

64. Deflection coefficients, being the deflections, y, in inches, at the
centers, of beams 1 inch square and of 1 foot span, supported at each end and
loaded at center with extraneous loads of 1 tb. — 6-8 the weight of the elear
q;>an of the beam itself.

Average

Cast iron, 0.000018 to 0.000036 0.000027

RoUed bar iron, 0.000012 to 0.000024 0.000018

. Rolled tool steel, 0.000010 to 0.000020 0.000016

White oak, well seasoned, 0.00023

Best Southern pitch pine, well seasoned, ) n nnA97

White ash, weU seasoned, J u.uuu^/

Hickory,^ well seasoned, 0.00016

White pine, wpII seasoned,
Ordinary j'ellow pine, well seasoned,
Spruce, well seasoned.
Good, straight-grained hemlock, well sea-
soned.
Ordinary oaks, well seasoned,

66. Caution. The deflections of timber of the same kind vary giieatiy
with the degree of seasoning, the age of the tree, the part from which th«
beam is cut, etc. The coefficients given above are avera|pe8 deduced from
our own experiments on good pieces, well seasoned, on which the loads wsre
allowed to remain for months. In all kinds, less than 2 per cent, of the
breaking load produced a permanent set in a few months. Several of the
sticks bore their breaking loads for months before actually giving way. The
vibrations and jars, to wnich all structures are exposed, in time increase the
directions.

66. Eccentric Concentrated Loads. Let Y, Fig. 12 (a), be the de-
flection, at the center of the span (t. e., at the point of apj^ication of the load)
of a beam supported at each end, due to a load, W^witnin the elastic limit,
at the center of the span. Then, if the same load, W, be placed ecoentriotdly
upon the same beam, as in Fig. 12 (6). the deflection, Y', at the point» c^ ol
application of the load, and due to the load, W, is

Y'- Y

16 m* n*

trhere

I — the span;

m andn — thesegmentsinto which the load divides the span

DEFLECnOKB.

485

67* TTnlf onn Ijoads. Let T be the deflection, due to any central ex«
traneous load (within the elastic limit), on a beam supported at each end.
Then the deflection, Y', of the same beam, due to the same load uniformly
distributed over the span, is

T' - I Y.

68* Inclined Beams* If the beam is inclined, use the horisontal pro*
jection ol it-s span, in place of the span, 2, in determininis its deflections.

68. Cylindrical Beams. Let Y be the deflection of a square beam
under any given load. Then, for a cylindrical beam whose diameter — sidi
of the square, the deflection, under the same load, is — 1.698 Y.

Strontpett

(«)

SUffest

(b)

Wig. 12.

Flff. IS.

70. ¥Us»' 13 (a) and (&) show. respeetiTely, the strongest and the stiffest
rectangular sections wmch can be cut from a given cylindrical log, of diame-

ter, D. In the strongest section Fig. ia),ae — -^, and h

stiffest section Fig. (b), a c « -7-, and d

-Vf

-i/|d2. In the

D2.

71* Maximum Pennissible Deflection. Under even a perfectly
safe load, a beam may bend too much for certain purposes. Thus, to pre-
vent the cracking of the plaster of ceilings, it is usual to limit the deflection

of beams to -^^ — ;^ inch per foot of span » 3} ins. per 100 ft. In long

snan
lines of shafting, for machinery, the deflection is usually limited to r^f^ "*

I inch per 100 feet of span; in highway bridges to ^~ -* _^ inch in 10 ft.;

d40 lO

in railroad bridges to ^^r-: — -r inch per 100 feet.

loUU 4

72* Let Y » the maximum permissible deflection, in inches per foot ol

span, in any given case :
V — the deflection coefficient, illf 60, etc.
L » the clear span of the beam, in feet;
v> *" the weight of the clear span of the beam, in lbs.;
W — the center load + \ w;
— ♦ (uniform load + tr).
Then, Y L — the deflection, in inches, for the whole span, L, and we have^
for the permissible load, W, and the required breadth, 6, and depth, <{, for a
leotangular beam (see f 61) :

Load - W - -

Breadth* - i - W.
Depth ♦ - /f - W .

L»v

d»Y*
L9y

L2y •

6Y*

♦ See foot-note to f 61.

486 STRENGTH OF MATERIALS.

Suddenly Applied Loads.

73. Suppose a load to be applied to a flexible beam suddenly, though
without falling oi^jarring ; as, for instance, if it be supported by a cord which
allows it just to touch the beam without bearing upon it, and the cord be
then suddenly cut in two. The deflection of the beam, in such a case, is
theoretically twice as great as when the same load is applied gradually, as by
very slowly relaxing the cord, or by dividing the weight into small fragments
and applying them at intervals, one by one. See Art. 5 (6), under
Strength of Materials. Hence the strenc^th of the beam (within the
elastic limit) is much more severely taxed in the former than in the latter
case. A heavy train, coming very rapidly upon a bridge, presents a con-
dition intermediate between the two.

Cantilevers and Beams of Uniform Strenjsth.

74. For equilibrium, the resisting moment, R, of any section, must bal-
ance the bending moment, M, at that section. Or,

S T

■^.I = M; or, S - M. j;

where S «=• unit stress in extreme fibers;

T — distance from neutral axis to extreme fibers ;
I "- moment of inertia of section.

75. In a beam of uniform cross-section, therefore, since T and I are uni-
form throughout the span, the unit stress, S, on the extreme fibers, varies
with the bending moment, M. For uniform strength against bending mo-

T
ments, the cross-section must so vary that y shall be inversely proportional

to M, in order that S may remain constant.

76. The following table shows, in elevation and in plan, the theoretical
shapes of rectangular cantilevers and beams of uniform strength against
bending moments, under concentrated and uniform loads. In practioe,
some of these shapes would of course have to be made stronger near their
ends, in order to provide a sufficient section to resist shear.

77. Notwithstanding the reduction in material which would be effected,
by using beams of uniform strength, their use is seldom economical, except
in the case of cast iron. In timber, the material removed would not be
saved; and, in steel, the saving in material would often be offset by the cost
of additional labor.

Moreover, it will be noticed that the deflections of beams of uniform
strength, under a given loading, are considerably greater than those of beams
of uniform cross-section.

In the table,

W ■- concentrated load;

w « uniform load per unit of span;

I -■ span;

X -■ distance from a support to any given seetiont

d "■ depthof beam at that section;

h •- breadth of beam at that section;

D — maximum depth of beam;

B — maximum breadth of beam;

S — unit stress in extreme fibers;

E ■- elasticity coefficient — .,-- — —r-l

unit stretch

• V — deflection, due to extraneous load, in beam of uniform strength;

Y — deflection, due to extraneous load, in beam of uniform cro9«»-«ee-
tion M maximum cross-section of beam of uniform strength.

UmFOBM STBKNOTH.

487

Cantilevers of Beetansrular CrossHsectlon and of Uniform
Strength. Profiles, Plans and Deflections.

For symbols, see 1 77.

OoBcentrated Load, — TF, at. end.

Breadth, b, constant.

Profile, parabola, with vertex at load.

V 86 B6D»

D — MaTrimiim depth.

]>epth, d, constant.
Flan, trlan^rle.

8dS

EBd> 8

-=-e- Y.

Uniform Load,= «o per unit of span.

Breadth; h, oonstant^
Profile, teiangtoi

d — «

8 10
8b

liSSS^

--<>^i>yy>i^>i<'^fi<yi^^.i

Depth, d, constant.

Plan, two parabolic curves, with vertices
at free end.

8 to g'

Sd«

/ 8W?8
Y =

EBd^

= a Y.

488

STRENGTH OP MATSRIALS.

Beams of Bectang^ular Cross-section and of TTniform Strehstb*

Profiles, Plans and Deflections.

For symbols, see opposite page.

Coucentrated Load,'-Tr, at center.

Breadth, b, constant.

Profile, two parabolic curves, with vertices
at supports.

V8W«

Y'-

8E6D'

D at center of span.
— ST.

Depth, d, constant. B—mazinmm width
Flan, two trlanfirles

b —

y'- '^'

8EBd> i

Uniform I««ad, — w per unit or span.

m

mm^.Kimmmm

Breadth, h, constant.
FrolUe, dlipae or aeml-elllpfle .

"-ViT ("-'•>

Depth, d, constant

Plan, parabolas with vertices at center of spaa

b ~

Sio

sT'C-— >

CONTINUOUS BEAMS.

489

Symbols in table opposite:

W -

I =

d =

D =

S =

E =

Y' -
Y -

uniform load per unit of span;
dist from a support to given sec ;
breadth of beam at that sec ;
maximum breadth of beam ;

concentrated load; w »

span ; x =

depth of beam at that sec ; b »
maximum depth of beam; B >-
unit stress in extreme fibers;

^ !.■ -J. m ' 2. ^i^it stress

elasticity coefficient — — r-— - — — l ;

unit stretch

deflection due to extraneous load in beam of uniform strength ;
deflection, due to extraneous load, in beam of uniform cross-sec-
tion -■ max cross-section of beam of uniform strength.

Continuous Beams.

78* A continuoTis beam is one which rests upon more than two supports.

79. The resistances and deflections of continuous beams, like those of
beams with fixed ends, are determined by means of the elastic curve, using
the calculus. The * more important facts, thus deduced, are indicated in
Fig. 14 and illustrated table, t 89.

80* Fig. 14 represents the general character of the deflections, and the
variations of the moments and of the shears, in uniformly loaded continuous
beams;

81. Moments. Fig. 14 (6). Ordinates drawn above the zero line, a' b\
represent potitive moments, or those where the segment of the beam, to the
left of any section, tends tcf revolve clockwise; and vice versa.

8/3. At each end of the beam, at one point, t (called the inflection point, or
point of contrary flexure) in each end span, and at two such points in each
remaining span, the moment is zero.

83. At another point, m, in each span, the positive moment reaches a
maximum for that span; while the negative moments reach their maxima
at the supports. Both the positive and the negative moments vary in the
different spans; but, if the spans are equal, then the moments, at any two
points eqwdistant from the center of the whole beam, are equal.

(a)

(W
Momenta

Shears

Tig. 14.

84. The moment diagram, between each support and the point, m, of
maximum positive moment on either side of it, is a -semi-parabola, with its
apex at m.

85. Shears. Fig. 14 (c). Ordinates drawn above the zero line, a' b*,
represent poeitive shears, or those in which the left-hand segment, at any
section, tends to slide upward past the right-hand segment ; and vice versa.

86. At the point, m, of maximum moment, in each span, the shear is zero.
Between each such point and the next support on the left, the shear is posi-
tive, and vice versa.

87. At each support the shear suddenly changes, by an amount = the re-
action of the support.

88. The shear diagram is a series of straight lines.

1
i

Contlnuons

Besnu.

1

I

3

+

1 I 1 1

1

ll

. 1 rf

1

li

i 1

i|.

1 1

OONTINUOtnS BEAMS.

491

89. The illustrated table opposite represents the conditions theoreti-
eally existing in uniformly loaded continuous beams of from two to five equal
spans. Only the left haJf of each such beam is shown, the right half being
symmetrical with it.

90. The Figs, show the amoimt of the maximum positive moment in
each span, that of the negative moment at each support, and the shear on
each side of each support.

91* The Figs, show, also, the coefficient, a, for the distance, a I, from the
left support of each span to the point of maximum moment in that span;
and the coefficient, x, for the distance or distances, x It from the same sup>
port to the inflection point or points in that span. In each central span, the
sum of the two values of x is » 1. In each end span, x "■ 2 a.

92* In each central span, the point of maximum positive moment is at
the center of the span. In other words, the deflection in that span is sym-
metrical, or a "" 0.5.

93. The numerical sum of the two shears, one on each side of a support,
is — the reaction of that support. At each central support, the shears, on ita
two sides, are equal.

In the Figs.,

IT ■" load per unit of span ;
/ « span;

m B> the coefficient for moment;
mwP — moment ; *

V -" the coefficient for shear;
vwl " shear;

a — the coefficient for distance to point of maximum moment;
a 2 -■ distance from left support of any span to point of maximum
positive moment in that span;
X -» the coefficient for distance to inflection point ;
a; Z — distance from left support of any span to either inflection
. point in that span.

94. Fig. 15 shows the values of m and of r in a uniformly loaded non-con-
tinuous beam. Comparine these with the corresponding values in con-
tinuous beams, as shown m the illustrated table, opposite, we see that the
continuous beam has considerable theoretical advantage. But see ^ 95.

(a)

J

m

IE II

:ri:iiEn::

INI

. rJ

(6)

(C)

-OJf

Tig. 15.

95. Certain practical considerations, however, materially redu^th^
advantages in many cases. Thus, in a continuous railroad bridge of 100 ft.

spans, so designed that the maximum deflection shall not exceed -^ inch, a

settlement of -|- inch, in an intermediate pier, would deprive the bridge of

the support of such pier, and thus practically throw two adjacent spans
into one, bringing upon their members stresses far m excess of those for
which they were designed. Again, with moving loads, the theoretical ad-
vantage may at times be much less than that due to a stationary load and
indicated in the illustrated table.

492

STRENGTH OF MATERIALS.

Cross-shaped Beam.*

96. In a cross-shaped beam, Fig. 16, of homogeneous material, loaded at
center, let

W = the load;
E » the elasticity coefficient —

unit stretch'
Y — the deflection at center;
L, Z =• the spans of
D, d — the depths of
T, t = the half depths of
I, t — the moments of inertia of
S, « » the unit stresses in the extreme

fibers of
P, p » the portions of W borne by

the two branches respee-
tively.

Fly. 16.

Then (see illustrated table, p. 480), since the deflection is necessarily the
same for both branches,

L« P P p

I.P

EI
P
P

t.L»'

and, since P — 4 .

fp T * *°<* p = 4 . -— ^, (see table, p. 474), we have

L . Li t . I

8

§ ii-y-

97. In other words, in order that both branches may be equally strong,
their depths (independently of their breadths) must be inversely as the
squares of their spans, or their spans inversely as the square roots of their
depths.

Beslstance of Plates.

98. The laws governing the resistance of plates, to pressures normal to
their surfaces, are but imperfectly understood; and formulas respecting
them must be used with caution and as probable approximations.

99c Bectangiilar plate, with central load, W.
t = the thickness of the plate ;

L
I

S

c

its longer span ;

its shorter span;

the maximum unit fiber stress ;

a coefficient. See p. 493.

2 1.2 + fJ <2 •

W

s

(L« + P ) £«

Chi

t'oT a square plate, L

■■ I, and

^ 4^ r-

w - 1 s '*

*See foot-note (t), p. 493.

STRENGTH OF PLATES. 493

100. Rectangular plate, uniformly loaded. Let

w -B the uniform load per unit of surface ; other letters as in ^ 99.
Then, according to Grashof,*

For a square plate, L » 2. Hence,

S --^C.La.-^; w-4S. ^

101. Value of C.

For uni- For cen-

If the plate is form load. tral load.

merely supported along its four edges, C = 1.125 C = 2.00
firmly secured along its four edges, C = 0.75 C «■ 1.75

103. Ctreular platet uniformly loaded. Let
iff — the load per unit of surface;
S — theunitnber stress in the material;

E - its elasticity coefficient - ^^^^i

r — the radius of the unsupported portion of the plate ;
t " the thickness of the plate ;
Y >■ the deflection at the center.

Then, according to F. Reuleaux,t if the plate is merely supported.
If the plate is firmly secured,

»-|s(|)': i-r^

r-vhs-

2 to „ to r*

3 • S • eEfi'

For strengths of cylinders, pipes, etc., see Hydrostatics, Art. 17.

TBANSTEBSE AND LONGITUDINAL STRESS COMBINED.

103. Although the combination of longitudinal and transvei^ stress in
the same piece is objectionable, it is often unavoidable. Thus, in a timber
roof, the rafters generally act both as columns and as beams.

In such cases, the total unit stress, S, in the extreme fibers, is the sum
of the uniform stress, So, due to direct compression or tension, and the
extreme fiber stress, Sb, due to bending moments only, under the action
of the transverse and longitudinal loads combined. Or S = So -f Sb.

Let Mb " the bending moment due to the transverse load; Me '* the
bending moment due to longitudinal load. P; and M = the total resultant
bending moment, "-Mb — Me when the longitudinal load is tensile;
— Mb + Me when the longitudinal load is compressive.

But Me — P <2, where P »■ the longitudinal load, and d — its leverage, «

P Sk
the deflection of the beam, due to all causes ; and (see t 57) d — .^ -. ;

where I » span, Sb '^ unit stress in extreme fibers, due to bending;
£ — modulus of elasticity; T — distance from neutral axis to extreme fibers,
and c -" a coefficient, whose values, for different cases, are given in H 57.

Hence. Ble - P £-^ ; and resultant moment M - Mb + P ^r^- The

Hi 1 C — ill i C

resisting moment, R (see If 10), is — Sb m ; and, for equilibrium, R » M.

I P &

Hence, Sb. tk =" Mb + P ^srnr-', whence we derive, for the extreme fiber

stress, Sb, due to bending only, under the action of the transverse and longi-
tudinal loads combined,

♦ "Theorie der Elasticitat und Festigkeit." Berlin, 1878.

t "Der Konstrukteur," Braunschweig, 1889. "The Constructor," trans-
lated by H. H. Suplee, Philadelphia, 1893.

494 STRENGTH OF MATERIALS.

a M * I where the longi- I Sb — * ) where the longi-
p 23 y tudinal stress is I F Pf tudinal stress is

I + -ig— J tensile | I — .= — | compressive

Besides this we have the unit stress. So, due directly to th^ longitudinal
p

load, P, and — -rt where A is the area of cross-section of the beam. Hence,
A

for the total unit stress, S, in the extreme fibers, we have

p MbT

S " So + Sb ■■ "T +

^ ^ II'

0;

Sb-

MT
I '

as in

IT 10;

and

When the deflection, d, is negligible, M - 0; Sb
a P . MT

^"A + ~r-

It is often assumed that the resultant unit stress, S, in the extreme fibers,
is equal to the sum of the longitudinal and transverse unit stresses, and the
piece is then so designed that the resultant unit stress, so obtained, shall
not exceed the permissible imit stress. *

STRENGTH OP PILLARS. 495

STBEHeTH OF PIIiI<ARS.

The foregoing remarks on crushing or compressiye strengtli refer to that of
pieces so short as to be incapable of yielding except by crushing proper. Pieces
longer in proportion to their diameter of cross section are liable to yield by
bending sideways. Tbey are called pillars or columns.

The law governing the strength of pillars is but imperfectly understood ; and
the best formul» are rendered only approximate by slight unavoidable and un-
suspected defects in the material, straigbtness and setting of the column. A
▼ery slight obliquity between the axis of a pillar and the line of pressure may
reduce the strength as much as 50 per cent; and difierences of 10 per cent or
more in the bkg load mav occur between two pillars which to all appearances
•re precisely similar' ana tested under the same conditions. Hence a liberal
Ikctorof safety should be employed in using any formulse or tables for pillars.

In our following remarks on this subject, the pillars are supposed to sustain
a eotukmi load; and the ultimate or breaking load referred to is that one which
would, during its first application, cripple or rupture the pillar in a short time.
But struts in bridges etc often have to endure stresses which vary greatly in
amount f^om time to time. Their ultimate load is then less.

Long pillars with rounded ends, as in Fig 1. have less strength than
those with flat ends, whether free or firmly fixed.

In steel bridges and roofe, the ends of the struts are frequently
sustained by means of pins or bolts passing through (acros^
them, at either one or both ends. These we will call hinged
"Fig. 1. ends. Pillars so fixed are about intermediate in strength between
those with flat and those with round ends. There is much uncer-
tainty about this and all such matters. The strength of a given
hinged-end pillar is increased to an important extent by increas-
ing the diameter of the pin.

The formula in most general use for the strength of pillars,

is that attributed to Prof. liewls Oordon of Glasgow, and

called by bis name. With the use of the proper coefiBcients for the given case^

it ffives results agreeing approximately with averages obtained in practice witk

piuars of such lengths (say from 10 to 40 diams) as are commonly used.

It is as follows

Breaking load in Iks per sq inch ^ /
- of area of cross section of pillar is

1 + -I-
in which *•*«

f is a coefficient depending upon the nature of the material and (to
some extent) upon the shape of cross section of the pillar. It is often taken,
approximately enough, as being the ult crushing strength of short blocks of the
gWen materiu. For good American wrought iron, such as is used for pillars.
40000 is generally used : for cast iron 80000. Mr. Cleeman* found for mild steel

il6 per cent carbon) 62000 : and for hard steel (.36 per cent carbon) 83000 lbs.
[r. C. Shaler Smith gives 5000 for Pine.

A, for wrought iron, is usually taken as follows : a =

when both ends of the pillar are flat or fixed 36000 to 40000

when both ends of the pillar are hinged 18000 to 20000

when one end is flat or fixed, and the other hinged... 24000 to 30000

For cast iron about one eighth of these figures is generally used ; and for pine
about one twelfth.

1 is the length of the pillar. If the pillar has, between its ends, supports
which prevent it firom yielding side-ways, the length is to be measured
between such supports.

r is the least radius of gyration of the cross section of the pillar. I and r
must be in the same unit; as both in feet, or both in inches.

^ 1^ ■ II I ,_ _ 111,

• ProoeedingB Engtneeri' Club of Phlla, Not 1884.

496 8TBENGTH OF PILLARS.

Radius of iryration. Soppose a bo4y tn9 to raTolTe around an axis whick
paases through it in any direction ; or to oscillate lilce a pendulum hung from a point
of BuspenBion. Then suppose in either case, a certain given amount of force to be
applied to the body, at a certain given dist from the axis, or from the point of sn^
pension, so as to impart to the body an angular rel ; or in other words, to canae it
to describe a number of degrees per sec. Now, there will be a certain point iti ttao
body, such that if the entire wt of the body were there concentrated, then tha samo
fbrce as before, applied at the same dist teom the axis, or teom the point of raspMi^
sion as before, would impart to the body the same angular motion as before. Thli
point is the center of gyration ; and its dist flrom the axis, or fh>m the point of sili*
pension, Is the Raditu ofgypraJtUmy of the body. In the case of arecu, as of croaai
sections of pillars or beams, the wirfwst is supposed to reToWe about an imaglMM
axis ; and, unless otherwise stated, this axis is the neutral axis of the area, whin
passes through its center of gravity. Then

Kadins of gryration — i/Homent of inertia h- Area

Square of radius of gyration — Moment of inertia -»- Area

In a circle, the radius of gyration remains the same, uo matter in what direc-
tion the neutral axis may be drawn. In other figures its length is different for
the different neutral axes about which the figare may be supposed to be capaUe
of revolving. Thus, in the I beauL page 898, the radius ox gyration about the
neutral axis X Y is much greater than that about the longer neutral axis A B.
In rules for pillars the least radius of gyration must be used.

The following formulae enable us to find the least radius of gyration, and tha
souafo of the least radius of gyration, for such shapes as aie commonly used for
pillars.

Shape of cross seotion I<eas« radius ii!S?^'^?£«

ofplllar Of gyration ^o^S^JiflST

side side*

Solid square

V^

12

Hollow square of uniform f^-hdl^ D> -f <|i

\ 12

thickness \/ 12""" — 12"

Solid rectangle

least side least side*

12

Hollow rectangle of uniform ^/ C^A — ^i^ 0»A — c»s
thicknen \12 ((JA — 00) 12(CA — •<

diameter diametei*

Solid circle

1«

Hollow circle of uniform /P* + d» D> -f- rfl

thickness A/ 15 — |g —

* r'iisftboMS.MAl.

STRENGTH OF PILLARS.

497

Tlie foUowlny are only approximate :

Shape of eross section
of pillar

.j[>.

f--*^-^

PboeDix column.
Carnegie Z-bar column.

I beam.

I^eiwt radios .|S5«din.

D X 0.3636

D2 X 0.1322

BX 0.590

B8X 0.348

F

F8

4.58

21

-F-^

Cbannel.

F
3.54

F«
12.5

l^-F

Beck beam.

6

36.5

Angle, wltb equal legs

5

F8
25

. Angle, witb unequal legs

■^^^

yj

F2/2

2.6 (F-f/) • 18 (F2 4-/«)

F-rf

T, with F »= f

f

I

Cross, with F « /

F
'<.74

22.5

32

498

STRENGTH OP IRON PILLARS.

The Toniiff engrtneer mast bear In mind that the breakg and the
lafe loads per sq inch, of pillars of any given material, are not constant quantities ;
but diminish as the piece becomes longer in proportion to its diam. If a yery long
piece or pillar be so braced at intervals as to prevent its bending at those points,
then its length becomes virtually diminished, and its strength increased. Thus, if a
pillar 100 ft long be suflBciently braced at intervals of 20 ft, then the load su8taine<^
may be that due to a pillar only 20 ft long. Therefore, very long pillars used ic.
bridge piers, Ac, are thus braced; as are also long horizontal or inclined pieces,
exposed to compression in the form of upper chords of bridges ; or as struts of any
kind in bridges, roofs, or other structures.

Mistakes are sometimes made by assuming, aay 5 or 6 tons per sq inch, as the safe
compressing load for cast iron ; 4 tons for wrought ; 1000 pounds for timber; without
any regard to the length of the piece.

But although the final crushing loads, as given in tables of strengths of materials,
are usually those for pieces not more than about 2 diams high, they will not be much
less for pieces not exceeding 4 or 5 diams.

Cautions. Remember a heavily loaded cast-iron pillar is easily broken by a
side blow. Cast-iron ones are subject to hidden voids. All are subject to jars and
vibrations from moving loads. It very rarely happens that the pres is equally dis-
tributed over the whole area of the pillar ; or that the top and bottom ends have per-
fect bearing at every part, as they had in the experimental pillars.f Cast pillars are
seldom perfectly straight, and hence are weakened.

Hollow pillars Intended to bear beavy loads should not be cut
with such mouldings as a a ; or with very

f rejecting bases or caps such as ^, Fig 19.
t is plain that these are weak, and would
break off under a much less load than
would injure the i(haft of the pillar. When
such projecting ornaments are required,
they should be east separately, and be at-
tached to a prolongation of the shaft, as
cd, by iron pins or rivets.

Ordinarily, it is better to adopt a more
simple style of base and cap, which, as at
b, can be cast in one piece with the pillar,
without weakening it.

Piir. 19.

^^-y^^^t^^^^

Fig. 20.

When a fltt-ended pillar, Fig. 20, is so irregularlv fixed, that the pressure
upon it pusses along its diagonal a a, it loses much of its strength. Hence the
necessity for equalizing, as far as possible, the pressure over every part of the
top and bottom of the pillar ; a point very difficult to secure in practice.

t In important cases both ends should be planed |MrflBctly true.

SH£ARINO AND TOB8ION.

499

BHEARIMO STRENeTH.

Sliearingf or detrnslon occurs when a body is acted npon by two
opposite forces in parallel and closely adjacent planes, tending to slide some of
tae particles over the others. In i'Hg 1, the two forces are (1) the downward
pressure of the weight, W, and (2) the upward reaction of the support, A.

In sinele sbear. Fig 1, the shearing area, a, = the section g g. In double
•hear. Fig 2, a = gg-\-oo^2Xff9' In Fig 8, a = 6 X cross section of
piece. In Fig 4 (single shear), a =» section c c. In panehiug rivet holes,
a = circumference of hole X thickness of plate.

In any case, if S => the ultimate unit shearing stress, Shearing strength = 80.

Fig. 3

Fig. 4

Ultlniate nait sbearlsfr stress, S, in lbs per sq^incb. The following
igures indicate the range of values of S in metals and in timber.
Metals. Wrought iron, 35,000 to 55,000 ; cast iron, 20,000 to 80,000 ; steel.

Fig. 1 Tig. «

figures indicate the range of values of S in'metals an

[etals. Wrought iron, 35,0
46,000 to 75,000 ; copper, 33,000.

r Spruce
I Wl

With the grain, Fig 4. Across the grain.
250 to 600 3,250

hite pine " 2,500

Timher: J Hemlock '♦ "

From our experiments : \ Yellow pine 4,300 to 5,600

J Oak 400 to 700

• • •••••••

4^400

White oak

TORSIOBT Ali STREBT OTH.

Torsion occurs when a body is acted upon by two couples or moments of
contrary sense and in different planes. Thus, torsion takes place in a brake
axle when we try to turn it while its lower end is held fast by the brake chain ;
and in shafting, when it transmits the motive power of an engine to tools. Sup-
pose such a body to be divided, by cross sections, into layers. Then each layer
tends to shear across from those next to it. Hence, in order to maintain equi-
Mbriuia, each two adjacent layers must exert, in the cross section between them,
an internal resisting moment equal to one of the two external and contrary
torsional moments.

Resistlii|ir moment in a circular cross section of a cylindrical shaft. Lei
P a the torsional force of one of the two external moments in pounds ;

I s. its leverage, = its distance from the axis of the shaft in inches ;

M =3 P / = external or torsional moment in inch-pounds;

T = distance from axis to farthest fibers, = radius of shaft in inches;

D s diameter of shaft, = 2 T in inches;

8 =s unit shearing stress in farthest fibers in pounds per sq inch ;

/ =3 distance from axis to any given fil)er in inches;

t = unit stress in said given fiber .in pounds per square inch ;

a =3 area of said given fiber in square inches;

F s total stress in said fiber in pounds;

r ss resisting moment of said fiber alx>ut the axis in inch-pounds;

B ■= internal resisting moment of the entire cross section •
= 2 r == sum of resisting moments of all the fibers in inch-pounds;

lip= polar moment of inertia* of the cross section
s moment of inertia of cross section about the axis of the shaft
= 2 /* a, =« the sum, for all the fibers, of fi a in inches.

*Tn any figure, the polar moment of inertia, Tp, is = the sum of the greatest
and the least moments of inertia of the same figure, about two axes lying in the
figure and Intersecting in its center. In a solid circle, each of these is a moment
of inertia about a diameter, and is => w T^ -^ 4. Hence, in such a circle.
I, - IT T* -^ 2.

500 STRENGTH OF MATERIALS.

Then the unit stress, in any given fiber, is « =» S f -«- T ; its total stress, F. is
msas = Sat-i-Ti and its resisting moment, ^is = Ft = 8afi-i-T. ¥ot
equilibrium, the internal resisting moment, R, of the entire section, must be >■
the external torsional moment, M.

Heuce, for the Internal Resisting^ moment, R, we have :

R = M = 2 r = 2 S a <2 -J- T = (S -5- T) 2 <« a = (S -5- T) Ip.

Hence, also, S = M T -i- L; M = S Ip -^ T; and F = M -5^^ = S Ip -s- {T I).

In a solid circle, L = ir T* -f- 2. Hence, S = 2MT-*-(irT*) —
2 M -i- (ir T8) ; M = S ir T^ -t- 2 ; P = S ir T« -5- (2 ^ ; and

Diameter, D, - 2 T = 2 X \jl^ = \l^-^ = 1.72 'y/|.

For approximate ultimate Talnes of S, for torsion, use the

yalues for shearing, p 499, with safety factors from 5 to 10.

Horse power of shafting. In one revolution, the force, P fi>8, de-
scribing a circle with radius = / ins, does a work = 2 n- / F inch-fi>s, and, in n
revolutions, work = 2 ir / F w inch-fi^s. If n be the number of revolutions per
minute, the horse power is :

H = 2ir;Fn-*-(12X 33,000) = 2 ir M n -j- (12 X 33,000) ;

or, since F I = M = R = SIp-5-T, we have :

H = S rrn Ip -f- (12 X- 16,500 T) ; and S = 12 X 16,500 T H -*- (irn Ip).

In a solid cylindrical shaft, I„ = tt T* -^ 2. Hence,
H=SirwirT44-(12X 33,000 T) = S n^n T» -4- (12 X 33,000) :
8 = 12 X 33,000 H -f- («-2 n T^) = 12 X 3,343 H -i- (n T«) :
n = 321,000 H ^- (Ds S) : and \

Diameter, D = 2 T = 2 X

® 112 X 3,343 H * /321,000 H * /

A' — s^ — == ^-k^- = «8 Vi

H^

Sn*

The higher the speed, the less is the force, and hence the less
is the strength of shaft, required in order to transmit a. given horse-potoer ; but
if the speed is increased by increasing the torsional ^rce, the horse-power
transmitted is thereby increased also.

Example. Given, a wrought iron shaft; let S = 6,000 lbs per sq inch;
P = 7,500 fba; I = 10 ins; M «= 75,000 inch-ros. Required the diameter, J).
Here, D = 1.72 X f aHTS = 1.72 X f TSiOOO'^T'pOO = 1.72 X f izis -
1.72 X 2.82 = 4 ins. Let the horse-power, H, = 25. Then n ~ 321,000 H -s-
(D3 S) = 321,000 X 25 -:- (4^ X 6,000) = 21 revolutions per minute. Checking,
D = 68 X f H -^ (Sn) = 68 X f 25 h- (6,000 X 21) = 68 X 0.068 = say 4 inches.

Rectangpnlar Sections. The foregoing equations are based upon the
assumption that the stress increases uniformly from the axis of the shaft out-
ward. It has been shown (notably by St. Venant*) that this assumption is not
applicable to square and rectangular sections. In a rectangle, let 6 = the longer,
* = the shorter side, and c = 6 -r- B. Then S = M (8 + 1.8 c) -j- (B ftS) ; and
F = S B 62 -A. [(3 _}. 1,8 c) I]. In a square, with side = b, this becomes : S «■
4.8 M -h b»: M = S 68 -s- 4.8. P = S &» -f- (4.8 0-

The ang'le of torsion is that described by one of the external torsional
moments, relatively to the other. Within the elastic limit, this angle is pro-
portional to the torsional moment, M, and (assuming / constant) proportional
to the force, P. Other things being equal, the angle is proj)ortional to the dis-
tance between the planes of the two contrarv external moments, and. in a solid
cylindrical shaft, is inversely i)roportional to D*. It is inadvisable to allow
the angle of torsion to exceed 1° in a length = 20 diameters, in shafts revolving
4n one direction. In reciprocating shafts allow still less. See Fatigue, p 465.

Practical Considerations. In many cases the diameter of the shaft
must be made greater than that required by the foregoing forranlas ; as in a long
shaft, in order to keep the angle of torsion wHhin permissible limits; in fly-
wheel and other shafts, carrying considerable bending loads in addition to the
torsion ; and, in most cases, to allow for additional moments due to alternate
acceleration and retardation.

*See Treatise on Natural Philosophy, by Sir William Thomson and
Guthrie Tait, Part II, New Edition, Cambridge, 1890, pp 236, etc

HTDROSTATICB.

501

HYDROSTATICS.

Art. 1. Hydrostatics treats of the pressare of water and of other
fluids at rest.
At any giyen point within a flaid tbe pressure is equal In all

direetioiis; and the pressure against any point of any surface, whettier
plane or curved, is normal to the pressed snrfoee (or to a plane tangent
to that surface) at that point.

The Intensity of the pressare is proportional to the depth of the
point below the water surface.

Pressure aspainst any plane sarfince.
Let

a = the area of the pressed surface ;

h = the vertical depth of the center of gravity of the pressed surface below
the free surface of the fluid ;

H = the total depth of the fluid ;

w — the weight of a unit volume of the fluid ;*

p =s the mean unit pressure on the pressed surface ;

P = the total pressure on the pressed surface ;

llien the mean unit pressare, p, is equal to the weight of a prism of
the fluid, whose base is = 1, and whose length is = A ; or

and the total pressare, P, is- equal to the weight of a prism of the fluid,
whose base is = a, and whose length is = A. Or

'P = ahw = ap.

In tbe diagri*ams of Figs. 1 and 2, the ordinates (supposed to be drawn
from, and normally to, the pressed surfaces respectively) represent the unit
pressures (as in fbs. per sq. inch, kilograms per sq. centimeter, etc.), and the
areOt opposite any given surface, repreeeuts the total pressure on that surface.
Thus, in Fig. 1 (a), the unit pressures at n, at (/, at c and at o, are represented
by n (= 0), by o' a', hj c & and by o g respectively; and the total pressure
on no is represented by the area nog, that on n o' by the area .n o' 9', that on
& ohy the area 0 g', and that on 0' c by the area ccf. ^

Tbe center of pressare upon any surface is opposite the center of
gravity of the area representing the total pressure upon that surface. Thus, in
FigB. 1, the center of pressure on no, is opposite the center of gravity of the

2
triangle, n 0 9, or at a depth, d, = — - H, below the water surface. See The Center

o

of Pressure, Arts 8, etc., also ff 133 etc., of Statics.

Tbe bydrostatic paradox. For a given depth, A, both the mean unit
pressure, p, and (for a given area, a) the total pressure, P, are independent
of the quantity of water. Thus, in Figs. 1, the walls sustain as great a pressure
from a vertical film of water only an inch thick, as if the water extended back

*For water, to = about 62..'» lbs. per cub. ft., = about 0.0362 fi>s. per enb. ineh. .^

HTDROfiTATICB.

for miles. In Fig. 2 (6), the eicem of weigiit of w»Ur, over that In Fl^ 2 (o),
lotil pre«ur« upon th« base is greater, add in Fig. 2 (ii)lesa, Chnii the weight ol

th. -af..r. hnl in .liihcr fou tha a I ni'li rs j|. Bum nf all tbS TertlOl prSBSUnS iS

' nelght of all the irat«

."^ti.

Il^d lElltgebraio snia of alVSe iori™

laid^

Nov l«t the lube, a o, 3S ft. h^ and of 0.2ST inch bore,

the lube, a r), U6 ft. high aj ,

— --- . n the tube alune,alihough veighlng odI; al

I pound, will cause an iddiUoual hunting preesurs of SON Bw. per sguare fool
or ssj 334 tons total, to Ik exeited upon (he lop, bottom and sides ot Uie box.

iressnre «n w>(er snrftace. In addition to ih« prewura of the
lel^ the free surfsoe of bh; bodt of water suslHioa also the prea'tire ot
^ about 11.7 lbs, per so. inch, 'iliis pressure (trausBiitted through the
the walls of the •easel) Is indiesied bj the disgrama <paralle)ogramB)
'air- in Fig. 2 (6). lu most eases, the pressure aeaioit a surDue, duo
r pressure oil Ibe water aurface, is counler-bAlauceab^ an equal n

that prenure loogitudinallr and thus a°old bmdiog momenu ; but nihsr ooo-
i> nl»rij''y"'^°'^the 'po^M wm^d'hS'vt^lo' be T^de lDo''rd'lStlery°long''; Mii

or the down-stream pos^would proJ«!t h» °^ihe''c^t''or'the dBm%ad would
thus he liable Co Injury by Ice or Vogs. etc.. tumbling orer Ibe dam.
Inasmuch a< the pressure increase uniformlj- from the water sorAce dowo-

Slmllarly, Ihe hoopa oD Uoks, if of unlfoim strength, are pUced cToser logeU^

HYDKOSTATIOa.

508

Horizontal and vertical components. In Figs. 1 (b) and (c), the
force triangle (Statics, ^46, etc.) gives us tlie horizontal and vertical componeDts,
li and y, of the total normal prtissure, P. Or, if n o be taken, in each case, as
r^resenting the total normal pressure, P, by scale, then H = the total hori-
aontal pressure. In Fig. 1 (a), with pressure against a vertical surface,
L = P, and V = 0.

In Fig. (b), the yertioal oomponent, V, presses the wall downward against its
base; but in Fig. (c) it tends to uplift and overturn it.

The depth, H, being the same in each of the three figures, Figs. 1 a, 6 and c,
the vertical projections of the three submerged surfaces are equal, and hence the
total horizontal pressures are equal in the three cases ; but the horizontal projec-
tion, and consequently the total vertical pressure, vary with the inclination of
the surface. Thus, in Fig. 1 (a), the horizontal projection and the vertical
pressure are each = 0.

Pressures in enbical and otiier wessels, ftall of water. Let

F
F — the weight of water oont«ined in a prismatic vessel; and / s= - = the

3
weight of that in a conical or pyramidal vessel of the same base and height.

In a cubical vessel, we have

pressure on base »=* F ;

F

one side = r- ;

2 *

u

it

II

li

base and four sides. together = F + 45 = 3F.

In a conical or pyramidal vessel, we have

pressure on base = 3/= F.

In a spherical vessel, we have

total-pressure = 3X weight of water.

Art. 2. Unequal pressures in opposite directions. In Figs.
^ let a = the area of that portion, n' o, which is subjected to pressure on both

Tig. 4.

■ides; and let H and h « the vertical depths of the center of gravity of n'o
below the two water surfaces, n and n' respectively. Then, in each Fig., the
large triangle, nog, r^resents the sum of the pressures of the deeper water on '
the left against the entire wall, no; the trapezoid, m a, represents the sum =
a H tr, of the pressures of the water on the left against the portion, i^o; and the
smaller triangle, n' o ^, represents the sum = a A tc, of the pressures of the shal-
lower water on the right against the portion, n' o. Then the parallelognuoa, n' o,
represents the excess of pressure, from the left, against ttie portion, n'«. This
exoess, due. to the diflSeveace, K-h^ between the two levels, is un^oi-m^y distributed
over n' o, the uniform exoess unit pressure being represented by the ordinate
»' m s 9<" g. . ,

The preseure coming from the right against the portion n' o, and represented
by the triangle n' o <p, is balanced by an equal portion (represented by the
triangle n'oi/^) of the total pressure, mo, from the left against the portion
s'o; and the centers of these two pressures, each being at a depth = % n' o
below n', are opposite. Hence these two pressures are in equilibrium. But the

s'^ coming from the left, acting through its center of gravity and therefore
tendtBg to move it bodily toward the right, without rotation.

KYDnOSTATlCa.

, 3. Hnrnicea, vert, ■■ A m e «, a h o t, Fla fl, or otberwlse, of

I wlilths, b m, a n; (!omin«iiclng M the level, banm,at

inter, but extenalaa; lo diffilepths, jnc.no, menBOred

'Ing tbe utBae IncItnnCiaii to th» snrr of (li«

- '-'-■ proportlansl lo the Bqnnra*

Art. 4. Tta« pressure of quiet wnlcr, tn na^ one rlTen di-
rectton, agniDat unr glisn plane mrbcs, wfaeUiar tbRIcbI, lioriHiaUl^Dr IncUntd,

RuLi. To And the pre* In lbs. mull togethBrthtuM
I H| rt or Uh bniMUtn ulin at rigtii kitiIBi tn ib fl?« dlnallH i Om
jrf deptb Id h of ihe«n or fn^r «f gi« pmrtd aurr Mo* U* Dpp«T iHrf

3 « i»MrwuH™«iit'w™pi^i3"l»»IMTf,

« reqd to And only

I n, lo ponndi, mnU logotfa

HTDKOSTATIC8.

505

a

n'A,

.11.

1^9

Therefore,

EotO

Again, let Fig 9 repreMnt a conical Teasel fall of water;

Ms base 6 e, 2 ft diam ; iu vert heifht a n, 8 ft; tben (be eixvumf of the base will be
1.3883 ft; the area of the base 3.1416 sq ft; the leugth of Its slant side a b ot a c, 8.16
fl; the area of iu ourred slaating sides wlU be ^-'^^^ X 8-16 _ ^^ ^ ^^, ^^ ^^^

2
Tert depth of the oen of grav of the slaDting sides will be at two-thirds of the vert "U <
height a n from the apex a, or 2 ft.

Here, to find the total pres agaiast the base, we hare by rale in Art 1, 8.1416 x 3
X 62.5 = bH&.Oi lbs. For the total pres against the slant sides, by the same rule,
9.83 X 2 X 62.5 = 1241.25 lbs. For the vert pres upward against the entire area of the
slant sides, we have giTon the area of the base (whioh is here the hor prcti^tioD of
the slant stden) = 8.1416; and the vert depth of the cen of grav of the slant sides, 2 ft.
3.1416 X 2 X 62.5 = 892.7 As, the upward vert pres.

Finally, for the hor pres in any given direction against the slant sides of on* half of the oone, we
have the vert projeotioa of that half, reiH«MBted by the trian^e a b e, with iti base 2 ft, and Its perp
heightSft; andoonsequenUy, with an areaof Ssqft. Thedepth of itseenof gr»T is 3ft; thweCon^
8X2 X 62.5 = 375 lbs, the reqd hor pres.*

In Fig 10, whioh represents a vessel full of water, the total pres
•gainst the semi-oylindrieal surf avemdk, and perp to it, must be
also hor, because the surf is vert ; but inasmuch as the surf is ettrvtd,
this total pres, as found by rule in Art 1, acts against it in many di-
reotions, which might be represented by an infinite number of radii
drawn from o as a center. But let it be reqd to find the hor pres in
Bke, In one direction only, say parallel to o e, or pefp to a d; whioh
voaid be the force tending to tear the curved surf away from the flat
fides a 6 n «, and d c « fe, by producing fractures along the lines a v
and d t ; or which would tend to burst a pipe or other cylinder. In
this oaee, mnlt together the area of the vert projection a d k v in sq
ft } the depth of the cen of grav of the curved surf in ft ; (whioh, in
the semi-cylinder would be half of e m, or of o < ;) and 62.5. Since
tbe resulting pres is resisted equally by the strength of the vessel
along the two lines a v and d ft, it is plain that each single thicliness
along those lines need only be sufiQclent to resist safely one ha^fot it ;
and M in the ease of pipes, or other cylinders, such as hooped cisterns
or tanks. See Art 17.

Should the pres against only one half of the ourred surf, as e d nt Jk
be sought, and in a direction parallel to o d. tending to produce frao-

tnree along the lines e m, and d k, then use the vert projection o e m < ; with the same depth ; and 6S.&
as before.

It follows, that if the faoe of ametalHo piiton be made ooooaye or oonTex. no more pres will be reqd
to force the platon thtoni^ any diet, than if it were flat; for the pna ecalMt the face of the piston,
in the direction In whleh it moves, mnst be measured by the area of a prtifcetlon of that faoe, takea
at right angles to said direction ; and the area of said pr<^ection will be the same in all three oaaei.

Rem. 2. If a bridgre piei** or otber constraction,
Fis* 10 ^ be fonnded on sand or gprairel, or on any kind of

foundation tnrough whioh water may find its way nndemeath, even in a very thla
■beet, then the upward pres of the water will take effect upon the pier ; and will tend
t« lift it, with a force equal to the wt of the water displaced by the pier; (Arts 18,
19. In other words, the eflfeotive wt of the tubm«rg«d portion of the pier,

vin be rednoed 62^ lbs per onb ft; or nearly the half of the ordinary wt of masonry.

Bat if the foundation be on roek, coyered with a layer

ef oement to prevent the infiltration of water beneath the masonry, no such effect
win be produced ; but on the eontrary. the vert pres downward, afforded by the-bat- .

tering sides of the pier, and bv iu ofltets. will tend to hold it dowa, and thus inorease its sUblllt/;
wbioii, in gniet water, will then actually be greater than on laud.

Art. O. To divide a reetangrnlAr surf,
irbether Tert nm abed, or inclined aa
fnno i>; Fiir 11, whose top a & or m n i»
IcTOl witli the snrf of the water, by a
hor line as 2, such that the total pres
asainst the part above said hor line,
irtiall e«inal that aicainst the part be-
low It. _ . V

Bias. Unit one half ef the length of 5 0. er mp, as the ease
may be, by the oonttant nnmber 1.4142; the prod will be fr 2,

**£.*' Let ft e= U ft. Then 6 X 1.4142 = 8.4862 ft ; or ft 2.
Lat i» » = 16 ft. Then 8 X 1.4142 = llJtl36 ft, or m x.

BasfT The Una a 2, thoe fonnd, moat not be ooufonnded with
the cen o/prea, which is entirely diff. See Art 8.

Art. 6- In a rectangrnlar snrf, whether vert as a bed, or in-
«tliif ed Mtnnop, Fig 11, whose top a b or mn coincides with
the snrf of the water, to find any number of points, as 1, 2, Ac,
thronsh which if hor lines, as 1#, 2 as, Ac, be drawn, t^ey will
diTide the siven surf into smaller rectangles, all of which
shall sustain equal pressures.

RiT« First fix on the number of small reetangles reqd. Then for point 1 from the top. mult the
«i2JKi:br«iM»Srofrectangk». Take the sqrt of the prod, Mnlt this sqrt by the entire length

K^io^

• Id a sphere filled with a fluid the total inside pros =x 3 timee wt of fluid.

506

HYDROSTATICS.

i«ormj>, uth«eaaemii7^ Dir tlia prod by th« nnmbar of reoUoglM. Tbe qaot will Iw (h« 41M
i 1. orn 1, an the case m*j be.

For the dist i 2, or n 2, prooeed io preoiwljr tbe Sftme waj; onlj inBteftd of the namber 1, a«9 th«
nnmber 2 |o be mult by the oamber of reotAogles : aod eo oae eaooessively the numbers 8, 4, 6, Ao,
If it be reqd to find that namber of polats.

Bx. Let 6 c = 10 ft ; and let it be reqd to find 2 points, 1 and 2, for dividing tbe rectancalar suf
mhcd into 3 rectangular parta, which shall sustain equal pressures. Here we hsTo for point 1,

1X8 = S. The sqrt of 8 = 1.782. Aud 1.782 X 10 (or » e) = IT.U. And

For point 2, we hare

2X8 = 8. Thesqrto

And so for any namber of poiuts.

17.82 _
Sreotaogtes

5.TT8fl=»L

2X8 = 8. The sqrt of 8=2.440. And2.449X10(or» e)= 84.48. And. **** . =8.168 ftsftt.

Sreotaagles

RlM.

1. This role will be found useftil In spaeinir tbe
l»»r« of lo€k-i:ates; tbe booiM around cyliudrieal elateross
and tbe props to a structure, like Fly 8.

Bui. 2. For dlTldlny any surf, as o ft 0 d, Fl|r 12, wblcb Is not

rectanirular, in tbe same manner,

with an aooarao^ sulBoient for moat practical purposes,
haps the following method is as coaveoieut aa any.

Bulb. First dir the surf, as in Fig 12, into several i __
hor parts, equal or not, at pleasure. Then by Bale in Art I,
find the pres on each part separately, as is sappoaed Io bu
done in tbe numbers on tbe left hand of the fig. The sum of
these (in this case lodlO) is the total pres against the eailr*
surf 9 6 e d. Now suppose we wish to div this surf la 4 parte
bearing equal pres; drat div 1S610 by 4=8878. Then begin-
ning at tne top, add together a nnmber of the separate
pressures anOicient to amoaat to 8878 ; by this means fla4
point 1. Then procoed with the addition until the suoi
amounts to twice 8878, or 7756. which will indicate point 2;
and in the same manner find point 8. by adding up to three
times 3878, or 11634. Then the hor dotted lines raled through

Gints 1. 2, and 3, will give the reqd dlviaiona approximately,
thia manner the hoops of conical, and other shaped Tea-
sels, may be spaced nearly enough for practical purposea.

IISO

16'iO

2060
250a

Total = 15510

Tig. 12,

Art. 7. Tbe transmission of pressure tbrouirl^ water. 'Wn«
ter. In eommon witb otber fluids, possesses tbe Importauit
property of transmlttlntc pres equaUy in all direetions. Thus,

suppose the vessel. Fig i8, to be entirely closed, and filled with water;

T and suppose the transverse area of T,C, D, and B, to be each equal too«M

C ' aq ineh. Then, if by means of a piston, or otherwise, a pres of 1 1^, 1

ton, or^any other amount, be applied to the one sq inch or area of T, Ot

f sq inch of the inner surf of the vessel, and of the pipe 9,

receive, at right angles to itaelf, an equal pres of 1 lb. «r

addition to the pres which it before suataiBed fkom th«

water itself; and this will occur if the veaael oonsiat of parts even mttos

asunder ; aa, for Instance, if T w«re miles distant from B ; and unitei

to it b/ a long aeries of tubes. If the vessel were a strong steam beUsr

ftall or water, a single pres of a tew hundred pounds at T, C, *o. would

burst it. See also Fig 2 (e) and paragraph above it.

Tbe bydrofltatle press aets on tbis prln-
eiple. Any body, within the ▼eeaet, would also reoeire
an equal additional pres on each sq inch of its surf.

If the top of T be open, the air will press upon tits sq inch of the exposed surf of water tp the extent
of nearly 16 lbs ; and the same degree of pres will also be transmitted to everv sq inch of the Ulterior
torf of the vessel, and its oonneeting tubes ; but no danger of bursting will result trom thia atmo-
iphertc pres, because the air also presses eTei7 sq inch of the outside of the vessel to the same exteat.

Air, and otber graseous fluids, transmit prjMi equally In all
direetions, like liquids; but not as rapidly.

r~i wn, or any otner

J L D, or B, every sq

/^ X . will Instantly reo<

a. / V^E Iton, *o: in add

Fig. IB.

Fig. 14.

•g, aoaato

Art. 8. Tbe eentor of pressure. Let Vig 14

represent a veesel tnU of water, and suppose the side P to be petJfcoUy
loose, so as to be thrown outward by the slightest pres of the water trmm
within. Now, there is but one single point, P, In eveiy swf ao preaaad,
no matter what its ahape may be, to whieh if we appfy a fbroe e^aal ta
the pres of the water, and in a direetlen opposite to said pvea. the sMeP
will be thereby prevented from yielding. Boeh point Is called the osm>
Ur of yrtmwr*. It must not be understood by this tbut the actual
amount of pres of the water agalnit that part of the snrfaee whiah la
above the hor dotted line passing through F, Is equal to that of Ike waMr
tMlow said line ; but that the sum of the products of the aevatml preaauiea
above It, mult by thtir several leverages, or vert disu from P, Is equal
to the sum of the products of the pressures below, mult by their levtr-
ages ; or, in other words, that the sum of the momsnte armmd the peibft
P, of the preaaurea above the line, is equal to tbe sum of tlie moMSMle
of those below It : so that if a hor iron rod 6 ft were passed entirely

dotted line, m showa to IM

through the tids P, at the same level m the
aa a hiafe for the side P to turn on, the side would have no teBdanay to tarn.

HYBBOBTATtCS.

507

Art* 9. To Anil il&e een of pros ^Jt a q«lei
AiiMt fisalnBf a plaae •urfiMse. Fig 15.

1. The oeoter of pressure of a quiet fluid against auy plane surface
wbose vidtb ia uniform throughout its depth, whether said surface be
Tertieal, as e o, or inclined, aa e a, (or iiioiiued in the opposite direction :)
and wbose top c, or «, coincides with the hor waiM* surif ; is dlNtant wrt
below the water surf, two>thirds of the vert depth, «s, from said water
surf to tbe bottom of th<3 plane ; as at n and i. Inasmuch as a hor line
at % of the depth of ax, intersecu boib ea and so at ^ of their length*
respectively, we might say at onoe thai the ceuter of pres against a plana

{>arallel(>grMm, with its top at tlie water surface, is at two-thirds of it*
ength betow the water surface.

Throughout Art 9 any measure, as yard, fck>t, or inch

&Ct may 06 used.

%. But if the hor top a, or o, Fig 16, of the rectangular plane af, er
• k, be covered to some depth vith water, then the vert depUi ««, af a*
MD of pre* d, or e, below the tnrf of the wal«r, wtU be aqMal ••

cube of « e — cube of « w

I

of

square of « c — square of « 10

where so ia the vert depth of the bottom, and « tr the vert depth of the
top, of the pressed aurf, below the water surf. Or, in words : Prom the
crabe of se, take the oatw of «w; and oall the rem a. Then, from the
square of s e, take tba square of »ic ; and imll the rem h, Mt a by ^,
•bA take ttf*-thirda of tbe q«ol for •••,

8. When a plane snrf of any shape whatever, whether
rectangular, trfangnlar, or drcalar, ^; whether rert as
op, Vig 17, or inolined as m», is entfrely Isimersed, ao as to
be pressed over the entire area of both sides : bat by 4^
tfMtlAe of water on its two sides ; then the oen of pre* soia-
ciaea with the «•» nfffmm of the pressed surf.

lb tbe 8 foregoing flgnres tbe supposed snrfttoes are shown
•dgewlae, 10 that their widths do not appear.

Fig. 15

Fig. 16,

Fig. 17.

4. In any triangplar plane sorf. whether right-angled, or
otherwise, as a 6 e. Fig 18 ; whether vert, or Inclined ; Uie 6cMe
a i of which oolnoides with the hor surf of the water ; the oen
of pros o, will be in tbe oenter of tbe line e r, which biseota the
base at.

6. But if tbe triangle, as a s e, vert, or tnelined, haw Its
I, at the sarf of the water ; and its base s e. iter ; then ae
of proa s, wiB alao be In the line am whlob blaeotaae bass;
as wUl bs Hf of am.

a If Mur plane triangle a& e, Fig 19, bate ap, and hor ; hare its base
• ft ooTored to some depth isd, with water; then ae oen of prea •, will
lis In ae line e« whieb biissts tbe baas ; and no will beeqtwla

mw« -t- (»ma X ma) -H 8ma»

(«l(i 4- Sma) X S.

Fig.lA.

7. TheosntsrofpvesagninstanT
flana rsetaajwlar enrfaoe, Fig aO.
wlisaer Tert aa mn, or InoUned as
|i0, or w«; having its top coinoidlnf
wia as snrf of the water; and
pressed br diff depths of water on
Its opposlls sides, as shown a tbs
4g: wlU bs sen below as
wnter snrf, a dist equal to

ifl/»srt

Fig.dO.

/•rsao/swi/nin,5.3^rt»l,\ / orsaqfsiw/ x?*!?** rb\_/jS»X JJI^c? X ''V**!*^

Umrtaiiftmrf y. jaVs/ab") — ( «'««qr««rf x Ml/^ rbV
a, or f o» or w» '^ ^ •/ • "^ Von, or s 0, or s X '^ '"^ ^ ' "V

508

HYDROSTATICS.

8. To find the center of pressure against either a circular or an elliptic sur-
face, pressed on one side only ; whether vertical or inclined ; and having its top
either coinciding with the surface of the water or helow it.
Let h 33 the vertical depth of the center of pressure below the water surface,
r = the vertical or inclined «emi-diameter of the surface.
d = the vertical distance of the center of the pressed surface, below the
water surface.
Then

In a vertleal eirele, with top at water surface, A i- 1^ X radius.

Art. 10. Walls for resisting' the pressure of qnlet water.

A study of our remarks on retaining-walls for earth, pp. 608, etc., will be of use in
this connection. It is of course assumed that the water aoes not find its war
under the wall ; and that the wall cannot slide. In making calculations for walla
to resist the pressure of either earth or water, it is convenient to assume the
wall to be but one foot in length (not height, or thickness); for then the num-
ber of cubic feet contained in it, is equal to that of the square feet of area of its
cross-section, or profile ; so that these square feet, when multiplied by the weight
of a cubic foot of the masonry, give the weight of the wall. In ordinary cases,
it is well, for safety, to assume that the water extends down to the very bottom
line of the wall.

Now, by Art 1, the total pressure of quiet water, against the rectilineal back of
a wall, whether vertical or sloping, is found in 0>s, by multiplying together the
area in square feet of the part actually pressed, (or in contact with the water ;) .
half the vertical depth of the water, in feet, (being the vertical depth of the
center of gravity of a rectilineal back, below the surface); and the constant
62.5 fi>s; and this total pressure is aXyraja perpendicular to the pressed area.

When the back of the wall is vertical, as in Fig 20K> this pressure p is of
course less than when it is battered ; and is also horiaonud; and it tends toover^
throw the wall, by making it revolve around its outer toe, or edge t. The center
of pressure is at c; cs being "% the vertical depth on; in other words, the entire

Eressure of the water, so far as regards overthrowing the Wall as one mass, may
e considered as concentrated at the point c : where it acts with an overthrowing
leverage 1 1. The pressure in JD>s, multiplied bv this leverage in feet, gives the
moment in foot-fi)6 of the overturning force. Tne wall, on the other hand, resista
in a vertical direction g a, with a moment equal to its weight (supposed to be
concentrated at its center of gravitv o), multiplied by the norisontal distance
a tf which constitutes the leverage of the weight with respect to the point < as a
fUlcrum. If the moment of the water is greater than that of the wall, the latter
will be overthrown ; but if less, it will stand.

In Fig 21 the overturning moment of the water is equal to its calculated pres-
sure p X its leverage 1 1 ; while the moment of stability of the wall is equal to
its weight X its leverage a t. By aid of a drawing to a scale, we may on this
principle ascertain whether any proposed wall will stand. For we have only to
calculate the pressurep, then apply it at c, and at right angles to the back ; pro-
long it to Z ; measure tlhy the same scale. Then calculate the weight of wall ;
find its center of gravity a ; draw g a vertical, and measure the leverage a t. We
then have the data for calculating the two moments.

If the water, instead of being quiet, is liable to waves, the wall-shoald be
made thicker.

KTDKOBTATIC8.

Art. 11. T* find the thiebness kt bnae Vf » w»U required to b<
■■r« ualnst mnfumiiia under the pres of quiet water level with Its tup, eM
preuidg igalnii IM eDtlre T«rt lucV CaBtlo>. See Art. IS.

<lst) TertiOBl wsU, Fig n.

TbIckaeM Height „ | Tmlorot latatr' Helglit the proper deoimil

in feet - in feel X ^ 3 ^ ap gray of V»Tl " '» «*' '^ '" 'oll<"'"lg t»ble

(2d) Utgttt sn^Ied trlBacnlar w»U, Fig 3S.

riiictnesa Height , ( Factor of Mfeir* He1g!il _ the proper dedmtt
at bate — ,„ fj^t ^ \2 X 'P gr»T of wall ~ '" ™*» '^ '" "IK'"'"!! "ble

— Ihlckaeu, n », of vertical wall X l-SSS-

ITotwl 1)1 standing Ihelr greater thickueu at hasc, auch Irianrular walla con-

Uiu, sssren by the flg, not much mnre than half the quintllTormainnr; reqd

for ren onea of equal itabilfty. This 1> uwina to the fact that their cent of

mils thrown farther liack ; thua locieailog tbe lererage by which the wtof

<8tl) Wall with werdenl bach and alAplnB lh««. Fig £4.

Tb Ick neaa ( (Ht',ftxf»etor'otaafetT*) + <batter4rtl,ftX«pgra'of wril)
Inf™ ~Af S X ■pedac grsTltT of wall

^ Height in feet X the proper deoiinal In tb« following table.

ri«.!w.

S..Of.

■•SE

1Ud..-..Sp™.

u«...~

bri-t-sm..

SSr

11

s?

i

i

I

fi

taln = I.Sp~.

B«l.l~lp.«.

Tig. M.

'.'tool

1^;™

B>I»

.1.""

n^

■"liM

s=-

"

11

M^

;l

i

i

1

1

13

HYDHOBTATICa.

Art. la. T»bl«aho«liiKli«wlIie ■lability of a wall samMlB-
i»g water la olEected by s clianc« In the form of tlie wall ;

eicetd^ ) (be hi. tbe BUMUtf
. CanUOB. Sk An. 13.

It If ttie bu

qanntlty of n

2 applyoElT
-erore Iccapil

nneqaal dlBtrlbnuoii of pressiire. rr.< —t —

llie slabUUy of m rigid vati reBEln^f upoD a t^rf baiie, aud tbereTDre incapftble
o[ ftiliire uoep' bji Bverlvming ai a whole, Ttae; abaw that the ilutiliiv ia

polD I w be re tbe resultant of all t be preuures go tbe base of thawallcitii the

baap, must not be ao near to either toe as to eudanger a cnuhiag of -ali or

of toiiudatlDTi. Tbli cciiBld«ratlaii ofua makei It beg) to let the wuer preM

igilnat t)ie vert back, Dutvithtuiiidint! the cooaequeat ]oe> In aUbillti.

Art. 14. Flg.M.howi, loac-ale.adamnallatPoona, Indta.deaigBBdbTMr.

Fife, C E., of Englaud. It i„ of inorl.r rubble, of lEO

. JbH per cub ft. Its total vert belgbt It 100 ft; IMcknMa

uvBt bane, 80 ft SlnB; Bttop,ri, 1)rt9lni. Tbe fn»t

ru slopes 42 fl in 100 ft; and Ite hack ib,B ft Iq 100 (t

Ihe water prcaws sgalmt fla mlwiback xv. Through

- ' - - m c, where tba

IkceGi.laj DlT

t of 1 ft leogUt

nbvK.

!i Saw G
oalof

» Ibe] wi

p ftlnl»nBthof ;

„ Ita diait e m iHpreaents the resultant of wl the pi-eenurei

** upon (he ban' uo, and cuts the biae at a, 20 ft back from

Ihe loo u. Doing tbe aatne with the 161,4 tooa prea p

■calnat ru, we get tbe resultant 0^, which la gmlec

Uan cm, and ciiU tbe base (at <] onlj 12.7 ft from tbs

inlj 2.Z (calling lbs o

HYDROSTATICS.

611

Art. 15. The points a and i, Fig 25, are called eeiit«i« of preMwve
upon the base, or centers of realstance of the base. If similar points, as
d and «, be found in the same way for other lines, as /A, by treating a part (as
rxhf) of the wall as if it were an entire wall; a slightly curved line joining
these points is called the line of pressure. Thus, &a Is the line of pres-
sure when the water presses against xv. Each point, as d, in 6 a, shows where
any Joint, as /A, drawn through that point, is cut by the resultant of all the
forces acting upon said joint, bi is the line of pres when the water presses
agidnst ru. These lines do not show the direotion of the resultants. Thus, at a,
the latter Is cin^ not ba. The angle between the direction of the resultant and a
line at right angles to the bed or Joint, must be less than the angle of friction
of the materials forming the Joint.

If from the end m or y of the resultant of the pressures upon any joint, we
draw «i2 oryl hor, then c2 or o 2 (as the case may be) measures the entire vtiti
pres on that joint: and m2 ory/ measures the hor pres against the back of the
walU which tends to cause sliding at the same joint. If the direction of the re-
snltant comes within the limit stated in the preceding paragraph, m 2 or yl will
be less than the nrictional resisiance to sliding, which last is — c2 (or ol) X the
coeir of friction for the surfaces forming the joint. Hence sliding cannot take
f^ce. Sliding never occurs in the ma$onry of walls of ordinary forms. Qo6d
mortar, well set aids to prevent sliding, bat it is better not to rely upon it. Bat
entire walls have slidden on slippery foundations.

Art. m. In California is tbis dam of a mining reservoir, built of
rough stone without mortar, founded on rock. Helght^TO feet; base, 50: top, 6;
water-slope, 90 feet; outer-slope, 14. TO present leaking the
water-slope is only covered with 8-inch plank bolted homon-
tally to 12 by 12 inch strings, built into the stone-work. All
laid with some care by hand, except a core of about one-fifth of
the mass, wuloh was roughly thrown In. Cost about |3 per cubic
yard. It has been in use since 1860.

Rem. If a dam is eompaetl jr backed with earth
at its natural slope, and in sufficient quantity to prevent the
water from reaching the dam, the pressure against the dam will
not be inereased.

Art. 17* To And tiie tliiclLness of a eyllnder to resist safely the
pressure of water, steam, Ac, against its interior. If riveteiL see next page.

Where the thiekness is less than one*thlrtieth of the
raditBS, as it is in most cases, the usual formula

Thiehness pressure ^^ ,, ^

(1) iDlaehe. - ..ferti^ngtfa X»diu«*

Is employed. It renrds the material as being subjected only to a direct tensile
attain, which Is Mfflciently correct in such thin shells.

For somewhat greater pressures and thicknesses. Professor
T. Benleaux (l>er Konstrukteur, p 52) gives

Thickness pressure / pressure \ s^ ^i •

(2) . in inches " gafe strength T ■•■ 2 X safe strength/ ^ ™^"''
For very yreat pressures and thicknesses, as in hydraulic

J cresses, cannons, Ac, Professor Beuleaux (Konstrukteur. p 58) gives Lamp's
6rmula :

Thickness / / safe strength + pressure -\ ^^ ., ^
(8) in inches - I if ,1.^ -.v 1 — 1 1 X radius.*

\ \ safe strength — pressure

The three formnbigive retnlts m foilew^ preBcnres and strengths in lbs per
■quare inch :

Diameter.

Radius.

Pressure.

Safe

tensile

strength.

Thickness, inches.

Formula (1).

Formula (2).

Formula (8).

90 Inches.

M
M

10 inches.

M

80

800
0OOO

10000

M

«

•05

.60
S.00

.050125
.5135

6.25

.05
.513
7.82

The thicknesses given by the fonnulte appropriate to the several pressures are
printed In heavy type. It will be seen that in these cases the results differ
bat ilightiy, except for vary great pressures.

■ 1 • * • • ■ ^^ • n - II

• In all three formulse tabe the mdiM Is imehet, and the pressure and strength

512

HYDROSTATICS.

lteia.2. Wantof nntft»rintf jr In the coollns of tblck castings makeb

them proportionally weaker than tbin ones, so tbat In order to reduce tbickneM In important oaaei
we aboald use only best iron remelted 3 or 4 times, by whicb means an nit oobealon of about 80000

lbs per sq inch' mav be secured. But even with this precaution no rule will
apply safely m practice to cast cylinders whose thickness exceeds either

about 8 to 10 ins, or tbe inner rad however small.

Under a pres of 8000 fbs per sq inch, water will ooa« throafffei cast
iron 8 or 10 InB thick ; and under but 260 fba per sq inch, through J5 inch.
Table of fhleknesses of slniple-ri voted wronvbt Iron pipes.

tanks, standpipes. fto, by the above rule, to bear with a safety of 6 a quiet pressure of 1000 ft head
of water, or iH4 lbs )>e? sq inch ; the ult ooh of fair quality plate iron being taken at 48000 lbs per sq
inoh, or at 8000 0>b for a safety of 6 ; wbioh is farther reduced to 8000 X M = 4480 lbs, to allow for

weakening by rivet holes; for sin vie- riveted cyls have but about .66 of the
strength of the solid sheet; and uonble- riveted ones about .7. With the

abOTe pres and other data, the rule here leads to thickness = .1016 X Inner rad in ins.

Dl.

The.

DI.

•The.

Dl.

The.

Di.

Ths.

Di.

Ths.

Di.

DI.

Tha.

Int.

Ins.
.025

Ins.
5

Ins.

Ins.

Ins.

Ins.
SO

Ins.

Ins.

Ins.

Ins.

10

Ins.

.6

.254

16

.818

1.52

60

8.05

120

6.00

1.0

.051

6

.805

18

.914

88

168

66

8.86

182

11

6.7«

1.6

.076

8

.406

20

1.016

36

1.88

72

8.66

144

12

7.81

3.0

.102

10

.508

22

1.117

42

2.1S

84

4.27

192

16

0.76

8.0

.152

12

.609

24

1.219

48

2.44

96

4.88

S40

ao

W.W

4.0

.206

14

.711

27

1.S71

64

2.74

106

6.40

888

84

14.68

For a less head or pressure, or for any safety less than 6, it is safe acd
near enough in practice, to reduce the thiokness of wrought iron cyls in tbe same proportion as a^
head, pres, or safety is leas than the tabular one.

Double-riveted cylinders, Fairbairn says, are about 1.26 times as strons
as single-riveted. Hence they may be one-fifth part thinner. Ijap«w^elded
ones are nearly 1.8 times as strong as single- riveted ; and hence may be only
.56 as thick.

Many continuous miles of double- riveted pipes in Call foam ia have

been in use for years with safetys of but 2 to 2.6. In one case tbe head ia 1720 ft, with a pres of 746 Iba
per sq inch ; dlam 11.5 ins ; thiokness, .84 inch.

I;

Cast iron city water pipes must be thicker than required by formula
1), in order to endure rough handling and the effisots of " water-ram *'

due to sudden stoppage of flow, see second Bem, p 613), and to provide against
irregularity of casting and the air bubbles or voids to which »U casUnga are
more or less liable. In the following table the ultimate tensile strength of east \
iron is takeu at 18,000 lbs per square inch. Column A'gives thicknesses by Mr.
J. T. Fanning's formula (Hydraulic Engineering, p 454).

Thickness ) ^ (pres, ft>s per sq in + 100) X bore, ins ^^^
in Inches / " .4 >< ultimate tensile strength "*■

(-

bore, in8\

100 r

These correspond with average practice. The addition of 100 lbs to the pres ti
made in order to allow for water-ram. Column B gives thicknesses by formula

il), taking coefficient of safelTy =« 8 (thus making safe tensile strain a 2260

l>s per square inch) and adding three-tenths of an inch to each thickness given
by the formula:

Head in feet 50

100

200

800

600

1000

Pressure,
lbs per sq in.

21.7

43.4

86.8

180

217

484

Bore, ins.

TliielLness of pipe. In Inelies.

A

B

A

B

A B

A B

A

B

A B

2

.86

.31

.37

.32

.38 .84

.89 .36

.42

.40

.48 .61

3

.37

.81

.38

.33

.40 .86

.42 .40

.45

.45

.54 .60

4

.39

.82

.40

.34

.42 .88

.46 .42

.60

M

.61 .71

6

.41

.83

.43

.36

.47 .42

.60 .48

Jil

.60

.75 .94

8

.45

.34

' .47

.38

.52 .47

JJ7 .65

.66

.70

.90 l.U

10

.47

.85

.60

.40

.56 .60

.62 .00

.74

.81

1.04 IM

12

,49

.86

.53

.42

.60 M

.67 .66

.82

.91

U8 IJS!

16

.55

.38

.60

.46

.70 .62

.79 .77

.98

1.10

1.46 2.00

18

.67

.39

.68

.48

.74 .66

.85 .84

1.06

1.21

1.60 t»

20

.61

.40

.67

.60

.79 .68

.91 .90

1.16

1.81

1.75 SJO

24

.66

.42

.78

.68

.87 .77

1.02 1.01

IJO

1.61

2.03 2M

80

.74

.45

.88

.69

1.01 .89

1.19 1.19

IJK

IM

i.46 &4I

36

.82

.47

.98

.66

1.15 1.01

1.86 1.87

IM

i.13

2M 4.11

48

.98

.58

1.18

.77

1.42 1.24

1.70 1.78

3.S8

IfS

S.7S «J8

HYDROSTATICS.

513

TAMe of thiekness of lead pipe to bear internal pressures with a

. nfe^ of 6; taking tbe ultimate oohesloa of lead at 1400 Aa per sq inch.

Hem. Although these thicknesses are ss(fe againstquiet pressures,the7 might not
recUt Bhooks caused by too sudden closing of stop -cocks agunst running water.

•

Heads in Peet.

Heads in Feet.

1

100 200 800

400

600

8

1

1-4

a

100

900

300

400

600

a

Free in lbs per sq inch.

Fres in lbs per sq inch.

1

43.4

86.8

130

174 217

£

43.4

86.8 130

174 217

&

1^

ThioknesB in Inohes.

ThiokneBB in Inches.

H

.026

.055

.088

.128

.171

1

.102

.221

.857

.611

.682

H

.088

.083

.134

.192

.256

1«

.127

.276

.447

.639

.863

H

.061

.111

.179

.256

.341

IH

.153

.888

.536

.767

1.02

H

.064

.138

.2-A

.330

.427

IH

.178

.887

.626

.895

1.20

^

.076

.166

.268

.383

.612

2

.204

.448

.714

1.02

1.36

.089

.193 1 .313

.447

.697

Rem. Tbe valves of water-pipes mast be closed slowly, and

the necessity for this precaution increases with their diams. Otherwise the sud-
den arresting of the momentum of Uie running water will create a great pressure against tbe pipes
lA all directions, and throughout their entire length behind the gate, even if it be many miles ; thus
endangering their bursting at any point. Hence stop-gates are shut by screws.

8

Fiff. 5S6.

Art* 18. Baoyancy. When a body is placed in a liquid, whether it float
or sink, it evideutly displaces a bulk of the liquid equal to tbe bulk of the im-
mersed portion of the body ; and the body, in both cases and at any depth, and
in any position whatever, is buoyed up by the liquid with a force equal to the
weight of the liquid so displaced. Thus, if we immerse entirely in water a
piece of cork, c, c. Fig 26, or any body of less specific gravity than water, the
cork will, bv its weisbt, or force of gravity, tend to
descend still deeper ; out the upward buoyant force of
the water, being greater than the downward force of
gravity of the cork, will compel the latter to rise. In
this case, the cork receives a total downward pressure
equal to the weight of the vertical column of water
above it, shown by the vertical lines in vessel 1 ; and
a total upward pressure equal to the weight of the
column shown in vessel 2. The difference between
these two columns is evidently (from the figs) equal to
the bulk of the cork itself; therefore the difrerence
between their weights or pressures (or, in other words,
the buoyancy of the water) is equal to the weight or

pressure of the water which would have occupied the place of the cork ; or, in
other words, of the water which is displaced by the cork. This difference^ or
buoyancj', will plainly be very nearly the same at any depth whiatever of entire
immersiun. Itiucreases slightly with the depth, owing to increase in the density
of the water; but, on tbe other hand it is diminished by compression of the
cork. Now tbe cork, if left to itself, will continue to rise until a portion of it
reaches above the surface, as in vessel 3. The downward pressing column
then ceases to exist; and the cork is pressed downward only by its own weight.
But, as it now remains stationary, toe upward pressure of the water must be
equal to the weight of the cork. But the upward piessure of the water arises
only from the shaded column shown in vessel 3; and this column is (as in the
ease of total immersion) equal to the bulk of water displaced. Therefore, in all
eagesy the buoyancy is equal to the weight of water displaced ; and when the
body Jloats on the surface, the buoyancy, or the weight of water displaced, is
also equal to the weight of the body itself.

If the body be of a substance beavler tban water, its weight is greater
than the buoyancy of the displaced water, and the body therefore sinks, with a
force equal to the difference between the two. Thus, a cubic foot of cast iron
weighs 450 lbs., and a cubic foot of water 62.5 lbs., so that the iron sinks with a
force of 450 — 62.5 = 387.5 lbs.

The same principle applies to otber llnlds. Thus, light bodies, such as
smoke, a balloon, etc., in air, all tend, like a cork in water, to fall; but the air,
being heavier, crowds them out of the lower positions which they tend to assume,
and pushes them upward.

Although a pound of lead and apouhd of feathers, weighed in the air, balance
each other, yet in a vacuum the feathers will outweigh the lead, by as much as
tbe bulk of the air displaced by them outweighs that displaced by the iruu.

514 BUOYANCY, FLOTATION, METACENTEB, ETC,

The downwd force of gmT may be regarded «s oonoentrated at the een cf

KaT G of a floating body. The apwd pree, or buoyancy ,t of the water may similarly
regarded as acting at the cen of gr W of the displaced water * W is also called
the center of pressare^ or of baoyancy, of the water; and a vert Him
drawn through it is called the axis, or Tertlcal, of buoyancy, or of flo*
tatlon. Ordinarily,^ W shifts its position with every change in that of the bodj.
Thus in L it is at the cen of gr of the rectangle oobb; and in N at that of the tn>
angle a a V,

When a floating t,

body, L, P or R, is at
rest, and undisturbed
by any third force,
as F, it is said to be
In eqailibrlnm,
and G and W are then
In the same vert line "^
1 1 Figs L and R, or "
e e Fig P ; which line
is called the axis.
or vertical, of
e«^illbrlnni.

when a third force, g as F, In N and O, causes the line, joining G and W, to
lean, as in Figs N, 0 and S, then if a vert line be drawn upwd from the cen W of
buoy, the point M where said line cuts said axis, is called the metacenter of
the body .J G and W are then no longer in the same vert line ;| and the two opp
imd vert lorces, grav and buoy, acting upon those points respectivelv, form a
"couple" and, when the third iorce F is removed, they no longer hold the body in
equilib, but cause it to rotate. If (as in Figs O and 8) the positions of G and W
are then such that the metacenter M is above the cen of gr G, tnis rotation will tend
to restore the body to its former position, and the body is said to have been (before
the application of the third force F) in stable eqnilibriani4 But if (as in
N) M is below G the direction of rotation is such aa to upset the body, by causing
it to depart further from its former position, and the body is said to have been in
unstable equlllbriuni.t

The tendenoy or moment in ft-lbs of a floating bodj eKher to apeet or to right Itself, is,

_ the wt of the body (or the eaaal ^ the bor diit between W M and O H,
upwd prea of the water) in lbs ^ Figs N, O and 8, In ft.

The third force F may of oonrse be so great as to OTerpower the tendenoy of the body t* i
self. Tbna, a ship may npset in a hurricane, although Jndloloaalj loaded and ballMted nr i
winds. A hor section vf a body at water-lioe is oalled Its plane of fl«tatl<m«

* Tbe body is In fact acted upon by other forces4 such as the hor

pressures of the water against its immersed portiooR ; but as all of these in any one given dtreotlea
are balanced by equal ones in the opposite direction, they have no efliMt upon the fbroes O and W.
It is also acted upon by the air, which prenses It downwards with a foroe of 14.76 lbs per eq inoh; bal
this Lb balanced by an equal pree of the surrounding air upna the snrfaoe of the water.

t Thiii buoyancy Is made up of the parallel upward pressurea of the
innumerable vert filaments of the displaced water as shown by Fig 26, and

the aria of flotation is their resultant, as In the case of parallel fbroes.

t The shape of a body (as that of a sphere or cylinder U) nay be loeh that the poaftion of iCs mm el
buoy W, relatively to that of Its een of gr O, Is not ehanged b.v the rotation off the body abovt a ghtm
axil (as anv axis of the sphere or the longit%idinal axis of the oyl), bat remains oonstantly la tkt
same vert line with O, so that the body, in rotating, remains in fqullib. Bach a body is aald to be

in indifferent eqnilibriuin about said axis. But if a cyl U be made to

route about iu traruverae axis x x, It plainly comes under the remarks on Figs R and 8, and nuH
(before rotating) be In either stable or unstable equilib about that axis aoeordlag to tho way la whMt
Its wt is distributed.
II This metaoenter shifts Its position on the line t ( aooording to the inclination of the latter*
^ Uneven loadiufp^ instead of a third force, may cause a vessel at rest t«
lean as at F ; and yet the vessel so leaning may be in oqailib : for ita axis e e of equilib may be verli
although not coinciding with the axis of symmetry of the vessel, as it doea at
1 1 in L.

f JnfioaHng bodies, this may sometimes (as In Fin R and S) be the ease oven when the Mm of

huoy W Inot the metacenter) is ielow the oen of gr O ; beoause, whan the body is ieroed to Isaai, W

move* to aootter point la i^ and Ihia point nay be iiioli as to Mag II above Q. W le always Mkrv

<i In bodies of uniform density, floattnpc at nut If any part of the body is above watar. Waaa aaoh

lodiea are entirelr submerged, W aod O ooinoide.

HTDBOSTATIC8.

515

Fig 27

Art. 19. A body Uvhter tbaii water. If placed at
tbe bottom of a vessel eontalninar water, will not
rise anlesB the water can yet under it, to bnoy it,
or press it upward, as tbe air pressfw a balloon or
smoke upward. Thus, if one side df a block of light wood,

perfeeUn flat and smooth, be placed upon the ■imilarty flat and smooth Dottom of •
▼essel, aod held there until the ▼esaei is filled with water, the downward pres wilk
keep it in its plaoe, until water insinuates' itself beneath through the pores of the
wood. But if the wood be smoothly Tarnished, to ezolude water fkvm its pores, it
will rem^n at the bottom.

On the other hand, a piece of metal may be pre^
vent<»d fW>m sinfeinir in water, by suLdecting it to a snffl*

oient vpwvrd pres only, while the downward pres is excluded. Thus, if tbe bottom
of an open glass tube, (, Fig 27, and a plate of iron m, be made smooth enough co b«
water-tiffht when plaoed as in the flg ; and if In this position they be plsoed in a
▼easel of water to a depth greater than about 8 limes the thickness of tbe iH»n, tht
upward pres of the water will hold the iron in tM place, and prevent its sinking*

beeania it is preaaed npwaid by a oolnmn of water heavier thao both tbe ooluain of air, and iU ow»

wvight, which preas it downward. On this principle iron shipa float.

BsM. 1. A retaininff-wall, as in Figr 28,
founded on piles, may be strong enough to re-
sist the pres of the earth s behind it. In case water does not find
Its way andemteth ; and yet may be OTcrthrowa if it does ; or
even if the earth • t around tbe heads of the piles becomes satn^
rated with water so as to form a fiuid mud. In either case, the
upward pres of the water against the bottom of the wall will rir^
tually reduce the wt of all such parts as are below the water surf,
to the extent of 63){ lbs per cob ft; or nearly one-half of the ot-
dinary wt of mbble masoniy in mortar.

Rbm. 1. Although the piles under a wan. as in Tig 38, may be
abundantly suffloient to sustain the wt of the wall ; and the wall
equally strong in iUeXf to resist the pres of the backing a ; vet if
thii son 1 1 around the pOea be soft, both they and the wall may be pushed outward, and the latter
ererthrown by the pres of the backing s. From this cause the wing-walls of bridges, when built
on nilea in rery soft soil, are f^vquently bulged outward and disfigured. In such cases, tbe piling,
and the wooden platfbrm on top of i^ should extend over the whole space between the walls; or else
some other remedy be applied.

Art. 20. Arauyht of vessels. Since a yioating body displaces a wt of liquid
equal to the wt of the body, we may determine the wt of a vessel and its cargo, by ascertaining how
many cob ft of water they displace. Tbe cub ft. mult by 62>^, will give the reqd wt in B>s. Snppose,
for instance, a flat-boat, with vert sides, 00 ft Inng. 15 ft wide, and drawing unloaded 0 ins, or .5 of
a fL In this ease It displaces 00 X 15 X .5 = 450 cub a of water ; whleh weighs 460 X 92H = SSlSft
%9 ; which consequently is the wt of the boat also. If the cargo then be put in, and found to sink
the boat a ft mors, we hare for the wt of water displaced by the cargo alone, 00 X 15 X S X 63^ ss
113500 fts ; which is also the wt of the cargOb So also, knowing beforehand tbe wt of the boat and
•arfo^aadthedimanalonaof theboa*,we«Mifl]idwiaatthedraeghtwiUbe.. Thus, If thewtas belbie

140625

be 140025 fts. and tbe boat 00 x 15. we have 00 X 15 X 02^ = 50280 ; and = 25 ft the required

60250
draught. In veaaels of more eomplex shapes, as In ordinary Bailing Tes«eli«. the oalenlatloD of the
amount of displacement becomes more tedious; but the principle remains the same.

516 HYDRAULICS.

HYDEAULIOS.

Flow or WaWr tbropih Pipe*.

Uuch or the ibeorj of hjdraiilLcB la Bllll milur of. dlipute. Tbls, iiDd tbe
rarely 'r^^iors ii'i^ppljlni; hydraulic rarrai'ilu.' Even dcv pipes are liable to

d°mlDlsbe3"tlie flov.

Both In theory Hnd In prnrllce It ■■ immnlprlnl mi prvnrilti
the veil Antl lh« quantity of water dlsehariEed, whelbpr tlie
pipe ts Inellneal downnanl. ns ro, ¥Ik 1; orhor, aavAi ar In-
clined opwartl. IM Id; provided the loUil head po. ond alma
the lenKth of theplpe, remnln nnchnnsed. If one pipe li longer

niTlalons of the Totnl UmmI. Id laij plpn, u t

"ArCt'l^'. The'vSoclty jiend l>"the"pighl ihroughVlifch ■

Artel's!' m Fig""fl'w'i'f M
fMctlon b«»d t ar the bud vli

HYDSAULIGS.

617

th« pipe as rapidly u it flowf throagh it, and thni keepiog th« pipe sopplled. If, bj ahortening tba
pipe, or bj smoothing Ita inner aarf, ire diminish the total friction, then a less friction head will be
required ; bat the vel will, at the same time, be iDcreatied, and this will reqaire a greater vel head,
maa entry head, so that the three together make up the total head, as before. Since the friction la
eqnal to the foree or head reqd to overoome it, it also is represented by wo.

Art. 1 e» Th« frictioii head may as in v o, « o, an d I o, Fig 1, be al I above the en trance
to the pipe, and therefore (mutde of the pipe : or, as in a pipe laid from « to o, it may be all htlom
ftbe entrance, and within the pipe; or. aa in ro and to, it may be partly above, and partly below, the
entrance: and therefore partly within, and partly without, the pipe. The vel ana disch, after the
pipe ie filled, are not affected by this diffsrence in position of the entry end ; but the preuurea in the
pipe, and the vela while the water it filling an emptg p^e, are affected by it, aa explained in Arts II
and 1 o.

Art. If, Bat It Is neeeMlary that tbe entry end of tbe pipe
slioald De plaeed so far below tbe snrf m i, that thero shall be left,

above the een of grav of the entry end, at least a head, i a, sufficient to perform the duties of the entrv
and Tel heads. If the entry end' of any of the pipes be raised above «. a portion of the vel head will
be in the pipe. In other words, the head m the pipe will be more tnan sufflcieot to overcome the
Teeistanees In tbe pipe ; and tbe surplus will act as vel head, and will give greater vel to the water
<n the pipe. The reduced bead thus left above the entry end will plainly be Intnfflcieiit to maintain
tiie anpply for the greater vel, and the pipe will run only partly fnll.

In ordinary eaaes of pipes of considerable length, the sum of the entry and Tel beads theoretically
veqnired, is but a small portion of tbe total head, and rarely exceeds a foo^ Indeed, in a pipe of
considerable diameter, the upper half of ita orosa section at the entry end may often be more than
enough to provide snflielent entry and vel heads above the oen of grav of said cross section: so thiic
the top of the entry end might, so far as these considerations alone are concerned, pn^Ject above tbe
•nrf of the water in the reservoir. But the end of the pipe should in practice always be entirely be-
low tbe surf; otherwise air and floating Impurities will be drawn into it, and oaiise obstructions.
Moreover, the water snrf of reservoirs is alwaya liable to oonaiderable ehaogea of height; and the
entry end of the pipe mnst be placed at aueh a depth that the water ean flow into It with snffleient
Tel when at Its Uneeet stages. As before stated, this will oaose no diminution or increase of diaoh.

Art. 1 a. To find tbe flrletlon bead reqd for any part of
m pipe; Knowing the fric head reqd for the whole pipe Since the friction, in a
pipe of uniform diam. Is (other things being equal) in proportion to its length ; and since w o. Fig 1,
lepreaenta the total friction, or reqd fHotion head, we have

Total length , Length of the
of tbe pipe • given portion

9r, having drawn w o by scale, < w hor. and a o;

Total length , Length of the ,
of the pipe • given portion •

A dlst, ail « ft. to be laid off
from a on a w

• •

W0

90

Tbe friction head reqd
for that portion.

A dist, aa a e. to be laid
off f^m ff on a e.

Or

aw

m--^^^^^w

\

0

r

I

V, \! J.

Then a Tert line, aa 6 e, drawn from b or «, and joining a w and a o, glvea by aoale the friction bead
reqd.

Art. 1 iu If the pipe Is straight, as r o, v o, { o, the Motion in nuy part b€giit»
ming at the reeervotr, aa 1 6 in the pipe I o, may be found at onoe by drawing a line 6 1 vert upward

ftnem the axla of the pipe at d. The line 9 8 will then give the friction in 1 6. It also gives the frie-

tlon in r 4, or in that part of e o whloh Ilea between v and the dotted line 1 6. It muat be remem'

bored that all the pipee Id Fig 1 are auppoaed to be of tiie
same actual length. They would thus end at different points
o, and strictly, a separate diagram must be drawn for each
pipe. In a part of the pipe not beginnijDg at tbe reservoir,
as In r o, V o, or I o, between points vertically under c and
z, the amount of friction ia giveu by tbe line d x, for it la
plainly — ysr — Ac.

Art. 1 J. If tbe pipe Is Tert, as v o.

Pig 1 A; let taton its axis io) represent, as before, tbe sum
of tbe vel and entry heads. From a, «, and o, respectively,
draw hor lines » w,v k, and o p, making oy^vo. Draw
the oblique line « y. Then, to And tbe friction In any part,
as » o, beginninfr at the reservoir ; from q lay off fd hor, and
equal to v o. and draw tbe vert line a a, crossing a v at 9.
Then b g will give the friction in v q.

Art. 1 k» If tbe pipe Is enrved^and

if the curvature Is uniformly oistributed along itn length, or
ao iilight that it may be neglected; tbe friction beada reqd
for the several portiona of the pipe, may be found in the
same way as for straight pipes, as in Art 1 H. Otherwise
they muat be found by proportion, as In Art 1 G.

. Art. 1 I. Wblle water is fllilnflr
an empty pipe, the excess of the total liead

above the requirements of friction, &c, glvea to the water a

g greater vel than it has after the pipe is filled;
ut this graduRlly decreases as the advancing water encoun*
tbe friction along the inoreaaed lengths of pipe filled'; and finally becomes least when the water
Alls tbe whole length, and begina to flow from tbe disch end. o. But if only the vel and entry
beads are left above the entry end, as in a pipe laid from « to o, there will plainly be no such exceif
>f total head, and. ooniequently, no such change of vel during the filling of the pipe.

M'

\i

Jrf\

\

\i

Fig.X A

y

518

HYDRAULICS.

Art. 1 *n-r.* Relation between dlsctaari^e, area, velocity and

gressnre. la Fig 1 B-D^ wber^ the jpip<3, b F, running full, receives water
om an unlimited .rcBervoir, B, at 6, and discharges through an orifice, F; the
Tolume of water, passing any given cross section of b F\ in a given time, is
constant and tKiuai to the rate of discharge at F. Thus ; — if the rate of discharge,
at Ff be (J cubic feet per second, then Q cubic feet will pass each cross section of
the pipe, o Fj p6r second.

o

i

tl

>'

eS"9

pwas^MP

i

i
1

L
i

3
2

4

1'

e

J

c

D^ra

*

>

2

5

. 1

' 9

1

r-

r

Jt

i

1

1

\

' y

\

—F

h?

\

l h —

>

'

f

Ci

N^J^Sv*'

Fl|r. 1 B-D.

Let a = the area of cross section, and r= the velocity, of the stream issuing
through the short pipe beyond F. Fis called the velocity of efflux.

Let Ai, Ai. etc., be tjie aiffereut areas of cross section of b F, and let vj, vj, etc.,
be the velocities at those cross sections respectively. Then Qs= a V == Aiv^

= Ai wj, etc.; or F = ^, ti =» 4^, t;2 = -^, etc. In other words, the veloci-

ties are inversely as the areas of cross section,
etc.

Q

Also, o = ^,

^1 =

Q

Vi^

A — "*

The losses of pressure, due to the velocities, respectively, are di = s^, dj = ~,

etc.; Rs represented by the ordinates between the line o o^ of static pressure, and
the diagram, ol23456ii^, of actual pressures. The difference, due to velocity,
between the pres heads at any two points, as ci and c^, where the velocities are

vx and vo respectively, is p^ — Pi ^^ ^i — ^a

Vt

v^

Vi* — Vi

1 *

2g 2g 2y

The remaining pressure head, pi, jd^, etc., at any point, is = static head in
reservoir — velocity head at the point, = H — di, It — d^, etc.

The loss of pressure head, at F, is {6 F) = p^ = ff — d^; and the pressure
drops to zero ; i.e., to the atmospheric pressure.

Art. 1 «. Open piezometers. If the lower ends of vertical or inclined
tubes, open at both ends, be inserted into a pipe, b jP, Fig. 1 J? D, as at Cj, e%
etc., the water surface, in these tubes, will stand at heights, pi. />s, etc., corre-
sponding to the pressure heads at the points where the tubes are inserted. Sadi
tubes are called open piezometers. In order that the water level mav be
observed, they are of glass, at least in those portions where that Itt^el is likely to
be found. An obstruction, in the pipe, between Co and F^ would raise the level
in a piezometer at Cs; while an obstruction between b and e^ would louter it.

*In Art. 1 m-^y for simplicity, we neglect all resistances, Induding those due
to the abrupt enlargements and contractions of the pipe.

HYDRAULICS.

519

Figr.l E

Art* 1 tm If we imagine any pipe, full of water, to be supplied with a narobti
of piezometers, then a line, Joining the tops of the columns of water in tlie several
piezometers, is called the liydraallc grade line.

Art. 1 ff . In a straight tube of uniform diam throughout, as r o, r o, or I o, Fig

1, ruDuiug foil and disoharging freely into the air, the bjd grade line is a strHiglit line drawn

from its discb end o to a point a immediately orer the entry end of the pipe, and at a depth below
the surf equal to the sum of the vel and entry beads.

If the oriflee at o be contracted, the hyd grade line must ho. drawn

from « to some point, as e, immediately over o, and depending, for its tasigbt, upon the amount of

contraction at o. Hut in this case
the point a will also be higher than
before, because the vel in the pipe is
reduced by the contraction ; and the
sum i s of the vel and entry heads
will be less.

if the disch at o is
nnder water, the effect
upon the position of the grade line
will be the same tu* that of a con-
traction of the orifice at o. The
point e will be on the surf of the
lower water, and immediately overo.

If the pipe, of uniform

diam, (whether discharging freely or through a con-
tracted opening ato, whether into the air or under
water), Is bent or cur vc^d, the hyd grade
line will still be straight, provided the
resistances are equal in each equal division of the hor
length of the pipe, as in Fig 1 R, where equal divisions
9 1«, tc x,/ic, of ',be total length, correspond with equal
divisions d a, a 6, Ao, of the hor length.

Rut in Pig I F, the hrd grade line will take the
shape 8 ao. For If. in accordance with Art. 1 G, we
divide a o into two equal parts, « m, m o. correspond-
ing with the two equal parts vr.ro, of the length of the
pipe, we obtain m c = a e for the head consumed in the
resistances in v r, leaving only r a for the pres head at r.

a very large vessel, the total head upon any point at the level
of the entrance / to a pipe loo' Fig 1 (t, is represented by «/, as already ex

FifiT.l F

Art i w.

I
I

8k-b\-

I \

plained but of this total head a portion, as is, is required to act as

velocity head and entry head for the entrance at I, leaving only .»/ as the* pres-
sure head upon a point in the pipe, immediately to
the right of I. Thus while the pressure, in pounds
per square inch, in the t^sel at /, is

p = ilXOAU .
that in the pipe'at I is

P = «^X0.434.
But now a portion, as *t;, of «/, is expended in
lo m balancing or "overcoming" the resistances
throughout that portion of the pipe ; and, in doing
this work, it gradually diminishes from sv fat/) to
nothing (at o) as indicated by the dotted line se.
Thus, Ht the point 6, a portion = fc c has already been
expended in overcoming the resistances in the pipe
between /and 6, leavine c6 as the pressure head at
6, of which c in must still be expended against resist-
ances in the wide pipe between 6 and o, leaving
'l^^r'vJ — <fo&8 the pressure head for a point just
to the left of the contraction at o. The pressure in
lo IS thus gradually diminished from */ (at /) to
eo = v/ (at o).

... , , u J /. XV ^^^ * portion ^^'of «o is required to act as ve-
tocity and entry head for the entrance o to the narrower portion o o' of the Dine •
because we need at o not onlv an additional ^ntry head to overcome the reskt-
ance due to the Muare shoulder formed by the contraction, but also an addi-
tional velocity head to give the increase of velocity which must take place as t.:e
water passes from the wide pipe /o to the narrower one oo': for, so lone as a
pipe runs /utf and the discharge remains constant, the velocitv in each part of
the pipe must be tnversely as the area of cross section of that part : because in
each second the same quantity of water passes each point; and this constant
quantity is = area X velocity. Hence, as the area diminishes, the velocity
increases. •'

There remains, therefore, ^o as the pressure head upon a point in the narrow
part just to the right of o; and this in turn gradually diminishes to nothing at

I
I
1

0
Fig.l Gt

6 I

520 HYDRAULICS.

w hydxAuLic grmdiei

Whpn Ihe pressure is thuB diinJDLahed by oveTcomlng realataQceB. or b^
celentldg >er«lly, the dimiuuLiDii is csJIed lOM ofheMl. Tbui we uy
wis ^u^ Ht Ibeentrftnc? A<vu friciloo bead Id to, et' at tba coDtrAcUoD 0

At (/f all tbe avallBbla bend^ i [, has been uted
therBfore eierts nn laiural prewure, so that ths a
iDdilsoapacitytiir forward pressure is due entirely 10 in BInetioepergy (energy

but this last is or course balanced by the air preeeure from without sgainst lbs
openiTig o". Where a grsat rerfuctlon ol cross eocliona] area id a pipe la (bllinrtij

all of tbe atmotpruric pressure od (he Bur^ce la tbe reservoir, thus causliig m
putlal oroomplBta laouurost (ha ooaatrtetloo. 3eeib« Veaturi Meter.

"nie ayphnn, orilBboo. Irosa lee a &of« bent tube nr pipe aba
Pll H. o( uy dUm^ULil wlik wuir. ud wllh l»[h Lu CDdi lUppM.

AJdOieTcrt d{itApu1b»kDtiti>ftkaHtarlKT*Bd«nn1ud]it.wUi

Ihenl* sEBipljibltilbaL boLh tbHSlsfi Bd.aftd bi.Mulnt iKsd wllb *ita-, fOa HitVa bJu

vooildaNd u flm u ■ ponioB vT LtiA fvan**-. aad not vT tbv inAoa,) Ik MhMn UiBt wlin Uu «u>
wr<SHrciiiDndrrDinibBaDdBovtda,(k*alrprw««qiul1rac>hiHlh«HHdii bolllHBruawt
b*sd of wUAT & ^tn ibt «i«r Iff bdp nmm udDil tba sir ■( e, with man IOff« tku lb* HafeU^vd

A Uie poBd li bi bs ndiHtd} any faTerreDi

In effecl two separate tubes opeD at ii>p ; and the wsler will fall In bglh. An ori-
fice at tbe escape will be needed Tor filllDg tbe syphon at the start ; iDd Us nre-
Tent the water thus Introduced, froDi ruDnlng out. stopcocks must be provided at
the ends. Bnd kept closed until tbe flUlD)! la eoupicttd.
Tbe greatest pains must be t«ten in mate all the Joints perfectly air-tight
Tbn motive power or bead which causes the Bow la s sypbuu. is Ibe
Tert dlst J 0, froDi Iheeurf of (be resorroir. to thedleeh end c: or Id other wordL
it is the diff. 1 0. between the theoretical lengths l> i and A a, of the two legs. Con-

HYDRAULICS.

521

■equently, the farther e is below « the more rapid will hp the flow ; and it is plain
that as the surf gradually sinks below «, the less rapid will the flow become. Hav-
ing this head, the entire length u^cof the syphon, and its diani, all in ft. the
diseh may be lound approximately by eiiher of the rules given in Art 2 for straight
pipes. These rules give 55^ galls per min, instead of the &% galls actually discbd.
by Col Crozet's syphon, with a head of 20 ft. See below.

In H true s^^lion, agnyo Fig U, free /torn air inside, and running full.
the total beaa po is measured vertically from the surface mt in the reser-
Toir to the center of gravity of the outlet o, as in Fig 1 ; the hydranlie
i^radlent (with the restriction named in Art 1 v) is, as before, a straight line

• sro drawn from the foot *

"^ '^f^ Jp. — — . — ,p of the combined entry and

velocity heads to the end
o; and the velocity and
discharge are the same as
they would be if all parts
of the pipe were brought
below sro. But see cau-
tions 1 and 2, below.

The pressure at
any f»oint, jr, n or p, is
then given by a vertical
line, gv,nr or yv, drawn
froin the point in question to sro: but for points, as n, situated abovf sro, this
pressure is negative or intoard; while at points where sro and the pipe are at the
same level, as at /and e, there is neither i)res8ure nor vacuum.

Caution !• But if the water be admitted to the empty pipe »t a, while the
end o is open, the pipe will not form a true syphon. The part a ^n will then run
full, and will have sen as its hydraulic (rraaient; but upon reaching, at n, a
portion no of the pipe with a much steeper grade, the water will run off, in n o,
with a velocity greater than that with whidi it arrives from a n. Hence the
stream in no will have a less area of cross section than in an, and therefore can-
not flll no^ but will run off in it as in an open gutter.

Caution 2. The tendency to vacuum at points above sro causes an accu-
mulation, at n, of particles of air that have been carrie<9into the syphon by the
water or have found their way in through imperfect joints, etc. ; and these
bring about a condition approaching that described in Caution 1; for their
expansive force, by reducing the negative pressure or vacuum nr at n, diminishes
the total head h r of the part agru while, oy practically reducing the croas-sec'
tion of the syphon at n, tney require that a portion pf the remaining head he
vsed at n, as entry head to overcome the resistance caused by the contraction,
tnd as velocity head to give the increase of velocity needed for passing the nar-
rowed section at n. Now since the friction head required for tne part agn re-
mains about the same, the velocity head in the reservoir is considerably dimin-
ished, and the water arrives at n too slowly to keep n o filled. The accumulation
of air at n thus retards the flow and disturbs the distribution of the pressures,
■0 that the^e are no longer correct! v indicated by vertical lines drawn to sro.

At Bine Rldire Tunnel, Virginia, Col. C. Cruzet constructed a drainage
syphon 1792 ft long of cast iron faucet pipes 3 ins bore. 9 ft Ion?. Its summit was
9 It above the surface of the water to be drained ; and its discharge end was 20 ft
below said surface, thus giving it a head of 20 ft. At the summit 570 ft fror.i the
inlet, was an ordinary cast iron air-vessel with a chamber 8 ft high and 15 ins
loner diam. In the stem connecting it with the syphon was a eut*off stop*
(soch I and at its t6p was an opening 6 ins diam, closed by an air tight screw lid.
At each end of the syphon waa a stopcock. To start the flow these end
cocks are closed, and the entire syphon and air-vessel are filled with water through
the opening at top of air-vessel. This opening is then closed airtight, and the two
end cocks afterwards opened; the cut-off cock remaining open. The flow then
begins, and theoretically it should continue without diminution, except so
far as the head diminishes by the lowering of the surface level of the pond. Biit
in practice with very long syphons this is not the case, for air begins at once
M disengage itself from the water, and to travel up the syphon to the summit,
where it enters the air-vessel, and rising to the top of the chamber gradually
drives out the water. If this is allowed to continue the air would first fill the en-
dre chamber and then the summit of the syphon itself, whore it would act as a
wad completely stopping the flow. The water-level In tlio Hlrchnmber
can be detected by tne sound made by tapping against the outside with a hammer.

622

HYDRAULICS.

To prevent tbto stoppage, the cut-off at the foot of the chamber is
closed before the water is all driyen out ; and the lid on top being removed the
chamber is refilled with water, the lid replaced, and the cu^off again opened.
The flow in the meantime continues uninterrupted, but still gradually diminish-
ing notwithstanding the refilling of the chamber; and after a number of refill-
ing it will cease altogether, and the whole operation must then be repeated by
filling the whole sypnon and air chamber with water as at the start.

At Col. Crozet's svphon at first owing to the porosity of the joint-caulking,
which was nothing but oakum and pitch, air entered the pipes so rapidly as to
drive all the water from the chamber and thus require it to be refilled every 5 or
10 minutes; but still in two hours the syphon would run dry. The joints were
then thoroughly recaulked with lead, and i)rotected by a covering of white and
red lead made into a putty with Japan varnish and boiled linseed oil. But even
then the chamber had to be refilled with water about every two hours ; and after
six hours the syphon ran dry, and the whole had to be refilled. In this way it
continued to worK.

Care in making the joints air-tight, and an outside and inside coating of the
pipes and air-vessel with coal pitch varnish are important precautions.

Art. 2. Approximate formulae for the Teloeity of wnter in
straight, smooth, cylindrical iron pipes, as ro, vOj lOf Fig. 1. Having the total
head p o, and the length and diameter of the pipe.

Approx
mean vel

in fi per sec

'.}

coefficient

m X
as below

4.

diam in ftX total head in ft
total length in ft + 54 diams in ft

Table of coefficients ** m '*•

Diam of pipe,

m

Diam of pipe,

m

feet

inches

feet

inches

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

1.2

2.4

3.6

4.8 *

6.0

7.2

8.4

9.6
10.8
12.0

23
30
34
37
89
42
44
46
47
48

1.5
2.0
2.5
3.0
3.5
4.0
5.0
6.0
7.0
10.0

18
24
80
86
42
48
60
72
84
120

63
57
60
62
64
66
68
70
72
77

For heads not less than 4 feet per mile, this formula gives results practicaU j
corresponding with those by Kutter's formula (p. 523) with coefficient n of
roughness = 0.012. But slight differences, as to roughness, etc., may cause cod«
siderable variations of velocity, especially in small pipes ; for, in such pipes, a
given roughness of surface bears a greater proportion to the whole area of surfaoe
than in a pipe of large diameter. Extreme accuracy is not to be expected in
such matters.

As in a river the velocity half way across it, and at the surface, is usually
greater than at the bottom and sides, so in a pipe the velocity is greater ar the
center of its cross section than sii its circumf. The iiiean ireloeltjr
referred to in our rules is an assumed uniform one which would give the
discharge that the actual ununifurm one does.

Hence

Dischargee _ Mean Telocity w

in cub ft per sec "" in It per sec ^

Area of cross section

of pipe in sq ft.

1 cubic foot = 7.48052 U. S. gallons

1 U. S. g:allon » .13368 cubic foot » 231 cubic inches.

• For intermediHte diameten. eto. take intermediata ooefflolents from tha (able bj elmple
H^rtion.

HYDRAULICS. 523

In the case of long pipes with low heads, the sum of the velocity and entry
heads is frequently so small that it may be neglected. Where

this is the case, or where their amount can be approximately ascertained, Knt*
ter's formula, although designed for open channels, may be used. This
formula is the joint production of two eminent Swiss engineers, E. Ganguillet
and W. R. Kutter, but for convenience it is usually called by the name of the
latter.*

It is, properly speaking, a formula 'for finding the coefficient e in the well
known formula,

Mean Telocity — e v^mean radius X slope

/diame

/diameter _ ,

- X slope

According to Kutter,

For Eng^llsli measure. For metric measure*

C =

41.6 + ^-1 + i:^ 23 + --^+l

slope n ,___ slope n

/ .00281 \ / .00155 \
1 , \ ' slope r J ^ ^V slope r

l/ineau rad in ftet i/meau fad in riv^es

See also tables of e, pp 566 etc.

The mean radius is the quotient, in feet or in metres, obtained by divid-
ing the area of wet cross section, in square feet or in square metares, by the wet
perimeter (see below) in feet or in metres. In pipes running full, or exactly half
rail, and in semicircular open channels running full, it is equal to one-fourth of
the inner diameter.

The wet perimeter is the sum, ah co Figs 28, 29, 30, of the lengths,

^bfhc^cOj in feet or in metres, found by measuring (at right angles to the lentrth
<>f the channel) such parts of its sides and bottom as are in contact with the
'^ater. In pipes running fUU, it is of course equal to the inner circumference.

Tlie slope is - >Hc<ton head too Fig 1, *__

length of pipe measured in a straight line from end to end.

M sine of angle wto. Fig 1. .

In open channels, this becomes

, a. ^^^^ ^^ water surface in any portion of the length of the channel

~. length of that portion

-B fall of water surface per unit of length of channel

» sine of the angle formed between the sloping surface and the horizon.

The number indicating the slope in any given case ia plainly the same for
English, metric and all other measures.

**n »» is a <« coefficient of roujjrhness" of wet perimeter, and of course
depends chiefly upop the character or the inner surface of the pipe. For iron
pipes in good order and from 1 inch to 4 feet diameter, n may be taken at from
,010 to .012; the lower figures being used where the pipe is in exceptionally good
condition.

If the diameter, or the mean radius, is in feet, metres etc, the velocity will be
in feet, metres etc, per second.

• See " Flow of Water," translated from Ganguillet and Kutter, by Rudolph Hering
•ad John 0. Trautwine, Jr., New York, John Wiley & Sons, 1889. $4,00.

524 HYDRAULICS.

<

The diameter or the slope, required for a slven Telocity,

may be found by trial as follows: assume a diameter, or a slope, as tbe case may
be ; take tbe corresponding c from tables, pp 566, etc. Then say

Approx l>lain required _ mean -^a^ /_If!2£!l^\* y. ^
for the given vel radius ^ "^ \ e l/slope/

Approx Slope required ^ / ^^^Qcity V / velocity \t
for the given vef \^ ^ i/mean radiul/ "^ \ ci^f^Si^ '

With the approximate diameter (or slope) and c, thus obtained, say

r' =3 c'l/'mesiri radilis X slope" If v' is near enough to the given velocity, the
assumed diameter (or slope) is the proper one. If not, try again, assuming a
greater diameter or slope than before if v' is less than tbe required velocity, and
vicetfersa.

Curves and bends do not greatly affect the discharge, so long as the total
beads, and total actual lengths of the pipes remain the same ; provided the tope
of all the curves be kept below the hyaraulic grade line ; and provision be maae
for the escape of air accumulating at the tops of the curves.

Relation between area, velocity, and discliargre.

Let g = rate of discharge (as in cubic feet per second),
V = mean velocity (as in feiet per second),
a = area of cross section (as in square feet).

Then : g = av: v = - ; a =-.

Relation of disciiarge to diameter *_and slope. If we assume

velocity = c i/meau radius X slope, or v = e \^ r s (page 523); and if the pipe
bet)f circular cross section, we have, for the rate, Q, of discharge through a pipe
of diameter, d, and area, A^ of croes section, running full : —

Q = Av=^.c-syi:= 8 -'^

or : Q is proportional to the ^/j power (square root of fifth power) of the diam-
eter, and to tne ^ power (square root) of the slope. For tables of fifth powers,
and of square roots of fifth powers, see pp 67-69.

EflTect of resistances.

The pressure bead of running water, upon any point in a pipe between

the orifice and the reservoir, is :

X. ^ . ^„ ,1 the head oonsnmed

^„t t^ ♦Ko the in overcoming re-

vpI ftt + ®°*''y + "istances in the pipe
Jvfo* l^*«f l»ead between the reaervoir
that point ^^^ the point.

Thus, at the point 6, in the pipe, I o, Fig 1, the pressure head is A » (8 6) » (1 6)
— [(1 2) + (2 3)1 ; where (1 2) =» « « = the sum of the velocity and entry heada.
At 4, in the pipe r o, A = (3 4) = (1 4) — [(1 2) + (2 8)].

In Fig 1, let the straight line, s o, represent the Actual length of the pipe,
whether straight, bent or curved, etc.; and s v the sum of the resistaooea
(supposed to be uniformly distributed) within the pipe. Then, the angle, « o v,
is called the hydraulic gradient, and a\ue s o v = s v -t- s o.

In the vertical pipe, v o, Fig 1 A, the pressure, at 9, is = ^ d.

* Diameter — 4 X mean radius, or d = 4 r (p 523).

f the total ]
-i head on > minus «
(that point)

HYDRAULICS.

525

TABIii: OF WEIGHT OF WATER CONTAINED TN 0»E
FOOT liENOTH OF PIPES OF DIFFERENT BORES.

(Original.)

Water at maximum density, 62.425 lbs. per cubic foot— 1 gram per cubic centi-
meter ; corresponding to a temperature of 4^ Centigrade = dld.29 Fahrenheit.

Weight = 0.340475658 X square of bore in inches.

Water.

Lbs.

0.005320
0.021280
0.047879
0.085119
0.132998
0.191518
0.260677
0.340476
0.430914
0.531993
0.643712
0.766070
0.899068
1.042706
1.196984
1.361902
1.537460
1.723658
1.920495
2.127972
2.346089
2.574846
2.814243
3.064280
3.324957
3.696273
3.878229
4.170826
4.474062
4.787938
• 5.112453
5.447609
6.149840
6.894630
7.681980
8.511889
9.384a58
10.299386
11.256973

Bore.

Ins.

Water.
Lbs.

12.26712

23

13.29983

24

14.38509

25

15.51292

26

16.68330

27

17.8%25

28

19.15175

29

20.44981

30

21.79044

31

23.17362

32

24.59936

33

26.06766

34

27.67862

35

29.13194

36

30.72792

37

32.36646

88

34.04756

39

37.53743

40

41.19754

41

45.02789

42

49.02848

43

53.19931

44

57.54037

45

62.05167

46

66.73321

47

71.58499

48

76.60700

49

81.79925

50

87.16174

51

92.69447

52

98.39744

53

104.27064

54

110.31408

55

116.52776

56

122.91168

67

129.46583

58

136.19022

59

150.14972

60

164.79017

61

Bore.
Ins.

Water.
Lbs.

180.1116
196.1139
212.7972
230.1615
248.2067
266.9328
286.8399
306.4280
327.1970
348.6470
370.7779
393.5897
417.0826
441.2563
466.1110
491.6467
517.8633
644.7609
672.3394
600.6989
629.5393
659.1607
689.4630
720.4463
752.1105
784.4557
817.4818
851.1889
885.5769
920.6469
95(1.3958
992.8267
1029.9386
1067.7314
1106.2051
1145.3598
1185.ll54
1225.7120
1266.9096

Bore.
Ins.

Water.
Lbs.

62

1308.788

63

1361.347

64

1894.588

65

1438.609

66

1488.112

67

1628.395

68

1574.869

69

1621.004

70

1668.330

71

1716.337

72

1765.025

73

1814.894

74

1864.444

76

1915.175

76

1966.587

77

2018.680

78

2071.453

79

2124.908

80

2179.044

81

2233.860

82

2289.358

83

2345.536

84

2402.396

85

2459.936

86

2518.157

87

2577.060

88

2636.643

89

2696.907

90

2757.852

91

2819.478

92

2881.785

93

2944.773

94

3008.442

95

3072.792

96

3137.823

97

3203.535

98

3269.927

99

3337.001

100

3404.756

Tbe weig^lit of water in a given length (as one foot) of any pipe or other
circular cylinder Is in proportion to the square of tne bore or

inner diameter. Hence the weight of water in 1 foot length of any cylinder of
other diameter than those in the table can be found by multiplying that for a 1
inch pipe, 0.340475558, by the square of the inner diameter of the given cylinder in
inches. Thus, for a cylinder 120 inches diameter : diameters = 120« = 14400, and
weight of water in 1 foot depth = 0.34047n558 X 14400 = 4902.848 lbs. Or, weight
for 120 ins. diam. = 100 X weight for 12 ins. diam. = 100 X 49.02848 = 4902.848 lbs.
Similarly, i^-^) * •= -^^ = 0.191406, and 0.340475658 X 0.191406 = 0.065169 lb. =-
weight in 1 foot of -^ Inch pipe. Here, also, ^^ = half of | ; hence, weight for
^ inch = onQ-fmrth of weight for J inch = one-fourth of 0.260677 = 0.065169.

Welgrbt of one square incii of water 1 foot iiig^ii, at 62.426 9>s.
per cubic foot = 62.425 -^ 144 = 0.433507 ft.

♦Actual. See noiuinul and actual diameters, foot note, p 526.

HYDRAULICS.

TABLE 8

Arei

-

""SSKT"

S,ssr'""'""'""s™,.

1^

?J!l

"1C°'

Z

i:

":

EK

Dlim.

DIUD.

"Vbft,

s

I
:
i

„

1
1

i

S

:
1

1

}

J
i

1

i

i

i

i

■1
ii

ii

1
J

so!*

1

1
1

s
1

Ii

tb

c»

ntio

(uaio

UM

M

%Z

bleso

"HI

S

■is

»

thed

Mnf.'^g

w, and

«'3

" dlnmetera. m]

lerglzaa Hpeclallr. tha uKOI
h ibe pipe wlioae ■■nomino''
T dIuneUT or fall quarlsriact.

HYIiRAULICS. 527

Art. 3. To find the total head required for a g^lTen velocity, ot

fiven dischari^e, througli a suaigbt, smuoth, cyhodrical iron pip« of
DOWD diauj aud length.

If the discharge is given, first find

mean velocity discharge in cubic feet per second

in feet per second area of crosit section of pipe iu square feet

Then

Vdiam X head^ _ mean velocity in feet per second
I

^ \ leugtl) + ^ diams the proper divisor an fullows

diam of pipe iu ft .05 .10 .50 1 1.5 2 3 4
divisor 40 43 46 48 51 54 58 61

(for intermediate diams, take intermediate divisors by guess.)

From table Art 2, take the coefficient m correspunding to this value oi

/ diam X head , .»^ ■ i- m.

Vl«ngth+54diuu.s- "^ "" "■" 8""" """"• ^'""'

Tot«l ra^^f is' X O^-K'" '- " + M di..... iD ft)
in Set m«Xdiaminlt.«i

To find the Frletion head. Weisbaclfs formula.

(.01716 \ Length Vel«in

.0144 + '- ly in teet y, ft per sec
^velinft p piani ^ 64.4""
per sec ' i^ feet

For the total head^ we have only to add together, the fVlctlon head

BO found, the velocity head^ taken from the next table, or from Table 10,
opposite the given velocity^ and the entry head (-= say half the velocity headX
The sum of the velocity head and entry head rarely amounts to a foot.

TABIjE 4 Of the vel, and discharge of water through straight, smooth,
cylindrical cast-iron pipes; with the -friction head required for each 100 feet in
length ; and also the velocity head. Calculated by means of Weisbach's formula, by
James Thompson, A M; and George Fuller, C E, Belfast, Ireland. The vel head
remains the same for any lengt/i of pipe ; being dependent only on the velocity of the
water in the pipe.

The entry head is equal to about half the vel head.

628

HYDRAULICS.

TABIiE 4

T«l. In

Vel-

Feet

head Id

perSeo.

Feeu

2.0

.06i

2.2

.076

2.4

.090

2.6

.106

2.8

.122

3.0

.140

3.2

.160

3.4

.180

3.6

.202

3.8

.226

4.0

.250

4.2

.275

4.4

.302

4.6

.330

4.8

.360

6.0

.390

5.2

.422

^.4

.456

' 6.6

.490

6.8

.525

6.0

.562

6.2

.600

6.4

.640

6.6

.680

6.8

.722

7.0

.766

Diam. Id Inohea.

Fr bead
Ft per
100 ft

Cub ft
per Min

.1.

.659
.780
.911
1.06
1.20
1.36
1.62
1.70
1.89
2.08
2.28
2.49
2.71
2.94
3.18
3.43
368
3.94
4.22
4.50
4 78
5.08
5.39
5.70
6.02
6.35

5.89
6.48
7.07
7.65
8.24
8.«3
9.42

10.0

10.6

11.2

11.8

12.3

12.9

13.5

14.1

14.7

15.3

15.9

16.6

17.1

17.7

18.2

1S.8

19.4

20.0

20.6

3H

Frhead
Ft per
100 ft.

Cub ft
per Min

.566
.669
.781
.901

1.03

1.16

1.31

1.46

1.62

1.78

1.96

2.14

2.33

2.52

2.72

2.94

3.15

3.38

3.61

3.85

4.10

4.36

4.62

4.89

5.16

6.45

8.02
8.82
9.62

10.4

11.2

12.0

12.8

13.6

14.4

16.2

16.0

16.8

17.6

18.4

19.2

20.0

20.8

21.6

22.4

23.2

24.0

24.8

25.6

26.4

27.3

28.0 i

Frhead
Ft per
100 ft.

Cab ft
per Min

.494
.585
.683
.788
.900
1.02
1.14
1.27
1.41
1.56
1.71
1.87
2.03
2.21
2.38
2.57
2.76
2.96
3.16
3.37
3.59
3.81
4.04
4.28
4.52
4.77

10.4
11.6
12.6
13.6
14.6
15.7
16.7
178
18.8
19.9
20.9
22.0
23.0
24.0
26.1
26.2
27.2
28.2
29.3
30.3
31.4
32.4
335
34.5
35.6
36.6

4H

Frhead
Ft per
100 rt.

.439
.520
.607
.701
.800
.906
1.02
1.13
1.26
1.39
1.52
1.66
1.81
1.96
2.12
2.28
2.46
2.63
2.81
3.00
3.19
3.39
3.69
3.80
4.01
4.24

Cub ft
per¥ln

13.2
14.6
15.9
17.2
18.5
19.8
21.2
22.5
23.8
25.2
26.6
27.8
29.1
804
31.8
33.1
34.4
36.8
37.1
38.4
39.7
41.0
42.4
43.7
45.0
464

Frhead
Ft per
100 ft.

Cabfl
perlOn

.396
.468
.647
.631
.720
.815
.916

1.02

1.13

1.26

1.37

1.60

1.63

1.76

1.91

2.05

2.21

2.37

2.63

2.70

■-'.87

3.05

3.23

3.42

3 61

3.81

16.3
18.0
19.6
21.3
22.9
24.5
26.2
27.8
29.4
31.0
82.7
34.3
36.0
37.6
39J2
40.9
42.6
44.2
45.8
47.4
49.1
60.7
62.3
64.0
56.6
67.2

Vel-

Dlam. In Inches.

Vel. In

6

1

7

8 1

9

1ft

Feet

head in
Feet.

FFhead
Ft per
100 ft.

Cob ft
per Min

Frhead
Ft per
100 ft.

Cub ft
per Min

^

per See.

Frhead
Ft per
100 ft.

Cub ft
per Min

Frhead
Ft per
100 ft.

Cub ft
per Min

FrheMl
Ft per
100 ft.

Onbfk
P«rMto

2.0

.062

.329

23.6

.282

32 0

.247

41.9

.220

63.0

.198

65.4

2.2

.076

.390

25.9

.:i34

35.3

.293

46.1

.260

. 68.3

.234

72.0

2.4

.090

.456

28.2

.390

38.5

.342

60.2

.304

63.6

.273

78.5

2.6

.105

.526

30.6

.450

41.7

.394

64.4

.360

68.9

.316

85.1

2.8

.122

.600

32.9

.514

44.9

.450

68.6

.400

74.2

.360

91.6

«.o

.140

.679

35.3

.582

48.1

.509

62.8

.463

79.6

.407

98.2

3.2

.160

.763

37.7

.654

61.3

.57.2

67.0

.608

84.8

.468

105

3.4

.180

.851

40 0

.729

64.6

.638

71.2

.567

90.1

.510

111

3.6

.202

.943

42.4

.808

67.7

.707

76.4

.629

95.4

.566

118

3.8

.225

1.04

44.7

.892

60.9

.780

79.6

.693

101

.624

124

4.0

.250

1.14

47.1

.979

64.1

.866

83.7

.761

106

.685

181

4.2

.275

1.25

49.6

1.07

67.3

.936

87.9

.832

111

.748

187

4.4

.302

1.36

61.8

1.10

70.6

1.02

92.1

.906

116

.814

144

4.6

.330

1.47

64.1

1.26

73.7

1.10

96.3

.981

122

.883

150

4.8

.360

1.59

66.6

1.36

76.9

1.19

100

1.06

127

.954

167

5.0

.390

1.71

68.9

1.47

80.2

1,28

106

1.14

132

1.03

16S

6.2

.422

184

61.2

1.68

8:».3

1.38

109

V23

138

1.10

170

6.4

.466

1.97

63.6

1.69

86.6

1.48

113

1.31-

143

1.18

in

6.6

.490

2.11

65.9

1.81

89.8

1.68

117

1.40

148

1.26

188

6.8

.526

2.26

68.3

1.93

93.0

1.68

121

160

164

1.35

190

6.0

.562

2.39

70.7

205

96.2

1.79

125

1.69

159

1.43

196

6.2

.600

2.54

73.0

2.18

99.4

1.90

130

1.69

164

1.62

906

6.4

.640

2.69

75.4

2.31

102

2.02

134

1.79

169

1.61

900

6.6

.680

2.86

77.7

2.44

106

2.14

188

1.90

176

1.71

916

6.8

.722

3.01

80.1

2.58

109

2.26

142

2.01

180

1.81

989

7.0

•

.765

318

82.4

2.72

112

2..V?

146

2.12

185

1.M

291

HTDEAUUCS:

529

TABI4E 4 ■

— (OontinQdd.)

Vel-
headin

Feet.

Diam. in Inches.

▼el. in
Feet

11

12

13

14

15

perSeo.

Frhead
Ft per
100 ft.

Oabft
per Kin

Prhead
Ft per
MO ft.

Cubft
perMin

Prhead
Ft per
100 ft.

Cub ft
perMin

Frhead
Ft per
100 ft.

Oabft
per If In

Frhead
Ft per
100 ft.

Oobft
per Mia

2.0

.062

.180

79.2

.165

94.2

.162

110

.141

128

.132

147

2.2

.075

.218

87.1

.195

103

.180

121

.167

141

.156

162

2.4

.090

.248

95.0

.228

113

.210

183

.195

154

.182

176

2.6

.106

.287

103

.263

122

.242

144

.225

167

.210

191

2.8

.122

.327

111

.300

132

.277

156

.257

179

.240

206

3.0

.140

.370

119

.339

141

.313

106

.291

192

.271

221

3.2

.160

.416

127

.381

151

.352

177

.327

205

.305

235

3.4

.180

.464

134

.426

160

.393

188

.365

218

.340

260

3.6

.202

.514

142

.472

109

.435

199

.404

231

.377

266

3.8

.225

.5«r

150

.520

179

.480

210

.446

243

.416

280

4.0

.260

.623

158

.571

188

.527

221

.489

256

.457

994

4J2

.275

.680

166

.624

198

.576

232

.534

269

.499

309

4.4

.302

.740

174

.679

207

.626

243

.582

282

Ui43

824

4.6

.330

.803

182

.736

217

.679

254

.631

295

.589

389

4.8

.360

.867

190

.795

226

.734

265

.682

308

.636

353

6.0

.890

.935

196

.857

235

.791

276

.784

821

.685

368

6.2

.422

1.00

206

.920

245

.850

287

.789

333

.736

383

6.4

A55

1.07

214

.986

254

.910

298

.845

346

.789

897

6.6

.490

1.16

222

1.05

264

.973

309

.903

359

.843

412

6.8

.625

1.22

229

1.12

273

1.04

321

.964

372

.899

427

6.0

.562

1.30

237

1.19

283

1.10

332

1.02

385

.957

442

6.2

.600

1.38

245

1.27

292

1.17

343

1.09

397

1.01

456

6.4

.640

1.47

258

1.36

301

1.24

864

1.16

410

1.08

471

6.6

.680

1.55

261

1.42

311

1.31

865

1.22

423

1.14

486

6.8

.722

1.64 209

1.50

320

1.39

376

1.29

436

1.20

500

7.0

.766

1.73 277

1.59

330

1.46

387

1.36

449

1.27

516

Vel-

bead in
Feet.

Diam. In Inohea.

▼eLin
Feet

16

1

Frhead
Ft per
100 ft.

7

18

19

20

ferSec.

Frhead
Ft per
100 ft.

Oabft
perMin

Cub ft
perMln

Frhead
Ft per
100 ft.

Cab ft
perMin

Frhe^
Ft per

100 ft.

Oabft
perMin

Prhead
Ft per
100 ft.

Oobft
perMin

2.0

.062

.123

.167

.116

189

.110

212

.104

236

.099

262

2.2

.075

.146

184

.138

208

.130

233

.123

260

.117

288

2.4

.090

.171

201

.161

227

.152

254

.144

283

.137

314

2.6

.105

.197

218

.185

246

.175

275

.166

307

.188

340

2.8

.122

.225

234

.212

265

.200

297

.189-

331

.180

366

3.0

.140

.255

251

.240

284

.226

318

.214

354

.204

393

3.2

.160

.286

268

.269

302

.254

339

.241

378

.229

419

8.4

.180

.319

284

.800

321

.263

360

.209

401

.255

445

8.6

.202 '

.354

301

.333

340

.814

382

.298

425

.283

471

3.8

.225

.390

318

.367

359

.347

403

.328

449

.312

497

4.0

.250

.428

335

.403

378

.380

424

.360

472

.342

623

4^

.275

.468

352

.440

397

.416

445

.394

496

.374

560

4.4

.302

.509

868

.479

416

.452

466

.429

519

.407

576

4.6

.330

.552

385

.519

435

.490

488

.466

543

.441

602

4.8

.360.

.596

402

.561

454

.530

509

.502

567

.477

628

5.0

.390

.642

419

.605

473

.571

530

.641

590

.514

654

6.2

.422

.690

435

.650

492

.614

551

.681

614

.552

680

6.4

.455

.740

452

.696

511

.657

572

.623

638

.592

707

6.6

.490

.791

409

.744

529

.703

594

.666

661

.632

783

6.8

.526

.843

'486

.793

648

.749

615

.710

685

.674

769

6.0

.502

.897

602

.844

567

.798

636

.765

709

.718

786

6.2

.600

.953

519

.897

586

.847

657

.802

732

.762

811

6.4

.040

l.Ol

636

.951

605

.898

678

.851

756

.808

888

6.6

.680

1.07

653

1.01

624

.950

700

.900

780

.855

864

6.8

.722

1.13

609

1.06

643

1.00

721

.951

803

.904

89«

7.0

.765

1.19

586

1.12

662

1.06

74^ ,

1.00

827

.953 1

»1B

34

HYDKAUUCe.

Exsmple of nsc «f dlnn-am. Olven n 6 Inch pipe, In &lr coi

In the column, on Uie right, helped ■' F^r." find dlam, 6 ina. Follonin
l«ft the direction of the ihort inclined line, preferably bv mems of a
BtrBight-wJgeQfpapcr, wefindtliBtitoolnoldtB nearlj^ with one of the ]
11n« which croffi the diagram. B7 roeana of the Intersections of this li

• ■'Old,"
Kutter'8 "

"fslr," and

F.'- [p. 664) 88

::z:^:: ™

™pond

"""

oxim.^. w

th TKlun of

Diameter...

Rlnch

«lnch

121

noh

BOlnoh

120tndi

BlDpe,lnft
perlOOOfI

10.0 1.0

10.0 1.0

10.00

0.04

1.000 0.02S

n »

. n

»

~

> »

n .

Old

F«lr„

Mew

0.012 0,013

0.010 0.010

O.0U 0,014

0.010 0.010

0,010

0.016

0.018 0.020

0.019 a021
0.013 o.ou

HYDBAULIC8L

Art. 4. To flnil tlie dlscliargre, q^ tlurong^li

rying diameter, Fig. 1 H.

pound plpe^ or pipe of yaxying

631

Ions com-

H

X-

- — li-^

Ir

-H*

i

^

B=!P

3C

Figr. 1 SL

Let

hi h, ki ^c. = the lengths of the several portions of the pipe ;

di, d^ dsj etc. = the corresponding diameters ;

vi, vg, va, etc. = the corresponding yelocities ;

Fi, Fj, Fa, etc. = the corresponding valaes of the resistance or *' friction »'

factor. See p. 530.

L =^1 + ^2 + ^8 + etc. — the total length of the pipe ;
H =» the total head (p. 616) ;

q = rate of discharge = %vd^V\ = K'^d^ ''a = ©te-
la a long pipe, the velocity and entry heads are usually negligible, relatively
to the friction head. Neglecting them, we have

H = total head =s friction head.

In each portion of the pipe, the resistance, and the corresponding "friction"
head, hf, are believed to be proportional directly to the length, l, of such portion

and to the velocity head, jr— , and inversely to the diameter, d; or

2 g

A/ — F • — • s-^.

Hence,

d 1g

H = Fr:^.^^- + Fg.^-^ + Fs^'jr^ + etc.;
^ di2 g "^ d^2 g "" dz2 g ' *

and, since V\ =

4 g 4 g .

«» *2 = -^a» etc..

we have, also.

2,H.r;Aii4 + F,Aii^, + etc.
div^d^ di n^d*

16^2

U

+ Fa

dl

+ etc.

whence

0

^TAFx^ +Fa^. +etc.

)

632

WATER-PIPEB.

Fig. 1.

Art. 4 a.* Tlie Ventnri Meter Is designed for the measurement of th«
flow of liquids in pipes of large dimensions, running full.

The meter proper, patented by Clemens Herschel, consists essentially
of a mere constriction in the area of cross-section of the pipe, with openings
in the pipe opposite its normal and its constricted diameters, for measurmg. by
piezometers or pressure-gauges, the pressures at those points: while tne
register, patented by Messrs. Frederick N. Connet and waiter W. Jackson,
is an elaborate mechanism, provided with clock-work and dials.

Tbeory.t Let Figs. 1 to 8
represent a Venturi meter tube,
with three piezometers in place,
viz.: No. 1, over the tube up-stream
from the constriction ; No. 2. over
the constriction itself; and No. 3,
over the tube down-stream from
the constriction. Let the unshaded
area W in Figs. 1 to 3, represent
the depths at which the water
stands above any assumed hori-
zontal datum plane 0-0; and let
the shaded area A represent the
uniform pressure of the atmos- "
phere, which, for convenience, we .^ j. x

may suppose to be converted into some liquid of the specific gravity of water,
but distinguishable, by its appearance, from the water.

The vertical distance, between the upper boundary of this latter area and
any given point in the tube, represents the combined pressure of air and water
at such point. ^ .^ . _

The velocities in the meter tube, at any instant, are of necessity inveraelj
proportional to the areas of cross section; and, as the heads corresponding to
the several velocities are proportional to the squares of those velocities, the
remaining or pressure heads must vary also, the smallest or lowest pressure
head standing over the throat, where the velocity is greatest.

. The increase of velocity, ac-

?[uir^ by the fluid in passing
jrom section 1 to section 2, is again
given up in passing from section
2 to section 3 ; and, in the case of
a perfect fluid, the pressure lost
between sections 1 and 2 would be
perfectly restored in passing firom
section 2 to section 3. In practice,
a small total loss occurs. This loss
is greater with high than with low
velocities.

For a given head in piezometer

O O jTo 1 and given diameter of pipe

at section 1, the expenditure of
head in velocity between sections
1 and 2 increases as the area of the throat is diminished and as the throat
velocity is thereby increased.! In Fig. 2 is shown the case where all of the
water head above the top of the throat is required to maintain the velocity
through the throat. , ^

In Figs. 1 and 2 the head, H, expended in the increase of velocity between
sections 1 and 2 is represented by the diflerence in level between the tops of the
two water columns 1 and 2, or between the tops of the two corresponding air
•columns. In Fig. 2 this diflerence is equal to the total vertical height of the
water column at section 1 above the top of the throat at section 2.

* Abridged from a description prepared by the writer as Chairman of a Com-
mittee of the Franklin Institute. Journal of the l^anklin InstituU, February,
1899.

t The Venturi meter, apart from its merits as a measuring device, embodies
important hydraulic principles. Heuce its theory is here stated more fully than
would otherwise be necessary.

t In a given Venturi tube the pressure and velocity at the throat mav be
varied also by modifying those at sectons 1 and 3, as by regulating the ofMnings
of the valves of influx to and of etflux from the meter tube, by changing the
total head on the system, etc.

Fig. 2.

WATBB-P1PE8.

In Fig. 3, tha loss of besd, due ut lniu«aw of i^oclly b* . .
ImH—hw + Aa — the entire arailable head of water, A., plus a portion, IU,ort)ie
■Uno^ilwric preeeure. The latter portion, h.^ la [requeoi]]' called "the

le topof th
at, it ii DO'

I of bead b; UkiDK the dllRr-

D« Eetw«n th.

eV^ir

f ihe w

degree'' of ■■vacuum" mBT be

Flo. 4.

found, ai shown In Fig. 4. by

a glam tube bent over and led

t^^wB^"orTe°rd'u™ Thi

height to which the water (or the

mereurj. convened Into /eel of

water) rises In thia tube, show.

Ibe eitent of the lacuum, or the

portion ft-, of the air presaure

*hich iw, i^a c^lcd lat« Wfice

in producing the high velocity
through the throat ^r adding

tola fi )W. we obt^n, u abov^

O the total low of bead S between

When the reductlo

nofe

rea at the throat harprI»aed«i"»o*far that the eoUn

irand ■!

r at section t ia reqnired. In order to main-

-J J

M

*-S?^

'i

--

~

_|.U'

]-* ""■"

-

i|

1

___

' ^;---—

tain the corresponding velocity tbrouEh tbe ibrost ii, e^ when the line repre-
(•Dting the upper aurRuM of tbe ait Ealli to the level ot the top of the throB(>

684 WATER-PIPES.

no further increase of throat velocitj can he secured (with a given totai head
oyer section 1) by still further narrowing the throat. If the throat is further
narrowed, the velocity through it will remain the same ; and, the rate of dis-
charge being thus diminished, the velocity through section 1 will be neces-
sarily reduced. In other words, throttling begins.

Let vi be the velocity in section 1, above the throat, and v^ the " throat veloe-
ity," or velocity in the throat or section 2.

Referring to Fis. 5, the velocity head at section 1, measured Arom an assumed
datum represented by the upper horizontal lines, is

and that at section 2 is

Neglecting resistances to flow, the loss of head, between sections 1 and 2,
or " the head on the Venturi," is equal to the increase in the velocity head, of
to the loss in pressure, between a^, and a^, or

If = Ao — A, = -t- — -T- = * ^ — *— = Pi — p.*

Hence, A, = -^ = H + Aj

5j«
2g

and tliroat veloeity s. », = V2^ (H + A^) = -%/ 2^ (fT + ^) •

In other words, the velocity at the throat is that corresponding to the ** head
H on the Venturi," plus the head correspondiug to the velocity of approach v^
in section 1.

But, since the velocities are inversely as the areas of cross-section a^ and a^

«i = „ «ai and r^a = -f^ v^\

Ox «!*

2g 2g 2g 2g
«i

and tbroat Telocity — v, = . ' y2gfr.

roj* — Oj" *^

The ratio — ^

between the area oj of cross-section at the throat, and that, a^, at the upper end
of the up-stream cone, is called the tbroat ratio. For a ratio of 1 : 9 we have

ax 9 ^ 9 /8i

or Va =• 1.0062^2^ -ff.

Tlie Tontarl tabe, for pipes not over 60 inehes in diameter, la fbnned of
8«iveral short sections of oast iron pipe, having the required taper, and Air-

* By Bernouilli's theorem, P\ -V h^^Pt ^r S*

WATER-PIPES. 635

nlshed with flanges, by means of which the sections are bolted together to form
ttie two truncated cones required.

In the smaller sizes, the shorter cone is generally in one section and the
loneer cone in two or more sections.

The throat section is generally made in a separate piece, and is either made
of bronze or lined with that metal.

The ends of the Venturi tube are furnished with either bell, spigot, or flanged
ends, according to the character of the pipe in which the tube is to be used.

For fitill largrer streams, such as those in masonry conduits or riveted
flumes, the Venturi tube may be made of wooden staves, sheet steel, cement
concrete, brick or other suitable material, metal being used for the throat piece
and where required by the pressure.

The tbroat piece ia surrounded by an annular chamber called the press-
are eliamber. which communicates with the interior of the throat by means
of several holes drilled radially through the walls of the latter at equal or
nearly equal distances around the circumference.

A similar pressure chamber is provided at the larger end of the short cone for
observing the pressure in the normal section up-stream from the throat; and,
if it is desired to ascertain the final loss of head due to the passa^ of the water
through the Venturi, a similar chamber must be provided at the larger end of
the longer or down-atream cone.

In deslsrnatiiiK tbe slse of the meter, the diameter of the pipe of which
it forms a part is us^, and not the throat diameter. Thus, a meter for use in a
6-inch pipe is called a 6-inch meter.

Tbe reipister gives periodic registrations, usually every ten minutes, in
which the heaA Jff=^ kz'— hit existing At the instant of registry, is recorded in.
terms of the total discharge in cubic feet since the last registry and as an in-
crease in the total number of cubic feet registered. In other words, the registry
involves the assumption that the average velocity, during the period between two
registrations, is equal to the velocity at the instant of the following registration*

The register may be placed at a considerable distance (not exceeding, say, 500
feet) from the Venturi tube. It must be placed at such a depth below the
hydraulic grade line that the pressures existing in the Venturi tube shall at all
times be transmitted to the register.

The pipe lines, connecting the Venturi with the register, must be covered, and
a shelter from weather and frost must be provided for the register.

The site and cost of the register are independent of the size of the Venturi.

Bebavior. From experiments by Mr. Herschel,* f IT bv the Bureau of
Water, PhilBddphia,f and by others,! it appears that the Ventun meter may ordi-
narily be depended upon to give results within 3 per .cent, of the true discharge.

With a 48 inch Venturi, Mr. Herschel f found a total loss of bead, due to
the passage of the water through the Venturi tube, of about 10.6 per cent, of the
head H on the Venturi. With two 54 inch Venturis, Professors Biarx, Wing, and
Hoskinsg t found a loss of 14.9 per cent., part of which, no doubt, was due to
the presence of a 42 inch gate valve in the down-stream cone. This last result
would add about 1.12 feet to the head required in pumping 20,000.000 gallons
dail / through a 48 inch main and a Venturi having a throat ratio ox 1 : 9.

The Venturi meter has been found to give perfectly satisfactory results in
measuring the flow of brine and very hot water.

Venturi tubes are made with throat ratios raneing from 1 : 4i (or 2 : 9) to
1 : 16. The former are adapted to high, and the latter to low velocities ; for,
where the velocity in the pipe is low, it is necessary to accelerate it greatly in
the throat in order to obtain sufficient loss of pressure to secure reliable in-
dications in the renter. These cannot be obtained where the throat velocity
ii less than about 3 feet per second. With a throat ratio of 1 : 16, this would
give a pipe velocity of ^^ ^^^^ P^^ second. On the other hand, a meter with
a high throat ratio, adapted to low velocities, would,with high velocities, exceed
the spper limit of the register.

Owing to its unobstructed channel, tree from moving parts, the Venturi
meter is far less liable to clogging than the forms of meter in common use.

The priees of the principal sizes of the Ventud meter are as follows :~on
board cars at Providence, B. I.

6 inch S600.00 24 inch Sl,130.00 48 inch 88,060.00

12 inch 770.00 36 inch 1,680.00 60 inch 4,890.00

These prices include the register, which, in the smaller sizes, constitutes the
principal item of cost. Discount, 1901, 10 per cent.

♦ Trans. Am. Soc. Civil Engrs., Nov., 1887, Vol. XVII., page 228.

t Journal of the Franklin Institute, Feb., 1899.

I Journal New England Waterworks Assn., Vol. VIII., No. 1, Sep., 1893.

f Trans. Am. Soc. Civil Engrs., Vol. XL., Dec, 1898, pp. 471, etc.

536

WATER-PIPES.

Art. 4 b. The Terrls-Pltot meter, invented and patented bj Mr.
Walter Ferris, of Philadelphia^ w desigDed to measure the flow of liquids in
pipes running full. It consists of a device for the regutration of the results
obtained by the Pltot tube, described on pages 661 and 6qk2, and of special devices
to prevent the clogging of the tubes and to permit their examination while
in use.

In Fig. 6 let P represent the level at which the water stands in the straight

Pitot tube, s. Then h == Aw*, or the difference in level

between the columns in the two tubes, is the head Fio. 6.

(theoretically =— J due to the velocity of the water

in the pipe as it impinges against the open up-stream
end of the bent tube, c. For a given velocity, v, this
difference, h, is constant, and is independent of the
pressure represented by P.

The Ferris register, like that of the Yenturi meter,
records the velocity (existing at the instant of registra-
tion) in terms of the total discharffe since the last reeiS'
try and as an increase in the total number of cubic ieet
registered. The registry thus involves the assumption
that the average velocity, during the period between
registrations, is equal to the velocity at the end of that
period. In the Ferris meter the registration is made
every two minutes.

Evidently the instrument measures the velocity at
only one point in the cross-section of the pipe, and it may thus be used to de-
termine successively the velocities at any number of such points, but the ve-
locity at such a point may or may not be equal to the mean velocity in the entire
cross-section. The instrument is therefore usually calibrated by reference to
some accepted standard, and the coefficient or coefficients thus obtained are
used in subsequent observations.

The recording mechanism is operated by a small hydranlic motor, driyen by
means of the flow of the water in the pipe itself. For this purpose a secona
pair of Pitot tubes, is inserted into the pipe ; and the current, flowing through
these tubes, drives the motor without loss of water, the water used for power
being returned to the pipe. If the velocitT in the pipe is less than 3 feet per
second it must be increased by means of a ^'reducer."

Experiments made by Mr. Ferris and by the Bureau of Water, Philadelphia,
indicate that the Ferris-Pitot meter will ordinarily register within 8 per cent
of the true discharge.

In general, the sise and cost of the registering apparatus are independent of
the aise of the pipe.

htdbai;lic&

637

Art. 5. BeslBtenee of carves and bends In water pipes.

Much uncertainty exists respecting these matters. WeislNM:li*B form-
ula,* for the resistance due to a circular curve, Figs. 2 and 8, is

A'-C

180 2

,^-[am + l.»47(^)«]^.|l.,her.

A » head in feet required to OTercome resistance due to corre or bend,

G » experimental coefficient,

A a angle of deflection, in degrees,

V s> mean Telocity of flow in pipe, in feet per second,

g « acceleration of gravity » 82.2 ft per aeo per seo,

r— = head theoretically due to velocity v,

D a inside diameter of pipe, in ftet,
r s inside radius of pipe, in feet,
B = radius of axis of curve, in feet.

Ifr + B=' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
then C ^ am 0.13S 0.168 0.206 0.2M 0.440 0.661 (kVJl 1.408

1.0
1.978

FiflT. 8.

Fiff.9.

Fi«. 4.

(See next page.)

Aoeotdlng to this formula, the reslBtanoe due to curvature decreases rapidly
M B increases from ^ D to 2 D : and hot little flirther decrease occurs beyond
B = 5 D ; but, fh>m very careful and elaborate experiments on cdty water mains,
from 12 to 80 ins diameter, in Detroit, Mich..t the investigators conclude that a
line of pipe with a curve of ihort radius B (down to a limit of B = 2^ D) causes
Us9 resistance than does a line of equal length and equal total angle A, wiUi a
curve of longer radius B. Their results were approximately as follows, where

H a resistance due to a section of 80 diameters in length, with a curve of

A = 9fp9X mid-length,
h a resistance in a tangent of length = 80 diameters.

IfB + Da- 1 2 2.5 8 4 5 10 15 20 25

then H -i- A » 1.85 1.14 LIS 1.14 1.18 1.24 1.60 1.66 1.80 1.93

They found also that the loss of liead, due to a curve, occurs not only
in tiie enrwe itself, but that head continues to be lost in the following
tangent, foi;some distance down stream from the curve.

Tlaeir experiments led to the inference that even very slicbt defleetions,
A, in the line, cause material losses of bead, and tnat care in securing
a straight alignment is therefore highly advisable. For bends, see next page.

*I>er Ingenieur, pp. 444, 445.

t Paper by Gardner S. Williams, Clarence W. Hubbell, and George H. Fenkell*
TranMctions, American Society of Civil Engineers, Vol. XLYII, April, 1902.

638

HYDRAULIC8.

For abrupt aiiirlM* Fig. 4, Weisbach gives : BcBistonee, in feet of head <

c |L = (0.95 sin* >^ A + 2.05 sin* >i A) |^

If >^A- 10°
then 0 — 0.03

20O

80«

40°

450

60<»

550

6OO

65°

7(P

0.14

0.86

0.74

0.98

1.26

1.66

1.86

2.16

2.43

li«.4.

In addition to tbe resistanise offered to flow, caires and bends In-
TOlye additional labor and expense in manufacture and in lajins ; and yertical
bends and curres lead to the formation of pookets of sediment at the feet of
slopes, and of air cushions at their summits.

a h

6'
Fiff. 5.

Art. 6. Although, in Fig. 5, the static pressures opon the equal bases, a h
and a' I/, of the two pii)es are equal (see Hydrostatics, Art. 1) : yet, in order to
pump water through either pipe, at a given velocity, an additional force is
required, in order to overcome resistances to flow ; and these resistanoes and the
additional force required in order to overcome them, will be greater in the longer
than in the shorter pipe.

HYDRAULICS.

689

Art. 7. Flow throajvli orifices. Tbeoreticall7 the Telocity, v, of a
fluid, flowing through a small orifice in the side or bottom of a very large vessel,
is equal to that acquired by a body falling freely in vacuo through a height
equaUto the head. A, or depth, measured vertically from the level surface of the
fluid in the veosel, to the center of gravity of the orifice ; or,

«. y^2ffh - |/64.4A «■ 8.08 |/A;

> 0.0155 V*.

and

This law applies equally to all flnids. Thus, theoretically, mer-
cury, water, air, etc., all flow with equal velocities firom a j^ven orifice under a
given head.

For deviations Id praotioe firom this theoretical law, see Art 9, etc.

Table 10.
Teloeities tbeoretieally due to ffiTon heads.

Head

Vel.

Hnd

Vel.

Head! Vel. 1

Head

Vel.

Head

Vel.

Head

Vol-

Head

Vel.

F«et.

Ft per
see.

Feet.

Ft per

■eo.

Feet.

Fiper
•ec.

Feet.

Ft per
aeo.

Feet.

Ft per
•eo.

Feet.

Ft per
nee.

Feet.

Ft per
see

.005

.57

.29

4.32

.n

7.0*

1.50

9.83

7.

21.2

98

42.5

76

69.9

.010

.80

.30

4.39

.78

7.09

1.59

9.90

.2

31.5

29

«t.2

77

70.4

.015

.96

.31

4.47

.79

7.13

1.54

9.96

.4

21.8

SO

43.9

78

70.9

.090

1.13

.32

4.54

.80

7.18

1.56

10.0

.6

22.1

31

44.7

79

71.3

.025

1.27

.33

4.61

.81

7.22

1.58

10.1

.8

92.4

S3

45.4

80

71.8

.030

1.39

.34

4.68

.89

7.26

1.60

10.9

8.

99.7

33

46.1

81

72.2

.0S5

1.50

.35

4.75

.83

7.31

1.65

10.3

.2

93.0

34

46.7

82

72.6

.040

1.60

.36

4.81

.84

7.35

1.70

10.5

.4

93.3

85

47.4

83

73.1

.046

1.70

.87

4.87

.86

7.40

1.75

10.6

.6

93.5

86

48.1

84

73.5

.050

1.79

.38

4.94

.86

7.44

1.80

10.8

.8

23.8

87

48.8

85

74.0

.056

l.t«

.39

5.01

.87

7.48

1.85

10.9

9.

34.1

38

49.5

86

74.4

.000

1.97

.40

5.07

.88

7.58

1.90

11.1

.9

34.3

39

50.1

87

74.8

.065

2.04

.41

5.14

.89

7.57

1.95

11.2

.4

24.6

40

60.7

88

75.3

.070

2.12

.42

5.20

.90

7.61

9.

11.4

.6

24.8

41

51.3

89

75.7

-075

2.20

.43

5.26

.91

7.65

9.1

:i.7

.8

25.1

49

52.0

90

76.1

.080

2.27

.U

5.32

.99

7.70

9.9

11.9

10.

95.4

48

59.6

91

76.5

.005

2.34

.46

538

.93

7.74

2.3

12.2

.5

26.0

U

53.2

92

76.9

UMO

2.41

.40

5.44

.94

7.78

9.4

19.4

11.

96.6

46

68.8

98

77.4

.086

2.47

.47

5.50

.96

7.82

3.5

12.6

.5

37.2

46

64.4

94

77.8

.100

2.54

.48

5.56

.96

7.86

9.6

12.9

19.

27.8

47

65.0

95

78.2

.106

2.60

.49

5.62

.97

7.90

17

13.9

.5

98.4

48

55.6

96

78.6

.110

2.66

.50

5.67

.98

7.94

3.8

13.4

IS.

98.9

49

56.2

97

79.0

.115

2.72

.51

5.73

.99

7.98

2.9

18.7

.6

99.5

50

56.7

98

79.4

aao

2.78

.59

5.79

IFt.

8.03

3.

13.9

14.

80.0

51

67.3

99

79.8

.IS

2.84

.53

5.85

1.09

8.10

3.1

14.1

.5

80.5

59

57.8

100

80.3

.180

2.89

.54

5.90

1.04

8.18

3.2

14.3

15.

81.1

53

58.4

125

89.7

.1S5

2.95

.55

5.95

106

8.96

3.3

14.5

.5

31.6

54

59.0

150

98.8

.140

3.00

.56

6.00

1.08

8.34

8.4

14.8

16.

89.1

55

59.5

175

106

.146

3.05

J»

6.06

1.10

8.41

8.5

15.

.6

89.6

56

60.0

200

114

.150

8.11

.58

6.11

1.12

8.48

8.6

15.2

17.

83.1

57

60.6

225

120

.156

3.16

.69

6.17

1 14

8.57

3.7

15.4

.6

83.6

68

61.1

250

126

.100

3.21

.00

6.22

1.16

8.64

3.8

15.6

18.

84.0

58

61.6

275

133

.165

3.96

.61

6.28

1.18

8.72

3.9

15.8

.5

84.5

00

62.1

300

139

.170

3.3!

.69

6.32

120

8.79

4.

16.0

19.

35.0

61

62.7

350

150

.175

3.36

.68

0.37

l.Ti

8.87

.9

16.4

.5

35.4

69

63.2

400

160

.180

S.40

.64

6.42

1.24

8.94

.4

16.8

90.

85.9

68

63.7

450

170

.185

3.45

.65

6.47

1.26

9.01

.6

17.9

.5

86.3

64

64.2

500

179

.190

3.50

.66

6.52

1.28

9.06

.8

17.6

21.

86.8

65

64.7

550

188

.195

».55

.07

6.57

1.80

9.15

5.

17.9

.5

37.2

66

65.2

600

197

.900

3.59

.68

6.61

1.39

9.91

.9

18.3

99.

37.6

67

66.7

700

919

.n

3.68

.69

6.66

1.34

9.99

.4

18.7

.6

38.1

68

66.2

800

927

.n

3.76

.70

6.71

1.36

9.36

.6

19.

93.

38.5

69

66.7

900

941

M

3.85

.71

6.76

1.38

9.43

.8

19.3

.5

38.9

70

67.1

1000

954

.94

3.96

.72

6.81

1.40

9.49

6..

19.7

94.

39.3

71

67.6

.96

4.01

.73

6.86

1.42

9.57

.9

20.0

.6

39.7

79

68.1

•* -

4.09

.74

6.91

1.44

9.63

.4

90.3

25

40.1

78

68.5

.27 •

4.17

.75

6.95

1.46

9.70

.6

20.6

26

40.9

74

69.0

.20

4.95

.78

6.99

1.48

9.77

.8

90.9

27

41.7

75

60.5

540

HYDRAULICS.

ibownby

Art, 8. On the flow of water
throuKta vertical openlnars far-
nlsbea with abort tabes. Tf^en water

flows from a reservoir, Fig 6, through a Tert partitloa

mm a a, the thickness a mat whieb is about '214 or 3 times

the least transverse dimen8ion of the opening, (whether

that dimension be its breadth, or its height;) or when, if

the partition be very thin, as it n, the water flows through

a tube, as at t, the length of which is about 2 or 3 times its

least transverse dimension, then the effluent stream will

entirely fill the opening, or the tube, as shown in Flg6 ; or,

in technical language, will run %rUh a fuUJIow: or a fiiU

bore ; and will diseh more water in a given time, than if

the tube were either materially longer or shorter. For if

longer than 8 times the least transverse dimeasioo, the

flow will be impeded by the increased friction against the

sides of the tabe ; and if shorter than about twice the least •

transverse dimension, the water will not flow in a full stream, bat in a oontraoted one, M

Fig 11. Tbis will be the case whether the tnb« be oirenlar, or reetiUaear, in its orote

To find approximately the actoal Tel. and disch into tlie
air, throofrh a tnbe, or openinur, either circular or reeti*
linear in ito outline, or crosa-seetion % and whose lengrth e iy
or e 0, in the direction of the flow, is about 2>^ or S times its
least transverse dimension ; when the surface-level, «• Fly e»
remains constantly at the same heiarbt; and which heiirht
must not be below the upper edge of the tube, or openiuir*

BaLB 1. Take out the theoretical vel flrom Table 10, oorresponding to the head measured vert

ft-om the oenter (or more properly, the oen of grav) e, of the opening, to the level water surf «. Mnlt
it by the coeff of disch .81. The prod will be the reqd vel, in ft per see. Mnlt this actual vel by the
transverse area of the opening, in sq ft. If oiroular, knowing iu diam. this area will be found in
Table 3. The prod will be the quantity of water dlsohd, in oab ft per see ; within, probably, S

or 4 per cent.

Bcu 3. Find the iq rt of the head In ft. Mult this sq rt by 6.6. The prod will be the aetnal
vel in ft per sec.

Ex. An opening c • ; or box-shaped tube e t, Fig 6, is 8 feet wide, by .25 of a ft high ; and its length
in the direction e < or e e in which the water flows is about .82 of a ft, or about SM times its Icaat
transverse dimension, or its height. The head from the oen of grav c, of the opening, to the oonatnat
surf-level «, is 4 feet. What will be the vel of the water ; and how much will be dlsohd per see 7

By Rule I. The theoretical vel (Table 10. ) corresponding to i head of 4 ft is 16 ft per see.

And 16 X .81 = 12.96 ft per sec, the actual vel reqd. Again, the transverse area of the opening, or of
the tube. Is 3 ft X .26 ft = .75 sq ft. And .75 X 12.96 = 9.72 cub ft ; the quantity dieohd per sec

By BuU 3. The so rt of 4 is 3. And 3 X 6.5 = 13 ft per see. the reqd vel. a* before; the very Mght
dlff being owing to the omission of small decimals in the coeffs.

Rbm. 1. If the short tube t projects partly Inside of the vert
partition n n, the disch will be diminuhed about ^ part. In that case, use .71

or .7 Instead of the .81 of Rule 1 ; or 6.7 instead of the 6.5 of Rule 2.

Rbm. 2. When the thickness a m of the vert partition mmaa; or the length e < of the tnbe (, Pig
6, is increased to about 4 times the least transverse dimension of the opening ; or of the diam. when
circular : then the additional friction against its sides begins appreciably to lessen the vel and dlaeh.
In that ea«e. or for otiU greater lengths, up to 100 diams, they may be fbund approximately, by nnlac
instead of the ooeiT of dieeh .81 in Rule 1, the fbllowing ooeflk, by which to mnlt the tbeoreUanl T«la
ef Table la

TABLE It.

Length of
Pipe

Length of

Coeff.

Pipe

Coeff.

in Diams.

in Diams.

4

.80

40

.63

6..«

... .76

60..

... .60

10

.74

60

.67

16...

... .71

70...

... .66

20

.69

60

.6a

2b...

... .67

90...

....60

ao

.66

lOO

.48

Rbm. 8. When the length of the opening or tube, in the direction in which the water flows. '
less than about twioe lu least transverse dimension, the disch is diminished; so thatferlengtbel
IH times, down to openings in a very thin plate, we may use .61, instead of the .81 of Rulel. For
such openings, however, see Arts 9 and 10.

Rbm. 4. But on the other band, the disch through such short openlngi and tubes as are sbovn In
Fig 6. mav be increase<l to nearly the theoretical ones of Table 10, by merelv rounding off neatly tk«
edges of_the entrance end or mouth, a* In Fig 7 ; which is the shape, and half aetnal siae ef one with

~ disobarge, when the bend was 10 ft; and .MS

which Weisbaoh obtained .075 of the theoretical vel and

HTDBAtJIilOS.

641

with « hMd Of one foot; oo ibMt la simUAr omoo, .975, ud .968 magr be wed tnitead of the eoeff .SI
in Bole 1.

Mgr.T.

B1ff.8L

Fig.QL

A« muoh u .92 to .94 may be obtained by widening the opening, m n, toward its onter month, e«,
fig, 8, making the diTorgenoe, or angle a. about 6° : or hj widening it toward its inner month, as at
i e. Fig 9; but Increasing the angle of divergenoe, at b, to from U9 to 16°. In all oases, we consider
(he small end as being the opening whose area must be multiplied bj the rel to get the diitofaarge.

In some experlaients made wlUi lar^e pyramidal wooden
troaiplis 9.5 ft long, with an inner mouth of 3.2 X 2.4 ft, and a discharging one
of .82 X -44 ft ; and under a head of 9)< feet, the disoharge was .96 of the theoretloal one, due to the
•mailer end. Therefore, .98 may be need in sneh eases, instead of the .81 of Rule 1.

Rnu. 5. By nting an adjutage shaped as in Pig 10, the disoharge may be inoreased to sereral
times that due to the head above the center o/^rvseMy • of the orifloe mn : oecanse in such oases, as
explained in Art 1 w, the true head at «, or the head oaaaing the rapid flow through the nar.

rowest portion mn, may be much
3 greater than the head above «.

3C -.r;?;;f::

8«e the Ventnri Meter, Art 4 a.

Flgr.ia

Art. 9. On tbe dlsch of water throairli openingrs In f bin
Tfirt partitions, wltb plane or flat faces, ee^orn n. Fig 11.* If the

face « «, orn n, Instead of being plane, and vert, should be curved,
or Inclining in diff directions toward the opening, then the disch
will be altered. When water flows from a reservoir. Fig 11, through
a vert plane plate or partition nn, which is not thicker than about
the least transverse dimensioD of the openlng.whether thatdimenslon
be its breadth, or its height o o ; t or when, i' the partUlon e e itself
is muoh thicker, we give the opening the shape shown at b, (which
evidently amounts to the same thing,) then the effluent stream will
not pass out with a, full flow, as in Fig 6, but will assume the shape
shown in Fig 11 : forming. Just outside of the opening, what is
called the vena contraeta, or contracted vein. In order that this
oontraotioQ may take place to ita fullest extent, or become oomplefe,
the inner sharp edges of the opening must not approach either the
surf of the water, or the bottom or sides of the reservoir, nearer
than about IH times the least transverse dimension of the opeoini;.
The oontracted vein occurs at a dist of about half the smallest di-
mension of the oriflce, from the orifloe itself. In a circular orifice,
St about half the diam diet; and ordinarily its area is about .62 or nearlv % that of the orifloe itself:
At this point the actual mean wl of the stream is very nearly (about »7)'the theoretical vel given bj
m^w- i« ^jj^ henoe the actual dUch* are but .62, or nearly % of the theoretical ones.

To find the actual discli Into air.t tfiroiifl-li either a

Fifir.ll.

Table 10,

Ocue 1. ^ _

Circular or rectilinear § opening: In a thin Terit iMane'parti-

Fiff.lS.

* We believe that these rules for thin plate are rIro sofliciently approximate
for most practicAl purposes, if the opening be in the bottom of the reservoir;
or in an inclined, instead of a vert side.

t When the side of a reservoir, or the edge of a plank, fto. over which water
flows, has no greater thickness than this, the water is said to flow through,

or over, thin plate, or thin partition.

t Should the disch take place under water, as in Fig 12. both eurf-levela re-
maintnff conatant. then the head to be uned is the vert diflT so. of the two
levels. After making the calculation with this head, we should, according te

Weisbaoh, deduct the i^ part; inasmuch as he states that the disch is that
mnch less when under water, than when it takes place freely into the air.
Other experimenters, however, assert that it is precisely the same in both oases.
^ If the shape of the opening is oval, triangular, or irregular, the head
must be measured vert from its oen of gray.

642

HYDBAULICS.

41oii, wben tbe contraction Is complete ; and when the sarf-
leirel, «, remains constantly at the same |ielirl><; water beincr
supplied to the reservoir as fast as It runs out at the open-
ing.*

RuLB 1. When the head, meuured vert from the center (or rather from the oeu of fraT) c, of the
opening, to the surMevel « of the reaervoir, is not leaa than 1 ft. nor more than 10 ft ; and when the
MMt traasrerae dlmenaion of the opening ia not leaa than an inch, malt the theoretical yel in ft per
aeo doe to the head, (Table 10, ) by the ooefflcieut of diach .82. The prod will be the aetaal

mean Tel of the water through the opening. Mult thia vel bj the area of the opening in eq ft; thf
prod will be the diach in cob ft per aeo, approzimatelj.

When the head ia greater than 10 ft, aae .6, instead of .62.

Bulb 2. Find the aq rt of the head in ft. If alt this aq rt by 6 ; the prod will be the vel in' ft pei
iCC ; whioh mult by the area aa before for the diach.

Ex. What will be the diach through an opening in complete contraction, whose dlmenaione are^
ins, or .5 ft vert ; and 4 ft hor ; the rert head above the oen of grav of the opening being constantly
• feet?

By Rule 1. The theoretical vel (Table 10, ) corresponding to 6 ft head, la li>.7 ft per aee. And

19.7 X -62 = 12.214 ft, the reqd vel. Again, the area of the opening = .6 X 4 = 2 aq ft; and 13.2U X
2 = 24.128 cub ft per aec ; the diach.

By BuU 2. The aq rt of • = 2.46 ; and 2.46 X & = 12.26 ft per sec, the reqd vel; and 12.26 X 2 =i
'24.6 cub ft per aec, the diach.

Both Tcry approz eren if the orifloe reachea to the anrfaoe of the isaaing water.

Rem. 1. The coef .62 Is a mean of results of many old experimenters

In 1874 Genl. T. G. Ellis of Masaaobaaetta conducted an elaborate aeriea (Trana Am Soc 0 B, Fe^
1876) on a large scale, the general reanlta of which, within leaa than 1 per ct, are given in the follow<
ing table. See also Rem 3. The aharp-edged orificea were in iron platea .25 to .5 inch thick.

Orifice.

Head aboVe Center.

Coef.

2ft8q.

2. to 3.6 ft.

.60 to .61

2 "long, 1ft high

1.8 to 11.8 "

.00 to .61

2 " long, .5 high

1.4 to 17.0 "

.61 to .60

2 " diam.

1.8 to 9.6"

.50 to .61

Rem. 2. Extreme care is reqd to obtain correct results; but for man^

purpoaes of the engineer an error of 6 to 10 per ct ia unimportant.

It will rarely happen that greater aoeuracy U required than may be obtained bj the foregoina
rules; but when such does occur, aid may be derlTcd ft-om the following table dedUCCfi
from the experiments of licsbros and Poncelet, on openinga « ins

wide, of diir heights, and with diff heada. Use that coeff in the table whioh applies to the caaa, in-
stead ef the .62 of Bule 1. In some of the cases in this table, the upper edge of the opening la
nearer the surf-level of the reservoir than IH times its least transverse umenslon.

TABIiE 12. (Toefliclents for rectangular openlngrs In thin
Tcrtlcal partitions In full contraction.*

Head

Head

The breadth in all the openinga = 8 inchea.

above cen.
•f grav. of

above oen.
of grav. of

HEIGHT or OYVNnra.

opening

opening

Ins.

Ins.

Ina.

Ina.

Ins.

Ina.

Ins.

in Feet.

in Inchea.

8

6

4

8

2

1

.4

.033

.4

.8

1

.70
.00
.08
.08

.0666

.06
.04
.64

.0838

.125

.61

.1666

2
2H

.60
.61

.62
.62

.64
.64

.08
.07

.2083

.69

.250

8

.60

.61

.02

.64

.07

.2917

SH

.67

.60

.61

.02

.64

.60

.8833

4

.68

.60

.61

.08

.64

.00

.8750

*H

.66

.69

.60

.61

.08

.64

.06

.4167

6

.67

.69

.61

.62

.08

.64

.00

■OODD

8

.69

.60

.61

.62

.08

.64

.06

1

12

.60

.60

.61

.62

.08

.68

.64

8

36

.60

.60

.61

.62

.62

.08

.08

6

60

.60

.60

.61

.61

.02

.02

.02

10

120

.60

.60

.00

.00

.00

.01

.61

Rbm. 8. Careftil experiments on openln^rs 4W^ft wide, and IS
Ins hlirh. under heads of from 6 to 15 ft, show that the coeff .62 will giro raoulta
correct within -^ part, for openinica of that wise also, under large heada ; although the thickness aS
the partition varied on Its diff rides, from 12 to 20 ins. It must be recollected, however, that nothlnf.
more than close ttpproxttiuUiotu are to be attained in such matters.

Rem. 4. It has been asserted by some writers, that when two or more
eontiifiMUS openlnirs are discharginff at the same time from the same reser-
Toir, they diseh less in proportion than when only one of them is open. Other experiments, hevr*
ever, seem to show that this is not the case ; it is therefore probable, at least, that the diff, if anTi
la but trifling.

* See first footnote bn preceding pags.

HYDBAULIC8.

643

Ckut 2. Tbe dlaebarse ibrooiTli tliiii vert partiUona in com*
plete contraction, when tlie ■urTace-Ievel, «n, Fiy IS, descends
MS tlie water flows out into tlie air. In this case, if the reservoir is
priamfttio, that U, if iu hor Motiont ar« evtrjwbere equal : and ir uo watar !■ flowing into the reser*
voir, to tapply the plaoo of that whioh flowi oat, then, to find the time reqd to disoh the reeervoir.

Bvis. Inasmaoh as the time in whioh eaoh a reaervoir ontirelj disohargea itaelf, is twice that in
wkiAh Um same quantity would flow oat nnder a oonaiaat head, aa in Case 1, therefore, oal-

ealate the diaoh In eab ft per aee by Bale 1, Art V ; dlv the namber of eub ft oon*
tained in the reaervolr, above the level g of the bottom of the opening. Fig 13, by
this diaoh ; the qaot will be the namber of eee in whioh a Tolame equal to that in
the reaerroir, to the depth a. woaM run out in Caae 1, of a eotutant head. And
twies this namber will be the seeonds reqd to empty the reservoir iu Caae 2, of a
varying head. .

Bbm. If it ahonld be reqd to find the time in whioh anch a prlsmatie reaerroir
would parUy empty itaelf, aa, for inatanoe. from m to », Fig 13. ftrat ealealate, by
the above rule, the aeea neoeaaary to empty it if it had only been llUed to « ; and
afterward ealealate as If it had been filed to m. The dlff between the two timea
will evidently be the tine reqd lo empty It from m to n. If the opening la not ia
oomplete oontraotlon, aee Arte 11, Ims.

If tiiediscli is into a lower reservoir, wiiose
surf-level remains constant, proceed in the same manner;
only use the dtff of level of the two aarfa aa the head, and afterward (aoeording

to Welabaoh) inoreaae the time -Jw- part.

Art. 10. Disch from a reservoir R, Fly 14, tlie snrf*level, «,
of whicb remains constantly at the same height; thronjirh
an openinfT, o, in thin vert partition \ and in complete con-
traction; but entirely nncler water; and into a prismatio
lervolr, n».

Fiflr.18.

Seconds required ^beight.o ^ hor area o

to diaoharge a qaantity = V ,„ ^ ^ m in aq ft
c d «, Ma UtHl a ramatnifM s: —

eenaCiU.

area of opening ^ «« v «i»
o in aq ft ^ •* ^ ^'^

^/height a e y hor area of ^ .

Becsonds rei|nired ^ ^ lafi ^twinaqft^*

a rmUt level In m from e to • ~

Seconds required (^^^r^Tft ) ^ «• i« 3q ft ''»

t» ruin laMi to « fh)meto = V botttoll/ ^h^ —

mvt oMer level, d. Area of opening n/ gv v gJi

• to aq ft '^ •• '^ ojw

[. 1. If it should be reqd to find the time of flllinff n», flrom
its bottom e, up to d, we may do so Tery approximately by calculating by
the flrat role to Art t. the time reqd fh>m « to the center of the opening o, aa if all that portion of
the diaeh took plaee into air; and aAerward, ftrem the oentor of the opening to d, by the rale Juat
given. This oaae ia aimilar to that of fllliag a leek from the eanal reaoh above, in whioh the anrf-
level may be eenaldered oenatant.

Bbm. 2. If the bottonk of ^e openiuir <>• should coincide with
the bottom of the reservoir, then the coeff will become greater than .62.
Bee Art 11, fbr obtaining ooeffa for imperfeet oontraotlon.

Rem. 3. If the opening, instead of beluff in complete con-
traction, is of any of the shapes Figs 6 to 9, then a reference to Art 8 will show
what eoeif muat be anbatltated for M.

Que 3. Disch from one prismatic reservoir, Vig 10, W, into
another, X, of any comparative sises whatever, throuirb an
openinfir o, in a plane thin vert partition, and in complete
contraction; when the water rises In X, while it fklls in W.

ji 2b JInd the Hme in wMch the water, flowing from W into X, Ikrou^k

-» — « .» o, wittfM ikrovigh the diet a a, ae oa to etmnd M the eame Unel a o. in

.^ _, _ hoth reeervoire.

■JB — a ■ In thia eaae, the water reqd to fill Z f^m a to 4, (d being the bottom

of the opening o,) flows out into the air ; and the time neoeaaary for it

S H C B to do ao, most be calealated separately twm that reqd above d, which

flows into water.

-»y ■ -»> H Bvui. First from a to d. Find the hor area of each reservoir, in

W i .J^ ■ iq ft. Mult the bor area of X, by the vert depth de in ft, for the onb

ft oontalned in that portion. Div these oub ft by the hor area of W.

The quot will be the dlst a m, in feet, tbrongh whioh the water in W

— 0 1 ■ most deaoend, in order to fill X to d.

Fiff.15.

Seconds
quired to low-
er f^m a to m, and
raise fh>m eVod.

Wii^ft^ V loft -^ inft /

Area of opening ^ «« w a ao
• in sq ft X .ea X 8.03

644

HYDRAULICS,

Seeonds required

to lower from m to «, and imiM
from d to 0. (Verj approx)

lor area of ^ tuiee tho hor ar«a ^ «/liMd m i
Xlnaqft ^ ofWlnsqft ^ ^ toft

ft/

Area of
opeoing
•uwift

(hor area hor
of W -^ of X
in iq ft

In eq

X .il X 8.W

Bx. Letthehorareaof WbelOOeqft: andthaiof X.eOwift. Let an be 80 ft; and mm 16 ft|
and the area of the opening o, 8 aq ft. la what time will the water deeoend tnm • to «, and xiw
Itame toe?

Inaamuoh as the method of finding the time for filling flrom « to d, by the water falling ftrtm a to
m, requires no farther exemplification, we will oonfine oureelvet to the addUional time neoessan for
filling from d to e. by the water falling from m to «. To find thia, we hare, the eq rt of the head

4 ^ 100 X 00 X X 4MI00

mn = 4ft;andthe.umofthe2an»M = 100+«0 = 160. Henee. j^—^j^^ -- = ^^=

ao.l see; the additional time reqd, Tery approximately.

NoTKl. If the opening, as (i,Fls^ 16, reaelies
to tlie very bottom of the reservoirs, we may

oonsider all the water flowing from R into T, aa flowing into water.
Therefore, using the head am, we at onoe oalcalate the time neoeaaary
for the water in the two reaervoira to arrive at the aame level « e, by
the laat prooens of the preceding role ; or, in other worda, by the pro*
oeaa given in the preoediog example. Bat in thia caae it muat be borne
in mind that the opening o is no longer in complete contraction, inas«
much aa the oontraotioa along ita lower edge ia aoppreaaed.

The diach will oonaequentiy be aomewhat increased; and a ooeff
greater than .62 beoomea neoeaaary. The method of finding thia, ia
given in the following Oaae 4. A refsrenoe to Art 8 will give the ooeflT
to eaae the opening ia ahaped aa Figa 6 to 9.

Art. 11. Case 4. The discharge throuarh openings in plane
thin vert partitions ; bat in ineompleie eontractlon.

The opening may be each that contraction will take plaoe
along one portion of ita perimeter, or at the top of the open-
ing a. Fig 17 ; while it la auppreeaed on another portion ; aa
at the bottom and two enda of the opening a; where aupprea-
•ion ia oauaed by the addition of thort aide and bottom piecea
0, 0, c. Or it majT be oauaed by the bottom, or enda, or both,
ooinoiding with the bottom and aidea of the reservoir. In
anoh caaea the diach will be greater than in those of complete
oontraotioQ ; but less than in those of full fiow ; inasmuch as
the opening now partakes somewhat of the character of the
short tubes of Art 8 ; and the ooeif Will rise from .62, or that
which u$ualln pertains to openings in full contraotion ; and
will approach .8, or that of rail flow, in proportion to the ex-
tent of perimeter along which contraction is suppressed : or
even to .9 or .96 by the use of such openings aa are ahowu by
Figa 7, 8, 9.

FiflT. 16.

. Fig. 17.

To And approx luiateiy a new eoeflT of dlsch; and the dladi
itself, in cases of Incomplete contraction.

RcLB. Firat find by the foregoing roles, what woald be the diaoh in the partieolar eaae that may
be under consideration, supposing the contraotion to be/)omplete. Then dlv that portion of the
parimeter of the opening on whioh oontraotlon is rappreesed, by the entire perimeter. Molt the onot
by the dec .152 if the opening is rectangular, or by .128 if oironlar. To the prod odd ani^, or 1. Oall
the sum, g. Then Ray, as unity, or 1, is to 9, so ia the ooefffor complete oontraotlon in ordinary eases
(usually .62) to the reqd new coeff. Finally, repeat the original oaloulaUon, only aubetituting this new
ooeff In the plaoe of .62.

Aooording to thia rule, we have the following ooeff of diaoharge for reetangnlar openings within pro-
bably 3 or 4 per cent, when oontraotlon is not suppressed on more than X of the perimeter. The tneo-
retieal discharge multiplied by the oorresponding ooeff will give the aetnal discharge. Vhea the eon*
traction is carried farther, the ooeff becomes extremely irregular, and ia probably u&ieterBinahle.

ForcompUt* contraetUm (ordinarily) fit

When contraction it ngi^ened on % theperimttor .64

., << al •• " a '* " 67

t> <4 tt li <i a^ It II ^ «n

'• " " *' entlrelif around the orOue'.., '..'..,..'. .90

Intermediate ones can be estimated nearly enongb, mentally.

Rem. 1. When, instead of a short spent, as in Fi|r 17, the"
opening is provided with an Indefinitely lonur i>or troofrh,

similarly attached, and open at top, there will be no practically appreciable diminotloa of dieeh Mow
that through the simple opening as at a, Fig 11 ; provided the head measured above the oen of grav
of the opening be at least as great as 2 or 2^ times the height of the opening itaeir. Therefore, under
such cironmstanoef the disoh may be calculated by the rules in ArtV. Bat with smaller heads the
disch diminishes considerably ; so when the head above the center beeomes bat as great as the height

of the opening, it will be but aboat ^ of the calculated one. With still smaller heads, the flow

becomes less maoh more rapidly ; bat has not been redaeed to any mla.

Bbx. 2. If, instead of belnff hor, the trough is IlTCIilH'EB

HYDRAULICS.

645

tm mneli as 1 in 10, the diach will be inereaaed very slightly, (some 8 or 4 per
eoBt) over tbat oftloalatad by the role* in Art 9, for the plain opening. Theeo reemlte were oMeiaed
bf experiments on a very email eoale ; and should be conildered ae mere approadmationB.

Art. 19. In a ease like Tig 18, where contraction is sappoaed
to be ■appresBcd at tbe bottom, and at botb Tcrt sides of tlie

opening o, in consequence of their coinciding
with tbe bottom and sidee of the reeenroir ; bnt where the
front of the reservoir, instead of being vert, is sloped as at/;
and when the water, after leaving the opening, flows away
over a slightly sloping apron, g, then the disch in cob ft per
MO may be approximately foand by Bole 1, Case 1, Art 9,
only safaatitnting .8 in plaoe of .62, when / slopes baek 45°,
or 1 to 1 ; or .74 when / slopes baek 6il°, or with a base of 1
to a rise of 2. In such eases of Inclined fronts, the height of
the opening must be measured vart, or rather at right angle*
to the Jloor 0/ th4 reemrvoir: and not in a line with the
sloping IVont.

Bnc. Wlien tlie firont, /, of tlie reservoir is vert, and a slopinsr
apron or trouKli, g^ is used, having its upper edge level with the bottom
of the opening, the disoh is not appreoiably diminishea below that whloh takes plaoe ftvely into the
■ir, provided the head above the een of grav of the opening is not less than fhua

18 to 24 ins, for an opening 6 to 9 ins high.
U to 18 " " " ** 4 ins high.

g M u i« 2 ins or less, high.

Art. 18. To find, approximately, the time read for the emp-
^IniT of a pond, or any other reservoir, as Fly 19, which Is
not of a pnsmatic shape; thronffh an opening, m, near the
bottom.

BuLx. First Moertain the exact shape and dimensions
of the reservoir. If large, and irregular, it must be care-
fully surveyed ; and soundings taken, and figured upon a
oorreot plan and oross-seotions. Next, consider the entire
boitr of water to be divided into a series of thin hor strata,
A, B, 0, D ; the top line of the lower one being at least a
few ins above the top of the opening n. It is not necessary

^- . that these strata should be of equal thickness; although

MTn \Q ^ — .. the thinner they are, the more oorreot will the result be.

^^ The depth of the lower one, D, will vary to some extent

with the height of the opening ; those next above it should
not easeed abont a foot in thickness, until a depth of 6 or 8 feet is reached ; then they may oonve-
lyently, and with sufficient aoonraoy, be inoreased to about 2 ft, for 6 or 8 ft more ; and se on; be-
eeming thioker as they approach the surf. Bv aid of the drawings, calculate tbe content of each
stratum in cab ft. Now, sinee the strata are thin, we may, without serious error, assume each of
them to be prismatic, as shown bv the dotted lines ; and may assume that the head under which each
stratum (exoept the lowest) easpties itself through n, is equal to the vert height from the eenter of
the opening to the center of the stratum. Thus, m n will be the head of A ; w n. the head of B ; xn,
the head of C. Then, for tbe stratum A, by Bnle 1, Art 9, (only using mn as the head instead of on,}
And instead of the ooefT .62 of that rule (whieh can only be used if n Is in complete contvaotion) using
.84, or whatever other coeff near the end of Art 11 applies to the case, calculate the disoh in cub ft
per see. Div the content of the stratum A by this disch, and tbe qoot will be the number of seo reqd
for dlsoharging A. Using the head wn, prooeed in precisely the same way with the stratum B ; and
■sing the head sn, do the same with G. Finally, for the lower stratum D, find by Bule 1, Art 9, (with
the same oantion as before respecting the proper coeff,) in what time ft would empty itself under a
•e«M<«n< head equal to yn, measured from its ntr/to the oenter of the opening. DovM* this time will
be that reqd to empty itself in the case before us. under its varying liead. Finally, add together all
tfeese separate times ; and their sum will bo tbe entire time reqd to empty the pond, or reservoir, ap-
proximately enoogh for practical purpoi

L

35

. HYDRAULICS.

far veaterD miolng 8tit«a. It it flxpr#fl«ed Ja ti:rDi9 at a btandv^ oriflcep usuallj

^ O^wL ™ fi = 0.188 U. sl^ls ^r"^!'™"?!!)?^ pe? niS"l')w"i''ri = llflfi
gals; per dav, 2,111 cu ft - 16,791) gala, UBnuIlt Ihs griOw is of flied deplh

MiDddbyaiK^b^^stment. Thenlitlon b«lw«D ireaand ibspeaf orlflee ami
thediBcaATgc perHq Id uf area Lb indicat«d by the following uble:*

Illicliarga, io cu n per mln, per square incli of apeniag, under fi Ids h«sd.
Leoglh of opflolDg, Idi
OpeniDg2iD»hj^h, . .

* CoDdenaed from olroi

if Palion Water Wheel Co.

HYDRAULICS.

547

14 (a). On the dlMsliarire of water over welm or over*
flills. The weir affords a very conrenient means for gauging the flow of small
streams, for measuring the quantity of water supplied to water-wheels, etc.

(b) A measuring weir is always arranged with ite back, or up-stream side, a 6,
Fig. 20, vertical, aud as nearly as may be at right angles to the direction of iiov
of the streauL The ends, a A, a A, Figs. 21 and 22, are vertical, and the crest a a
is horiaontal.

(e) End eontraetlons. When the weir a a extends entirely across the
diannel of approach, as in Fig. 21, so that its ends a A, a A coincide with, or form
portions of, the sides «« of the channel, contraction (Art. 9, p. 541) takes place
only on the lop and bottom of the sheet of water passing over the weir, as at
m e and at a, Fig. 20, and is entirely " suppressed " at the emit, so that the water
flows out as shown in Fig. 21 a. Such a weir is called a stii^pressed weir,
or a weir witbont end eontraetion. But when, as m Figs. 22 and 22 a.

JB:ig.2Sa

the ends a A, a A are at a distance trom the sides stot the channel or reservoir,
oontraetion takes place at the ends of the weir, as shown at a and a, as well as
over the crest. Such contraction diminishes the discharge. A weir of this kind
is called a weir wttk end eontrnctionfi.

Other things being equal, the extent of the contraction, and its effect upon the
discharge, increase with the head H. When the length a a or L of the weir ex-
eteds about 10 times the head H, the effect of the enacontractions upon the di8<
•barge Is nearly imperceptible ; but as the length diminishes in proportion to
the head, the effect of the contraction increases rapidly. Mr. Francis (Art. 14 m)
fMind that when L »= only 4 X H, the discharge was reduced 6 per cent, by com-

eete end contractions. In view of the uncertainty as to the effect of end eon-
actions, it is better tcHivoid them and to use weirs, Uke Fig. 21, where the con-
traction is suppressed ; but if end contraction is permitted at all, it must be made
complete;* for the coefficients given do not apply to cases of incomplete
eontraetlon, <.«., with contraction only par^j/ suppressed.

(<Q In a weir without end contmetlon, care must betaken that the
air has fne aeceu to the space {to. Fig. 20, or 22 h) behind the
fiUling sheet of water. Otherwise a partial vacuum forms there,
the sheet is drawn inward toward the weir, and the discharge
is greatW modified. At the same time, the sheet should be pre*
▼ented from expanding lateraUy as it leaves the crest. Botn of
these obiects may be attained by prolonging the upper portion
only of both sides of the channel a little way down-stream
beyond the crest and the upper part of the falling sheet, as in r^A^

Fig. 22 b. Mr. Francis found that such projections, by confining Wig,texD
the sheet laterally, diminished the discnai^ about 0.4 per cent.

(e) Ordinarily the crest is " in thin plate '* or " In thin partition " (see foot-
note t, p. 541), so that the sheet passing over the weir touches it only at the very
corner, a, Fia. 20. A rmtnded corner increases the discharge, as does the round-
ing of the edges of an orifice (Art. 8, p. 541), and a crest sufficiently wide to de-
flect the falling sheet diminishes the discharge (see coefficients for this case in
Table 16, p. 564), but both forms introduce much uncertainty, and should there-
fore be avoided.

♦The contraction is said to be "complete" when it is practically as great as It
could be made by any further increase of the distance a «, Figs. 22 and 22 a ; and thia
is believed to be attained when a « is made equal to the head H.

548 HYDRAULICS.

(/) The lengtli I< of tbe erest. Figs. 21 to 22 a, shoald be at leut three
times the head H, in order to reduce the eflfect of friction of the sides s s and
that of end contractions where such exist. Tbe helg^ht p, Fig. 20, of the
rertical back a 6 in contact with the water should be not less than twice the
^ead H; for, in order to reduce the velocity of approach (see Art. 14 u), the
crossHsection of (lie clmnnel leading to the weir should be large in propor*
tion to that of the stream a c. The cross-section of the channel of approach
should be as regular as possible.

(g) The weir should be stoutly built, as Tlbrations of the structure may
seriously modify the discharge.

(h) Theoretically, the head is the vertical distance H', Fig. 24, from the

crest a to a point o' where the water is perfectly still, and the surface therefore
horizontal. But in fact the head is usually measured from the crest a to a point o
a few feet back from the weir, where the water is only oompcu-cUively still, the
velocity of approach being perceptible. (See Art. 14, u.) The difference between
the head H actually measured and the head H' to stiU water is usually very
slight. It is gr^tly exaggerated in the figure.

The correct nieasarement of the head is a delicate matter, the dis-
charge being increased or diminished about 1^ per cent, by 1 per cent, of in-
crease or diminution of the head. Waves or ripples and other disturbances of
the surface, and capillary attraction, are the chief sources of error..

(i) To avoid the latter difficulty, tbe hook-graave is used for measuring
the height of the water surface in important cases. This consists of a long grad-
uated rod, provided at its foot with an upturned hook or point, and sliding
vertically (^by means of a screw motion) in a fixed support, to which is attached
a vernier indicating on the scale the neight of the point. The sliding rod is
first run down until the point is well below the surface, and then gradually
raised by means of the screw until the point just reaches the surface, which it
indicated by the first appearance of a " pimple " in the water surface imme-
diately over the hook, tfnder favorable circumstances a good hook-gauge may
be read within from .0002 to .0005 foot.

(J) To avoid inaccuracies due to the dlsturbanee of tbe surface by

the current, by wind, etc., the level is sometimes taken (with the hook-gauge oz
otherwise) in a side chamber which communicates with the main channel of
approach. The surface in the chamber maintains the same level as that in the
channel itself, but is comparatively tree from disturbance- Or a bucket oom-
municating with the channel by means of a pipe, can be made to serve in the
same way. Either may of course be sheltered from the wind. Caation*
Messrs. Fteley and Stearns found that when the bucketor chamber communicated
with the water w.ar t/te bottom and close behind the weir, the head thus obtained was
generally somewhat greater than that found by measurement near the surCaoe
and 6 fe.et back from the weir. But Mr. Francis found^^the difference scarcely
perceptible.

(k) Great care is necessary in adjasting^ tbe boob-grangre for tbe

beigpht of the crest; for any error in this affects all the subsequent experi-
ments. The hook is usually adjusted to the height of the surface when the latter
just reaches the level of the crest ; but this method is rendered inaccurate by
capillary attraction at the crest. A more accurate method is to have, in addition
to the hook-gauge, a stout ^6d hook, pointing upward, the level of which, rela-
tively to that of the crest, may be ascertained by means of an ensineer'B level,
holding the rod on the crest and also on the point of the fixed hool. The water
surface is then allowed to fall slowly until a " pimple " just appears over the fixed
hook. It is then kept at that level and the hook-gauge adjusted acoordin^y.
Or if the gauge-hook is a stout one, the levelling rod may be set at once upon its
point without having recourse to a fixed hook. It is better to adjust the hook-
gauge .so as to read zero for the crest level, which is thus made we datum ; for
the reading of the hook-gauge for the water surface then gives the head H at
once, and without subtracting the height of the crest

HYDRAULICS.

549

1) Fonnalv for weir <llaeli«iv«»

ffl

^ = the actual discharge orer the weir, in cubic feet per second ; *

a' -B the theoretical discharge orer the weir, in cubic feet per second',
[ =11' t = the vertical distance or head a tn^ Fig. 24, p. 556, in feet,* measure
from the crest a to the horizontal surface o' of tHU water up-stream from
the weir;
= the length a a of the weir, In feet,* Figs. 21 to 22 a ;
"= the acceleration of gravity = say 32.2 feet * per second per second^

_ - ^ - ,, , _ actual discharge Q

= ooefllcient of discluurge «* rr r- — . «, . = ^, ;

^ theoretical discharge Q'

2

-8*»

I.

» —

3

o ^Tg» m yTg — say 5^ « — say 8.025 m.

t

Then, tor the theoretleal dlscliar|:e» we hare

4|'-|LH|^?irH;< (II

and for the actual dlsebaiv^*

= |cLHi/?7H . . . . .

= m L H V^FB.

» X L H /ff = X L >/H«"*

hI

(2)

(8)
(4)

See foot-BOtea

For tlie value of the eoeflielent («, m, or x t) we have recourse to
experiment, measuring the actual discharge and comparing it with the theoret-
ioal one, as in ttie following articles.

* The formulsB apply equally to any ^stem of meaaorea, as the Xuglish, the metric,
ete. It is lequisite merely that the units, of length, of time, etc., used, be the
same throughout. In metric measure, g «» 9.81 meters per second per second.

t For thepretent we suppose the head to be measured to ttHl water, so that H s= H'.
When this is not the case, see " Velocity of Approach," Art. 14 («), etc.

X It will be noticed that the formnlse (2), (3) and (4), with their corresponding coef-
ficients, «, m, and x, are really identical, differing only in form. The last is the most
convenient in practice, but all are met with in works on hydraulics.

g When water issues, under a head H, from a horisotUal orifice In the bottom of a

vessel, the theoretical velocity (Art. 7, is = V^lfS; and this may be regarded

as true aLw for vertical orifices in the $ide$ of vessels, provided the head H to the
center of gravity of the orifice is at least two or three times the vertical dimension
of tibe orifice ; for in both cases the theoretical velocities through the several parts
of the orifice may be taken as equaL But when a vertical orifice is nearer to the sur-
fitoe, or when it reaehet to the surface as in the case of a weir, we must take into con-
sideration the differences in the velocities with which the water issues from points at
different depths.
Theoretically, the particles pass the oblique plane ao\ Fig. 23, in horizontal lines,

with velocities (= i/TJT^ 8.025 |/S)
proportional to the square roots of
fheir several vertical depths h (not
Indicated In fig.) below still water
furfiBkce at o'. Ijierefore if from a m
we imagine horizontal lines a a\
d d\ vr/^ecfy etc., etc., to be drawn,
r^reeenting all these velocities to
any scale, then the outer ends a\
4f, v', e\ etc., etc., of these lines
wiU form, with a m and aa\ a
parabolic segment amc' a\ the area
of which is :

2 2 2

area » — area of rectangle a n (see Parabola, p. 192) = -- amX^ aa' = H ^TgW;

3 So

and this area In square feet, multiplied by the thickness of the escaping sheet of

FiS.23

650

HYDRAULICS.

(m) Mr. James B. Franeis * experimented at " the lower looka,'* Lowell,
Mass., in 1852, with weirs lu feet long, 5 feet and 2 feet high, under heads from 7
to 19 inches. To apply his results, tne following conditions must exist:

The head H, Fiff. 20, must be between 6 and 24 inches. The height i> of the
yertical back of tne weir above the bottom b of the channel must be at least
twioe the head H. The crest a must be " in thin partition " (foot-note t p. 641),
and its length L, Figs. 21 to 22 a, must be at least 3 times the head H. The ends
ah. ah must be vertical, and, when there is end contraction, " in thin partition."

When there is endcoutraction. Mr. Francis first deducts from the actual lengUi
L of the weir one lenlh-\ of the head H for each end where contraction oocur&
Thus, if N = the number of end contractions (two in Fig. 22),

Q-.(L-n-|)HKH-x(L-n-^)H*t (5)

InFIg.22,Q-«(L-y) KVKf ^t ^L-y) H*

But within the limits speoliied above, the formula is very approximate vfWkout
correction for end contraction, provided the length L of the weir Is at least 10
times the head H ; and within 6 per cent, of the truth when L is = 4 H. When
there is no end contraction, of course no such correction is required, and the

4
formula remains Q a*x L H yKX =* « L H *

Mr. Francis yiwes x = 3.S3 tor feet; § or

».< «. a — V. A _xi- number of ^ head H\ ,, _# ^

»i.eta.rc..a.»x (length - ,„d eontr«^ns ^ -~1^) ^ ^'^

the mean of his 88 experiments being 8.3318. The least value of x obtained bj
him was 3.3002, or 1 in 112 less than 3.33; and the greatest was 3.3617, or 1 in 105
more than 8.88. Hence, with x » 3.88. the formula will give the discharge for
each of his experiments within 1 per cent. In 67 oat of the 88 expertments
X ranged between 3.32 and 3.35, and in 53 between 3.82 and S.84. When s is 3^
m is = 0.415, and c is => 0.622.

The height of the surface was measured six feet hack from the weir by two
hook-gauges, one on each side of the channel ; and the mean of their readinge
was used In calculating the coefficient x.

water, or length L of weir, in feet, gives the theoretical discharge In cubic feet per
second. Or

Q' - LX arMiamo'a' - L X -|h i^STS:

8

Hence, area a m e' a' in sq. ft. reprtsents the theoretical disoh. In enb. ft. per eea over
1 ft. length of wei r, under head H. The theoretioal metm vet. through the section a e' ie
_. , theoretical discharge Q' 2 _ _. ^ -_ 2 _

q' tn, ^ or two thirds of the theoretical hoii*

mg.tis

V^5^

Bontal velocity a of of the partMee
passing immediately over the weir.
As in the case of orifices (Art 9,
p. 541), the actual vel. at the •moBerf
eecUon of the sheet after passing tha
weir (corresponding to the ^'vena
contracta ") is probably very nearly
equal to this theoretical Telocity.

♦"Lowell Hydraulic Experi-
ments," Van Ncstrand, New "l^)rlL
1883.

t In Messrs. Fteley and Btearns' experiments this fignre was not constant at 0.1flL
but varied between 0.061 and 0.124, generally increasing as the head decreased.

X We here suppose the head to be niessured to the surface of «M} water, so that H
and H' (see Art 14 fc, p. 648) are the same. See Velocity of Approach, Art. 14 (a),

g Since 1 meter — 3.2808 ft, the value of x for metric measure corresponding to Mr.
frands* 8.33, Is «i 8.83 ^ ^8.8806 >« 3.83 -i- 1.8118 'm 1.88&

HYDRAULICS.

551

Table )3.* ]>isoliarare in enbic feet per seeenil Utv eaeli foot
in lennrtli of weir in tmn plate and without end contraction, by the Francis

formula: Bischar«^e, Q =-- 3.33 L fit = 3.33 L H ^ff. .

Very approximate also when there is end contraction, provided that L is at
least = 10 H ; and but about 6 per cent, in excess of the tfuth if L » 4 H. Mr.
Francis limits the formula to beads H from 0.5 foot to 2.0 feet, but no serious
error will result from using the table for any of the heads given. For irelrs
of other leng^tlis than 1 foot, multiply the tabular discharge by the actual
length in feet. .01 foot = 0.12 inch = scant ^ inch.

Head, H,

Gab. fU

Head, H,

Gab ft.

Head. H,
in ft.

Gab. ft.

Head, H.

Cob. ft.

Head, H.

Cob. ft.

iBft.

p«r8«Q.

inn.

per seo.

per aeo.
3.380

in ft.

per wo.

In ft.

per MO.

.01

0.003

.51

1.213

1.01

1.61

6.179

2.01

9.489

.02

0.009

.52

1.249

1.02

3.430

1.52

6.240

2.02

9.560

.03

0.017

.53

J.285

1.03

3.481

1.58

6.302

2.03

9.631

.04

0.027

.54

1.321

1.04

3.532

1.54

6.364

2.04

9.708

.05

0.037

.55

1.358

1.06

3.583

1.66

6.426

2.05

9.774

.06

0.049

.56

1.395

1.06

8.634

1.66

6.488

2.06

9.846

.07

0.062

.57

1.433

1.07

3.686

1.67

6.661

2.07

9.917

.08

0.075

.58

1.471

1.08

3.737

1.68

6.613

2.08

9.989

.09

0.090

.59

1.509

1.09

8.790

1.69

6.676

2.09

10.062

.10

0.105

.60

1.548

1.10

3.842

1.60

6.739

2.10

10.184

.11

0.121

.61

1.586

1.11

3.894

1.61

6.808

2.11

10.206

.12

0.138

.62

1.626

1.12

8.947

1.62

6.866

2.12

10.279

.13

0.156

.63

1.665

1.13

4.000

1.63

6.980

2.18

10.862

.14

0.174

.64

1.705

1.14

4.053

1.64

6.994

2.14

10.426

.15

0.193

.65

1.745

1.15

4.107

1.65

7.058

2.16

10.498

.16

0.213

.66

1.786

1.16

4.160

1.66

7.122

2.16

10.571

.17

0.233

.67

1.826

1.17

4.214

1.67

7.187

2.17

10.646

.18

0.254

.68

1.867

1.18

4.268

1.68

7.251

2.18

10.718

.19

0.276

.69

1.9Q9

1.19

4.323

1.69

7.316

2.19

10.792

J»

0.298

.70

1.960

1.20

4.877

1.70

7.881

2.20

10.866

.21

0.820

.71

1.902

1.21

4.432

1.71

7.446

2.21

10.940

.22

0.344

.72

2.084

1.22

4.487

1.72

7.612

2.22

11.016

.23

0.367

.78

2.077

1.23

4.543

1.78

7.677

2.28

11.089

M

0.392

.74

2.120

1.24

4.598

1.74

7.648

2.24

11.164

Q416

.76

2.168

1.25

4.654

1.76

7.709

2.26

11.239

.26

0.441

.76

2.206

1.26

4.710

1.76

7.776

2.26

11.814

.27

0.467

.77

2.250

1.27

4.766

1.77

7 842

2.27

11.389

Ji»

0.^3

.78

2.294

1.28

4.822

1.78

7.908

2.28

11.464

.29

0.520

.79

2.338

1.29

4.879

1.79

7.976

2.29

11.640

.30

0.547

.80

2.383

1.80

4.986

1.80

8.042

2.30

11.616

Jil

0.576

.81

2.428

1.31

4.993

1.81

8.109

2.81

11.681

.32

0.603

.82

2.473

1.32

5.050

1.82

8.176

2.32

11.767

.33

0.631

.83

2.618

1.33

5.108

1.83

8.244

2.83

11.843

.84

0.660

.84

2.564

1.34

5.165

1.84

8.311

2.34

11.920

.35

0.690

.85

2.610

1.35

6.223

1.86

8.379

2.35

11.996

.36

0.719

.86

2.656

1.36

6.281

1.86

8.447

2.36

12.073

.37

0.749

.87

2.702

1.37

5.340

1.87

8.616

2.37

12.150

.38

0.780

.88

2.749

1.38

6.398

1.88

8.684

2.38

12.227

.39

0.811

.89

2.796

1.89

6.467

1.89

8.662

2.89

12.304

.40

0.842

.90

2.848

1.40

6.516

1.90

8.721

2.40

12.381

.41

0.874

.91

2.891

1.41

6.676

1.91

8.790

2.41

12.459

.42

0.806

.92

2.939

1.42

6.685

1.92

8.869

2.42

12.586

.48

0.989

.98

2.987

1.43

5.694

1.9.S

8.929

2.48

12.614

.44

0.972

.94

3.035

1.44

5.754

1.94

O.UVo

2.44

12.692

M

IjQOS

.95

8.083

1.45

5.814

1.96

9.068

2.46

12.770

AS

1.089

.96

3.132

1.46

6.876

1.96

9.138

2.46

12.848

.47

1.073

.97

3.181

1.47

6.985

1.97

9.208

2.47

12.927

.48

1.107

.98

8.231

1.48

5.996

1.98

9.278

2.48

13.005

.49

1.142

.99

8.280

1.49

6.057

1.99

9.348

2.49

18.084

JiO

1.177

1.00

3.330

1.50

6.118

2.00

9.419

2.60

13.168

* Table 13 is an extension of the " original ^ table published in our first edition,
1872. Most of the values now given are taken, by pemiiasion, from a table published
by Messrs. A. W. Hunking and Frank S Hart, of Lowell, Maas., in May, 1884.

652 hydbauucb.

(n) NeMT*. A. Ftcler Mid F. P. Mcama • eTperimented at Burton,
Miaa., in 187T-79, upcn vein 9 feet BDd 19 feet long, 3 teafi Incbeg sado feete^
inches bigli, and under besda fioco O.a luch u> 19 Inehas. Forwairsin thin par-

u9|i«oifled In (») and (d)), ll

niHharKe, Q - S.31 L hS + 0.007 L 1 ,,,

- 0.«!8 L H jTsH + 0.007 L /' " ■ ■ ■ ^'''■
lu thair expflrimenta, the hasda were meaAured six feet bulk from the weir.
The total TariBtion in the rtlmt of the coefficieuU obtained «ea about 2^ per
cent. Compare foot-not* | below.

(o) H. Basin teiperlnienl«d at Dijon. Prance. In ISSO-BS, with vein from
t about lUtoSUfeol lone, from about B Inches to .1 feat 9 Incbea
S high, and under heode^omaU to 21 Inches. Thelapoftha
w^ iBBbown In Fig. 2^ s. The weirs were placed at iiirrRnnt

While Mr .__ ___ ,

for the effect of velocity of approach |aee Art U vr and vj tij
uodifjing the measured A«ui 11, M. Bailn inelodei it in the
eoglclaa mia the formulaQ ^nLH yTeH.

S Let M ^ the value or m for the case where Telocltr of ap-
~proacb = D. Then, vei'y approxiuialaly :

When Telocity of approach la to be taken into account:

m ^ M fl

,„ „,„„,„ H la the head actusllr measured (o running water,

andplstheheight aAoftbewelr. Fia. 20. Haodprnusti^
course both bemeaiured In the aameunlt, as both InmeWrB.orhoth In [<iet,ett

M. Basin bellevea that eicept In the case of Terj low weirs (which sfaouM ba
STolded) the values of »i given b; formula (7) and In Table U calculated Tnm
It, will be found within 1 per csni. of the truth for welts in thin partition and
without end contraction. If the conditions of his eiperiments aneiactlf repro-
duced, and provided especially that the sheet of wsierls not allowed to eiiMnd
laterally after pasaing the crest (Art. 14 )<f)) and that the air hag free acseaa !•
the space m, Ffg. W, behind the falling eheet of water.

For heads between 4 Inches anil 1 fOot, M. Bailn glvei, as •ulBelenUy »p>
pr«sl|nate,

when there ia no velocity of approach, U •> 0.42S j

w fbr velocity

of approaeh, m = 0.«5 + 0.11 (o^ )?

• TranucUon^ American Society of Civil EnglnHTi, Jan., Feb. and March, IBIa.

IlipCrloncra iHuvclles sur Pfcoulem^nten diveraolr. •Exttall deeAnnales da
Fonts et Chauw»e^ Oct., 1888. Paris, Vve Ch. Sunod, 1888. Tnnslatlon by A.
Harlchal and John C Tiautwine, Jr., pmented to EDgineen' Club of Phibtdelphia,
In 18^ for publication in Its Proceedings.

; This would make 1-3,41 |slncs i ^ « ^2^ - S.OtS n) ; wbemig Mr. Fram^
gfres z = 8.33, which agrees very well with Hessra. Ftelsy and SIsuna, within the
nmits of Art. It (m). Tel M. Basin meaennd the head IS feel back horn tbt
weir, while the other eiperinienti-is meesnred it only 8 feil back, and the sttrhl
Increaae of head Ibus obulntd by H, Badn would of Itself have maJe his eoendut

approach, which In bis chh was mim M to 700 feet long, rectangular and regular in

HYDRAULICS.

553

Table 14» Talnes of Basin's m, in the formula:

Q = m L H 1/2 fjf H.*

2 2 0

The. coefficient, m " ~ c " — . ^,p. 549, being a mere ratio, is independent

of the unit of length adopted ; but BazMs M and m include correction for velocity
of approach. They therefore depend upon the unit in which H is expressed.
Seep. 662.

Using Bazin's m, as given in the table below, we have, for the dlscbarg^e
per second :

Cubic meters - m X length L in met X head H* in met. X 1/2 X 981 H in met;

Cubic /c«< - TO X length L in feet X head H* in feet X V^ X 32.2 H in feet.
It will be noticed that below tbe beawj lines the head H is greater than
^ height j9, and thus exceeds the limit laid down in (f ) and (m).

HMidH,Fig.94,p.66«.

•fa*nra

approximate

■OTBra.

Ibet.

inehe*.

.05
.OS
.07
.06
JOB

.164
.197
.280
.363
.386

1.97
2.S6
2.76
8.15
8.54

JO

.838

8.94

.12
.14

Mi
.458

4.72
5.51

.16
.18

.535
.561

6.S0
7.09

M

.656

7.87

.733
.787

8.66
9.45

.36
.28

.858

.819

10.24
11.02

.80

.964

11.81

.83
.84
.86
M

1.060
1.116
1.181
1.247

12.80
ISM
14.17
14.96

.40

1.SI2

15.75

.43
44

.46
.48

1.378
1.444
1.500
1.575

16.54
17..S2
18.21
18.90

M

1.640

19.69

M
M
M
.58
.60

1.708
1.773
1.887
1.903
1.989

30.47
21.36
32.05
33.88
88.63

Height, p. Fig. 20, or crest of weir above bed of np-stream obanoel.

^■' ■ ..I « I , ■„

. meters 0.20 O.SO 6.40 0.50 0,60 0.80 1.00 1.50 2.00
g ( feet 0.656 0.964 1.SI2 1.640 IMO 2.624 8.280 4.920 6.660 . 8
a(ilMhes7.87 11.81 15.75 19.68 88.62 81.60 89.86 59.07 78.76 °i|

mmmmmmniinin Mf

.456 .453 .451 .450 .449 .440 .449 .448 .448 .4481

.466 .450 .447 .445 .445 .444 .443 .448 .443 .4437

.455 .448 .445 .448 .443 .441 .440 .440 .439 .4891

.468 .447 .448 .441 .440 .488 .488 .437 .487 .4868

.457 .447 .443 .440 .488 .486 .486 .435 .484 .4840

.466 .447 .443 .438 .487 .486 .484 .488 .488 .4332

7B2I .448 .442 .488 .486 .488 .482 .480 .480 ^391

.466 1 .4M .443 .488 .485 .433 .480 .428 .428 .4267

.47lT3ff
.475 .456

.444 .488 .485 .481 .429 .427 .426 .4246
.445 .489 .435 .481 .428 .426 .435 .4839

.480 .458 .447 .440 .496 .481 .428 .426 .428 .4215

.484 .462 i4J»| .442 .487 .431 .428 .424 .428 .4208

.488 .466 .4531 .444 .486 .483 .428 .424 .422 .4194

.440 .433 .438 .434 .422 .4187

.441 .438 .429 .tt4 .433 .4181

.482 .468 .455

.496 .472 .467 .448

.fOO .475 .460 .4501 .443 .434 .4S0 .434 .431 .4174
.478 .468 .452 TST

.481 .464 .454 .446
.488 .467 .466 .446
.486 .468 .468 .448

.436 .430

.437 .431

.438 .482

.438 .432

.424 .421 .4166

.424 .421 .4162

.424 .431 .4166

.424 .421 .4160

.489 .472 .469 .451 1 .440 .433 .424 .^21 .4144

.491 .474 .461 .452 TOT

.494 .476 .488 .454 .442

.496 .478 .466 .456 .443

.480 .467 .467

4.^4 .425 .421 .4139

.486 .425 .421 .4184

.433 .425 .421 .4138

425 .481 ^132

.482 .468 .468 .4461 .437

.483 .470 .460 .446 .438

.485 .472 .461 .447 .436

.487 .473 .468 .448 .439

.489 .476 .464 .449 .440

.490 .476 .466 .451 .441

.426 .421 .4118

.436 .421 .4112

.426 .421 .4107

.427 .431 .4101

.427 .421 .4066

.427 .431 .4092

Owing to the wide range of the head H and of the height p in these experi-
ments, we find in them a wider dlTergrenee in the values of the coefficient
than resulted from the earlier investigations. Thus, the smallest value of m
above the heavy lines is 0.4092, or about one nineteenth less than the mean,
0.4S25; and the greatest is 0.459, or about one sixteenth more than 0.4325.

* In these experiments, the head H was measured at a point 5 meters (16.4 ft ) back
from the weir. The correction for velocity of approach is contained in the coefficient m.

t M if the value of m when there is no velocity of approach ; i. e., where the croes-
«ection of the channel of approach is indefinitely great compared wilh that of the
•tream of water passing over the weir.

554

HYDRAULICS.

(p) From a comparison of a number of experimental data, €he Anilior
deduced the following /

Table IS. Approximate Talaes of tbe coefficient m in the

formula :

4| = mLHy2JH,
for weiH of several different shapes and thicknesses. (Original.)

sn. tbkk;

Head, U.

/ihwpMai.*

S Inohe*

tu«k.

smooth; alop-

ing outward

snd downward,

8 ft. chiek:

smooth: aikt

level.

Feet.

Inches.

from I in 12 to

llnlS.

m

m

m

m

.0833

1

.41

.87

.32

.n

.1066

2

.40

.88

.34

.30

.25

8

.40

JS9

.84

.81

.3333

4

.40 .

At

.86

.81

.4166

5

M

At

.85

M

.5

6

.39

.41

.85

.88

.0888

7

.89

.41

.85

JS2

8

.89

Al

.84

.31

.8833

10

.38

.40

.34

.31

1.

12

.38

.40

.38

.31

2.

24

.37

.89

.32

.30

8.

86

.37

.39

.32

.30

(9) To And the bead H, approximately; having the discharge a
According to formula (3) and (4), Art. 14 (/).

Hence

H

■V,

^

m*L*2g

(8)

or

Head

ai^roximately

» H, ^ /square of discharge of stream, in cub, ft. per eec.
aately " \ m* X length* X 64.4

-V

sq. of discharge

z* X length*

The coefiBcient m or a; itself Taries somewhat with the head ; but the formula
tnay be usefully employed as an approximation by taking, for sharp-crested
weirs, m = 0.415 (m« «= 0.172) or « « 3.33 (x* = 11). For other shapes, see Table
15, aboye.

(r) Submerired weirs. Fig. 23 6, are those in which the surface of the

<iotoi»-stream water at A, after tbe constniotion
of the weir, is higher than the crest a.

In a weir discharging freely into the air, as
in Fig. 20, Mr. Francis found that with a head
of 1 foot the discharge was diminished only
about one thousandth part by placing a solid
horizontal floor about 6 inches below and In
front of the crest of the weir for the water to
fkll upon. Also, when the head was 10 inches, and the water fell freely through
the air into water of considerable depth (as in Fig. 20), the quantity disoharm
was the same whether the surface or the down-stream water was about 8 ine&ea
or about 13 inches below the crest a.

In experiments by Mr. Francis and by Messrs. Fteley and Steams, with air
freely admitted underneath the falling sheet of water Just below the crest Ottlie
discharge was not appreciably affected by s submergence of A » from 0.017 H to
0.023 H. When air was only partially admitted, the discharge was afiteted {i$^
ereeued) by less than one per cent, while h remained less than 0.16 H.

Fi4:.23b

* These values are lower than those gjven in Art. 14 (m) and (n), and much low«
than those in (o).

HTDBAUUCS.

666

IHibuat** fl»raiiil» f»r submerged welnu Let

SI and /« = the heads measured yerticallf from the creet a of the weir to th«
surface of still water * up-stream and down^stream from the weir,
respectlTely.

«l — H— A = their diflference = the diflbrence in level between the up-streaa
and down-stream sur&oes of still-water ;*

• - coefficieDt of dl«>li«g« - -,J!^*^*^ .

** theoretical discharge
Then

4|-«L(*-f.| «l)y57ir;t (9)er:

Aetaal dtaeb»rffre „ ^ X >e?«th^ x «.025 y^InltX (h in ft. + 1 din ft.),
in cub. ft. per sec. weir in fL ' V 8 /

(«) Messrs. Ftoley and Steants $ experimented st Boston in 1877 with
•■biergsd weirs under up-stieam heads U from about 4 to 10 inches : and
Bb*. Fnsnets S at Lowell in 1888 under heads from about 1 foot to a ieet 4
inches.

From these experiments we deduce the following

Table lA, of approximate values of the eoafllcieAi e in the IbrmnU for
4iaeharfo ever snbieryed weirs*

Deduced firom experiments by Fteler and Steame and bj J. 6. Francis. In
Mr. Francis' experiments, the value oi e for a given value of A -i- H generally
tecteaeed as H inoreessd.

Fteler and ateama.
(H--0.ra$ to 0.816 feet)

J.B.Frsneis.

(H -i 1 to 2J2 feet)

k-fU

«

«

M

J28 to J82

ao

JBO te .610

JM to J80

M

' .618 to .628

.610 to J25

JO

J80 to .618

.698 to .616

.40

J90 te -608

J86 to .610

JBO

J86 to J85

J86 to .607

.80

J88 te JI06

J86 to .607

.T8

JS80 to J88

J86 to .607

M

J81 to J»l

JW6 to .607

S9

J80 te J08

M

.610 to .610

• For velocity of spproech, see Art 14 (») etc.

t In deducing this formula, the water that passes over Uie vrdr between e and b is
assumed to flow as over a weir with its crest at b, and with free discharge into the air,
ae over the crest a in Fig. 20; and tar this portion, by formula (8) In Art 14 (I), the
discharge would be:

Qj-«L|dy'27T;

wMle ttie water that passss throogh the lower portion between b and a Ib regarded as
flowing tfapough a suMneraed vertical «r»/Ice whose height Is 6 a »• A, under a head
Mi d. For this lower portion, therefore, the discharge would be:

It is assumed that the coefficient of discharge e is the same for the upper section
e » as ibr the lower one a h* Henoe,'adding these two discharges together, we obtain,
for the entire disharge:

1 Transactions, American Soclely of Civil ISngineen, March, 188S, p. 101, etc
{ Transactions, American Society of Civil Xngineeri, Sept, 1884, p. 896^ eta.

566

HYDRAULICS.

(t) Mr. Clemens Hersehel,* comparing these experimentt with some
earlier ones by Mr. Francis, gives the following :

Having ascertained the depths H and A of the crest below the still- water levels
np-stream and down-stream respectively, divide A bv H. Find the quotient, as
nearly as may be, in the column headed h-rHin Table 17. Take out the cor-
responding coefficient a, and multiply it by the up-stream head H.f

The proauct a H is the head which would cause the cdven weir to discharge
the same quantity freely into the air, as in Fig. 20. Find the discharge into air
ever the given weir with the head aH ; and this discharge will be approximately
the same as that of the actual submerged weir under the up-stream nead H and
against the down-stream head A ; or (H being the actual up-stream heaud. on the
submerged weir) the discharge is

Q—inLaH}^2yaH»s LaHf^oTH". (10).

TABUB 17.

A+H

a

h + H

a

A-hH

a

.10

1.000 to 1.010

.45

0.894 to 0.930

.72

0.762 to 0.784

.20

0.9r6 to 0.996

.60

0.874 to 0.910

.74

0.747 to 0.769

.25

0.960 to 0.984

.55

0.868 to 0.889

.76

0.732 to 0.752

M

0.94a to 0.973

.60

0.829 to 0.863

.78

0.713 to 0.733

M

0.928 to 0.960

.65

0.803 to 0.888

.80

0.693 to 0.718

Mi

0.912 to 0.946

.70

0.775 to 0.799

(u) Teloeity of approacb. See Fig. 24. It is generally impractieable

to measure the head H' to perfectly still
water o' up-stream. The head is usually
measured at a point o, from 2 or 3 to 6 or
8 feet or more up-stream from the weir,
according to the size of the latter. At
such points the velocity is generally ap-

f)reciable, and the surface therefore a
ittle lower than at o'. Hence a formula
using the smaller head H so measured,
instead of H', and coefficients based upon
H', will give too small a discharge. Mr.
Francis found that a current of 1 foot
per second, or nearly 0.7 mile per hour,
at the point o to which the head was
measured, increased the discharge but about 2 per cent, when the head was
18 inches; and a current of 6 inches per second increased the discharge about 1
per cent, when the head was 8 inches.

If, howeyer, the velocity of approach is such as to require consideration, pi*-
oeed as follows : For the approximate mean velocity of approach, we have :

m^.24

approximate discharge

3.33 L

hI

area of entire cross section of stream at o

area at o

•sd, for the bead due to this velocity, ^'^n^
Then, for all practical purposes, we may say : H' — H -f- A ; or
Q =- m L (H -»- A) y2giH + h) « a? L (H -I- A)i

(11)

lathough, strictly speaking, the difference of level between c' and a is really
(as shown in Fig. 24) somewhat greater than A, or than v* -i- 2 ^, because some
nead is lost in friction between </ and o.

• Tmnsactions, American Society of Ciyil Inglneers, Ma3% 1885, pp. 180, etc.

t Mr. Herschers table, from which ours is condensed, gives a for every 0.01 foot of
A -I- H ; but the values of a intermediate of those we have selected may be taken tnmk
our table almost exactly by simple proportion. The range in the coefficient a In the
table for each yalue of A -»- H is that indicated bv the experiments, which varied
similarly. We are not instructed how to select between these extremes; bat ia
most caaes their mean value is probably nearest right

HYDRAULICS.

657

(«) Messrs. Fteley and Stearns make H' = H + 1.5 A for suppressed
weirs, and H' = H + 2.05 h for weirs with complete end contractions, as averages ;
and Mr. Hamilton Smith, Jr.,* after comparing their experiments with others
by Lesbros, Castel and Mr, Francis, gives a.' = a.-^l%h, and H' = H + 1.4 A, for
the two cases respectively.

(tr) On the other hand, Mr. Francis' formula, as modified for velooity of
approach,

Q = a: L t (,/(H + AP - i/P^ = m L fr^{^(S. + h)» - |/H») J . . . (12),
makes the effect of H' less than that of H 4- A.

(po) Messrs. A. W. Slunklnir »n^ Frank S. Hart, Civil and Hy-
draulic Engineers, have substituted for the expression (y'(H + A)* — yTP) in

formula (12), the equivalent one K y'H^, in which K is a coefficicDt deduced from
the former expression, and therefore depending upon the -relation between H
and A, or, ultimately, upon that between the cross-section a s Fig. 24 at the weir
and the entire cross-section of the stream at 0.

Having found the area of cross-section at 0, divide it by (^"''^^ )> '^Uch

is the length of the weir corrected for contraction. See Art. 14 (m). Call' the
quotient D.g Divide the measured head H by D. Find this last quotient in

the oolumn :=r of the table. Multiply the approximate disoharga, Q » s^

L — n — j H2,by the corresponding coefficient K ; or

Aetnal Biscbarire Q = S.a3 K /l — » ^ ^ ni (13)

Table 18. €oeffieient K in formula (18),

(

H

H

H

H

H

K

K

K

K

K

D

D

D

D

D

.01

1.0000

.09

1.0020

.17

1.0072

.24

1.0143

.31

1.0239

.02

1.0001

.10

1.0025

.18

1.0081

.25

1.0166

.32

1.0254

.03

1.0002

.11

1.0080

.19

1.0090

.26

1.0168

.38

1.0271

.04

1.0004

.12

1.0036

.20

1.0100

.27

1.0181

.34

1.0287

.06

1.0006

.13

1.0042

.21

1.0110

.26

1.0196

.86

1.0806

.06

1.0009

.14

1.0049

.22

1.0121

.29

1.0209

.36

1.0322

.07

1.0012

.15

1.0056

.2«

1.0182

.30

1.0224

.87

1.0341

.08

1.0016

.16

1.0064

• " Hydraulics," John Wiley & Sons, New York, 1886.

t If there are end contractions, L here becomes f L — ** TT ) •

See Art 14 (m).

X This formula is deduced as follows : Let the area of the parabolic segment a s a\
Fig. 24, represent the theoretical dischax^e over a weir one foot long (as explained in
foot-note I p. 549) under the measured head H = a s, as though there were no current
at o. Let m$=h = v'-i-2g. The theoretical velocities of the particles passing the
oblique plane o a under their actual heads, will now be represented by horizontal lines

8^', a a", etc., etc., drawn from every point in a a to the outer curve s" a" ; the line « ^'

representing v = velocity of approach =» V2gh, and a a" representing y2g{B. -\- hy.
Then, area a tf' a!' a

2 2
— area of rectangle an — — area of rectangle $ Ic

3 3

= area maf'a — area ms" a =

2^
3

( H + A ) v^g(n-[-h) - 1 h i/jrp; = f »^2g ( iWT^' - v^ ) ;

and the actual discharge is
Q =1 c X length of weir X area**'' a'^a = ch^ V^ (vWrW— V^)

L y2T/v'(H + A)» — v^^) = « L (^yj;Br+~xf' — |/pV

m

2 In a weir without end contraction. D = H + P«

558

HYDBAULICS.

1 / H \ *

K is very approximately =" ^ + "k ( ^ ) • Henoe

» [3.33 + 0.83 (^y](L-nf)H»

(M)

See Journal of the Franklin Institute, Philadelphia, August, 1884, firom whick
we condense the above table.

(p) M. Bazln, see Art. 14 (o), provides for the velocity of approach by modi*
fying the coefficient m instead of the head H, making m = 0.425 + 0.21 I T7 ) ;

while by Messrs. Hunking and Hart's method (based upon Mr. Francis' experl*

/H \«
ments) m becomes = 0.415 + 0.10 i w j •

Art. 15. Inclined weirs. If the up-stream face of the weir, Instead
of being vertical, as in Fig. 25, is inclined up-stream, as in Fig. 25 a, or down-
stream, as in Fig. 256, the character and amount of the discharge are modified.
With an up-stream inclination (Fig. 25a) the lower side of the sheet of water
passing over the weir leaps higher, and tends more and more upstream as the

Fio. 25 a.
Inclined np-stream.

Fig. 25.
Tertical.

Fig. 366.
iBclined iown-stream.

inclination is increased. With a down-stream inclination (Fiff. 15 b). on the oon-
trarr, as the inclination increases the upward leap of the sheet deereasea, its
profile becomes more and more flattened, and the curve of the upper surfkoe,
due to the fall, extends farther up-stream from the crest of the weir.

An up-stream inclination (Fig. 26 a) decreases, and a downnstream one (Fig.
26 b) increases, the discharge, as is indicated by the following coeffieiMita <^
tained by M. H. Bazin :*

For tlie diseliarire over »n inelined weir, having ascertained the
discharge over a vertical weir of the same height and head and under similar
londitions in other respects, multiply the discnarge over the vertical weir bj
ihe following approximate coefficients:

Inclination. 1

Angle

Coef.

Horizontal.

Vertical.

with hor.

with vert

fioieirt.

r

1

46«>

4BP

(KM

Weirs inclined up-,
stream, Fig. 25 a....

*
*

660 19'
710 84'

880 41'
18© 26'

0.94
0.96

Vertical weirs. Fig 25...

0

90°

OO

1.00

'

i

710 34'

18« 26'

1.0i

Weirs inclined down-
stream. Fig. 256.....'

*

1

66° 19'
460

380 41'
480

1.07
1.10

'

2

1 1 26° 34'

63° 26'

1.12

* "Experiences Nouvelles sur T^coulement en D6versoir,*' 2e Article; "Annalea
des Fonts et Chauss^es," January, 1890, translated in Prooeeding$t Engineen'' CUib ^
VhXUxdAlphiix^ vol. Ix., 1892.

HYDBATTIilGS. 669

Th« cUacluurM will be inoraasad also if the inner comer or edge of the crest
be roundea o£^ instead of being left sharp ; or if the sides of the reservoir con-
verge more or less as they approach the weir, so as to form wings for euiding
the water more directly to it ; or if a &, Fig 20, be less than twice a m. Indeed*
so many modifying circumstances exist to embarrass experiments on this and
similar subjects that some of those which have been made with great care are
rendered inapplicable as other than tolerable approximations, in consequence
of the neglect to take into consideration some local peculiarity which was not
at the time regarded as exerting an appreciable etfect. Unless, therefore, cir-
cumstances admit of our combining all the conditions mention^ in Art. 14 (d),
(/) and (m), pp. 547, 548 and 590, thereby securing very approximate results, we
must either resort to an actual measurement of the discharge in a vessel of
known capacity ; or else be content with rules which may lead to errora of 6,
J6, or more per cent, in proportion as we deviate from these conditions. Fre-
quently even 10 per cent, of^error may be of little real importance.

Bbhabk 1. When the water, after passing over a weir, Fig. 26, instead of fall-

Fig.2e.

ing freely into the air, is carried away by a slightly inclined apron or trough, T,
the floor of which coincides with the crest a. of the weir, then the discharge li
not appreciablv diminished thereby when tne head a m, is 15 inches or mor*.
But if the head a m is but 1 foot, then the calculated discharge must be reduced
about one-tenth ; if 6 inches, two-tenths ; if %\i inches, three-tenths ; and if 1
inch, five-tenths, or one-half, as approximations.

Bemakk 2. Professor Thomson, of Dublin, proposed the use of triangular
netohea, or weirs, for measuring the discharge ; inasmuch as then the peripneiy
alvavs bears the same ratio to the area of the stream flowing over it ; which is
not the ease with any other form. Experimenting with a right*angled triangular

Fig.2SA.

aotoh in thin sheet-iron. Fig. 26 A, with heads of flrom 2. to 7 inches, measured
vertioally from the bottom or the notch to the level surface of the fuiet vxUer^ he

found discharge in cubic feet per second = .0051 X Y fifth power of lEeiil in inchee.

iB 2M X f^fifth power of head in feet.* Or, in general, if m =» coefficient of
contraction (Art 9, n. 541) T »= tangent of half the angle of the notch = width
of water-surface ■*- the depth in the notch, g «= the acceleration of gravity <= say
32.2 feet per eeoond, and A => the head, measured as above ; then

Discharge - -^ T yJg3 * - 4.28 m T /»•*

Remark 8. In constructing the irrigating canal, Canale YiUoresl, near Milan,

in 1881-4, the Italian engineer, Cesare Cippoletti,t adopted a trapezoidal

4 vertical
notell, with its bottom edge horizontal and Its ends sloping at — r — r-j, in

order to avoid the necessity of either suppressing or allowing for end contrac-
tions. (See Art. 14 e, p. 547, and tn, p. 860i ) The contraction was found to affect
only the triangular spaces over the sloping ends of the weir, and the effective
length of the weir thus remained constant (and equal to the length of the bot-
tom edge) for all heads. In using these weirs the contraction is complete along
the bottom as well as at the ends.

* For such roots see p. 68.

f See his work, Ccmale ViBoreri ; Modulo per la Diepensa della Acqtta^ etc., Hllan,
1886 ; published by Societa Italiana per Condotte d* Acqua. Results summarized by
li. O. Carpenter, in Bulletin No. 13, Agricultural Experiment Station, Fort Collins,
Colorado. October, 1890.

560 BYDBAUUGS.

ON TIf £ FI.OW OF WATER IN OPEN CHANN;;BIJL

Art. 16. The mean velocity of flow is an imaginary uniform oney

which, if given to the water at every point in the cross section, would give tht

tame discharge that the actual ununiform one does. Or

_ .. volume of discharge

mean Telocity « ? jS.

area of cross section

In channels of uniform cross section, tbe maximum velocity is found
about midway between the two banks, and generally at some dist below the sur-
face. Tbis dist varies in diff streams; but, as an average, it seems to be about
one third of the total depth. Where the total depth is great in proportion to
the width, (say ^ the width or more), the max vel has been found as deep aa
midway between surf and bottom; while in small shallow streams it appears to
approach the surf to within from .1 to .2 of the total depth. Many experiments
upon shallow streams have indeed indicated that the max vel was at the surf.

The ratio l^etween the velocities in different parts of the
cross section varies greatly in diff streams; so that but little dependence
can be placed upon rules for obtaining one from the other. With the same surf
yel, wide and deep streams have greater mean and bottom vels than small shal-
low ones. In order to approximate rooi^hly to the mean vel when
the greatest surf vel is given, it is frequently assumed that the former is s |
(or .8) of the latter. But Mr. Francis found, in his experiments at Lowell, that
surface floats of wax, 2 ins diam, floating down the center of a rectangular flume
10 ft wide, and 8 ft deep, actually moved about 6 per cent alnwer than a tin tube
2 ins diam, reaching from a few ins above the surf, down to ^vithin 1| ins of the
bottom of the flume; and loaded at bottom with lead, to insure its maintaining
a nearly vert position. While the wax jmrf float moved at the rate of 3.78 ft per
sec, the rate of the tube (which was evidently very nearly the same as that of
the center vert thread of water) was 8.98 ft per sec. Also, that in the same flume^
with vels of the center tube varying from 1.55 to 4 ft per sec, the vel of the tube
was less than that of the mean vel of the entire cross section of water in the
flume, about .j .96 to 1, for the lesser vel ; and .98 to 1 for the greater veL
While, in another rectangular flume 20 ft wide and 8 ft deep, with vels varying
from 1.16 to 1.84 ft per sec, that of the tubes was grtaUr than that of the entire
mass of water, about as 1.04 to 1. In a flume 29 ft wide, by 8.1 ft deep, with vels
of about 3 ft per sec, it was as 1 to .9 ; aud in a flume 36^ ft wide, by 8.4 ft deep,
with vels of about 3^ ft per sec, as 1 to .97.

Charles Ellet, Jr, € E, found in the Mississippi "at diff points
on the river, in depths varying from 54 to 100 ft; and in currents varying from
8 to 7 miles an hour that the speed of a float supporting a line 50 ft long, is al-
most always grea er than that of the surf float alone." The same resulta were
obtained with lines 25 and 75 ft long; the excess of the speed of the line float*
being about 2 per cent over that of the simple floats: and Mr. Ellet conclude^
therefore, that the m'ean vel of the entire cross section of the Mississippi, insteaa
of being less, is absolutely greater by about 2 per cent, than the mean ntr/" veL
He, however, employed .8 of the ffrecUest surf vel as representing approximately,
In his opinion, the mean vel of the entire cross section of water. In shallow
streams, he always found the surf float to travel more rapidly than a line float.

European trials of the mean vel of separate single verticcUSf in tolerably deep
•ivers, nave resulted in from .85 to .96 or the surf vel at each vertical. Tbe mean
of all may be taken at .9.

Bottom velocity. In streams of nearly uniform slope and cross section,
there is a great reduction of vel near the bottom. As a very rough approxima-
tion, the deepest measurable vel, in streams of uniform slope etc, appears to b«
firom ( to I or the mean vel.

Art. 17. To measure the snrflace velocity, select a place where

the stream is for some dist (the longer the better) of tolerably uniform cross
section ; and free from counter-currents, slackwater, eddies, rapids, etc. Ob-
serve, by a seconds-watch, or pendulum, how long a time afloat (such as a small
blocjc of wood) placed in the stoifiest part of the current, occupies in passing
through some previously measured dist. From 50 feet for slow streams, to 150 n
for rapid ones, will answer very well. This dist in ft, or ins, div by tbe entire
number of seconds reqd by the float to traverse it, will give the greatest surf vel
in ft or ins per sec.

The surf vel should be measd in perfectly calm weather«

BO that the float may not be disturbed by wind ; and, for the same reason, tha
float should not project much above the water. The measurement should b«

HYDRAULICS.

661

repeated several times to Insuve accuracy. In very small streams, the banka
and bed mav be trimmed for a short dlst, so as to present a uniform channel*
way. Tbe noat should be placed in the water a little dlst above the point for
commencing the observation ; so that it may acquire the full vel of the water,
before reaching that point.

I^vice a •
reloeltw*

Art. 18. To
lay meauas at Its weFoeltT'. Saleot

a place where the cross^ectjon rt* mains, for
a short distance, tolerably uniform, md
free from connter-currents, eddies, still
water, or other irregularities. Prepare a
carefnl crose-aection, as Fig. 27. By meana
of poles, or buoys, n, n, divide the stream into sections, <t, b, e, Ac. Plant two range-
poles, K. R, at the upper end, and two others at the lower end, of the distance
through which tike floats are to pass ; for observing by a seconds watch, or a pendu-
lum, the time which they occupy in the passage. Then measure the hmom velocity of
each section a,b^et Ac, separately, and directly, by means of long floats, as V L,
reaching to near the bottom : and projecting a little above the surface. The floats may
be tin tubes, or wooden rods; weighted in either case, at the lower end, until they
will float nearly verticsl. They must be of different lengths, to suit tbe depths of
the diffisrent sections. For this purpose the float may be made in pieces, with scrsw-
jtrints. The area of each separate section of the stream in square feet, being multi-
plied by the observed mean velocity of its water in feet per second, will give the
discbarge of that section in cubic feet per second. And the discharges of all the
separate sections thus obtained, when added together, will give the total discharge
of tbe stream. And this total didcharge, divided by the entire area of cross-section
of the stream in square feet, givesi the mean velocity of att the water of the stream,
in feet per aecond.

Bern. If the ehannel is in common eartli, especially if sandy
the loss by soakage into the soil, and by evaporation, will frequently abstract so
much water that the disch will gradually become less and less, the farther down
stream it is measured. Long canal feeders thus generally deliver into the canal
but a small proportion of the water that enters their upper ends.

Tlie double float is used for ascertaining vels at difi" depths. It consists
of a float resting upon the surface of the water, and of a heavier body, or *' lower
float ", which is suspended from the upper float by means of a cord. The depth
)f tbe lower float of course depends upon the length of the suspending cord
^which may be increased or diminished at pleasure until the lower float is be-
Heved to be at that depth for which tbe vel is wanted), and upon its stralght-
ness, which is more or less affected by the current. Owing to this latter circum-
ttanoe, it is difficult to know whether the lower float is really at the proper
depth. Moreover it is uncertain to what extent tbe two floata and the string
Interfere with one another's motions. In deep water the string may oppose a

Csater area to the current than the lower lioat itself does. It thus becomea
nbtful to what extent the vel of the upper float can be relied upon as indicat-
ing that of the water at the depth of the lower one.

Art. 19. Casteili's quadrant, or hydrometrie pendnluml

consisted of a metallic ball suspended bv a thread from the center of a graduated
are. The instrument was placed in the current, with the arc parallel to the
direction of flow ; and the vel was then calculated from the angle formed be-
tween the thread and a vert line.

Oanthey*s pressure plate was a sheet of metal suspended by one of its
ends, about which it was left free to swing. The plate was immersed in the
stream, with its face at right angles to the current. The vel was estimated by
means of the weight required to make the plate hang vert in opposition to the
force of the current.

Pitot*s tube was originally a simple glass tube. Fig. 27 A, open
at both ends and bent in the shape of the letter L. One leg of the
L was held horizontal under water, with its open end facine the
current ; and the velocity v at the point o where it was placed was

measured by the vertical height h (theoretically = o~ I ^ which

the water rose in the other leg above the surface of the stream.

3e

Fig. 27A

HTDRADLICB.

bj M. D>peT BBd by Pr*r. H. W. Kol

rudalr InilWad In Fig. ft B, Pllol's tube
Uitlf of nco boriEOBlnrgluB or maUl lub
Tfirr imall bore, placed aide bf afde Id tt
pointed up-strenm. Tube a raceives the

: Lvo Bailble pipes ma; be Joined logetbac Into ana double

epe. Br BuckiDg througli > >lop-oinik T «t lbs lop. wata
dnwD up Ig sdj coavaalaat beighl in tbe two legi of
I (htgauge. WbenlherelBnocurreul.tbelwocolumnaof
I ODune lUnd at tbe aama Height ', but ia ■ curreul, the dU-
reronoe A in thatr heighle i> eush that v — yTg'h, no oor-
nctlie coefficiept being requirad. Tbe Instrainenl ia re-
marksblj flimple and aecurate, and can be uaed in Tery

In pmitiM, a and b are fli»d togrthec in uue piece, and placed, whea In uaa,

f , either lipon a wire paiiikg thi
ests upon lh«_ bottom and keej-- '

ri^'tiiriowi/e'nd 0 f "w"i

with a plummet which rests upon the bottom and keepe tl

/i_ -M-JL. — -i._n ,K-.^ ^k^»* m f^^t- .,«.« ^ T«rUC«l 1

upon a

B la prOTlded with a long vane Tor keeping the Instrument haded iu»>
Id either case, means an proTlded for sliowlDg the depth la wbieh tbe

Bt making the ga^ge scale aiUuaUble Tertlcallr, and pludog it (at each cbuq*
of depth of InttriimenO with its Eero oppoalM the top of the lower eolnnu^ m

for the reading of Che upper column alone then glres the head it at once.

iu.i-uL,.,.intu-uuuim « auu «re'fjli"V.in ine uniei w«u a. <i«>

meter if often made nelf-reKl'terlns ; the wheel, at each fB»olntion. aalo-
matically breaking and re-establlshlBg a gaWanic current generated br a bat-
tery. The wire carrying this current Is thus made lo operate Morse telegrsphie

velocities UdtObren't deptba maj'be msde~iind registered.

Meters are usubIIt so arranged as to iwlug freely about the long Ta-U^ pola
til which tber are damped, and arc proiided eacb with a rane or lall similar lo
that of ■ windmill, for keeping the wheel in the proper poailion as r^arda tbe

lerbiiUnce> „

Journal friction due to It, Meiers proilded with electrical regis lering
apparatus lametlmce ba>e tbe gearing and Indices, etc., encluaed in a glus cast,

A wtieel meter la ml«d by moilng it at a known TelooltT throagh etlU
water, and noting the eOect produced. In this way a coeffloient la obtklnel for
each meter, which, when multiplied by the number of reTolutioua reoordad iB
any giieu oaM, glfM the Teloolly for that case.

•See Tan Mutruid'B Uagailne. Uansb. 1S7B, and Augiut, 1(84.

HTDBAUUG8. 663

Art. 81. Kntter's formula for the mean vel of water flowing in open
cbanuels of uniform croaa section and slope throughout.

Caution. The use of all such formula Is liable to error arising from the
difficulty of ascertaining the exact condition of the stream as regards roughness
of bed, surface slope,* etc.

Bkm. 1. Care must be tiCken tbat tlie bottom vel is not so
i^reat as to wear away the soil. If there is any such danger artificial
means must be applied to protect the chauuel-way ; or it mar be advisable to
reduce the rate or fall, and increase the cross section of the channel ; so as to
secure the same disch, but with less vel. A liberal increase should also be mad^
in the dimensions of sucli channels, to compensate for obstructioDs to the flow,
arising from the growth of aquatie plants, or deposits of mud from rain-
washes, etc ; or even from very strong wiuos blowing against the current.

Bbm. 2. Water mnnluir in a ebannel with a horisontal bed,
or bottom, eannot baTe a uniform Tel, or deptb, tbrouyb-
out Its courses because the action of gravity due to the inclined plane of a
sloping bottom, is wanting in this case ; and the water can flow only by forming
itsjiMTace into an inclined plane; which erideptly inTolves a diminution of
depth at every successive dist from the reservoir.

Fig. 29.

Theory of flow. It is generally held that the resistances to the flow et
water in a pipe or channel are directly proportional to the area of the bed sur*
face with which the water comes in contact (i e, to the product of the "wetted
perimeter" as a 6 co Figs 28, 29, 80 mult by the length of the channel, or of the
portion of It under consideration) ; and to the square of the vel of the flowing
water; and, inasmuch as the resistance at any given point in the cross section
appears to he Inversely as the dist of that point from the bottom or sides, we
conclude that the total resistances are inverselv as the area of the cross section ;
Inesose the greater that area, the greater would be .the mean dist of all the par.
tklw from in« bottom and sues. The resiatanee is indepmuleni ^f the pnuvn.

In short, the resistances are assumed to be in proportion to

vel* X wet perimeter X length iApl

area of cross section a

and the head h" in feet or in metres etc, required to overcome those resist,
ances, is

resistance a coeflBlcient^^ vel* X wet perimeter X length ... ^n^ p I

"^■** C area of wet cross section a

, / area of wet

« /I . . •% / cross section ^ ,

resistoe
head

perimeter ^ length

• " In iD«Mnrins the ilope of % large rlT«r, the ordinary errors of the mogt careftil leveling are a
larfe proportion of the whole fUl ; the variation of level In the croiv section of the aurfaoe is often aa
gnat aa the alope for ten mllea or more ; the ezaot point where the level should be taken is often
uncertain : the rise and fttU of the water makes it extremely difBcnlt to decide when the levels ahould
be t«ken st the upper and lower points ; waves of translation may affect the inclination to a great
knd nnoertain degree, and may even make the surfboe slope the reverse war." Genl T. O. Bills.
Trans Am Soc Civ Bngrs- Ang 1877.

564 HYDRAULICS.

But ^!^reB_otj;>^t^^ss^tAon ^^ a j^t^e "hydraulic radius" of
wet perimeter p

" mean depth " or '^ mean radius," B, t>f the cross section ;
and r?«i«taiice_head ^^ *! j^ ^y,^ inclination or slope, S, (fre-

length (

quentlv denotea by "I") of the hydraulic grade line, or the sine of the

anffle wso Fig 1|. In open channels, il is <-i the fall of the surface per

nnit of length.

We therefore hare Telocity =- -X^—^ X l/mean radius X slope

or, by using a coeff (e) =■ a/-^*

Telocity =» coefficient c X V^mean radius X ilopo

or v =- « 1/1^

The earlier hydraulicians gave (each according to the results of his tuTestiga-
tion8)yized Talaes for the coeflTe, (generally about 95 to 100 for channels
in earth or gravel, hs in our early editions), making it, in other words, a con-
ttanlf and independent of the shape, size, slope and roughness of the channel.
But more recent investigators have shown that the coefficient c is affected by
differences in any of these particulars.

According to the formula of Ganguillet and Kutter (generally called, for con>
Tenience, '"Kutter's formala''*) the value of c is:

For Enirliftli measure. For metrie measure.

^, ^ . .00281 . 1.811
*l-6 + - ,-— +

_ slope n

C = —

23 +

.001S5 1
slope '*' n

1 +

(--^)x-

/^, ^ , .00281 \^,

V'uieau rad in feet V^mean rad in m^ret

Tables s^tIiik Talnife of e for diiT grades, mean radii and degrees of
roughness, and for English and metric measures, are glTcu on pp 066, ete.

Here n is a ** coefflclent of ronsliness" of sides of channel as giTos
below. These values of n were obtained from experience, by averaging a large
number of experiments made under very different circumstances. They there-
fore embrace all the disturbing effects arising fh>m obstructions existing upon
the bottom and sides of the channel in the cases experimented upon. In small
artificial channels of uniform cross section and slope, these obstructions may be
said to consist entirelv of the comparatively minute roughnesses of the maUritU
of which the bed of the channel consists. But in rivers and earth canals, even
where the general direction, slope and cross section are tolerably uniform, (as
they were in the cases upon which our list is based), there are still many con-
siderable irregularities in the sides and bottom ; and these exert a much greater
retarding effect upon the mean Tel than the mere roughneu of the material of
the banks. We tnerefore find larger values given for n in such cases than for
small regular artificial channels, although the material of the sides ete was is
many cases smooth mud; and we must not apply to such comparatively IrregU'
lar channels the small values of n obtained by exneriments with small and car^
fully made straight flumes of uniform section ana slope, even if we suppose tlM
bottom and sides of the former to be made as smooth as thoetr of the latter.

No general formula is applicable to cases of decided bends in the oourss
of a natural stream, or of mariied Irrei^nlarltles In the erosa aeo*
tion. Such cases would require still higher coefficients a than those here giiTes
for rivers and canals : but they would have to be ascertained by experiment for
each case, and would oe useless for other cases. For such streams we must there*
fore depend upon actual measurements of the velocity, either direct or by means
of the diflcb.

* See " Flow of Water," translated from Ganguillet and Kutter, bT Rudolph
Hering and John C. Trautwine. .)r., New York, John Wiley A Sons, 1889. $4.oa

HYDRAULICS. 565

There is much room for the exercise of judgment in the seleetion of tlie
proper eoeffielent n for any giyen case, even where the cundition of the
efaannel is well known. It may frequently be necessary to use values of n inter-
mediate between those given ; for careless brickwork may be rougher than well
finished rubble; side slojpes in **Tery firm eravel " may have very difT degrees
of roughness ; etc etc. The engineer should make lists of values of n from bis
own experience, fully noting the peculiarities of eac^ case, and calculating n
from the tables.

A given diff in the deg n of roughness exerts a mooh sreaier effect upon the
eoefGicient e, and thus upon the velocity, in small channels than in larger ones.
It is therefore especially necessary in small channels that care be exercised in
finding (by experiment if necessary) the proper value of r» ; and, where a large
disch is desired, the sides of small channels should be made particularly smooth.

Table of n, or eoelBeient of ronirbness.

In any given case the value of n is tbe same wbetber tbe mean
radius Is griwen in Enylteli? metric or any otber measure.

ArtiUcial ebannels of uniform eross section.

ffidet and bottom of channel lined with n »

well planed timber 009

neat cement (applies also to glaeed pipes and very smooth iron pipes). .010
plaster of 1 measure of sand to 3 of cement ; (or smooth iron pipes). .Oil

unplaned timber (applies also to ordinary iron pipes).. 012

ashlar or brickwork :01i(

rubble « 017

Cbannels subject ^ irregularity of cross section.

Canals in very firm gravel.. ^.......u 0^

Canals and rivers of tolerably uniform cross section, slope and direction,
in moderately good order and regimen, and free from stones ana
weeos»«»».. —.«——»»»....#»««>»».—»»»»»».»»».■»....#»«»■»«»»»«»—.««»«...»«»»»»»».. ».....».. .ozo

havinff stones and weeds occasionally «... 030

in bad order and regimen, overgrown with vegetation, and strewn
with stones and aetritus..................M..M.. 03S

Art* 22. Tbe following tables glire walues of tbe coeflicient

• as obtained bv Entter's formula for diff slopes (8) mean radii (B) and degrees
of roughness (n).*

Caution. Diflbrent values of e most be used with English and with metric
measures. We give tables for both measures.

1st. Having the slope S, the mean rad B and the deg n of roughness; to
find tbe coeff c. Turn to the division of the table corresponding to the
Kiven slope S. In the first column find the given mean rad, B. In the same
Bne with this B, and under the given n, is the proper value of c*

Sd. Having the slope S, the mean rad B and either c or tbe actual or reqd
Tel «; to find tbe actual, or tbe g-reatest permissible, deg: n of
rouflfbuess of channel. If the vel is given, and not c, first find

e =* ■■ ^ — . Turn to the division of the table corresponding to

l/'slope X niean radTus
the given S, and in tbe first col find the given B. In the same line find the
yalue given, or Just obtained, for e; over which will be found the reqd n.*

8d. Having the slope S, the deg n of roughness, and tbe actual or required
yelv; to find tbe actual or necessary mean rad, R. Assume a
mean rad; and from the division of the tahle corresponding to the given S take
oat the value of e corresponding to the given n and the assumed B. Then say

t/ = c so found X (/assumed mean, radius X slope

♦ It is often necessary to interpolate values of S, B, n and c iutermediate oi
those in the tables ; this may be done mentally by simple proportion.

666

HYDRAULICS.

If this xf is the Mine as the giTen Tel, or near enough to It, take the assumed B
as the proper one. Otherwise, repeat the whole process, assuming a new B,
greater than the former one if ir is leu than the given vel, and viee verta,*

4tli. Having the dimensions of the wetted portion {abeo Figs 28, 29, 30,) of
the channel, the deg n of roughness, and the actual or reqd Tel; to And m«
•ctnal or neeeamary slope^ S :

_. ,^. , _ area of wet cross section

Find the mean rad, B — s — -rr — r = 1 — -; — r—

' length, aft CO, of wet perimeter

Assume one of the four slopes of the tables to be the proper one. From the
correspondins^ division of tBe table take out the value of c corresponding to the
given R and n.

If B is 3.28 feet, or 1 metre, the value of e thus found is the proper one (be*
cause then c, for any given n, remains the same for all slopes) ; and the slope, fi^
maj be found at once, thus;

\cX Vmean raiSIus/
But if B is greater or leas than &28 feet, or 1 metre, say

v^ '^e thus found X l/mean radius X assumed nSpk

If this i/ is near enough to the given vel, take the assumed S as tiie proper oiieu
Otherwise, assume a new 8, greater than the former one if v' is leae than the given
vel, and vice veraa; and repeat the whole proeesa.*

• It la often neoeinry to interpolate vmlDee of 8, B^ « and «
This woMj be done aMatellj bgr simple praportloe.

lotenaediate ef theee la the

Table ofeoeflleieiit e« for mean radU imfteL

a

Mean
radB

Coefficients n of roughness

Mean
rsMlB

«i4
1

feet

.009

.010

e

.011

e

.012
C

.018
e

.010

e

.017

e

.020
e

.026

e

.000

e

.086
e

.010

feet

t

Si

e

•

.1

65

67

60

44

40

38

28

23

17

14

12

10

.1

^

87

75

67

69

58

46

88

81

24

19

16

14

.2

.4

111

97

87

78

70

59

51

42

82

26

22

19

.4

S

127

112

100

90

81

69

60

49

88

81

26

22

.6

?s.

.8

138

122

109

99

90

77

66

66

48

86

80

26

.8

0tf

1

148

131

118

106

97

88

72

60

47

88

82

28

1

H-

1.5

166

148

133

121

111

95

88

69

65

46

88

88

IJi

2

179

160

144

181

121

104

91

77

61

60

48

87

2

3

197

177

160

J 47

135

117

108

88

70

69

60

44

8

8.28

201

181

164

151

189

121

106

91

72

60

62

46

8J8

4

209

188

172

158

146

127

113

96

78

66

66

48

4

0^

6

226

206

188

174

161

142

126

108

88

74

64

67

e

OS

8

238

216

199

184

171

151

185

117

96

82

71

68

s

1^

10

246

225

207

192

179

159

142

124

102

87

76

68

10

12

253

231

214

198

186

165

149

129

107

92

81

72

IS

m

16

263

242

223

208

195

174

157

138

116

100

88

79

16

i

20

271

249

231

^15

202

181

164

144

121

106

94

84

20

80

283

261

243

228

215

198

176

167

188

117

104

96

80

50

297

274

257

241

228

207

190

170

147

180

117

107

60

m

76

306

284

267

251

288

217

200

180

167

140

127

117

75

. 100

812

290 273 1

257 244I223>207I187I168I

147 184 '1841

100

i

HTDRATTLICS.

667

Table of eoefficlent c, for mean radii In /ee^.<~GoiiTiNUED.

r

Mean
radR

OoefBciente n of roughneM

Mean
radB

a

9*

feet

.009

.010

G

.011

e

.012 .018|.016

.017

e

.020

e

.026

e

.080
e

.036
e

.040

. feet

1

c

e 1 e

C

e

.1

78

67

59

62

47

39

83

26

20

16

13

11

.1

g. .

.15

91

79

69

62

56

46

89

31

28

19

16

IS

.15

or leni
rmile.

• J2

100

87

77

68

62

61

44

35

26

21

18

16

.2

.8

114

99

88

79

71

69

60

41

81

25

21

18

.3

.4

124

109

97

88

79

66

67

46

85

28

24

20

.4

as.

.6

139

122

109

98

90

76

65

53

41

33

28

24

.6

.8

150

133

119

107

98

83

71

59

46

87

31

27

.8

SI

1

158

140

126

114

104

89

77

64

49

40

84

29

1

ll •

1.5

173

154

139

126

116

99

87

72

57

47

40

84

1.6

2

184

164

148

135

124

107

94

79

62

61

44

88

2

^■^

8

198

178

161

148

136

118

104

88

71

59

60

44

8

8.28

201

181

164

151

139

121

106

91

72

60

62

46

3.2S

© ••

4

207

187

170

156

145

126

111

95

77

64

66

49

4

^s

6

220

199

182

168

156

187

122

106

85

72

68

66

6

m

8

228

206

189

175

163

144

129

111

91

78

68

61

8

10

234

212

196

181

169

149

134

116

96

82

72

64

10

12

238

217

200

186

178

163

138

120

99

86

75

68

12

0

16

245

223

206

191

180

160

144

126

106

91

81

78

16

20

250

228

211

196

184

165

149

131

110

96

85

77

20

8

30

257

236

219

204

192

172

157

139

118

103

92

84

S9

50

266

245

228

213

201

181

165

148

127

112

101

93

60

76

272

250

233

218

207

187

171

153

133

119

108

99

75

100

275

254

237

222

210

190

176 158

137

123

112

104

100

.a

f -1

90

78

68

60

54

44

37

30

22

17

14

12

.1

J2

112

98

86

76

69

67

48

39

29

23

19

16

.2

.3

125

109

97

87

78

65

56

45

34

27

22

19

.8

fa

.4

136

119

106

95

86

72

62

60

38

31

26

22

.4

«fe

.6

149

131

118

105

96

81

70

57

44

35

30

26

.6

:aS.

.8

158

140

126

114

103

88

76

63

48

39

33

28

.8

^%

1

166

147

132

120

109

93

81

67

52

42

86

81

1

21

1.5

178

159

144

130

120

103

89

75

59

48

41

86

1.6

B

2

187

168

151

138

127

109

96

81

64

63

45

89

2

8

198

178

162

149

137

119

104

89

71

69

61

46

3

3.28

201

181

164

151

139

121

106

91

72

60

62

46

3.28

4

206

186

169

155

143

125

111

94

76

64

55

49

4

^^ — **

6

215

195

178

164

152

134

119

102

84

71

61

64

6

is

8

221

201

184

170

158

139

124

107

88

76

66

69

8

10

226

205

188

174

162

143

128

111

92

78

69

62

10

■a

16

233

212

195

181

169

150

135

118

98

&5

76

68

16

r

20

237

216

200

185

173

154

139

122

102

89

79

71

20

30

243

222

206

191

179

160

145

128

108

95

84

77

30

50

249

227

211

197

185

166

151

134

114

100

91

83

50

tt

100

255

234

218

204

191 172 ' 158 1

140

121

108

98

91

100

^

r -1

99

85

74

65

59

48

41

32

24

18

16

12

.1

tft^

.2

121

106

93

88

74

61

52

42

31

25

21

17

.2

fi**^

.8

133

116

103

92

83

69

59

48

86

29

24

20

.8

*- -

.4

143

125

112

100

91

76

66

63

40

82

27

28

.4

^s

.6

155

138

122

111

100

85

78

60

46

37

31

26

.6

;aa

.8

164

145

131

118

107

91

79

65

50

41

34

29

.8

0 V

1

170

151

136

123

113

96

83

69

64

44

37

32

1

Za

1.6

181

162

146

133

122

105

91

77

60

49

42

86

IJS

H

2

188

170

154

140

129

111

97

82

64

54

46

40

2

CM

8

200

179

163

149

137

119

105

89

72

59

61

46

3

4

205

185

168

155

143

125

111

94

76

68

66

48

4

fi

6

213

193

176

162
167

150

132

117

100

82

69

60

58

6

8

218

198

il^

155

137

122

105

87

73

64

67

8

H o

10

222

201

170

158

140

125

108

89

76

67

60

10

•5

16

228

207

190

176

164

145

131

113

96

82

72

66

15

••a

20

281

210

194

180

168

149

134

117

98

86

78

68

20

S^

30

285

216

198

184

172

154

139

122

103

89

80

73

30

Sr

60

240

220

208

189

177

158

143

126

108

94

86

78

60

OD

100

246

224

208

194

182 1 163 1

148

181

113

99

90

88

100

568

HYDRAULICB.

Table of eoefficlent c, for mean radii In /«tf.— CovrnruxD.

-I

^ S

as
S3

t{

Ill

S

Mean

rad R

feet

.009

JOIO

e

C

.1

104

89

.16

116

101

.2

126

110

.8

138

120

.4

148

129

.6

157

140

.8

166

148

1

172

154

1^

188

164

2

190

170

8

199

179

4

204

184

6

211

191

10

219

199

20

227

207

50

285

215

[ 100

289

219

GoeffldeDtfl n of roaglmess.
009 jOlO .011 .012 .018 .015 .017 .020 .025 .030 J>85

e

78
90
97
107
115
126
133
138
148
154
162
168
175
183
190
198
203

c

69
80
87
96
104
113
121
125
135
141
149
154
161
168
176
184
189

e

62

71

78

87

94

103

110

115

124

130

138

142

149

167

164

178

177

e

50

59

66

78

79

87

93

98

106

112

119

124

130

138

146

154

158

e

4S

50

54

62

68

75

81

85

93

98

105

110

116

123

131

189

143

e
84
40
44

60

65

62

67

70

78

83

89

94

99

107

115

123

127

e

26
29
32
37
42
47
61
55
61
65
71
76
81
88
96
104
108

e

19
23
25
30
88
38
42
45
50
54
69
68
69
76
88
91
96

e
16
19
21
24
27
81
86
87
42
46
51
66
60
66
78
82
87

.040

e
13
16
18
21
23
27
80
82
87
40
46
48
68
69
66
76
80

rad]

feet

.1
.15
.2
.8
.4
.6
.8
1

1.5

2

8

4

6

10

20

60

100

«5
is.

Si

II.S
• II

fita

Ss

.1

110

94

.2

129

113

.3

141

124

.4

160

181

.6

161

142

^

169

150

1

175

165

1.5

184

166

2

191

171

8

199

179

4

204

184

6

211

190

10

218

197

20

226

205

60

232

212

100

236

216

88
99
109
117
127
134
139
149
155
163
168
174
181
188
196
200

78
89
98
105
115
122
127
136
142
149
154
160
167
176
182
186

65
81
89
96
104
111
116
124
130
138
142
149
155
163
170
174

54

45

86

27

21

17

14

66

67

45

34

27

22

18

74

68

61

89

30

25

21

80

69

56

43

34

28

24

88

76

68

48

39

82

27

94

82

68

62

42

36

30

99

86

71

66

45

88

33

108

93

78

62

60

48

87

112

98

88

66

64

46

40

119

106

89

71

69

61

45

124

110

98

76

68

64

48

130

116

99

81

68

69

52

136

122

106

87

74

65

58

144

129

113

94

81

72

65

161

187

120

101

89

79

72

155

141

124

105

94

85

77

.1

o

.3
.4
.6

1

1.5

2

8

4

6

10

20

50

100

feoo

or"
I"

f -J

110

95

.16

122

105

J2

130

114

^

143

125

A

151

133

.6

162

143

.8

170

151

1

175

156

1.5

185

166

2

191

171

3

199

179

3.28

201

181

4

204

184

6

210

190

10

217

196

20

225

204

50

231

210

I 100

235

214

83
93
100
111
119
129
135
141
149
155
162
164
167
173
180
187
194
197

74

83
90
100
107
116
123
128
136
142
149
151
154
160
166
173
181
184

•75
81
90
98
106
112
117
125
130
138
139
142
148
154
161
168
172

64

62

67

76

82

90

95

99

107

112

119

121

46

52
57
64
70

77
82
87
94
99
105
106

123 109

129
136
143
150
153

115
121
128
135
139

86

42

46

52

57

64

68

72

79

83

89

91

93

99

105

112

119

122

27
81
84
39
44
49
53
56
62
66
71
72
76
81
86
93
100
104

SI

17

14

24

20

17

27

22

19

81

26

22

85

29

24

89

83

28

43

86

31

45

88

38

51

48

37

66

46

40

59

61

45

60

52

46

68

66

48

68

69

52

74

66

58

80

71

64

87

78

71

91

82

75

.1

.16
.2
.3
.4
.«
J
1

1^
2
8

8.28

4

6

10

20

60

100

For slopes steeper tlian .01 per unit of length, » 1 In 100 b 62.8 feet
per mile, e remains practically the same as at that slope. But (he veiocitg

(being = eX 1/mean radius X *l^) of course oontiaues t» IncreMe m tiM
slope becomes steeper.

HYDRAULICS.

TaMe of eoeOketeut c, for mean radU in

i

Mean
jradR

Coefficients

n of roughDeaa.

uetres

.009
e

.010
e

.011 .012

.013
e

.016 .017

.020
c

.025

e

.030 .03£

c

e

e

:

e

c

C

.025

34

29

25

22

20

17

14

11

9

7

6

.05

44

38

83

30

27

22

19

16

12

9

8

.1

58

50

44

40

36

30

26

21

16

13

11

||

.2

72

63

56

51

46

39

34

28

21

18

16

.3

82

72

64

58

53

45

39

33

25

21

17

tta •

.4

89

79

71

64

59

60

44

37

29

23

20

»oooa

-11

.6

99

88

80

72

67

57

50

42

33

2S

28

1.

111

100

90

83

77

67

59

50

40

83

28

IJBO

121

109

100

92

85

74

66

57

46

38

88

•

2

127

115

106

98

91

80

71

61

50

42

87

1

1

8

136

124

114

106

99

87

78

68

66

48

42

4

142

130

120

111

104

93

83

73

61

62

46

6

149

137

127

119

111

100

90

80

67

68

61

10

158

146

185

127

120

108

98

88

76

66

69

s

15

164

151

141

133

126

114

104

94

81

72

64

20

167

155

145

137

180

118

108

98

85

76

68

30 .

172

160

160

142

185

128

118

103

90

81

74

^

r J02S

40

85

80

26

24

20

17

18

10

8

7

^

.05

52

44

89

34

81

26

22

18

18

11

9

^

a

66

67

60

44

40

84

29

24

18

14

12

1

fl .

.2

79

69

62

.56

61

48

87

80

28

19

16

1

2s

.8

87

77

69

62

67

48

42

35

27

22

18

1

• o

.4

93

83

74

67

62

63

46

38

80

26

21

1

M a

.6

102

90

82

74

69

69

52

43

84

28

24

2

1.

111

100

90

83

77

67

59

50

40

88

28

2t

2*^

IJi

118

107

97

90

83

73

66

55

45

88

88

2}

2

128

111

102

94

87

77

68

59

48

41

36

31

8

129

117

106

100

93

83

74

64

63

46

40

3£

lU

4

188

121

112

104

97

86

77

68

56

49

48

3fi

«s

6

138

126

117

109

102

91

82

72

61

63

47

42

Sr

10

143

131

122

114

107

96

87

78

66

58

52

47

e

16

147

135

126

118

111

100

91

82

70

62

56

51

B

20

150

137

128

120

113

103

94

84

72

64

68

63

80

152

140

131

123

116

106

97

87

76

68

62

67

^

.025

47

40

85

31

28

22

19

15

11

[ 9

7

6

■*•

.05

59

50

44

40

35

,29

25

20

15

12

10

8

•So

.1

72

62

55

50

45

37

32

26

19

16

13

11

S V

.2

84

74

66

60

54

46

39

32

25

20

17

14

1=

.8

91

81

73

66

60

51

44

37

28

23

19

17

.4

97

86

77

70

64

65

48

40

31

25

21

18

8*^

^9 II

.«

104

92

83

76

70

60

53

45

35

29

25

21

1.

111

100

90

83

77

67

59

60

40

33

28

25

1.5

117

105

96

88

82

72

64

54

44

37

32

28

2

120

109

100

92

85

75

67

57

47

40

34

30

4

128

116

107

99

92

82

73

64

53

46

40

36

Is

6

181

119

110

102

96

85

77

67

56

49

43

39

Pk2

10

135

123

114

106

100

89

81

71

60

53

47

43

15

187

126

116

109

102

92

83

74

63

55

60

46

OD

80

141

129

120

112

106

96

87

78

67

69

64

60

.025

62

46

40

85

31

26

21

17

12

9

8

6

sS

.050

63

56

48

48

89

82

27

21

16

12

10

8

^ iO

.1

76

66

59

63

48

40

84

27

21

16

13

11

5? o

.2

87

77

69

62

67

48

41

84

26

21

17

15

SiH

.4

99

88

80

72

66

67

49

41

82

26

22

19

^1i .

.6

104

93

•84

77

71

61

63

45

86

29

25

22

%^

1

HI

100

90

83

77

67

59

60

40

33

28

25

It

2

118

107

98

90

84

74

65

66

46

89

84

30

4

124

118

104

97

90

79

71

62

61

44

89

35

0^

10

180

119

110

102

96

85

77

67

67

60

45

40

8*5

80

186

124

114

107

100

90

82

78

62

66

50

46

670

HYDRAULICS.

Table of eoefllclemt

Cy for mean radii In metres.— CosTTsumy.

•

a

Hean
radR

Goefficieots n of ronghness.

Mean
radR

^

^

meters

.009

.010

.011

.012

e

.018

e

.016

e

.017
e

.020
e

.025

e

.080

e

.086

e

.040

metres

o

«

e

"5

.026

66

47

41

87

88

27

22

17

18

10

8

7

J025

0eS

.060

66

68

61

46

40

88

28

23

17

13

11

9

.060

.1

78

68

61

66

60

42

86

28

21

17

14

12

.1

.2

90

80

70

64

69

49

42

86

27

22

18

16

J2

.8

96

86

76

70

68

64

47

89

80

24

21

17

.8

.4

99

89

80

78

67

67

60

42

82

27

22

20

.4

j6

106

94

86

78

72

62

64

46

86

80

26

22

.6

•

1

HI

100

90

88

77

67

69

60

40

88

28

26

1

1

2

117

106

97

89

88

78

66

66

46

88

84

80

2

1

4

128

111

102

96

88

78

70

61.

60

48

88

84

4

6

126

114

106

97

91

81

72

68

68

46

40

86

6

10

128

117

108

100

98

83

76

66

66

48

48

89

10

OB

80

182 1 121

112 1 104

98

87

79

70

60

62

48

481 80

.026

67

50

48

88

84

28

28

18

18

11

9

7

.025

Ox

sS

.060

69

69

62

47

42

84

29

28

17

18

11

9

.060

p* ^j

.1

80

70

68

66

60

42

86

80

22

17

14

12

.1

S.a

J2

90

80

72

66

60

60

48

86

27

22

18

16

.2

"*0

27

.8

96

86

77

70

64

64

47

89

80

26

21

18

.8

.4

100

89

81

74

67

68

60

42

88

27

28

19

.4

Si-

••d

.6

104

94

86

78

72

62.

64

46

86

80

26

22

.6

1

111

100

90

88

77

67

69

60

40

88

28

26

1

lU

2

116

106

9'7

90

88

72

64

66

46

88

88

29

S

•s

4

121

111

102

94

87

77

69

60

60

42

87

88

4

5^

6

124

118

104

97

90

80

71

62

62

45

40

86

6

So

10

127

116

106

99

92

82

78

64

64

47

42

88

10

«

. 80 180

119

110

102

96

86

77

68

68

61

46

42

80

^A

.026

69

60

44

89

86

28

24

19

14

10

9

7

.026^

.06

69

60

63

48

43

86

29

24

18

14

11

9

.06

.1

81

71

63

57

61

43

36

80

22

18

16

12

.1

.2

91

81

72

66

60

50

44

36

27

22

18

16

.2

S,-2

.8

97

86

77

71

65

65

48

40

81

25

21

18

.3

s.i.

.4

101

90

81

74

68

58

60

42

83

27

23

20

.4

.6

106

96

86

78

72

62

64

46

36

80

26

22

.6

:i

1.

111

100

90

83

77

67

59

60

40

83

28

26

1.

1.6

116

104

94

87

80

70

62

53

43

86

31

27

1.5

It

2

117

106

96

89

83

72

64

56

45

38

33

29

2

4

121

110

101

93

87

76

68

69

49

42

87

88

4 .

s^

10

126

114

105

98

91

81

78

64

58

46

41

87

10

L 80

129

118

108

101

95

84

77

67

57

60

46

41

80

For slopes steeper than .01 per unit of length. s> l in 100, the «>•
efficient e remains practically the same as at that slope. Tne velocUy^ howeTer,

being — cX l/mean radius X <A)pe, continues to increase as the slope beeomei
steeper.

To constrnet a dlagrram, fig 30 A, from which the Talnes ylTon

by Kntter's formnla may be taken by inspection.

Draw »z her, and say from 2 to 4 ft long; and oy yert at any point o within
say the middle third of xz. On oy lay oflT, as shown on the left, the Talues of «
for which the diagram will probably be used. If a scale of .06 inch, or .08?
metre, per unit of e be used, and be made to include e — 250 for £nglisn meas-
ure, or 150 for metric measure, oy will be about 1 ft long. For the sake of
clearness we show only the larger divisions in this and In what follows.

On o;i! lay off, as shown on its Upper side, the sqnare roots of all the Talnes of
the mean rad B for which the diagram is to be used. One inch per ft, or jM
metre per metre, of sq rt, is a convenient scale. Mart the dividing points wiUi
the respective values of the mean radii themselves.

Having decided upon the jiatle^ slope to be embraced In the diagram, saj

10 — 41.6 +

.0028

flattest slope per unit of length ^O' B»«li»h messuis.

»r

For CMb (nine of » to be ambriced In the dlifrmm, nj

To euh Tilusof r — H, iddu. thui obtaiolDK Tsluea of f. We like JMO0»
per DOtt of length u the lUUMt slope,* aud .01, .02, .03 end .04 for n.f Heno

(nelDg SnfUgfa n

LRU l.eil I.KIl

572 HYDRAULICS.

and y — 181.1 + 153.6, 90.5 + 153.6, 60.4 + 153.6 and 45.3 -f- 153.6;

or 334.7, 244.1, 214.0 and 193.9 respectively. Lay off these values of y on oy in
pencil, as at y^ y\ y'\ and y"\ using the scale already laid off for c on oy.

From each point, y, y' etc, draw a hor pencil line yt^ y't' etc, and mark on it,,
in pencil, the value of n used in determining its height oy etc.

Next say a; = «; X grecUesl value of n. Make ox = « by the scale of sqrti o(^
on 0 2. In our case ox— 153.6 X -04 — 6.144 by the scale of sq rts of B, or °» 6.14^
— 37.75 by the scale of R. • .

Divide ox into as many equal spaces (4 in our case) as .01 is contained in
greatest n. Mark the dividing points witn the values of n, as in our Fig.

From each dividing mark on ox erect a perpendicular, (xt^'^ etc) in pencil, to
cut that hor line {^y'V etc) which corre8i>onds to the same value of n. The
intersections are points in a hyperbola. Join them by straight lines H" t"y ff,
<'<etc.

From r in oz (corresponding to a mean rad of 3.28 ft, or 1 metre) draw radial
lines, rt, rt\ rt" etc. Mark them " n = .01 ", " n — .02 *' etc, the same as their
corresponding lines yty y' if etc.

For each slope (S) to be used in the diagram (except the flattest, for which
this has already been done) say

(0028 \
41.6 + -j — j X greater n, for English measure.

(00155 \
23 -h '—z j X greatest n, for metric measure.

Thus, our slopes are = .000025, .00005, .OOOi and .01 per unit of length. Henoe,

"'-(41.1

6 + ^) X .04 - 1.675.

Lay off each value of x', x" etc from oy on a separate hor pencil line </af etc,
using the scale of sq rts of B as on oz.

Mark each line o'ir etc in pencil with the slope used in fixing its length.

Divide each dist o'x' etc into the same number of equal parts as ox. From
the dividing points (which, like those of ox, represent the values of n) erect perpa
to cut the radial lines rf", rt" etc, each perp cutting that radial line which cor-
responds to the value of n represented by the point at the foot of the perp. The
intersections corresponding to each line o'x' etc form a hvperbolic curve. Mark
each curve with the slope of its corresponding line, ox, o'x' etc.

The drawing is now in the shape proposed by Mess Ganguillet and Kutter, and
is ready for use in finding either c, n, R or S when the other three are given.
Thus :

1st. Having R, S and n, to find c. For example let R >« 20 ft, S — .00006,
n — = .03. From the intersection d of slope curve .00005 and radial line n — .05,
draw* d-20 to the point (20) in oz corresponding to the given R. At «, where
d-20 cuts oy, is the reqd c, = 96 in this case.

2d. Having R, S and e, to find n. For example let R =« 20 ft, S = .00005,
c = 96. Through the points R = 20 in oa, and c = 96 in oy, draw* d-20 to cut
curve .00005. n (= .03) is found by means of the radial lines nearest to the in-
tersection, d.

3d. Having S, v and c, to find R. For example let S =s .00005, n = .03,
e == 96. Find curve .00005 and radial line n = .03. From their intersection d
draw d-2fi through the point e showing c = 96. Its intersection with oz shows
the reqd R, 29 in this case.
■ ■ ■ ■ ' - . ■ ■

* Instead of draining these lines, we may use a fine black thread with a loop at one end. Drira a
needle either into one of the points R or into one of the intersections, d ete. Slip the loop over the
needle. The other end of the thread i« held between the flnnfers. and the thread is made to cat th«
other points as reqd. The diagram nhould lie perfectly flat, and the siring be drawn tight at eaeh ob-
servation, in order that fyietion between string and paper may not prevent the utring from fnrmlnga
straight line. Or the free end of the rtring may rest on a pamphlet or other ohjecr about M Ineh thiek,
to keep the string clear of the diagram. Special care must then be taken to have the eye perp ever
Ike point observed.

HYDRAULICS. 673

4 til. Having B, e and n, to find S. For example let B — 20 f t, c —96,
n — .oa. Through R =- 2U and c — 96 draw d-20. S (.00005) is found bv meaoi
of the curves uearest to the point d of intersection of d--20 with radial line
% — .03.

The following addition to Kutter's diagram, proposed by Mr Rudolph Herinf,
Civil and Sanitary Kngiueer, Philadelphia,* enables us to rea^ tlie veloe"
Itjr from tbe dlaf^rBin.

Find the sq ri of the reciprocal of each slope to be embraced in the diagram

— '%}—. ., - . -r . Lay off these so rts on the right of oy, using

\ slope per unit of length ^ b y. s

tbe scale of c already laid off on its left. In our fig we have so proportioned the

c 15

two scales that — zrzzrzzr: ^ ~t"* -^^w* 'ho dividing points with tbe slopa

1/recip of 8 ^
per unit qf length.

On o» lay off the vels to be embraced in the diagram, using the scale of sq rts

vel c

of B already laid off on o«, and making ■- — ■

1/E 1/recip of S

1st. Having B, S and n ; to find ir. For example let B — 20 ft, 8 — .00005.
• — .03. From B — 20 draw d-20 to the intersection d of curve .00005 with radial
line n->.03. d-20 cuts oy at e, where c — 96. With a parallel ruler join B

— 20 with 8 — .00005 on oy. Draw a parallel line through o ■« 96. It cuts os at
in, giving the reqd vel, 3.03 ft per sec.

2d. Having B, 8 and v; to find n. For example let B — 20 ft, 8 — .00005,

V — 8.03 ft per sec. With a parallel ruler Join B •=» 20 and slope .00005 on oy.
Draw a parallel line through v — 3.03. It cuts oy at e, where e =» 96. Through
B — 20 aud c — 96, draw <^-20 to cut curve .00005. The *point d of intersection,
being on radial line n ^ .03, shows .03 to be the proper value of n.

Any line drawn to the curves from B — 3.28 ft or 1 metre, is one of the radial
lines used in making the diagram. It therefore necessarily cuts all the slope
turves at points showing the same value of n.

Sd. Having 8, n and v ; to find R. For example, let S — .00006, n — .03.

V » 3.03 ft per sec. Assume a value of B, say 10 ft. Find curve .00005 and radial
line n — .03. Join their intersection d with B i— 10 ft. The connecting line cuts
oy at 0 »- 82. With a'parallel ruler join c -• 82 with v =— 3.03. Draw a parallel
line through slope » .00006 on oy. it cuts o« at B — > 27.3, showing that a new
trial is necessary, and with an assumed B greater than 10 ft.

If B thus found is the same as the assumed one, the latter is correct. If they
«re nearly equal, their mean may be taken.

4tli. Having B, n and v ; to find S. For example, let B =>° 20 ft, n -= :08,

V ■- 3.03 ft per sec. Assume a slope (say .0001). Find its curve, and radial line
n "— .08. Join their intersection with B — 20. and note the value (89) of c where
tbe connecting line cuts oy. With a parallel ruler join c ■— 89 with v — 3.03.
Draw a parallel line through B — 20. It cuts oy at slope .000058, showing that
a new trial is necessary, and with an assumed a flatter than .0001. If B is 3.28
ft, or 1 metre, the diagram gives the correct^ at the first trial, no matter what
8 was assumed at starting. With any other B, if tbe diagram gives tbe same 8
as that assumed, the latter is correct. If the two differ but slightly, we may take
their mean.

• TnuiMkotfoDt of th« AMflrioan Society et Clril BBgiaeera, jABvary lt7t.

574

VELOCITIES IN SEWERS.

Table of Tels in Circnlar Brick Sewers when tuddIiir ftill, by
Kntter's formula, but taking n at .016 instead of his .013, in consideration

of the rough character of sewer brickwork generally*

When rniininir only linir full the vel will be the same as when full,
but this is not the case at- any other depth whether greater or less. At greater
ones it increases until the depth equals very nearlv .9 of the diam, when it ii
about 10 per cent greater than when either full or half full. From depth of .9 of
the diam the vel decreases whether the depth becomes greater or less. At depth
of .25 diam the vel is about .78 of that when full ; and then diminiahes maoh
more rapidly for less depths. All this applies also to pipes.

The vel for any fkll or diam intermediate of those in the table can be found by
simple proportion. OriginaL

FaU

in ft

per

mile.

.1
.2
A
.6
.8
1.0
1.25
1.50
1.75
2.0
2.5
8.0
8.5
4.
5.
6.
7.
8.
9.

10.

12.

15.

Ig.

21.

24.

27.

80.

85.

40.

45.

60.

60.

70.

80.

90.
100.

3

.19
.90
.46
.59
.69
.79
.89
.98
1.06
1.15
1.32
1.44
1.58
1.68
1.90
2.06
2.2
2.4
2.5
2.7
2.9
3.3
3.6
3.9
4.2
4.5
4.7
5.0
5.4
5.6
5.9
6.5
7.0
7.4
7.9
8.4

8

IMeaieten In ftet.

6

8

12

16

Teloeltiee in ftet per aeeoiid.

.27

.35

.50

.42

.53

.74

.65

.80

1.08

.81

1.00

1.35

.95

1.17

1.57

1.07

1.82

1.77

1.21

1.49

1.98

1.33 .

. 1.64

2.18

1.44

1.78

2.34

1.55

1.91

2.63

, 1.78

2.18

2.85

1.94

2.38

3.2

2.10

2.58

3.4

2.2

2.7

3.6

2.5

3.1

4.1

2.7

3.3

4.4

3.0

3.6

4.8

3.2

3.8

5.1

3.4

4.1

5.4

8.5

4.3

6.7

3.9

4.8

6.3

4.4

5.4

7.1

4.8

5.9

7.7

5.1

6.3

8.4

5.5

6.8

8.9

5.9

7.2

9.5

6.2

7.5

9.9

6.7

8.2

10.8

7.1

8.7

11 :5

7.5

9.2

12.2

8.0

9.7

12.8

8.7

10.7

14.1

9.4

11.5

15.2

10.1

12.8

16.2

10.7

13.1

17.2

11.3

13.8

18.2

.64
.93

1.39

1.70

1.94

2.16

2.42

2.64

2.85

3.1

3.5

3.8

4.1

4.4

4.9

5.4

5.8

6.2

6.6

6.9

7.6

8.5

9.3
10.0
10.8
11.4
12.0
18.0
13.9
14.8
15.5
17.0
18.4
19.7
20.9
22.0

.89
1.26
1.81
2.22
2.66
2.84
3.17
8.5
3.8
4.0
4.6
5.0
5.3
6.7
6.3
6.9
7.5
8.0
8.5
9.0
9.9

11.0

12.1

13.0

13.9

14.8

15.6

16.8

18.0

19.1

20.1

22.1

23.9

25.5

27.0

28.6

1.10

1.56

2.20

2.70

3.08

3.43

3.8

4.2

4.6

4.8

5.4

6.0

6.5

6.9

7.6

8.3

9.0

9.7

10.3

10.8

11.9

13.3

14.5

15.7

16.8

17.9

18.8

20.4

21.7

28.0

24.2

26.6

28.6

31.0

32.3

84.1

20

1.84

1.84

2.60

8.18

8.60

3.96

4.5

4.9

5.3

5.6

6.3

6.9

7.4

7.9

8.7

9.6

10.4

11.1

11.8

12.6

13.6

16.3

16.7

17.9

19.2

20.4

21i>

28.2

24.8

26.3

27.7

80.3

82.8

86.0

87.1

89.1

FaU

In ft

per

100 fL

.0019
>OQM
.0076
.0114
jD161

.orn
sum

.03M

.om

.0979
.0479

.0662
.0768
.0947
.1196
.1325
.1514
.1708
.1894
.2278
.2841
.8409
.8876
.4546
.6109
.5682
.6629
.7676
.8528
.9470
1.186
1.826
1.516
1.706
1.894

A vel of 10 fit per see =« 600 ft per minute = 86000 ft, or 6.818 miles per
hour. About 5 ft per sec is as great as can be adopted in practice to prevent the
lower parts of the sewers from wearing away too rapidly by the deoris carried
along by the water.

HTDRAULIC8.

676

Art. 83. The rate mt whieh rain irater readies a sewer or

culvert, etc. ^urlLti'^iegler Fonunla. See "European iSeweri^
Systems/' hj Kudolph Uering, C. E., iu Trans. Am. Soc C. £., Nov. 1881.

Cub. ft. per ^ ^^^ ^^ ^^ ^ ^^ rainfall

according X per second per acre,

to judgment during heaviest fall.

second per
acre, reach-
ing sewer

M

Av. slope of ground
in feet per 1000 ft.

No.of acres drained

C«»eflicleiit, for paved streets, 0.75 ; for ordinal^ cases, 0.625 ; for suburbs
with gardens, lawns, and macadamized streets, 0.81.

Note that 1 inch of rainfall per hour may be taken as equivalent to 1 cubic foot
per second per acre. See Conversion Tables, pp. 285, etc.

Example. If an area of 8100 acres (nearly 5 square miles), with an average
slope of 6 feet per 1000 feet, receives a maximum rainfall of 8 inches per hour,
then, assuming a coefficient of 0.6. the rate at which the water would reach th«
mouth of a sewer at the lower end of the 8100 acres would be

0.6X8X

V

6 .

= 0.6 X 3 X 0.203 => 0.305 cubic feet per second per acre;

or 0.306 X 8100 = 945.6 cubic feet per second, total.

Let the grade of the intended sewer be say 4 feet per mile; a/Id, to avoid
excessive wear of its brickwork by debris swept along by the water, let its.
velocity be limited to 6.8 feet per second, which may be permitted on occasions
as rare as rains of 8 inches, per hour, although, for tolerably constant flow, where
liable to debris, it should hot exceed about 5 feet per second.

Find, in table opposite, the diameter, 14 ft., corresponding, as nearly as may
be, to a velocity of 6.3, and to a grade of 4 feet per mile. The area is 164
square feet. Hence, 154 X 6.8 :£» 970 cubic feet per second = capacity of sewer.
Tx> allow for deposits in the sewer, make the diameter say 14.5 or 15 feet

Table of least Telocitles and grades for drain-pipes and
se^rers in cities, in order that they may under ordinary circumstances keep
themselves clean, or free from deposits. (Wicksteed.)

Grade.

Orade.

Dlam.

Tel. In ft.

Onde,

Feet per

Dlam.

Yel. in ft.

Orade,

Feet per

In luohM.

per Min.

lin

Hile.

in iDohee.

per Min.

lin

Mile.

4

X40

86

146.7

18

180

294

18.0

6

S20

65

81.2

21

180

843

15.4

7

220

76

69.5

24

180

892

13.5

8

220

87

60.7

SO

180

490

10.8

9

220

98

5S.9

S6

180

588

9.0

le

210

119

44.4

42

180

686

7.T

11

200

145

36.7

48

180

784

6.8

I'i

190

175

S0.2

54

180

882

6.0

15

180

244
1

S1.6

60

ISO

980

5.4

lireiflrlit per foot run of arlaaed terra cotta pipes for drains, etc.;

g trices per foot run adopted l)y the United Sewer Pipe Makers of the United
tates, March, 1887. For discounts, see Price List.

Drain pipe, with socket joint

Sewer pipe, with sleeve joint

Bore

Wt

Price

Bore

Wt

Price

Bore

Wt

Price

Bore

Wt

Prict

ins

lbs

S

ins

lbs

$

ins

lbs

9

ins

lbs

s

2

4

0.14

6

18

0.30

15

45

1.25

80

150

5.60

8

7

0.16

8

22

0.45

18

65

1.70

36

195

7.00

4

10

0.20

10

30

0.65

21

89

2.50

42

203

8.50

5

12

0.25

12

33

0.85

24

100

3.25

48

230

10.60

The joints are filled with cement mortar; or. when used for drainage only,
with clay. Drain pipes (3 to 12 ins bore) are about { inch thick. A bend or
branch costs about as much as from 3 to 6 feet of pipe. The 48-lnch pipes are
about 2 ins thick.

Art. 24. Wben the area of cross section of channel is re-
duced at any point, as by a dam (Fig 83, p. 576) or by narrowing it, either
at its sides (Fig 32) or by placing in it a pier etc« Fig 34; a portion at least of
the force of grav (which would otherwise be giving vel to the water up-stream
from the point where the obstruction takes place), csLuaea pressure against the
dam etc. Thia pres maintains the up-stream water at a nigher level than it

676

HYDRAULICS.

would otherwise have. Said water is then practically in a rtservoli", i e, it hu
less Tel aud greater pres than before. If the reservoir has uo outlet, there is no
vel ; and all of the head, or force of grav, acting on the water is expended in pres.

But if there is an outlet, as over ihe dam, or between the piers etc, a portion
eo, Figs 31. 33, 34, of this pres ur head, is expended in giving vel (ur an accelera-
tion of vel) to the water escaping by that outlet; after which only so much
head (in the shape of surface slope) is needed as will overcome the resistances
of the channel oown-stream from the obstruction, and so maintain uniform the
vel given to the water by the head co.

Where a large canal, such as chose intended for navigation, is fed from a reser-
Toir, the fall co in feet is approximately

B* mean velocity* in canal, in feet per secoud, X .017 ;

and in smaller canals, such as miircourses,

= mean velocity* in canal, in feet per second, X •02.

The abruptness of the fall may be diminished by rounding off or sloping ihe
edges of the piers, or the corners at the sides of the channel (Fig 32) or the
approach to the dam

Fig 33 is a cross section of Clecg's dam, across Cape Fear River, N. C. It
is from measurements made by Ellwood Morris, C £: by whom they were com-
municated to the writer. The dam is of wooden cribwork ; and its level crest,
8 ft 5 ins wide, is covered with plank ; along which the water glides in a smootl*
sheet, 6 ins deep, (at the time of measurement). At the upper end of thUir

sheet, and in a dist of about 2 ft, a head co of 9 ins forms itself, as in the fig.

Fif) 33

EigSl

Vi6 34

HTDBAULTCa o"

Art. SO. Scaup. Id ■ cfaianal of nnUDrni Mid e«Da(ut iloiH *iid cnm

•Botlon, tba Tel uf itae puiicia at irnUr ImmedlBtsly adjoining tta« bottom and

itlaa in the tlope orc^rou hcIjod occur, u iu tlielaM ■rCicle.lhaicoui iBgmtlf
increaoed io Ineir imnaediata nvlgbborhood.

dace an almwt Incredible Jmoal^ot'cout. ff IbTlAltom'fs'il itf o'^'a'jIelifiDii
Beaurinc action !■ aappoBdil to bs ■■ Bqaure of v«l.

To r«dnce Incbe* pep sec, to fert per minii(«, muutplr bj t.

S.'ffi!.?

n (wllh many ci>r»ctloail from Nicbolaoa'a

Of heada produced by obctpuctiouB to atPC

■^|A|>| tl'IMH M*

i pradaeed at aad bj

laeed at aad by

tbeae vela moat, aooorq^

Pnpoition of Am of VtUr«t;, Moiptad bj ths Obitnuitloni.

A I A I t I * I t I i' I i I t I i

.n

TalodtT prodnood at tho Obatenetioii in Fan per Seoond.

578

HYDRAULICS.

Airt. 26. The resistance of water afralnst a flat sarfaee moT-
IniP tlii*oa«rli it at riirlit anffles, is nearly as the squares of the toI ; and,

aocording to Button, its amount in fi>s per sq ft approx » Square of vel in ft per
sec. Or like the pre* of a rnniiiiiar stream agunst a perp fixed flat

BorfMe, it la = wt of a eol or water wboM lMuie = pr6MM sarf, and wbOM bt=:h«ad dae to Um t*!

The realit of a ■phere li to that of iti great circle about aa 1 to S.9.

When th« moring rarf. Instead of being at right angles to the direction in whloh it morea, formi
another angle with it, the resistance becomes less in about the following proportions. Therefcwe,
«rhen the sarf is inclined, first calculate the resistance as if at right angles ; and then mult bj the
bllowing decimals opposite the angle of inclination :

90°.... 1.00

60^. ... .oo

40° 68

90O 16

oil • • • • •Wf

56 88

96 46

16 10

70 96

80 76

SO 34

10 06

66 . • • • •99

40 ■ • a • «Q0

3v • • « « kSSv

6 09

Tbe scour, or abrading power of moving water is considered to

to as the square of its TtL

Art. 27. To calealate tlie borse-power of flslliugr water, on

the ordinary assumption that a horse-power is equal to SSOOO lbs lifted 1 foot rert per min. That of
average horses is really but about % as much, or 22000 lbs, 1 foot high per rain. Mult together the
anmber of cub ft of water which fall per min ; the vert height or head in feet, through which It faUa;
and the number 02.3, (the w^ of a cub ft of water in lbs :) and div the prod by 98000. Or, by formula.

»aa« TV " wa •• wma/ i« v« n ^wca aaa «va %^ ■aaa^* v

cub ft ^ vert v^ Us
Th4 numb«r of __ per min ^ height in/t '^ 6'i.8
*orse-j>ot0«ri

83000.

lin. Haw

Over a fall 16 ft in Tert heifbt, 800 enb ft of water are disehd per
powtM doM the Call aflbrd 7

•nbfl ft As
800X16X6a.8_ 7OT440

■•^ 88000 nooo = **•" ^'^'

Watei**WbeelS do not realise all the power inherent in the watar, as temaA by
ml*. That, underanots realise but from }ito^i breast-wheeils, M ; oversbott, from H*»9ii tvr>
bines, K to .86 of it ; aocording to the skill of design, and the perfeetien of workmanship. Ktob '

the wheel revolTes in a close-fitting casing, or breast, elbow buckets give considerably mora powi
than plain radial or oenter- buckets. Of the power actually received by a wheel, part is ezpanood In
friction, Ac ; while the remainder does the ue^ul or paifiiig net work of raising water, griadinc
grain, sawing, Ac.

Observations by Oeifti Hanpt, in 1866, gaye the following results for a
small hydranlic ram. Head of water to ram = 8.812 ft ; diam of driTe-pipe »
IH ins; length 16 ft. Diam of delivery -pipe =r H ioeh; length 300 ft. Tert height to whiah tha
water was raised by the delivery-pipe, 68.4 feet. Strokes of ram per min, 170. Quantity of waMr
which worked the ram = 768 cub ins, = 8.81 galls, = 37.78 Iba per min. Quantity raised 68.4 ft Ugh

per min, = 48 cub ins, = 1.786 %b. Hence the power expended per min, wma 97.78 X 8.813 = SiiJk

Bm watar ft ft-lbs

And the uaefU efSMt, was 1.736 X 63.4 = 110.06. Hence the ratio whioh the ue^ful efect bears to tha

110.06
power in this instance, is ■ , or .46. The oeltMri power of the ram is, however, greater than tUa,

inasmneh as it has to overcome the fHotion of the water along the delivery-pipe.*

To find the horse-power of a mnniniT stream.

with simple float-boards.t instead of b '

Water-wheels
' backets, are sometimes driven by tbe mere force of the ordiaarr
natoral current of a stream, without any appreciable fall like that in tha foregoing case. In aoeh
eaHCs, we must substitute the virtual or theoretic head ; which is that which would impart to it tha
same vel whioh it actually has. This virtual head may be taken atonoefrom Table, n. 689. Tbua, a
stream has a vel of 2.886 miles per hour; or 310 ft per min ; or 3>^ ft per see ; and in the column of
heads in Table 10, opposite to 3.6 vel per sec, we find the reqd head .190 of a ft. Having thus twamA
■the head, we must now find the quantity of water which passes any given area of the stream ia a
min. Thus, suppose that the immersedTpart of a float when vert is 6 ft long, and 1 ft wide or deep;
then the area of this part whleli receives the force of the current, ia 6 X 1 = 6 square feet. Haaaa,

6 sq ft X 310 = 1060 cub ft per min. Having now the cub ft per min, aad the vert height or haei,
the number of horse-powers of the etream of the given area, is found by tha foregoing rule, or formula.

* A committee of the Franklin lustitote, in 1860, gave .71 aa the

ooeflMent for a ram at the Girard College, in which the diam of drive-pipe was 3H ins ; its laagth,
160 ft; fall, 14 ft. Delivery-pipe, 1 inch diam ; 2360 ft long : vert rise, or height to which the wstv
was raised. 93 ft. No details of the experiment are given. Some large rams In France give a ue^ftU
effect of from .6 to .66 of the whole power expended. It is an exoellant machine for many porpaaw;
and is sometimes used for filling railway tanks at water stations.

t Snch wheels, for floatinsr mills, in Europe, rarely ezoeed 16 ft
diam. Whatever the diam, they may have about 18 to 30 floats. The floaU are fk«m 8 to If ft last;

and about ^ to ^ as deep aa the diam of the wheel. They should not dip their anllra depth late

the water, but nearly so. They should not be in the same straight line with tha radii ; bat ahooM
InoUne from them 309 up stream, to produce their full effisot. All thaaa remarks apply to whaala
moving freely in a wide or indefinite channel ; as in the ease of a floattag mill, hailt oa a aoew, aai
anchored out in a stream : but not to wheels for whioh tbe watar is dammed op, aad aeta with a prat*
^oal fall. No great exaotaess is to be expected ia rules on this suhjMt. The beat vri tar Mm wkMl
-Aeat .4 that of the stream.

HTDBAUIilCS.

tmbftper wUn ^ ««r( ht teik ^ (&« »

No of ^ 1050 ^ .180 ^62.S 12420 J

A Pow, ~ 33000 ~ 33Q0Q — .877 o/ a A fbw.

But in practice the wheels actually realize but about 0.4 of t^iis power qf
the stream. Therefore, the actual power of our wheel will be but .877 x .4 ^
0.1508 of a horse power; or 33000 X .1508 = 4976 ft-Ibs per minute. Making
a rough allowance for the friction of the machine at its journals, &c, we
should have about 4400 ft-fi>s of tts^vZ power per minute ; that is, the wheel
would actually raise about 440 lbs 10 it high ; or 44 lbs 100 ft high, &c, pet
min. The vel of the stream must not be measured at the surface : but at
about 14 of the depth to which the floats are to dip, or be immersed. This,
however, is necessary chiefly in shallow streams, in which the depth of
the float bears a considerable ratio to that of the water.

Tills power of a ranning^ stream, (for any ariven area of
transverse section.) increases as the cnbes of the tcIs: for, as
we have seen, the power in tt-fbs per min is found by mult together the
weight of water which passes through the section in a min, and tne virtual
head in ft ; and since tnis weight Increases as the vel, and this head as the
sauare of the vel, the prod of the two (or the power) must be as the cube
01 the vel. Therefore, if the vel in the foregoing case had been 10.5 ft per
sec, or 3 times 3.5 ft, the power of the wheel would have been 27 times as
great, or .1506 X 27 == 4.07 norse powers.

\

I>BEDOIXa.

in

S?

DBEDGINQ.

iUDQiHG is generally done by skilled contractors, who own the requisite machiaea,
W8 or lighters, Ac ; and who make it a specially. It Is necessary to specify whether
.e dredged material is to be measured in place before it is loosened ; or after being
deposited in the scow: because it occupies more bulk after being dredged. It was
found, in the extensive dredgings for deepening the Rirer St Lawrence through the
Lake of St Peter, that on an arerage a cub yd of tolerably stiff mud in place, makes
1.4 yds io the scow ; or 1 in the scow, makes .716 in place. Also stipulate whether the
remoTal of bowlders, sunken trees, Ac, is to constitute an extra. These often require
sawing and blasting under water. The cost per cub yd for dredging varies much
With the depth of water ; the qiutntity and character of the material ; the dist to which
it has to be removed ; whether it can be at once discharged from the machine by
meatas of projecting side-shoots or slides ; or must be discharged into scows, to be re-
moved to a short dist by poling, or to a greater dist by steam tugs ; whether it can be
dropped or dumped into deep water by means of flap or trap doors in the bottom of
Uie hoppers of the scows ; or must be shovelled from the scows into shallow water, (at
say 4 to 8 cts per yd ;) or upon tond, (at say from 6 to 10 or 20 cts for the shovelling
alone, or shovelling and wheeling, as the case may be ;) whether much time must be
consumed in moving the machine forward frequently, as when the excavation is
narrew, and of but little depth ; as in deepening a canal, Ac ; whether many bowI->
ders and sunken trees are to be lifted ; whether interruptions may occnr from waves
in storms ; whether fuel can be readily obtained, Ac, Ac. These considerations may
make the cost per cub yd in one case firom 2 to 4 times as great as in another. The
actuai cost of deepening a ship-channel through Lake St Peter, to 18 ft, from its orig>
inal depth of 11 ft, for several miles through moderately stiff mud, was 14 cts per
cub yd in place, or 10 cts in the scows ; including removing the material by steam
tugs to a dist of about }^ a mile, and dropping it into deep water. This includes f
pairs of plant of all kinds, but no profit. It was a favorable case. When the buckets
work in deep water they do not become so well filled as when the water is shallower,
bscause they have a more vertical movement, and, therefore, do not scrape along as
great a distance of the bottom. Hence one reason why deep dredging costs more
per yard ; in addition to having to be lifted through a greater height. Perhaps the
following table is tolerably approximate for large works in ordinary mud, sand, or
gravel ; assuming the plant to have been paid for by the company ; and that common
labor costs $1 per day.

Table of actaal cost of drediplngr on a lariro scale; inelod*
lofT dropplnir tl>« material info scows, alongside: or into
side-slioots, on board. Common labor $1 per day. Repairs
of plant are included ; but no profltto contractor. (Original.]

Depth
in Ft.

Cta per Yard,
in place.

CU per Yard,
In soow.

Depth
in Ft.

Cts per Yard,
in place.

CU per Yaad,
in aoow.

Lmi than 10
10 to 15
15 to JO
aOtott

8.4
9.8

ii.a

14.0

6

7

8

10

S5toS0
80toS5
85to40

18.1
2S.S
S6.0

18

18

r

For towing of the SCOWS by steam tugs to a distance of »»< mile, and dropping
the mud into deep water, add 4 cts. per yard in the scow ; for }i mile, 6 cts. :
for % mile, 8 cts. ; for 1 mile, 10 cts. Add profit to contractor. On a small
scale work is done to a less advantage ; and a corresponding increase must
be made in these prices. Also, if the contractor himself furnishes the
dredgers and plant, a still further addition must be made. It is evident
that the subject admits of no great precision. Small iobs. even in favora-
ble material, but in inconvenient positions, may readily cost two or thrde
times as much per yard as the above : and in very hard material, as In
cemented gravel and clay, four or five times as much'for the dredging. The
cost of towing, however, will remain as before, if wages are the same.

The cost of dredgers, tugs, &c., will vary of course with their capabilities.
strength of construction, style of finish, whether having accommodations
for the men to live onboard or not, &c. "When for use in salt water, the
bottoms of both dredgers and scows should be coppered, to protect them worn
sea-worms ; and if occasionally exposed to high waves, both should be
extra strong. The most powerml machines on the St. Lawrence cost about

DREDGIKQ. 581

$45^000 each ; and removed in 10 working hours on an average about 1800 cubic
yards in place, or 2520 in the scows. Good machines, capable, under similar
circumstances, of doing as much, may, however, be built for about $25,000
to $30,000. To remove this quantity to a distance of ^ to 1 mile, would require
two steam tugs, costing about $6000 to $10,"000 each ; and 4 to 6 scows (some
to be loading while others are away), holding from 30 to 60 cubic yards each,
and costing from $800 to $1500 each at the' shop. Scows with two hoppers
are best. Such a dredger would require at least 8 or 10 men, including cap-
tain, engineer, fireman, and cook ; each tug 4 or 5 men ; and each scow 2
men. The engineer should be a blacksmith ; or a blacksmith should be
added. In certain cases a physician, clerk, assistant engineer, &c., may be
needed.

Dredgers are often built on the principle of the Yankee Excavator, with but
a single bucket or dipper, of from 1 to 2 cubic yards capacity. Hull about 25
by 60 feet. Draft 3 feet. Cy Under about 7 or 8 inches diameter ; 15- to 18-inch
stroke ; ordinary working pressure 60 to 80 lbs. per square inch, according
to hardness of material. Cost $8000 to $12,000. Will raise as an average
day's work (10 hours) from 200 to 500 yards in place, or 280 to 700 in the scow,
according to the depth, nature of the material. &c. Require 5 or 7 men in
all aboard, including cook. Burn V^ to 1 ton or coal daily. Tolerably large
bowlders and sunken logs can be raised by the dipper.*

When the material is hard and compacted, the buckets of dredgers should
be armed with strong steel teeth projecting from their cutting edges. On
arriving at such material every alternate bucket is sometimes unshipped:
By arranging the buckets so as to dredge a few feet in advance of the hull,.
low tongues of dry land may be cut away : the machine thus digging its-
own channel. The daily work in such cases w ill not average half as much
as in wet soil.

On small operations, dredgrers worked by two or more borses,

instead of by steam, will answer very well in soft material ; or even in
moderately hard, by reducing the size and number of the buckets. A two-
hone machine will raise from 50 to 100 yards of ordinary mud in place, or
70 to 140 in the scow, per day, at from 12 to 15 feet depth.

Soft material in small quantity, and at moderate depth, may be removed
by the slow and expensive mode of the bag^-Bcoop, or bag^-Bpooii.

This is simply a bag B, made of canvas or leather,
and having its mouth surrounded by an oval iron
ring, the lower part of which is sharpened to form
a catting edge. It has a fixed handle A, and a
swivel handle ». One man pushes the bag down
into the mud by h, while another pulls it silong by
the rope g; and when filled, another raises it by the
rope c, and empties it. If the bag is large, a wind-
lass may be used for raising it. The men may woiit
from a scow or raft property anchored. Or a long-
handled metal spoon, shaped like a deeply-dished hoe, may be used by
only one man ; or a larger spoon may be guided by a man, and dragged
forward and backward by a horse walking m a circle on the scow, <Sec., dec.

Tbe weiirbt of a cubic yard of wet dredged mud, pure sand, or
gravel, averages about IV^ tons; say 111 lbs. per cubic foot; muddy gravel,
mil 114 tons ; say 125 &»s. per cubic foot. Pure sand or gravel dredges easily ;
also beds of shells. Wet dredged day will slide down a shoot inclined at
from 5 to 1, to 3 to 1, according to its freedom from sand, Ac. ; but wet sapd
or gravel will not slide down even 3 to 1, without a free flow of water
to aid it ; otherwise it requires much pushing.

* The writer has seen cases in which a circular saw for logs in deep
water would have been a very usefhl addition to a dredger. It should;[be
worked by steam ; and be adjustable to different depths. It would cost but
about $500.

582 FOUKDATIONS.

FOUNDATIONS.

A TOLUMK might be occupied by this important subject alone. We have space for
•nly a few general hints ; leaving it to the student to determine how far they may
be applicable in any given case. In ordinary cases, as in culverts, retaining walls,
kc, if excavations, or wells, &c, in the vicinity, have not already proved that the soil
is reliable to a considerable depth, it will usually be a sufScieut precaution, after
having dug and levelled off the foundation pits or trenches to a depth of 3 to 5 ft. to
test it by an iron rod, or a pump-auger ; or to sink holes, in a few spots, to the depth
of 4 to 8 ft farther ; (depending upon the weight of the intended structure ;) to ascer-
tain if the soil continues firm to that distance. If it does, there will rarely be any
risk in proceeding at once with the masonry ; because a stratum of firm soil, from 4
to 8 ft thick, will be safe for almost any ordinary structure ; even though it should
be underlaid by a much softer stratum. If, however, the firm upper stratum is ex-
posed to running water, as in the case of a bridge-pier in a river, care must be taken
to preserve it from gradually washing away; or from becoming loosened and broken
up by violent freshets ; especially if they bring down heavy masses of ice, trees, and
other floating matter. These are sometimes arrested by piers, and accumulate so as
to form dams extending to the bottom of the stream ; thus creating an increase of
Telocity, and of scouring action, that is very dangerous to the stability both of the
bottom and of the structure. When the testing has to be made to a considerable
.depth, it may be necessary to drive down a tube of either wrought or cast iron, to
prevent the soil from falling into the unfinished hole. If necessary, this tube may
be in short lengths, connected by screw Joints, for convenience of driving ; and the
•urth inside of it may be removed by a small scoop with a long handle.*

Borlng^s in common soils or clay may be made 100 feet deep in a day
or two by a common wood auger 1^ inches diameter, turned by two to four
men with 3 feet levers. This will bring up samples.

in starting the masonry, the largest stonte should of coarse be placed at the bot-
tom of the pit, so as to equalize the pressure as much as possible; and care should
be taken to bed them solidly in the soil, so as to have no rocking tendency. The
next few courses at least should be of large stones, so laid as to break Joint thoroughly
with those below. The trenches should be refilled with earth as soon as the masonry
will permit; so as to exclude rain, which would injure the mortar, and soften the
foundation. It is well to ram or tread the earth to some extent as it is being deposited.

If the tests show that the soil (not exposed to running water) is too soft to support
the masonry, then the pits should be made considerabl}- wider and deeper; and afte]>
Ward be filled to their entire width, and to a depth of from 8 to 6 or more ft, (de-
pending on the weight to be sustained,) with rammed or rolled layers of sand, gnvel,
or stone broken to turnpike size ; or with concrete in which there is a good propor-
tion of cement. On this deposit the masonry may be started. The common practice
in such cases, of laying planks or wooden platforms in the foundations, for building
upon, is a very bad one. For if the planks are not constantly kept thoroughly wel»
they will decay in a few years ; causing cracks and settlements in the masonry.

Some portions of the brick aqueduct f for supplying Boston with water gKW
a great deal of trouble where its trenches passed through running quicksands aad
other treacherous soils. Concrete was tried, but the wet quicksand mixed 'itself
with it, and killed it. Wooden cradles, Ac, also failed ; and tne difficulty was finally
oyercome by simply depositing in the trenches about j;wo feet in depth of strong
gravel.) Sand or gravel, token prevented from ^reading sidewayK^ forms one of the
best of foundations. To prevent this spreading, the area to be built on may be sur^
rqunded by a wail; or by squared piles driven so close as to touch each other; or in
less important cases, by short uheet piles only. But generally it is sufficient simply

* Subterranean caverns in limestone regions are a frequent source of trouble,
against which it is difficult to adopt precautions.

t The Cocbituate aqueduct, built 1846-48 ; egg-shape, 6 feet 4 inches X 5 feet,
with semicircular invert.

X Smeaton mentions a stone bridge built upon a natural bed of gravel only
about two feet thick, overlying deep mud so soft that an iron bar 40 feet long
sank to the head by its own weight. C>ne of the piers, however, sank while the
arches were being turned, and was restored by Smeaton. Although a wretched
precedent, for bridge-building, this example illustrates the bearing power of a
thick layer of well-compactM gravel.

FOUNDATIONS. 683

to give the trenches a good width ; and to ram the sand or gravel (which are all
the better if wet) in layers; taking care to compact it well against the sides of
the trench also. Under heavy loads, some settlement will, of course, take place,
as is the case in all foundations except rock. If very heavy, adopt piling, Ac.
See Grillage.

Wlien an unreliable soil OTei*lie« a firm one, but at such a
depth that the excavation of the trenches (which then must evidently be made
wider, as well as deeper) becomes too troublesome and expensive; especially
when (as generally happens in that case) water percolates rapidly into the
trenches from the adjacent strata, we may resort to piles. When

making deep foundation-pits in damp clay, we must remember that this
material, being soft, has, to a certain degree, a tendencv to press in every direc-
tion, like water. This causes it to bulge inward at the sides, and upward at the
bottom. The excavations for tunnels, or for vertical shafts, often close in all
around, and become much contracted thereby before they can be lined ; there-
fore they should be dug larger than would otherwise be necessary. The bottoms
of canal and railroad excavations in moist clay are frequently pressed upward
by the weight of the sides. I>ry clay rapidly absorbs moisture from the air,
and swells, producing effects similar to the foregoing. lU expansion is attended
by ^reat pressure ; so that retaining-walls backed with dry rammed clay will
be m (danger of bulging if the cluy should become wet. It is a treacherous
material to work in. t or concrete foundations, see pp. 946, Ac

A« to the grreatent load that may safely be- trusted on an earth founda-
tion, Rankine advises not to exceed 1 to 1.5 tons per square foot. But experi-
ence proves that on good compact gravel, sand, or loam, at a depth beyond
atmospheric influences, 2 to 8 tons are safe, or even 4 to H tons if a few inches of
settlement may be allowed, as is often the case in isolated structures without
taremors. Years may elapse before this settlement ceases entirely. Pure clay,
especially if damp, is more compressible, afd should not be trusted with more
than 1 to 2.5 tons, according to the case. All earth foundations must yield some^
what. Equality of pressure is a main point to aim at Tremor in-
creases settlements, and causes them to continue for a longer period, especially
in weak soils. Great care must be taken not to overload in such cases, even if
piled. Foundations in silty soils will probably settle, in years, at the
rate of from 3 to 12 inches per ton (up to 2 tons) per square foot of quiet load,
if not on piles.

Figure 2 shows an easy mode of obtaining a foundation in certain cases. It
is the ** pierre perdue " (lost stone) of the French ; in £nglish, '* ran-
dom stone," or rip-rap.

It is merely a deposit of rough angular quarry stone thrown into the water ;
the largest ones being at the outside, to. resist disturbance from freshets, ice,
floating trees, Sx. A part of the interior may be of small quarry chi| s, with
some gravel, sand, clay, &c. When the bottom is irregular rock, this process
saves the expense of levelling it off to receive the masonry. I- or 2 or 8 feet
below the surface of the water, the stones may generally be disposed by hand, so
as to lie close and firmly. Small ^pawls pack^ between the larger ones will
make the work smoother, and less liable to be displaced by violence. Cramps
or chains may at. times be useful for connecting several of the large stones
together for greater stability. Rip-rap, boivever, is apt to settle.

If the bottom is so yield! n«r as to be liable to wash away in

freshets, it may, in addition, be protected, as in Fig 2, by a covering of the same kind

of stones, as at c : extend-
ing all around the struc-
ture. Or the main pile
of stones may be extend-
ed as per dotted line at d;
so that if the bottom
^^ _^ - a \ should wash away, as pef

W^^^ft4^^^t^^!^!^^^;!^^^?^^ dotted line at o, the

rU • i;^ -• stones d will fall inta

V * Pi ri 2 '^' ***® cavity, and thus pre-

■^^V ^ vent further damage.

Sheet-piles, * », may be
driven as an additional precaution. For greater security, the bed of the river may
be dredged or scooped under the entire space to be covered by the main deposit as
(•r dotted lines in Fig 3, to as great a depth as any scouring would be apt to react

rommATioKB.

■b>1 sIhl win 11 it guvTillT wnrM !■ pHlUn. ml IhM ink br Uniriu iUh t*H ■ »*
UorTlba hiwtr (diiif U> «Ui, H u iw u pniat Um crib IniB HUllllli^llf luHifH^ Hi

(•^ *ft IMj »■! I»oo™«l» •bc.rt or In Bin -lib plul.or pIiuIihT mj UhmiIm Hmtlfci
«Hd bj trvft Kirapi, Ad. ia dcetwaltr. ■ rbua^tloD uajrte nKnla pvll^tf m4(H ttHfl. h ta
PlftludAiiDdoD Lcror ibltKubiioahaGFlb. *1lb lu wpjiboulirtMDdtrlow wabir,uft^v«

On nnevCB rocb iMttnni It mi^ be necwmrf to tcrltn tb* bulion of th»

'K''''' '■•*J""f*'°L»^"'"- ""•'I ■""•■ iinjbitm»iiliiMlli«i«lM»d nllivM wtef

FOUKDATIONB. 585

A crib wltli only want ontalde row of eells for Binking it may ho

ballt } and tbt Intarior ohamber m»j be fUted with eonorate ondrn- water. The maaoDry may thea
reet on the oonorete alone. If the onh reiti upon a foandatlon of broken stone, the upper interetloee
ef thii Btoue should first be lerelled off by small stone or coarse gravel to reoelre the oonorete of the
inner ohamber.

Or a crib like Fir. 4 may be rank, and piles be driren in the cells, which

mar afterward be filled with broken stone or oonorete. The maaonrr owy then rest on the piles only,
which in tarn will be defended by the erib. If the bottom Is liable to sooar, plaoe sheet-piles or
rtp-rap aroand the base of the orib.

By all means avoid a crib lilce a» Fig: OU, much higher at one part
than at another, if the mpentructure. a ig to rest on tJw. timber of the crib instead of on

Cles, or en oonorete independent of the timber ; for the high part of the crib will oompress more under
I load than the low part , and will thus caune the snperstraotare to lean or to craok.

A crib either straight sided or circtilar, with only an outer row of cells for pnd«
diinir may be used as a cofl^rdam (see coffeidam8,p. 5M). The joints
betweeaihe eater timbers should be well eanlked ; and eare be taken, by means ef oatside pile-planks,
graTol,^, to prevent water from entering beneath it.

The east-iron Brid||e acroM the Schnylfcill at €be«tnnt St,
Pbila, Mr. Strickland Kneass, Engineer, affords a striking example of crib
foundation. The center pier stands on a crib, an oblong octagon in plan ; 31 by 87 feet at baae ; SA
by 80 ft at top ; and (with its platform) 29 ft high. ItN timbers are of yellow pine, hewn 12 ins
■qnare : and framed as at Fig 6. The lower timbers were carefully cut or scribed to conform to the
Irregolarities of the tolerably level rock npoQ which it rests. These were asoertalned (after the 6 ft
depth of gravel bad been dredged off) in the usual manner of mooring above the site a large floating
wooden ^attorm, oomposed of timbers corresponding in position with all those of the lower oonrso
of the intended orib, both longitudinal and transverse. Soundings were then taken olose together
along all these linee of timber. Most of the oella are about 3 by 4 ft on a side, in the clear. A few
of them had platforms at the level of the second course from the bottom, for receiving stone for sink.
ing the crib ; the others are open to the bottom.

The erib was bnilt in the water ; and waa kept floating, during ita eonatmotion, with ita unfiniahod
top oontinually jnst above water, by gradually loading it with more atone as new timbers were added.
The atone required for thia parpose alone waa 300 tons. When the erib was towed into position, and
moored, liSO tons more were added for sinking it. All the cells were afterward filled with roagh dry
stone, and coarse gravel sereenings ; making a total of 1866 tons. A platform of 12 by 12 ineh squared
timber eovered the whole ; iu top being 2^ ft below low water. The pier alone, whiob stands on this
erib, weighs 3266 tons; and during ita eonatrnetion tt eompreaaed the orib 6^ ina. The weight of
auporatrnotnre resting on the pier, may be roughly taken at 1000 tons more.

All ordinary ealmM>n Is merely a stronn; seow, or a box with-
out a lid ; and with sides whieh may at pleasure be readilv detached from its bottom. It is built on
land, and then launehed. The masonry may first be built in it, either in whole or in part, while
afloat; and the whole being then towed into place, and moored, may be sunk to the bottom of the
river, to rest upon a foundation prerionsly prepared for it, either by piling, if necessary ; or by
ply levoliiag off the natural surface, &e. The bottom of the caisson constitutes a strong timber

Elatform, upon which the masonry rests ; and
I so arranged, that after it is sunk, the sides
may be detached from it, and removed to be
rebottomed for use at another pier, if needed.
This detaching msy be effected bv some such
oontrivance as that shown in Fig 6, where
P P w is the bottom of the caisson, to whieh
are firmly attaobed at intervals strong iron
eves (; whieh are taken hold of by hooks d, a*
the lower end of long bolts I! n, reaching to
the top timbers S of the crib, where they are
oonflned by screw nuts n. By loosening the
nuts n, the hooks d can be detached, from the
eyes t; and the sides can then be removed
from the bottom ; there being no other connec-
tion between the two. These hooks and eyes
are usually placed outside of the caisson ; the
serew nuts n being sustained by the projecting
ends of cross pieces, as (f, Fig 9. 1 he im-
proper position given them in oar Fig was
merely for convenience of illustrating the prin-
eiple. It will sometimes be necessary to have
oao sMo doUobable trem the others, in order to float the caisson away clear from the flnished pier ;
vniesa it be floated away before the masonry has been built so high as to render the precaution use-
less. Fig 6 shows one of many ways of constructing a caisson ; with sides consisting of upright
eorner-posts, I; cap pieces S, un top ; and sills g at Iwttom, resting on the bottom platform P P w/
intermedlate uprights T, framed into the oaps and sills; the whole being covered outside by one or
two thicknesses of planking B, which, as well aa the platform, should be well calked, to prevent
leakiBg. Tarpaulin also may be nailed oatside to assist in this. The greatest trouble from leaking
is where the sides join the platform. On top of the platform is firmly spiked a timber o o, extending
all aroond tt Just inside of the Inner lower edge of the sides of the caisson. Its use is to prevent
the sid«s f^om being forced inward by the pressure of the water outside. The details of construction
will of eourse vary with the requirements of the case. In deep caissons, inside oroso-braces or struts
lk«m side to side, as at e e. Fig 7, will be required to prevent the sides from being forced inward by
the pressure of the water, as the vessel gradually sinks while the masonry is being built within it.
▲s the masonry is carried up, the strata are removed; and short ones, extending from the sides of
the oaisson to the masonry, are inserted in their plaoe. When the caisson is shallow, only the upper
eourse of braces will be required^ they also support a platform for the workmen and their materials.
In deep caissons, in order not to be la the way of the masons, the outer planking of the sides may.
In part, be gradually built up as the masonry progresses. It may sometimes be expedient to build
the masonry hollow at first, v^th *hiA trrASVerse wails inside to stiffen it if necessary ; and to com

586

FOUNDATIONS.

pleta the interior after ■InklDK ^* o&tuon. Indeed, muonry or brickwork* In cement, may thus %,^
bailt boUow at first, resting on tbe platform ; tbe masonry itself forming the sides of tbe caisson.
Or the sides may oonsist of a water-tight casing of iron, or wood, of the shape of tbe Intended pier,
Ac. This easing being confined to the platform, becomes, in fact, a mould, in which the pier muj be
formed, and sunk at the same time by filling it with hydraulic concrete. For
concrete foundations, see pp 946 Ac.

Ob roek bottoat the under timbers of the platform maj be cut to init tbe irregularities
M alnady stated nnder ** Orlbe." Or the bottom may be leTcUed op by first depositing lai^ stones
arennd the area upon which the caisson is to rest ; and then filling between these with smaller stones
and gravel; testing the depth by sounding. Or a level bed of oeuient concrete may, with care, be
deposited in the water. If there are deep narrow crevices in the rock, through which the concrete
may escape, they may be first covered with tarpaulin. Diving bells may often be used to advantage,
in all such operations. But in the case of very irregular rock, it will oftm be better to resort to cof-
fer-dams.

Talves for the admission of water for sinking the caisson are
usually introduoad. If, after sinking, it should be neoepsary to again raise the whole, it is only
necessary to close the ralves, and pump out the water. Guide piles may be driven and braced along*
side of the caisson, to insure its sinking vertically, and at the proper spot. Or it may be lowerod by
screws supported by strong temporary framework.

Assamtng the uprighto I, T, Ac, Fig 6, to be sufflelently bnuwd, M at oe, Fig T, the following tabto
will show the thickness of planking necessary for different distances apart of the uprights, (in the
•lear,) to Insure a safety of six against the pressure of the water at different depths ; and at the
same ttme not to bend inward under said pressure, more than -t^tt P*rt of the distance to which
they stretch fi-om upright to upright; or at the rate of ^ inch In 10 ft stretch ; H loch In 6 ft, *•.
Bnoh a table may t>e of use in other matters.

Table of tlilclLness of wblte pine plank required not to bend
more than 7^ part of lt« clear borlsontal stretcb, nnder
different beads of water. (Original.)

1

HEADS nr yxar.

Stretch

in Ft.

40

90 20 10

5

Thickness in Inches.

8

SH

8

2«

3M

IH

4

iH

4

SH

2«

2^

6

6?i

6

b%

s^

6

9

8

7

^H

*H

10

UH

10

SH

1

&H

I'i

liH

1214

lOH

6H

^H

15

UH

16

13

10^

6H

ao

22 J4

ao

1T«

14

11

Ooffer-dams are enclosures from which the water may be pumped out, lo m
to allow the work to be done in the open air. Their construction of course varlei
greatly. In still shallow water, a mere well-built bank of clay and gravel ; or of
bags partly filled with those materials when there is much current, will answer
•very purpose ; or (depending on the depth) a single or double row of sheet-piles ; or of
■quarMl piles of larger dimensions, driven touching each other; their lower ends a
few feet in the soil ; and their upper ones a little above high water, and protected
outside by heaps of gravelly soil or puddle, (as at P in Fig 7,) to prevent leaking.
The sheet-piles may be of wood; or of cast iron, of a strong form.

The sufficiency of a mere banlc of well-packed earth in still
water, is shown by the embankments or levees, thrown up in all countries, to pre-
Tent rivers A:x)m overflowing adjacent low lands. The general average of the leveee
along 700 miles of the Mississippi, is about 6 ft high ; only 3 ft wide on top ; side-
dopes 1^ to 1. In floods the river rises to within a foot or less of their tops : and
frequent^ bursts through them, doing immense damage. They are entirely too slight.

The method of a single row of 12 by 12 inch squared piles, driven in contact with
•ach other, (close piUs J a.nd simply backed by an outer deposit of impervious soil,
is very effeotive; and with the addition of interior cross-braces or struts, like ce. Fig
7, to prevent crushing inward by the outside pressure of the water and puddle when
pumped out, has been successfully employed in from 20 to 26 ft depth of water, in
which there was not sufilcient current to wa«h away the puddle. The crooa-bracee
are inserted successively, as the water is being pumped out; beginning, of course,
with the upper ones, 'rhe ends of these braces may abut on longitudinal timben,
bolted to the piles for the purpose. Another method is a Atronv crib* com-
posed of uprights framed Into caps and sills ; and covered outside with squared
timbers or plank, laid touching each other, and well calked ; as in the caisson, Fig
6 ; but without a bottom. Between the opposite pairs of uprights are strong interior
struts, as c e, Fig 7, reaching from side to side, to prevent crushing inward. Th«

rODNDATIOMB.

rhv orib bft^lng bHD biult Od ldua« ia Uunchod, Ukini tg itB
plIlDC noDH Dn a Hmporu; pLitfDrm nating on the croia-sl
itnam b>TJDS bees pieiloualj levelled ol^ if nMseur;, fari

687

li tmlj It UwHoliUr ftdftpHd w

B«>7

PUn «t one end..

(.drlTnarui
UabiUtT M 1(

1 TWIiMl ebHt-pllea

IS ft loDg, 12 liu eqiBre, 12^ ft

two apper ones mi A ft In Uie dear; gndiullT df mlnlihlnf to IS Ina between the

oppoeite pdJT 0

, anaecoDBt of tk* InewMiil plwure of Uio water In demeiidlag, Ob

ooMda of the i^ilcbt*, und uppmite tbo endi of the bno«, wen bolMd lonfi-

h ■hMt-pltlne «1.
lo tb* lop of th«

^u.fle.uito.1

The ihect-pl las will driie in m (ki more r»ubr and utliractory DOBDner, wtlh tbn
unniansntebownlnFigaB. EenooiirelheuprlgbisjccsnpiinofloDKlludlntil

588 50UNDATION8.

pieces, notched and bolted to the uprights, near both their tops and their feet; and
at as many intermediate points as may be desired. The sheet-piles I, are inserted
between these ; and of coarse are guided during their descent much more perfoctlf
than in Fig 7.

When the cnrrent is too strong to permit the use of outside paddle, P, Fig 7, th«
principle of coffer-dam shown in Fig 9, is generally used ; in which both sides of the
paddle are protected from washing away. The space to be enclosed by the dam is sur-
rounded by two rows of firmly-driven main piles pp, on which the strength chiefly
depends. They may be round. In deciding upon their number, it must be remeni'
bered that they may have to resist floating ice, or accidental blows from vessels, Ac.
With reference to this, extra /encl«r-pilee may be driven. A little below the tops of
the main piles are bolted two outside longitudinal pieces to to, called waits; and oppv
site to them two inner ones, as in the ng. The outer ones serve to support croa^
timbers 1 1, which unite each pair of opposite piles, and steady them ; and prevent
their spreading apart by the pressure of the puddle P. The inner ones act as guides
for the sheet-piles « «, while being driven ; after which the heads of the sheet-piles
are spiked to them. In deep water these sheet-piles must be very stoat, say 12 ins
square ; to resist the pressure of the compacted puddle.

A S'Anvway m, is often laid on top of the cross-pieces 1 1, for the ase of the
workmen in wheeling materials, Ac. The puddle P is deposited in the water in the
space, or boxing, between the sheet-piles. It should be put in in layers, and com-
pacted as well as can be done without causing the sheet-piles to bulge, and thus open
their Joints. The bottom of the puddle-ditch should bs deepened, as in the fig, in
case it consists, as it often does, of loose porous material which would allow water to
leak in beneath it and the sheet-piles. This leaking under the dam is frequently a
source of much trouble and expense. Water will find its way readily through almost
any depth and distance of clean coarse gravelly and pebbly bottom, unmixed vriUi
earth. Sand is also troublesome ; and if a stratum of either should present itself ex-
tending to a great depth, it will generally be expedient to resort to either simple
cribs. Fig 4 ; or to caissons ; with or without piles in either case, according to di^
cnmstances. But if such open gravel, or any other permeable or shifting material,
as soft mud, quicksand, Ac, is present in a stratum bat a few feet in thickness, and
underlaid by stiff clay, or other safe material, leaking may be prevented, or at least
much reduced, by driving the sheeting-piles 2 or S ft into this last ; and by deepening
the puddlo-trench to the same extent. It may sometimes be better, and more con-
venient, to dredge away the bad material entirely from all the space to be enelond
by the dam, and for a short distance beyond, before commencing tiie construction of
the latter. If the dam. Fig 9, is (as it should be) well provided vrith croos-braces^
like c c. Fig 7, extending across the enclosed area, the thickness or width o o of the
puddle, need notbe more than 4 or 6 feet for shallow depths ; or than 6 to 10 ft for grsaft
ones: because its use is then merely to prevent leaking. But if there are no bncaa,
it must be made wider, so as to resist vptetUng bodily: and then, with good paddle,
o n may, as a rule of thumb, be ^ of the vertical depth o I bdow high water; ezespt
when this gives less than 4 ft: In which case make it 4 ft; nnlera more ahoidd be
required for the use of the workmen, for depositing materials, Ac. Or If the ezoaTi
for the masonry is sunk deeper than the puddle, the dam must be wider ; el«e it
be upset into the excavated pit.

Tbe excavAted soil may be

raiiad Id bdoketa by wlndlauM, or br ^Mid. in PLAW

raooMaive itegn. Tbe pumps may M worlted ^^^^^^^^^^

by band, or by ttaam, •• the cane may roqolre;
as alM (he windlaaeea generally needed for
lowering mortar, etone, ko. More or lem leak-
ing may always be antioipated, notwithstanding
tvtrr precaution.

Wbere a eoffer-dam is exposed to a violent
enrrent, and great danger from ioe, ftc, tbe ex-
pensiTO mode sbown in Figs 10 may baeome
necessary. The two blaek rectangles e e, repre-
sent two lines of rough oribs filled with stone. ,___^_,^^___^_^^__^^
and sunk in position; one row being enclosed ^MUlil^ |W*' ^^|j^SaU '""'**
by the other ; with a space several feet wide be- PP p-p

tween them. Sbeet-piles p p are then driren -CT-ir

around tbe opposite faces vf the two rows of
cribs ; and tbe puddle Is deposited within the boxing thus provided for it, as shown in Hie fig.

Where the current is not strong enough to wash away gravel backing, we may. on rock espcaiaDy,
aooloee the spaoe to be built on, by a single quadrangle of oribs sunk bv stene; aad nflcr aAeaSH
precautions to prevent tbe gravel from being pressed in beneath the mibn', apply the baaklag.*

Figs 10^ show the plan, outside view, and transverse section, to a scale <4' 90 ft to
an inch, of a coffer-dam on rock, in 8 to 9 ft water, used successfhlly on the Schaylkll!
Navigation.

'^S^^BI^^p'^^^

* X pure dean coarse gravel is entirely onflt for sneh purpose*. A eeaeiderable
"^ Is esaentlal for preventing leaks.

FOUHDATIOm. OoE

w«v ■ddlUbDkJ llv-b^np" It- -_ , ,, --.,--

HGHHI7 rK pnTeaUD« Un oaiildi EM*d rrw eioflvdlDf 9 fu 1^ leorOLi or lb? ap

Tbe OKKtrlnsr of Inrxe calssoiu or crlba, preparatory <o •InklDC
lary lo driTO cLumpa of piles ; or lo lempuraril)' sinll rough crIbB tUiti with atom,

«f nsTlgatlon, QrotWvUeobJactloui>bl«; luuiuuuhaa tlie niiitermig are lureli wai^
Aa eipeiue of rsmoyal. Bui if remo.od, the pile, should not bo drawn out of tti*
groond; hut bo ru( -ijT close to river bottom ; for If dni»n, llifl water enlsring their
Soles niftj aofK.n the soil under tlie maaonrj. It la often eipedieut to drive two

or eran (or homos and carls; or'for aJailway foflhe'ofsj S^WTofiar^iWw *:!
Conir-dKiaa may be annb tbrouKh ■• aoft M s flrm soil, in

llupt af 4 IjDX of cribvrork, eithar rsctanvainr or oironlar. and nlehnnt n. bottom
mi being atTonglir pal togetliar, and '
bnuios. (to b« gndDdlT remoTed as tbe
■nd aner b«lDg loadad bo as to nst on th

■rft utarUI ftmo Inatde. Additional lo .,, - -

comlluUietrlctianortlieHllag^Dsttbsautalde; or it may even become uoceasarj
to dredge avaf Kime of the outer lualeriai also. On ruck it loay at timet be
expedient to drill holeeinde^p water, for rocolviii^ the enrlq of piien^uroriron roda,
Ac. This ni&¥ be dnne by moans of Ioiir drlll-rDda, worhln)^ in an iron tuiw or pipa

bell may be used. Or m cyll niler of stayee 4 to 1 2 [nchse thick, long enough lo
reach abo to the aurtace, aud having a broad tarpaulin Sap or apron around iu lairer
edge, to ba eoverwl with gt«isl lo pra*anl leaklDg: may be annk, and the walet
punped out, lo allow a workman to deecend. and work In the open air.

Plica. Whan drlTau in close onnlact.as In Fig II, for prrrenting leakage; for
ODDflnlng pnddle In a coflkr-dam ; or (or enclosing apiece of soft or sandy groond. to

590 FOUNDATIONS.

around them, Ac, they are called slieet-pllea. Generally these are thlnpev

thai> tliey are wide;

but frequently they are

square ; and as large as

bearing piles ; and are r

then called clofie j

S»lles. To make them ^

rive tight together
foot, they are cut ob-
liquely as at /. Occa-
sionally, when driven ^ - -
down to rock through X X
soft soil, their feet are
in addition cut to an
edge, as at i, so aa Figrll
to become somewhat

bruised when they reach the rock, and thus fit closer to its surface. Their heads
are kept in line while driving, by means of either one or two longitudinal piecea
a and o, called wales or stringers. These wales are supported by gaugerfMes.
or guide-piles, previously driven in the required line or the work, and several
ft apart, for this purposa See Figs 8.

A dogr-iron d, of round iron, may also be used for keeping the edges of the
piles close at top to those previously driven, both during
and after the driving. Its sharp ends, c c, being driven f^i

into the tops of the wales w tr, (uio wn in plan, ) it holds * j\.

the descending pile o firmly in place. At n, d, jb. Fig zl——.—.

11, are other modes occasionally used for keeping the ^ < T i > ^

piles in proper line. At p, the letters s s denote small iiiig«agaBiiaiBBa
pieces of iron well screwed to the piles, a little above W 1 -* t w

their feet, to act as guides ; very rarely used. At m ' '

are shown wooden tongues t <, sometimes driven down Fig 12

between the piles after they themselves have been

driven ; to assist in preventing leaks. In some cases sheetHpiles are emplo^r^d
without being driven. A trench is first dug to their full depth for receiving
them ; and the piles are simply placed in these, which are then refilled. Closer
joints can be secured in this manner than by driving.

When piles are intended to sustain loads on their tops, whether driven all their
length into the ground, or only partly so, as in Fig 3, they are called bearinfp
piles. They are generally round; from 9 to 18 ins diam at top; and should be
straight, but the bark need not be removed. White pine, spruce, or even hem-
lock, answer very well in soft soils ; good yellow pine for firmer ones ; and hard
oaks, elm, beech, <&c, for the more compact ones. They are usually driven from
about 2^ to 4 ft apart each way, from center to center, depending on the char-
acter of the soil, and the weight to be sustained. A tread-wheel is more
economical than the winch for raising the hammer, when this is done by men.
Morin found that the work performed by men working 8 hours per day, was
3900 foot-pounds per man, per minute by the tread-wheel; and. only 2600 by a
winch.

Alter piles have been driven, and their heads careflilly sawed off to
a level, if not under water, the spaces between them are in important oases filled

up level with their tops with well rammed gravel, stone
spawls, or concrete, in order to imnart some sustaining
power to the soil between the piles. Two courses or
stout timbers (from 8 to 12 ins squar& according to the
weight to be carried) are then bolted or treenailed to
the tops of the piles and to each other, as shown in the
Fig, forming what is called a 8ri*mairo* On top of these is bolted a floor or
platform of thick plank for the support of the masonry; or the timbers of the
upper course of the grillage may be laid close toirether to form the floor. The
space below the floor should also, in important cases, be well packed with gravel,
spawls, or concrete. If nnder water, the piles are sawed off by a diver, or
by a circular saw driven by the engine of the pile-driver, and the grillage is
omitted. Instead of it the masonry or concrete may be built in the open air in
a caisson, which gradually sinks as it becomes filled ; or on a strong platform
which is lowered upon the piles by screws as the work progresses. Or a strong
caisson may first be sunk entirely under water, and then be filled with concrete,
up to near low water; the caisson being allowed to remain. Or the caisson may
form a cofferdam, to be first sunk, and then pumped out. If the ground is liable

FOUNDATIONS. 591

to wash away from around the piles, as ia the case of bridge piers, &c, defend it
by sheet-piles, or rip- rap, or both.

The cost of a floating^ steam pile driver, scow 24 ft by 60 ft, draft
18 ins, with one engine for driving, and one (to save time) for getting another
pile ready ; with one ton hammer, is about S6000 ; and 9500 more will add a cir*
cular saw, Ac, for sawing off piles at any reqd depth. Requires engineman, cook,
and 4 or 6 others.- Willburn about half a ton of coal per.day. Driving 20 feet
into gravel, and sawing off, will average from 15 to 20 piles per day of 10 hours.
In mud about twice as many. On land about half as many as in water.

In the gunpowder pile driver invented by the late Mr. Thomas Shaw,
of Philadelphia, the hammer is worked by small cartridges of powder, placed one
by one in a receptacle on top of the pile ; and exploded by the hammer itself.
It can readily make 30 to 40 blows of 5. to 10 ft per minute; and, since the
hammer does not come into actual contact with the piles, it does not injure their
heads at all; thus dispensine with iron hoops, Ac, for preserving them. When
only a slight blow is required, a smaller cartridge is uscmcI. To drivd a pile 20 ft
into mud averages about one-third of a pound of powder ; into -gravel, 4 times as
much. This machine does not assist in raising the pile, and placing it in
position, as is done by ordinary steam pile drivers ; the latter, however, average
but from 6 to 14 blows per minute.

Piles have been driven by exploding small charges of dynamite
hid upon their heads, which are protected by iron plates.
Meam-hammer pile drivers^ operatliif os the prlnelple of that dsritea

by If asmjrth about 1860, are economical in driving to great depths in difflciitt
soils where there are say 200 or more piles in clusters or rows, so thai the machine
can readily be moved from pile to pile.

The steam cylinder is upright, and is confined between the upper ends of two
vertical and parallel I or channel beams about 6 to 12 ft long and 18 ins apart,
the lower ends of which confine between them a hollow conical ^ bonnet east*
In V,** which fits over the head of the pile. This easting is open at top, and through
it the hammer, which is fastened to the foot of the piston-rod, strikes the head of
the pile. Each of the vertical beams encloses one of the two upright guide-timbers,
or ** leaders," of the pile driver, between which the driving apparatus, above de^
scribed, is free to slide up or down as a whole.

When a pile has been placed in position, ready for driving, the bonnet casting is
placed upon its head, thus bringing the weight of the beams, cylinder, hammer, and
casting upon the pile. This weight rests upon the pile throughout the driving, the
apparatus sliding down between the leaders as the pile descends.

The steam is conveyed from the boiler to the cyl by a flexible pipe. When It is
admitted to the cyl, the hammer is lifted about 30 or 40 ins, and upon its escape the
hammer fells, striking the head of the pile. About 00 blows are delivered per min*
ate. The hammer is provided with a trip-piece which automatically admits steam
to the cylinder after each blow, and opens a yalve for its escape at the end of the
ap-stroke. By altering the ac^nstment of this trip-piece, the length of stroke (and
thus the force of the blows) can be increased or diminished. The admission and
escape of steam, to and from the cyl, can also be controlled directly by the attendant.
The number of blows per minute is increased or diminished by regulating the sup-
ply of steam.

In making the upstroke, the steam, pressing against the lower cyl head, of course
presses downward on the pile and aids its descent.

The chief advantage of these machines lies in the great rapidity
with which the blows follow one another, allowing no time for the di8turl>ed earth,
•and, Ac, to recompact itself around the sides, and under the foot, of the pile. This
enables the machines to do work which cannot be done with ordinary pile drivers.
They have driven Norway pine piles 42 ft into sand. They are less liable than
others to split and broom the pile, so that these may i«e of soiter and cheaper wood*
The bonnet casting keeps the head of the pile constanCily in place, so that thp piles
do not " dodge " or get out of line. Their heads have, in some cases, been set on fire
by the rapidly succeeding blows.

These machines consume from I to 2 tons of coal in 10 hours, and
require a crew of 5 men. They work with a boiler pressure of from
60 tu 75 lbs per sq inch.

F0UNDATION&

Sales for thti SaBtAlBlnS' Power of Piles.

k« ■kjtlHLoq'of waHrlwtvsn IhapltoHd tbvoZ^; t:

IbipUH, lauDtlupci, nppsr1i4Hn4r (b4 Loi
M piUft. 4H^, La var; «V*J *oilt, tbxn Lb at

In the Bnp London brldc« urotg th« Ttismei, •ach pl1» nnder eone of ths
plin iDBUms Lh<' <erf h«vy ted of SU toDa. Tbef ire driiea but -^ bet Into tbt
■tlir, Uaa LapdoD clny ; &nd an plsctd UMrlj4 ft apart from caatsr to sanur ; wblch
b too much for Bucb plen Had mhes. At 3 ft apart acant, thej would haTs had but
M tons to suatoia. f hay are 1 ft In aiam at tie middle of their length. Uglf aet-

BlBckb-tKTS Itrtdge. in tbe tame tIgIbI ty, eiblblta the la

— . , a, perlupsbyoipllluTaDaanof thapIlatlienualTea; orparhapib]
leaklnir. It now, bowarar. ba owlBg in part to tbe cnublng of the pIotftirmB on
topof Ihepiles;ortoabodUr>ettlementaf the entire manof pUed claT.lnto
the unplled clar beneatli. iiader tbe Immenaa load that rMtanpon It. Tbubara
amouuutoS^^tonaperaqraot ofareacoTeredb^apier; and la probably toe
to trust upon damp claf, when eren tbe slighteet Bloklog 1* prejudlolaL

MbI J. HMBden, li. S. Engi, eiperimenti '
mnd ', and gare the following Id the Jour. Frai
loadforB

bj tbe small elating s't each blow in ine. Mult the qnot by tbe wrtght of tl
hammer, ram, or monkey. In Ions or pounds, ae the oaM may be. BItMb tl
prod by S. Ele doe« not atate any speeifio coemclent of saf&ty.

kataldaaib

At'tka tbSi Bl ehfclatfMa. alLia*L» I

■»d- Td Had, Lt If nDt BrfraaHIr t^ pnw
u laa Baril BardeT Brl£», laaii

4rltlat,lbB£ABuLaualBi vaiiabitltwad. ■rl(hn«CB- ^b«j nH dr1r«s utiflb^aa
taellMrUDV, aodiralTOtlk a»akaj, raOLu tt n. Tbaj iirppbl Id hni aaflh. Oar ral

gtaibldtiyonclrlriBf, HumrT tka pllMwtn Mnn aMU^lHT Hat (l^^^aa iDit JCtwlS
Uawa:lHlwaaaalilAathn'UMrwara>BTHilir. ar firUHrIa ibe tiaaal tbu Mm tlHSkal™
•alnaanljaaaafthaia:aalDsvMa>aH&t»nwprauaIiDB>anilkiaaidaai. —>•«■'
JbnHBa OfaripHtS (UTI) H Pklladi, a (riaL plH waa driiaD II II Lau ion rim Big. ki a

FomrDATion.

Ii|rsMdlSv"i.'ui!i5ri^°";,M"tl?»i*™.''^^ .is^M

i IT. ^S«Tt Irtol ■!)«, iboui i: Ini HI, gilnn a n ibrcsfb

wl^ IM»>niBui IkoHud^, drrnqnlH u naUlj Id inr.

L!JjK"tta'«i

There la ~ ■ the peaetrsMllte of mflsnu

Sun u pkn* lli ud Mr. BKdd fHEl IhU h P|"iiidii£^U

«• ■ III! •ti?!' LiHdH ud BlukMan brldiu nn IobkM :

_d Isu akUB 11 ru » n wULHt ipHlal dUBmlv.

Uik IST'inMn > tamrnlr vT K Kb (lUO Si) hllini a H, hi a tr; hhbi (nidad, ikiiund tU
Btbai aHiCI loDhtUM tthl'llkin (Ul, drtn Uam BditutgMlj. Umllimim be dUiIii
to> UDU ll»wLlh>lo>r>ll: ullUiiliH tw Ulna tnttniUlusiiMputlUairinnniit the ptM

WaLbi Ita Loud, fend Ifacnbj nliVTlni (li« pmpurt apoD Ihe r«t. A ptl* ^^J rvl ai»D nek. ud
Jtt banrr vfrt; Tiirlrdrlrtn tbraoril »erj »ft tolK all tlie preuarv la boriMb; Ihetltup polDIT

a I, S&Hiu^l^ Inn Puiui. IhaDchiCHllilbaiiltHDHdTiryUulaabHItaDiDrf iDdBvd,

MHK »■ trinn uMau iiariar (n* tau a>d dm.

lEaastle' MsetleB off (be eeu"™ bw™l»own u°wi^ Aiitln pUed uua
. . . -" --^u»rtla«,briDaUM.wat«dllapo=.
•oil, Mpselill J ir rtonj ; or

B pninJl'lind

be protaciad b; ■boes of eitbsr wroughi
Iroo, u kt a. j,ud b, Figi 13 ; ipiked Is th«
plla b; meuia of Ilia Iron Blrapa n. torgHl

bASAof t(hlcb afTorUfldgood baulDgfor the
tM botlom of ths pilft-point. Tbs dotted

^ii hold! the ibo* to the pile. Bognlu

!gl

594

POUNDATIONB.

«»«af hMroB alMM will geowally wvlgh 18 to 80 Iba ; bat sheet iron may be nted wben the eotl li bvt
uoderatelj oompaet ; plate iron when more so ; and solid ifon or steel points, from t to 1 Ins sqnai*
at the bntt, and 4 to 8 ins long, when rery compact and stony. Holes may b<B
drilled In raek for receiving the points of piles, and thus preventing them
fW>m slippinf ; bj first driving down a tnbe. as a gnide to the drill, after the earth Ls cleaned oat of
the tube. To preserve tlie heads to some extent from splitting under the
blows of ttae hammer, they are nsnally sarroonded by a hoop h. Fig d; fhim M to 1 inch thick ; and
1^ to 8 ins wide. These are, however, sometimes bat imperfeet aids; for in hard driving the head
will orash, split, and bulge oot on ail sides, fk^uentl.v for many feet below the hoop : moreover, th«
hoops often split open. The heads, thmrefore, often have to be sawed, or pared off several times
before the pile ia oompletelj driven; and allowanoe mast be made for this loss in ordering piles for
any given work ; especially in hard soil. Oapt Tamball, U 8 Top Kng, states that at the Potomae
aquedaot, his plleheads were preserved ftom iAJary by the simple expedient of dishing them oat to a
depth of about an inch, and covering them by a loose plate of sheet iron ; as shown in section at s.
Pigs 18. A. rtrr slight degree of brooming or emshlng of the head, materially dimlnlaiies the foroe
of the ram. Piles may be driven through small loose rubble without mueb labor. Shaw's driver
doee not injnre the heads. Piles which foot on sloping rook may sUde when hMded.

To drive a pile bead below water a wooden panch, or follower, m

aMp, Figs 18, may be used. The foot of this punch fits into the upper part of a casting //, ronad er

Snare, according to the shape of the pile ; and having a transverse partition o o. The lowerparl
the casting is fitted to the head of the pile t; and the hammer falls on top of the puneh. Wh«i
driving piles vertically in very soft soil, to support retaining-walls, or other strnetnres expoeed te
borisontal or inclined forces, care must be taken that these forces do not push over the piles them-
selves ; for in such soils piles are adapted to resist vertical forces only, nnleae they be driven at aa
inclination corresponding to the oblique force.
A broken pile may be drawn oat, or at least be started, if not yery

firmly driven, by attaching soows to it at low water, depending on the rising tide to loosen It. Or a
long timber may be used as a lever, with the head of an adjacent ?ile for its falorum. Or a crab
worked by the engine of the pile driver. In very difllonlt oases the method devised tar Mr J. Moarea,
0 B, may be used. A 4 inch gas pipe 1& ft long, shod with a solid steel point, and having an oatar
shoulder for sustaining a eircular punch, was thereby driven close to and S or 8 fl deeper thaa ««•

Eiles driven 12 ft, in 87 ft water, and broken off by ios. Four pounds of powder were then deposited
1 the lower end of the pipe, and exploded, lifting the piles completely oat of plaee. It will often be
host to let a broken pile remain, and to drive another elose to it. May be drawn by hydranlle prsas.
lee adheres to piles with a force of about 30 to 40 lbs per sq Inch, and in
risiag water may lift them out of plaoe if not sufllciently driven.

Iron piles and eyllnders. Cast iron in rarious shapes has been mach
ased in ■nrope for sheet piles : especially when intended to' reasain as a foelng for the proteetioa eC
concrete work, filled ia behind and against them.* Cast-iron eyiindars, open at both eads, may be
used as bearing piles ; and may be oleaned out, and filled with eonorete, if required. The fkietioa ia
" " . . -y - . ththef

driving is greater than in solid piles, inasmuch as it takes place along both the inner and the oami
sarfaoee. This may be diminished by gradually extraeting the Inside soil as they go down. They
require much care, and a lighter hammer, or less fall than wooden ones, to prevent breakiag ; ^
which end a piece of wood should be interposed between the hammer and the pile : or the ram maj be
of wood. But it is better to use them in the shape of serew eyllnders» whioh,
moreover, gives them the advantage of a broad base, as in the following.

Bronel's process. He experimented with an open cast-iron cylinder, 8 f|
eater diam ; 1^ ins thick ; in lengths of 10 ft, connected together by internal socket and joggle jeint^
■soured by pins, and ran with lead. It had a sharp<edged hoop or cutter at bottom ; and a Uttls
above this, one turn of a screw, with a pitch of 7 ins, and prejeeting one foot all around the outalds
ef the cylinder. By means of capstan bars and winches, he screwed this down through sdff elay an4

Id, 58 feet to rook, on the bank of a river. In desoending this distance the cylinder made
revolutions ; sinking on an average about 5 ins at each. The time oocnpied in actaally screwing was

48)4 hours; or aboat 1^ ft per hour. There were, however, many long intervals of rest for oleaa

ing away the soil in the inside. After resting, there was no groat difllealty in roetarHng. The neH
fig will give an idea of the arrangement of the screw.

Tbe screw-pile of Alex. Mitchell, Belfast, consists usually of a rolled iro^
shaft A, Figs 14, from 3 to 8 ins diam; and having at its foot a cast-iron screw
8 8 S, with a blade of from 18 ins to 6 ft diam. The screws used for light-hoasei^
exposed to moderate seas, or heavy ice-fields, are ordinarily about 8 ft diam, havf
l^ turns or threads, and weigh about 600 lbs. The round rolled shaft* are ttom
5 to 8 ins diam. They are screwed down from 10 to 20 ft into clay, sand, or coral, by
about 30 to 40 men, pushing with 6 to 8 capstan bars, the ends of which describe a
circle of about 30 to 40 ft diam. For this purpose a platform on piles has freqaentli
to be prepared. In quiet water, this may be supported on soows ; or a raft well
moored may lie used when the driving is easy ; or the deck of a large scow with t
well-hole in the center for the pile to pass through. Roughly made tamporary
cribs, filled with stone and sunk, might support a platform in some positions. The
platform must evidently be able to resist revolving horizontally under the greal
pushing force of the men at the capstan bars ; and on this account it is dimoult
to drive screws to a sufficient depth, in clean compact Hand, by means of a floatinf
platform. The feet of the piles must be firmly secured to the screws, to prevant

* Cast Iron, Intended to resist sea-crater, should be close^Kraioed,

hard, white metal. In such, the umall quantity of edntaiaed carbon is obemleally oombloed with the
metal ; but in the darker or mottled Irons it Is mechanically combined, and such iron seen beessMS
■oft, (lomewhat like plumbago.) when exposed to sea-water. Hard white iron has been proved li
resist for at least 40 years without any deterioration : whether constantly under water, or afteraaMy
wet and dry. Oopper and bronte are but slightly and sopcrfloially aflketed by sea-water ; but ('
tive galvanic aotlen takes place if dlff metals are in contact.

FODNDATtOHS.

It iMlDB lifted out of (h

>;rTl>tn<,utiiaa"St

lD( n n, enclodDg tbe cadi of
udia piooed through thtpilM,

icnw b fnuo 2 to 10 bmni lii *
IK lb> ■FutjwlH lltUbin, H

ftrnwntbHlOn. ltotbHiil>a«»^laHndlarvuhBd«
•MtmluBrnddnlorowd. IndtAbeBrlBv ^irer ■■ Uin

4 n dIkB. Thn pui Unnfl^ araftU btobft luntt ftad oon] nnB wlltlHWI DHh diaooLv : KDd vUL
HBb uidt bovrihn dT UDdvnic ilu. OrdlnuU;. ala; nr iftitd will pr»«Dt DO (THl obilnHtMaj

aUnODlLJ. Al lUB BnodjHlDB flbnl Llia dri'lDC VU Bided hj ft ipur Ud plQlOO plHBd U kMTHtb*
kmlrr pBTBiLUM ; uid EM b-Bn i>qn vorhd b; » HOD. Tbfl duw pT twjitblv off tlu Itakft la

roandtd bj N «&«» or fi lu dluB, l

;.=:»

Mt borlncB alioald be mitdc 1

Br Bi*»iis Af nie« ot w»Mr forclblj IohkIIMI thnngb ■ tqbo br ■ fom

Ev of B>n>'|illH, or >oida OBK (r nn Iko IvnnQlladHi. ba inall) lullluud! In tpyruw-
ment |*ler Mt €3in>e MolepeM In n<tj comput und, la wbich A oni of T

SnHM Onnl/ b; > or 1 lam ohlli lu |>1ia •» Mof Hmod io>i

naxd tbopll- Id .f mln.M. .rwU.^ «n •<•

A» TcniuH Blvrr, AlMlminK, H

fnoloalni Ul«, Id ««i> llibt iklTUDf tDsd. Ibo M ••• tonod In mill

TKtMTj psDipsf MO 10 MO nrolsUoga w mlosH. tbrxitbKuru boH 1 loa dlim,

dnnii" Thli apuTUu m niBBd and !»«•/ kc •'l|'l UikI and iliia^ and b; It

■ Bapon 8*0 of War im. * Jnhn W. OIbdd.C B, Van Noilrvi^, Jujh Iff

I Olk»cl«l Jifdail, 0 B i Trua Am Soa C 1^, Feb 1814.

596

FOUNDATIONS.

At the I^evan Tladuet, Mr James Branlee, Bngland, in a light

■andy marl of great depth, iiank hollow oast iron cyliodem of 10 ins oater diam, to a depth of 30 fl,
hj means of a Jet pipe 2 ins diam pasning down Inside of the ojlinder, and through a hole in ita base,
which was a oast Iron disk SO ins diam, and 1 inch thick, strengthened bj outside flanges. The eon-
neetlng flanges of the cylinder sections are otuHde, thus impeding the deeoeot, as did also the broad
bottom disk ; still S or 4 hours usually sufficed for the sinking of each, to 30 ft depth. Actual trial
■bowed that their safe sustaining power was about 5 tons per sq ft of bottom disk.

At Itoek Ken Tladnet each pier consists of two cylinders, open at both
ends; of calst iron, 8 ft in diam ; l]/^ ins thick ; in lengths of 6 ft, weighing 4 tons
each ; and bolted together by inside flanges, with iron cement between them. The
cylinders stand 8 ft apart in the clear ; and are in 36 ft water. " A strong staging
W118 erected ; and 4 guide-piles driyen for each cylinder. The several lengths being
previously bolted together, these were lowered into their places. £acfa cylinder sank
l)y its own weight one or two ft through the top mud, and then settled upon the sand
HuJ gravel which form the substratum for a great depth. Into this last they were
Hunk about 8 or 9 ft farther, by excavating the inside earth under water, by means
of an inverted conical serew-pHn, or dredger, of ^ inch plate iron. This was
2 ft greatest diam, and 1 ft deep ; and to its bottom was attached a screw about 1 ft
long, for assisting in screwing it down into the soil. Its sides had openings for ths
entrance of the soil ; and leather flaps, opening inward, to prevent its escape. From
opposite sides of the pan, 3 rods of |1 inch diam projected upward 4 feet, and were
there forged together, and connected oy an eye-and-bolt joint to a long rod or shaft,
at the upper end of which was a four-armed cross-handle, by which the pan
screwed down by 4 men on the staging.**

" When a pan was full, a slide which passed over the Joint at the bottom was lifted; and iha^^
was raised by a taokle. This pan raised about 1 cub ft at a time. ▲ smaller one. of only 1 ft diam,
and 1 ft deep, raising about yi cub ft, was used when the material was very hard. By this meana
the cylinders were sunk at the rate of from 2 to 18 ins per day. The slow rate of 2 ins waa oaased
by stones, some of them of 50 lbs. These were first loosened by a screw-pick, whieh was a bar of
iron 3 ft long, with circular arms 13 ins long projecting from the sides. After being loosened by this,
the stones were raised by the pan. The expense of all this apparatus was very trifling ; and the ax-
oavation wm done easily and cheaply. AfMr the excaration was finished, and the cylinder sunk,
before pumping out the water, concrete (gravel 2, hydraulic cement 1 measure) was filled in to tlie
depth of 13 feet, by means of a large pan with a morable bottom ; and about 12 days were left it u
harden. The water was then pumped out, and the masonry built in open air. In some of the cylin-
ders, however, the water rose so fast, notwithstanding the 12 ft of concrete, that the pumps could not
keep them clear ; and 6 ft more of concrete had to be aidded in those. Finally random-stone, or rough
dry rubble, was thrown iu around the outsides of the cylinders, to preserve them from blows and
nnderosining." * The masonry extends 20 ft above the cylinders, and above water.

Tbe Taeaam and plennm processes. We can barely allude to
the general principles of these two modes of sinking large hollow iron cylinders. In
the vacuum process of Dr. Lawrence Holker Potts, of London, the cylinder

e. Fig 16, while being sunk, is closed air-tight at top, by a
trap-door, opening upward. A flexible pipe p^ of India-
mbber, long enough to adapt itself to the sinking of the
cylinder, and provided with a stopcock x, leads from the
cylinder to a vessel v ; which may be placed on a raft, or a

scow, or on land, as may suit circumstances. The cylinder

^^^RCSSR^ssssSSs^ss^BgSBMi being first stood up in position, as in the flg, the water is
JPiqX'b pumped out, and the interior soil removed if the cylinder

•^ has sunk some distance by its own weight. The cock

M is then closed, and the air is drawn out from the vessel v
Iby an air-pum)). The cock is then opened, and most of the air in the cylinder rusher
into the void ve«Bhl v; thus leaving the cylinder comparatively empty, and therefore
less capable of resisting the downward pressure of the external air upon its top
'T^is pressure, as is well known, amounts to nearly 15 fts on eveiy sq inch ; or nearty
1 ton per sq ft of area of the top. Consequently the cylinder is forced downward in
the bed of the river, by this amount of pressure, in addition to its own weight. At
the same time, the pressure of the air upon the surface of the water is transmitted
through the water to the soil around the open foot of the cylinder ; so that if this
voil be soft or semi-fluid, it will be pressed up into the nearly void cylinder, in which
is no downward pressure to resist it. The descent varies from a few inches, to 4 or 6
ft each time. The process is then repeated, by admitting air again into the cylin-
der, opening the trap-door, removing the water and soil, as before, Ac. Addittonsl
lengths of cylinder may be bolted on, by means of interior flanges.

Ills atlapted on ly to soft sol Is, and to wet sandy ones : but is not sufficient-
ly powerful in Tery compact ones ; nor does it answer where obxtruetions from bowlders, logs. See, oeenr :

*Hollow Iron Piles either east or wrought with .•talid pointed feet, to be drifen by the hamaMr
fklling wuide of ihem and striking agaiuHt the top uf the itolid foot, are a recent device of greai as* 1b
many cases. They are made in seetioun of which eunufth can be gradually united to rvaoh aay
required depth. They avoid the danger ctf bending which Attf>udR striking the top. The iroo Itot i
■welted outwardlv a little to diminish earth-fnotion sgainst the pile above "

FOUNDATIONS.

697

ElglT

) ramoTil of wbioh nqaires men to enter the cylinder to its foot ; which ther einnot do in the r&reflad
air. The pipep should be of sufflcieut di»m to allow the air to leave the cylinder rapidly, so that the
enter pressure may act upon the top as suddenly as possible. '

At the Goodwin Sands light- bouse, England, hollow cylinders SH f^ in diam, were sunk 84 ft into
MBd by this process, in about 6 hours ; where a steel bar could be driven only 8 ft by a sledge-ham-
■er. Others, I'i Ins in diam, have been sunk 16 ft into sand within less than an hour. In this laat
lastanoe the air-pnmp had two barrels, i^i ins diam, 16 inch stroke, worked by 4 men. The pipe 9
was of lead, and only yi inch diam.

Tbe plenam proeens, indented by Mr Trifrer,
of Fnoce, consists in forcing air into the cylinder
0 C, Fig 17, to snch an extent as to force out the
water, compelling it to escape beneath the open foot,
into the surrounding water. The interior of the cylin-
der beiDg thus left dry to the bottom, men pass down it
to loosen and remore the soil at and below its base. When
this is done, they leave ; the compressed air is allowed to
escape ; and the cylinder, being no longer sustained by
the upward pressure of the compressed air beneath its
top, sinks into the cavity, or the loosened material at its
foot. Fig 17 shows the simple arrangement by which
workmen are enabled to enter or leave the cylinder,
without allowing the compressed air to escape ; as well
M the general principle of the entire process.

L L is a separate small ehamber, A« mlr-lo«k« whieh is
removed when a new length of pipe is to be added ; aad afMrward
leplaoed aad firmly bolted on. This ehamber has a small alr'tlght
door d, bv whioh it can be entered fkt>m without ; and another, o,
opening into the cylinder. The flaps, I, Jk, of both doors, open in-
ward, or toward the cylinder. This ehamber also has two stopcocks ; one, a, in Its floor, eonuBmnl*
eating with tbe cylinder ; and one a, above, communicating with tbe open air. At « is a bent tnbe,
also with acock, whieh passes air-tight through the side and the bottom of the air-lock. Throngh
it the compressed air is forced into tbe cylinder, by an air force-pump or condenser ; and throngh It
the same air is allowed to escape at a later period. A siphon is shown at nnn. A drum w is used
tor foisting theeicavated material from tbe bottom, to the air-lock ; its axle i i passes air-tight through
stafling-boxes in the sides of the lock ; the hoisting being done by men outside. This is tbe general
arrangement employed by Mr W. J. IfcAlpine, C E, of New York, at Harlem bridge ; and from his
deserlption of it, ours has been condensed. Tbe cylinders were there 6 ft diam, 1^ ins thick, and in
lengths of 9 ft, bolted together through inside flanges /, as the sinking went on. The air-look is 6 ft
diam, by nearly 6 ft high ; with sides of boiler Iron ; and top and bottom of oast iron.

Now suppose the cylinder CO to be lot down, and steadied in position, as in the fig; and the air-
lock L L to be adjusted on top of it. The next process is to force in air through the curved tube « ;
tbe flap t of the lower door o. and tbe cock a, being previously closed. As the eompressed air accu-
mulates In the cylinder, It forces eat the water ; which escapes partly beneath the bottom of tbe eyl-
jBder, and partly by rising throngh the siphon nn, and flowing out at g. The door o being already
eiosed. and that at d open, the air In the air-lock Is in tbe same condition as that outside ; so that
workmen ean enter it readily. Having done so, th^ dose the door d, and the cock e ; and open thb
eoek a, through which oondenaed air fWnn the cylinder rushes upward, soon filling the air-lock.
When this is done, the flap t Is epeaed. aad the men desoend throngh the door o by a ladder, or by a
1»aekot lowered by the drum w, to the bottom. Here they looeen aad ezoavate the material as deep
ae they ean ; and, filling it Into a bucket or bag, they signal to those outside, who raise it to the air-
lock. When done, they ascend to tbe air-lock, close the door o, and the cock a; and open the cock «,
throngh which the eoodensed sir in the lock soon escapes, leaving the internal air the same as that
•■telde. The door d is then opened, tbe buekets of earth are removed, and the men go out. Finally
tbe eoek at « Is opened, the oondenaed air in the cylinder eseapes throngh It to the onuide air, and
the ovllnder sinks by its own weight into the cavity and loosened soil prepared for it at iu base, and
whieh Is now forced np Into the cylinder by the rush of the returning water. The proeess la thea
repeated. The ciaklng will often vary fh>m 0 to 10 or more feet at one operation. Until depths of
40 or 60 ft, most men oan endnre the pressure of the condensed air ; but as the depth increases this
beoomes more diOicnlt, and positively dangerooa to life. Cast-iron cylinders 1& ft diam ; and great
calasona, Fig 16, have been thoe snlik ; but at times at great expense and trouble.

f be eyliuder slioiild be gruided in its descent by a strong frame, which

may be supported by piloe. Otherwise it will be apt to tilt, and thus give great trouble to setUe it
apon its ezaot plaoc Have been sunk in deep water by divers undermining inside.

Tbe plenam process as applied at the South St bridge, Philada.
by Mr. John W. Murphy, contracting engineer, difl'ers materially from that described above ; and
moreover deserves notice on account of the great simplicity and efficacy of his plant. This consisted
partly of two canal boats, decked, eaoh 100 ft long, by 17^ ft wide, and 8 ft depth of hold. They
were aaehored parallel to eaoh other, 1& ft apart. Supported by the boats, and over the space between
thea, waa a strong fonr»legged shears about 50 ft high ; at the top of whioh was attaohed taokle for
handling the cast iron cylinders. In the hold of one of the boats was a BarlMcb
Compressor having two pistons of 10 ins diam, and 9 ins stroke ; together with
its boiler. On thedeck of the same boat stooda vertical air-tank, or reirnlator,
3S ft long, by 2 ft diam. made of quarter inch boiler iron. This served to maintain a supply of eom-
presaed air In the submerged oylinder in case of an accidental stopping of the compressor ; whieh
otherwiee would probably be fatal to the laborers in the cylinder. The condensed air flowed from
this air-tank to the air-lock of the cylinder through a hose 4 ins diam, made of gum elastic and oaa-
▼aa, aad so long, and ao placed, aa to extend Itself as the cylinder went down, thus maintaining the
eommonleation at all times. Entirely, across both boats, and aoroes the interval between them, ex-
tendMl two heavy wooden clamps, each 3 ft wide by 18 ins high ; each composed

598

FOn2in)ATIONfiL

of thr«e piMes of 12 X 18 inoh timber ■trongly bolted together. At the oentera of tbeie otampe the
tvo inner rertioal sldee wbieh fheed eaeh other were hollowed out to the depth of a foot by ooaoavl*
tie* oorrsBponding to the carve of the oylindere. The distance apart of the clampe was regulated by
tTo etroDg iron rod*, having ■orewe and note at their ends for that parpoee. Thoe when a eeotioB
of a cylinder was hoisted by means of the shears into its position over the space between the tw«
boats, the two concavities of the damps were brought into contact with it. and the nnts being then
serewed up, the cylinder was flrmly held in place by the clamps. The shears could then be used to
raise another section of the cylinder to its place upon the first one, that the two might be bolted to>
gether. By repeating this process the height of the cylinder would soon become too great to allow
the shears to place another section upon it ; in which case the nuts of the screws were alightly
loosened, and the cylinder was allowed to slip down slowly into the water until its top was bnt a
little above the surface. The screws were then again tightened, and the cylinder again held fait
until other secUons were added and bolted to it. When there was danger that the upward pressure
•f the oondeneed air might lift a ovlinder, the clamps were raised by the shears olear of the boata ;
then tightened to the cylinder, and a platform of planks laid upon them, and loaded with<stone.

The Air-lock was so arranged as not to reonire to be remoTed when a new mo<
tion was to be bolted on. This was effected as follows. Seotlons of the oylindar were bolted together
in the manner just described, until its foot rested on the bottom, with its top a few feet above hLj^
water. A heavy cast iron dlaphraffm 1^ inches thick, to form the floor of the
air>Ioek, was then placed on top. Then was Mded another 10 ft high section of the cylinder, to form
tte chamber of the air-lock. These were bolted together ; and then another diaphragm was added
at top to form the roof of the air-lock. These diaphragms were furnished with openings, and with
doors and valves corresponding with those shown in Fig 17, and remained permanoitly in the

eylinders when the work was finished. If the depth of soil to be passed through before reaching
roek Is so great as to require other sections of cylinder to be bolted on above the top of the air-look,
this may be done to any eatent, inasmuch as it is immaterial whether the air-lock Is under water or

not. To keep the cyllDder both air- aod water-tlyht the faces of
the flanges before being bolted together were smeared with a mixture of red andwhite lead and eo^
ton fiber.

At the Sonth St brldfre the cylinden were 4, 6, and 8 ft diam ; in lengtbi
or sections 10 ft long. They were all 1^ inch thick. Inside flanges 29i ins wide, IH thick, with bolt-
holes \}i inch diam, by 5 ins apart from center to center. The bottom edge baa no flange. A 10 ft
seetion of an 8 ft cylinder weighs 14000 lbs ; of a 6 ft one, 10800 ; of a 4 ft one, 6800. An 8 ft dlS'
pbragm, 3800 As ; 6 ft, 1600; 4 ft, 78S. The rock under the soil was quite uneven in places ; bat was
levelled off as the cylinders went down. These were then bolted to It by oast iron brackets.

The work went on, day and nlirht, snmmer and winter: with no inter-
ruption from the tides, floods, or floating ice ; and the thirteen columns were Muk, fllled with oca*
orete, and completed in 11 months ; much of which was consumed la levelling off the roek, and bolt-
ing the brackets. The want of guides caused much tilting, trouble, and delay.

Rise and fall of tide about 7 ft. Greatest depth of soil, gravel, mo, passed through, 90 ft : least, €
ft. Depth of water about 25 ft. The work was under charge of John Anderson, a very sklUfal and
energetic superintendent of such matters. The entire neat cost of the cylin-
ders in place, and fllled with hydraulic concrete, was approximately 883 per foot of total length
for the 8 ft ones ; $64 for the 6 ft ; and $40 for the 4 ft diams. There were three gangs of workmen ;
and each gang worked 4 hours at a time. See a full and very instructive descnptioD wiUi eagraT*
lags, IqrD. M. Sunlfcr, Saperiatending lagi»s« fbr fehe elty, la the Jenmal «C the PraakMa laal^
Not., 18T2, From it the above few Items are taken. Mr. AndMwm'a firm (Anderson A Bair. Tritaa*
BnUding, ir. Y.) have since aneoearftally snak a aoiabsr ef soeli fjUadart, iaeluding (1884i«) fear of
WToughVlron, 8 ft diam, 66 ft long, at an ang^ of 48^ with the her. Inteadad as atrau to preveat tia
aiovement of oae of the abut iders ef Chestnut 8t bridge, Phlla.

Oast iron cylinders haweeraeked throiiir^(^'o°°<^ ^^^^''"''^^'^^^^'^^■'i*
feranee, In aiaay parte of the U. 8. in very eold weatherf ewlag to the diff of ooatraetloa «r the
nwB, and ef the eonerete filling. Ignorant aae of them aiay be atieaded by great danger.

The shaded part of Fig 18 shows a transverse section of the ealMlon of yelled w«
pine timber and cement, for the Brooklyn tower of East River (N Y)

•aspeneion bridge, of 1600 ft clear span. It is 168 ft long at bottom, and 102 ft wide.
A longitudinal section resembles the transverse one, except in being longer, and in
showing more shafts J. Of these there are 6, arranged in pairs, for expedition and as
a precaution against accident. Namely, two water^hafts J, each 7 ft by 6^ ft acroM,
for remoTing by bncketa and hoisting apparatus, the material excavated beneath the

caisson; together with such
water as may accumulate at
o o; two air^hafts of 21 ins
diam, through which air la
forced (h>m abore, to expel
the water flrom the chamber
0 S 8 D below the caisson, to
as to allow the laborers to
work there at undermining ;
the expelled water eecaping
under the foot 0 D of the oaie<
son, into the river ; and two
supply shafts of 42 ins diam,
for admitting laborers, toola,
Ae. Hie several shafli of course have air-chambers on top, on the same principle •■
Vif 17, to prevent the escape of the compreised air in 9 f .

jouin>ATiON& 699

TiM ■hafta are of ^ Inoh boUw tron. TIm fiool O D, nla* ttmben high, li eontinaoat, •ztendini
antfarely aroand the oalMon ; Ita bottom It shod with OMt Iron : ita foar oornen ar« ttroogtbonod bf
wooden kneee 20 ft long.

From the bottom, up to the line M, N, 14 ft, the oeision is built of horliontal layer* of timbers one
Ibot square ; the lajers eroasing eaeh other at right angles ; and the timbers of each layer touching
eaoh c«her. well forced and bolted together ; and all the Joints filled with pitch. To aid in prerentiiig
leakage, the nuts and heads of the screws hare India-rubber washers ; also all outside seams, as well
•a all the seams of the lairer of timbers N, N, are thoroughly calked; and a layer of tin, enclosed
between two layers of felt, is placed outside of eaoh outer Joint ; and over the entire top of the layer
next below N, N.

When the caisson was built up to N, N, on land, it was launched, floated into poaition. and anchored ;
after which were added for ainUng it, fifteen conraea of timbera one ft aqnare ; and laid one ft apart
fa the clear ; with the interrala filled with concrete. The top conrae A B ia of aoUd timber, to aerre
M a floor for anpportlng machinery, 4e. It waa rank aome feet below the very bottom of the
liver, in order to avoid the teredo.

Criha are annk ontaide of the caiaaon, to form temporary wharTca for boata carrying away ezcaTated
material ; and for veaaela bringing stone, 4c.

When the caisson was sunk, and the water forced out from the chamber or space CSS D. workmen
began to ezoavate unilbrmly the enclceed area of rirer bottom, so as to allow the caisson to descend
■lowly onttt it reaiMied a firmsnbatratum. The apace 0 8 8 D, aa well aa the ahafta, waa then filled up
•olid with oonorete maaonry. ▲ oo0Br*dain waa built on top of the oaiason i and in it the rcgnlar
masonry of the tower waa started. The total height of this tower including the caiaaon, ia about SQO
ft For ftall detaila eee report, 187S, of W. A. Boebllng the chief engineer.

Hollow cylinders, or otlier forms of brickwork or mi^
■Olirir« with a strong curb or open ring of timber or iron beneath them, may bs
gndnally annk by undermining and exoaTating from the inaide ; and form very atable fonndaUona.
under water thia may be done by properly shaped sooope, with or without the aid of the diving-bell,
aooerdiag to the depth, 4c On land it will often be the meet eeonomieal and aatiafaetory mode,
•eneoiaUy in firm aoUa. The deecent may be aaaiated by loading them, if, aa aometlmee happens, the
mistien of their sides against the earth outside prevents their sinking by their own weight. A brick
oylfnder, 4$ ft outer dlam, walls S ft thick, has been sunk 40 ft in dry sand and gravel, without any
dOffleoltj. It was bailt 18 ft high, (on a wooden corb U ins thick,) and weighed 300 tons before the
■inking was begun. The interior earth was excavated slowly, so that the sinking was about 1 ft per
day ; the walla being built up aa it aaak. Tannei ahafta are at timea so sunk.

On tbc Rbine for a ooal shaft, a brick cylinder 2SV^ feet diam was first fhns
sank by its own weight 76 ft throogh sand and gravel ; then an intenor one, 15 ft diam, waa annk in
the aame way to the depth of 356 ft below the anrfaoe: of which depth aU the 180 ft below the flnt
^tinder was a running auieksand. At 256 ft fHction rendered the cylinder immovable. The quick-
■and was removed by boring ; no pumping waa done ; but the water was permitted to keep the oyl fhll.

The entire foundation for a large pier of masonry has been aunk in thia manner, In a single maae ;
a sanoleat number of vertleal openinga being left in it for the workmen to descend, or for tools to be
inserted for undermining. This is generally a verv slow and tedious operation, especially under
water. It may often be expedited by diviug-bells or by diving-dresses. It will generally be better to
■lake the mass wider at bottom than above it, so as to diminish friction against the outside earth.
On land, watv may at times be used for softening the bottom earth. By keeping the interior of sucb
hollow masonry dry, it may even be buUt downmewd from the anrfhoe ; by undermining only a por-
tion of ita droumference at a time, filling said portion with masonry, and then removing and filling
the other portion ; and so on in saooessive stages of 2 or 8 ft downward at a time. This mode may be
adopted also when friction has stopped the sinking of a masa by its own weight when undermined.

The sand pamp as used at the St Louis bridge will often be of service in rais-
tag sand f^m cylinders while being sunk in water. With a pomp pipe of 8.5 ins bore, and a water
i«| nndor a praaanre of 150 lbs per sq inoh, 20 cub yds of sand per hour were raised 125 feet. A jet of
•ir haa also been •oooe^^tally nBedintheaama way, aaatthe Kast River, N T, auapeaaion bridge, 4o.

Fnsctnes. On marshy or wet quicksand bottoms, foundations may be laid by
ftnt depositing large areas of layers of fascines, or stout twigs and small branches,
•trongly tied together in bandies from 6 to 12 ft long, and from 6 ins to 2 ft in diam.

The layera or atrata of bundles should cross each other. A kind of floating raft or large mattress
!■ first made of theee, and then sunk to the bottom by being loaded with earth, gravel, stones, 4o.
In thia manner the albutments and piers of the great suspension bridge at Kieff, in Bussia, with spans
/■f 440 ft. were founded in 1852, on a shifting qnloksand. There the fkseine mattressea extend 100 ft
b^oad the baaes of the masonry which rests upon them.

Paaolaei may be used in the same way fbr sastaining railway embankments, 4c, over marshy
groand, but they will aettle eonaiderably.

Snnd-piles. We have already alluded to the lue of sand well rammed In layers
Into trenches or foundation pits ; but it may also be used in soft soils, in the shape
<rf piles. A short stout wooden pile is first driren 5 to 10 feet or more, according to
the case. It is then drawn out, and the hole is filled with wet sand well rammed.
The pile is then again driven in another place, and the process repeated. The inter-
Tals may be from 1 to 3 ft in the clear. Platforms may be used on these piles as on
-wooden ones. If the sand is not put in wet, it will be in danger of afterward sink-
ing from rain or spring water. In this case, as with fascines, it is well to test the
foundation by means of trial loads. Some settlement must inevitably take place'
nntil all the parts come to a full bearing ; but it will be comparatively trifling. The
fame occurs in every large work to some extent ; as in a roof or arch of great span,
whether of wood, iron, or masonry ; so also with all tall piers, walls, Ac, Ac. Sandy
foundations under water should be surrounded by stout well-driven sheet- piling, tu
prevent the enclosed sand from running out in case the outer sand is washed away ^
•ad should slso bs defended by a deposit of random-stone.

600 BOCK DRILLING.

On bad bottoms under water, small artificial Islands of good soil have
been deposited ; and the masonry founded upon them. Canal locks and other
structures may at times be advantageously founded in this way in marshy soils.
If necessary, a depth of several feet of the bad soil may be dredged out before the
firmer soil is deposited ; and the latter may be weighted by a trial load to test its
stability.

The mode of laying a foundation under water, by building the masonry upon
a timber platform alMve water, upheld by strong: screws, and lowered into
the water as the work is finished in the open air, a course or two at a time, has
of late been much employed with entire success, in large bridge-piers in deep
water. It however is uot new. It was suggested more than 100 years ago by Belidor.

Piles are driven 6 to 10 ft apart arouud the space to be occupied by the pier;
having their tops connected by heavy timber cap-pieces. These last uphold the
screws, which work through them. The whole is oraced against lateral motion.

A CLUMP OF PILES WELL DRIVEN ; aud then enclosed by an iron cylinder sunk
to a firm bearing, and filled with concrete, is an excellent foundation. The piles
may extend to the top of the cylinder, and thus be enclosed in the concrete. Such
an arrangement has been patented by S. B. Cushing, C. E., Providence, B. I. The
cvlinder and concrete serve to protect the piles from sea-worms, and from decay
above low water ; and are not intended to support the load above them.

STONEWOEK.

Where work is done on a large scale, blasting can sometimes be done at from 10
to 20 percent less cost per cubic yard by means of maclilne drills and
dynamite, than by band drills and snnpoivder. Ordinarily, how-
ever, tbe cost is about tlie same^ and the advantage of the newer methods
consists rather in economy of time, convenience, and naving the work more
entirely under control In ordinary railroad work in average hard rock, and when
common labor costs 91 per day of ten hours, the cost per cubic yard, for loosening,
will ordinarily range between 30 and 60 cts, including tools, drilling, powder, Ac

Holes for blasting, drilled by band, are generally from 2>^ to 4 ft
deep ; and from 1}^ to 2 ins diam. Cbnrn-drilling^ is much more expeditions
andeconomicai than that hj jumping ,mentioned below. The phurn-drill is merely .
a round iron bar, usually about l]^ ins diam, and 6 to 8 ft long ; with a steel cutting
edge, or bit, (weighing about a fi>, and a little wider than the diam of the bar.)
welded to its lower end. A man lifts it a few inches ; or rather catciies it as it
rebounds, turns it partially around ; and lets it fall again. By this means he drills
from 5 to 15 feet of hole, nearly 2 ins diam, in a day of 10 working hours, depend-
ing on the character of the rock. From 7 to 8 ft of holes 1% ins diam, is about a
fair day's work in hard gneiss, granite, or compact siliceous limestone; 5 to 7 ft
in tough compact hornblende ; S to 5 in solid quartz ; 8 to 9 in ordinary marble
or limestone ; 9 to 10 in sandstone ; which, however, may vary within all thepe
limits. When the hole is more than about 4 ft deep, two men are put to the drill.
Artesian, and oil wells, in rock, are bored on tbe principle of the churn-drill.

Thejumper, as now used, is much shorter than the chum-drill. One man (the
holder) sitting down, lifts it slightly, and turns it partly around, during tbe intov
vals between tbe blows from about 8 to 12 9> hammers, wielded by two other labor-
ers, the strikers. It can be used for holes of smaller diameters than can be made
by the churn-drill ; because the holder can more readilv keep the cutting end st
the exact spot require^l to be drilled. It is also better in conglomerate rock ; the
hard siliceous pebbles of which deflect the churn-drill from its vertical direction,
so that the hole becomes crooked, and the tool becomes bound in it. The coal
conglomerates are by no means hard to drill with a Jumper. The Juniper was
formerly used for large deep holes also, before the churn-dnll became estaolisbed.

Either tool requires resharpening at about each 6 to 18 inches depth of hole;
and the wear of the steel edge requires a new one to be put on every 2 to 4 dsys.
With iron Jumpers, the top also becomes battered away rapidly. As the bole
becomes deeper, longer drills are frequently used than at tne beginning. The
smaller the diameter of the bole, the greater depth can be drilled in a given time ;
and the depth will be greater in proportion than the decrease of diam. Under
.similar circumstances three laborers with a Jumper will about average as much
depth as one with a churn-drill.

The band-drill, in which the same man uses both the hammer and the short
drill, is chiefly used for shallow holes of small diam. With it a fair workman
will drill about as man^ feet of hole from 6 to 12 ins deep, and about ^ inch diam,
as one with a churn-drill can do in holes about 8 ft deep and 2 ins dlaro, in the
same time. Only the Jumper or the hand-drill can be used for boring holM
which are horizontal, or much inclined.

COST OF STONEWORK. 601

Cost of qnarryinff stone. After the prelfanlnary ezpenees of pnrctaMliig
the site of a good quarry ; cleaning off the surface earth and disintegrated top rock ;
and providing the necessary tools, trucks, cranes, Ac ; the total neat expenses for
getting out the rough stone for ma^nry, per cub yard, ready for delivery, may be
roughly approximated thus : Stones of such sises as two men can readily lift, meas-
ured in p^, will cost about as much as from ^ to ^ the daily wages of a quarry
laborer. Large stones, ranging from }^tol cub yd each, got out by blasting, from
1 to 2 daily wages per cub yd. Large stones, ranging from 1 to 1}^ cub yds each, in
which most of the work must be done by wedges, in order that the individual stones
■hall come out in tolerably regular shape, and conform to stipulated dimensions ;
from 2 to 4 daily wages per cub yard. The smaller prices are low for sandstone,
while the higher ones are high for .granite. Under ordinary circumstances, about
1^ cub yds of good sandstone can l)e quarried at the same cost as 1 of granite ; or,
in other words, calling the cost of granite 1, that of sandstone will be^; so that
the means of the foregoing limits may be regarded as rather fuirprices for sandstone;
rather scant ones for granite ; and about fair for limestone or marble.

Cost of dressingr stone. In the first place, a liberal allowance should be
made for wnste. Even when the stone wedges out handsomely on all sides from
the quarry, in large blocks of nearly the requiied shape and size, from 3^ to ^ of
me rough block will generally not more than cover waste when well dressed. In
modeiate-eiaed blocki, (say averaging aboat ^ a cub yard eachj and gat oat faj
blasting, frt)m ^ to ^ will not be too much for stone of medium oharacter as ts
straight splitting. Aoout the last allowance should also be made for well-8cabble<l
rubble. The smaller the stones, the greater must be the allowance for waste in
dressing. In large operations, it becomes expedient to have the stones dressed, a»*
fitf as possible, at the quarry ; in order to diminish the cost of transportation, which,
when the distance is great, constitutes an important item — especially whenby land^
and on common roads.

A Stoneoatter vlU Am take oat of vind; and then fUrly patont-haiBiDer droH, aboat »
to 10 iq ft of plain fmoe in hard granite, in a day of 8 working honra; or twioe aa moeh of aach infe-
rior dreasing as it nsoally beetowed on the beds and jointi ; and generally on the faoeo alao of bridge
masonry, ko, when a Tery fine finish Is not required. In good sandstone, or marble, he can do about
% more than in granite. Of Jkutt hammer finish, yran««, 4 to 5 sq ft.

Cost of miisonrjr. Every item composing the total cost is liable to much
variation ; therefore, we can merely give an example to show the general principle
upon which an araroximate estimate may be made ; assuming the vrafpes or a
laborer to be 92.00 per day of 8 working hours ; and $3.50 for a mason. Tike
monopoly of qnarrles affects prices very much.*

Cost of RSblar fiaelnfr nkasonrjr. Average size of the stones, say 6 ft
long, 2 ft wide, and 1.4 thick ; or two such stones to a cub yd. Then, supposing tha
stone to be granite or gneiss, the cost per cu^ yd of masonry at sncb wages
will be. Getting ont the stone rn»n the qnarry by blasting, allowing M 'or waste in

AvHlag; l^enbyds, attS.0Oper yard 94.00

DresaincUaqftof faoeat86cu f.M

•« 53 <• beds and JoinU, at 18 eu t-Se

Keatooetof the dressed stone at the qaarry 18.M

HaoliBff, say Intlle; loading and anleadiag 1.^

Mcrtar, say. • *^

Laying, inolnding soalTold, hoisting machinery, superintendence, ke 8.00

Nea*eos( 21.86

Profit t» eontraolor, say Ift per et ^M

Total cost SS.14

Dressing will cost mere if the faces are to be rounded, or moulded. If the stones are smaller, tiiaa
we have assumed, there will be more tq ft peroub yd t» be dressed, *c. . , ,. .^ w ,. .w

If in the foregoing oase, the stones be eer/ecUy well dressed on all sides, including the baek, the
eoat per cub yd would be increased about |lO ; and if some of the sides be enrred, as in aroh stones,
say 913 or 914; and if the blocks be carefully wedged out to given dimensions, 916 or 918; thoa
making the neat cost of the dressed stone at th9 quarnf uy 938, 931. or t»5 per oub yd.

• The blocks of granite fbr Bunker HUl monument averaging 2 oub yds each, were
quarried by wedging, and delivered at the site of the mouument, at a neat actual cost of 95.40
per enb yd : by the Monument Assooiatlon ; from a quarry opened by themselves for the purpose. The
Awoelation reoeived no profit ; their services being voluntary. The average contract offers for the
same were 924.801 The actual coat of getting ont the rough blocks at the quarry was 92.70. Load-
ins upon truoks at quarry, about 16 ots. Transportation 8 miles by railway and common road, 92.S5.
Totair96 *0. In 18tt to 1845 ; common nnskiUed labor averaging 91 per day.

602 COST OF STONEWORK.

Th« item or Imyioc will be mooh increased if the stone has to be raised to great heighta; or if tt hae
to be mooh handled ; as when carried in scows, to be deposited in water-piers, ita. Almost erer\
large work presents oertain modifying peoaliarlties, which must be left to the judgment of the engi>
neer and eontraotor. The percentage of oontraotors' profit will nsoally be less on large works ttaaa
•a small ones.

Cost of asblar fkclnsr masonry. If the stone be sandstone

with good natural beds, the getting out may be put at $3.00 per onbio yard. Faoe' dressing at 36 cts
per sq ft : say $3.64 per cubic yd. Beds and JoinU IS oto per sq ft ; say $6.76 per cub yd. The neat
oost, laid, $17.00.

And tbe total oost oflarire well scabbled ranfrod
sandstone masonry in mortar, may be taken at about $10 per cub yd.

Cost of lari^e scabbled grranlte rabble, such as is generally used as
backing for tbe foregoing asblar ; stones averaging about }^ cub yd each :

Cost per
I<abor at $1 per day. eub yd of

masonry.
Getting out the stone trom the qnarry by blasting, allowing K for waste in

soabbling; iX cub yds at $3.00 $S.4S

Hauling 1 mile, loading and unloading 1.20

Uor tar ; (2 cab ft, or 1.6 struck bushels quicklime, either in lump or ground ;

and 10 cub ft, or 8 struck bushels of sand, or gravel ; and mixing) 1.50

Scabbling ; laying, including scaffold, hoisting machinery, Ac 2.50

Neat oost 8.63

Profit to contractor, say 15 peret 1.30

Total oost 0.93

Common rabble of small stones, the average size being such as two
men can handle, costs, to get it out of the quarry, about 80 cts per yard of pile ;
Dr to allow fur waste, say $1.00. Hauling 1 mile, $1.00. It can be roughly scabbled,
and laid, for $1.20 more ; mortar as foregoing, $1.50. Total neat cost, $4.70 ; or, with
15 per ct profit, $5.40, at the above wagetfor lcU>or,

Wltb smaller stones, such as one man can handle, we may say, stone 70 ets; banting $1 :
laying andsoaffold, tools Ac, $1; morUr $1.50. Making the neat oost $4.20; or with 15 per ct profit, $4.83.
Neat scabbled irregular range- work costs from $2 to $3 more per yd than rubble; according to tbe charac-
ter of the stone Ac. The laying of thin walla costs more than that of thick ones, such as abutments fte.**

Tbe cost of plain 8 inch thick ashlar faclnvs for dwellings Ac ia

Philada, in ISM, is about as follows per square foot showing, put up, including ererytbing. Sand-
stone, $1.60 to $2.25. Pennsylvania marble, $2.50. Mew Ingland marble, $2.75 to $3.25. Granite.
$2.25 to $2.75. If6 ins thick, deduct one-eighth part. First ClaSS artificial S tone
could be made and put up at one-third the prioe. North Rlver bine StOne

flaffS, S ins thick, for footwalks, pat down, inolnding gravel ke, 70 cU per sq foot. Belgian
Street pavement, with gravel, complete, $8.50 per sq yard in Eastern cities:
When dressed ashlar facing is backed by rubble, the expense per cub yard of the
entire mass will of course vary according to the proportions of the two. Thus, if
ashlar at $12 per yd, is backed by an equal thickness of rubble at $5, the mean cost
will be ($12 + 95)-i-2 = $8.50 : or if the rubble is twice as thick as the ashlar then
($12 -f $5 -f $5) -i- 3 = $7.33, <&c. Such componnd walls are weak and
Apt to separate in time, as also walls of cut stone backed by concrete, or by brick ;
from unequal settlement of the two parts.

At times the contractor must be allowed eztrs in opening new qnarries; in forming
abort reads to his work ; In digging foundations ; or for pumping or otherwisa draining them, when
aprings are unexpectedly met with ; for the centers for arches, Ac ; unless theaa items are axpreaaly
iaoluded in the eontraot per cub yd.

RETAIXTNO-WALLB.

EETAINIITG-WALLS.

«-w»ll. which i>

J aadiaturti4d luturu poaiuou^ aa in imicii & Tort or tncUDM

pm, ud Iherefan ttia wUl Eu; fsiiocBlij be tblnnar Ifaan ■

Itla, tkH ••> ■• lalKta. Uk> np. w UMn. uS
•OUiiHtba lUdlM^bi rock inui ud tkaa Mn*

ki HI to mttSBbj am •akK of rmi iim tM dv
nil. A*T«irU«a wall hu both es

WI>«ii ttas baekln^ la depoaltMl l«anBly, ■■ iua«I, aa iiA«
AnHpEil^rim airti, enri, <fa:.
TRiU (^ CH^fto<w« DT v/fitli-doM largt rangwi rttbblt,

i%mi,rtat....Ji.b. 3&0jtU miMW rtrl luifU d b.

" ffood contvin tcabbUd mortar-mbblt, or brick, A '^ '' " "

" vteU-Kobbitd dry ruUV" -"—'-.-'.'- -...,- ^ " '■ »» '»

lolfdatMl In li«r Iityera,

TbabMot lanii.ii^actkabfifkitd. lo ika KnfoiM

tor to Ht properl;, before the huklng Is depoglted bobiud It. The otject of In

604 RETAINING-WALLB.

ing the courses, is to place the Joints more nearly at right angles to the direetton
/F, Figs 6, 7, and 8, of the pres against the back of the wall ; and thus diminish
the tendency of the stones to slide on one another, and cause the wall to bulge.

When the courses are hor, there is nothing to pre-
rent this sliding, except the friction of the stones, one upon the other, when of dry
masonry ; or friction and the mortar, when the last is used. But if; as is frequently
the case, (especially in thick and hastily built walls,) this has not had time to harden
properlT, it will oppose but little resistance to sliding. But when the courses are
inclined, they cannot «2t<je, without at the same time being lifted up the inclined
planes formed by themselves. In retaining^walls, as in the abuts of important
arches, the engineer should place as little dependence as possible upon mortar ; bat
should rely more upon the position of the Joints, for stability.

An ot^Jeetion to thli UioUning of the joinU in dty ( wtthont morUr) walls, ia that rain-water, fUIinc
on the tettared hoe, is thereby oarried inward to the earth hacking: wblcb thai beoomea eoft, aaS
Bettlea. This maybe In a great measure obviated by laying the enter or faee-oonrsea hor; or by
uaiug mortar for a depth of onlv about a foot from tbe faoe. The top of the wall aboald be proteeted
by a ooping e d, Fig 1, whloh had better prqjeot a few ina in front. After the maaonry nas been
built np to the anrfaoe of the gronnd, the fonndation pit aboald be ilUed up ; and it ia well to opb-
■olidate the filling by ramming, Apeeially in front of the wall.

The b»ek dbot the wall shoold be left roncli. In brickwork it

would be well to let erery third or fourth oonrae pnqeot an ineh or two. Thia inoreaaea the fHetkn
of the earth againat the baok, and tbua cauaes the resultant of the forces acting behind the wall to
become more nearly Tert ; and to fall farther within the baae. giring increaaed atabillty. It also con-
ducea to atrength not to make each oonrae of uniform height throughout the thickneaa of the wall ;
but to hare aome of the atonea (eapeciallv near the back) aufBciently high to reach np through two or
three oonraea. By thia meana the whole maaonry becomea more effectually interlocked or bonded
together aa one maaa ; and therefore leaa liable to bnlge. Very thick walla may oonaiat of a facing
•f masonry, and a backing of concrete.

Rni. S. It la the pree itaelf of the earth agalnet the baolc, that creates the fHotton, which in tnm
Bodlflea the action of the prea ; as tbe wt or prea of a body upon an inclined plane prodocea fHoti<»
between the body and the plane, aufflcient, perhapa, to prevent tbe body from eliding down it. A re-
lainlng-wall ia overlArvwn by being made to rcTolTC around iu outer toe or edge e. Fig 1, aa a fU-
smm, or toming-point ; but in order thua to reToIve, its back must first plainly rise ; and in doing
so must rub against the backing, and thus encounter and overcome this friction. The
friction exists the same, whether the wall atanda firm or not ; aa in the case of tbe
bodT on an inclined plane ; the only diff is that in ime case it jmeweiUs motion ; and
in the other only retard* it

Where deem fireexliiir oeears the back of the wall should

be aloped forwarda fw 8 or 4 ft below ita top aa at e o, which ahonld be qaite amootk
■o aa to leeaen the hold of the f^oat and prevent displacement.

. 4. When the wall is too thin, ft will generally fail

by bnlfrlnv outward, at about ^ of its height above the '

ground, as at a, in Fig 2. A slight bulging in a new wall

'Fid* Q, ^^' ^^^ necessarily prove it to be actually unsafe. It is

^ generally due to the newness of the mortar, and to the

greater pres exerted by the fresh backing ; and will often

cease to increase after a few months. It need not exoite

apprehension if it does not exceed ^ inch for each foot in

^sasSJigSi^i^iSiSk thickness at a. See Remark 3, Art 7.

Art* 2* The yvnng engineer need not in practice concern himself partlcnlaily about the i
ar SKAV ov nn BAOxms, or about the anoli ov supb at whloh it will stand ; for the material whldt
he deposits behind his wall one day, may be drr and incoherent, so as to slope at IH to 1 ; the next
day rain may convert it into liquid mud, seeking its own level, like water ; the next tt may be lee,
capable of sustaining a considerable load, as a vert pillar.

Moreorer, he cannot foretell what may be the nature of his backing; for, as a general mie, thta
must consist of whatever the adjacent excavation may produce from time to time ; sand to-day. rack
to-morrow, Ac. Betainlng- walls are therefore usually built before the engineer knows the character
af their backing; so that in practice, these theoretical considerations have cMnparatlvely bat Utile
weight. Theory, uncontrolled by obaervation and common sense, will lead to great errore in every
department of engineering ; but, on the other hand, no amount of experience alcoe will eompeneats
for an ignorance of theory. The two most go hand-in- hand.

Again, the settlement of the backing under Its own wt, idded
by the tremors produced by heavy trains at high speed ; its expansion by frosty or
by the infiltration of rain ; the hydrostatic pressure arising from the admission of
the latter through cracks produced in the backing during long droughts ; as well a»
its lubricating action upon it, (diminishing its friction, and giving it a tendency to
slide,) Ac, exert at times quite as powerful an ovortiiriiing tendency as the legitlmat#
theoretical pres does. The action of these agencies is gradual. Garefhl observation
of retadning-walls year after year, will often show that their battered fitces are be>
coming vertical. Then they will begin to incline outward ; and eventually the wall
will fidl. Theory omits loads that may come on backing increasing its prea.

BBTAnnNO-WALI&

605

Airamhig llie theoretical Tlews ftdTamoed by Profearar Moseley to be oomot ee
ttieories, the thickneapee which we have recommended in Art 1, for mortar walla,
eorreapond to from 7 to 14 timep ; and for dry walla about 10 to 20 times, the ^rea
aaaigned by him; and we do not consider onra greater than experience haa shown
to be neceaaary. See Table 3. Betainiug-walls deaigned by good engineera, but in
too oloae accordance with theory, (which aaanmea that a realatance equal to twice
the theoretical prea ia anl&cient,) have failed ; and the inference ia fair that many of
thoae which atand have too amall a coefficient of aafety.

The flMt U, (or at teut m U tmpwn te as,) ther* matt be d«heU In th« theoretleel annnptloBa of
some of tho most prominent writers who five praotleal miss on this sut^eot. Thns Poneelet, who
oartainly is«t thMr head, statM that his tables, for praatieal use, gire thicknesses of base for sos-
(alBiog 1 X. timse the theeretleal pros ; and this he eonsiders amply safe. Tot, for a vert waU of eat

granite, his base fbr sustaining dry sand level with the top, as in Fig 1, is M of the rort height;
and for brick. .46. Bat the writer found tha^is*«n not tui^Jeet to trtmtor, a wooden model of a vort
wall, weighing but 18 lbs per eub ft, and with a base of M of its height, balaaeed perfeetly dry saad
•loping at 1^ to 1, and weighing 89 lbs per onb ft.

How, THB BBSISTAIIOB OF SIini.An WALLS, OF TBB SAMB DI1IB1ISIOM8,

WAJOBS AS THBiB sPBcinc •KATiTiBs ; and, since granite weighs about 166
lbs per cab foot, or 6 limes as much as our model, it follows, we concelTS,
that a wall of that material, with a base of .86 of its height, must have
• resistance of 6 times any lr«i4 tlUor«Hoal pree, instead of only 1.8
tiases ; and that his brick wall must hare aboat 6 times the mere bal>
MMing reelstaace. Our experiments were nude in an apper room of a
•SroBgly built dwelling ; and we found that the tremor pnMuoed by pass-
ing vehicles in the street, br the shutting of doors, and walking about
the room, snlDced to gradually produce leaning in walls of considerably
more than twice the mere balancing stability while quiet; and it appears
ta us that the injarious effects of a heavy train would be oomparatiTcly
quite as great upon an actu|| retaining- wall, supporting so uoohesive
a material as dry sasd.

Since, therefore, Ponoelet's wall Is in this instance suffleiently stable
for jiraclics, it seems to us that his theory, which neglecu the effect of
tremors, ftc, must be defsetlTB. He also gives X of the height as a snf-

floiently safs thickness for a vert granite wall supporting atigemrth; but
we suspect that very few engineers would be willing to trust to that pro-
portion, when, as usual, the earth is dumped in fhtm carta, or cars ; espe-
•ially during a rainy per.od. If deposited, and consolidated in layers,
theory could scarcely assign any thickness for the wall ; for the backing thus bCQcmes, as it were, a
mass of nnbumt brick, exerting no hor thrust ; and requiring nothing but protection ttom atmoapbene
Influetice, to insure its stability without any refa<n<n 9- walh It is with great diffldenoe, and distrust
in our opinions, that we venture to express doubts respecting the assumptions of so profound an tai'
Tostigator and writer as Poneelet ; and we do so only with the hope that the views of more comp^
tent persons than ourselves, may be thereby ellolted. Our own have no better foundation than ex-
periments with wooden and brick models, by ounalves ; combined with observation of actual walls.

Art. S. After a wall aheo^ Tig 3, with a Tert back, haa been proportioned by
OUT rule in Art 1, it may be coiiTerted Into one witli an oflhetiea

baelL, aa a i n o. This will present greater resistance to orertumlng; and yet con-
tain no more material. Thua, through the center t of the back, draw any line t n;
from n draw n «, Tert; divide « < iuto any even number of equal parts; (in the fig
there are 4 ;) and divide « n, into ont mnrf. equal parte ; (in the fig there are 5.) From
the points of division draw hor, and vert lines, fbr forming the ofTseta, aa in the fig.
In the offsetted wall, the cen of gray ia thrown farther back from the toe «, than
in the other, thua giTing it increased leTwage and resistance; but within ordinary
practical limits, the diff is very small ; and since the triangle of supported earth ia
greater than when the back is rert, its prea is also greater; so that probablv no ap-
preciable advantage attends that consideration. Tbe inereaso of thick ■
near tbe iMwe, dimlnlsbes, boweirer* tbe
leTerafre v a, Fife 8, of tbe pres/P^ of tbe
eartb against the back. The center of pressure of
this pres is in both cases at ^ the Tert height, meas-
ured from the bottom ; and it is therefore plain that
the fitrther back fh>m the front it is applied, tbe shorter
mnat v a become. Moreover, in the offaetted back, the
direction of the prea becomea more nearly Tert than
when the back is upright. It is to these causes, rather
than to the throwing back of the cent of grav, that
the ofbetted wall owes its increase of stability 0T3r
oae with a Tert back.

Art. 4. Wben, as in Flff 4, tbe backinip is blgrber tban tbe
wall, and slopes away firom its inner edge d, at the natural slope d », of Ij^ to 1, we
are confident that the following thicknesses at base will at least be found sufficiem

RETAIHIHO-WALLS.

Hitb bJkiai

S^'

.g m lnch« lo n fa«t.

Tit

TABLE 1. (Origliial.)

1

«\

WUl

«

'^

J£..

s.

WJU

a,-;*

BlUk.

c.'.!r

ST

h

How.

SSiS?

1 ^

TDbk>

•• •>&»,!

MIUX

e a

>k.b.l|k>.

■Wbilthl.

:u

in

M

.u

^

M

.M

:»

.M

■"

.11

""^

■"

■"

■"

TABI.E 2.

• will, u In Fill iuds,ii» Willi! sarelisrK*<)>

e tliri«py of r«tMnliis>wsll».

RBTAINING-WALI*.

bt aSihu

Ml; to Hi < iTia ■ nn bad, if Uh ■

■upaor iKui.«>a°ii'.i>uii tn.iD«i>=,ta™sr'is'^ ud .''^'* =. 38° B'.ttaeeor-
reapondlnc ui(I« « m t of utux. pre*.

tUtna. Tk* Dvmbtr 4f '«(& ft Qf viJI, or of ljHklD(. la Lhn iqiflftl u thai frf U14 ■^■n bn la

baftlring ta be pwfoclJj dir, *Dd devoid af MhfifllDD, {or teadflDcj Id stick to SAch
ndd«nl7 reiDOTBd, then tha trinnglfl of mrtb cmt, comprlfed bMWBUk tb« ^op* ■■ I
•(iBaipn(,iitidtligTBrtbMkefiifirtb*wall,nK 6, voald ilide down, nndu thtln-
aiieiica of ( force whicb mi} ba repnaanted bj y P. acting In ■ dinclum y P. >1 right
knglH to tbetuse en at liie4TiBn«l» of eulbi (or In othar ooids. nl right anglH

Muid c, meuDivd Iromttw bollom; ud IM amoiint eijukl toBitbaraf tha&JIowliig:

Sol. "^^"^ ^^td^lhZ^ f"| Sea

So 9.

Id Ttew of (he great nncanalntj inTolved In tha matter of the actual preHnra 01
Mitb againet rBl«lniuK-«all> in practice («ee Art 2. ), and tn order lo fnnrilh

k aimpiaiule irbiol., althccgh enlirely uniupporled b j theory, li elill (In the wrller'i

that No 1 of the two foregoing formulBe aupliea ntar'aumigh'to 'walla with Id-
cllnad baeliB e ni, also, u Fip V and ». (preciwlj as rbey are lellered,) at leait
until tbe bnch of (be wnll Inclines ronrnrd h mucb a» 6 Ina
kap, to 1 foot veFt, or at an angle cmo of ■IS' 34'. tVtiMt folio wa on
reUunlnc-wtUla will Involve tills tnrorrect nan ntnp lion, nnd
moat be reffapded merely aa v'vlDKanfc apprnzlmntlon.

SETAIN I NO- W ALU.

idwiilL Tli»( lii,ff »w«ll were to begin to QfBi

^<iil; cdcnli

give lU tba amaunt cf tli^ir niulunt ; wblcta (• the
>ppr*x alnKle (beoretlOBl fsr« — '

MHOniitaiidInd' — — —■-.-'

n dlreellon. which thew«U

But thll f0IO0,/P, ii Mita h1w»i equal to Ihs pn
. fon» II P, malt W the ut hc iJ (be uigle jr P/ <^
tbe will friction ; (or dlTided bj lie ut cMln«) ud a(
coune Dwj be Mceitained thna:

caDfH f P - vtndipa-a ~ «»Trtxoiii

tllT, if It ti uenmed. u ws do Ihrnughonl, llwl the esrOi ia pertKllf dr; (la

:: triciloD sre tben ucb 33° *V or l.S u 1, then In Figi B, T lod S, ir the uuis
betwMD tha beck c m ud tbe lerl d m doaa not eicssd ftbont 2*° Si' «• tuy

I tnclndee Ibe kIIod of tbe friction or the eutb afi^Jimt tbe back or tlu w^I.

Km. 2. How M and botb (ke eTcrtaFBlBr toadener »r th«
MtrUl, >nd the mletADoe of (be wall Mninet belnn oierlnnied sniuad^te t« 11 h

■ ralcruD), first Asd the cen of (rai g of ths well >nd through It dnw ■

Terl lines'!. Prolotig/F lowardi eand dniiaEp«rpto It. By ujr eula muke
10 = w( of trail, ud 1 1 = calculated prvi /P. OiiiDp1e[e the parallalogma oii'i.
uddnw ill diagonal in, Hhlcb will U tbe reinllaot of tbe prea/P and of the «t
of the wall ; and ab-^uld for aafetT be such that oj be not lea than about one-fifth
of a m.even vtUfi ^eV moMmry and unyidding toU- Otherwise tho irrcat preuurfi ao
neu thR wo a mi.J either fracture tbe wall or compress the soil near that point
■o that the *al> :r|]| lean forward. In waUs buili b/ our rule, Art 1, or bj £abl«,
p «(J, o ) will be Jiore than one-flfth of a m. The pree / P if mult bj l(a leterue
aawillgiva the moment of tbe preeabouta; and the wt of the wiQ multbjlU
levomge e a will glTe that of tbe wall. The w»U Is safe from overturning in pro-
portion u lia momsut eiceeda that of the prea. It li asaumed ia be aafe ngilDft
tliiti:ii, (.rtoWnsf, or "Uliag into the ™i].

>r wall fr|.

B«a>. 4. ir the tortk Blnpea dowannrd rrom C, aa c

»t A or B, ImUad of baEng hor u In Figs 6, 7, 8, use the wl of Ibe Mv* A
earth cntn iiiBleiulor cm'.nin belag theBlope of max presiuie. M/\

In A the paint of upllcatlon wtll still be at P <at oae-third of M/ J:::^

nrtsslD e.7, 8;but In B it will be a little higher At> explained m
below for Fig 8.

SarrhnrKed walla are thoae ia which Uie earth bsekiog
eilenda aboie the tops of the walla.

According to IheorT, when as in Fig 9, there is a aurcharge

height. DO additional una la Ihcrel^
tbrown against tU« back nf (he waif.

then (ha alopa m (f of max pros mint ,
extend Dp to meal thia other slope. I

TIm »p|»rox I maM a>ioun(

Borcharged, (as In any of the Figs 4.

tuew7of°BatbB'e»rSd'.mi,l''lg'4. "
d mi r, Fig 5, or c d m. Fig 3 (If Ihe

between the slope ■■ d. Fig >, m ^Figi 1 and (i, of max prea. the back of the wall, and
the front slope ; omitting any whiob, like if c n, Fig f, leata on the lop of (be vail
(and ID 111 add. nits stabilll)-) when the elope sWru in fron[ of c. Haying found

•pp'^SST™ JlelT } - W» of tta. e^h X .MS,

iDcloding tbe action of the friclloa of the earth against the back of the wall; n<w
enoagh (In tbe writer's opinion) for praotlcal pnrpMea Id so uncertain » matter;
but eHenllHlI; ein|ilrlcal.

The direction of tbeproaaaretbua foandwlll be the sameaa when the

peip to tbe back cm, whetlier >ert or inclined^^ Then draw another line' ei Pf,

be 33°ll',or 1.6 tol. Then P/ will gite tbe direction of tbe pressure. Hut Us
polol of application will not alwavs be at P (one-[blrd of the height of Ihe wall

bicheroneng *, where the back la cutbjallneiPore&,Plgfl, drawn from tbe

third tbe height of the wall oulj »tieii the sustained earth I c m or d tr n forniB a

Smplete trlaiiBle, one of whose aniileBia at the iuusr top edge c of the wall.
■U other caaa said line for a lurcharse will strike aboie P.

610

BETAININGhWALI^.

Art. 7. On page 603, Fig 1, we recommend that the base o < at the ground-
line of well bnilt Tertleal walls should not be less than .^, or .4, or .6 of the
height cffl above said line, depending on the kind of masonry. But a wall with a
battered (inclined) front ur face as found by Art 8, (by which the following

table was prepared), will be as strong, and at the same time contain less masonry
than a vert wall, although the batteredone will have the thickest base os.

Table 8, of thicknesses at base o «, Figr 1, and at top e d, of
walls with battered faces, so as to be as strong as vertical
ones wbicli contain more masonry.

For tbe cub yds of masonry above o s per foot run of wall, molt the
sqaare of the vert height d « by the number in the column of cub yds. Then
add the foundation masonry below o s.

(Original.)

Ail the walls below have the fame strength
as a vert one whoae base o<, fig 1= .35
of itshtda.

Batter, in
ina to aft.

0

3

4

6

Triangle

Cut stona.

Base, in Top, In

pts of pte of

ht. ht.

.350
.352
.366
.359
.364
.371
.379
.389
.400
.426
.429

.860
.810
.270
.234
.197
.163
.129
.096
.066
.007
.000

0 yds per
ftinin.

All the walls below have tbe
same strength as a vert one

whoiie base
of Itshtds.

o «, fig 1=.4

Mortar rabble.
Base, In Top,

.01296
.01226
.01168
.01098
.01039
.00989
.00941
.0U89S
.00863
.00800
.00794

pts of
ht.

.400
.401
.408
.408
.413
.419
.426
.436
.446
.468
.490

pts of
ht.

.400
.369
.320
.283
.246
.210
.176
.143
.110
.051
.000

G yds per
ft run.

.01482
.01407
.01339
.01280
.01220
.01166
.01111
.01070
.010-28
.00961
.00907

AH the walla below have the
same strength as a vert
one whose base o s, fig isr
.5 of its ht d s.

Dry ral>bl3.

Base, in

Top, in

ptoof.

pts of

ht.

ht.

.600

.600

.601

.460

.603

.420

.606

.881

.610

.343

.616

.308

.622

.272

.528

.236

.687

.204

.666

.188

.612

.000

0 yds per
ft ma.

.01852
J01778
.01709
.01648
.01580
.01526
.01470
.01415
.01872
.01288
.01188

Moseley and others qaote G^adroy, for a dht sahd slopdio at 2I0. It wonld be better to oease from
eircalatinic f>uch evident mistakes. Dry sand will stand at no less angle for a savant than for aay<.
body else. For praotfeal purposes, we may say that dry sand, gravel and earths, slope at S3o 4] or
m to 1 ; as abundant experienw on railroad embkts proves. Ponoelet gives tables for waUs to sap-
port dry earth sloping at 1 to 1, or 45°; but as we do not believe in the existenoe of sueh earth w»
emit such tables. Sand, gravel, and earths may be moistened to diff degrees, so as to stand at any
angle between hor and vert; and by moistening aud rammiuR, the earths may be converted into oom*

{>act masses, exerting little or no prea : and may even si* continue after they become drv : beinc then
n fact, a kind of air-dried brick. It is sometimes dtlQoalt to know whether earth or sajid is pwfeetlv
dry or not; and an exceedingly small degree of moisture will eanse them to stand at 1 to 1 in mah
heap*, such as have probably been observed by the authorities on the subjeet. The writer found tha4
fine sand fj-om the seashore, and under cover, would stand at 1 J^ to 1 during warm dry weather, and
^ ^***H "''•'?/•»* ''^ T,^„*J!""P\. ^®' °° **'' whatever in its degree of moisture was peroeptlbleto
the feeling. Its snsoeptibi ity to dampness was of course owing to salt. A f^sw handftals of dry evS
may perhaps be ooauetied into •tandfng at 1 to 1 on a table ; but so far as our observation ekteods!
when it is dumped in large qaantities ft-om carts and wheelbarrows, its slope is about IW to 1 • and
this we consider the proper one to be used in practical calcnlaUons, where safety is the oonsideratioa
of paramoant importance. ' w««.«w»»«i»

«inT?f« ?S2?iJIl? ?I'**K*l?'*®? ***« firreater Is the pren: and since th«
wS«n L lwi,!^il *B* ^**'¥°« ^ perfectly dry, (omitting of course its condition
when so absolutely wi^ as to become partially fluid,) we have, on the score of safetT

Tus^'tlThlf \hoti^.7^*"V°«- ''l'''''^ ^° Artl, we cannot recommend d^S^aJ:

TaSonl^Ve^'pose^dTn pub7ic"wo?is"''" "' ^^"^''^' *^* "*"^^ *^'**'"«"* *^ -»***^^

«.?w K?ir^*?!F ^xf***** »lon» dangrerons preclplees, we should
rather be- tempted at times to make thicker walls. We Imaging, for instance that

dan^:S,'nf ^^ ^T^^'J \**^*^ *™^"' ^^^■'""^ »^""°<» » sharTcui^e'cSuvex ou th«
dangerous side, should not be overlooked in designing walls for such localities. Thlt
force 18 hor; and is applied near the top of the wall : and, consequently, its levenura
may be considered as equal to the height: whereas the theoretical pres of the earth
18 oblique ; and is applied at J^ of the height from the bottbm ; so that its leveraire
abput the toe of the wall is very short. Moreover, the simple Wfight of the train, pro-

JS^^^^f*? i'^K^^ri ^^^. Y*" L^ T" ** ***** ^^ *^® backing. All such considerationi
are omitted by theorists. The dangerous pres caused by tremors. Ac, cannot be

BET AIMmo-W ALLS.

Char^g: bill is Bpt In becnmo aaturetarl vhh vtUt, Mpecially brlow l(iit.»»ler
level; apd tliqa lo eiert a very kivbI pr» sgainel the walls. Iiloreovc^r, ILe water

miueqnfntJy Jta acabllitj. The Mise fsum of course dlmjoiahps the frklioD of the

Is smoolli. BDd borHoDtBli ind have doue to'eieii wbeo the roundatlou h^Aoi?

•gitlBii II. It otHDt K lo « or fu tulgtil .Iw.i mnnd, [n Rent fieei, piniiiiitn. buiu-fim nl

Art. S. To cbftuce a vert rrtalnlUK-wnll, Into one nitti a
bsttered face, which shall preiieut nn eqiml reslBlAiie*
B^alust overtnruluci «lthoug;b retialrlna; lean masoury.

TUs ii someliinea tetmed * (rkBBroriDBtlon of proBle. (Ongiusl.)
Letaftoi,Flal0.bethevertwi.ll. Mult its hue

612 EETAINING-WALLS.

BsM S^ If OBBom, WHXir ooimoir icoxtab u iTsni> withodt av Anuarumm ov obmbht, whloh it never
■boald be, in retaining- wails, wiiere durabllitT is au otdeol, a great batter is olyeo*
tionable ; inasmaob as tbe rain, combined witb frost, fto, soon destroys tbe mo^
tar. In snob oases, tberefore, tbe baiter sbould not exceed 1 or l}i ins to a ft ; and
eren tben, at least tbe poindng of tbe joints, and a few floet in beigbt of botb
tbe apper and tbe lower oonrses of masonry, sbould be done witb oemunt, or
eement- mortar. We bare obserred a most marked diffin tbe corrosion of tbe mor>
tar, wbere, in tbe same walls, witb tbe same exposure, one portion bas been built
witb a Tert face ; and anotber witb a batter or but 1^ incb to a foot. Commoo
mortar will nerer eel properly, and oontinue firm, wben it is exposed to mois-
ture f^m tbe eutb. Tbls is very observable near tbe tops and bottoms of
abuts, reMlning- walls, fto; tbe lime-mortar at tbose parts will generally be
found to be rendered entirely wortbless. A profile somewhat like Fljfc 12, may
at times prove servioeable, instead of tbe triangular. Tbls is tbe form of tbe
Gothio buttress ; wbiob probably bad its origin in tbe cause Just spoken of. ^

Art. 9. A retalninip-irall may slide, wlthont
!?• X I Q loslnff Its vertleality ; and, indeed^ without any danger
JClQ l/C of being oyertumed. This is very apt to occur if it is built upon
^ a hor wooden platform ; or upon a level surf of rock, or clay.

. without other means than mere firiction to prevent sliding. This may be obviated
by inclining the base, as in Fig 1 ; by founding the wail at such a depth as to pro-
Vide a proper resistance from the soil in front ; or in case of a platform, by securing
one or more lines of strong beams to its upper surf, across the direction in which
sliding would take place. On wet elajr* friction mav be as low as fh>m J2 to
% the weight of tne wall ; on dry earth, it is about % to jf^ : and on sand or gravel,

about % to %. The friction of masonry on a wooden
fcl fcl fcT platform, is about JL of the wt, if dry ; and % if wet.

L I CoanterfortS, sbown in plan at e e e. Fig 18, oonsUt Is

an increase of tbe tbiokness of tbe wall, of its hack, at r^ular inber*

~ri> A 9 vale of Its length. We conceive tbem to be but little better than a

XTCI X6 waste of masonry. Wben a wall of tbis kind fails, it almost in*

J vuiably separates ftom its eounterforts ; to wbiob it is connected

merely by tbe adhesion of tbe mortar ; and to asligbt extent, by the

bonding of the masonry. The table in Art 7 shows that a very small addition to the base of a wall. Is

attended by a great increase of its strength ; we therefore think that the masonry of counterforts

would be much better, and more cheaply employed in giving the wall an additional thickness, alonf

its entire length ; and for the lower third of Its height. Counterforts are very generally need in

retaining- walls by European engineers; but rarely, if ever, by Americans.

Buttresses are like counterforts, except that they are placed <n/Vv(U of a wall instead of b*<
hind it ; and that their profile is generally triangular, or nearly so. They greatly increase its strength|
but. being unsightly, are seldom used, exeept as a remedy when a wall is seen to be failing.

liaod-tles, or long rods of iron; have been employed as a makeshift for upholding weak re*
taining-walls. Extending through thb wall flrom its face, the land ends are eonneeted with andiors
of masonry, oast-iron or wooden posts ; the whole beiqg at some dist below the snrbee.

Retaining' wails with cnrTed profiles are mentioned here merely to ea«-
tion the young en^neer against building them. Although sanctioned by tbe practice of some hick
authorities, they really possess no merit sufficient to oompensate for the additional expense aadtexm-
ble of their construction.

Art. 10. Among military men, a retaining- wall is ealled a revetment. When tha

earth is level with the top, a scarp revetment; when above it, a connterscarp

revetment, or a demt-rto^trntnt. Wben the face of the wall is battered, a aioptng; and when the bMB
48 battered, a comUmrOoptfig revetment. Tbe batter is oaUed the talns>

Art. 11. The pres against a wall Fig 6, from sand etc level with its top, is not
diminished by reducing the quantity of sand, until its top width e a becomes lees than
that (c t) pertaining to the angle cm I of maximum pres. The pres then begins to di-
minish, but in practice tlie diminution if not appreciable wtUU the width it reduced to about
one sixth of that (c a) pertaining- to the angle cms of natural slope, or about half of
1 1. The pres then begins to decrease rapidly as the width is flirther ceduced.

Table 4, of contents in cnb yards for each foot in lenprth
of retainlnff-wallSy with a thickness at base equal to .4 of the vert height,
if ther back is vert. If the back is stepped according to the mle In Art 8, tiM

proportionate thickness at base will of course be increased. Face batter, 1^^ inches
to a foot ; or ^th of the height. Back either vert, or stepped according to the nil«
in Art 3, Fig 3. The strength is very nearly equal to that of a vert wall with a
base of .4 its height. Experience has proved that such walls,

when composed of well-scabbled mortar rubble, are safe under all ordinary circum-
stances for earth level with the top. Steps or offsets, o e, at foot, Fig 1, are not here
indnded.

STONE BRIDGES.

613

TABIiE 4. (Original.)

fit.

Cob.

Hi.

Cub.

Ht.

Gab.

Ft.

Yds.

Ft.

Yds.

Ft.

Yds.

1

.013

lOX

1.88

20

5.00

H

.028

11

1.51

H

5.25

a

.050

H

1.65

21

5.51

H

.078

12

1.80

H

5.78

9

.113

H

1.95

22

6.05

H

.163

18 .

2.11

»*^

6.33

4,

.aoo

H

2.28

6.61

H

.253

U

2.45

H

6.90

.6

.813

H

2.63

24

7.20

H

.878

15

2.81

H

7.50

6

.450

H

8.00

25

7.81

H

.528

16

8.20

H

8.13

7

.«13

}S

8.40

26

8.45

H

.703

17

8.61

H

8.78

8

.800

H

8.83

27

9.12

H

.903

18

4.05

H

945

9

1.01

H

4.28

28

9.80

H

1.18

l»

4.51

H

10.2

10

1.26

H

4.75

29 1

10.5

Ht.
Ft.

29 Ji

80

81

32

33

34

35

36

37

38

S»

40

41

42

48

44

45

46

47

Gab.

Ht.

Cab.

Yda.

Ft.

Ydii.

10.9

48

28.8

UJI

49

80.0

12.0

50

31.3

12.8

51

32 5

13.6

52

83.8

14.5

53

85.1

15.3

54

86.5

16.2

55

37.8

17.1

56

39.2

18.1

67

40.6

19 0 •

58

42.1

90.0

59

43.5

21.0

60

45.0

22.1

62

48.1

S8.1

64

51.2

24.2

66

54.5

25.8

68

57.8

26.5

70

61.3

S7.6

72

648

Ht.
Ft.

74

7ti

i8

bO

82

84

86

88

90

92

94

96

98

100

102

104

106

Cub.
Ydi.

Gb.5

72 2

76.1

80.0

84.1

88.4

92.6

96.8

101.3

105.8

110.5

115.2

120.1

125.0

130.1

136.2

140.5

STONE BBIDQES.

Art. 1. In an arch sts, Fig 1, the dist eo is called its span ; t'a its rise ; t its
erown ; its lower boundary line, ft a o, its soffit, or Intrados ; the upper one,
Ttr, its baefc, or extrados. The terms sofBt and back are also applied to the
•ntire lower and upper curved gurfaces of the whole arch. The ends of an arch, or
the showing areas comprised between its intrados and extrados, are its faces ; thus
the area itiaiaa face. The inclined surfaces or Joints, re^rn^ upon which the/e«^
of the arch rest, or f^om which the arch springs^ are the SKewbacks. Lines
level with e and
edges df its feet,
blocks of which the ;

The center one, to, is the keystone; and the lowest ones, <«, the sprlnfpers.
The term archblock might be subetituted fo^ Yonssoir, and like it would apply to
brick or oflier material. as well as to stone. The parts <r, <r, are the haanebes ;
and the spaces trl,trb^ above these, are the spandrels. The material deposited
in these qpaces is the spandrel ffllling^ ; ft is sometimes earth, sometimes ma-
■oniy ; or partly of each, as in Fig 1.

Ib large artshei, it often oonsiata of sereral parallel aPANDsn-wALui, (I, Fig 2yi, nionlng length wiie
flf thenwdway, or straddle of the aroh. They are eoTered at top either by small arches from wall to
wall, or by flat atonea, for aupporting the material of the roadway. They are also at times connected
togetbier by rert oroas-walla at Intervals, for ateadying them laterally, as at 1 1, Fig 2>j. The parte
gpen, gpont Fig 1, are the Aavrusm* ot the arch; en, on, the /aeea; gp, gp, the backs; and
pn,pnt the haam of the abnta. The baaee are asuAUy widened by /««(, ttepa, or offaett, d d, for dis-
tribnting the wt of the bridge over a greater area of foandation ; thns dlminlahing the danger of set- ,
tleoDMnt. The diataaee t oia any aroh-atone, la called iu depth.

The onlv arches In common
use for bridges, are the circular,
(often called segmental); and
the elliptic

Art. 2. To find the
deptli of keystone for
llrst-elass ent* stone
arebes, wbether eir-
ealar or elliptle.*

Find the rad e o. Fig 1, which
will touch the arch at o, a, and
c Add together this rad, and
half the span o e. Take the sq
rt of the sum. Dir this sq rt

1^ 4. To the quot add ^ ot %
K, Or bj formula.

• Inasmneh as the rates whioh we glre for arches and abnta are entirely original and noyel. it may
not Iw analM to state that they are not altogether empirical i bat are baaed upon aooorate drawings

614

STONE BRIDGES.

Depth of key VRad -f half span i a «> a^#
infest ™ 4 -1-u.z/oor.

For second-class work, this depth may be increased about ^th part; or
for brick or fair rubble, about ^d. See table of Keystones.

In large arches it is advisiable to increase the depth of the archstones toward the
springs ; but when the span is as small as about 60 to 80 or 100 feet, this is not at all
necessary if the stone is good ; although the arch will be stronger if it is done. In
practice this increase, even in the largest spans, does not exceed from J^ to ^ the
d^pth of the key ; although theory would require much more in arches of great rise.

Beh. To find the rad o o, whether the arch be circular or elliptic. Square
half the span e o. Square the whole rise i a. Add these squares together; div the
sum by tvoice the rise i a. Or it may be found near enough for this purpose by the
dividers, from a small arch drawn to a scale.

Amonnt of pressure sustained bT arelistones. In bridges of
the same width of roadway ; If all the other parts bore to each other the same propor-
tion as the spans, the total pres would increase as the squares of the spans, while the
pressure per square foot would increase as the spans. But in practice the depth of the
archstones increases much less rai^dly than the span ; while the thickness of the
roadM'ay material, and the extraneous load per sq ft, remain the same for all spans.
Hence the total pressures, at key and at spring, increase 'hss rapidly than the squares
of the spans ; but more rapidly than the simple spans; ai do also the pressures per
sqwwe foot. Thus in two bridges ^f the same width, but with nians of 100 and 200 ft,
with depths of archstones taken from our table and uniform from key to

spring; supposed to be filled up solid with masonry of 160 lbs per cub ft, to a level of
about 16 inches, above the crown, (including the stone paving of the roadway); with
stf extraneous load of 100 lbs per sq ft; the pressures will be approximately as fol-

lows:

•

Span 100 ft.

Span 200 ft.

AT KEY. 1

1 AT 8PBWO. 1

AT KEY. 1

1 AT BPBINO.

For 1 ft in

width of

Its entire

depth.

Per sq fk

Por 1 ft in
width of

Its entire
depth.

Per iq ft.

Fer 1 ft in
width of

its entire
depth.

Per aq ft.

For 1ft in
width of
its entire

depth.

Per sq ft.

Bise.
%

Tons.

Tons.
18«

Tons.
68

Tons.
18H

Tens.
128

Tons.
29J<

Tons.
179

Tons.

i

86H

Wk

5T

19

lis

«7M

181

44

8

18

11
9
6X

57K

e7«

SO

25 1

97

80M

67X

SI

188
S07
230

♦7H

It will be seen that with the same span, the pres at the key becomes less, while that
at the spring becomes greater, as the rise increases. Also that %hen the archstones
are of uniform depth, the pres at either spring of a semicircular arch is about 4 times
as great as at the key ; whereas when the rise is but one-sixth of the span, the pres ac
spring averages but about one-third greater than at the key. These proportioiis yary
somewhat in different spans.

The greater pres per sq ft at the springs may be reduced by increastng the depth of
the archstones towards the springs. This however is not necessary in moderate spans,
Inasmuch as good stone will be safe even under this greater pres.

By nsinar parallel spandrel walls, see Fig 2^ or by partly fill-

ing with earth Instead of masonry, the pres on the archstones may be dioalnished,
Bay, as a rough average, about \ part

sad ealonletions made by the writer, of lines of pres. ta, of arches ttom 1 to SCO ft span, and of every
rise, from a semloirele to A of the span. From these drawings he endeavored to find proportioat
Which, although they might not endare the test of strict oritidsint would atiU apply to all tlw<
With aa aoour«oy suffloieat for ordinary practical porpoees.

BTOMG BRIDOEB. 615

Tkble 1. or Mtme ezlatlnR arehpa, nlth both thdr utnal mi (heir

liilljjljii III UAUiii

iM

qEe3S.c,.3SSo2iBGa3S3i:sx$ss:^5_„!;::ass.^sn^

i t%isi s S p3 a eiie s ess ^SSsssaScs Sss

^oSft^iS iS S 'SS ' -SS^ 2 ^S** i: B

>i SS^I^ S i S^S = 'SSS ^ '^i^' S«5iBSXSS3 iSS

■III a:

!f fc .-1

mil,

ni

3,^

till' ^

jijiji;

616

BTONE BRIDGES.

Experimental Arch at Sonppes, Franee. See Table.
Bpan = about 18 X rise.

Span.

Rise.

Badius of
intraduB.

Depth of arch-stones

at spring.

at key

Width.

on faces.

betw&ces.

Meters

Feet

37.886
124.30

2.126
6.97

86.5
280.62

1.10
3.61

1.10
3.61

0.80
2.624

3.5
11.6

Arch of granite. The centers rested (for four months) on sand in 16 cylinders, 1 ft
diameter, 1 ft high, of ^inch sheet iron. The unloaded arch settled 15 millimeterB
(0.59 inch) on striking the centers. The additional settlements under extraneoiu
loads were as follows:

Extraneous load.

Increase of settlement.

EilogramB.

Pounds.

Millimeters.

Inches.

Distributed

Center

367000

4975

132600

809000

11000

292000

21
0.3
1.2

0.8
0j012

Distributed

0.0i7

With the distributed load of 867000 kilog, a load of 4976 kilog, falling 0.3 m (11.0
ins) on key, caused vibrations of 2.8 mm (0J.1 inch). Amudea dn Pont» et Cha»u»£e$,
1866Par<2,1868i\ir<2.

The veh on the BooBSomiAB Railway, i« probably the boldest :• and nn Oabut Jomr Mman, Vf
Oapt, now Gcn'l M. O. Meigs, IT S Army, the graodest stone one In existence. Poht-t-Pktdd, &
Wales, is a common road bridge, of very rude constraction ; with a dangerously steep roadway. It
was built entirely of mbble, in mortar, by a common country mason, in 17&0: and is still in pcrfeet
condition. Only the outer, or ahowing arch-stones, are 2.5 ft deep : and that depth is made up of twa
stones. The inner aroh*stones are but 1.5 ft deep ; aud bui from 6 to 9 inches thick. The stone quar^
ried with tolerably fair natural beds ; and received little or no dressing in addition. Tbe bridge is •
fine example of that ignorance which often passes for boldness. Poht Napolbom carries a railroad
across the Seine at Pans. The arches are of tbe uniform depth of 4 ft, from crown to spring. Tbey
are composed chiefly of atnall rough quarrif chip: or apawU; well washed, to free tbem from din
and dust ; and then thoroughly beaded in good cement ; and grouted with the same. It is in fact aa
arch of cement-concrete. The Pont db Aula, near it, and bviU in the some wag, has elliptic arche*

of fh)m 136 to 141 ft span ; with rises of l- the span. Key i.9 ft. These two bridges, considering th«

want of precedent in this kind of construction, on so large a scale, must be regarded as very bold;
and as reflecting the highest credit for practical science, upon their engineers, Darcel aud Couche.
Some trouble arose from the unequal contraction of the ditftrent thicknesses of cement. They shoW
what may be readily accomplished in arches of moderate spans, by means of small stone, and good
kudratMe eemeHt when large stone flt for arches is not procurable. In Pont Napoleon the depth (rf
arch is less than our rule gives for aeoond class out-stone.

Art. 3. Tbe keystones for largre elliptic arcbes by the best en-
gineers, are generally made about ^ part deeper than our rule requires ; or than is
considered necessary for circular ones of the same span and rise ; in order to keep the
line of pres well within the Joints ; although the elliptic arch,with its spandrel filling^

has slightly less wt; and that wt ha«
a trifle less leverage than in a circular
one ; and consequently it exerts lesn
pres both at the key, and at the skew-
back. See London, Gloucester, and
Waterloo bridges, in the preceding
table.

Rbm. Tonng engineers are apt to affect shallow arcfa-fltoneg; bat it would be fiur
better to adopt the opposite course ; for not only do deep ones make a more stable
structure, but a thin arch is as uuHlghtly an object as too slender a column. Aooord-
log to our own taste, arch-stones tully % deeper than our rule giyes for fitst-clssi
cut stone, are greatly to^be preferred when appearance is consulted. Bspedally
when an arch is of rough rubble, which costs about the same whether it is bulH up
as arch, or as spandrel filling, it is mere folly to make the arches shallow. Stability
and durability should be the objects aimed at; and when they can be attained even
to excess, witiiout increased cost, it is best to do so.

* Built Uke that at Souppes.

STONE BRIDGES.

617

Table 2. Depths <»f keystones for arebes of first-class cut stooe,
by Art 2. For second class add full one-eighth part ; and for superior brick one*
fourth to one>third part, if the span exceeoa about 15 or 20 ft. OrigibaL

Biae, in parts of the span.

8pah.
Feet.

i

*

i

i

Key. Ft.

Key. Ft.

Key. Ft.

Key. Ft.

3 •

.55

.56

.58

.60

4

.70

.72

.74

.76

«

.81

.83

.86

.89

8

.91

.93

.96

1.00

10

.99

l.Ol

1.04

1.07

15

1.17

1.19

1.22

1.26

30

1.32

1.85

1.38

1.43

25

1.45

1.48

1.53

1.58

90

1.57

1.60

1.65

1.71

85

1.68

1.70

1.76

1.88

40

1.78

1.81

1.88

1.95

60

1.97

200

2.08

2.16

00

2.14

2.18

2.26

3.S5

80

2.44

2.49

2.58

2.68

100

2.70

2.75

2.86

2 97

130

2.94

2.99

8.10

8.22

140

8.16

8.21

8.38 .

8.46

160

8.36

8.44

8.58.

8.72

180

3.56

3.63

8.75

8.90

200

3.74

8.81

.8.95

4.12

230

8.91

4.00

4.18

4.30

240

4.07

4.15

4.80

4.48

360

4.2S

4.81

4.47

4.66

380

4.38

4.46

4.63

800

4.58

4.62

•

4.80

i

Key. Ft.

.61

.79

.92
1.03
1.11
1.30
1.48
1.64
1.78
1.90
2.08
2.25
2.44
2.78
8.09
8.35
8.60
8.87
4.06
4.29
4.48

*

Key. Ft.

.B4

.83

.97
1.09
1.18
1.40
1.59
1.76
1.91
2.04
2.18
2.41
2.62
2.98
3.32
8 61
3.87
4.17
4.38

Key. Ft.

.68

.88
1.03
1.16
1.26
1.50
1.70
1.88
2.04
2.19
2.83
2.58
2.80
3.18
3.55
8.88
4.16

Art. 4. To proportion tlie abuts for an areh of stone or
brick, wbetber clrenlar or elliptic. (Original.)

The writer ventures to offer the following rule, in the belief that it will be found
to combine the requirements of theory with those of economy and ease of applica>
tion, to perhaps as great an extent as is attainable in an endeavor to reduce so com-
plicated a subject, to a simple and reliable worklni: rule for prac-
tical brldjpe-bnllders. This is all that he claims for it. Notwithstanding its
simplicity, it is the result of much labor on his part. It applies equally to the smiillest
culvert, and to the largest bridge ; whatever may be the proportions of span and rise ;
and to any height of abut whatever. It applies also to all the usual methods of fllliug
above the arch ; whether with solid masonry to the level v/, Fig 2, of the top of the
arch ; or entirely with earth ; or partly with each, as represented in the fig : or with
parallel spandrel-walls extending to the back of the abut, as in Fig 2^^. Although
the stability of an abut cannot remain precisely the same under all these conditions,
yet the diff of thickness which would follow from a strict investigation of each par-
ticular case, is not sufficient to warrant us in embarrassing a rule intended for popu-
lar use, by a multitude of exceptions and modifications which would defeat the very
object for which it was designed. We shall not touch upon the theory of arches,
except in the way of incidental allusion to it. Theories for arches, and their abuts,
omit all consideration of passing loads ; and consequently are entirely inapplicable
in practice when, as is frequently the case, (especially in railroad bridges of moderate
spans,) the load bears a lai^e ratio to the wt of the arch itself Hence the theoretical
line of thrust has no place in such cases. Our rule is intended for common practice :
and we conceive that no error of practical importance will attend its application to
any case whatever ; whether the arch be circular or elliptic.

It fflves a thickness of abut, which, without any baeklnir
of earth behind It, Is safe In Itself, and In all cases, ayalnst
the pres,w^hen the bridge Is unloaded. Moreover, in very large arches,
in which the greatest load likely to come upon them in practice is small in comparison
with the wt of the arch itself, and the filling above it, our abuts would also be safe
from the loaded bridge, without any dependence upon the earth behind them ; but
as the arches become less, and consequently the wt of the load becomes greater in
proportion to that of the arch, and of the filling above it, we must depend more and
more upon the resistance of the earth behind the abuts, in order to avoid the neces-
sity of giving the latter an extravagant thickness. It toiU therefore be understood
throughout thai, except when parallel tpandrel voaUs are weed, over rule's svppone that
after the bridge i$ finuhed^ earth toill bi deposited behind the abuts, uud to the neight
wff the roadwajff a» usudU

618 nOITE BSIDGSi.

la niU bilil«« »b1 Uct* oalncti (K tnt elH nllnadi, MlfM* •> Ik* JntK
tl hMiy tn<B> •• kl(b qMdi, tk* ooHpantln (knpMa vllk vkkk u amrnm
■tavDithowi UiH b* flna U iBpsrtut itngtana, kw lad, Jo Mun ohm, M tfe*
«• a iDdMhsu rrtin oM-lMnti te MM-telf tttokw ikM by tk^Mlmiw nli,

Sf of rODKb rubble adds iHBto insure full thlckDBBB In «very part,
ITricfcion ofaiut at ipHnffl

.. , . .._ . ^ .JDB Hceitftlned. Nm., , _

parallel to a A, draw ths iodaflnlta Wnb gnp or tbe abut. J>q tbe nme witb the

taw Snd bj- trW the point j, Fig 2, it which ths thlckneH > p ■■

W»n til*. '52."*!?"'*'^

STOKE BRIDOBB.

61^

thirds of the corresponding vert height ojr, and draw sp. Then will the thiclmtgt
on or ey be that at the springing line of the given circular or elliptic arch of any
rise and span ; and the line gp will be the b<ick of the abut ; provided its height os
does not exceed l}^ times <p; or in other words, provided tpia not less than ^ of
o s. In practice, o a will rarely exceed this limit ; and only in arches of considerable
rise. Bat if It should, as for instance at oq, then make the base qu equal to sp, added
to one-fourth of the additional height tq; and draw the back uio, parallel to ^p ;
and extending to the same height, sc, as in Fig 2. If, however, this addition of ^
of sq should in any case give a base a u, less than one-hoi/ tlie total height oq, (which
will very rarely happen in practice,) then make q u equal to half («aid total height ;
drawing the back parallel to gp, and extending it to tbe'same height as before. The
additional thicknesses thus found below sp, have reference rather to the pres of the
earth behind the abut, than to the thrust of the arch. In a very high abot, the inner
line a p would give a thickness too slight to sustain this earth safely.

When the height o 6, Fig 'i, of the abut is less than the thickness dn at spring, a
small saving of masonry (not worth attending to, except in large flat arches) may be
effected by reducing the thickness of the abut throughout, thus : Make o k eanal to
on, and draw kl. Make ot equal to ^ of on, and draw l». Then, for any neight
obof abut less than on, draw oo, terminating in { «. lliis b v will be sufficient baise,
if the foundations are firm. The back of the abut will be drawn upward from v,
parallel to gp, and terminating at the same height as y or w.

Rkm. 1. All the abuts thus found will (with the provisions in Art 6) be safe,
without any dependence upon the wing-walls; no matter how high the embkt may
extend above the top of the arch. If the bridge is narrow, and the inner faces €f
the wing-walls are consequently brought so near together as to alfurd material as-
sistance to the abuts, the latter may be made thinner; but to what extent, must
depend upon the Judgment of the engineer.

We, boirever, eaatloB the yoang practitioner to be oaretal how he adopta dimeniions len tbaa tboee
given bj our rale. Tbcre are certain practical couitiderations. siich as earelesaDe^fi of workmannhtp :
newneM of the mortar; danger of nndne strains wben remnvtug the centers; iiabilitT of d^rnnge-
nent daring the process of depositing the earth behind the abuts, and over the ard ; ki, which must
not be overlooked; although it ia impoesible to reduce them to caloulntton.

Whenever Heaa be done, die eeatere rImuM remain In piece until the embkt Is flnlshed; and fnr
■ome time afterward, to allow the mortar to h«i well. Km 'nr more on this see Rem 4 p. 6SIS.

Bm . f . A goed deal of Hhertr la aometlmea taken in reducing the qnantity of masonry above the
itprlnglng line of arehes of oonsiderable rise, and of moderate xpanw. Wb<>D cnre i« taken to leave
the centers sunding until the earth fltltng is completed above the arch, and behind iu abuu, so tha»
it may not be deranged br accident during that operation ; and wben good cement is nsed Instead o\
common mortar, such experiments may be. tried with oomparatlTe safety; especially with culvert
arches, in which the depth of arob-stones is great in proportion to the span. They must, bowever. be
left to the Judgment of the engineer In charge ; as no speci6o rules can be laid down for them. They
ean hardly be regarded as legitimate praotlee, and we cannot recommand them. We have known
nearly semicircular arches, of SO to 40 ft span, to be thus bailt snceessftaUy, with scarcely a particle
of masonry above the springs to back them. Such arobea, however, are apt to fall, if at any future
period the earth filling is removed, wlthont taking the preeantion to first build a center or some other
■npport for them. Bven when the embkt can be finished before the centers are removed, we cannot
raoommend (and that only in small spans) to do U»* than to make n g. Pig 2, equal to )^ of the total
height l( of the arch ; and from g so found, to draw a straight line touching the back of the arch aa
high up aa possible.

Bxx. 3. We have said nothing about batterlngr the flieefli of the Abnts,

because in the crossing of streams, the batter either diminishes the water-way ; or
requires a greater span of arch. Such a batter, however, to the extent of A*om "^
to 1^ ins to a ft, is useful, like the offsets, for distributing the wt of the structure,
and its embkt, over a greater area of foundation ; ''specially when the last is not
naturally very firm ; or when tne embkt extends to a considerable height above the
arch. In our tables, Nos 3 and 5, of approximate quantities of masonry in semi-
circular bridges of flrom 2 to 50 ft span, the faces are supposed to be vert

Art. 5. Abntment-piera. When a bridge consists of several arches, sus-
tained by piers of only the usual thickness, if one arch should by accident of flood,
or otherwise, be destroyed, the acescent ones would overturn the piers ; and arch
after arch would then fall. To prevent this, it is usual in important bridges to make
some of the piers sufficiently thick to resist the pres of the adjacent arches, in case
of such an accident ; and thus preserve at least a portion of the bridge from ruin.
Such are called abutment-piers.

Rad ' Rise
Our formala of ~.~ -H To + ^ '^^ '**** ^* thickness at spring ; with the hack battering aa heitort,

at the rate of JU- ot the span to the rise ; fkce vert ; will of itself {without amg modifieation fifr great
MMgha) give a perfeetiy safe abut-pier. for any unloaded bridge ; and to any height whatever ; doe
regard being had, however, to the consideration alluded to in the next Art. Thus, for an abut-pier
as high as o q, Fig 2 ; or of any greater height ; it is only necessary first to find the thickness o n at
■pring aa before ; and then draw the battered back gn p : extending it down to the base at B ; with-
Mt adding H of the additional height t q. This addition is made in the case of abnta, that they

STONE BBIDOEB.

BTONE BRIDGES. 621

The eUiptto form U plainly vntevorabto for nniting the ■xeh'itones with the inollned muonry near
the springs, so aa to receive the thrast properly ; or about at right angles to its resaltant. In ordi-
nary oases this difflouUy may be OTeroome by making the Joints of only the oatside or showing arch-
stones to conform to the elliptic cnrve; as between e and a; while the Joints of the inner or hidden
ones, may have the directions shown between g and u, nearly at right angles to the line of thrust. It
will rarely happen, however, that the young engineer will have to oonstruot elliptic arches of suffl-
otent magnitude to require either this, or any equivalent expedient. For spans less than 50 ft, with

rises not less than about 4 of the span, notiiing of the kind is aotnally necessary, if the mortar is

good, and has time to harden.t

In order to incline the masonry of any abut with sufBcient accuracy, it would
be necessary first to trace the curved line of pres of the given arch,

BO as to arrange the bed joints about at right angles
to it at every point of itscdurse; but we- offer the following process as suflBcing for all
ordinary practical purposes ; while its simplicity places it within the reach of the com-
mon mason. In actoal bridges the 4irection of the actual thrust changes as the load
fs passing ; therefore, in practice no given degree of inclination of the abut masonry
can conform to it precisely during the entire passage. Consequently, any excess of
refinement iu this particular, becomes simply ridiculous ; especially in small spans.

Rale for Inclining tbe beds of the masonry in the abuts.

Add together the rad cm, Fig 4; and the span of the arch. Div the sum by 5. To
the quot add 3 ft. Make o t, on the rad, equal to the last sum. Then is t a central
point, towud which to draw the directions of the beds, as in the flg. Draw 1 1 hor.
and fh)m < as a center, describe the arc oy; o being the center of the depth of the
springers. From y lav off on the arc the dist y n, equal to one-sixth part of ty: draw
f n a. It will never be neceiaary to incline the masonry below this tna. Neither
ne$d the inclination extend entirely to the face mi of the abut ; but may stop at e,
i^ut half-way between i and n. From e upward, the inclination may extend for-
ward to the line e m.

Ceokewkt ahoald

be fireely used, not only im the arches themselyes, and in the masonry above them, as a
protection fh>m rain-soakage ; but in abuts, wing- walls, retaining- walls, and all other
Important masonry exposed to dampness. The entire backs of important brick arches
■hould be covered with a layer of good cement, about an inch thick. The want of it
can be seen throughout most of our public works. The common mortar will be
found to be decayed and falling down from the soffits of arches; and ttom the joints
of masonry generally, within f^om 3 to 6 ft of the surface of the ground. The mois-
tore rises by capillary attraction, to that dist above the surf of the nat soil ; or
descends to it (torn the artificial surf of embankments, Ac; therefore, cement-mortar
should be employed in those portions at least. The mortar in the faces of battered
walls, even when the batter is but 1 to 1^ inches per foot, is far more iBJured by rain
and exposure, than in vert ones ; and should therefore be of the best quality. See

MORTAS, Ac.

We have, however, seen a quite firee percolation of surface water through brick
arches of nearly 3 ft in depth, even when cement was freely used. In aqueduct
Ividges, we believe that cement has not been found to prevent leaks, whether the
arches were of brick, or even of cut-stone. May not this be the effect of cracks
produced by settlement of the arch ; or by contraction and expansion under atmos-
pheric influence ? Cement at any rate prevents the Joints from crumbling.

r The fset of both elliptic and Bemietreutar arches are always made bor ; but it Is plain from Fig
4^. that this praotiee is at variance with correct principles of stability in the ease of the ellipse. It
fs the same in tbe semieirele. In ordinary bridges of the latter form, the vert pres, or weight resting
•a each skewbaok, Is (roughly, speaking) nsuallT about from S^ to 4 times the her pres on the same :
and the total pres is about 4 umen as great as tbe pres on the ke.vstone. Therefore, iheoreticaUy, tbe
■kewbaok should usuallv be about 4 times as deep as the keystone; and its bed, instead of being hor,
' ~ be inelined at the rate of about I vert to 4 hor.

, To and the length Iii, Fig T)

tram bc« to tec« or B culvert.

Art. 7. The rollowinc tebles. S, 4, Hid S, of qnMntltlea, v!ll

biidga, mooaured from «nf[ Co vsd
(fKe to face) of Ihfi arch propar;
and iDcJDdlDR oblf lh« ferch and Ita
abnle. m ihowD in Fig 1 ; or In t)w
, half MKBon "pmgn In FlgS; t»-
clDdlne footlDgi to Iha abati, but

tb^ ■pudnl-wolft (I), F<» I ud
1^. At Ihafbot of eKhcolnnin lithe tpproilmata Danlem in cob jidi of tfa* Im

Tbe esnlentB flf the ftinr wlnc-wnllH. of uhloh ni» A, nt fl, la ona,

Tert. Wa tun algo added a lab)s (No. fi) for complste •emlclrcolar cnlnrta of
mrtom Isogtba, loclndlng tbelc ipaodral and vlng walla.

STONE BRiDaias.

623

RxM. 1. Although the thlckneaa of wing-walls increafles in all parts with their
height, they are not made to show thicker at nj than at tt, Fig 6 ; but (as seen in the
fig) are offsetted at their back ^ n, a little below their slanting upper surf ijy so as
to give a uniform width for the steps or flagstones, as the case may be, with which
they are covered. In the fig the covering is supposed to be of flagstones ; but steps
are preferable, being less liable to derangement. To prevent the flagstones ft-om
sliding down the inclined plane jt, the lower stone i should be deep and large, and
laid with a hor bed. The flags are sometimes cramped together with iron, and bolted
down to the wall. Steps require nothing of that kind, as seen at «, Fig 11.

Rix. 2. The tables bIiow the Inexpediency of too much con-
tractlngr the width of water-way, with a view to economy, by adopting
a small span of arch, when a culvert of greater span can be made, of the same total
height.

For the winga matt be the aame, whether the ipan be great or small, provided the total height is
the tame in both eases ; and since the wings constitute a large proportion of the entire qnantitj of
masonrr, tn culrerts of ordinary length, the span ^tself, within moderate limits, has comparatlTely
little eflect upon li. Thus, the total mastmry in a semicircular culvert of 8 ft span, 8 ft tot^ height,
and 60 ft long between tbe faoes of the arch, is, bj Table 5, 161^ cub jds ; while that of a 5 ft span,
of the same height and length, is 152.4. A semicircular bridge of 25 ft span, 24 ft total height, and
40 ft between the faoes of tbe arch, oontalns 1081 cab yds ; while one of S5 ft span, of tbe same height
and length, oontaini 1184 yds ; so that in this oase we may add nearly 50 per cent to the water* way,

by inereaaing the masoniy of the bridge bat -]^th pert.

Rm. 3. Partly for the same reason, and partly because the cnlverta for a
double-track road are not twice as lonir as those for aslnnple-
track one, the quantity of culvert masonry for the former will not arerage more
than about from ^ to ^ part more than that for the latter ; so that it frequently
becomes expedient to Jlnish the culverts at once to the ftill length required for a
double track, although the embkts may at first be made wide enough for only a
ringle one, with the intention of increasing them at a fature time for a double one.

Thns, the aTorage else of enlrerts for a single traok may be roughly taken at 6 ft span, 80 ft long
fh>m fkoe to taee, and 10 ft total height; and sneh a one oontains, }ay Table 5, 140 oub Tds. For a
double treek, it would require to be about IS feet longer; and we see by Table S that this will ad^
3.67 X 12 = 82 cub yds; making a total of 172 yds instead of 140; thus addinc rather lass than }£
part. When the euWerts are under very high embkts, and eonsequently muob longer, Ihe addition
Ibr a doable track becomes comparatively quite trifling.

Table 3, of approximate numbers of cub yds of masonry

SBr foot run, contained In the arches and abutments only, as
own in Vig 1 (omitting wings, and the spandrel-walls over the faces of the arches)
of semicircmar culverts and bridges, of from 2 to 60 ft span, and of different total
heights, h /, Viz 1, or o e, Fig 6. It will be seen that in many cases, a bridge of larger
apan oontains less masonry than one of smaller span, when their total heights arc Ac
There ig a Ub«ral allowance for footings or offsets at the bases of the sbats

TABI.IS 8. (OriginaL)

Total
Height.

Span
Sft.

Span
8fl.

Span
4 ft.

Span
6ft.

Span
6ft.

Span
8ft.

Span
10 ft.

Spac
12 ft.

Span
16ft.

Feet.
3

Cub. y.

.42

.60

.19

.99

1.38

1.6S

S.01

2.46

2.94

Cub. y.

Cub. y.

Cub. y.

Cub. y.

Cub. y.

Cub. y.

Cub. y.

Onb. y.

s

.88
1.04
1.38
1.68
1.96
tJ8
2.86
8.38
8.98

.67
.87
1.08
1.28
1.65
1.91
2.81
2.76
8.26
8.82
4.42
5.06

4

.92
1.15
1.87
1.64
1.95
8.29
2.72
8.19
8.72
4.29
4.90
6.67
6.80

.97
1.21
1.46
1.72
1.99
2.27
2.67
8.12
8.62
4.17
4.7T
5.42
6.12
6.87
7.69

6

A

1.58
1.86
2.18
2.42
S.77
8.16
8.57
4.10
4.67
6.80
5.97
6.70
7.48
8.82
9.20

1.69
1.97
2.26
2.56

2.87
8.19
8.52
4.02
4.57
5.17
6.82
6.52
7.27
8.07
8i92
9.82
10.8

1

8
9
10
11
IS
18

2.12

sje

2.66
2.W
8.2S
8.5S

8.8«
4.41
5.01
5.5«

7.01
7.71
B.6i
9.46
10.S
11.3
12.8

*'s!62**
8.84
8.67
4.01
4.S6
4.72
6.09
5.69
6.84
7.04
7.69
8.49
9.84

10.2

11.1

12.1

18.2

14.S

14

16

16

17

18

:::::::: ::::::::

19

to

SI

SS

SB

M

S6

St

Contents of
1 2.9

the two 1

1 8.7 1

pandrel'
4.4 J

trails, on
1 5.2

it tbe two ends of
1 6.8 i 7.9 1

the arch
9.8 1

, in cn^ f
12,

rds.

16.

STONE BRIDGES.

J3S..

T,t.

K

H.Utl. kS.

.'aUclK

!«>.

i

1
1

0.b. ,,

F«u

1

1

1

i

1
1

s

i

Si

s

a

i i

Art. 8. Tho

, ori9D°.withttH

U^ed It (UDh pttrU giifflcleiillT bir Uwl parpoM. Sw BMUuk ^ Thla lupp«iu oatj

ioicim top t^tckoeH of & ft. Ths muoonr li
morur nbbls. Tho taeigbt eItsu In lh> lint

M the bue uf U|0 vlngt; LU th«4 Jkre rrequeullf omltt«d ip wlngl DU food foDlldA*

STOKE BBIDOE8.

625

■

tioM. In taking ont quantities fh>m the table, bmr in mind that the height of the
wings is usually a little greater than that of the cutvert itself.

The plan shown at C is the common one, but that at D is greatly preferable for
culverts; for the shoulders at oo in Fig. C, apart from their greater liability to catch
branches of trees, etc., floating d«)wn stream, offer of themselves a much greater re-
sisttmce to the flow of the water into the culvert than do the mere corners at o o,
Fig. D.

Table 4, of approximate contents, in enb yds, of tlie four
wing-walls of a culvert, or bridjure. (Original.)

The heights are taken where greatest ; as aijw^ Fig 6

Height

Length

Gab. yds.

Height

Length

Cob. yds.

of

of

in

of

of

in

wing.

one wing.

4 wings.

wing.

one wing.
Feet.

4 wings.

Feet.

Feet.

Feet.

6

1.73

4.04

SO

48.3

818

7

S.46

8.85

82

46.8

997

S

5.20

14.6

84

50.3

1192

6.93

21.5

86

63.7

1414

10

8.66

80.2

88

67.2

1661

11

10.4

40.9

40

60.7

1928

12

12.1

63.7

42

64.2

2220

14

15.6

85.2

44

67.6

2562

16

19.1

128

46

71.1

2912

18

22.5.

183

48

74.6

3806

ao

26.0

247

60

78.0

8741

2t

29.6

329

56

86.T

4942

u

32.9

426

60

95.3

6404

»

36.4

541

66

104

8131

28

39.8

672

70

113

10156

To redaoe oub yds to perches of 26 cab ft, malt br 1.080.
To reduce perches to oub yds, mnlt by .926, or div by 1.08.

The contents for heights intermediate of those in the table may be found approximately by simple
proportion.

Rem. 1. It is not recomnxended to actually prolong all wings until their dimen-
sions become as small as shown at E, in Fig 8. In large ones it will generally be
more ecoinomical to increase their end height m m, a few feet. The contents, how-
ever, may be readily found by the table in that case also. Thus suppose the height
of the wings at one end to be 30 ft, and at the other end 8 ft ; we have only to sub-
tract the tabular content for 8 ft high, from that for 30 ft high. Thus, 818 — 14.6 =:
808.4 cub yds required content.

Rem. 2. It might be supposed that inasmuch as the wings of arches often have to
sustain the pressure from embankments reaching far above their tops, they should,
like ordinary retaining-walls, be made much thicker in that case. But the ihct that
they derive great additional stability from being united at their high ends to the
body of the bridge or culvert, renders such increase imnecessary when proportioned
by our rule ; no matter how far the earth may extend above them ; as shown by
abundant experience.

Relying npon this aid. we may Indeed, when the earth does not extend above the top, reduce the
base at o to one- third of the ht, as shown at o (; and by dotted line t *. Experience shows that we
laaj also do the same even when the earth reaches to a great height above the top : provided that
the wings, instead of being splayed or flared out, as at o n, o n, merely fhrm straight proloDgations
of the abutments of the arch, as 'shown by the dotted lines Atogto. In this case the pressure of the
earth against the wings is less than when they are splayed. We have known the tbiokness at o
to be reduced in such cases to less thau one-third the height, when the wings were 15 ft high, and
the height of the embankment aiove their tope 16 feet in one case, and 36 ft in another. In another
instan<S7 similar wings 25^ ft high, and with 29 ft of embankment above their top, had their bases
at o rather less than X of tho height. In all these cases, the uniform thickness at top was 2.5 feet;
backs vertical. We mention them because this particular subject does not seem to be reducible to
any practical rule The last wall appears t« us to be too thin ; especially if the earth is not deposited
in layers : and after allowing the mortar full time to set. The labor, however, required in compact-
ing the earth carefully in lavers, may cost pore than is therebv saved in the masonry. The young
practitioner must bear this in mind when be wishes to economize masonry by each means ; and also
that the thin wall may balge, or fail entirely, if the earth backing is deposited while t^e mortar li
lanperfeetly set.

40

626

STONE BRIDGES.

Table H. Approxlmatfe comtente in cubic yard*, of com*
plete semi circular cnl^erto and bridipee of from 8 to 50 feet
span I including the 2 spandrel walls ; and the 4 wings ; all proportioned by the
foregoing directions ; and taken from the two preceding tables. The height in the
second column, Is from the top of the keystonn to the bottom of the foundation. The
wings are calculated as l>eing 2 ft higher than this, including the thickness of the
coping. The wings are frequently carried only to the height of the top of the arch;
thus saving a good deal of masonry. Table 4, of wings alone, will serve to make the
proper deduction in this case.

The several lengths are fh)m end to end, orfh>m face to foce, of the arch proper.
The contents for intermediate lengths may be found exactly ; and those for inter
mediate heights, quite approximately, by simple proportion. In this table, as in
No. 3, it will be observed that wfien tfu heig/Us are the same in both castSy a largei
span frequently contains less masonry than a smaller one. A semicircular culvert
or bridge contains less masonry than a flatter one, when the total height is the same
in both cases ; therefore, the first is the most economical as regards cost ; but it does
not afford as much area of water-way ; or width of headway.

(Original.)

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Cab.T.

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7

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337

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396

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167

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380

484

487

541

594

12

146

164

200

236

308

881

453

526

698

670

743

815

14

206

219

277

325

420

516

611

706

803

897

993

1088

16

7

281

811

878

434

556
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679

801
215

928

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124

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294

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879

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173

228

284

339

895

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605

561

61ft

12

147

'«

300

236

306

379

450

522

598

664

736

807

14

206

JTO

3T6

323

416

510

603

696

790

883

977

1070

16

281

810

370

430

549

669

788

908

1027

1146

1366

1965

18
8

867

406

480
108

554

704

854

1003

1158
811

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74

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176 1 221

366

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4M

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150

179

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861

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689

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301

236

806

877

447

618

588

658

729

799

14

397

229

375

321

412

504

506

686

778

869

961

1063

16

am

809

368

426

542 '• 659

•776

891

1006

1124

1241

1867

18
10

a«6

no"

402

475

548

603 I 839
242 1 301

964

1139

1275

1430

1566

1711

125

154

183

359

418

476

535

594

663

12

161

168

204

239

310

381

452

523

694

665

786

807

12

14

906

228

272

317

405

498

581

669

758

846

964

1033

16

279

806

862

418

529

640

751

862

974

1066

1196

1801

18

864

399

469

540

680

820

960

1100

1341

1381

1631

1661

90

_470

613

598

684

855 : 1026

1197

1368

1640

1711

1883

1068

STONE BRIDGES.

62?

fable 5— (Continued.) (OrigiQal.)

•

s

•

1

Ft

is.

in

i>i

u

ii

in

a

Is!

5-

5£'

u

s-

js

|a

js

Cab.T.

J8

88

l§

JS

S9

SI

Ft

Cub.T.

CnUY.

e»b.r.

C»b.T.

Cab.T.

Oab.T.

Cub.Y.

Cub.T.

Cub.Y.

Cub.Y.

Cub.T.

12

162

182

222

262

342

422

502

688

663

748

828

90S

14

215

239

286

333

427

622

616

711

805

889

994

1U68

15

16

285

813

370

427

541

664

768

882

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1110

1223

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18

869

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474

545

686

828

967

1108

1249

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1530

1671

20

4T3

515

600

685

855

1024

1194

1364

1534

1704

1673

2043

22
U

595

237

646
264

748

850
871

1054

1258

1462
693

1666

1870

2074
1015

2278

2482

817

478

586

801

906

1123

1230

16

ao4

83ft

397

458

582

706

829

953

1076

1200

1324

1447

20

18

881

416

486

556

697

888

97»

1119

1259

1400

1541

1681

20

479

520

601

682

844

1007

1169

1332

1494

16S6

1819

1981

22

596

646

741

837

1028

1220

1411

1608

1794

1985

2177

2368

24

739

795

908

1021

1247

1478

1609

1925

2151

2377

2603

2820

16

327

360

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496

631

766

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1036

1172

1307

1442

1577

18

403

441

517

594

746

898

1050

1202

1355

1507

1669

1811

25

20

500

543

629

716

887

1059

1231

1403

1575

1747

1919

2091

22

614

663

760

857

1061

1246

1440

1635

1829

2023

2218

2412

24

751

807

919

1031

1255

1479

1703

1927

2151

2375

2599

2823

26

909

974

1104

1234

1494

1754

2014

2274

2534

2794

3054

8314

28
22

1086

1160

1310

1460

1760
1207

2060

2360

2660

2960

3260

3560

3860

686

743

859

975

1489

1671

1903

2136

2367

2699

2831

24

817

8f«>

1007

1134

1388

1642

1896

2150

2404

2658

2912

3166

35

26

969

1038

1181

1309

1686

1861

2187

2418

2689

2065

8241

8517

28

1130

1205

1356

1507

1800

2111

2413

2716

3017

3319

3621

3923

SO

1327

1408

1571

1734

2060

2386

2712

8038

3364

3690

4016

4342

82

1549

1639

1820

2001

9S6S

2725

8087

8449

3811

4173

4535

4807

35

30

1946
1494

9064

2271
1796

9488

9929

3366

8790

4224

4668

6002

6526

6960

1594

1996

2398

.2600

8202

3604

4006

4408

4810

5212

32

1711

1819

2035

2251

2683

3115

3547

3979

4411

4843

5275

5707

34

1956

»71

2302

2533

2995

3457

3919

4361

4843

5305

5767

6220

50

86

2228

2350

2597

2844

3338

3882

4326

4820

6314

5808

6302

6796

88

2519

2650

2913

3176

3702

4228

4754

6280

6806

6332

6868

7384

40

2835

2975

32S5

3536

4005

4656

5215

6775

6335

6895

7455

8015

42

8197

8347

3647

3947

4547

6147

5747

6347

6947

7547

8147

8747

46

8818

3991

4337

4683

6375

6067

6759

7451

8143

8835

9527

10219

60

5063

5281

5717

6153

7026

7897

8760

9641

10513

11386

12257

13129

Art. 9. Especial pains should be taken to secnre an nnyleldlnff foftn*
4 atlon ft»r ealTerte and dralnn nnder taiyli embk.t» } otherwise
the superincumbent weight, especially under the middle of the embkt, may squeea^
them into the soil below, if soft or marshy; and thus diminish the area of water-
way, or at least cause an ugly settlement at the midlength of the culvert. Also, in
soft ground, the embkt may press the side walls closer together, narrowing the
channel. This may be prevented by an inverted arch, or a bed of masonry, between
the walls. A stratum from 3 to 6 ft thick, of gravel, sand, or stone broken to turn-
pike size, will generally give a sufficient foundation for culverts in treacherous
marshy ground ; or quicksand, with but a moderate height of embkt. It should ex-
tend a few feet beyond the masonry in every direction, and should be rammed; the
sand or gravel being thoroughly wet, if possible, to assist the consolidation. Piling
will sometimes be necessary. If the masonry is built upon timber platforms, or a
smooth surface of rock, care muq^t be taken to prevent it from sliding, Arom the pres
of the earth behind it. This same pres may even overthrow the piles, if they are
not properly secured against it.

Art 10. Drains.

Drains of the dimen-
sions in Fig 11, con-
tain 1 perch, of 26
cnb ft ; or .026 of a
cub yd, per ft run.

Tbey are freqtieiiilv
built of dry aoi^bblea
nibble, Bod paved with
■p»wl8. When there is
mnob wash through
them, with a oonaider-
able slope, it ia better to
sontinae the fonndattea

628

STONE BRIDGES.

■olid olesr aarou. This ia often done without tbow oauaea, inumoeta as the additional maaonrr ia «
mere trifle : and the excavation of a single broad foundation-pit is less troubleaome than tliat or two
narrow ones. A deep flag-atoue/at the entrance, and others at short diats of the length, may be in*
troduced in both d^^ains and culverts, to protect from undermining.

Theae drains extend under the entire width of the embkt, from toe to toe ; and may terminate ia
Btepa, as in the side view at S. Thej are of course better when built with mortar, with an admlxtue
of oement to prevent the water when fall from leaking into and softening the embankment.

Sometimes two or three sueh drains mav be placed parallel to each nther. instead of a oalvert.
When two are so placed, they contain only 1^ timet the masonry of one ; still their nse will generally
involve no saving of maaonry over a culvert. , A man can crawl through Fig 11 to olean Ifc.

Art. 11. Tlie drainage of tlie roadways of stone bridges of aeversl
arches, is generally effected by means of open gutters, which descend slightly from
the crowns of the arches, each way, until they reach to near the ends of tlie re*
spective spans.

There they diaoharse into veitloal iron ptpee bnilt into the maaonry. The upper ends of the
pipes should be covered by gratings. When inconvenience would reanlt fkt>m the water falling upon

£ arsons passing under the arches, these pipes may be carried down the entire height of the piers;
ut when such is not the case, they may extend only to the soffit, or under face of the arch ; allowing
the water to fall freely through the air trom that height.

Table 6, of approximate eontentn. In eub yds, of a solid

Eler of masonry, tf ft by 22 ft on top; and battering 1 inch to a ft on each of
I 4 faces. The contents of masonry of such forma mnat be ealculated by the prlamoldal formnla:
and not by taking the length and breadth of the pier at half ita height aa an average length and
breadth, aa is sometimes done. This inoorreot method would give only 64M onb yda as the eontent
of the pier 200 ft high ; instead 7178 yds, its true content. High piers may for economy be built bo^
low, with or without interior cross- walls for strengthening them, as the case may require; and the
batter is generally rednoed to M hioh or lesa-to a foot. HoUow pien require good well- bedded

■°'*'"y- (Original.)

Ht. ^«*»»

XAIM

Ft.

at
base.

6

23.

7

.17

8

.33

9

23.5

10

.67

11

.83

12

24.

13

.17

U

.33

16

24.5

16

.67

17

.83

18
19
20
21
22
28
24
25
26
27
28
29
80
31
32
33
84
35
36
38
40
42
44
46
48
50

25.
.17
.33

25.5
.67
.83

26.
.17
.33

26.5
.87
.83

27.
.17
.33

27.5
.67
.83

28.
.33

• .67

29.
.33
.67

30.
.33

Bdth

at
base.

Onbic

yards.

7.

32.5

.17

38.6

.33

44.9

7.6

51.3

.67

58.

.83

64.8

8.

71.7

.17

79.

.33

86.4

8.5

94.

.67

102

.83

110

9.

118

.17
.33
9.5
.87
.83

10.
.17
.33

10.5
.67
.83

11.
.17
.33

11.5
.67
.83

12.
.33
.67

13.
.33
.07

14.
.33

127

136

144

153

163

172

182

192

202

212

223

234

245

256

•268

280

292

304

329

356

383

411

441

47-2

504

Ht.
Ft.

52
54
.56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
102

lot

106
108
110
112
114
116
118
120
122
124
126

Lgth

at
base.

30.67

31.
.33
.67

32.
.33
.67

83.
.33
.67

34.
.33
.67

36.
.33
.67

36.
.33
.67

37.
.33
.67

38.
.33
.67

39.
.33
.67

40.
.33
.67

41.
.33
.67

42.
.33
.67

43.

14.67

16.
.33
.67

16.
.33
.67

17.
.33
.67

18.
.33
.67

19.
.33
.67

20.
.33
.67

21.
.33
.67

22.
.33
.67

23.
.33
.67

24.
.33
.67

25.
.33
.67

26.
.S3
.67

27.

Cubic

Ht.

Lgth

at
base.

Bdth

at
base.

yards

Ft.

637

128

43.38

27.3:r

670

130

.67

.67

605

132

44.

28.

641

134

.33

.33

679

136

.67

.67

717

138

46.

29.

757

140

.33

.33

798

142

.67

.67

840

144

46.

80.

884

146

.33

.33

928

148

.67

.67

973

150

47.

31.

1021

152

.38

.38

1070

154

.67

.67

1120

156

48.

82.

1171

168

.83

.33

1224

160

.67

.67

1278

162

49.

33.

1384

164

.83

.33

1392

166

.67

.67

U.nl

168

60.

34.

1510

170

.83.

.33

1569

172

.67

.67

1631

174

61.

35.

1695

176

.33

.33

1761

178

.67

.67

1829

180

62.

86.

1899

182

.38

.33

1968

184

.67

.67

2041

186

63.

87.

2116

188

.88

.33

2191

190

.67

.67

2269

192

64.

38.

2346

194

jaa

.33

2424

196

.67

.67

2.')04

198

66.

39.

2.587

200

.33

.33

2672

202

.67

.67

Cubio
yards.

2769
2848
2940
9032
3126
3222
8320
8420
3621
3623
8728
88S6
39M
4060
4168
4284
4402
4620
4640
4763
4887
6014
6143
6276
5400
6546
5680
6820
5962
6106
6262
6t0(.
6552
6704
6850
7016
7178
7380

BRICK ARCHES. 629

Art. 12. Brick Arches. Since even good brick fit for large arches has
fikr lees crushing strength than good granite or limestone, and is inferior even to
good sandstone, while Its weight does not difibr yery materially ftom stone, it is
plaiq that it cannot be used in arches of as great span as stone can. Some of
those already built, and which have stood for many years, have a theoretical co-
efficient of safety of but about 8 ; whereas the authorities direct us not to trust even
stone with more than one-twentieth of its crushing load. This last, howaver, ap-
pears to the writer tp be one of those hasty assumptions which, when once aa-
mitted into professionid books, are difficult to be got rid of. It is his opinion that
with good cement, and proper care in striking the centers, one-tenth of the ulti-
mate strength is sufficiently secure against even the abnormal strains caused by
the settling at crown, and rising at the haunches when the centers are struck, it
is useless to attempt to fix limits of safety for bad materials poorly put together.
Rem. 1. The common practice of building brick arches in a series of con*
centric rlngs^ as at a c e e, Fig 12, with no other bond between them than

that afforded by the mortar, is censured by
authorities, on the ground that the line of
pressure in passing firom the extrados to
the intradoB tends to separate the rings,
and thus weaken the arch by, as it were,
splitting it longitudinally. The reason
for using these rings, instead of making
the radial joints continuous throughout
the depth m n of the arch, as at b, is to
avoid tne thick mortar-Joints at the back of
the arch, and shown m the Fig. If the
center of an arch built as at & be struck
too soon, the soft mortar in these thick
joints will be so mndi compressed as to cause greaX settlement at the crown,
uirowing the arch out of snape, and creating such inequality of pressure as
might even lead to its fall, especially if flat. As a compromise between rings
ana continuous joints, thej are sometimes employed together, so as to get rid of
some of the long radial joints ; and at the same time to break at InterraUi
the continuity of the rings. Thus in Fig 12, which is supposed to be brick-and-
>ha]f deep, beginning at the abutment a, we may lay half-brick rings as far as
say to e 0 s; then cattlnf^ uwHjr the brick o to the line « «, we may lay firom
e e to m n a block of bricks with continuous radial joints, the same as at 6; and
then start again with three rings; and so on alternately. A still better, but
more expensive, mode would be to fill « e, m i» with a regular cut-stone voussoir.
The proper intervals for changing from rings to blocks will depend upon the
number of the rings and the depth e a of the arch ; reference being also had to
reducing the amount of brick cutting as much as possible.

These points can be best decided on from a drawing of a portion of the arch
on a scale of 8 or 4 ins to a foot. Generally the rings are made only half-brick, or
about 4 to 4i( ins thick, as at a 0 ; and in Brunei's Imddenhead viaduct of two ellip-
tic brick arches of 128 ft span, and 24.25 ft rise ; the boldest brick arches yet at-
tempted; but which have Deen estimated to have a co-efficient of safety of but
three against crushing at the croWn.

So many others of firom 70 to 100 ft span have been successfully built entirely^ in
rings of either half or whole biick thick, as to justify us in attaching but little weijjht
to uie above theoretical ol]{)ection, provided first class cement be used, and time
allowed it to become nearly or qulto as hard as the bricks themselves, 'before
striking the centers. Under such circumstances we should not object to a series
of rings even -1,5 bricks thick, laid alternately header and stretcher, as at b.

If tlie bricks were wouMM^ir-sliaped, that is, a little thicker at one
end than the other, then rings a. whole-brick thick could be used without any in-
crease in thickness of mortar-joint at the back of each ring. Still with more
than one ring, the radial joints would not be continuous, as at &,but broken as at
ac. Such bricks however would be more expensive to make ; and moreover, in
order fully to answer the intended purpose, they would have to be made of many
patterns, so as to conform to the many radii used in arches; and even to the
radii of the different rings, when the depth of the arch required several of them.

630

BBICK ABCHE8.

Rem. 2. Wet the brieliLa before lajing.

Rem. 3. Whun tlie ends or fr^cee of a brick arch are to be finished with ent*
•tone Tonssoirs. these had better not lie inserted until some time after the
completion of the brickwork, the hardening of the mortar, and a partial easing
of the centers ; lest they be cracked or spawled by the nneqoal settlements of them*
ielTes and the bricks.

Rem. Brick arches, from their great number of Joints are apt to settle

much more than cut stone ou*'» when the centers are removed, and thereby to
derange the shape of the arch, and at times, without due care, even to endanger
its safety, especially if it be large and flat. When the span exceeds about 30 to 35
ft, and particularly if flat, use onlv brick of superior quality in good cement
mortar. With even best materials and work we advise the young engineer not
to attempt brick arches for railroad bridges of greater spans than about the fol-
lowing. Considerably larger ones than some of them havp lie<:n built, and have
■tood ; bat their coefiB of safety are not in all cases satUrfkctory. In thli teble the
IJM is in parts of the span.

R.

.6

.4

.36

8. 1

100
97
93

R.

'A

88
82
75

R.

.225

.183

68
60
55

; R.

8.

50
46
40

R.

.134

85
80

On the Filbert Street Extension of the Penn» R R, In Phlla,

are four brick arches of 50 ft 1 inch span, and with the V017 low rise of 7 ft. They
are 2 ft 6 ins thick, except on their showing faces, where they are hut 2 ft. The
joints are in common mortar, and about % inch thick. These four arohes, aboat
200 yards apart, with a large number of others of 26 ft span, form a viaduct, nio
piers between the short spans are 4 ft 3 ins thick. Those at the ends of the 60-fk
spans, 18 ft 6 ins. The springing lines of all the arches are about 6 to 8 ft above tha
gronnd. One of the 60-ft arches settled 8 ins upon prematurely striking ths
centers; but no further settlement has been observed, althoufch the viaduct has,
since built (1880) had a very heavy freight and passenger traffic, at from 10 to 91
miles per hour. Koadbed, about 100 ft wide, giving room for 9 or 10 tracks.

OENTEBfi FOR ARCHES.

631

OENTEBS FOB AEOHES.

Arte 1« A eenter Is & temporary wooden itnicture (built lying flat, on a ftill
•ise dravring, on a fixed platform, nnder cover or not) for ■upporting an arch
while it is being built* It consists <^ a number of tronee or flraines»>;/. Fig. 1,
placed firoro 1 to 6 ft apart ttom. cen to oen, and covered with a flooring 1, 2, of
rough boards or planks, usually laid close, and called the slieetiiBK or lair-
flrlnil^, immediately upon which the archstones are laid. In Fig 3, the lag-

gng 18 not laid close. There is no great economy in placing the frames very
r apart, on account of the greater required amount of lagging, the thickness
of which increases rapidly. For the thickness of lagging see Bern 9.

Figsl.

The centen rest by the ends of
their chords, c, upon wooden
■trlMlnic wedges w. Fig 1,

supported Dy stondardB com-
posed of posts p, whose tops are
connected by caiHpieees o;
and whose feet rew on wtrkng*

em «; the whole being braced
diagonally as shown.

Ii the ffronnd is very firm, and
the arch light, the standards may
Test on it, with the interposition
of a«Uiuitinir-MoelKS, n, be-
low the stringer, to accommodate

irregolarities of the surfkce of the ground, as in the Fig. These blocks should
be somewhat double-wedge-shaped, so that br driring them the standard may
be raised at any point in case it should settle a little into the ground. But for
iieaTy arches the standards must rest on a much firmer foundation, such as short
blocks of brickwork sunk a fbw feet into the ground, or some other device
adi^ited to the case. Frequently projecting offtots or footings, or at times re-
cesses, are provided in the masonry of the abutments and piers for this expre^
ptirpose ; and with a view to this it is well to design the center at the same time
as uie arch.

Up to spans of 50 or 60
ft a single row of posts (one under each end of
each frame) will suffice ; but for much larger ones
two or three rows, 2 or more feet apart may be-
come expedient, as in the lower Vlg 2.

The BtrllKliicr or lowerinir*wedir^s

before alluded to are for striking or lowering the
center after the completion of the arch. They
consist of pairs of wedge-ehnped blocks, w w, at A,
Figs 2, of hard wood, from 1 to 2 11 long, aliotit half
as wide, and a qnarter or more' as thick, (safflcicnt
to lower the center firom say 2 to 6 or more inches,
According to span and other circumstances,) rest-
ing on the cap o, of the standard, while the chord
c of the frame rests on than. When the end of a
frame is supported by two or more posts p, as at ^
Fig 2, instcvtd of upon one, the striking-wedges are
flometimes made as there shown ; and where B «
Is one long wedge at right angles to the abutment,
and acting as (our wedges which may all be low-
ered together by blows against the end B

Up to snanp of 6U or SO ft. all the frames majr rest

oo but two wedges Iik«« B

632 CENTERS FOB ARCHES.

each 80 long as to reach traiifiTemely across the eDtire archi Then all the
frames can be lowered at one operation, as described near end of Art 9.

If we had to consider only the friction of dry wood against dry wood, the taper
of these wedges might be as steep as 1 vert to 3 hor, without any danger of their
sliding upon each other of their own accord ; and they would then require Tery
moderate Mows to start them, or even to entirely separate them, when the center
had finally to be lowered. But it is of the utmost importance, especially in large
arches, that the centers should be lowered very slowly « otherwise
the momentum acquired by so heavv a body as the arch in descending saddenly
even but 2 or 8 ins, might pessibly affect its shape, 6r even its safety.

Therefore the wedges should not have a taper steeper than about 1 in 6 or 8 for
arches of less than about 50 ft span ; orthsm 1 in 8 or 10 for larger spans. VertlcaJ
lines at equal dists apart should be drawn on the long sides of the wedges as a
guide for lowering them all to the same extent at a time ; and this should not ex-
ceed in all about naif an inch a day in intervals of about an eighth of an inch, for
60 ft spans ; or about .1 to .26 of an inch per day in all, for spans over 100 ft.
Slowness is especially to be recommended in brick arcbes, not only because
their greater number of joints exposes them to greater derangement of shape,
but because even good brick has much less than the average crushing strength
of good granite, limestone, or sandstone, and therefore is uur more liable than
they to crack, or even to crush (as the vnriter has seen) when the strains are
thrown almost entirely upon their edges, as described in Art 3.

At Oloocester Brldve, England, of first class cut stone, span 160 ft, rise
86 ft, the centers were entir^v struck within the very short snftce of 8 hours : and
the crown of the arch descended 10 ins! At Orosvenor jBridpre, England,
of first class cut stone, span 200 ft rise 42 ft, such care was taken m easing the
centers that the crown of the arch settled but 2Ji ins. This ease however was
marked by two or three peculiarities, all of which contributed to this favorable
result. Namely, the center Instead of being a series of frames supported as usual
by their ends, and of course involving an appreciable, although small, degree of

sagging or settlement, consisted
essentially of vertical and in-
clined posts or struts, see Fig 8,
footing on four temporary men
of masonry, 7 or 8 feet thick, DuUt
in the riyer, parallel to the abut-
ments, and as long as they. These
piers supported six frames (or
rather six series) about 7 ft apart
cen to cen, of such stmts, footing
on cast iron shoes. Fig 8 shows
half of one series. Each fhone
or series consisted of four £ui-like
setsof posts, all in the same m^
tieal plane. The long horizontal pieces seen extending from side to side of the
arch were bolted to the struts to increase their stiffness ; and other pieces for the
same purpose united the six series transversely. Here each strut sustains its own
share of the weight of the archstones, and transfers it directly to the unyielding
foundation of the pier ; whereas in the usual trussed centers, tne entire load nets
upon the frames, and is finally transferred to the comparatively- unstable support
of the posts at their ends.

The tope j9 of the posts of a series varied about from 6 to 8 ft apart cen to cen;
and were connected by a continuous curved rib, rr, of two thicknesses of 4 inch
plank, bent to conform approximately to the curve of the arch. On this rib were
placed pairs of striking-wedges to like Fig 2, about 16 ins long, 10 to 12 ins wide, and
tapering 1.5 ins, so near together (varying about from 2.5 to 3.5 ft cen to cen) that
there was a pair under each joint of the archstones, a a. On these wedges, and ezf*
tending over all six of the frames, were the lagging pieces /, 4JS ins thick.

FtgsS

lowered together, without giving an opportunity to rectify anv slight derange-
ments of shape or inequality of bearing that may have occurred in the arch durmg
its construction. This center, designed by Mr. Trubshaw, admits of lowering
either the whole equally, or any one part a little more or less than the others.
He had much experience in large arches, and stated that during the striking he
found that he had an arch under better control, or could humor it better, by keep-
ing the haunches a little dow^ nnd the crown a little up; until near the end of
'^he operation.

CENTEB8 FOR ABGHEB, 633

•

Rem. 1. Instead of plem of masonnr for supportiog the feet of the
poets, wooden cribs or piles may often be used if the arCh is over water.

The principle of sapportins even trussed firames by strata
at points of the cnord as far from the abutments as circumstances. will admit of
(in addition to those at the very ends) should always be applied when possible,
in order to reduce their sagging to a minimum. Steps or offsets m tlie
masonry of the abutments and piers may be provided for receiving the feet
of such struts, when they are inclined.

Rem. 2. Screws may be used instead of wedges for lowering centers. At
the Pont d' Alma, Paris, ellipse of 141.4 ft span, and 28.2 ft rise, the frames were sup-
ported by wooden pistons or plungers, the feet of which rested on sand eon-
fined in plate-iron cylinders 1 ft in diam and height, and having near
the bottom of each' a plug which could be withdrawn and replaced at pleasure,
thus regulating the outflow of the sand and the descent of the center. This de-
vice succeeded perfectly, and is well worthy of adoption under arches exceeding
about 60 ft s^an. When much larger than this tne driving of the wedges on
striking requires heavy blows, and hecomes a somewhat awkward operation, re-
quiring at tmies a battering-ram, even when the wedges are lubricated. In rail-
road cuttings crossed by bridees, tlie eartb under tlie arcii has been
made to serve as a center, by dressing its sarfkce to the proper curve, and then
embedding in it curved timbers a few xeet apart, and extending from abut to abut,
for supporting the close plank lagging.

Rem. 8. All centers must yield or settle more or less under the wt
of the arch, especially when supported only near their ends ; and since the arch
Itself also settles somewhat not only when the centers are struck, but for some
time after, it is advisable to make them at first a little higher than the finished
arch is intended to be. This extra height, when the supports are at the ends,
may be from 2 to 4 ins per 100 ft of span for cut stone arcnes (according to time
of striking, character of masonry, workmanship, etc.), and about twice as much in
brick ones.

Rem. 4. Tbe proper time for striliins centers is a disputed
point among engineers, some contending that it should be done as soon as the
arch is finished and sum^ently backed up ; and others that the mortar should
first be given time to harden. It is the writer's opinion that inasmuch as in
cut-stone arches the mortar joints should be verv thin ; and since, in such, the
mortar is at best of very little service, it is of no importance when they are struck;
provided the masonry backing, and the embkt up to y n Fig 2, p 618, have been com-
pleted; but that in brick or rubble, the numerous Joints of both of which require
much mortar, (which fur hardness should cousist largely of cement,) 3 or 4 months,
or longer, if possible, should be allowed it to harden sufficiently to prevent undue
compression smd consequent settlement when the centers are struck. The ooxk>
tinuance of the centers need not interfere with traffic over the bridge.

Art. 2. The pressure of archstones against a center is very trifling until alter
the arch is built up so far on each side that the joints form angles of 26° or 309
with the horizontal. Theoretical discussions on this pressure make no allowance
for|aocidentaljaiTingsin laying the archstones, or by the accumulation of material
ready for use, laborers working on it, <&c. Without going into any detail, we merely
advise on the score of safety not to assume it at less than about the following pro-
portions or ratios to the weight of the entire arch, namely, in a semicircular arch
.47; rise .35 span, .61; rise .25 span, .79; rise .2 span, .86; rise .167 span, or less, 1,
or equal to the wt of the arch. This gives the pressure of a semicircular arch
upon its centers rather less than half its wt. Tlie wt of the centers
tnemselTes when supported only near the ends must be considered as part
of the load borne by them.

Art. 8. We have seen that as an arch a a a Is being gradually built upward on
both sides, after passing the points e, e. Fig 4, where its joints form angles a « e, of
about 9(P with the horizontal a a, the arch begins to press more and more
upon the centers ; thereby tending to flatten them at the haunches, as shown at A
in the dotted line ; and consequently to raise them at the crown, as shown at c.
But as the building goes on stillhigher, the added stones press much more heavily
upon the centers than those below had done, and thereby tend to a final derange-
ment of the centers just the reverse of that caused by tne lower ones ; namely to
depress them at the crown a, at' at o; and consequently to raise the haunches as
at n ; and this the more because the upper stones actually tend to lift or ease the
lower ones from the lagging. In some cases where this tendency has been in-
creased by.forcln§^ the keystones into place by too hard driving, the lagging
under the haanehea could be drawn out without any trouble before the centers
were eased at all. On striking the centers this tendency to sink at crown and

634

CENTEBS FOR ARCHES.

rise at haunches is rery apt to exhibit itself more or less dangerously in the arch-
stones themselves, as in Fig 5, causing those near the crown to press very hard
together at the oxtrados, and to separate from each other at the mtrados ;* while
near the haunches the reverse takes place. Hence the angles of the stones are
frequently split and spawled oflf near c and A by this unequal pressure. These

fiiflr.4.

derangements are of course much more likely to be serious in high arches than
in flat ones, especially if their spandrels are not sufficiently built up before
lowering the centers.

In the Grosvenor bridge, before alluded to, of 200 ft span, this dangerous excess
of pressure near c and h was prevented by covering tne skewback joint of the
springing course at each abutment with a wedge of lead 1.5 ins thick at the in-
trades ofthe arch, and running out to nothing at the extrados. Beside this a
strip 9 ins wide of sheet lead was laid along the intrados edi^e of every joint until
reaching that point at which it was judgea that the line of pressure would pass
from the intraaos to the extrados ; after which similar strips were laid along the
extrados edges of the joints, up to the crown. Hence when the centers were
struck, this excess of pressure merely compressed the lead, and was thus enabled
to distribute itself more evenly over tne entire depth of the joints. See Trans
Ins Civ Eng London, vol i. See nl«50 top of p 921.

At tlie bridi^e at Hen lily, France (of 5 elliptic arches of 120 ft span,
and 30 ft rise), the centers were so radically defective in design that the arches
sank 13.25 ins at crown during the time of building; and 10.5 ins more during
and immediately after the striking ; or say 2 ft in all. Their construction made
the striking very tedious and hazardous ; greatly endangering the lives of the
workmen and the existence of the arches. Some of the joints at the extrados
at the haunches opened an inch each ; and those at the intrados of the crown .25
of an inch. By the exercise of great care and humoring in lowering the center;},
these openings were much reduced.

Rem. 1. Chain ferlng^ the edg^es of the archsto.nes diminishes
the»danger of their spawling off from unequal pressure ; as does also the MSran*
Ingr out of the mortar of the Joints for an inch or two in depth be-
fore striking the centers.

Rem. 2. It is evident that in order to prevent, or at least to diminish the
alternate derangements of the center, those of its web members which at first
acted as stmts near the haunches, Fig. 4. to prevent them Arom sinking as at
h, must afterwards act as ties to prevent them from rising as at n; while those

which at first acted as ties near
the crown a, to prevent it fi*om
rising as at c, must afterwards
act as struts to prevent it flrom
sinking asat o. In other words,
the principle of ••anter-
braelnsf must be attended
to as weirin a tnme or truss
for a center, as in one for a
bridge.

Art. 4. From the foregoing it is plain that a simple nubraeed wooden

CENTEB8 FOB ABCBEB.

635

areb, or curyed rib is, on account of its great flexibility, about as unfit a form
as could be cbosen for a center, except for very small spans, where a great propor-
tional depth of rib can be readily secured. Still the writer has seen it used mr a
cut-stone semicircular arch of 85 ft span, with archstones 2 ft deep. Fig 6 shows
one rib r r, and the arch, a a, drawn to a scale. Each rib consisted of two thicknesses
of 2 inch plank in lengths of about 6.5 ft, treenailed together so as to break joint,
as at B. Each piece of plank was 12 ins deep at midale, and 8 ins at each end ;
the top edge being cut to suit the carye of the arch. The treenails were 1.25 ins
in diam ; and 12 of them showed to each length. These ribs were placed 17 ins
apart from cen to cen, and steadied together by a bridging piece of inch board, 18
ins long, at each joint of the planks, or about 3.25 ft apart. Headway for traffic
being necessary under the arch, there were no chords to unite the opposite feet
of the ribs. The ribs were coyered with close board lagging, which also assisted
in steadying them together transyersely. As the arch approached about two-
thirds or its height on each side, the ribs began to sink at tne haunches, as at A,
¥ig 4 ; and to rise at the crown, as at e. This was rectified by loading the crown
with stone to be used in completing the arch ; which was then finished without
ftirther trouble.

A still more strtlLtiiar example of the use of a simple unbraced wooden
rib, was in theold National Tumnike bridge over Wills Creek, at Oumlierland, Md.
. This bridge, of which one arch with
its center is shown in Fig 7 drawn
to a scale, consisted of two elliptic
cut stone arches 26.5 ft wide acroes
roadway, and of 60 ft span, and 16
ft rise. The archstones were 3 ft
deep at crown, and 4 ft deep at
skewbacks. Eacli frame of
tl&e center was a simple rib 6
ins thick, composed of three thick-
nesses of 2 incn oak plank in different lengths (about 7 to 15 ft) to suit the curye,
and at the same time to presenre a width of about 16 ins at tne middle of each
length, and 12 ins at each of its ends. The thicknesses were well treenaUed to-
gether, breaking joint and showing from 10 to 16 treenails to a length.

Here, as in Fig 6, there were no chords, owing to the yiolence of the floods in
the creek. These ribs were placed 18 ins from cen to cen, and steadied against
one another by a board brioging-piece 1 ft long, at eyery 5 ft. These were of
course assisted by the lagging.

When the archstones had approached to within about 12 ft of each other near
the middle of the span, the sinking at the crown, and the rising at the haunches
had become so alarming that pieces of 12 X 12 oak, 00, were hastily inserted at
Interyals, and well wedged against the archstones at their ends. The arch was
then finished in sections between these timbers, which were remoyed one by one
as this was done.

1. 8ncli instaneea of partial Ikllnre are yery instractiye.
It is indeed hj such, rather than by theoretical deductions, that the proper dimen-
sions are arrived at in a vast number of cases pertaining to en^neering, ma-
chinery, 6iG.* Thus we might with entire confidence of no serious mishap, apply
ribs of the foregoing dimensions to spans only half as great.

Rem. 2. Assuming the rib-planks to be 12 ins wide, it would, as a matter of
detail, be better to make them about 10 ins wide at the ends instead of the 8 ins
in Fig 6 making top curve 2 ins. To secure this, their lengths, depending on the
radios of the rib, must not exceed those' in tlie following; table s

Bad
of Arch.

Greatest Length.

Bad
of Arch.

. • ■ ■ 1_—

Greatest Length.

Feet
5
10
15
20
26

Feet and Ins.

2 " 5

3 « 4

4 " 2

5 " 0

6 " 9

Feet.
30
85
40
45
50

Feet and Tns.

6 ** 4

7 " 0

7 « 6

7 " 10

*8 " 2

• TlM yottnf engineer ■hovU make and preieire Ml notes in detail of all looh as may fall within
klfl aotlee

! to BdS

636 CENTERS FOS ARCHES.

If cut lU limes (le long is this Ubl«, ther will be •er
nld« at ends i or each wlfl od Up curve t Ins.
Art. 5. In oast

daring [he building of the Brub, aa in the two forgoing ones, . . ._.

' -'le eipedlenl mdelT Ulustraied by

■ — '- *- -■-<«• ibe eent«ra

iiul«d of below

la Gompleled in aeo-

iatteadof lAwerlnc Iheceu-

Ti."'n 7 Fl«Bla» (rsnBTsrse section through mirt

riqo. of (hecentw, uidoribeBn:b so. Hera

^ rc,rc,re, mre ^meaof the center my Gor6

ftapart; andof any depth and couitmcttDn

ever that may be Decesaary \o insure absolute safety ; uid M la the laf^likCi

ig built the aj^hfrDiu BbuUDentlaabatineDtliiaseilesotsectioiua,(i,a,De-

j? -' --a foot or more by the deep frames, TB may take the oenten

le nu-row Inlennediate sectloDS upon a lagglDe suspended
oy iron rooa irom tne already completed sectiona Good concrete might be used

FT ff ff" sBI
nl Irani li^nl In --"—

^1 II g ^1 11 5 II n "»,"•"■'• «V.-iiii«.'

' separated say a foot or more by the deep frames, '

id then all In the nar '-' "-' "

* ' om the alread

Bllng the lagglDg on
F embedded Id the mi

Inlng embedded Id the maionry; and the upper part ai

both flangea remoied after Oie arch is Bulsbed.
Art. A. f;ei>t«rswlthhorcb

standing their strength) In large spans i

what ia shown on the le^ hand
oftheFlg, HerealruM/ehortar
and ahaSower than thai on tba
right hand, la aubBtltuted for the
latter. Atltsendsprovialonmust

iUelf. but the archalones below
it. As the pressure of these low*
er arcbstonea la comparatiTelv
unan,thlsmaynstiaUybeelftiit«)
by reetlng the end of the trttat
/ upon another and shallower (nune o a. This may In btge spans be aided by
ellber Inclined or Tertical alruta, either elnile or braced together ; orasthetnsUa

throughout ill entire length. Theetrlklng-wedgcefor these Tarious supports may
be placed at cither their tops or their feet, as may bo most convenient.
itaea of 10 feet clear apan, a

If tbe crntera have to b« moTMl ft^in place to plav*.

' " ir arche&, then, to preserie them from injury in handllng^elr le<

each Inune a cbor^ piece of at

CBNTEIia FOR

Inch baud ', and also a isrtlcal piece or ptecea of tbe sune siH (raiD (be ceoter
Df the chord to the lop at the tnmo.
ETen wbeD thejr BPe not to b« moved, (he chord pleccg are useful

iDg (he reet of the tibsto^ve iroubla by sprendlng autwudaod pKaalu^ against

For apitiu.

■pan, the^ollovlagdlineii

See Fig 11. PortliebOH^

plank froiH 9 t« 12 Ins widn

10 IsB at each end, well spiked
together breaking loiat as at B.
FCgt. FortHeehOrdctoo

nia as the bow at lis middle:

placed on Dut«ldes of bow, and

well Bplked to Its ends. A

vcptlesl e, In one piece as

wide H a how plank, and twice

as thick. Its (op Is placed under thcbofr, and Is conflned (o i( b; (wo pieces, o, o,

of how plsjih twice aa long as the bow plank la deep, and spiked to both c and tbe

spiked (o th— "- ■-" ."■'-— — — - — >■ -' .1' -..-.-• ^ ^ —

plank, ouHlt __.

(o bow and*. Thees with ndlrtde tho how .

Bern. 1. The abave dimensions are suitable lo a rise of one slKh. If the
tIbo Is one fourth, the tlilehness only of the plsnks may be reduced one (bird
part; and for a rise of one third or more, we may reduce to one half.

Bern. 2. If !n the larger of thae spans (ho stnils t should show an; Incli-

oftwo; Ihusdiylding thebowinloaparU,siatlB(l>ideof Fig.ll. Forapansof

Art. 8. For snaiu crenter tbon oboat SO n, tbe wdler belleTu
that as a general rulo (liable to modifications according to the Judgment of the

left side of Fig 9. The how to rest on Ihe ehord, and eacTi lo bo o^ a sInilB thick-
ness. The weo members (especially In large spans) tobealaocf ulngle thlcknens,

ta act as either ties or struts. la smaller spaost^e web members mar each' bo In
two tliickneases,aue bolted orireeuallcd to each side of tliehow&ndchord. Other
modes win soneit themselves : but we have not apace for such details.

Or xrebofffie Hove, or of tbe Pnitt e jsum, as on (he ilghi side of Fig 9 1
used. Bat In referenca to both of these It may be remarked that lite u

■ In «eiit«f« of large BpLos Is high); objcctiouahle, owing
' I between Iron and wood. Therefore If It

it rates of exnaoslon betwe

___, _,_,__ ,_-BBhoiJdbeofwood. Tlie lattice may beused-

Ereu when the rise of the mrtb exceeds ^ of the span, 11 is better not 10 let
that of tbe «i>iitcra exceed tbat limit ; bat adopt tbe expedient shown at
the left Sid* of Fig 9, with a rise of about one slilh of the span.
Bern. 1. To Hz on Itae number of web trlstngles In a Warren

tenrti of the apany ufvidTtheir aum'^r'a^nd call the 'quotient b. DlTide the

decimal, tise tbe whole number neareatto it, as adlslance in feet to be stepped off

points thua found on the chord, are the places for tbe feet afthe triangles.
Neil, trom balf wbf between each [wo of fiieae points, draw >erHcal lines (o the
bow. Tbe points thus found along the bow, arc the pUces of tbe tope of the
trioagles. This rule will be used in connection with the following Table of Areas
of Bows, as the two are depecident oD each other.

In Urge arches tbe timber of tbe bow ahoald >ot be wnated by
trlmmingltsupper edges to tbe curreof tbe arch, but should he left straight ; and
Bep*nte places so trimmed, like b in Fig. 10, should be spiked ou top of them.

638

CENTERS FOR ARCHES.

Tlie transTerse area of tbe bow, in square inches, may be taken fitom
the following table : and may in practice be assumed to be uniform throughout
its entire length ; wnich in fact it is quite approximately. See Rem 2.

TABI.E FOR BOWtSTRIllfO CENTERS.

Table of areas in square inches at the crown of each Bow, of property
trnssed Bowstring frames for centers of stone or brick arches. The frames to
be placed 5 feet apart from cen to cen. With these areas, the combined weiriits
of arch, center (of oak), and lagging, will in no case in the table strain the Bow
at crown of the greatest spans quite 1000 lbs per square inch ; diminishing grad-
ually to 600 or 700 lbs in the smallest spans, which are more liable to casualties.

Although centers of moderate span are usually made of whit« or yellow
pine, spruce, or hemlock, all of which are considerably lighter than oak, we have
tor safety assumed them to be of oak, in preparing our tabla
For spans of flrom 10 to 20 leei use the same sizes as for 20 feet.

Original.

Rise in parts of tbe Span

t

.5

.4

.35

.3 .25 .2

.15

.1

Span

in feet.

Areas of transverse section of Bow,

in square inches.

20

14

17

19

21

24

29

88

59

26

18

22

25

28

83

40

53

80

30

28

28

. 32

87

43

51

71

108

35

28

34

40

45

64

64

87

125

40

84

41

48

55

65

77

106

160

45

40

49

57

66

76

92

126

176

50

47

67

66

76

89

107

146

208

65

53

64

75

87

102

121

166

288

60

60

73

85

99

115

185

187

263

66

68

81

95

110

129

151

209

294

70

75

90

105

122

143

168

233

325

75

83

99

115

133

157

184

256

867

80

91

108

125

145

171

201

279

390

85

99

117

136

157

185

218

302

423

90

108

127

147

169

199

236

325

467

96

115

136

158

181

214

252

^8

490

100

128

146

169

194

229

270

372

624

110

133

166

191

219

260

307

420

602

120

156

187

213

246

291

345

470

660

130

172

208

237

274

323

384

620

140

190

230

263

803

357

424

672

150

209

252

289

333

393

466

160

229

276

315

365

430

509

170

250

299

343

399

469

180

272

323

373

435

511

190

294

847

403

472

200

318

372

435

609

>

Rem. 2. The square root of any of these areas gives in Inches the side of
a square bow of that area. The distances apart of the triangl& which form
the web of the frame, having first been found by Rem 1 (for said Rem and this
table are dependent on each other), the above areas for bows 5 ft apart from cen
to cen, suffice not only to resist the pressure along the bow, but also, as sauare
beams, to sustain with a safety in no case less, than about 5. the load of' arch-
stones resting upon them between the adjacent tops of two triangles : and with
very trifling deflections. It is therefore unnecessary to deepen the rioB for that
purpose: although it may be done (preserving the same area) in case consider-
ations or detail should render it desirable.

As before suggested, it will generally be best, in spans exceeding 80 or 40 ft, to
give the bow a rise not exceeding abuui one fifth or one sixth of the span ; and
to support the frames as at/. Fig 9.

Tbe sise of tbe cbord may be the same as that of the bow ; and like it
uniform from end to end ; care however being taken that it be not matorislly
weakened by footing the bow upon its ends ; or (when too long for single tim-
bers) by the splicing necessary to prevent its being stretched or pulled •furt by

CENTSRS FOR ARCHES.

63«

the thrust of the bow. When, howeyer, the chord can be placed at, or a little ,
below the springs of the arch, all danger of this kind may be avoided by simply
wedging its ends well against the faces of the abutments.

As to tbe sise of tbe web members, when a bowstring truss is
AlUy loaded on top of the bow, (as is approximately the case with a center
and its archstones,) the strains on the web members are quite insignificant, and
arise chiefly from the weight of the center itself; bat wbile It Is belnv so
loadedi they are not only greater, but are constantly changing, not only in
amoant, but aJso in character — ^being at one period compressive, and at another
tensile.

Hence it would be very tedious to calculate the dimensions of the web members.
Fortunately the necessity for doing so is in a great measure obviated by the fact
that a center being but a temporary structure, the timber composing it is not ulti-
mately wasted if a greater quantity of it is used than is absolutely required.
Moreover facility of workmanship is secured by not having to employ timbers
of many diflferent sizes.

Hence the writer will yenture to suggest, entirely as a rule of thumb, to ytve
eaeb web member balf tbe trnnsverse are» of tbe bow»
taking care to make each of them a tie-strut.

Rem. 3. As to details of Joints, we refer to the Figs on pages 785,
736 ; merely suggesting here tbe use of long and wide iron shoes where timbers
are subjected to great pressure sideways.

Rem. 4. To prevent the thrust of the bow when its rise is small, from split-
ting off* the ends of the chords, the two may be united bv many more bolts than
are employed in roof trusses, &c, where only one is generally placed near each end
of the chord. But they maV when required be inserted at intervals extending to
many feet £rom the ends. Thev should have strong large washers ; and may have
about the same inclination as the shortest web member.

Another way of securing the same end in smaller spans, is by completely en-
casing the two sides of the bow and chord, to a distance of a few feet from their
ends, in short pieces of board or plank spiked to both of them, and having about
the same inclination as just suggested for bolts.

Rem. 5* Build up both sides of the arch at once, in order to strain the cen-
ters as little as possible.

Rem. 6. When a bridge consists of more than one arch, and they are to be
built one at a time, there must be at least two centers ; for a center must not
be struck until the contiguous arches on both sides are finished, for fear of over-
turning the outer unsupported pier. Therefore if there are but two arches, they
must be built at once, requiring two centers.

Rem. 7. Alwajrs use supports either vertical or inclined (and pro-
vided with striking- wedges) under the firames, and intermediate of the end sup-
ports, when possible ; even if they can extend out but a few feet from the abut-
ments, as at the left dde of Fig 9.

Rem. 8. Tbe welsbC of largr® centers and their lagging is greater
for flat arches than for hign ones of the same span ; and also approaches nearer
to that of the supported arch.

Rem. O. TblclLness of lag^lngr. The following table gives thicknesses
which will not bend more than an eighth of an inch under the weight of any
probable archstones adapted to tbe respective spans ; and generally not
so much.

TABI«I1 OF liAOOIBTO.— Original.

Distance apart

of firamea,
in tbe elear.

Feet
6
5

4
8
2

Span of center In feet.

lO.

50.

lOO.

150.

200.

Thickness of close lagging not to bend more than V^ inch.

ins.

Ins.

Ins.
5

2

Ins.

4
3
2

IV^ltb tblcknesses three quarters as great as these, the bending may reach
a ftiU quarter inch ; which may be allowed in dists apiurt of 3 or more ft.

Rem. 10. Centers are ft*amed, or put together, (like iron bridges) ou a
firm, level temporary floor or platform, ou which a fiill-Hize drawin^r of a fntme is

640

OENTEBS FOR ARCHES.

lint
ftbuts.

Am «ttch fhtme is flolahed, it ii removed to its place on the piers ol

L?J Fig. 13 \M

ArU 9, Tbe Wiasahlckon Bridg^ of the Reading R R, at PhilRdelpiiia,
has five arches of 65 ft span, 23 ft rise, 'M ft wide (archstones 3 ft deep, with dressed
beds and Joints, in cement mortar) ; with four cutstone piers 9.5 ft thick at top, and
from 35 to 50 ft hi(rh. It contains about 15400 cub yds of masonry.* Eaell center
consisted of 7 frames or trusses of hemlock timber, of the Bowstring pattern, with
lattice web-members ; and as nearly as may be, of the same span and

rise as tlie arches. They were placed 4.5 ft apart tvom center to center; and were

supported near each end /, Fig 13
(a transverse section to scale) by
a hemlock post p^ 12 ins square.
The bow was of two thicknesses
b 5 of hemlock plank, 6 ins apart
clear, in lengths of 6 ft, with their
upper edges cut to suit the curve
of the arch. Each piece was 4 ins
thick, by 13.5 ins deep at its middle,
and 12 ins at its ends. These pieces
did not break Joint ; but at each
Joint were four^ inch bolts, with
nuts and washers, uniting them
with chocks or filling^in pieces.
The bow, bb, footed on top of the ends of the chords /; and the angle formed by
their meeting (seen only in a side view) was (for about 2.5 ft horizontal and 5.5 ft
vertical) filled up solid with vortical pieces, to aflbrd a firmer base for resting the
frame on n ; beyond which it extends (in a side view) about 18 ins.

The cliordis /were of two thicknesses of 4 X 12 hemlock plank, 6 ins apart
clear, and most of them In two or three lengths ; breaking Joint, and with two ^
Inch bolts, with nuts and washers, at each Joint, for bolting them together, and to
fliling-in pieces. The web membem of each frame were 26 lattices, o, of
8 X 12 iuc^^ hemlock, crossing each other about at right angles, at Intervals of about
8.5 ft from center to center, and passing between the two thicknesses 6 6 of the bow,
and //of the chords. A few of the lattices were in two lengths, and the Joints w^ere
not at the crossings. The lattices were connected at each crossing by two hard wood
treenails 9 ins long, and 2 ins diam; and one such, 18 ins long, passed through the
intersection of each end of a lattice with a bow or chord. The first lattice foots
about 4 ft from the end t f a chord. They do not extend above the top of the bow.
All the spaces between the two thicknesses of bow or chord, where not occupied by
the ends of lattices, were completely filled by chocks, well spiked.

Each A*aine contained about 360 cub ft of timber; and weighed about f
tons. They were very flexible laterally until in place, and braced together by 4
transverse horizontal planks spiked to their chords; and by 5 others above them,
spiked to the lattices.

Until the keystones were placed, all the Joints of the frames continued tight, under
the pressure from the arch, and from the unfinished backing to the height of about
14 ft ai>ove the springing line; but after the keystones were set^ all the Joints of the
chords alone opened from .25 to .76 of an inch ; and at the same time the lagging un*
dsr the haunchea of the arches became slightly separated from the soflSt of the masonry.
Each center sanh but a fall inch at the middle, under the preeenre firom
the arch and 14 ft of backing.

The portion of the bridge above the piers was about two thirds completed before
the centers were struck.

There was one wedye to, to, (82.6 ft long, of 12 X 12 inch ooXr) under eaoh
end of a center. It was trimmed to form 7 smaller ones to, to, each 4.5 ft long, and
tapering 7 Ins ; one under each end of each fhtme /. They played between tapered
blocks a, a, of oak, 2 ft long, 1 ft wide, let 1 inch into the cap c, or into the piece «,
on which last tbe frames /, /, rested. The sliding surfaces were well lubricated witk
tallow when put in place.

The weoKea were strnck with ease, at one end of a center at a time, by aa
oak log battering-ram 18 ft long, and nearly a ft in diam, suspended by ropes, and
swung and guided by 4 men. They generally yielded and moved several inches at
the second blow with a 3 or 4 ft swing. Although each wedge was loosened etUirtlf
within 2 or 3 minutes, thus lowering the centers very tuddmlff, yet on account of the

• This bridge, finished without socIdeDt, in 1882, rvfleot-i much credit on the late William Loreos,
Bsq, Gb. Eng; on Mr. Ctaarlea W. Buohbolft, A«»tsUnt in Charge: and on the tkilful and eMrcMlo
•ontraoton, William k James Nolan, of Reading, Penna. These last most eordiallj aaalited U^
writer in making obaervations during the entire progresa of the work.

CENTEBS FOR ARCHES. 641

good character of the masonry, not the slig^hten cracK of a mortar Joint could atten
wards be detected in any part of the work. After three days the average sinking of
the keystones was only .35 of an inch ; the least was i^; and the greatest ^ of an
inch. The heads and feet of the posts p compressed the hemlock caps c, t^nd tlie
sills, about % of an inch each, showing that for arches of this size the caps am! sills
had better m of some harder wood, as yellow pine or oak ; although probably the
compression was {JBicilitated by the large mortices, 3 by 12 ins, and 6 ins deer

41

TIHBEB DAUS.

Prlmnrr rMalalles, tn tha ereclion of dftms, are, > roundalioD auffi-
illeallf ana u> pnveal tbem fram letlling, snd thus leaking; the preTeuiioD
ot leaks thro Ligh their backs, or uader tbeir bues; aod the prsveDtlou of weal

nter. For the first purpose, bard iKvel rock bottoai is of coutso the beat; and
■hould be choaea, if poaaible. In Ibal cue, thick plaoka, II, Fla 6, (single or
doable.utbe cue may be,) cloaelf jointed^ and reacbing from tbe ores I, £, to
tba back lower edge w, (wbete Ibej atould be aeribod down lo the rock ;) with >
good baeklng, b, of gravel, will auace to preieat leaki. Gnial, or rather rerr
graTsIlT soil, ii Tar better Ih&a earth for thia purpose; Ibr If the water abould
chance to form a void in ll, the graiel falli and itopg It. To prevent this bak-
ing ^om being disturbed near the ci^t of the dam, br floatlDg bodies swept

aho^n in F^ T%bould be added fDr B width of about ID M 30 feel : or umlf' lis

In Fig l,(a dam on tbe Scbuflklll
close Jofntod and laid touching, su as i

Joint, to a depth of several feet, lo prev
the base of the dim. Frequeatl; but oi
soft or open for a depth ofonlfs few feet. It

J tbe nnper timberm, e, are
Ire planklDB Id addition.
.. . Is graiel or eanh. there muat In addition __
>t sheet piles, p, Fie 2, Ac, oloM driien, brsaklng
Bt, lo prevent leaking through the soil beDoatfi

hrough the i

■ used. Iftb
imes belter to remoT* It. ana
iweier, oiili^ tb* ibMt pUea.
remared from tha base. Id
.' 10 aupport tbe dam eattrelr

^i" si"^' — ' —

'^Jt GRAVEL

nard rock, or of medlniD roc* protecled by a eonalderable deplh ot water. Tha*
4am, Fig 1, WW buUt upon a tolerably firm mloaoeona gnetas In nearly vertical
sim»,oo«iadbjabout3(eWorwaleriDnrdiiiar7«tai9«.InMjB«citharockwM

■*• <!»>>> OB Cape Fenr Kl ver ) tieliht oTdun, IS ft ; ftvui Tart:

UBbir, laid a4«* Mgirtiri uMDdiBj obdv
•■ppKHd H bt l»ll«d ^ Bbpn pitn If . drim

R ICK
• nt wn*<t«n 4MmB are mnnj' j (s»e chf i

644

DAMB.

Figr.8.

olher, fbrmlng in plan a Mries of netangles with tides of about T to 12 ft. Thej are not voteliel

together, but simply bolted by 1 inch square bolts (often ragged or Jagged) about 2 to 2H feet long,

through two timbers at every intersectioD. These are not found to rust or wear seriously, even when

exposed to a current. Square bolts hold best. Bound logs are flattened where they lie upon eaoa

other. Experience shows that firmer but more expensive connections are entirely unnecessary. Th»

oribs are usually, but not always, filled with

rougb stone. In triangular dams, disposed

as in Figs 1, 2, and 7, this stone filling is v

not so essential as in other forms ; because

the weight of the water, and of the gravel

backing, tends to hold the dam down on its

base. Still, even in these, when the lower

timbers are not bolted to a rock bottom, or

otherwise secured in plaoe, some stone may

be necessary to prevent the timbers from

floating away while the work is unfinished,

and the gravel not yet deposited behind it. 1ju_ m

On reck, the lowest timbers are often bolted J?!^* !•

to it, to prevent them from floating away PP

durttiy corutmction; and when the water

is some feet deep, this requires coffer-dams. Or. the oribs may be built at first only a few feet high ;

then floated into place, and sunk bv loading them with stone; for the reception of which a rough

platform or flooring will be read in the oribs, a little above their lowest timbers. The bolting to the

rook may then be dispensed with. The water may flow through the open oribwork as the building

higher goes on ; attention being paid to adding stone enough to prevent it floating away if a fk«shet

should happen. Or, cribs shown in plan at c c, Fig 8, loaded with

stone, may be sunk, leaving ooe or more intervals, llkeHbat at o o o o,

between them, for the free escape of the water. These openings to

be finally closed by floating into them clo^ing-crlbs shaped like n.

The workmanship of a dam in deep water can of course be much
better executed in cofier-dams, than by merely linking cribs. The
joints can be made tighter: the stone filling better packed ; the sheet
piling more closely fitted, Ac.

When a very uneven rock bottom in deep water, or the introduce
tion of sluices in the dam, or any other considerations, make it ex-
pedient to build dams within coffer-dams, both should be carried on
in aectiona; so as to leave part of the channel-way open for the es-
cape of the water. Commencing at one or both shores, tbe first (lection of the enffier-dam may reaoh
say quarter way or more across the stream. In the section of the dam iuelf built within this'enclos-
ing coffer-dam, ample sluioes should be left for the water to flow through when we oome to bnild the
eloeinrj section of the coffer-dam. When the dam has 6een finished, these sluices may be closed
by drop-timbers*. Before rerobving one section of coffer-dam. the outer end of tbe enclosed
section of dam itself must be firmly finished in pueh a manner as to constitute a part of tbe inner
endof the next section of coffer-dam. It is impossible to give details for everj contingency ; the en-
gineer must rely upon bis own ingenuity to meet tbe peculiarities of the case' before him. In some
cases of shallow water, mere mounds of earth may answer for coffer-dams; or rough stone mounds,
backed with earth or gravel.

After tbe water has passed beyond the crest, c in the figs, there is no necessity for preventing its
leaking dowii among the crib timbers: on the contrary, the thick sheeting planks, (or squared tim"
hers, as occasion may require.) el, Figs 4 and 6, which form the slopes along which tbe water then
-flows in some dams, are usually not laid close together, but with open joints of about ^ inch wide be-
tween them, for tbe express purpose of allowing part of the water to fall through then^ so as to
keep tbe timbers beneath them partially wet ; which, to some extent, renders them more durable. In
Figs 1. i, 6, and 7, tbe water of the lower pool flows ffeely back among the crib timbers, and rongh
-quarry stones with which tbe cribs are filled either partly or entirely. In Figs 4 and 6, these stonea
are not shown. In the dam, IMg 1, none were used. In Fig 2, tbey were as shown.

A substantial, and not very expensive dam of the form of Fig 7, may be built ^f rongh stone In
oement. Some hewn timbers should be firmly built horisontally into the masonry of the sloping
back c nu7, at a few feet apart, with their tops level with the surf of the masonry. To these must bo
well spiked close-jointed sheeting-plank envo, for protecting the masonry from the action of the
water, and of floating bodies. The gravel backing b, may be omitted ; but tbe sheet piles j», and an
apron in front of the dam, will be as indispensable in yicQding soils, as if the dam were of timber.

Figs 1, 2, 4, 6. and 7, are sections drawn to a scale, of existing dams in Pennsylvania, that have
stood successfully the force of heavy ft'eshets for a long series of years, t These f^sneu at times carry
olong large bodies of ice, trees, houses, bridges, ko. ; and have risen to 11 ft above tbe ereats. Fig 1,
on the Sch Nav, was built in 1819, and served perfectly for 39 years, until in 1858 the decay of mucb
of its timber, especially of tbe close-laid top ones, e, rendered it necessary to build a new one Just in
f^ent of it. It was of extremely simple construction ; and was never filled with stone. The bottom tim-
bers, o 0, 10 ft apart, were bolted to tbe rock ; and immediately over each of tbem, was such a series of
inelined timbers as is shown in the flg. The top ones, e, however, were close-jointed, and laid touching
so as to form tbe top sheeting, instead of thinner planks. The short pieces at t were lai«i in the sama
way. No coffer-dam was used ; but tbe bottom pieces were first bolted to (he rook ; 10 ft apart ; then
tbe Rtringers and the sloping pieces were added. Tbe close covering («) was carried forward trom
each end of the dam, until at last a space of only about 60 ft was left in the center, for tbe water te
pass. The close covering for this space being then all got ready, a strong foroe of men was aet to
work, and the space was covered so rapidly that the river had not time to rise sulBciently high to
Impede the operation.

* Timbers ready prepared for closing an opening through which water is flowing ; and saddealy
dropped into plaoe oy means of grooves or guides of some kind for retaining them in poeltion. Sev-
'jral such timbers may at times be firmly framed together, and then be all dropped at onoe ; elosiBg
the opening or sluloe at one operation ; especially when it is of small eiae. In aomo ciaeee, m erio
may be sunk on the up-stream side of such an opening, for oloaing it,

t Those on the Schuylkill Navigation were obligingly furnished by James F. Smith, Bsq, chief
engiii<>er and RMperintendent of that work. Other valuable infons»tl«a from (he some aonroe wUI
be found ill different parts of this volume.

DAMS. 645

Fig 2 ia a o«nal feeder dam on the Juniata. Here « • are timben ■tretohing elear aorosi the Btream,
fcbout 300 ft.) and tustalning the apron a a, of ■tout hewn timbers laid toaohlng. This dam waii filled
riftfti atrOD*. for the retention of wbieh the f^ont abeecinf planka were added.

Fig 6 la oa the Soh Kav ; waa bailt in 1855. It la a form maoh approved of on that work, ftir auch
Uoatione ; namely, firm rock foundation, with a conaiderabte depth of water in front. The highest
Am (82 ft) on the Soh Nar, ia very similar to it ; built in 1851. All the dama ou this work are of
i*«rn timber, obiefly white and yellow pine. The water oooaaionally rnna from 8 to 12 tM deep over
heir create ; and iiien overflows and aurroanda many of the abate. The vertioal back allows the
•▼erflowing water to leak down among all the lower timbera of the dam, and thua tend to their
ireaerTatlon.

Fig 4 shows the dams on the Monongahela alaokwater navigation ; W. Mllnor Roberta, eng. They
kre of round logs, with the bark on : flattened at croaaings. The longest onea in the fig are 10 feet
ipart along the length of the dam. Experience ahowa that auch dams poaaess all the atrength neoea'
lary for violent streams. On rook, the lowest timbera are bolted to lU

Vig 7 has been anooessfbUy nsed to heights of 40 ft.*

Fig 8 ia Intended merely as a hint fbr a very low dam on yielding bottom. Its main aupporta are
pilea f <, troxa i to 8 ft apart, aooording to the height of the dam; and other oiroumatanoea ; and tt
are abort piles for sustaining the apron dd. It may be extended to greater heights by adding braces
In front ; wbloh may be eovered by stent planks, to form an iaoUned slide for the overfalllng water.
ICany eflbetlve arraagementa of piles, and sloping timbers for dams on aoft ground, will auggeat ihem-
aelToa to the engineer. Thua, at intervals of aeveral feet, rows of 8 or more piles may be driven trans-
versely of the dam ; the top of the outer pile of each row being left atthe intended height of the crest,
while those behind are auooeasively driven lower and lower; ao that when all are afterward eon>
neotod by tranaverae and longitudinal timbera. and covered by atout planking, and gravel, they will
form a dam somewhat of the trtangnlar form of Fig 7. It woirid be well to drive the piles with an
inclination of their topa up stream.

There is muoh scope for Ingenuity both in designing, and in oonstruoting dama under varioua clr-
enmatanoea ; and in turning tba oourae of the water from one channel to another, by means of dltohea,
pipes, or troaghs, fto., at diff heights; aided at times by tow temporary dama or mounda of earth ; or
of sheet piles, ko; or by oeffsr-dams ; so as to keep it away from the part being built. Baeb locality
will have ita peonliar features ; and the engineer must depend on his Judgment to make the moat of
them.

Abatments of dams as a general rule shonld not contract the natural
width of the stream : or, if they must do so, as little aa poaaible ; for eontraetlona inereaae the height,
and violence of the overflowing water In time of fi^eshets ; during which a great length of orerfall ia
espeelally desirable. They ahouid be very flrmlv connected with the ends of the dama; and should,
if the aeotion of the valley admits of it, be so high, and carried so far inland, that tbe high water
of fresheu will not sweep either over them, or around their extremities ; and thua endanger under-
mining, and destruction. In wide, flat valleja they, cannot be so extended without too muoh ex-
pense; and the onlv alternative is to found tbem so deeply and securely aa to withstand such
aetiott ; making their height snob that they will, at least, be overflowed but seldom. Their ends
adjnoent to tbe dam, should be rounded oflf, so as to facilitate the flow of the water over the crest.

They are best built of large stone In cement; for although sufllcient strength may bs secured by
timber, that material decays rapidly in such expoanres. If of earth only, they are very apt to be
earried away if a freahet ahouid overtop them.

Sluices should be plaeed in every Important dam, in order thav

all tbe water may be drawn off. If necessary, for the purpose of repairs ; or of removing mud deposits ;
or finding lost articles of importance, Ac. They may be merely strong boxings, with floor, sides, and
top of squared timbers ; and passing throa-^h the breadth of the dam. Just above the bottom. To pre-
vent trees, Ae. f^om entering and atioking fast in them, aome kind of atrong acreen is expedient. In
common cases a sluice ahouid not exceed about 3V^ ft by 5 ft in cross-section; otherwise it becomes
hard to work. Two or more such openings may be used when mncb water is to be voided. They
should be near the abutments. The gates or valves for opening and shutting them, should be at thi-
up-stream end; for if at the lower one. aooumalations of mud, ko, will fill the sluices, and prevent
them from working. They are usually of timber; and slide vertically in rebates; being raised and
lowered by rack and pinion ; but in very important dams they may be of oast iron. Two sets of slnioet
are desirable; that one may be alwavs ready for use if the other la atopped for repairs.

The part of the apron in f^nt of the ainloe should be partioularly firm, ao aa not to be deranged by
:,be water ruahing out under a high head.

Danui are sometimes, but rarely, built In the form of an
arcb ; convex up stream. This form is strong; and when tlie shores are of rock
it may be expedient to uae it ; but if the banka are soft, they will be exposed to wear by the ourreni
thrown against them at the abuts of the arob.

At times dams are built obliquely aeross tbe stream, with

the ohieot of increaaing the length, and consequently rednolng the depth of water over the crest in
times of freibeu. The argument, however, appears to the writer to be of but little weight, inasmneh
aa tbe redaction of depth would extend but a trifling diatanoe up atream from the dam ; and wonld
tberefnre voiircely have an appreciable eflhct in diminishing the Iqjnry to the overflowed district above
Moreover, the inoreased expense is probably always more than commensurate with any advantage
gained.

* Cost of crib dams. With common labor at $1.50 per day ; lumber, 920 per
1000 ft. board measure, delivered ; stone for ailin|p, Si per oub. yard ; gravel 50 cents per cub. yd. ;
irenlbrbolUi, etc., iota, per lb.,— aucb dams in shallow water usually oost, complete- ^m 9 to 12
Mats p«r sable tbot, or 93.43 to 98.24 per eubio yard of orib.

FIe3. 9 nai ID are dealgwa for aiual I ni«iWMrlnK weirs, giillable An

■hsIloiF gtreaug up lo Hy lUO feel wide ; I'igg. 9 for eanh or gtaxnt bottoui, Bud

liy 1^1 Ld b IriiK Line. Ths nllls should eitend stj Irom S (o 10 (eel luu eacti
bank af llie Btieaiu. Tougued and ei.^ved slieet piling P.atSX 10 inch hant-

jftgg. O.—Measnrtnq WpIt on Earth or Gravel Bottom,

ock 1< then driven close behind the upper sill S, loadepth of rroin two to four
eet, and Bniked to S,. A third sill, Sj, of (he »»mii leogih u s, and S,. Is then
RLdbehiuSthpsheelpnipgi and the twoallla S, and S, and the iheet plIioR P
i-e then seoiited together, as »ho»«. by 1 inch bolts, spaced about '2 feut apart
llie tops ofthe sheet pUlQgpiolwjl about a ool above Ihenllla.anU are stilfeoed
ly 4 K 4 inch timbers ic. bolted in from of them and resting upon the Booring
'of IX ID itich spruce. rhisHoaring, like the sillB, extends eeianl feel beyond

leav'^ones. Any apaces left oBdernealh it hy unevennew of the bottom should
lIbu U leieled up with stones or Kravel

A 10 X '0 inch yellow pine post \1. :( feet high. Is tenoned between sills S, and
>, at each end of the nverltair, and braced by an h ■,< in Inch > eilow pine stmt
< tenoned lo II and totheslUS,. Beyond these posts the sheet piling Peitends
1^ hinh as the lou of the posts, and is carried, al that heieht Inlo the b«nk ; the
ops u( the piles beintr held iu line by tvu ^ X 8 Inch waling pieces ■■ ~ boiled to
hem, one on each side.

InFlgalO.thehemlockiUIa.S, orinxiOliwh.aBdS,ar<l:

Ketween the two sills are boiled n|

post M is tenoned belvveen the sllli

In both rigs, the crest-piece n, is of 2 X 8 inch ask, bevel>4 s" as to ieaie • .
horiiontal lop face \4 Inch wide. The crest-piece is let in flush with the hack of
the piles or boarda P, lo which it is boiled, and Is let Into the rnd posls M about

the length of the o'erfalT^y flash-boards placed along the reM of the dan.

leas subject to abrasion by drift passing o'er the weir. The /op edge, and Ihe abut-
ting ends of ths eevBrnllenglhs, should be planed amoath and square: IhaformM

ordu lo foid IcLkage, A> a further precaution againac leakage, a atclp or butC'
Mnjiof S X ^incti iron, about a foot lone, may be let iu,teliuem the crut-piec*
atd lie thetl pUmg, oppMlMeait joint oi the former, and oiarlapplng both Ibe
lh4Jolalng anda, (hepuing b«ia(( cut ana; ^Incb dceperat those poiDte, In order
10 accvmiuoilBte them, buch butt-atraps, If ulaoed ou the n^tream aide of the
flrest-plecflr would breaL tbe continuity of tbe Bheet of rater paving over the

Iron 'is obUinable in any coinmercial center, la lennhB of about IB ^°f ' B xli
■elgln t% pounds per running foot; 8 X J4 3!^ pound..

I apply tbe usual weir rormuln
w vertical for a daptb p below

Ftft. 10 — Xeatvrinig VHn «

tbe middle of Id length, and then In turn (Jected at Che point where they enlaied,
thiu traTdlug back and forth along the ipace bsblod lh« Bbeet.

648

DAMS.

Tremblliitps In Damn. Pams over which the water falls In a long^
smooth, unbroken sheet of considerable height, are more or less subject to
tremblings, caused apparently by alternate compression and rarefaction of the
air by the falling sheet, especially in the space (W, Fig. 20, p. 547) behind the
sheet, where a partial vacuum is often formed, because the air there is entangled
in the falling water and given oif again by it down stream in the shape of foam.

Such treraolings sometimes cause a rattling of windows hah a mile or more
away. We have known this to be stopped (in one case unintentionally) by build-
ing a well-covered wide crib apron, a lew feet high, against the front of the dam,
for preventing the abrasion of the bottom. In other cases a series of oblique
timbers placed against the front of the dam, and part way up it, at a slope of
about 11^ to 1, and covered with plank, has been perfectly effective in stopping
it. In short, any device.which admits air more freely benind the falling sheet,
or destroys the continuity of the latter (such as flash boards of different heights
or placed at intervals along the crest), or which reduces its height and its con-
tinuous length, ought to diminish or obviate the trouble.

The proper time for baildinip dams is of course at the longest
period of low stage of water.

Table ofthlckneflMi of white pine plank required not to benci
more than ^h part of Its clear horisontal stretch, undes
different heads of water. (Original.)

Stretch
inFU

8
4

.•

8
10
12
16
20

40

Heads in feet.

30

20

10

Thickness in Inches.

% 3
4

6

8

10

15
20

6

WATEB StJPPLY. 649

WATER SUPPLY.

Consamption of water. Owing largely to the proper extension of the
nse of water in dwellings, the quantity required in cities increases faster
than the population. In other words, the per capita consumption increases.

Use. Abundant experience shows that a supply of 50 gallons (or say 7 cubic
feet) per capita per day is abundant for all the needs and luxuries ot well-to-do
families iu American cities. The manufacturing consumption, of course, bears
no -fixed relation to the population. In cities it is generally much less than
the domestic consumption.

Waste. Iq American cities, the waste often amounts to two or three times
the quantity really need. Of the 116 gallons per capita per day, delivered in
New York in 1899, Mr. Freeman * estimates that from 81 to 56 gallons were used,
10 unavoidably wasted, and from 50 to 75 avoidably wasted.

In Philadelphia, iuvestigations by means of the Deacon waste-water detectot,
on 142 modern seven-room, two-story dwellings, with bath, etc., on two inter-
mediate streets, showed that, of 222 gallons per capita per day, furnished through
782 fixtures, 192 gallons, or 86.5 per cent, were wasted, and only 30 gallons, or
13.5 per cent., were used. The City is now building enormous works for the
purpose of pumping, filtering, conveying, repumping, storing, and distributing
the water wasted, as well as the smaller quantity used. Of the total C08t,f less
than half would have sufficed for the water used and unavoidably wasted.

Sources of waste. The waste is caused by heedlessness ; by allowing
water to run to waste in order t-o prevent it from freezing in winter and in order
to get cooler waier in summer; by leaky and otherwise defective fixtures; by
ansuspectecl leaks in mains and service pipes, etc.

As a "K«esa, tempered by judgment," Mr. Freeman* classifies the 50" to 75
gallons per capita per day, wasted in New York, as follows :

Leaks in mains 10 to 15 gals per capita per day.

*' service pipes 10 to 15 " " "

" defective plumbing 15 to 25 " •' "

Careless and wilful waste 14 to 17 " " **

The avoidable waste is usually perpetrated by a small fraction (say from one-
fifth to one-third) of the population, the remainder using water reasonably. In
the Philadelphia case, above cited, of the 782 fixtures, 22 were found to be
" leaking slightly," and 32 " turned on continually."

Waste restrtetion. Water meters. Waste is best restricted by
making its avoidance a pecuniary object to the consumer; and this is best
accomplished by the use of the water meter, at least on all services (domestic,
industrial, and public) where waste is found to be going on. The meters should
be owned and maintained by the corporation supplying the water.

Hinlmam ehargre. In order to encourage the liberal use of water, while
discouraging its waste, and thus avoid undue economy (tending to uncleanliness)
each consumer should be charged a minimum periodical rate, sufficient to cover
amply all the water he can possibly use and enjoy.

Mr. Freeman* estimates the average cost of domestic meters, for New York
and Brooklvnj mostly 5-8 inch and 3-4 inch, with a few of larger sizes, at $12.60
each, and the cost of installation by the city, working systematically and on a
large scale, at $2.50 each, or a total of $15.00 each. He assumes " the average
life of the ordinary domestic meter, of a good type, well carad for, and with
occasional repairs and renewal of worn parts," at "not far from 20 years"; and
annual expenses as follows :

Providence, R. I. New York,
Actual, approx. Assumed,

Interest on cost of meter and setting $0.50 $0.45

Depreciation and renewal of meter (life assumed

20 years) 0.75 0.75

Maintenance and repairs, testing and resetting 0.46 0.70

Heading meters and computing bills 0.42 0.60

Total annual cost, per meter $2.13 $2.50

* Report upon New York's Water Supply, made 'to Bird S. Coler, Comptroller,
by John R. Freeman, Civ41 Engineer, 1900.
fThe total cost may reach $20,000,000.

650 WATER SUPPLY.

Free water for fire protection. Cities sometimes give to manufac-
turers a free supply of water through special connections, to oe used for fire
protection only; the manufacturer giving bond not to use such connection for
any other purpose, and the city placing a met«r on the connection for the detec-
tion of any illicit use of the water for other purposes.

Wftter for city use shonld not be drawn fkwm die vory- bot-
tom of tlie reservoir, because it will then be apt to carry along the sedi-
ment ; which not only injures the water, but creates deposits within the pipes;
thus obstructing the flow. In fixing upon the necessary capacity of a reservoir,
this must be taken into consideration ; inasmuch as all the water below the level
for drawing off, must be regarded as lost. When circumstances justify the ex-
pense, it is well to curve up the reservoir end of the service main, so as to pro-
vide it with valves at different heights: for drawing off only the purest stratam
that may be in the reservoir. With this view, the valve-tower gen-

erally has such valves communicating with the water in the reservoir ; and by
this means only the purest is admitted into the tower: and from it, into tha
city pipes. This refinement, however, is rarely practicable. Such valves must
of course be worked by watchmen.

Art. 1. Beservoirs. In im^rtant reservoirs of earth, for storing water
to moderate depths for cities, experience appears not to sanction dimeusiona
bolder than 10 fact thick at top ; inner slope 2 to 1 ; outer slope 1^ to 1.* A top
width of 15 feet to 20 feet, and inside slopes of 3 to 1, are adopted in some im-
portant canes ; with outer slopes of 2 to 1. Both slopes, however, are at times
made only 1^ to 1. The level water surface should be kept at least 3 or 4 feet
below thH top of the embankment ; or more, if liable to waves. In a large
reservoir, a quite moderate breeze will raise waves that will run 3 feet (meaenred
vertically) up the inner slope. A low wall, or close fence, w. Fig. 37, is some-
times used as a defence against them. The top and the outer slopes should be
protected at least by sod or by grass. To assist in keeping the top dry, it
should be either a little rounding, or else sloped toward the outside.f The soft
soil and vegetable matter should be carefully removed from under the entire
base of the embankments ; which should be carried down to noil itself imper-
vious to water, in order that leakage may not take place under them. To aid in
this, a double row of sh^t piles, or a sunk wall of cement masonry, carried to
a suitable depth below the bottom, may be placed along the inner toe in bad
cases. If there are springs beneath the base, they must either be stopped, or
led away by pipes. The embankment should be carried up in layers, slightly
hollowing toward the center, and not exceeding a foot in thickness; and au
stones, stamps, and other foreign material, such as clean gravel, sand, and de-
composed mica schists, (fcc, that may produce leakage, carefully excluded. These
layers should be well consolidated by the carts; and the easier the slopes are,
the more effectively can this be done. The layers, however, should not be dis-
tinct, and separated by actual plane surfaces: but' each succeeding one should
be well incorporated with the one below. Tnis has sometimes been done by
driving a drove of oxen, or even sheep, repeatedly over each layer : in addition
to the carting. Boilers are not to be recommended, as they tena to produce
seams between the layers. This might possibly be obviated by projections on
the circumference of the roller.

Gravelly earth is an excellent material, perhaps the best. The choicest
material should be placed in the slope next to the water ; and should be de-
posited and compacted with special care in that portion, so as to prevent the
water from leaking into the main body of the dam, and thus weakening it. It
is not amiss to introduce a bench, b. Fig 87. in the outer slope, to diminish
danger from rain wash by breaking tne rapidity of its descent.

If the bottom of the reservoir itself is on a leaky soil, or on fissured rock,
through the seams of which water may escape, it must be carefully covered
with from 1^^ to 3 feet of good puddle; which, in turn, should be protected from
abrasion and disturbance, by a layer of gravel ; or of concrete, either paved or
not, according to circumstances.

* The writer suggests that a top width equal to 2 feet + twice the square root
of the height in feet, will be safe for any height whatever of reservoir properly
constructed in other respects.

t Some engineers slope the top toward the ingidi.

RESERVOIBS. 651

Reserroira constracted with the fore^^olng dimenfllons, and with care, mm

emain aafe for an indefinite period; but where seriouB damage would lesuit

roiu failure, the following addltlonAl precantlona should be taken.

^he inner slopes should be carefully faced up to the very top, with at least a

ilose dry rubble-stoue pitching, not less than 15 to 18 inches thick ; as a protec-

Aon against wash, and against muskrats. These animals, we believe, always

MHumenoe to burrow under water. If the slopes are much steeper than 2 to 1,

;bi8 dry pitching will be apt to be overthrown by the sliding down of the soft*

sned earth behind it, if the water in the reservoir should for any cause be

irwwn down rather suddenly. It will be much more efRsctive, but of onurse

more costly, if laid in hydraulic cement; and still more so if la d upon a layw

a few inches thick of cement-and-^rav<'l concrete ; especially if tnis last be

anderlaid by a layer about 1^ to 3 feet thick of vood puddle, spread over the

Caoe of the slope ; the great object being to protect the inner slope fk'om actual

eontact with toe water. If this can bt» effectually accomplished, slopes as steep

as 1^^ to 1 will be perfectly secure ; for the danger does not arise from any want

of weight of the earth for resisting overthrow. Speeial care should be

toeMtovred umb tlie iMiier toe 9€ ^ke alope. to prevent water fkom

finding its way beneath it, and softening the earth so as to undermine the stone

Bitching. Near the top, mSerenoe ehottld be had to danger of derangement by

lee, frost, rain, and waves. Flat inner slopes tend not only to prevent the dis-

plMement of the pitching; bnt increase the stability of the embankment, by

causing the pressure of the water (which is always at right amries to the slope)

to become more nearly vertical ; and thus to hold the embankment more firmly

to its base than if there were no water behind it. (Sometimes the toes of both

the inner and outer slopes abut agaiiut low retaining-walls in cement. This

gives a neat finish, and tends to preservation from injury.

Many engineers, in order to jMrevent leaking, either tnrough or beneath the
•mhankment, eonstmet a pnodle-w^l, p, JPig. S7, of well-rHmmed imper-
vious soil, 'grave ly clay is the
best,) reaching from the top
to several feet oelow the base.
Tills wall should not be less
th»n ^ or 8 feet thick on top,
tor a deep reservoir; and
should increase downward by
offiietg (and not by slopes, or
Fig. 87. batters) at the rate oi about

1 in total thickness, to B or 4
In depth. Other engineers object to these puddle-waUs ; and contend that leak-
age should be prevented by making both the inner slopes, and the bottom of the
reservoir, water-tight, by means of puddle, concrete, and stone facing in cement,
as just alluded ta They argue that if the raaibankment is well constructed, it
is itself a puddle-wall throughout.

Near mmi Fmneisco, €al, are two eartliem reaerwoir dams
built about 1864. one 96 fset high, 26 on top, inner slope 2.75 to 1, outer 2.5 to 1.
The other 98 high, 25 on top, inner slope 3.6 to 1, outer 3 to 1. In each the pud-
dle-wall is carried 47 feet deeper than the hase. No stone facing.

It is dtfltonM to prewent water ander Itisli pressure from
liiidiBflr its way throiifrii considerable dlatanees alonfp seams
where earth is in contact with smooth rock, wood, or metal ; as, for instanoe,
al0D$( the surfaces of iron pipes laid under reservoir embankments ; or along
the tie-rods sometimes used through the puddle of cofferdams ; and the same
is apt to occur under the bases of embankments which rest on smooth rock,
gpeeiel care should be taken that the earth used in such positions is not of a
poTOus nature ; and that it is thoroughly compacted all along the seam ; and the
etiaigfat continuity of the s<>am should be interrupted or brok(>n as frequently
as possible by prqiectinns. Faucets or flanses do this to a limited extent in the
case of iron pipes ; and something similar, but on a larger scale, should at short
intorvala be constructed in the snape of collars or yokes of cement stonework,
in the case of rook or masonry.

It is usually advisable to divide reserwoirs into two parts, so that
while the water in one part is beins: drawn off for use, that in the other may
purify itself by settling its sediment. Also, one port may remain in use. while
the other is being cleaned or repaired. Manv davs, or even two or three weeks,
sometimes, are required for the complete settlement of the very fine clayey par-
ticles in muddy water ; dependi ng on the depth of the reservoi r. One or moce
fiifl^ts of steps to the bottom of the reservoir should be provided.

Awl In Beoervoirs. The reservoirs of the New River Water Co, Lon-
don, jEiiglandi were unoleaned for 100 years, during which mud 8 feet deep was

652 BESEBYOIBS.

ieposited, or about an Inch annaally. At Philadelphia it is about .25 inch per
annum from the Schuylkill, and 1 inch from the Delaware River. At St. Louis,
Missouri, about 3 to 4 feet per year ! Vegetation is apt to take place in shallow
reservoirs and near the edges of deep ones, especially in very warm weather;
and the plants, on decaying, injure the water.

Wator flowlngr tbroufrli marsh lands is sometimes unfit for drink-
ing purposes. That, for instance, in some sections of the Concord River, Massa-
ohnsetts, was reported by the eminent hydraulic engineer, Loammi Baldwin, of
Boston, to be absolutely poitonotM from tnis cause.

The construction of a large deep reservoir is not only a very costly, but a
very hazardous undertaking. With everv watchfulness and care, it is almost
impossible entirely to prevent leaking; although this may not manifest itself
for months, or even years. Should a break occur, especially near a city, it
would probably be attended by great loss of life and property. If the water
once finds its way in a stream, either across the unpaved top, or through the
body of the embankment, the rapid destruction of the whole becomes almoct
certain.

Art. la. Storine Reserwolrs. The entire annual yield of a stream
may be much more than suMcient for supplying a certain population with
water; and yet in its natural condition the stream may not be available for this-
purpose, because it becomes nearly dry in summer, when water is most needed;
while, at other seasons, the rains and melted snows produce floods which supply
vastly more than is required ; and which must be allowed to ran to waste. A
storing reservoir is intended to collect and store up this excess of water, so that
it may be drawn off as required during the droughts of summer, and thus
equalize the supply throughout the entire year. This, when the locality per-
mits, is effected bv building a dam across the stream, to form one side of the
reservoir; while the hill-slopes of the valley of the stream form the other sidee.
The stream itself flows into this reservoir at its up-stream end. When the
stream is liable to become nearly dry during long summer droughts experience
■hows that the capacity of the reservoir should be equal to from 4 to &
months' supply, according to circara stances. During the construction of the
dam, a free channel must be provided, to pass the stream without allowing it
to do injury to the work. If the dam were built precisely like Fig 37, entirely
of earth, it would plainly be liable to destruction by being washed away in case
the reservoir should become so full that the water would begin to flow over its
top. To provide against this we may, by means of masonry, or of cribs filled
with broken stone, or otherwise, construct either the whole, or part of the dam,
to serve as an owerfall, or a waste-welr. Or a side channel (an open cnt^
pipes, or a culvert, Ac) mar be provided at one or both ends of the dam, and in
the natural soil, at such a level as to carry away the surplus flood water before
it can rise hign enough to overtop the earthen dam. Besides these, and the
pipes for carrying the water to the town, there should be an outlet, with a valve
or gate, at the level of the bottom of the reserv<dr ; in order that, if necessary
for repairs, or for cleaning by scouring, all the water may be drawn off. The
entrances to the city pipes should be protected by gratings, to exclude fish, Ac

To facilitate repairs or renewals of all walwes, Ac, wblcii
are nnder water, the reservoir ends of the pipes or culverts to which tbev
are attached, may be surrounded by a water-tignt box or chamber, which will
usually be left open to the reservoir ; but may oe closed when repairs are re>
quired. Access may then be had to them by entering at the outer end, after
the water has flowed away from inside. In case the outlet is through a long
line of pipes which cannot thus be entered, a special entry for this purpose may
be cast in the pipe itself, near the outer toe of the efubankment; to be kept
closed except in case of repairs. Sometimes a better, but more expensive meana
of access to such valves, is secured by enclosing them in a valwe-tower of
masonry. This is a hollow vertical water-tight chamber, like a well ; but near
the toe of the inner slope; having its foundation at the bottom of the reservoir;
whence the tower rises through the water to above its surface. This chamber
is provided with valves or g^tes usually left open to the reservoir; but which
may be closed when repairs are needed ; and the water in the tower allowed to
escape from it through the open valves of the outlets. This done, workmen can
descend through the tower by ladders from the aperture at its top.

At times the oatlets for the discharge of surplus flood water are, like those for
scouring, placed at, or just above, the level of the bottom of the reservoir. In
order that these ynay work in case of a sudden flood at night, Ac, they must be
fhrnished with self-acting valves, which will open of their own accord when the
flood is about to rise too nigh. This may be enected by attaching them to floats,
the risinff of which, when the water is high, will pull them open. All such oat-
Itta ahouid be large enough to let men enter them for repairs. They should ^y

VATEK-PrPE8, 653

meana t>e laid tfanugh tha ■rUflnlsl eartben body of ths dim Unat, wlilurA
.ng supported upon qmiwiirT reictdng dgWD Id > Quo nnunil louDditlon ;
lerwiae Lha j art tery apt w be broken by the subaideBce of Iha embankmeDl.
la uaually oafi^r to carrj tbem tbroogb tbe GrpL Datural boJI Dear one eitd of

aa to leave tbo outleie ihtmBaii^v uenaJLj empW, for loBpection ; but it la
Iter to haTf) two valvea, ao tbat one mAj be naed when the other tieede repair;

>pped wheD tbejare filled to the proper belgbt. Laroe atoring reaervoin
weaaarlly lubiBerKe mora or leat Und, wbioh bag tbereiure vt be purcbaMd.
r IntOTceptlng the deacending water, thay fteguaotly present slicing floodi
am Iniurlng Taw lendg farther dowq Btresm. If tben are mills down stream
am tbe reserToir. they would eiidenll; be deprived of water for diiTinx Ihem,
iileaa a portion of that slored iu tbe resenoir be deioted lo tbi^ purpne.

oiDiHniftauon water i and tba rtaerToir, a tempBualHig one.

Arl. lb. DtatrlbDlInK reserTAlni. ErniQentlTaTaltayfltforaBlorlni

■«. *Tbit la calM, by w» ^ dIKIaolJoD, a AcMbHItod mFrvDlrl baoMfiff ftioD llLba vubth altw*
Kvii^ flawed lota it rrom Ue itorlai TaHFHlr.llirabaa iha loDf ««vIrP4" *blah «H»De«< tins, b
.triliiiMd ini.rloui Hmstloai iDrcnA tin Wwa.brmaai •({ho imel m^da", ar plpti. Tbia
nalt raMTTirfr flbgaid koldaMpal^aUBeiast atrfaatforafawdara; a frw wnU «oaH bi baltarj
ad (be end or tbe aappljrblin «bH ttrvleatn laiL ibmldlM pnrlded vLtb iTalve CtribBUIaa
raap^y pipe wliboatdaprivTav tba lain afwalar la tba BUB tiaa. inik aTlairwiaoh-rflpalni »
'411 a« to HODriDi eat aadlaieal frenl (a« ■rin>lT Tripe, ttila Tin ihoaU be pmrlded with OatwC
ralvea atTirlaai lewpnlnB >bm> Ibe enUra InlemlMlwMa UiamnHiToliii eipadalljal

In flxinr upon Ike dlimis «f pipe* fbr suppljing citiea, it la oeceaaanr
db«rlnmlDd~ibullftrtbtgreaiirpgra<ia -roae u B^nn' >tcld U utaiilr d»>a Itmh ib«
lonilA tb. diiilv ■uppLj ia muob iiiiUia M bour.. «iiia, durLD|>baW> laoiBec miulbi, uuoh
a«iawaairlaB«dlluiii(iortBgtli««lBlareBaa;aBdlblioooildmUoaa«o«illaMatUHiKiniJLa».

TABLE A. (Orlgina

I

PIPI

A

„

"i'bT

^T

i^

■3

™jj

i

s

n

;3

E

It"

11

ijis

ir5

iSSffl

ii5

1 ^

ij^

s^

'St

ailaS

»s

SSIf

a

s;i

lOHWO

654

WATER-PIPES.

It iB well to aUow in addition flrom ^ Inch to 1 inch, or more, (depending on
the character of the water,) to each diameter ; for deposita and concretlona.

The water, after reaching the city tbrongh one or more large main pipee from
the reservoir, must be distributed through the streets hj means of smaller
mains branching from the larger ones. The diameters of these smaller ones
also may be found by Table A. Thus, if a street, with its alleys, Ac, contains
about 6000 persons, flbe rate of head being, as before, not less than 50 feet to a
mile at any point or the system,) then we see by the table that a 10-inch pipe
will answer. Ft would be well to lay no city street pipes of less than 6 inchea
diameter.

Mains whicli eross eikeli otiiei* sliould be coiia««te«l at some
of tbeir Intemeetlons, to allow the water a more free circulation through-
out the entire system ; so that if the supply at any point is temporarily cut olT
from one direction by closing the valves for repairs, or is diminished by exces-
sive demand, it may be maintained by the flow from other directions.

Awoid dead ends when possible, as the water in them becomes foul and
unwholesome.

RuLB 2. With the same diameters^ differead rates <if head tpiU supply the propof^
Honate populations in column 3 of Table B, Or^ to find the diameters which at aiffereiU
rates of head wHl supply the same populations given in the last column of Table A^
, multiply the diameter given in Table A, by the corresponding number Id col-
umn 4 of Table B; or (approximately) do as directed in column 6.

TABI.E B.

(Original.)

•

OOL. 1.

OoL. 2.

Ck>i.. 8.

OoL.4.

Coi.6.

Bate of Head,

Rate' or Head,
compared wi:h
that in Table A.

Proportionate

Proportionate
Diani. to anpply

Kei&u*k0«

In Feet per MUe.

Populations.

the Popalationii
in Table A.

5

,1

.82

1.58

10

.2

.45

1.37

1*2^

.25

.50

1.32

Add one-third.

16

.3

.55

1.27

Add full one-fourth.

30

.4

.84

1.20

Add one- fifth.

25

.5

.71

1.14

Add one-aerenth.

80

.6

.78

I.ll

Add one-ninth.

85

.7

.84

1.07

Add onefoorteentb.

87>i

.76

.87

1.06

Add ooe-sixtecntb.

40

.8

.90

1.05

Add one-twentieth.

46

.9

.95

1.02

Add one fiftieth.

50

1.0

1.00

1.00

75

1.5

1.23

.92

Dednet one-tbirteentli.

100

2.0

1.41

.88

Deduct one eighth.

126

2.6

1.59

.83

Dednet tun ene-iixtta.

150

8.0

1.73

.80

Deduct one-flfth.

200

4.0

2.00

.76

Deduct nearly one- fourth.

250

5.0

2-25

.73

Deduct nearly two-aeTeiulia.

300

60

1 246

.69

Deduct three-tentha.

400

8.0

' '2.83

.66

Deduct full one- third.

500

10.0

S.I8

63

Example. By Table A we see that with the rate of head of 00 teet per
mile, a 30>inch pipe will supply a population of 915»0; but with three times that
rate of head, or 150 feet per mile, we see by column 3, Table B, that the saftie

>ipe will supply 1.73 times as many persons, or 91580 X 1.73 ss 158433 persons.

)ut if, at this greater rate of head, we still wish to supply only 91580 persons,
then we find in column 4. Table B, that we may diminish the diameter of the pipe

roin 30, down to 30 X .80 » 24 inches; or, by column 5, we have 80 — 6 — 24

nches.
Again, after the water has reached the citv by the 80-lnch pi le of Table A,
if we wish to distribute it through the city by say eight branches or smaller
mains, we see by column 6, Table A, that each of them must have at least 1^
inches diameter. From these eight, other smaller ones may branch off lnt4> the
cross streets, alleys, Ac ; and in estimating the supply required, for any partic-
ular street main, we must evidently add what is required also for such cross
streets, dbc, ^, as are to be fed from said main.

If certain limited parts of a city pipe svstem have considerably less rates of
head than most of the remainder. It may become expedient to supply the former
by a spt>cial separate main of larger diameter; which may start either directly

WATER-PIPES. 655

rom the TV«er?olr ; or m a bniiek from the grand taidliig aMin which feedi llit
>wer part8» aooordiog to clreamstancea.

It must be remembered, that although by IncreRsing the diameters, an aba»i
Ant supply may be obtained under a small rate of head, as well as under a great
me. yet the water will not rise to as great a height in the service pipes for sup<
flying the different stories of dwellings, Ac. Even with the diameters in Table
kj the water, under ordinary use, will not rise in these pipes to the full height
n the surface of the reservoir; and if an unusual drawing-off is goinson at
;lie aanae time at many parts of the system, as in case of an extensive fire, or
rrequently during the hot summer months, it may not rise to even one-half of
that height.

A.rt. 3. Tke followlsy bas been fonnd wery efltectiwe fbr
preventiniT eeneretions in water pipes.. Formerly in Boston, cast-
TrOB city pipes, 4 inehes diameter, became closed up in 7 years ; and those of
larger diameter became seriously reduced in the same time. But later, during
6 years, ilk which this Tamish was used, no concretions formed.*

C<»al-piteb wamisb to be applied to pipes and eastln«,
made for tbe Water Department of Pblladelpbia, under
tlie ffollowinv conditions:

First. Every t>fpe must be thoroughly dressed and made clean, free from the
•arth or sand which dings to the iron in the moulds ; hard brashes to be used
In finishing the process to remove the loose dust.

SeeomL Everr fripe must be entirely free from nist when the varnish is ap-
plied. If the pipe cannot be dipped immediately after being cleansed, the snv*
nee must be oiled with linseed oil to preserve it until it is ready to be dipped:
ao pipe to be dipped after rust has set in.

Third, The ooal-Ur pitch la made from coal tar. distilled until the naphtha
Is entirely removed, ana the material deodorised. It should be distilled until it
has about the consistency of wax. The mixture of five or six per cent of linseed
oU is recommended. Pitch which becomes hard and brittle when cold, will not
answer for this use.

fburlh. Pitch of the proper quality having been obtained, it must be care-
fully heated in a soitable vessel to a temperature of 800 degrees Fahrenheit, and
must be maintained at not less than this temperature during the time of dip*
ping. The material will thicken and deteriorate after a numoer of pipes have
Men dipped ; fresh pitch must therefore be frequently added ; and occasionally
the vessel must be entirely emptied of its old contents, and refilled with fresh
pitch : the refuse will be haid and brittle like common pitch.

F^lh, Everv pipe m nst atlain a temperatu re of 800 degrees Fah ren h eit, before
It is removed from the vessel of hot pitch. It may then be slowly removed and
laid upon skids to drip.

All pipes of 20 Inches diameter and upward, will require to remain at least
thirty minutes in the hot fluid, to attain this temperature ; probably more in
cold weather.

aieeth. The application most be made to the satisfaction of the Chief Engineer
of the Water Department: and the material be subject at all times to his ex-
amination, Inspection, and rejection.

Seventh, Payment for coating the pipes will only be made on such pipes as
are sound and sufficient according to the specifications, and are acceptable inde*
pendent of the coating*.

Eighih. No pipe to be dipped until the authorized inspector has examined It
as to cleaning and rust; ana subjected it thoroughly to the hammer proof. It
may then be dipped, after which, it will be passed to the hydraulic press to meet
the required water proof.

Mm, Tbe proper coating will be tough and tenacious when cold on the
pipes, and not orittle or with any tendency to scale off. When the coating of
any pine has not been properly applied, and does not give satisfaction, whether
from defect in luaterial, tools, or manipulations, it shall not be paid for; if it
scales off or shows a tendt^ncy that, way, the pipe shall be cleansed inside before
it can be reooated or be receivable as an ordinary pipe.

*llr. Dexter Brackett, of Boston, informs us, 1892, that while tubercles form
there in uncuated pipes to a thickness of about mree-quarters of an inch, rendering
4-inch pipes of Mttle or no value for Are supplj', yet no actual stoppage has been
known to occur from this cause during the twenty-three .vears of his connection
with the City Engineering Departmeut. He states also that even their coated pipes,
taken up after being in the ground for ten or fifteen years, are generally found to be
pitted on their Inner surfaces.

656

WATER-PIPES.

Art. 4. The pipes are laid to oonform to the rertlcal andulations of the street
surfaces. The tops of the pipes are laid not less than 3^ feet below the surface of
the street; but in 3->inch pipes the water has at times been Anozen at that depth.

In Phllada., in 1889, tbere were about 784 miles of street
pipes; or about 1 mile to every 1100 inhabitants. The population was about
860,000; residing in about 150,000 dwellings. Berlin, 1837-8; 1,400,000 inhab-
tants, in 20,000 houses (average 70 persons per house). Mean consumption per head,
17 U. S. gallons per daj; maximum, 24; minimum, 12^; all approximate. 25,000
wheel meters in use.

Ho calvanie action has been observed where lead pipes or brass unite witk
east-iroD ones. STo pipe less than 6 inches diam should be laid in cities; and
•van they only for lengths of a few hundred feet. Their insuffloiency is chiefly felt in
ease of fire. 8 ins would be a better minimum. No more leakage occurs in winter
than in summer; except from the bursting of privateTmrvic0^^M by freezing.

To compact the earth thoroughly against the pipes exoludas air, and greatly im*
pedes rust. Pipes may be corroded by the leakage of gas through the body as well ai
through ^he Joints of adjaeont (pas-pipes.

WEICIKT OF CAST-IBOir WATER-PIPES,

As used in Pliila^ and tested by hydraulic press before delivery to an internal
pres of 300 lbs per sq inch. This table includes spigots, and faucets or bells. The
pipes are required to be made of remelted strong tough gray pig iron, euch as may
se readily drilled and chipped ; and all of more than 8 ins diam to be cast vertically,
with the bell end down. Deviations of 5 per cent above or below the theoreti-
oal weights, are allowed for irregularities in casting, which it seems impossible te
avoid.

The pipes are in lengths from 3 to 3i< ins longer than 12 ft ; so that when laid they
measure 13 ft from the mouth,/. Fig 38, of one bell to that of the next.

Dlun.

Thiok-

DPBS.

Wtpar

length.

Diam.

Thick-
nesi.

Wt per
length.

Diam.

Thlok-
ness.

Wtper
lensth.

In*.

Ins.

Lbi.

Ins.

Tni.

Lb*.

Ina.

Ina.

Lb«.

3

"fk

158

16

%

1322

36

XA

4884

4

%

211

20

n

1654

36

IJL

4862

6

Is

386

20

H

1798

36

IJL

5866

8

'*

460

80

1

3313

48

^za

7282

10

^

667

30

.5

3G10

48

laz

86«7

12

890

80

1

3964

48

iH

9878

The followin$c sizes of lap-welded ivrong^bt-iron water-pipe are

made by the National Tube Works Co., McKeesport, Pa., and fitted with their
** Converse patient lock-Joint.'' One end of each length of pipe has the
lock-Joint permanently attached (leaded) to it at the works before shipping. The
. " weights per foot" include these Joints. The weight of *' lead per joint" ^ven ta
that required to be poured in laying the pipe, or that for one side only of the Joint.

Outer diam, ins 2 3 4 6 6 8 10 12 16

Weifflit per ft, lbs 1.86 3.48 5.26 7.33 8.76 13.20 17.08 2&.12 47.7t

I^ad per joint, lbs % 1^ 2}4 ^14 ^^ ^ t 9^ 1M

Average car load :

Number of lengths 800 880 275 146 126 128 80 56 4ft

** ** feet 11500 6600 4500 2600 2000 200Q 1200 800 630

The pipes are tested for a bursting pressure of 500 lbs per square inch, or higher
If desired. They are furnished either coated with asphaltum, or *'kala*
meined;** or, if desired, first kalameined and then coated with asphaltnnu
Kalameining consists In ** incorporating upon and into the body of the iron a non-
iorrosive metal alloy, Utrgely composed of tin." The surface thus formed It ool
•racked by blows, or by bending the pipe, either hot or cold.

WATER-PIPJfiJ. n r 7

b* Joints or oonpltnir. Is of cast'iron, and h*s iiit0rDal recesBM which recslTs aii4
I lugs on the outside of eacii length of pipe, near each of its ends. The joint Is
1 poured with lead in the usual way (see next page), eitlier with clay collars, or
1 a special pouring clamp furnislied by the Co. This clamp resembles the
inter,** Figti '69 &c, except that it is in two rigid semi-circular pieces, connected
>ther by a hinge-Joint, and ftimished with handles like those of a lemon-squeezer,
has a hole in one side for pouring. The coupling forms a flush inner surfacs
ti the pipe at the Joint, thus avoiding much of the resistance of cast-iron pipes
loMT. For oases where it may be necessary to make frequent changes, the coup-
^ are made in two pieces, which are bolted together by flttnges.
¥roiiarl>^t'Oii» for pipes, has the great Hdvaniaipres over cast-iron
lightnees, toughness* and pliability. The lightness of wronght-iron pipes ren-
3 them easier to handle, and cheaper per foot notwithstanding tluit their cost per
is about 25 p«tr cent greater. They are not liable to breakage in trunspoitatton
from rough handling, and they miiy be bent through angles up to al*out 25^.
iy therefore require no special bend castings for such angles. The National Co
iply bending machines, to be worked by two men. One machine can, by changing
I dies, be used in l>euding all sizes of pipe. The pipes are in lengths of from 15 to
feet, instead of 12 feet, as in the case of cast-iron, so that fewer Joints are

aiiire«l per mile.
e Co furnish special ''serviee clamps*' and tapping machines for attaebinir
rvice pipes to mains. This may be done (as in the case of the Payne
ichine, while the main is under pressure. The service clamp is a cast-

n saddle, which, before the main is tapped, is attached to it by means of a U
It, and which remains permanently so attached after the tapping. A sheet-lead
sket is placed between clamp and main. The clamp has a tapped cylindrical
lining through it, into which the corporation stop is screwed before

9 pipe is tapped. The drill of the tapping machine passes through the stop, and
rough the cylindrical opening in the clamp, and drills through the lead gasket
d through the side of the main.

J'he Co furnish also pipe>cntting machines, and special castings (reducers,
Dsses, Ac, Ac) fitted with the Converse Joint.

A rt. 9. Wronn^tit-iron pipes eorrode much more rapidly than cast.

A fpatta-percha plpe^ ^ inch thick, and % inch bore, has sustained safely
Internal pres of more tban Sob fts per sq fneh; equal to nearly 600 feet head. li merely swelled
gtally at 387 Ib.t. In 1851 a tube of tliat material. 2yi ins bore, about h ineb thick, and 1850 ft long,
18 Aunk in the East River, New York, to carry the Croton water to Blackwell's Island. It was held
WD by weights. It proveo nnsaUsfaotory owing to abrasion cauned by tidal currents, and Injury
)m the auchora of dragging vessels. A wrapping of canvas, confined by spun yarn, was useful ia
eventing the former, but not tbe latter. This pipe was replaced In 1870 by wrought-lron pipes.

Ball's patent iron and eement pipe, is made by The Patent Water

id Gas Pipe Co, of Jersey City, N. J. It Is formed of nveted sheet-iron, and eaeb length Is dipped
to, and opated with, a hot mixture of coal t&r and asphalt. Tbe lining of hydraulic cement is then
)plied. This ranges, In thickness, ft-om H inch for I2-inoh pipes to 1 inch for 30-inch pipe.. Thia
.pe is made up to diams of 86 ins. It Is laid In a bed of cement mortar, and completely oovereid with
le same. Suitable means are provided for making all tbe attachments, Ac, required In city pipes
ir water and gas. More than ISOO miles of It are In use in various towns, some of it for 35 years ;
ad It appears to give general satlsfhotlon. Tubercles do not form In these pipes, as tbey are apt to
» in east-iron ones. There is every reason to suppose that they are durable. The trebohea being
ug, the Jersey City Co furnish pipes and lay them (iucluding the eement).

A. WyckoflT k Son, Elmira, N. Y., make wooden water pipes. For
ressures of 15 to SO lbs per sq tneh, they furnish either plain pipes, 8M to 7 Ins souara externally,
nd from IH^* ios Internal diam ; orround pipes, 1 inch to 16 ins bore, coated SKternallj wiib
sphaltnm cement. At their ends, both the square and the round pipes are banded with iron. For
iressure* from 40 to 160 Iba per sq Inch, the round wooden pipes, before being coated with eement,
re spirally wrapped, bv steam power, with hoop Iron, which Is first passed through a preparation
if coal-tar. The iron is wound so tightly as to be imbedded In tbe pipe, leaving lu outer surface
lush with that of the wood. The ends of each length of pipe receive extra banding. The a^haltum
«menteoating is then applied. These pipes have been extensively and auocessfully used for both
rater and gas. Sniteble arrangemeute are provided for Joinu and connections.

Water pipes of bored oak and pine lovs, laid in Philada 60 to 60

rears ago, are frequently .quite sound, and still fltfor uxe, except where outer sap wood Is decayed,
i^hen this is removed, many of these old pipes have been relaid In faoteries, Ac. Clay well packer
iround wooden pipes, excludes tbe contact of air, and thus contributes greatly to their durability.
Loose porous soils, such aa gravel, Ac, on the contrary, are unfavorable.

Pipes made of bitnminised paper, prepared under great pressure,

have been used for both water and gas. They are much less liable to break than oast-Iron, and ds
Dot weigh or ooit more tban about half as much. Pipes of 6 ins bore and H inch thick, have resiatel
tent strains of 330 lbs per sq iueh { equal to a water bead of 60T fU

42

658 WATEH PIPES.

Coata of wMer pipe luid Inj'InK. Tbe Iblloirliig BviirH us dedond
train > [able kiDdl; RirDtiihed b; Mr. Allen J. FuUar, Ocuenl SuperlntenflanL
Burr^u ot Wiler, fblladelptilii. Tbe; renietent iierage canditlODi far straiibl
^pe, Uld in einb, in lIui citT' TbecoBi.in au; giieu cuse, lu; differ msierullj

"LaviDg" lacli'idei iJl bundling of msMrUlu, iitler tbeii dcllvarr on tho
ground, lor liying tliem in tho treneh, making Jnliil*. calking, elc, Calken
rw«lve KM, lend mun n.oa, and laborers tl.TS nei da; of B bniin.

TbecwIBof miurlaisaretakeiissrallows: Plp«<vIiDga,l.3cta.; lead, Seta,;
gaskel, aS^eU,; cuke, 0 27 cla., per ponDd; blocking, 1.7 cts. per ft. B. M.

■loiiallon. add'ltlonaL deplb required for Irencb, wear and tear of tooli, and
otdlnarr repavliig. but not incliidtug damsces, mpball repaiiag, or trestUng,

Diameter or pipe 4 < 8 10 ' I! 16 (o «8 Inches.
Add W ;o W 60 SO 40 per cent.

Flp«.

t™,i

Earth irork.

per llpaal foot.

Slua.

Thlek-

W,..b,^,.

EiciYUian.

Back fill 1
■nd '

Top.

Bot-

"(At'

Cubic

"ZX'", ^"^

Id*.

Ina.

yardi

»

^

%

2.M

2Si

4.50

1

1

Ibo

460

0.0»

all

8

i«

7.00

4.50

6,S0

1.47

oIm

0,»

0.M

V4-TBB PIPES.

esd

JSUvuxQ, Laying, RscATitvukTioV'

Pipe.

H

lua.

Hauling

1

per

ineal foot, at

75 cents per ton.

Laying,

per

Ifneal

^S 1

i

• g s

i

foot

s

s .

s

«

9

9

0.01

0.01

0.02

0.03

0.01

0.01

0.02

0.04

0.01

0.01

0.02

0.04

0.03

0.01

0.04

0.05

0.04

0.01

0.05

0.06

0.07

0.01

0.08

0.08

0.0d

0.02

0.11

0.08

0.11

0.02

0.13

0.08

0.13

0.02

0.15

0.08

0.16

0.02

0.18

0.09

0.22

0.03

0.25

0.12

0.26

0.03

0.29

0.12

KecapitulatioD of Cost,
per lineal foot.

9 60
«5

»

•

$

0.24

0.11

0.41

0.11

0.65

0.11

1.00

0.16

1.64

0.20

2.63

0.30

8.65

0.40

4.38

0.40

5.10

0.62

6.17

0.52

8.68

0.99

10.10

0.99

0.02
0.02
0.02
0.04
0.05
0.08
0.11
0.13
0.15
0.18
0.25
0.29

3-

0.03
0.04
0.04
0.05
0.06
0.08
0.08
0.08
0.08
0.09
0.12
0.12

0.40
0.58
0.72
1.25
1.95
3.09
4.24
4.94
5.85
6.96
10.04
11.50

660

WATER-PIPES.

Art. 6. Cast-iron Pipe Joints. Philadelphia standard. The clear
distance, d, between the spigot and the faucet, is nearly uniform for all sizes
of pipe, varying only from A inch for 4-inch
pipe, to /b inch for 30-iDch pipe. The depth,
m n, of tne faucet varies from 3 ins in 4r-inch
pipe, to 4 ins in 30-inch pipe.

The small beads at « and m, *' and mf on the
spigot end of the pipe, project aboujt }^ inch ;
and prevent the calking material from entering
the pipe. The calking consists of about 1 to 2
ins in depth of well-rammed, untarred gasket,
or rope yarn; above which is poured melted
lead, confined from spreading by means of clay
plastered around the joint. The lead is after-
wards compacted by a calking hammer.

The lead is poured through a hole left in the
clay on the upper side of the pipe. In large
pipes, two additional holes are left in the clay,
one at each side of the pipe, and lead is first
poured into the side holes by two men at once,
one man pouring into each side hole until the
oint is half full. The side holes are then •

stopped, and, after the lead already poured has iu-uu — .scale c)F IN9HE8, ^__^

hardened, the two men finish .the pouring by Ol*8*o«78
means of the top hole. This course is necessary, Fie 38

because the great weight of melted lead in the

entire large joint would press away the clay at the lower side of the joint, and
thus escape.

The moisture in the clay is liable to freeze in cold weather, and to render it too
hard to be used. It is also liable, at all times, as is also any dampness in the pipe,
to be converted into steam by the heat of the melted lead. The steam sometimes
breaks out, or '* blows " through the clay, allowing the lead to escape.

Art. 7. The Watkins patent *' Pipe Jointer" avoids these difficulties by
dispensing with the ring of clay. It consists of a ring R, Figs 39 and 40, of square

cross-section, and made of packing composed of alternate layers of hemp cloth
and India rubber. This riug is encircled bv one or more thin strips of spring
steel, which are riveted to it at intervals, as shown. E E are iron-elliows riveted
outside of the steel bands. After the gasket has been rammed into its place, the
ring is placed around the spigot near the faucet, in the position shown in Fig 40,
and is held loosely by the clamp, Fi^ 41, one point of wnich enters a small pit in
each of the elbows, £ E. The ring is then, by means of a hammer, driven close
up against the end,>^ of the faucet, Fig 88 ; the screw of the clamp is tightened
somewhat, so as to oring the ring close to the spigot; a small dam of clay is
placed in front of the aperture between the two elbows, EE; and the joint is
ready for pouring. After the lead has hardened, the "jointer " is removed, and
is ready for use at another joint. Upon its removal the lead is foand snfooth,
requiring no chipping. One can be used for several hundred Joints. They dis-
pense with the services of the men who prepare the clay collars, and supply theoi
to the pourers. Thos. Watkins, Johnstown, Pa.

WATEB-FIPE9. CGI

rt.fl. As a riirther prerentiie sgslDst (he escape of Any of (be KM'

; InM Uie pipe. » ring of J«ad pipe is BomeiiiiieB placed in Hie joint
reiiiegsBtet isinseried, Tliis lead pipe is of sued diamcier that itcaaiust
uibcil ltirou)ih tlie space, d. Flc 38, betHnea tbe spimi'iid Ibebuoet; and
ieli length S2 Just la eiicirolo tSe water-pipe. It la driven »a closelj as pos-

Dd tbe lead puiired, as ueuaL

rl. ». In John F. W»Fd'a flexible Joint, Fig. i2. tor cast-iron
1 Utid asrosi llie irregulu beds of utreaiuB, a poriioD. i o, uf tbe inside of tbe
B ti accuntelf turned loform tiie middle zone of a epbete. wiili venter at C,

Ke former. The lead ia poured
e two ad]a«ul lenitha of pipe,
aa Hiluble Ttfiela or B«iU,
■tialght*

and is held in pla« on the spitrol

J ,._jj ...... ^^j jo

ut funhereara

Suitablaipparatuiisuied for lowering

large plpea Into deep water wiihou t dd-

dueatrainonthejornts. Tbejointper-

mils a deflation of IS^ «b shown ; Tjut

further dtaection, which would be iia-

" ble to split tbe bell, is pre»eoted bj the

v|— A9_ Stops at eo on the bell and vv on the

^' spigot. In some oiaee prelim in arydredg-

iST be eipedient, (o dimlniah abrupt irregularl Lies of the botroni. Over

' lines of pipe furnished with this Joint bays be«a snccewfull; laid, of

t. lo. In Figs 43, A is s double bmnclii which la ■ pipe having, in

pe is m Blnsle branch. Tbe pipe is etronger when these eili-a faucets
arilaend than if tber were atiUmiddle. In a longline of pi|>es, for the sake
edltlon,dia'erentgangsof men are frequently laying detached portions some
ea apart; and when two endsof diBferent portions are brought near enough
er to be united, as h and r, Fig C, their Junction cannot be efiteled hy l£e
ipigot-and-fancet joint. In iblscaBesfaat-lronBleeve, ( ' '

Fiir>* o

If the crack ia too long, or otherwise

e~U broken to plecee : and'the lead joinla at iia ends melted out, so as to
r its removal. Tbeo, since an enllrB Hew pine cannot now be inserled,
o the overlapping of the spigot-and-fancet ends, two short pieces must be
ited for it. One end of each of these is lead-jointed to tbe pipes already
hile the other two ends, which will probably be a few incbcs apart, are

662

VATEB' PIPES.

Lnte4 be temporaiilr repaired Jd an eiii«r^fliicj, by ■ wrapping
aBpFrnlbandingof thin boo- P"" r «" V

parts, B S, Fig 4*, and cianjped locellier br acrei
'( pipes

e fur

[ Fig. -44 1

ling by s.

s pnasEng

fiirnlshed "llh flBnges, m m, bnlted over tbe opeuini,. .

through female itrewi lapped in the thickness of tbe pipe. If Ilia new pipe la

be made ofat, vith the longest diameter in tbe direction of tbe UnffUi of tiia

Art. 13. Air TBlvea. Air Isapt lo culiect graduallV at the higb pOinli

Bow. Thismar bepreTented byanalrvaiTS, P'lg44A. This eoDellts of • niM-

Iran box, ccdd, oonflned to the main pipe mm, by screw-bolts puslng tbron^

posed 'in tbe Bg; or Ac. 'Ibis float has a epindle'or item it, faetto it; whicb

to rise and Tali ^ly, but preventliig "f™" moving'sidewaya. Wheii the ^pe

elem "i bas'tixcd to it a vaire"v, whicb i^sea and^fsils with It and Ib° Boat.
Suppose Ihe i^lrc m m to he empty, and consequently the float and the vilto n
down. Then, If water be admitted into the pipe, it will rise and ail also tbe box
airarupasr; tnd In doing aowililirt tba float/, and iheTalvei.Is tbepoajtion
In the &g ; thus preTenling egreas to the outer air by ciosing the opening at »,
Now, atr carried along by tbe waUr, will, on accoant of Ita jigtataew, aacand (a

WATEB-PIFBB. 663

ence, when such air arrives under the opening aa, it will rise throueh it,
ascend to «; the closed Yaive preTenting it from going farther. Thut
tessive portions of air ascend, and in time accumulate to such an extent
gradually to fgrce much of the water downward out of the box. When
takes place, the float, which is held up odIt bf the water, of coarse de-
ds also; and iu doing so, pulls down with it tne valve v. The accumulated
hen iostantlf escapes through the openings at v and n, into the atmosphere ;
the water in the pipe mm, immediately ascends again into the box, carry-
with it the float ; and thus again closing the valve v. The valve, and tM
e^eat e, are faced with brass, to avoid rust, and consequent bad fit. The
le is protected by an iron or wooden cover, reaching to the level of the street.

Ir valves are no loniter lised in city pipes; their place being
ilied by the fireplugs at average distances of about 150 yards apart. Thes^
g placed as much as pousible at the summits of undulations In the lines of
i, for convenience of washing the streets, and being frequently opened
bat purpose, permit also the escape of accumulated air.

fee escape of compressed air tbronsli an air Talvey or
er opeiilngr,.bas been Ikuowfi to prodbee liarsilns oC tlio
m pipes; for the escape is instantaneous, and permits the columns qf
r iH the pipes on both sides of the valve, to rush together with great
8, which arrest each other, and react against the pipes.

r»Te8seis« Motion Is Imparted to the water in a line of pipes, by the
iftd stroke of the piston of a single-acting pump: but during the backward
e, this motion is stopped ; and the water tn the pipect comes to rest. There-
at the next fbrwara stroke, all the water has to be again set in motion ;
he force that must be exerted by the pump to do this is much greater than
1 be required if the motion previously imparted had been maintained
ig the time of the backstroke. The addition of an air-vessel secures this
tenance of motion, and thus effects a great saving of power ; besides dimin^
i the danger of bursting the pipes at each forward stroke. It is merely a
nd strong air-tight iron box, usuallv cylindrical, strongly bolted on top
e pipes just beyond the pump, and communicating freely with them
gh an opening iu its base. It is full of air. The forward stroke of the
1 then forces water not only along the pipes, but also into the lower part
) air-veasel throogli the opening in its base; thus compressing its con-
1 air. But during the backstroke, this compressed air, being relieved from
'essure of the pump, expands: and in so doing presses upon the water in
pes, and thus keeps it in motion until the n^xt forward stroke ; and so on.
r-vessel also acts as an air-eushion; permitting the piston to apply its force
i water in the pipes gradually: thus preserving both the pipes and the
from violent shocks. The air in the vessel, however, becomes by degrees
led and taken away by the water; and its action as a regulator then
. To prevent this, fresh air must be forced into the vessel from time to
!>y a condenser, or forcing air-pump. A dovbU-ading pump does not n
need an air-vessel. Tbere is no particular rule for the size or capacity of
»el8. In practice it appears to vary from itbout 5 to SO^tijnes that of the
; with a height equal to two or more times the diameter. A stand-pipa
slow) is sometimes used instep of an ilir-vess^I.

ton€l-pipe is aoifietira^s used for the ^ame purpose as an air-vesftei (see
. It is a tall pipe, open to the aii* at top; and comthunicattng freely at
t witb tbe water<pi|)e. in the same manner as in an atr-ves^el. Its top
>e aomewhat higher than that to which the pump hds to force the water
:b tbe system of pipes; otherwise the water would be wasted by flowing
8 top. The area of its transverse section should be at least equal to that
pipe oc pipes which eonduet the water /rom it; hot it is at times better
a it macb larger, as a stbnd-pipe may then answer, especially ih a smal|
w a re^f^rwir, if the pumping should cease for a few hours. A stand-pipe
be cylindrical, not oonickl ; for if thick ice should form on top of the
lo 8 conical one, a sudden forcing of it upward by the pnmp might strain
knd-pipe aerioualy. The stand-pipes connected with the Philadelphia
Worka are from 125 to 170 feet high : 5 fbet diameter ; and made of riveted
iron about % inch thick near the base, and about ^ inch near the top.
ave no protection from the weather ; nor are the^ braced in any manner ;
ain their positions by their own inherent strength, although axpoead at
o violent winds.

WATER-PIPES.

Art. U. Tbe ■«rvl««-Blpes for Bappl^lac alncle dwelllDitai

B n. Fig 4S, by H IrnMB Rrrale,/, hen abowD st

IK It abon 1 1^ ioa ; snd Ihe joint aoldend, I.
IWckmw MBt/, iiforgiiintt propm ibapi

AfI. 13. Tlif »-«Ited '> corporation fltopB" or " corpomtlon cocki*

■n inHrled into tba pipe by * apecGil mscliine, ?Ib 4cI. Their gnat adienlage onr

ther«Tule,Tlg4fi,laihuUicycBDbelnHeri«dlB(oftplpe wben(k«

■alter Is fnll of water nn-" ^-- n..ij_

tn^ter H. Pa J Be £
Co., Foelorlt, (Mo. EuA

Tb« cIuId I> tlghUnw
•r,CC(H>la*hlcjb«ai

C>L bflmda>mefl to tbe prvper pcHltlon.ltlH H(Dpp«il bj
■ IDE Inilds oT tbe evl. The drttl le Ibeo ImmedliiwlT OTsr the center or i let^a
cirFutar opeatng In tbe base of Ibe cyl.C G, uid o>er a elmilar Dpeaing, Ihrtmgh tba

touchc'ijaldplpe. The tatobet-wreBoh, W W.li then ist dd tbeiquarn head of the
drill-ahsnk, K; the feeder-Jake, T, witb feed icrew, F. li pat In pDaltion u ibonB ;

.epip^lfn,

idb7<r<

worklagtbe latter, tba tap l> nowwlthdnwD from the taole.bnt nmalDdn IfaecTt
Thecyl hcadli dow reTOliedeoai Id raven* lb* pnaltloDiaf Sand It tbe lug In-
altle of the cjl itopploff the bead wben tbe rt<ip li Immedlautj OTer tba bole. By
meana or the ralchet-wrenpb, applied 10 the Bqaam hud of tbe mandrel, H, the atop^
S(tAe vuiH iiftcMch mmilbeclctid}. it aowKtewed Into the hole.bDI only far anonth

WATEK-P1PE8.

665

The mandrel, M, is made In two lengths (one of which screws Into the other) I|

ier that the upper part may be out of the way of the wrench-handle while drills

r. It has three or more diff threads at its foot, to suit diff sizes of stop. Stops,

Mte to Btiit the machine, are ftirnished as wanted.

The naachine can work in any direction radial to the pipe, and can therefore be

ed for tapping: a pip© in any part of its circumference.

After tlie stop is inserted, the service-pipe is atrached to its outer end by a coup-

ig nut pausing over the threud there shown.

The machines are guaranteed to tap under a presBore of 600 lbs per^quare inch.

Art. 19 au Tlie pneumatic dome Figs. 46 a. and 46 b, invented by

r. N. Monroe Hopkins, of Washington, D. C, is designed to prevent the burst-
iK of wat«r-pipes in freezing weather.

In unprotected pipes, the water, in freezing, is unable to expand longitudi-
allv, and therefore frequently bursts the pipe in expanding lat-
raliy. The domes, being placed in the pipe, as shown, at intervals
f about 12 feet, where freezing is to be apprehended, permit the
>ngitudinal expansion, which pushes the ice in both directions
3ward each dome, where it compresses the air-cushion there pro-
ided. In the horizontal dome. Fig. 46 a, the double inclined planes,
ast in its lower side at c, compel the two horizontal columns of ice
o rise into the dome, instead of merely abutting endwise against
ach other.

In order to insure that the domes in a system
such as those for a house or mill or on a bridge)
ihall not be deprived of their air by the flow
>f the water in the pipes below them, an in-
ipirator is placed in the pipe at the entranre
to the system. The inspirator consists essen-
tially of a constriction in the pipe, which in-
creases the velocity of flow at tnat point, and thus causes an indraft of air
through a valve provided for the purpose (see Venturi meter, page 532). The
air, thus introduced, is carried along the pipe in bubbles between the surface
of the water and the top of the pipe, and is entrapped by the domes. When,
by closing faucets, etc., tne flow in the pipe is checked, the increase of pressure
closes the valve.

Severe tests of both large and small pipes (4-inch ^d ^-inch), protected by
these domes, have shown them to be always effective in preventmg rupture.

Fig. 46 a.

Fig. 46 h.

BTOP-VALVES.

10 priiictpiil cnilEngaohEch compose Ibe boi or cover nni lialted togMbrr by
m»ns of nt,vgm, g. Thoiolnt (ata of Ihi cMllngs u-e CBrefully gniMiUitd ; wd >

frcm doting p«tftcUy. Tho lalTe g»u tn bcod nftb Babbitt tneUI. Althtlop
■t tbu pulDt, ytry cimful workmanBhlp is required tbrovghont.

8T0P-VALVEB.

Ban.

WL

Bore.

Ibg.

Bore.

Wt.

Bora.

I*"

2

a2

10

19S

13
18

800
843

loeo

U75

SO

i

t. 17. FIj 4* tbon ■!■ .n
irerlDg ths ntre. Htra ihe

idpmeot Willi oulaM«

Ihro ugh «b tell Ibe

ItlM

cunnot move Terlitilly.
Art. 18. A ronr-#Br stoker foDr-wa|^

«M. TbenDtlil
lU it, BDd th* wl

top, or foDr-waf

tbeitrsw. ^'"^ "P K

OMnlnxwbJcb leadi to tbeflr4>.h jdriLnt, the t4It« It not
Ibble to cloKgioi tbrongh tbis od». The fln-hj-

aijiB « fulltr supply thMn would be pi*

BTOP-VALVES.

AFt.19. Wli*t««rlho It, -„_ _

prol«c(ed b)' a surround I nc bwx.

le ^tHBf eflpedmlljof

rJi6D (Uuch«d to th« plM
(s of the plpet lo or iVom

u the widtlii of Itie

ach mlla at pipe ; or 1 1

:. Flre-plnK*, Figs

»n. rir^-plncit.F^

Flv« Hjdraiita.

FIEB-HTDBANTe. '

Art.30. PlgaSrepreBnUftcommoii street flre-pIacorlirc.brdrH

Saij|!i'. I, on the rod.aixl Itina ulluw

open louer esd or the tVost-
Jneket. Sj. "bich i. a hollQw

TlicwiMer,l«ft in iha bidram cMsiifUircloBinetlia'a1<B,e««apeBtti
■ eiUndrlul hole, d. ntoul % Inch diam, bored through the sulip.s. Bud e
Tilihoi.isBlinchlhluto tiBJuil sboie Iho lopof Ihe Loom plaW,p wbl

ETM^iililBetFnthsiXicMtlDV When lh™il.'» LSna'\^t\!i"bfhol
■loHd I17 the |iUte,)i, undraoiHiimiotuitll Ihenlta l>>c»ln fliitJnl]isloHd.

670 TB8T BO^lINOp.

Test Borings. Fij^ 1 and 2 show a tool for |»orlns |n|o |M»Ufl9

elay^ Mutd, or gravely eren when quite iDdurated, or when frozen, ft will

not bore througti nard rock, or ttiroash large bonlders. It
consists of two sheet-iron cylindrical segments B 8, called
'' pods,'* having their lower or cutting edges shod with steeL
These edges project (as shown in Fig 1) Iteyond the sides of
the anger, and thus make the hole larger than it, so that il
cannot 6ind or stick. The two cutting edges are equidistant
. from the Tert cen line of the tool, and this insures a straight
I and vert bole. At a the auger is attached to the lower end
of a Tert boring rod compost of a number of l}^-inch sqnara
iron bars, or 2^-inch iron tubes, about 10 to 15 ft long;
jointed together at their ends by means of square sockeV
joints. At the top of this boring'rpd is a swivel-hook, by
means of which the entire apparatus is hung to the end of •
rope, which passes over a pulley at the top of a derrick or
tripod, and down tu a drum worked by a windlass and geaiv
Vg. By means of this drum and rope, the anger and boring-rod (which at first con*
sists of only one bar) are lifted, and suspf'nded over the intended hole. The auger
is then lowered, and rotated hor by two lueu or one horse, working at the ends of
levers wliich grip the boring-rod a few ft above the ground. The swivel at the top
of the boring-rod permits this rotation to take place without twisting the rope.
The shape of the auger is such that its rotation feeds or screws it into the g^und;
and the man at the windlass has, during the boring, merely to keep the rope tight^
so as to prevent the auger from boring too fast, and becoming clogged. In about S
revolutions the auger fills with earth. By means of the windlass it is then raised
to about 2 ft above the ground ; and by unkeying and removing the band b the auger
is opened like a pair of tongs, and the earth emptied into a wooden box which has
In the meantime been placed over the hole. The box is then remored and emptied,
and the boring proceeds as before. When the boring has reached a depth of about
10 ft, a second bar must be added to the top of the rod. For this purpose the rod
and auger are raised a few inches; u slight frame-work of boards is placed on the
ground, close to the boring-rod and surrounding it; and a flange is clasped tightly
to the rod Just above, and close to, the framework. The framework and flange now
support the rod and auger; the swivel-hook and rope are removed, and attached to
the npper end of the second bar, which is then raised, and its lower end is fastened
into the socket-joint upon the top of the flrst one. The rope is then drawn tight;
the flange removed ; the auger lowered to the bottom of the hole ; and the boring
resumed. Additional lengths of boring-rod are attached in the same way firom time
to time, as required by the descent of the auger.

The borers may be made from 6 to 18 inches in diameter, or larger. If desired,
the boring may be made from 24 to 36 ins diam by attaching a reamer to the
auger. This uuger will bore to a depth of 100 ft or more at tlie rate of from 6 to
20 ft per hour, it removes stones as large as half the diam of the hole. In dry
soils a bucketful of water is poured into the hole each time the auger is raised.

This burer may be advuntugeously used in boring the holes for sand piles,
and at times, instesMd of drivings wooden piles, it may be better to
plant them (butt down if preferred) iu holes bored by this auger : ramming the
earth well around them aiterwards. This will save adjacent buildings from the
jarring and injury done by a pile driver. .

If sand, mud, or loose gravel is reached in boring with this tool,
the hole is reamed out 4 ins larger, and a tnbiniT o^ ii^c^ boards is inserted
into the hole, and driveu into and through the sand or gravel, which is then
removed from within the tubing by means of a sand-pump, consisting of a
hollow iron cylluder, about 5 ins diam X 30 ins long, with a valve at its fooL
opening upward. The sand-pump is lowered to the bottom of the hole; covered
with water to a depth of 2 to 4 ft, and churned ouickly up and down 4 to 6 ins,
by hand, 20 or 80 times, during which the sana fills the pump, which is then
drawn up and emptied. From 10 to 20 ft in depth of sand, mud. &c, per hour
can thus be taken from a 6 to 18-inch hole. This puthp is also used for removing
broken earth. &c, from a hole bored iu compact earth by the borer first described.
Tbe cost, with derrick, boring-rods, rope, sand-pump, &c, dc, complete, it
about $175. The angler welehs from 150 to 200 lbs, according to site.
Boring-rod IJ^ ins sq, 3)^ lbs per fx. Derrick, 150 fts.

The sana-borer, r igs 3 and 4, like the sand-pump iust described, is used
inside of tubing, and for the same purpose, llie hollow iron cylinder C. 10 ins
diam X 30 ins long, slides vertically on the rod, but the screw is fast to the rod.
While boring, the sand below and around the cyl keeps it in the position shown in

ABIESIATI VfEhh BOBINO.

ilntlona of Ilia nxl (sd K

g cft wllh MBd. Tberodlsthea

.--described. _-,_--

Steel proa|>eGtliis aucer*, from ! to 4 Ins dUm. and 2 ft
JIB, are used for borinn holea from 2Ji to 6 Ins dinm, and u, depths
r 10 to 50 n, iQ<o claj-, sand, or One Kr"v«l, of bII ot »hich
HIT brlug up Buuplea. Thev ve turned b; wreuchee, «ud b; man
r bone power, bee also p 874

The boplns teol ahewn tn vert section by Flv ■-

sd Id liar cron Bsdian by tlf fi, l> idrj literal (ttr borlPB Bni
Loir holes by hand tbronrh SDrAiee miila. cis

irlnglng up Moiplet. The borer proper conBlsla ofa cyllndpr - ■

■ - - 'oiigth of gM-p'ip* "hich Mrit ,,^. ,,_._,_

e iSH itriket the Broiind. thr beteled ihatH of iw ciiltlnKedf^-' , '■iSS'"
I .Itehllj.aod-hfn the Jo-nw«rd pr« I. ™5/"*'i3&C

-'iJ^S^^

'Nh Mtb Ih« lowHl ono ots HTli

Tfii» io length trmn 26 to 60 ft, see

BF tao Iw UiHii Ihit'ftf th» hnl.-. ll'/'Z
memtorTi alvsyi ■"ruiwwckH." Flg4 .
■r tbt iDDportlng ropa coble jh Htuiht '
■ h«t lever, wlilch, lij meiina ot ■

(MUa|4dc> ^ Ui» bit U lU lower cod,

inutelf lifted trom i to

672 ARTESIAN WELL BORING.

let fiill, from 30 to 50 rimes per miuute, and so drills Ha way Into the rock or eartk*
From 4 to 10 ft in depth of water are kept in the hole, to facilitate the drilling an4
the removal of debris. After water is reached, tlie drilling may be continued, evev
if the hole is full of water ; but a great depth of water of course diminishes the forot
of the blows of the bit. A suitable arrangement must be provided for paying
out tll<» rope as the boring tool descends. A clamp is attached to the cable ;
and the man in charge, by turning the clamp, twists the rope, and thus turnfli
the bit horlBontally about oue-fifth of a revolution after each stroke, until
six or eight complete revolutions have been made in one direction. He then re*
▼erses the motion, and makes an equal number of turns, at the same rate, in tbe
opposite direction.

After drilling a few feet, the string of tools Is lifted out of the hole by means of
the cable, to allow the removal of (lie debris which has accumulated in the
hole. This is done by means of a saud-piiinp, which is a sheet-iron cylinder,
say 4 ins diam, and 4 to 6 ft long, provided, at its foot, with a valve opening upward.
The pump is lowered to the bottom of tbe bole, and filled with the mixed water and
debris by churning it up and down a number of times. Sometimes, in addition to
the valve, the pump is fitted with a plunger, which is at the foot of the pump when
the latter is let down to tbe bottom of the hole. The plunger is then drawn up into
the pump, and the debris follows it. In either case, tbe pump, when filled, is lifted
out of the hole and emptied; the string of tools is again lowered into the bole, and
the drilling resumed. The debris' must be removed after every 3 to 5 ft of drilling.
Otherwise it would interfere too greatly with the action of the bit.

Wells are usually drilled fVoni 6 to 8 ins diam. For dlams less
than 6 ins, the touls are so slender that they are liable to be broken in a deep hole.

The same apparatus is used for drillinfr tlirouf h the earth above
the roek, before the latter is reached. This is callea'* spudding." In this case
the sides of the hole must be prevented from caving in. For this purpose a wrought-
iron pipe of such diam as to fit the hole closely, and ^ inch thick, is inserted into
the hole, and is driven down from time to time as the drilling proceeds. The pipe
is driven by means of a heavy maul of oak, or other hard wood, 14 to 18 ins square,
and 10 to 16 ft long. This maul is attached, by one end, to the lower end of the
same cable which, during drilling, supports the string of tools. It is thus repeat-
edly lifted, a:;a dropped upon the head of the tube, which is protected by a cast-iron
** driving-cap." The foot of the tube Is shod with a steel cutting-edge ring, or " staol
shoe." When the tube has been driven as far as it will readily go, the maul is re-
moved from the end of the rope; the string of tools substituted; and the drillini^
resumed within the pipe.

The pipe is put together in lengths of from 8 to 18 ft, and the drilling and pipo-
driving proceed alternately until the rock is reached, and the foot of the pipe forced
into it to a depth of a few ins, or far enough to shut off quicksand or sarface water.

If quicksand is encountered, the string of tools is removed, and ths
sand-pump is used inside of the pipe.

For reamlnur out, or enlarg^ing^, holes, or for stralflfhteninip
crooked ones, Ac, special tools, such as reamers, Sec, are substituted in place of the
boring bit.

Special care must be taken to have all the rabbinfT surflsoes thor*
onarhly lubricated. The pulley in the mast-head, and the pinion-wheels
of the horse power (if such be used) should be well oiled every two or three hours.

In very cold or wet weather, a shed of roug^h boards, or a covei^
lag of canvas, about 8 ft high, should be erected, to protect the men ; and, if steam
is used, 2 or 3 boards should be used as a covering for the belt, which will slip if wet.

The following description is based upon tUe uiacuines made by the Fienie
Well Engineering and Supply Co.*

For holes from 200 to 1000 ft deep, portable drillinir ma*
chlnes,t worked by horse or steam power, are used, in these luuchiues, tbe
drill-rope, extending from the string of tools up out of tbe hole, passes over a sheave
at the top of a wocden mast; down to, and around, a pulley fast to the working
lever; and thence, by way of a pulley fixed at the foot of the mast, to a dram upon
which it is wound. To this drum a friction and ratchet wheel is attached, for pay*
ing out the cable as the tools descend.

The mast is hiniped six feet above its foot, so that its upper part may be
laid hor when the machine is to be moved. When at work, it is held in position by
two timber struts or braces, bolted to it near its top, and having their lower ends
Ikstened to the *^ drill-Jack,'* which is a light and strong framework, 0 ft lon^
S ft wide, and 4 ft high, at the foot of the mast, conUining the working lever whick

♦See Business Directory, No. 484. fSee Price-list, 8.67.

ABTE8IAM W£LL BOBtKG,

BriI<i»n.oulaf reacbot timet

One of tiieH« portabli; mnctilnes requires

p, ADd comnieiicfi diillliig, In two hourt; *Dd, qd-

reLutded ia tb« wafoa In two boqn.

o 4 anow tbe UHtSk used with these init-

For lb« diSonot BJiH or macblne Ihej differ
aridwelgbls.

Fig 1 Bbona llisdrtlllnrblt,w:

■ng, and welghn about IM Iba, '
ige Ijj e l»s Joug, 1» top ia K

n, if ft long, ni welgE. i

iritlong... .

f awoiglit,Klvlii8iiadltloUBl force tu tbBblo»» of the bit.
Is tap l9 •crewsdlDlD tlie fool of tbe '•drIII-jBrB," Fig
: >Dd to ilie Wpof ibose le Ktewed ih« " rape^svchet,'' --
Ig 4, to «hlcb the drilling obleiettlucbod; irtheUit,
■r suKer^tena, beconaea wedved In the hole
nr an y meano, Ibe i^raluc elops tbe oburnlngmotlnn

aiU tb« Dpper link U of Ihe^drill-lan, Fig 3, to' elide dowD
buui 12 Idi In the ilat B In their lower iTnk. Tbs oburn-

IMOlbt TbejooetTromWOntotlSOOisclusii's of power.

**-- — H 1 — m^ be worked by bon« power. A hone

nut 800 Vm., nod coiti about •-' "

iiiund, the shift an*
oot of Uw matt, uul

I'

eniln*, 1600 Id 3600
For walla tti

tTB. etum
fiota 'iobo Ut 3000 Aet d««p. m

TIM in', It <tg foot, U

II'"'iiid'"'"aBr^''"^' ^'°-'- Fio.3.
d into the main Bill ot Uu micblDe, wblcb is 18

674 ARTESIAN WELL BORING.

The motiye power is a 15-hp ateam-engine, which, by means of a belt and pnlley,
crank and pitman, working at one end of the walking'beam, gives to the latter its
see-saw motion. To the other end of the beam, and immediately over the well, is
suspended, by means of a hook, a " temper-screw." This last is composed of two
bars of iron, about ^ X 2 ins, 5 ft long, hung 2 ins apart, fastened together at their
top ends, at which point there is an eye, which is suspended on the walking-bean?
hook. At the bottom of the two bars there is a sleeve-nut, and between the two
bars and passing through the nut, is a screw 5 ft long, at the bottom of which there
is a head, which carries a swivel, set-screw, and a pair of clamps. These grasp the
cable, 2 or 2^ ins diam, which carries at its lower end the strings of tools*
This, for a 2000-ft hole, consists of a steel bit, 3 or 4 ft long, weighing 200 to 400 IbB}
an auger-stem of 4 or 5-inch round iron, from 24 to 30 ft long, and weighing from
1200 to 2100 lbs; steeMined drill-jars 8 ft long, weighing 600 to 700 lbs ; a sinker-bai
of round iron of same diam as the auger-stem, 12 to 15 ft long, and weighing from 600
to 1100 lbs ; and a rope-socket, 2}4 ft long, weighing 200 lbs. Total length of string
of tools, 50 to 60 ft, total weight, 3000 lbs ; or, for an 8-inch hole in the hardest roc^
4000 lbs. Tbe sinker-bar is added to give additional wt, and thus to assist in
pulling the cable down through the water, either in lowering the string of tools or
in working the drill-jars. The shapes of the other tools are given by Figs 1 to 4.
Special tools are used for recovering articles that may be accidentally dropped
into the hole.

Tbe drilling^ cable Is wound on a drum, called a bull-wheel shaft, at th«
foot of, and inside of, the derrick. While drilling is going on, it passes from th«
bull-wheel shaft loosely over the sheave at the top of the derrick, and down to the
clamps at the lower end of the temper-screw on the end of the walking-beam. Am
the drilling progresses, the temper-screw is. turned or fed out by the man in chaiv%
who also, by means of a clamp, twists tbe rope, so as to change the position of we
bit after each stroke.

When the tools are to be lifted out of the hole, the cable is disengaged from the
clamps on the temper-screw, and is wound upon the bnll-wheel shaft, which, for this
purpose, is thrown into gear with the steam-engine; the pitman being at the same
time removed from' the crank-pin, so that the walking-beam is at rest. As in the
portable machines, the SJand-pump is also raised by the same power which does the
drilling.

About 10000 ft b m of rongrb Inmber are reqd for the derrick, walk*
i;ig-beam, sills, Ac, and about 3000 ft more for sheds over the boiler, engine, and belt.

In^ordinary hard limestone rock, such a machine will drill about 1^.^
per hour under the most favorable circumstances. Two men are required ;
ene to attend to the boiler, sharpen the bits, Ac, and one to operate the machine.

In Pierce's machine* for test-boring, mineral prospecting and well
boring, the pipes are driven by an iron ram, like that of a pile driver, but
bushed with hard wood on its lower or striking end. The ram is worked by a
hand winch. Tbe pipes are in lengths of 5 to 10 feet. After each length is
driven, water.f under pressure, is forced, by a hand pump, through a hollow
drill rod, into the bottom of the hole, while the drill rod is churned up and
down by liand. The water forces the drillings (mud, sand, gravel, etc.) to the
surface. The smallest machine drives 2 to 3 inch pipes; the largest, 2 to 8 inch.
The machines are in detachable parts, weighjug from 10 to 65 tbs each. Four
upright iron pipes, which carry the head cast*ing and crown pulley, act as guides
for the ram, their ends fitting into sockets in castings at tneir heads ana feet
The driving rams are made in sections which are bolted together. In the
smaller machines the weight of the ram may thus be made from 100 to 200
pounds, and, in the larger raachine.s, from 100 to 2,000 pounds, as rel^uired.
Borings can be made to depths of 100 to 400 feet. These machines have been
extensively used in Nicaragua by the Isthmian Canal Commission, If desired,
the machines can be famished with special tools for boring in rock and for
taking out solid cores (as with the diamond drill), with others for taking out
dry cores in earth, and with sand-pumps and mud sockets for bringing up mud,
fine sand, gravel, and detached pieces of rock and miuerals.

♦Business Directory, No. 484.

t In Alaska, where the frost extends to great depths, boiling or hot water is
used. This is obtained by melting ice or snow in iron tanks about 4 ft square
and 2 ft deep.

MACHINE BOCK-DRILL8. 675

MAGHINE BOGE-UBILLS.

*t» !• Machine Rock*drills bore much more rapidly than hand drills;
acre ooonomically, provided the work is so great as to justify the preliminary
;. They drill in any direction, and can often be used in boring holes so located
they could not be bored by hand. They are wprked either by steam directly ;

air, compressed by steam or water power into a tank called a " receiver," and
i& led to the drills through iron pipes. The air is best for tunnels and shafts,
ise, after leaving the drills, it aids ventilation.

rt« 2. Sacli drills are of Iwo.ltinds: rotating^ drills and
eaBslon drills. In the former, the drill-rod is a long tube, revolving about
xis. The end of this tube, hardened so as to form an annular cutting-edge, is
in contact with the rock, and, by its rotation, cuts in it a cylindrical hole, gen-
y with a solid core in the center. The core occupies the core-barrel. Art 8.
ilrill-rod is fed forward, or into the hole, as the drilling proceeds. The debris
moved from the hole by a constant stream of water, which is led to the bottom
le hole through the hollow drill-rod, and which carries the debris up through
larrow space between the outside of the drill-rod and the sides of the hole.

percussion drills, the drill-rod is solid, and its action is that of the
n drill.

rt. 3. In the Brandt (European) rotary drill, the cutting-edge at the
Df the tabular drill-rod is armed with hardened steel teeth. It is pressed against
-ock under enormous hydraulic pressure, and makes but from 5 to 8 revolutiona
ninute.

rt. 4. Tlie Diamond drill is the only form of rotary rock-drill exten<
y used in America. In it, the boring-rod consists of a number of tubes jointed
Uy together at their ends by hollow interior sleeves.

rt. 5. The borinip-bit. Fig 1, is called a "core-bit.** Its cutting-edge
imbedded in it a number of diamonds as shown. These are so arranged as
reject slightly from both its inner and outer edges. Annular spaces are thus
between core and eore>barreI, and between the latter and the walls of the hoI«.
se spaces permit the ingress and egress of the water used in removing the debrla
1 the hole, and, at the same time, prevent the core from binding in the barrel, or
latter in the hole.

OOKB BITr COBJC LIFTJfia. liOBINQ KUAD.

Lrt. 6. Just above the ''core-bit,** the ^^ COre-lilter^^' Fig 2, is screwed to
barrel. Ihii te a tube abont 8 ing long and of the same onter dlam as tha
TeL Inside it ia slightly coned, with the base of the cone upward, and fvof'
hed with a loose split-ring, R, toothed Inside, and similarly coned. While the
Uing is going on, this ring encircles the core closely, and remains loose from the
AT cylinder; but when the drilling is stopped, ana the drill -rod begins to be
Bed, the ring is canght and raised by the outer cylinder; and, by reason of its
reled shape, is pressed hard against the core of rock, which is palled apart close
its foot by the power which lius the drill-rod.

%.rt. 7* This power is supplied by a rope«drnm« xaso^ned to the top of tim
.me»«nii}h supports the drill and worked oy the same <iMp>ie irAfch rutates the
ill-rod. The rope from the drum passes up to a pulley at the top of a derrick,
d thence down to the upper end of the drill-rod. The considerable height of the
Tick enabiss from M to 60 feet of the drill-rod to be removed in one |iieca»
iirt. 8* Above the ** core-lifter *' is the *^ eore«l>arrel.«> This is a wron«k^
n tube from 8 to 16 ft long. It is spirally

30ved outside, to permit the ascent of the water and debris flrom the hole ; and is
Betimes set with diamonds on its outer surface, to prevent wear. The bit, lifter,
d barrd are of nniibrm outer diam, a little less than the diam of the hole. The
terd'sm of the dxill-ved faiisafirom abovt 17^ ius for ^inch barrel to b\^ ins for Ift
cbtiamL

676

MACHINE ROCK-DMLLS.

Art. 9. Where It Is not desired to preserve the core Intact, a ^^1»orliifp«
head,*' Fig 3, may be used instead of the "core-bit," Fig 1. This is a solid Ut
(except that it is perforated with holes vrhich allow the wafer to pass out from
the drill-rod), and is armed with diamonds, some of which project beyond its circnm*
ference.

Art. 10. The drill-rod revolves at a B|iee<l of from 200 to 400 revokitions
per minute. The eng^ine, by which it is rotated, consists nsimlly of two cylin-
ders, either fixed or oscillating, opei-ated by steam or compressed lur, and working
at right angles to each other. By means of cranks they turn a shaft, which com-
municates its motion, through bevel gearing, to the drill-rod. The latter is fSnl
down, as the hole progresses, either by other bevel g^eavliiif driven by the
same engine ; or by being attached to a cross-head which connects the piston rods of
2 hydraulic eyllnder», the piston rods being parallel with the drill rod.

Art. 11. The diamond drill boretf perfectly circular holes. In StnMLfpht
lines and In any direction, to great depths; from 300 to 1500 feet
being not uncommon. This, with the fact that It hrlngjS up unhrohen
cores, from 8 to 16 ft long, which show the precise nature and stratification of tlie
rock penetrated, renders it very valuable in test-boring, prospecting of mines, Ac
They are also furnished of sufficient size to bore holes from 6 to 15 ins diam, for
artesian wells. The roundness of the holes bored enables the use of casing of
nearly as great diam as that of the hole ; and their straightness is advantageous in
case a pump has to be used.

Art. 12. lu soft rock a bit may drill through 200 ft or more without resetting.
On the other hand, in very hard rocks, similar drills will wear out in 10 ft or less.
In 1883-4. a diamond drill by the Am*n Pbrnoud Rock Boring Oo, wjeighing com.
plete about 1400 lbs, and costing about $2800, bored, in 1428 hours of actaal boring,
53 holes of 2 ins diam, and aggregHting 9141 lineal ft. Average length of hole 172.1
ft. Average rate^ 6.4 lin ft per hour; greatest, 12.8. Average total tiost.
about 96 cts per lin ft. The rock was principally limestone, with some quartz asf
iandstone. The holes were bored at angles varying from OP to 46P with the vertieaL

As a rough average we may say that in ordinary rocks, as granite, lime'
stone, and hard sandstone, these drills will bore deep holes, 2 to 3 ins diam, at irons
1 to 2 ft per hour, and at a cost of from $1 to $2 jper It-
Art. 13. These drills are made of many widely different sise», and with
dilKerent niountinirs. depending upon the nature of the work to be done.

They are sold under restrictions as to the location and extent of the territory
in which they ure to be used. The prices depend, to a great extent, npoa the
nature of these restrictions. The card prices for some of the leading sijEes, are
as followa; Discount, see price list.

Diam

of
hole.

Diam

of
core.

Depth

of
hole.

Boiler

H. P.

required.

Card price.

Drill.

Pump

Ins.

Ins.
2

'^
1

II

Feet.

4000

1600

1000

600

400

H. P.

25

15

12 to 15

10
hand

S

4000
2500
1900
1400
425

1

8400
2800
1900

Art. 14. In percussion drilling machines, the drill-bar is driven
forcibly against the rock by the pressure Of Steam or of compressed

air, acting upon a piston, P, Fig 4, moving in a cylinder, CO, Figs 4 and 6; and
makes about 300 strokes per minnte. The rotation of the drill-bar is accomplished
automatically, as explained in Art 27.

Art. 15. The cylinder. C G, is free to slide longitudinally in the fixed
frame or shell, 8 S, Fig 5, to which it is attached, and which, in turn, is fixed to the
tripod or other stand (see Arts 18 and 19) upon which the machine Is supported.

Art. 16. The drill-rod, R, corresponding to the chnrn drill, Is

fastened, by an appropriate chuck, K, to the end of the piston-rod, 0. The drilHng
is begun with a short drill-rod, and with the cylinder as far from the hole a^ the
length of the shell, 8, will permit. As the bit penetrates the rock, the cvlinder li
fed forward,* either automatirslly or by hand (see Art 28), as fkr as the l<>ngth of

• By

forward* or doWDwmHI, we omui
or apward, /rom (be hole.

Itmmrd Ike hole whieh to beinc drilled. By

MACHINE ROCJ^-DRILI^. . 677

II permits. The drilling i« then stopped, by shutting off the stnani/* and th«
r w run back, by reversing the motion of the feeding apparatuis. Tbt* short
T is then removed, and, if the drilling is to l>e continued, a longer one is sqb*
I in its place, and the process repeated.

. 17. Inasmuch as the act of drilling wears the edges of the bit, thus reduo*
dlam somewhat, the taiole will of course be tapering:, or of
' less diam at bottom than at top. The second bit must therefore be of
' leas diam than the Urst; say from ^ to ^ inch less; the third must be less
le second, and so on. On the other hand, in long holes, the drill'bar will
be in a perfectly straight line, so that the bit, instead of striking always in
e spot, will describe a circle, and thus enlarge the bole.

18. The sbell, S, in which the cylinder slides, i« provided with an arrange-

Y which it may be clamped, either to a tripod, as in Fig 5, or to a long
' colamn, along which it may slide. The column, if hor, may rest upon
rs of legs ; or it may be braced, in any position, against the opposite sides of

V cut, or against the floor and ceiling of a tunnel-heading, Ac, in which case
;8 ends is provided with a screw which 1$ run out so as to cause the two ends
ol to press firmly against the opposite rock walls ; or rather against wooden
rhich are alwavs placed between each end of the col and the rock. In any
) supports of the drill are so Jointed that it can bore in any direction.

19. Frequently the drill is claimped to a abort arm, which, in
clamped to the column, and projects at right angles from it The arm may
lengthwise of the column, and may be revolved around it, and thus may be
n any desired position, and there clamped.' This give» the drill a greater
'motion, and enables it to bore holes over a greater space than would other-
possible without moving the column.

20. In tunnels, one or more drills may be mounted upon a drlll-car«
travelling upon a railroad track running longitudinally of the tuaiML

is track the carriage is moved up to the work, or run back f'-om it when a
to be fired. The gauge of the track may be made wide enough to admit of

I track, of narrower gauge, running underneath the drill-cakriage. Upon
rower track the cars are run which carry away the debris. Drill-carria^
commonly used In this country than in Europe.

21. Tbe pressure used in the cylinders of percussion drills Je
'rom about 60 to 70 lbs per sq inch. In an bour, one will drill
MQ 1 to 2 ins dism, and from 8 to ID ft deep, depending on the character of
and the sise of the maohine at from 10 to 25 ets per lln ft with labor at
ly. A bit requires sbarpenlnir at qbout everj 2 to 4 ft depth of
ne blacksmith and helper can sharpen drills for 6 or 6 machines.

22. 'Tbe bits are of many different shapes, varying with
re of the work to be done. Vor uniform hard rock, the bit has two cuttings
rming a cross with equal arms at right angles to each other. For seamv

arms of the cross are equal, but form two acute and two obtuse angles with
9r, as in the letter X. For soft rock, the cutting-edge sometimes has the
the letter Z.

I83. Eaeh drill requires one man to operate it. Two or three men
red for moving the heavier sizes from place to place. One man can attend

II air-compressor and its boiler.

24. Figs 4 and 6 represent fhe ** Eelipse ** percnesion drill of the Inger-
»nt Drill Co, Havemeyer Building, New York. Fig 6 allows the di:ill,

(as is most frequently the case) upon a tripod. Fig 4 is a longitudinal seo-
ugh the cylinder, yalve-chest. and piston.

IQ. The cylinder, G,is provided at each end with a rubber cusblon,
ulenlng tbe blows of the piston, which, in all percussion drillfv, is liable, at
strike either cylinder-head. The side of each cushion nearest the piston is

by a thin iron plate. Tlie cushions hflve to be renewed from time to time.
66, Tbe Talve, V, is shaped somewhat like a spool. The bolt, B,
>aely through its center and guides it. Steam is admitted from the boiler
am-chest, and occupies all of the space between the two end flanges of the
sept tt. It drives the valve alternately from one end of the valve-chest to
•, and back, according as one end or the other is relieved from opposing
by being put into communication with the exhaust, E^ by way of the pas-
D' and F F^ D and D' communicate with the ends of the steam-chest
passages not shown; while F aad F' communicate, through similar pas-
h the exhaust, £. The piston has an annular channel, L L', encircling it.
• the position of the piston, one of the passages, D or IV, is always, by means
lannel, in CQUunjinication with its corresponding passage, F or F', leading

'*!Sf ^*5*°£?' Y^^^ ""^ ^' ^^^^ (rteapi to aignUy «ith«r tUam or compTMiMl o^
MppeiM to iM nssfl.

MACHDn BOaC-DBIU&

„..,^ „ "cjl.C,(r

Arc 37. Tbe rotation of tbe piHloii, 4Dd, wJifa It, ibit ur th« drliv
Iwr, il eVecMd Ibnal Tll« Bplrallj-graoisd. cjlliidrlcsi ««] tnr. A, cmlled ■ rlSe-
h«r, P*"™ IhroDgb iDd warka in. ll>e rlfle-DBt, II, wblcb i> flrrnly filed la

tlw croon* 0» Uw Hflo-bw, cus« It, md, wilh it, tbe ritthst-wheel, to reiolrt

, leU.erinKiWblow,
>iid<Dc;arib»riH*-

, Bul.onlheijistrokB,

tbeu

borK

Dd »tcbet.»hee] Id

>Dd>

bT th« pawb. tbs rii(*

tarn

EiDiiDi liatoBorv, V

rbUo tb«I«ltoB,pil(«-

rod,,

Art. SB. Tbe Fm

l»«d.

, .t m upjKT end. to

tbe fl»d frame, a U

maiin^ iDDgitnilJnDlT

3l tbe cnnk filed to In

IDP.

in lower IDd Horki

in«BUt.T.fl,edtoU«.

tliln,

the cnDk la tnnicd

bi^ drill! >re freqMntiT fniDlihfld vlth sD MtttONtoMfl
aient In uddltlon to the hand'Cniili. In IbU urniuKiiiuc
(■nulns mdtng tiiw»rd,uil wbaii,coueiiiuiittj, tbe pUtoi

HACHINE ROCK-DSILLS.

679

brward limit of Its stroke, the piston presses against a cam projecting into t1i«
tear the forward end, and presenting an inclined plane to it. The motion oi
cam, by means of an exterior axle, running alongside of the cyl and furnished
I top with a dog, turns a ratchet-wheel fixed to the feed-screw. When desired,
Ritomatic feed may be thrown out of gear, and the feed moved by hand.
rt. 29. Tbe tripoti leg« consist of wronght-iron tubes, Vf W. These are
ved at their upper ends into sockets, XX. At their lower ends, they receive
pointed and tapering steel bars, T Y, about 2 or 3 ft long. The legs may be
thened or shortened by turning the set-screws, Z Z, thus regulating the distance
bich the bars, T Y, can enter the legs. The clamps, b 6, hare L-shaped hooks
I inch to I inch round iron forged to them. On these hooks tli<e weights,
ire hung, which hold the machine down against the upward reaction of its

8.

rt. 30, The following table gives the principal dimensions of these
s, with the dlams and leng^ttas of boles to which each is adapted.
H is used for submarine work, heavy tunneling, and deep rock cutting. G
F for tunneling, street grading, quarrying, and sewer work. E, D, and C
;eneral mining purposes. B is adapted only for very light work. In asking
estimates on drills and compressors, give the fullest possible description
^mpanied by a sketch) of the work to oe done, stating its present and pro-
d extent. State whether the work is on the surface or underground. State
far the steam or compressed air will be carried. Give depth of holes to be
ed, nature of rock, ^c. Percussion drills are sold without reetrictiOQ as to
mrpose or extent to which they are to be used.

r diameter of

nder ins.

;th of full

ke "

th of feed **
thof

hine* "

f machine,,
lounted, lbs.
f trip>od,
loutwts. •'■
rs wts for
od legs, "
fcolumn,
^ clamp"
I of hole

ed Ins.-

depth of
thole. ft.

Letter d(

Qsignat

A

B €

^%

2>^

^A

8
12

4
20

6
24

86

84

86

80

155

195

125

125

X '

250

250

^

200

280

Kto%

%tolH

lto2

. X

4

8

8

6
24

40

230

125

250

280

lto2

10

E

8%

6
24

42
250
125
250
280
lto2

12

F

«

8K

4H

26

7
34

53

60

345

605

150

275

350

400

420

420

lKto2>i

2to4

16

80

7
34

60
670
275
400
420
3tod

40

rom top of handle of feed-crank to lower end of piston at the end of the
stroke.

or greatest advisable Igth of hor holes, deduct one-fourth from these depths,
achine A is mounted on a small frame.

t. 31. Tbe drills of different makers differ ebiefly in the

ods by which the valve is operated. In some this is done, as in the IngersoU
ipse " drill, Art 26, by the pres of steam. In others, the valve is moved
lever or tappet, which projects into the cylinder so as to come into
ct with, and be moved by, the piston at each stroke. As these strokes are
with great force some 300 or more times per minute, such valve-gear is
sarily subject to great wear.

t, 83. In the ^« Ottle Oiant Brill,'' made by the Rand I>rill
the valve, V, Fig 6, is slid backward and forward, in the same direction in
I the piston is moving, by the tappet, T, which is pivoted at p. The
led lower corners of this tappet ride up as they come, alternately, lu contact
the shoulders, s «, of the piston.

t. 83. In the «« Economiser '' and the *<SIucrgrer'' (Rand Drill
he valve, as in the IngersoU " Eclipse" drill, is moved c»y steam, but upon

UACBIME BOCK-DRILLB.

le difi^rent principle.

the point of cut-off 19 fii(5 when

Art. 34. In the Improied
Burleigh dilll, the ralve. V,
Fig 7, i3 moved bf two tHppoU,
T l-, Bhioh Ate allernalely struck
hv the ends of tbc plslon, F. ,

Art. SS. lu Ihe-'^namic" '
roclE-drill, inieoledbrProfDe I
T«lson Wood, Uie TaLve is i

Fig Blanda) bj slesm pressing upon

each^^^ward atrol.e°br ufe oouJcS'
Burfsco of the pislfln, F, presslni

aiantly up and down, carrying lb
T«l™, a, with it. By luruing Ih
ping, n, by menna of^ihe adJnaUnj

occupy'ahigberor lower point in the

'nie'admlBaion *^- *•

edbys small buiIUut'b'^o "- ^ I"'
in the spiral groOTesbown

Is made Ityhler Uikii in

„. „ jder the piston for the pMsani*

on the up stroke, and, consequeDtlr, greater lifting power. HUj
IS uieiui ouen (he driirsllckn In the hole.
Thetrlpodlegaueof barlrOD. Tbdr leogth is adju't'ble.

MACHINE ROCK-DBILL8. 681

Art. 36. The Pierce liand roefc-drlU is a percussion drill
d bya crank which turus a disc about 2 ft iu diam. The disc has a semi-circular
which works the arm which raises the drill-rod. This arm, in rising, compresses
ipriog, which, ou the down stroke, drives the drill against the rock. An iron
eighing 30 lbs or more, is furnished with each machine. This ball may be
d to the top of the drill-rod, for giving greater force to the blows of the drilL
Jl may be used without the spring, by (usengaging the latter,
drill makes about 40 strokes of 10 ur 12 ins per minute; and bores holes from
^ Ids diam. It can be arranged to drill to depths of 30 ft and over. For
omg the bits, it has an emery wheel attached, which is turned by the crank«
;ter, at such times, is thrown out of gear with the disc.
i oHll is moanted on a rectangular two-leggjpd frame, about 5 ft high
; wide, made of iron tubes. To the top of this frame a third leg is attached,
isting which the angle of the drill-iod with the vert may be obuigedo Iiiko
•ereassion drills .worked by hand-power, this one ceases to work to adyantaga
laid aQgle exceeds about 46^.

. 37. ChannellBa: consists in making long, deep, and narrow cuts in
k. In this way large blocks can be gotten out without blasting and the con-
'. danger of fracture. This is ordinarily done by boring a row of holes aboat
I aput in the clear, and then breaking down the intermediate spaces by
)f a blunt tool, called a broach* This Is called broacb cbanneliiiy.
9 purpose a steam drilling machine is mounted upon a hor bar resting upon
in of legs. The hor bar is placed oyer the intended row of holes, and the
slid along npon it from one hole to the next. In using the broach^ the rotat-
taratus is thrown out of gear, so that the edge c^ the broach maintains its

I in line with the row. of holes.

86. Tbe Sanaders patent cbannellnar macblne, of the

II Go, consists of a rock-drilling machine, haying, in place of the usual drill-
a gang of tools consisting of a number of chisels, clamped together side by
d thus forming a cutting tool about 7 ins long by % inch wide. This tooi
many cutting-edges (each as long as the tool is wide) as there are chisels,
chine is supported upon a carriage, moving on a track parallel with the

to be cut. The tool is of course not rotated; but the rifle-bar. A, Fig 4, is
>d to move the carriage along the track about an inch after each blow. The
I remains stationary while a blow is being struck. Under fkvorable circum-
tbis macbine bas cut from 80 to 100 sq ft of channel per day of ten
[ts ireliTbt, including carriage, is about 5000 lbs.

^e is provided, by which, if desired, the steam may be sbvf oflT ftom
m on the down stroke, so that said stroke may be made with only the toeighi
Iston, rod, and drill.

89* The Ingersol! Oo have a special appliance, designed by Mr. W. L.
s, G £, I6r drlUlnv and blaatlny recbs under water, even
ey are covered by a considerable depth of mud.

40. Air compressor for rock-drills, as made and used in this conn-
mostly hor, direo^acting engine*. That is, the axes of the steam- and air>
i are hor; and the piston-rod passes directly from the steam-cylinder into
ylinder. A fly-wheel !s attached, by a crank and connecting-rod, to the
•d. Sometimes the steam-engine is separate from the compressor, and the
conyeyed to the latter by belts or gearing; or water-power may be used in
» way. Tbe air is forced into a reoaiver, which la generally a plate-4ron
8 or 4 ft in diam, and 6 to 12 ft long.

air- or pnmping-cylinder of the compressor is so arranged as to take In air
troke only, and force it out into the receiver upon the return stroke, it it
e*ACtlD|F«'' It at each stroke, it both takes in and forces out air, it it
le-actingf*** If the compressor bas only one air-cylinder, it is ^'8in«
it it has two, and thus practically consists of two single compressors, it is

fcl'Ves may be either ^* popjpet ** valves, held In place by springs, and
by the pressure of the air itself; or slide. valves, operated by eccentrics
as in steam-engines.

npression of the air develops beat. Thia is removed either by causing
r to circulate through the air-piston, and through Jackets surrounding the
er ; or by injecting it into the air^sylinder in the form of spray. Or both
nay be oaed together.

682

MACHINE ROCK-DRILLS.

Art. 41. The fallowing; partial list of Clayton compressors, compiled from
data given by the makers, shows the dimensions and performanee of
each. We give also a list of their receivers.

CliATTON DO UBIiE- ACTING AIR-COMPRESSORS. Partial List

Duplex Rlreot-actlnff* Compressors.

Diam of steam-cylinders ;lns.

« air " ins.

Length of stroke ins.

Number of revolutions per minute

Cub ft of ftree air compressed per minute Actual.

Approximate wt of compressor lbs.

Approx number of rock-drills with 3-inch cyls sup-
plied with air at 60 to 80 lbs per tjq inch

Single Dlreet-actlng* Compressors.

Mam of steam-cylinder ins.

** air •* ins.

Length of stroke ins.

Number of revolutions per minute

Cub ft of free air compressed per minute... Actual.

Approx wt of compressor ~

Approx number of rock-drills with 3-inch cyls sup-
plied with air at 60 to 80 lbs per sq inch

Number, designating the sixe
of the maehine.

8
8
12

ri2o

i to

(140

136

9000

8
8

12
(120

1 to
(140

68

1650

m

10

10
18
100
to
180
210
7000

14

U*

15
100

to

120

438

15000

8

10

U

18

10

u

18

13

15

24

100

100

80

to

to

to

130

120

00

106

210

450

3850

8260

13760

18
18
24
80
to
90
900
25000

18

9

* The prioe of a oompreBaor alone, to be worked by a separate steam-engine or water-power. Is ^
turse less than that or the above compressor and engine oombined.

eourse

Alr-R«o«lTeTS| vertlosl and Itorlaontal.

Diameter
inches.

Length,
?eet.

Approximate
weight, lbs.

Diameter,
Inches.

Length,
Feet.

Approximale
weight, 1»B.

88

30
36

40

7
8
6

700

890

1560

1600

40
40
40
40

8
10
11
12

Iiil

The Air-Receivers have brass-face pressure-gauge, glass water-gauge, safety-Talrf^
blow-off valve, try-cocks, flanges and connections to automatic fcM on oompreMor.

TBACTIOK.

683

TRACTION.

.'raetlon on cominoii roads, and canals ; or the power read to draw
olM and bmta aloog them. In oonneotion with this auhjeot rsad the preoeding and the following

le following Uble shows tolerable approximations to the foroe in lbs per ton, na& to draw n Bt«M
i and puseagers, up asoents on the Holyhead tompike road in Bngland, (a fine road,) by borMT
ioeruined by means of a dynamometer. The entire weight was 1)i tons ; bnt in the table, the
Its are given per single ton. From the nniure of suoh oases, no great aoeoraoy is attainable.

Proportional

Asoent in Ft.

At 4 Miles

At 6 MUes

At 8 Miles

At 10 Miles

Asoent.

per Mile.

per Hoor.

per Honr.

per Hour.

per Hoar.

Lbs.

Lbs.

Lbs.

Lbs.

lin l&yi

840.7

310

816

325

240

1 " ao

384.

196

103

313

339

1 " ae

908.1

155

160

166

175

1 " 80

176.

187

143

147

154

1 " 40

183.

114

130

134

180

1 " 64

83.6

100

116

130

136

1 " 118

44.7

103

107

118

130

1 " IS8

88.8

90

106

109

117

1 •• IM

88.9

96

101

106

113

1 '< 245

21.6

98

.96

101

107

1 •' 000

8.8

81

65

91

96

Level.

0.

76

80

66

91

I following results, most of them with the same Instrument, are also in lbs per ton ; with a four-

ed wagon, at a slow paoe, on a level ; and the roads in fair oondition.

naonbioal Mock pavement 83 lbs per ton.... ••••••A> M.

MoAdam road, of small broken stone 63" " " probably to 75.

< prnvelroad 140 " •< <•

' Telford road, of small stone on a paving of spawls 46 " " •' " ** 75.

bnAen stone, on a bed of oemeat oonorete 46" " " " '* 75.

' oommon earth roads 300to800. On a plank road 80, to 60 lbs.

le traetlTo powor of a horse dlntlnlslies as bis speed In-
ises ; and perhaps, within certain limits, say from ^ to four miles per hour,
in Inverse proporUon to it. Thus, the average traotion of a horse, on a level, and aotnally
I for 10 hours in the day, may be assumed approximately as follows:

Miles per hour. Lbs. Traotion. Miles per hour. Lbs. Traotion.

^ 888.88 3^ lU.U

1 850. i\i 100.

IH 300. 3K 90.91

IH 166.66 8 88.88

IH 1*3.86 SH 71.43

2 125. 4 62.50

works for a smaller number of hours, his traotion may Increase as the hours diminish ; down
It 5 honrs per da; and for speeds of about trtna l}i to 8 miles per hour. Thus, for 6 hours per
I traotion at 2H miles per hour will be 200 lbs, Ao. Wlien asoending a hill, his power dimin.
» rapidly, ftom having partially to raise his own weight, (which averages about 1000 to IIUO
at up a slope of 5 to 1, he oan barely struggle along without any load. On such an asoent,

he must exert a force equal to 439 lbs per ton, or of 196 lbs for
0 lbs of his own weight. Assuming that on a level piece of good turnpike, he would when faaul-
trt nod load, together weighing 1 ton, have to exert a traotion of 60 lbs ; then on asoending a
4° inolination, (or 1 in 14.3 ; or 869^ ft per mile,) he would have to exert 156 lbs, against the
of the 1 ton : and 67 lbs, against that of his own weight ; or 228 lbs In all. He may, for a fe«
xert without Injury, about twice his rogular traction. This calculation shows that up a hill
.n average horse is fully tasked in drawing a total load of one ton ; and should, thereforo, be
, in such a case, to choose his own gait ; and to rest at short intervals. A fair load for a single
rith a oart, at a variable walking pace, working 10 hours per day, on a common undulating
good order. Is about half a ton, in addition to the cart, which will be about half a ton more.
ro horses to this same oart, the load alone may be about lii tons.
Since the action of gravity la the same on good roads and bad ones, it follows that

Bta become more objectionable tbe better tbe road is.

n an aaoent of 2P, or 184.4 ft per mile, gravity alone requires a traction of 78 lbs per ton ;

J

684

TRACTION.

which la aboat 10 times that on a level railroad at 6 miles an hoar ; bat only about equal to that on a
level common tarnpike road, at the same speed. Therefore, (to speak somewhat at random,) It would
require 10 locomotives instead of 1 ; bat only 2 horses instead of 1 . A grade of I in 35 ; or 160 ft to a
mUe; or 1° 38', is about the steepest that permits horses to be driven down a hard smooth road, in a
fast trot, without danger. It should, therefore, not be exceeded except when absolutely necessary,
especially on turnpikes.

On canals and otber waters, the liquid is the resisting medium that

takes the place of friction on level roads. But unlike friction, its resistance varies as the squares of
the vels; at least from the vel of 2 ft per sec, or 1.364 miles per hour; tc

that of 11^ ft per sec, or 7.84 m per h. As the speed falls below 1 Ji m per h, the resistance varies less
and less rapidly ; and this is the case whether the moved body floats partly above the surface ; or is
entirely immersed. In towing along stagnant canals, &e, the vel is usually from 1 to 2^ m per h;
for freight most frequently from l^ to 2. Less force is required to tow a boat at say 2 m per h, where
there is no current, than at say l}i m per h, against a current of ^ m per h, because in the last caae
the boat has to be lifted up the very gradual inclined plane or slope which produces the oorrent.
The force required to tow a boat along a canal depends greatly upon the comparative transverse
sectional areas of the channel, and of the immersed portioa of the boat. When the width of a oanal
at water-line is at least 4 times that of the boat ; and the area of its transverse section a»great as at least
6^ times that of the immerted transverse section of the boat, the towing at usual oanal vels wUl be
about as easy as in wider and deeper water. With less dimensions, it becomes more difficult. (t)'An-
buisson.) Much also depends on the shape of the bow and otber parte of the boat ; and on tiie propor*
tion of its length to its breadth and depth. Hence it Is seen that the mere weight of the loa4 is by no
means so controlling an element as it is on land. The whole subject, however, is too intricate to be
treated of here, itorin states that naval constructors estimate the resistance to sailing and steam
>esaeie at sea, at but from abont .5 to .7 of a lb for every sq ft of immersed transverse section, when
the vel is 8 ft per sec, or 2.046 miles per hour. It is far greater on canals.

On the Scbnylktll STaTigratlon of PennsylTania, of mixed canal

and slackwater, for 108 miles, the regular load for 3 horsevor mules, is a boat of very fall build ; and n«
keel ; 100 ft long, 17^ ft beam ; and 8 ft depth of hold ; drawing b}i ft when loaded.* Weight of boa»
about 66 tons; load 176 tons of coal, (2M) lbs;) total weight 240 tons, or 80 tons per horse or mule.
On the down trip with the loaded boats, for 4 days, the animals are at work, actuaUy towing, (except
at the locks,) for 18 hours out of the 24; thas exceeding by far the limits of time usually allowed for
oontinuons effort.

On the canal sections, (which have 60 ft water-Hue ; and 6 ft depth,) the speed Is 1^^ miles per hour $
and on the deep wide poole, 2 miles.

On the up trip with the empty 66-too boats, the average speed is about 2^ miles per boar. The
empty boats draw 16 to 18 ins water ; and frequently keep on without .stopping to rest day or night
through the entire distance of 108 miles. The animals generally have 2 or 3 days' rest at each end of
the trip ; but are materially deteriorated at the end oLthe boating season.

If our preceding assumption of 143 lbs traction of a horse at 1^ miles per hoor, is oorrect, Uia

143 lbs

traction of the loaded boats on the oanal sections is -— r = 1.83 lbs per ton.

80 tons

The intelligent engineer and superintendent of the Sch Nav, James F Smith, gires as the results
ef his own extensive observation, that one of these large boats loaded <240 tons in all) may, without
distressing the animals, be drawn along the canal sections, for 10 hours per day, as follows : By one
average horse pr mule, at the rate of 1 mile : by two animals, at 13^ miles ; and by three, at I^ miles
per hour. When four animals are used the gain of time is very trifling. At a time of rivalry among
the boatmen, one of them used 8 horses ; but with these could not exoeed 2}^ miles per luoor in tbe
eanal portlens. Two or more horses together cannot for hours pull as much as when workinc sepa-
rately.

If our preeeding short table of the traction of a horse at diff vels for 10 hours is oorreot, then tbs
traetion of the above loaded coal boats (240 tons) on Che eanal sections' of the navigation, is as follows:
The last column shows the traction in lbs per sq ft of area of immmnd transverse seotion where largest;
viz, about 95 sq ft.

Horses. Miles per He«r. Lbs. |MrT<»a. Lbs. per 84 Vt.

1 1 ffj 1-®* • «•«•

2 -...IH ftJ 1-8* »•«>

8 1« tH 1"8 *-50

Son pools 2 f J^ 1.56 S.9S

8 2« fJJ.. 8.38 8.41

Sup-trip 2H W 4.61 1S.50

I^achine Canal, Canada, 120 ft wide at water-line; 80 ft at bottom ; depth

en mitre sills 9 ft ; 6 horses tow loaded schooners with ease.

Before the enlargement of the El*ie Canal,f its dimensions were 40 ft watcr.Une ; 28 ft bottooi ;
4 ft depth of water. The average weight of the boats was about SO tons. With 76 u«s of load, or 105
tons total, they were towed by 2 horses, at tbe rate of about 2 miles per hour ; which by our table gives
a traction of nearly 2.4 lbs per ton. The boats were about 80 ft long ; 14 ft beam ; full 8^ ft draof hfe
loaded ; hence the traction by our table would be about 5.7 lbs per sq ft of immersed transverse seetton.

4, (Schuylkill Canal) about f 1800. Annual repairs about
I. Length. 102 ft) beaai, 17M ftf draft, IHtobHtil sspsoitj. 18t
»eed, with 8 mules, I9i miles per kour.

* Cost of boats, 1884,

t8&. Boats last 16 to 20 years.

tons; weight, about 68 tons; speed, , _ ,.,

t Length 868 miles ; oost tlMW) per mile. The enlarged osnsi has 70 ft| 49 ft t and T ft of water }
and cost $90800 per mile for tbe enlargement only. The sest of ths ssTsral osnsls la ^•nnsjlTaolf
has ranged between $23000 and $60000 perlnile*

I

ANIMAL POWER. 685

Hiile, for 82-ton loaded boats on a smaller canal, (the boats nearly toochlng l^ottom.) tbe traction at
^ miles, wou]d be Z}4 lbs ^ei* ton ; or about twice as great as tbe abore I.Ts lbs. It also woald be 5.7
18 per 8q ft of immersed section.

ANIMAL POWER.

Art. 1. So far aa regards hordes, this subject has been partially considered
ider the preceding head, Traction. All estimates on this subject must to a certain extent be vague,
ring to the diff strengths and speeds of animals of tbe same kind ; as well as to tbe extent of their
sinlng to any particular kind of work. Authorities on the subject differ widely ; and sometimes
press themselves in a loose manner that throws doubt on their meaning. We believe, however,
at the following will be found to be as close approximation to practical averages as the nature of
• case admits of with our present imperfect knowledge. We suppose a good average trained horse,
dghing not less than about H » ton, well fed and treated. Buch a one, when actually walking for
hfeurs a day, at the rate of 2^ miles per hour, on a good level road, such as tbe tow-path of a cantJ.

a circular horse-path,* Can exert a continuoas pull, draugrbt, power,
r traetlon, of 100 lbs.

Now, 2H miles per hour, is 220 ft per min, or S% ft per see ; and sinoe 10 honrs contain 600 min,
i day's work of actual hanling on a level, at that speed, amounts to

min ft Dh

600 X 220 X 100 = 13 200000 ft-lbs per day.

, 22000 ft-Ibs per min, or 366^ ft-lbs per sect Which means that he ezerto foroe enough during the
r to li/t IS 200000 lbs 1 foot high ; or 1 S20000 lbs 10 feet high ; or 1S2000 lbs 100 ft high, to. Hd may
)rt this force either in tr«kcMon (hauling) or in Ufting loads. If he has to raise a small load to a
!at height, the machinery through which he does it must be so geared as to gain speed, at the loss
mmonly but improperly so expressed) of power. Whether be lifts the great weight through a
all height, or the small weight through a great height, he exerts precisely the same amount Of
oe or power.
Experience shows that within the limits of 5 and 10 honrs per day, (the speed remaining the sane,)

>e draft of a borse may be increased In alN»nt tbe same pro*
»rtlon as tbe time is diminisbed ; so that when working from 5 to io hours

day, it will be about as shown in the following table. Hence, tbe total amount of 13 200000 ft-lbs
day may be accomplished, whether the horse is at work 6, 6, or 8, &o, hours per day.} This, of
rse, supposes him to be actually lifting or hauling «M the time; and makes no allowance for stop-
es for any purpose.

Table of draft of a borse, at 2V^ miles per bonr, on a le^el*

Honrs per day. Lbs. Hours per day. Lbs.

10 100 7 U24

» mi 6 166Ji

8 125 5 200

Sxperience also sbows tbat at speeds between % and 4
lies an bonr, bis force or dranirbt will be inversely in pro*
•rtion to bis speed. Thus, at 2 miles an hour, for 10 hours of the day, his
ight will be

inilM miles Tte fits

2 : 2^ : : 100 : 125 draught.

b IH miles, it would be 166^ lbs ; at 3 miles, 83^ 9>s ; and at 4 miles, 62^ lbs ; as per table la

;tion.

lerefore, in this case also, the entire amount of his day's work remains the same ; § and within

To enable a horse to work with ease in a circular borse-walb, its diam
lid not be less than 25 ft; SO or 35 would be still better.

A nominal borse-power is 33000 ft-lbs per minute ; this being the rate

med by Boulton and Watt in selling their engines ; so that purchasers wishing to substitute
n for horses, should not be disappointed. Their assumption can be carried out by a very strong
e day after day for 8 or 10 hours; but as the engine can work day and ni^ht for months without
ping, which a horse cannot, it is plain that a one-horse engine can do much more work than any
luch horse. Hence many object to the term horse-power as applied to engines ; but since every-

understands its plain meaning, and such a term is convenient, it is not in fact objectionable.
ton and Watt meant that a one-horse engine would at any moment perform the work of a very
ig horse. An average horse will do ba< saOOO ft-lbs per min.
k is plain that although the d<m's labor will be the same, that of an hour, or of a min, will vary

the number of hours taken as a day's work. It must be remembered that a working dag eft a
1 number of hours, by no means implies, in every case, that number of hours of actual work;
nclndbas intermissions and rests.

rbis remark about speed will not apply to loads towed
■onarb the water. Thus, if his draught at 2 miles an hour be 125 fi)s ; and
niles, 62H Ihs ; he will on land draw loads in these proportions ; but in hauling a boat through
€U«r at the greater speed, he has to encounter the increased resistanoe of the tottter itself; which
taooe at 4 miles is much more than twice as great as at 2 miles ; probably 4 times as great,
sfore, at 4 miles on a oanal, bis draught of 62^ fis would not suffice for a load half as great ■>
nid tow with his draft of 125 lbs at 2 miles. -^

J

686

ANIMAL POWER.

all the foregoing limits of boars and speed, may be praotioally taken to be about IS 200000 A-lb« fm
day ; or 2'2O00 ft- lbs per min of a day of 10 hoars. ,But it' does not follow that the horse can alway*
In practice actually lift loads at that rate ; because generally a part of his power is expended la
overcoming the friction of the machinery which he puts in motion ; and moreover, the nature of tbf
work may require him to stop frequently ; so that in a working i<m of 8 or 10 hours, the horse nuiy
not actually be at work more than 5, 6, or 7 hours.

As a rough approximation, to allow for the waste of force in overcoming the friction of hoisting
machinerv, and the weight of the hoisting chains, buckets, &c, we may say that tlie USOAll

or pnyinft: dally net work of a horse, in liolstlnir by a eom-

mon ipin, is about lOOOOOOO ft-n>s. That is, he will raise equivalent to 10000000 lbs net of
water, or ore, Ac, 1 foot. The load which he can raise at once, including chains, bucket, and an
allowance for friction, will be as much greater than his own direct force, as the diam of the hors»
walk is greater than that of the winding drum ; and it will move thac much slower than he does.
His own direct force will vary according to the number of hoars per day that he may be required to
work, as in the foregoing table. With these data, the size of the buckets can be decided on ; and of
these there should be at least two, so that the empty one at the bottom may be filled while the full on*
at top is being emptied ; so as to save time. The same when the work is done by men.

Art. 2. A practised laborer hanllnfp aloni^T ^ level road, by
a rope over his shoulders; or in a circular path, pushing before him a
hor lever, at a speed of from 1^ to 3 miles per hour, exerts about % part as much force as a horse;
•r 2200000 ft-Ib!) per day ; or 3606^ ft-Ibs per min of a day of 10 hourjs of actual hauling or pushing.

But laborers frequently have to work under circumstances less advantageous for the exertion of
their force than when hauling or poshing in the manner Just alluded to ; and in saoh cases they oannot
do as much per day. Thud in turning a winch or crank like that of a grindstone, or of a orane, the
continual bendiug of the body, and motion of the arms, is more flitigning. The Sise of A
winch Shoald not exceed 18 ins, or the rad of a circle of 3 ft diam; and against
it a laborer can exert a force of about 16 0>s, at a vel of i]>i ft per sec, or l&O ft per min, making very
nearly 16 turns per min ; for 8 hours per day. To these 8 hours an addition must be made of abooi

H part, for short rests. Or if a working day is taken at 8, or 10, fto, hours, 4 part must generally be

taken from it for such rests. On the foregoiog data an hour's work of 60 min of aduoL hoUtUtg
would be

Dm ft min

16 X 150 X 60 = 144000 ft^lbs;

or, deducting \ part for rests, 115200 ft-Ibs per hour of lime, inatudiag rests. In practice, however-
a fhrther deduction must be made for the fHo of the machine, and for the wt of the hoisting ohalmi;
and in case of raising water, stone, ore, &o, ftrom pits, for the wt of the buckets also. As a roagk
average we may assume that these will leave but 100000 ft-lbs of paying, or useful work per hour;

that is. that a man at a winch will actually lift equivalent to
100000 lbs of water, ore, Ac, 1 foot hi^rh per hour's time, in-
dud inic rests. This is equal to ie66f< ft-Ibs per min of a day of 10 hours, inolading rests.
Therefore, in a day of 10 working hours he would raise 1 000000 lbs net, 1 foot high ; Or JUSt JL

Eart of what a horse would do with a «rln tn the same time. We have
fore seen that in hauling along a level road, he can at a slow pace perform about % of the dally
duty of a horse. He may also work the winch with greater foroe. say up to SO or even 40 fts: bat
he will do it at a proportionately slower rate; thus, aocomplisbing only the same dally daty.
With a ffrin, like those for horses, but lighter, with 2 or more baoketa, a prw>-
Used Uiborer will in a working day of 10 hours, raise from 1200000 to 1400000 ft-B>s net «^ water. vr%
Ac. With a shallow well or pit, more time is lost in emptying buckets than in a deep one; bat tbe
deep one will require a greater wt of rope. To save time In all such operations on a large scale, there
should be at least two buckets ; the empty one to be filled while thj full one is being emptied. It is
also best to employ 2 or more men to hoist at the same time, by winches, at both ends of the axis;
and the men will work with more ease if the winches are at right angles to each other. Each wtnoh
handle may be long enoui;h for 2 or 3 men. An extra man should be employed to empty the bnoketa.
He may take turns with the hoisters. The name remarks apply in some of "the following oases.

On a treadwheel a practised laborer will do about 40 per cent mor^ daily
duty than at a winch ; or in a working day • of 10 hours, including rests, he will do about 1 400000 ft-
lbs. And he ean do this whether he works at the outer ciroumf of the wheel, stepping upon fbot>
boards, or tread-boards, on a level with its axis ; or walks inside of it near Its bottom. In both cmam
he acts by his wt. usuallv about 130 to 140 lbs ; and not by the muscular strength of his arms. Whea
at the level of the axis, his wt acts more directly than when he walks on the bottom of the wheel;
but in the flnt case he has to perform a slow and fatiguing duty resembling that of walking ap a
oontinuous flight of steps ; while in the second he has as it were merely to ascend a very slightly in-
clined plane ; which he can do much more rapidly for hours, with comparatively little fatigue : and
tills rapidity compensates for the less direct action of his wt. Therefore, In either case, as experlenoe
has shown, be accomplishes about the same amount of daily duty. Treadwheels may be from 5 to 2b
ft in diam, according to the nature of the work. They are generally worked by several men at onoe,
and may at times be advantageously used in pile-driving, as well as in hoisting water, stone, Ac.

By a good common pump, properly proportioned, a practised laborex

will in a day of 10 working hours, raise about 1000000 ft-lbs of water, net.t

BailinHr with a liirht bucket or scoop, he can accomplish about

100000 rt-lbs net of water. By a bUCkct and SWape, (a Inng lever reeking vertloallyt

and weighted at one end so as to balance the full bucket hung mtm the other; often seen atoooatry

*The working day must he nnderstood to inoliide nt^oessary rests, and such Intermissions as the
nature of the work demands ; but doex not include timp Innt at meals. A worldng day of 10 hours
may, therefore, have but 8, 7, or A. Ac hoars of aetual labor. This will be understood when we here*
after speak of a working day, or simply a daj^

t OeaaguHar'8 astlmatee of dally work of men nod horiee ezeaed the above, but are entirely too great

ANIMAL POWER. 687

velli,)600000to800000. In the last h« haa only to pall down the empty bucket, and thereby raise tb«

MDnterweigbt. By 2 boekets at tbe ends of a rope saspended over

a pnlley, SOOOOO to eooOOO. Here he work* the bnekets by pulling the rope by band.

Bw a lymiNin, or tympannm.* worked by it treadwheel, about 1 200000
loiiooooo.

By a Pereian wheel.f a cbaln-pump, a cbaln of back et«4 or
air Archimedes screw, all worked by a tread wheel, from 600000 to 1000000
It-fiw.' or tbese four, the flnt three lose aaefal effect by either spilling, leaking, or the necessity for
raising the water to a level somewhat higher than that at which it is discharged.

Fhen say of the five foregoing maohines are worked by men at winches, the result will be about
i leas than by treadvheels. They are all frequently worked also by either steam.waier, or horse-power.

By walking backward and forward, on a lever whicb roeks
»n Its center, a man may, according to Robisou's Mech PhiloBophy, perform a
inch greater dntr than by any of the preceding modes. He states that a young man weighing 136
«, and loaded with 30 lbs in addition, worked fn this manner for 10 hours a day without fatigue ;
id raised 9^ cubic feet of water, 11^ ft high per min. This Is equal to 3 984 000 ft-lbs per day of 10

>ani; or 66M) ft- lbs per min ; or nearly -j^ of the net daily work of a horse in a gin.

A laborer standing still, can barely sustain for a few min, a load of 100
I, by a rope over his shonldcr, and thence passing off hor over a pulley. And scarcely as much,
ten (facing the load and pulley) he holds the end of a hor rope with his hands before him. He can-
t posh hor with his hands at the height of bis shoulders, with more than about 30 lbs force.

Velsbeob states fyom his own observation, that 4 praetieed men raided a dolly (a wooden beetle
rammer, of wood; with 4 hor prqjeeting round ban for handles) weighing 120 lbs, 4 ft high, at the
e ot 34 times per min, for iii min ; and then rested for 4^ min ; and so on alternately through
10 hours of their working day. Therefore, 6 of these hours were lost in rests ; and the dnty pei%
ned by each man during the other 5 hours, or 300 mina, was

120X4X84X800^^^^^.^^

n tbe old mode of driving piles, where the ram of 400 to 1200 Ibt
tended trom a pulley, was raised by 10 to 40 men palling at separate oords, from 86 to 40 lbs of the
were allotted to each man, to be lifted f^m 12 to 18 times per min, to a height of 3^ to 4^ fcet
time, for about 3 min at a apell, and then 3 min rest. It was very laborious ; and the gangs had
j changed about iiourly, after pOTforming but H an hoar's actual labor.

[anllnflT by bomes. See Traction. When working all day, say 10 working
i, tbe average rata at which a horse walks while haaling a ftill load, and wbile returning with
mpty vehicle, is about 2 to 2^ miles per hour; but to allow for stoppages to rest, Ac, it is safest
te it at but about 1.8 miles per hour, or 160 ft per min. The time lost on each .trip, in loading
nloadlng, may Usually be taken at about 15 min. Therefore, to find the number of loads tbat can
ul^d to any given diet in a day, flr»t find tbe time in min reqd in hauling one load, and return-
npty. Thus: div twice the dist in ft to which the load is to be hauled; or in other words, div
igth in ft, of the round trip, by 160 ft. The quot is the number of min that tbe horse is in mo-
irJng each round trip. To this quot add 15 min lost each trip while loading and unloading ; the
I the total time in min occupied by each round trip. Div the number of min in a working day
In in a day of 10 working hours) bv this number or min reqd for each trip ; the quot will be the
r of trips, or of loads hanled per day.
How many loads will a horse haal to a dist of 960 ft. in a day of 10 working hours, or 600 min T

1920
SOX 3=: 1920 ft of round trip at each load. And -— — = 12 min, oocupied in walking. And

160 goo min in 10 hours

in loadioff, *o) = 27 min reqd for each load. Finally, — - = — ; — = 22.2, or

27 min per trip

-f pa , or loada hanled per day.

»le of number of loads banled per day of 10 worklnjr

s. The first col is the distance to which the load is actually hanled ; or half
h of the round trip. The oost of hauling per load, is supposed to be for one-horse carts ; the
Ing the loading and unloading ; rating the expense of horse, cart, and driver at $2 per day.

ympaii rewolwea on a hor shaft: and is a kind of large wheel, the spokes, arms, or radii of

I ffuttera» troughs, or pipes, which at their outer ends terminate in scoops, which dip into

Aa the water la gradually raised, it flows along the arms of the wheel to its axis, where

d. The aeoop wheel is a modlfleation of it. It is an admirable machine for raising large

of water to moderate heights. We cannot go into any detail respecting this and other
maohines.

of large wheel with backets or pots at the ends of its radiating arms ; revolves on a hor
bargee at top. The bucketii are attached loosely, so as to bang vert, and thus avoid spill-
le^ arrive at the proper point, where they come into contact with a contrivance for tilting
Ing them. The noria is similar, except that the hnokets are firmly held in place, and thus
water. Tt is therefore inferior to the Pernian wheel.
lexs revnlvin«c rert chain of bnokets. D'Auboisaoa and some others erroneously eall this

It ia aui effeotive maohias.

688

ANIMAL POWER.

DUt.

No. of

Coat per

Dist.

No. of

Coat per

Dist.

No. of

Cost per

Feet,

Loads.

Load.

Feet.

Loads.

Load.

Miies.

Loads.

Load.

Gti.

Cta.

Cts.

SO

38

6.26

1500

18

11.11

1

88.57

100

37

6.41

2000

15

13.33

m

S3.S3

200

34

6.88

2500

13

15.39

IH

40.00

300

32

6.25

3000

11

18.18

2

50.00

400

30

6.67

3500

10

20.00

3

66.67

600

27

7.41

4000

9

22.22

4

100.00

1000

22

9.09

6000

»

88.&T

9

200.00

If the loading and unloading is such as cannot be done by the driver alone; bat reqalrea the help
of cranes, or other machinery, an addition of from 10 to 50 ois per load mar beoome necessary. Haul-
ing can generally be more cheaply done by using 2 or 3 horses, and one driver, to a vehicle. The neat
loaid per horse, in addition to the vehicle, will usually be from ^ to 1 ton, depending on the condition,
and grades of the road. From 13 to 15 cub ft of solid stone ; or from 23 to 27 cub feet of broken stone,

make 1 ton. Iti estimating^ for baalini^ rongrli quarry stone for

drains, CalTertS, Ac, bear in mind that each onb yard of common soabbled mbbl*
masonry, requires the hauling of about 1.2 cub yds of the stone as nsualiy piled up for sale In th«
quarry ; or about Hot a. cub yd of the original rook in place. A Cnb yd Of SOlid Stoae,

wben broken into pieees, nsnally oeenpies about 1.9 cnb yds

perfectly loose ; or about IH when piled op. A strong cart fbr stone hauling, will ynlgh
about % ton ; or 1600 0>b ; and will hold stone enough for a perch of rubble masonry ; or say 1.2 pen
of the rough stone in pHes. The average weight of a good working horse is about H a ton.

Morin ipives'tbe folio wiuflr results ft>om careful experiments made by
him for the French Government. The draft of the same wheeled vehiole on a road, may in practice
be considered to be,

1st. On hard turnpikes, and pavements; in proportion to tb^

loads : inversely as the diams of the wheels ; and nearly Independent of the widta of tire. It inoreaaea
to uncertain extents with the inequalities of the road ; the stiffness (waht of spring) of the vehicle ;
and the speed ; (considerably less than as the square roots of the last.)

9d. On soft roads, the draft is less witli Wide tires tban
iritb narrower ones; and for farming purposes he recommends a widtb c^
i ins. With speeds nrom a walk to a fast trot, the draft does Bot vary seniib^y.

TBTTSSES. 689

TRUSSES.

INTBODUCTION.

General Principles.

Fmss Design a Specialty. The design, construction and erection
3ses have become a specialty, to which persons confine themselves more
exclusively, and thus attain a degree of expertness beyond the reach
general engineer.* The latter, however, should have a knowledge
subject, sufficient at least to enable him to form a well-grounded opin-
the general merits of a design and to guard him against the adoption
involving serious imperfections. In a volume like this we can discuss
eneral principles.

The Truss Principle. Theoretically, a truss consists of a number
ight bars, joined, near their ends, by perfectly flexible joints, loaded
t these joints, and so arranged that all its internal stresses are sus-
by its members, and only the vertical f pressures, due tf> the weights
:russ and its load, are transmitted to the abutments.

Hstinction between Beams and Trusses. When a solid beam
, If 7, Transverse Strength) bends, under its own weight or under that
oad, all the fibers above the neutral axis are compressed, while all
»elow are extended; and the resulting change of length, in each fiber,
ortional to the distance of the fiber from the neutral axis; but, in a
;he loads ^including the weight of the truss itself) are theoretically
)d as divided into portions which are concentrated at the joints be-
the members and which act through the cens of grav of their cross-
3. So placed, the stresses caused by them could not act transversely
aembers, as in a beam, causing so-called secondary stresses, but must
l^itudinally or axially of the members, and must be uniformly distrib-
ver their entire cross-sectional areas. This is the distinguishing
of all trusses.

L such a truss the material would be used most economically, and the
in each piece and in each part of such piece could be readily and
sly determined.

the truss of a well-designed bridge or roof, this ideal con'dition is
mated by using, for the principal members, straight and rather
pieces, and by so distributing the extraneous load that it shall be

only at the joints between the members, thus subjecting them

0 forces acting at their ends and in the directions of their lengths,
onnected trusses (see t 175) the joints are practically flexible.

of the trusses m common use consist of two long members,
horizontal (but see H 49), called chords, extending throughout

1 and connected by web members, which are sometimes all in-
i,nd sometimes alternately vertical and inclined. Inclined web
s are called diagonals.

es and Struts. A member sustaining tension is called a- rod or
e sustaining compression is called a strut or post. One capable of
ig both tension and compression is called a tie-strut or a strut-tie.

3 dimensions of a truss are usually measured along the center lines
smbers; and, in pin-connected trusses, the pins are placed at the
ions of these lines. Hence, the measurements are usually made
»nter to center of pins."

a plate girder, the flanges are usually regarded as performing
ion of the chords of a truss, and the web as performing that of the
ibers of a truss.

«

oad companies and municipal corporations frequently prepare their
ge specifications; but the general proportions, number of panels,
>f ten left to the judgment of the bidders.

ere suppose the truss to be loaded vertically. If the load is other-

ied, as in the case of the wind pressure upon a horizontal bracing

pressure on the supports may be horizontal, or otherwise inclined

rtical, but all the internal stresses are still sustained by the truss

690

TRUSBEa

Loading.

9. Dead and Live Load. In bridges, we distinguish between the
"dead" and the "live" load; the dead load comprising the weight of the
permanent structure — i. e., of the bridge itself, with its trusses, bracing and
floor system ; while the live load comprises any temporary and extraneous
loads, such as engines, cars, horses, vehicles, foot passengers, etc., which
may come upon the bridge.

10. The dead load is usually distributed uniformly along the span, but
the loaded chord (that carrying the roadway) of course usually receives a
greater share of.it than the unloaded chord. ^ The live load comes only upon
the loaded chord. In determining stresses, it is usual to consider the weight
of live load and of floor system as being on the loaded chord, and the rest of
the dead load as divided equally between the two chords. It sometimes hap-
pens, however, that both the upper and the lower chords carry roadways.
They must then, of course, both be treated as "loaded," though not neces-
sarily equally loaded; for one may carry a railway while the other carries
only a highway.

Unsymmetrlcal Loading. Counterbraclng.

11. Unsymmetrical Loading. In Figs. 2 to 10, the loads are sup-
posed to be placed symmetrically.

IS. If this could be the case in practice, the compression members would
never be called upon to resist tension, or the tension members to resist
compression; and the. trusses in Figs. 2 to ^0 would suffice (supposing each
member to have sufficient strength), even though the compression members
were incapable of resisting tension and vice versa. Thus, the tension mem-
bers mi^ht be flexible chains, and the compression members might be posts,
merely abutting against supports at their ends.

P d

13. But in a truss, Fi^. 1 (a), with a flexible tie in the panel, n, as shown,
the load W, unsymmetrically pl«,ced, would cause failure, as indicated.

14. Coiinterbraclng. To prevent this, those members which, under
moving loads, may be subjected alternately to both tension and conopression,
may be so constructed as to be able to resist both kinds of stress. That is to
say, the tension members may be so stiffened as to be capable of acting as
posts, and the ends of the compression members so connected to the chorda
that those members can also act as ties. This is the expedient usually em-
ployed in trusses without vertical web members.

15. Counters. In trusses with rectangular panels, the distortion. Fig.
1 (a), caused by unsymmetrical loading, is usuaUy prevented by the intro-
duction of additional members called counterbraoes, or counters, in distinc-
tion from the "main" members, which last are designed to resist the normal
stresses due to uniformly or symmetrically distributed loads. Thus, in Fig.
1 (b) the unsymmetrical load, W, tends to convert the rectangle, p, into a
rhomboid, by lengthening its diagonal, W d: and this wa^ be prevented by
the introduction of an oblique tension member (counter) in the line of that
diagonal, as shown by dotted line. For a similar reason, such a counter is
inserted also in the corresponding panel, x d.

16* Triangles* It will be noticed that the introduction of counters
reduces the truss to a framework made up exclusively of triangle:

17. It might at first sight appear that the several parts of a bridge tnisB
must be most strained when covered from end to end with its maximum
load ; but this is true only of the chord and of the main diagonals and verti-
cals near the ende of the truss. The other web members may be more
strained by a part of the load, placed unsymmetrically on the truss; so that,
although correctly proportioned for a fuU load, they may be too weak for a

BRACING. 691

I one. If all be made as strong as the end ones, they will, it is true, be
r a passing load ; but this would require an expense of material that
be justified only in the case of moderate spans, especially of wood, in
the additional trouble and expense of gettmg out and fitting together
of many different sixes may more than counterbalance the saving in
&1.

In large bridges, where the live load is small, relatively to the dead
ut little counterbracing is needed, and that at and near the center
whereas, in a very light bridge, the coimters should extend from the
where they are most strained, to near the ends, where the strain upon
3 least.

Cross-bracing.

Bracing between Trusses. Advantage is taken of the proximity
wo or more trusses of a bridge, standing side by side, to connect them
9-bracing, thus giving to the entire structure far greater lateral stabil*
1 would be possible in the single trusses.

Thus, lateral bracing. Fig. 39, consists of horizontal trusses placed
I the two upper chords of the main trusses, or between the two lower
or both ; the chords of the main trusses acting also as the chords of
;ral trusses. The lateral bracing prevents latend deflection of the

Iway bracing, Fis. 64 (c) (called also diagonal, cross, vibration and
acing), consists of short trusses (usually vertical) crossing the bridge
*sely and thus connecting the two trusses. The sway bracing has
chords, but uses parts of the posts of the main truss as its end posts.

*ortal bracing. Fig. 54 (a), consists of sway bracing (usually in an
plane) joining the tops ox the end posts in trusses of sufficient depth
it its use. The portal bracing, with the end posts, forms a portal
which trains, etc., enter the bridge.

Types of Trusses.

he simplest form of truss consists of a single triangle, Figs,
d (b). In Fig. (a) the load produces compression in the rafters,
in the chord or tie rod,* and compression ( •- the tension in the
stween the heads of the rafters ; in Fig. (6) vice versa.

be truss shown in Fig. 2 (a) is in common use for roofs of small span,
filings. In practice, it is of course loaded along the rafters, and not
he apex as in Fig. (a) ; but, in calculating the stresses in truss mem-
commonly first assume that the loads are concentrated at the
ons of the member^. The eff.ect of their actual distribution cUong
bers is then determined separately, treating the members as beams.

WQ W

#<J7^

Tig. 2.

Fig. 3 (a) (called a King truss), the vertical tie (improperly called
•st), and in Fig. 3 (fc) the vertical post, simply carries the weight of
o the apex, i, where it produces the same effect as in Figs. 2 (a)

nee, neglecting the weights of the vertical tie and other members,
is, caused by a given load, W, in the diagonals, and in the horizon-
;. 3 (a), are the same (not only in character, but also in amount)
•roduced by an equal load, W, in Fig. 2 (a). Similarly, those in
correspond with those in Fig. 2 (6) .

•8. 2 to 12, and 14 to 17, double or heavy lines indicate posts or
I light lines indicate ties.

692 TBX7B8E8.

97* Pigs. 4, 6,; and 6, giving modifieations of the simple forms ahowi
in Figs. 2 and 3, illustrate in principle most of the bridge trusses in com-
mon use for spans up to 300, 400 or even 500 feet. See Figs. 7 to 10, 11 3fi,
etc.

28. In Figs. 4, 5, and 6, there is an upper chord, in compression, and a
lower chord, in tension;, the shorter chord sustaining the compression be-
tween the heads of the rafters. Figs. 2 (a) and 3 (a), or the horisont. 1 tcnaon

29. Figs. 4 (a) and 4 (6) may be regarded as showing Figs. 3 (a) and 3 (&)
respectively, with the vertical member, as well as the load, split in two, and

S laced symmetrically, so that the horizontal pressures, Fig. 4 (a), or tensions,*
ig. 4 (b), on the two ends of the shorter chord, are equfu, the two diagoniii
oounters in the center are unnecessary .

FI9. 5.

ao. Howe and Pratt Systems. In Fif. 5 (a) the vertical wA
members are in tension, and the diagoxials are m eompression, embodying
the "Howe" principle, used in bridges with wooden diagonals; while in Fig.
6 (6) the verticals are in compression, and the diagonals in tension, embody*
in^ tht "Pratt'' principle, used in bridges with metal diagonals. Insudi
bndges long compression members are objectionable.

(6)

dXQW <y\^^

Tig, 6.

81. Warren or Trlanirular Trusses. In Fig. 6, illustratinir the
"Warren" er "triangular" truss, the web members are all diagonal, aiid are
alternately in tension and in compression. They divide the truss profile
into iaoacelea triangles.

32. Through, Deck and Pony Spans. Figs. 4 (a), 5 (a) and 6 (a),
with the roadway on the lower chords, are called "through" spans, and Fus.
4 (6), 6 (b) and 6 (b), with the roadway on the upper chord, are called "deeir'
spans. The deck span permits the use of sway bracing (see t 21) between,
and throughout the depth of, the two or more trusses forming the bridge,
while the through span of course does not ; but the use of the through spaa
is often required, in order to give sufficient head-room for boats, floods,
trains on crossing roads, etc., below the bridge. A truss, loaded on the lower
chord, but too shallow for lateral bracing (see f 20) between the upper
ehords, is called a "pony" truss (or "pony through" truss).

33. Panels. The points where the vertical web members meet the
ehords, in Figs. 4 and 6. are called panel points; and the rectangular
spaces, a Tit need, etc.. Fig. 5 (a), between the verticals, are ealled panels.

34. The Warren truss, Fi^. 6, has no verticals, as essential parts of it. See
It 45 and 46. Its subdivisions are called simply triangles: and a panel is a
length of truss equal to the width of a triangle. A panel of either obord.
however, is that portion of it between two panel poinU.

TYPES.

693

^

35. Further modifications of these deeims, with more numffirous panels,
are shown in Figs. 7 to 10. Figs. 7 (a) ana 8 (a), with verticals in tension,
represent the Howe truss of Figs. 4 (o) and 6 (a), while Figs. 7 (6), 8 (b),
9 (a) and 9 (6), with diagonals in tension, represent the Pratt truss of Figs.
4 (b) and 5 (b). Figs. 10 represent the Warren truss of Fig. 6.

36* Fig. 8 (a) represents simplv Fi^. 7 (a), lowered so as to become a deck,
instead ofa through bridge; while Fig. 8 (6) represents Fig. 7 (b) converted
from a deck to a through span by being carried on vertical end posts.

37* In Figs. 8, the vertical end posts, and the horizontal piece at each
end of the loaded chord, form no part of the truss proper. The latter
simply act as beams, supporting the load during its passage from the abut-
nent to the truss and vice versa. The end post in Fig. (a) supports only
)ne end of this beam, while that in Fig. (&) supports half the tmss.

/MAfflNN% ^KKTsg/j/j/iT^

Throngb Howe.

(ft)

Flff. 7.

^22!^^^V^

Deck Howe.

Tbroaarli Pratt.

Fiff. 8

y\M^AA/\^ ^^^^^^^ ^

(a)
]>eck Pratt.

(W
Tlironirb Pratt.

Tig. 0.

^^^,^SZ^S2^

(a)
]>e«lK MTnrren.

ib)
Tltrongrli Warren.

Flir- 10.

D Figs. 8 the middle vertical carries no part of the load. Theoret-
serves merely to prevent deflection of the two unloaded middle

nels under their own weight ; but in practice such members are often
for the purpose of obtaining convenient connections for lateral

ich as floor beams.

Figs. 9 (modifications of Fig. 7 (b)) and in Fig. 10 are shown, in
, the most common forms of metal bridge truss, used as deck and
h spans respectively.

Fig 9 Co)., as in Fig. 8, the vertical end posts and the horizontal
!;be ends of the loaded chord form no part of the truss ; and in Fig.
D Figa. 8, the middle vertical supports only the unloaded middle
lels.

694

TRUSSES.

41. Intersections. In deep trusses, two or more sets of web members
are sometimes combined in one truss, with one pair of chords. Thus the
two simple Pratt trusses shown in Figs. 11 (a) and. (6) combine to make the
''Whipple*' or "double intersection Pratt" truss, Fig. 11 (c), recently
general use.

m

'•',j\/\/\/\4.

iPW^

Figr. 11.

(6)i

^

K ,.,b<;xxxxxx>i

Figr. 12.

42. Similarly the two simple Warren trusses, in Figs. 12 (o) and (6), com*
bine to form the double intersection Warren of Fig. 12 (c).

43. A combination of four systems is called a "quadruple intersection *
truss. See Fig. 59 (jt).

I t 11 V 1

^3l

LATTICE

Wn

la
A

Figr. 13.

44. The old Towne " lattice '* truss. Fig. 13, consisting of planks
crossing each other (usually at right angles) and bolted or tree-nailed to-
gether at their intersections, may be regarded as a combination of several
Warren trusses.

V

i m

'^\ywy\i/'<j7^

(a)

Fi§r. 14.

(*)

45. Sub-verticals. In deep trusses, where the horizontal spread of the
panels is considerable, sub- verticals, v, Figs. 14 and 15, are often used, esj
cially in Warren trusses, to support the segments of the loaded chord,
also Figs. 69 (i), (r), and («).

^ V V V V V V V V ^ }f^ V V V V V V ^

(a) (fr)

In the "Baltimore" truss. Fig. 15 (&), each diagonal is braced, at its
middle point, by a short diagonal strut inclined in the opposite direction, and
a sub-vertical is suspended from their junction. With very long panels,
sub-verticals are sometimes used for the panels of the unloaded chord also.
See Fig. 15 (c).

TYPES.

695

46, Collision struts, or collision posts, S, Figs. 50 (k), (rn), (o).
and (0, and 73 (a), are used for bracing long diagonal end posts against a
blow from a derailed train.

(a)

Fiflr. 16.

47. Fink and BoUman Trusses. Figs. 16 show two obsolete modi'
ficationsof Fig. 3 (6), vis. : the Fink, Fig. 16 (a), and the Bollman, Fig. 16 (6).
The large bridfge over the Ohio River at Louisville, Ky., completed 1870, is
of the fmk type. The Bollman was largely used on the Baltimore and Ohio
JRailroad years ago.

48. In the Fink and in the Bollman truss there was but one chord, as
ihown. This chord usually carried the roadway. Where the roadway was
placed lower, it gave the truss the appearance of having two chords. Under
iniformly distributed loads, in the Fink, and under all circumstances in the
Mtman, the stress in this chord Was uniform throughout. In the BoUman
see Fig.), the longitudinal stresses in the chord were all applied at its ends.
)aeh type may be regarded as a combination of several suspension trusses like
ig. 3 (6). In the Bollman, the simple trusses were all of the same span and
3pth; and each vertical post, except the central one, divided its simple
uss eccentrically. The Fink principle is still largely used in metal roof
usses. See Figs. 26.

Fig. 17.

9. Curved Chords* Trusses with curved or "broken" chords. Fig. 17,
frequently used for long spans. The members themselves, between panel
Its, are always straight. In the bowstringT. Fig. 17, the panel points
!ie upper chord lie in a curve, convex upward. In the crescent truss,
lower chord also is convex upward. The bowstring truss has the ad-
;age, over those with horizontal upper chords, of making all the chord
web stresses more nearly equal, tnus simplifying the construction and
cing the weight of the trusses. It has the disadvantage of permitting
irernead bracing near the ends of the span. If the curve of the upper
I is made parabolic, the dead load stress is uniform throughout the lower
U and in each vertical (uow in tension) the stress is equal to the dead
>n the lower chord. The diagonals receive no dead load stress, but
.lied into action only by eccentric loads.

Figr. IS.

le JSunr tmss^ Fig. 18, at one time much used for wooden bridges^
tbination of a Howe truss and an arch.

696

TBUSSEB.

Camber.

51. Camber. In practice, the members of the upper and lower ehords
of bridges are not placed perfectly in line, butso that the chords curve slie^tly,
with the convex side upward. This curve is called the camber. Its object
is to prevent the truss from bending down below a horizontal line wnen
heavily loaded. When the chords are cambered (see y • and c d. Fig. 19),

they become approximately concentric arcs of two large circles, of whieh the
center is at t; and the upper one plainly becomes longer than the lower.
The verticals, instead of remaining truly vertical, become portions of radii of
the arcs mentioned ; and, although their lengths remais unchanged, yet their
tops are farther apart than their feet; and this renders it necessary to
lengthen the diagonals. See ^^ 211-214.

Cantilevers.
52. The cantilever principle is shoWn in Fig. 20, where A and B

T

Flff. 30.

represent counterweights or anchorages. Fig. 21 shows the Niagara canti-
lever bridge. It consists of two cantilever trusses, ab, a* h\ connected by
an ordinary truss, 6a', which is suspended, by a vertical link at each end,
from the ends of the cantilever trusses. The weight of the truss is counter-
balanced by anchorages, A and B, or by weights, or both. The principal
advantage of the cantilever is that it may be built outward frooi the
piers across the channel. It thus greatly facilitates erection in oases where
false-work cannot well be used.

3Iovable Bridges.

53. Movable bridges, including draw, swing and lift bridges, are of
three general classes ; one in which the movable part slides horisontally, one in

Fiic. 22.

which it swings horisontally, and one in which it swings vertically. Ordi-
narily, the movable span is pivoted near the middle, and swings horisontally
on the central" swing" or "pivot" pier, as in Fig. 22. In such cases it M

SKEW BBIDOE8.

697

usually mounted on a central pivot, or on a nost of rollers or wheels running
00 a circular track. Such a bndge must be so designed that, when it is swung
epen. or if it is not brought to a bearing at the ends when closed, it shall
sustain not only its own weight, but also any other loads that may come upon
it. In addition, each half must be able to act as a bridge supported at both
wik, with all possible live loads; for, as an unbalanced live load comes on
either end, that end will be brought to a Iv^aring. In elaborate bridges, pro-
vision is made for raising the ends cf the draw span, when dosed, thus
bringing both ends to a firm bearing, and the floor flush with that on the
adjacent abutment or fixed span. This raising is usually made sufficient
to relieve the middle pier of only a portion of the load. The bridge then
acts like a "continuous " girder (see Transverse Strength, If^ 78, etc.) sup-
ported at three or at four points, depending upon the arran^sement of the
beanng on the pivot pier.

54* Drawbridges in which the movable part swings vertically, may either
voive about a pivot, or they may roll, as m the ScbiBrser rolling lift bridge,
9. 23.

5S, Skew bridges are used where a channel, road, etc., is crossed ob-
uely, and where it is inconvenient to have the abutments perpendicular to
I trusses. For simplicity in making floor connections, etc., the truss is
lally so designed as to Imng the nanel points opposite each other, as in
s. 24 and 25. Where the skew is out slight, this necessitates a difference

Plan

□

»'4^^SS^

Fiff. 94.

lination between the two end posts, as in Fig. 24, involving compliea'
1 the conneotions for the portal bracing. But where the skew is greater,
r be possible to make it just equal to one or more even panels, adjusting

Plan

^T^

(ft)^^

Eltvation
Flgr. 25.

mm

el lenfsth. to suit, and thus leaving each truss symmetrical, as in Fig.
ea.cb figXiTe, those members which belong only to the farther truss
»er one in th.e plar.) are shown by dotted lines.

698

TRUSSES.

Boof Trusses.

56* Roof trusses are made in a great variety of forms. Those shown in
Figs. 26 are common. In Fig. 26 (fi), part of load, at d, compresses the rafter
from d to a, while the remainder compresses the strut, d h, and pulls the
rod h i and the part-chord h a. Similarly, part of c passes through c a to
a, and the remainder through c k d h % to the apex i. Thus each load
is eventually carried by the mem'bers, part to the apex and part along a
rafter to an abutment. It will be seen that the greatest stresses in the rafters
and in the chord occur near the ends.* Sometimes the members shown

Flgr. 26.

vertical in Fig. (a) are inclined, or the lower chord is "broken," being usually
convex upward. Roof trusses are often composed, as in Figs. (6) and (c), of
two Fink trusses, inclined, and leaning against each other, their feet being
held in position by a tie, m n, and the rafters forming the upper chords of the
Fink trusses.

STRESSES m TRUSS MEMBERS.

General Principles.

57. Conditions of Equilibrium. In trusses, as in beams, it is neces-
sary and sufficient, for equilibrium, that the internal stresses, and their
moments, shall balance the external forces and their moments. The exter-
nal forces (viz., the loads and the end reactions) and the resulting moments
and shears, are discussed under Statics, ^1f 286, etc. We here discuss the
determination of the internal stresses. For the fundamental distinction
between beams and trusses, see Trusses, 1 3.

58. In general, the stresses in the members are found by means of the
principles of moments (Statics, If 301, etc.), and of Shears (Statics, If 325,
etc.), making use of the force parallelogram (Statics, 1[t 35, etc.) or force
triangle (Statics, tif 46, etc.), the force and cord polygons (Statics, 11^ 72,
etc., 86, etc.) and the influence diagram (Statics, 1ft 339, etc.).

59. A very convenient method, and one in common use, is that described
more fully in ft 67, etc., below, where the truss is considered as being cut
through by a section. We then seek to ascertain what stresses, in the mem-
bers so cut, would be required to preserve equilibrium.

60. Before the stresses can be calculated, and the truss proportioned to
those stresses, its weight must be known; for this constitutes a load, and there-
fore affects the stresses. But, on the other hand, we cannot learn its weight
until we know the sizes of its different members. In this dilemma we must
assume for it an approximate weight, based upon our knowledge of some-
what similar trusses already built. This becomes the more necessary as the
truss increases in size, so that its own weight becomes greater in proportion
to that of the load.

* If the diagonals were parallel, their stresses, and those in the verticals,
would be greatest at the center of the span, and least at the abutments.

STRESSES.

699

1. To dtotlnsrnlsh between ties and struts; from the point, o.
}. 27, where the force is applied, draw o c to represent the applied force,
le direction in which that force tends to move the point, o; and upon o c
diagonal construct the force parallelogram, a 6. Through o draw 1 1 jparal-
0 the other diagonal a b. Then, if a piece be on the same side of i i with
it is a stmt; while, if it be on the opposite side, it is a.tie.

o -•'•

(«)

rtgr. 27.

»)

. Ties and struts may often (as in Fig. 27) be readily distin^ished b^
ction, by imagining tne piece to be flexible, like a rope or cham. If it is
that it would then resist the force acting upon it, the member is a tie;
, it is a strut. Or, suppose that the piece is not secured at its ends.
3n, it is seen that it would resist the force acting upon it, the member is
it; if not, it is a tie.

Or we may proceed as follows: In Fig. 28 (a), representing joint a,
gin with the known net vertical reaction, R * ; and find the unknown
es in the chord and in the end post by means of the force triangle,
tg their arrows follow the known direction of R. Transferring these
3 to the respective truss members, Fig. (cI), we find that the chord
iway from a, and is therefore a tie ; while the end post pushes toward a,

therefore a strut.

M

Flff. 28.

n Fig. (6), representing joint b, we draw P upward to represent the
3 of the end post toward b; and the other two sides of the force tri-
ve the pressure in the chord member, Q, and the tension in the tie, T.

n Fig. (c), representing joint e, we know T, M, and the load, W, and
in the tension, N, and pressure, S, in the corresponding members.

imucli as half of each end panel rests directly up>on a support, and
is nothing to the stresses in the members, we must, in determining
resses, use only the

net reaction "■ reaction — half panel load.

700

TRU8SE8.

66* Tentile sitresses, because they tend to elongats a member, are con-
ventionally regarded as po$itive, and designated, oy +, while compreaaiv^
stresses are regarded as negative, and designated by — .

Method by Seetlons*

67* Let Fig. 29 (a) represent a roof truss, with three equal loads, W. of
2 tons each, applied at a, e and b, respectively, and let it be required to find
the stresses produced, by those loads alone, m the members a e and a iL
Suppose the portion shown in Fig. 29 (6) to be separated from the rest 6f the
truss, as shown, by cutting throu^^h the members a e and a d. The lower
portions of those members, shown m (6), are, however, suppoeed to be held
m their original positions by the stresses &« and S^, exerted in these mem-
bers themselves. Taking moments about the right supix>rt( fr. Fig. 29 (a),
we have, for the upward reaction of the left abutment, a,

K - y.

68. We have, then, at a. Fig. 29 (b), four forces, as follows: two known
foroe^ vis.: W, vertically downward, *■ 2 tons, and R, verti«dly upwaid,

» —K-** uid two unknown forces, &« and S«. Now S« makes a known

angle, A, and Sd a known angle, B, with the vertical. The vertical foroes,
W and R, have, of course, no horiaontal resolutes (see Statics, ^% 54, etc.) ;
and their vertical resolutes are the forces thMnselves.

60. The horizontal resolutes of the inclined forces, S, and S4, are, re-
spectively: S<,.sinA, and Sd-sinB; and their vertical resolutes are:
Se.cosin A, and Sd.cosin B.

70* We see, by inspection, that the stress, So» in the rafter, a e, is com-
pression, and that the stress, Sd, in the lower member, is tension; but, for
convenience, we may at first assume, in advance, that all of the unknown
stresses are tensions or +. Then those which finally appear as + are known
to be tensions, and vice versa. Their horizontal resolutes, in this case, are
therefore both taken, for the present, as being right-handed, or positive; and
their vertical components upward or positive also. It will be remembered
(see \ 66) that we regard tensions as positive, and compressions as negative.

71* Now, in order that the four forces at a, viz.: W — 2 tons, downwardg

3W
B — -^, upward, S, and Sd, may be in equilibrium, it is necessary:

(1) that the sum of their horizontal resolutes be aero, or
S..sin A + Sd.sin B -= G;

(2) that the sum of the vertical resolutes be zero, or
R — W + S,.cosinA -f- Sd.cosin B - 0.

Thus, let A

45*», sin A

0.707 J cosin A — 0.707.

B " 75", sin B - 0.966; cosin B - 0.259.

Then 0.707 S„ + 0.966 Sd - 0 ;

R — W + 0.7O7 S, + 0.259 Sd

0;

— 0 966 Sd — 0.259 Sd — R + W
^ " 0.707 " "" 0.707

0.966 Sd — 0.259 Sd - 0.707 Sd - R — W.

CHOBU STBESSES.

701

70. Again, in Pig. 30, with section uv, stress in srf-iWi — R — 6 —

15 «■ — 9. With section vy, stress in e/ •« — ' — r- — - — s-

cos 0 cos 0

W, + W, — R —3

tionux,stnaa'mgd

With sec-
It will be seen that these

cos B cos B'

forces, all acting downtpord on the part truss to the left of the section,
give tension in ea, and compression in ef and ^d.
With section uz we out <um> web membeFS, gd, and i^e; but the stress in gd

has already been found «■ 3. thQ vertical component of which is » 3.

cos o

Hence, stress in flc •■ Wi + Wj + Wj + 3 — R — 6.

73* It is, however, evident from inspection that the middle vertical bears
nmp^ the middle load, W3 = 6 ; for, cutting the truss by a curved section,
as at c, and examining the small portion thus cut out, we see that we have
but two vertical forces — viz., the central load, Wg, and the stress in the
Vffl^ical member; and, for equilibrium, these two must be equal.

FIff. 80.

Chord Stresses, Moments.

74* For the chord stresses, Fig. 30, let P -> panel length - 10 ft. Then
he bending moment at the panelpoint, d, is

M = 2RP — WiP
=» 15 X 20 — 6 X 10
= 300 — 60 «- 240.

Cutting the truss by section uv, we find that, of the three members cut,

ly the upi>er chord member, eg, has a moment about d. Call its stress S.

(leverage is the depth, D, of the truss, •- 12; and, for equilibrium, S D •■

„ c, M 240 -^

Hence, S «=■ g — -jn "" 20.

fS» Similarly, taking moments about e, we find the stress, in the lower

240
rd member, / d, cut by the section, u t;, to be -r^ "» 20, or the same as

stress in the upper chord panel cut by the same section. Inspection
▼s the correctness of this result ; for the diagonal strut, e f, evidently
t^ers to the up{>er chord panel, « (7. a compressive stress or "chord incre-
t " (see if 77) =- the tensile stress which it delivers to the lower chord
J» / d.

•• If the chord members are inclined, their lever arms must of course be
'ured perpendictUarly to them; and we can no longer use the vertical
1 of the truss as the lever arm.

, Cliord Increments. Fig. 30. Each diag delivers a comp stress to
pper chord, and, in trusses with parallel chords, an equal tensile stress
9 loiter chord. Find the shear, or vertical component, Vi, Vs, etc., of
ress in each diagonal, beginning with the end post. Then the "chord
aents," hu h^» etc., or the stresses in the chord members, 0/ and me,
I e0, etc.» due to the several diagonals separately, are

Ai - Vi tan B
^ = V2 tan B
As =« Va tan B

or the total stress in each -chord member, we have, Hi »» hi;
vi -i- hal Hb — Ai + Aa + A«; and so on.

oluDge. Thus, iQ

the vertloaJ, lu; thai, on me iFii

Uw loads ioduilw the diixouai.

1 Bhoar diacram applies lo all thass tnemben throutA
i, up to the partdf point where the shear underepes a
31 the ahear diagram oaths right of the Fig. includes

/mxixi/i/\

Fig. 81.

79. Shear Influence DIaRram. Bee Statics. H 325, eta. to a
fe^ui. Fie;. 32, the ordinates, if e. etc., to the line a' b' (ooaatruded u in Fis.
156, 8tatii!3. 1 349), give the left end reactiont: and those, rf h. eW., to the

resulting "AMrs tor s load, W (not ahown], at onu pamii point: but the
theart in a potxl, cd, for a load, W, hctwten the panel points, are modified bf

Sinta, aa indicated by the inQucnce hnen, qk, etc., inr the several paiuU^
lus. vith W at c and at d. respectively, the shear, in the Panel cd. is nipre-
anted respectively by e'ft (negative) and by d'q (positive) ; and, as the load

^ d. the shear in the panel or

% to d't.

A

Z^

\

X

\

a'

1

^A^

\

c

Q

a

But when W Is placed at
distributed between the panel

SO. Thus, drawing, for this
Fig. 16B). the panel influence II
d into the panel cd, sa to e. the ti
(ram d't to 4'k 1 but at the san

r the whole beam in Btatiea,
e see that, as W moves from
ia thereby atightly increased
oE W, represented by s't, is

WEB STBESSES.

703

aarried by the stringer to e, where it diminishes the shear e'/e (due to the truss

reaction at a), leaving tk as the value of the shear in the panel. As we place

W successively at other points, farther from d, and approaching o, the load

carried by the stringer to c, and represented by the ordinates from c'd* to

/<f, continue to increase faster than does the left end truss reaction, R,

represented by the ordinates from a'b' to a"!/ ; and the resulting shears in the

panel are represented by the ordinates from id* to iq. At o, the part load,

(/j, carried to c, is » the left truss reaction, and the shear in the panel is zero.

With the load between o and c, the part loads carried to c, and represented by

the ordinates from </ c', to if, are greater than the corresponoing left end

truss reactions: and the result is a negative shear in the panel, indicated by

the ordinates from i g to if. It will be noticed that tne resulting shears

throughout, both positive and negative, are indicated by the ordinates from

'/d' to hg*

Keversing the process, a similar argument may be ap{)lied to the panel
afluence line &», oeginmng with the load at c, with negative shear in panel
= i/h, and supposing it moved across the panel to d, where positive shear
1 the panel becomes =- d'q.

8U In the case of a uniform load* extending on to the span from the
ght support, b, the point o is the position of head of load for maximum pqsi-
ve shear in the panel, cd; for, in the case of a uniform load, the shear, with
»d of load at e, is represented b^ the area (sum of all the ordinates)
nqf/e'; and manifestly this area mcreases as the head of the load ap*
oaches o; but when it reaches o, the area above a'&' can increase, no further,
d when it paeeee o, the negative shears, represented by the ordinates from
' to 0% begin to reduce the resultant positive shear.

^2» Having found, by any method, the maximum shear, d'g, due to a
icentratftd load at d, for the diagonal, d n. Fig. 32, and the reverse maxi-
m shear, c' A, due to the same load at e, we may draw an influence
i. h g, which gives, as before, the point, o, of position of head of uniform
1 for maximum stress in the diagonal, d n, from which (as above) we find
corresponding position of the head of a series of concentrated loads.
Q practice, the influence line for shear is of value chiefly in thus
ing the position of load producing maximum stress, and the resulting
sses, in trusses with curved chords, such as Fig. 17. In such a truss,
ig to the inclination of the members of the upper chord, those members
some of the shears in their respective panels, and the stress in the diag-
is therefore less than the shear in the panel.

Graphic Determination of Dead Load Stresses.

• Construct first a diagram of the truss, as in Fig. 33 (a), lettering the
fl between the members, and those between the arrows representing
sad loads. Call the end post, 1-3, between A and B. ' ' AB, '' the stress in
>," the load at 2, **cd/* etc., using capital letters for panels and truss
lers, and small letters for loads and stresses. Adopt a suitable scale of
, and construct the diagram, Fig. 33 (b), as follows:

j5 <y 1

L '-f J

0 I

r J

.

X

X

/'

\

o *S J> ^

^ M. \

' X i

i s ^

\ ^

Flff. 33.

Diisider first the point 1, Fig. 33 (a). There are here three forces
3riura, viz., <xc, ab and be. Find the net end reaction, R — ac, and
up^ward Cslnce it acts upward on 1) from any convenient point, a, to
' (20* From a draw an indefinite line ab parallel to AB and from e

Since fh^ io', qd' and k^ are parallel, me' « ktf and ^h «» gf.

704

TRUSSES.

draw ch parallel to BC, obtaining the force triangle acb of the pomt 1. The
lengths of cb and 6a then give the stresses by scale.

85. In Fig. 33 (6), the arrow on ac indicates the upward direction of that
force. Following around the triangle, we affix arrows (in the same direction)
to cb and ba. Supposing these arrows now to be transferred to the corre-
sponding members m Fig. 33 (a) we see that b e pulls from the point 1, show-
ing that & c is tensile, or +, while b a pushes toward 1, showmg that 6 a is
compressive, or — .

86. The characters of the stresses may be found more guiekly as follows:
Draw a circle. Fig. 33 (c), and place on it arrows pointing around in the
direction (counter-clockwise in this case) followed around the truss in
constructing the load line. See % 92, below. Then consider any panel point.
Fig. 33 (a), and follow the letters in the spaces around that point in the direc-
tion of the arrows on the circle. Note the order of the letters, and follow
the corresponding equilibrium polygon, Fig. 33 (&), around in the same direc-
tion. This will give tne directions in which the forces respectively act on that
point.

87. Thus, consider the panelpoint 2. Following around 2 in the direc-
tion of the circle, we read B, C, D, E. Turning now to Fig. 33 (6), and
reading b, c, d, e, we find that on be we go from right to left (or opposite
to the direction indicated by the arrow drawn for point 1) ; heuce be acts
to the left on 2, and BC is therefore in tension, aud its stress 2m; is +.

(

SG 1

H J

O J

r 1

,

X

\ J

X

/■

X

JUL C \

Irg J> >

V K 1

[ X

[ s \

1 T

(a)

<c)

Flgf. 33 (repeated).

88. Given now the stress, be, in BC, construct, on &c, the force polyison
bcdeioT the four forces acting on the point 2. Thus, from c lay off tM down-
ward, to represent the dead load on the lower chord at 2. Since he acts as a
pull from tne left on 2, and since the forces must follow each other around the
polygon, cd must evidently be drawn downward from c and not from 6.
From d draw an indefinite line parallel to DE, and from b another, parallel
to BE. They will intersect at some point, as e, and eb and de will then repre-
sent the stresses in BE and DE.

89. Inspection would show that be *» cd, since cd is the only force acting
on 2 with a vertical component, and that be " de ; but the oonstructionm
the force polygon bade is necessary for the completion of the diagram.

90. Having now found the stresses in DE, BE, and AB, and knowingthe
panel load ( = g a) at the point 3, construct the polygon g ah e f g. This
gives e f and / g, and from these the process may be continued and the dia-
gram completed.

91. It will be noticed that, in some cases, a point on the dia£^m. Fig.
33 (&) , is given more than one letter. Ordinarily this is simply a comcidenoe,
arising from overlapping of the force polygons. In some cases, however, the
coincidence of the letters shows that the stress in the member is zero.

92. In practice it is usual to construct first the entire load line <i. thus:
draw first the net reaction, ac, upward; then, following around the truss
counter-clockwise, draw all the other exterior (dead load) forces in their
proper order, thus cd, dk, kl, U, st, tv, vp, po, oh, kg, ga. The stress diagram
may then be constructed, as before.

UVK LOADS.

706

Live Lioads.

93. It might at first be supposed that each member of the truss would
receive its maximum stress when the train completely covered the bridge;
but this is true only of the chord nLenibers. In the truss shown in Fig.
33 (a) each web member receives its maximum stress when the greatest
possible shear occurs in a section cutting that member.

Tig. 34.

94. In Fig. 34, the main diagonals to the left, and the counters to the right
of the center, C, are shown. Any one of these members receives its maxi-
mum stress from a uniformly distributed load when the load extends from it
to the right support b, with head of load at a point, o, Fig. 32 (a), found
as in ^ 81 ; and vice versa {or the diagonals inclined in the opposite direction.
Each vertical receives its maximum stress when the load extends from the
farther support to a point, o (see 1[ 81), in the panel beyond the vertical.
This statement must be slightly modified when the concentrated wheel
loads are considered. See 11197, etc.

Wig, 35.

95. Assumed Uniform Llye Load. As a crude approximation, the
engine and train are sometimes considered as a uniform load crossing the
bridge, Fi^. 35; but this method, ignoring, as it does, the great concentration
of weight m modern locomotives, is ai>t to be either unsafe or wasteful of
material. This assumption is< proper in connection with wind pressure on
train. See t 121.

7K

Mllll.o.lllllh- >:illllMI lllllll 'llllllll

¥//

FIgr. 36.

96. Concentrated Excess Loads. Again, to provide for the locomo-
tive loads, one or more concentrated excess loads. Fig. 36, are sometimes
employed. The stresses due to these loads may be computed separately, and
added to the stresses produced by the uniform live loads. To produce the
maximum chord stresses, the excess loads should be in the middle of the span,
and the train load should cover the entire bridge. This method is fairly
approximate, and engineers are divided as to whether this method of con-
centrated excess loads, should be used, or that of the actual or "typical"
locomotive wheel loads as explained below.

97. Standard or Typical ** Wheel" Loadings. In the method of
wheel loads, the actual stresses, produced by the heaviest engines likely to
cross the bridge, are considered. Even in engines of nearly the same weight,
the loads may be dififerently spaced, and spaced at intervals of odd fractions
of an inch, rendering computation very laborious. For this reason, and in
order to provide for the use of heavier engines in the future, it is customary to
consider an imaginary or "typical" engine, with loads and spacing given in
round numbers, the stresses from which shall at least be equal to those pro-
duced by the heaviest engines likely to be used during the life of the bridge.

The live loads are ordinarily taken as consisting of two typical locomotives
with their tenders, followed by a uniform train load. See Digests of Speci-
ications.

4.5

706

TRUSSES.

98. The following is an example of the computation of live load stresses by
the method of locomotive wheel loads :

Fig. 37 (6) represents the loads on one rail, corresponding to Cooper's
Standard,* Class E 40, which consists of two coupled consolidation loco-
raotives, followed by a train considered as equivalent to a uniform load of
4000 lbs. per linear foot. In the diagram. Fig. 37 (a), all loads are figured
in thousands of pounds, moments in millions of foot-pounds, and distances
in feet.

(W

MX

2 S

n

s

QD.

6 7 8 9 to

11 12 IS H

OQOO

isie n 18 ^

Qo.Qo r

10 9029 9020 12 in *31S 10 2020 20 20 iaWTsW

Zioads in thouadnds of pounds

Tit;. 37.

9 pet foot

00. Live Load Web Stresses. The maximum live load streflses

will occur in the web members of any panel of the truss in Fig. 34 or 38,

when the live load produces the maximum shear in that panel. It can be

W

shown that this will occur when P = » where P = the live load

n

on the panel cut by the section; W » the total live load on the truss,

and n = the number of panels in the truss. This equation is called the

criterion for maximum shear.

100. The following table is based upon this relation. The seoond col-
umn is obtained by adding successive wheel loads to P. In this case,
W — 6 P, since our truss has 6 panels. Let any wheel be at a panel
point. Then, by moving the wheel a little to the left or right, it will be
included in or excluded from P. Hence P and W have each a mimmam
and a maximum value for each wheel at the panel point.

No. of wheel at any

given panel

point.

1
2
3
4
5

Value, P, of load on

panel to left of

given point.

0 to 10,000
10,000 to 30,000
30,000 to 50,000
60,000 to 70,000
70,000 to 90.000

Corresponding value ci

W tor maximum

shear in panel.

Oto 60,000

60,000 to 180,000

180.000 to 300,000

300,000 to 420.000

420,000 to 540.000

101. The correct position of live load, for maximum uhear In any panel,
is found by successive trials. When the correct position is found* the

♦••Transactions Am. Soo. Civ. Engrs.," vol. xui, No. 868, Dec., 18W,
p. 227. See Digests of Specifications.

LIVE WEB STBESSE8.

707

moment »bout the right support is computed, and from this the shear is
obtained. For example, see below.

103. These operations may be performed by computation, with or with-
out the aid of graphic methods. As the ixMthod of computation alone is
rather tedious, particularly when the form of the truss is complicated by
curved chords or sub-panels, and as the graphic method is abundantly
accurate for all practical purposes, and has the advantage of direct appeal
to the eye, only the latter is given herewith.

103. The "wheel diaeram." Fig. 37 (a),* gives (1) a stepped "load
line" or "shear diagram/* and (2) a curved "moment diagram" or
"equilibrium polygon." See Statics, H 359, etc.

104. The load line gives the total live load to the left of, and including,
any point.

105. The moment line gives, at any point, the (left-handed) live load
momenttSibout that point, of all loads to the left of and including that
pmnt. Thus, to the left of and including wheel No. 5 we have

Wheel.

1
2
3

4
5

Load.

10,000
20,000
20,000
20,000
20,000

Distance from
wheel 5.

23
15
10

6

0

Moment about 5
in ft. -lbs.

230,000
300,000
200,000
100,000
0

Total, 90,000

830,000

and the ordinates, ab to the load curve, and ac to the moment curve,
under wheel 5, measure 90.0 and 0.830 respectively.

106* Fig. 38 represents the truss, to the same scale as Fig. 37. We may
call this a " truss diagram." f

B

c

D

E

F

\

\

\
\
\
\

\

\

\
\
\
\
\
\
\
\

N

» I

i t

» CI

I i

» i

g

Fig. 88.

»

107* Example. To compute the maximum shear in the panel he, Fig.

38, first find that position of the load which will produce that maximum

shear. As a guess, place the truss diagram. Fig. 38, t with its point c under

wheel 2, Fig. 37. Examining the load diagram, over the right end, g, of

the span, we see that we now have a total load, W, of 284,000 lbs. on the

span; and the load diagram, over wheel 2 (placed at point c) shows (see

also table, 11 100) that the load, P, on the panel, h c, is now somewhere

between lO.CJOO and 30,000 lbs.; but, for maximum shear in the panel, 6c,

the load, P, on that panel must be (see It 99 and 100) = W -5- n =

284,000 -5- 6 > 30,000 lbs. Hence, P must be increased by moving the

train diagram, Fig. 37, to the left (or, which is the same thing, by moving

the truss diagram. Pig. 38, to the right) until wheel 3 is over c. We now

have W = 292,000 lbs.; P = anywhere between 30,000 and 60,000; and

required value of P, for maximum shear, = W -i- n = 292,000 -r- 6 =

^8,667 tbs. Hence, the conditions are satisfied, and panel fee receives its

♦Method published bv Ward Baldwin, "Engineering News." vol. xxii,
Jept. 28, 1889, p. 295. See also letter, " Eng. News," Dec. 28. 1889, p. 615.

f For the following discussion it will be found convenient to make a
lopy of Fig. 38, or simply of the lower chord, on a separate piece of paper
irhich may be applied, in different positions, to Fig. 37.

708

TBUaSEB.

maximum shear, when wheel 3 is at c. The moment diagram shows, ver-
tically over g, the live load moment about the right support => 17,516,000
ft.-tbs. ; and the moment at c = 233,000 ft.-tbs.

108. Let M = the (left-handed) live load moment at the right abutment,

due to all the loads on the span.
L = the span.

m = the (left-handed) live load moment at the panel point on
the right of the panel in question.
I = the length of the panel.
V = the shear in the panel.

Then V = -^ _ -^ = 17.516.000 _ 230.000 ^ ^^^^^ ^^,

m

160

25

10, »)ltQ20»> V5/3 t3t3 1(t i020 20 80 1313 13 f 3

Xioacts in thousands of pounds

Figr. 37 (repeated),

iper/ooP

109. The maximum live load shears in the other panels, similarly com-
puted, are as follows, the load being, in each case, so placed as to give
said maximum shear:

Panel, No. Mom. M Mom. m
and Posi- at Rt. End i at Rt. End

tion of
Wheel.

ab 4 at b
2>c 3 at c
cd S &t d

de 2 tit e

ef 2 at /

of Truss.
Ft.-tbs.

27.176.000
17.516.000
10,816,000

4.936.000

1.743.000

of Panel.
Ft.-lbs.

480.000
230.000
230,000

80.000

80.000

Shear,
Pounds

L

162.000

107.600

63.000

29,700

8.400

Stress.
Pounds.

s = V

COS •!

—217.200
+ 144.200
-63.000
+84.500
-29,700
+39.800
+ 11.250

M

m
I

* Because --— = the reaction of the left support, a, and —7— == so much

of the panel load as goes to the left end of the panel.
1 9 •" angle between diagonal and vertical.

LIVE CHOKD STRESSES.

709

110. The Ifve load stress in the htp suspender, B&, is due entirely
to loads upon the two lower-ohord panels, a b and b e. Thus, with wheel
4 at b, panel length =.ab = 6c => 25 ft., we have:

On ab

On 6c

Stress
Dist, d, on B6 =
Wheel. Load, W. from a. M>d-l-25.

Wheel. Load, W.

Dist, d,
from e.

Stress

onB6 =

wd -i- 25.

1 10.000 7 2,800

2 20,000 15 12,000

3 20,000 20 .16,000,

5 20,000

6 13,000

7 13,000

20

11

6

16,000
5,720
3.120

Total, 50,000 30,800

Total; 46,000

24,840

Total load on a 6 = 50,000

6c= 46,000

Wheel 4 = 20,000

Stress in B 6 from ab =
** «« " "6c =1
" wheel 4 =

30,800
24;840
20,000

Total load on ac = 116,000

•« <i *«

Total.

75,640

111. For any given set of loads on a e; the maximum stress in B 6 oc-
curs when the load on a c is equally divided between ab and 6 c; and this
ordinarily occurs while some wheel (to be found by trial) is passing 6.
Thus, with wheel 4 just to the rio?U of 6, we have, on ab, wheels 1, 2 and
3, » 50,000; and. on 6c, wheels 4, 5, 6 and 7, =» 66,000 lbs.; but, with
wheel 4 just to the left of 6, we have, on a 6, wheels 1, 2, 3 and 4, ==
70,000; and, on 6 c (neglecting wheel 8, which now enters 6 c), wheels 5. 6
and 7, =» 46,000 tbs. Hence, while wheel 4 is passing b; there is an in-
stant when the loads on a 6 and on 6 c are eoual. and at that instant the
stress in B& reaches its maximum (75,640 lbs., see f 110) for the given
set of loads.

112. Live liOad Chord Stresses. The criterion; for position of load
for -maximum bending moment in any section, and nence for maximum

stress in the chord members at that section, is — -=-, or Z =- L.-r^^r- ;

iff I W '

where W ■• total load on the tmss, v> — load to the left of the section, L =»

span of bridge, and 2 — length of segment to the left of the section.

113. To find the position of load for laaximum moment in any panel, by
means of the moment diagram^ Fig. 37? place a wheel, say wheel 2, at the

1)anel point at the right of the given panel. From the intersection on the load
ine (usually coincidmg with the X-axis) vertically over the left support, lay a
Tuler or stretch a thread to the intersection of the load line with a vertical
from the right support. If the line so constructed recrosses the load line at
a point vertically over the section in question, the position is a correct one;
if not, it is inoonect. To facilitate this work, it is well to use a truss diagram.
Fig. 38, drawn on a sheet of tracing paper, with the verticals carefully ex-
tended from the panel points as far up as the load or moment lines are likely
to extend.

114. It will often be found that morethan one position satisfies the cri-
terion, and that some one of these may give greater moments than the others.
Hence it is well to look for all possible i>ositions. When these are found, de-
termine the moments, thus: On the moment curve find the two points
corresponding (vertically) with the left and the right support respectively,
and join these points by a straight line. When the head of train has not
reached the left support, the point corresponding with the left support is in
the :B-axis, produced.

115* The required moment is measured by the vertical ordinate distance
along the section, between the moment curve and the straight line j,iist con-
structed. The stress in the chord members affected is equal to the moment
divided by the depth or the truss. Using these methods, the following re-
sults are obtained:

t

Members.

ab -" 6c
cd BO

—CD - -DE

Section.

WheeL

Moment, ft.-tbs.

Stress, lbs.

B&
Cc
Cc

lyd

4

7

8

11

12

4.049,333
6,211,667
6,207,667
7,044.000
7.056,500

144,600
221,800
Not max.
Not max.
252,000.

710

TRUSSES.

Wind Loads.

116. A complete bridge is subjected not only to vertical loads, due to dead
load, to live load, and to impact caused by inequalities in track and in rolling
stock, but also to horizontal loads. These horizontal loads are due to the
thtnsverse action of wind, or of centrifugal forces produced by the train in
passing around a curve on the bridge, and to the longitudinal traction or
"drag caused by stopping or starting a train on the oridge. Hence it is
necessary to supply horizontal bracing, which, with the two upper chords
or the two lower chords of the two vertical trusses, form horizontal trusses,
known as the upper and lower lateral systems. Figs. 39 (a) and 39 (6), and
sway and portal brading, tH 21 and 22.

117» The wind is considered as blowing at right angles to the bridge.

118. The wind prodjuces several effects, and these must be ascertained
separately, and their joint effect then determined. Among these effects arc:

(1) Direct stresses in both the upper and lower lateral systems, by pressing
wlirectly upon the chords; acting horizontally as a uniformly distributed load.

(2) Additional direct stresses on the lateral system of the loaded chord
when a train is on the bridge, owing to pressure of wind against the train.

(3) An overturning moment upon the bridge as a whole, thus increasing
the dead and live load stresses in the leeward and diminishing those in the
windward vertical truss.

(4) A similar overturning effect upon the train and its wheels, which nmi-
larly modifies their pressures upon the floor beams and thus the stresses in
the main trusses.

-fnp-

-I «B njp

Vig. 39.

119. The wind load, acting directly upon the bridge, is assumed to be
equally divided between the upper and lower chords, and between the wind-
ward and leeward trusses.

ISO. (1) The direct wind stresses in the lateral bracing, due to the pres*
sure of the wind on the truss, are found as are the stresses in the main trusses,
due to dead load; the horizontal transverse struts of the lateral bracing
corresponding to the verticals of the main trusses.

121* (2) Direct stresses in lateral system of loaded chords. Fig. 30 (b),
due to wind on train.

Examining any panel, as c J, let

w => wind pressure^ in lbs. per lineal foot of train ;
p » panel length, m feet;
tp p -» wind presisure, in lbs., per panel fully occupied by train ;
n »• number of panels in span ( =- 6 in this case) ;
I ^ np '^ span, in feet;
m = number of panels from left support, a, to and including the panel
c d, under consideration;
m p = distance, a d, in feet ;

X » length, in feet, of that portion of the panel, c d, which is occu*

pied by train ;
t — (n — m) p -\- X = that portion of the span which is occupied
by train, in feet ;
» wind pressure on train for a pressure of 1 lb. per lineal foot ;
R = truss wind reaction,* at a, ■- to ^ -*- 2 Z ;
r — panel wind reaction,* at c, — to x* -i- 2 p ;
S -= wind shear * in panel, cd, — R — r^wfi-i- 21 — i0a:'-(-2p.

* See foot-note (t). H 2.

WIND BTBESBES.

711

The horisontal Atiat raaotion,* at a, due to a coneentrated horizontal
pressttie, » 1, actini; at any distance, y (not shown), from <2, ia «

^^-^ — ^-2 SLj and the horizontal panel reaction, at c, due to the same

pressure, is » -. The maximum wind shear,* in the panel, c d, due to wind

on train, occurs (see t If 79 to 81) when the head of the train reaches that
point, 0, at which, if the concentrated load be placed, these two reactions

will be equal, or ^ - C^-^^)P + y With head of train at o, we hav«
, p np

(n — m) p

'-"•• n-l •
Under ally conditions, the wind shear,* S, in the panel, is •• K — r, where

2 n p z p

Substituting here the value of ;r, just found, for maximum shear, we ob*
tain, as the maximum value of the wind shear * in the panel,

10 p

^umx

2(n— 1)

(n — m)*.

,122. (3) Stresses in main truss members, due to overtumins moment of
wind on truss.

Overturning moment — (wind panel load at top chord) X (number of
panel points in span) X height of truss.

_- ^. , ,. ^ ^1 overturning moment

Vertical reaction at one support - ^ width between trusses'

Since the upper lateral system carries all wind loads to the ends of the
bridge, the end posts and the chords (which take the horisontal components
ol the end post thrusts) are the only main truss members affected.

r

I
I

t

I__L

H

(a)
Fiff. 40.

tZ3» (4)Stress in main truss members, due to overturning moment of wind
GkD train, fig. 40. Let h — height from center of gravity of lateral system
cf loaded chord to center of pressure of wind on train, p -• wind pressure per
linMil foot of train, w » width between centers of gnetvity of trusses, m »
overturning moment, per lineal foot of train, v «- added vertical load on

leeward truss, per lineal foot of truss. Then m «- A p, and v — — .

. w

Impact, Etc*

124. The effects of impact, due to inequalities of the track; those of
"drag," due to the starting and stopping of trains; and those of centrifugal
force of the train on curves, are not susceptible of rigorous calculation,
and. engineers differ in their requirements respecting provision for them.
See ^Digests of Specifications.

♦ See foot-note (t), If 2.

712

TRUSSES.

Determination of Maximum and Minimum Stresses.

1!S5. Where specifications make allowable unit stresses depend upon tbe
relation between the maximum and minimum stresses in any given member,
both must be computed.

In computing the maximum and the minimum stress in any member,
bear in mind that a condition which, of Itself, would have a certain efiFect
upon the stress, may bring with it other conditions which produce a greater
effect of the opposite kind. Thus^ although the action of wind on train
would, of itself, reduce the stresses m certain members, this action can take
place only with train on bridge, and the vertical action of the train load
would ordinarily increase those stresses more than the wind action would
diminish them.

In computing minimum stresses, although the live load is usually to be
neglected, we must of course not neglect the dead load, which 'is always
present.

Curves on Bridges.

128. When the track on a bridge is curved, it is usually so laid that
the center line of the bridge bisects the niiddle ordinate, m, of the curve.
See Fig. 40 (a). The center of gravity of a panel load, P, at the center of the
span (supposing it to stand over the center of the track) is thus thrown out a
distance -» ^ m from the center line of the bridge, or a distance » i b +
i m from the inner truss, where b = width of bridge between centers of
trusses. Taking moments about the center of the inner truss, we have,
therefore, for the load, W, on the outer truss, due to P,

W

^b + i m

6 4- m
26

It is customary (see Di^sts of Specifications) to proportion the outer
truss on the safe assumption that its share of the live load, at each panel
point, is determined by the formula just given, and to design the inner truss
like the outer one.

• n w

SeeUan

T~1»

Figpi. 40 (a) and (b).

Fig. 31 (repeated).

Counterbraoing.

121 • In a truss of any ordinary form Gike that in Fig. 3}), under the action
of a uniformly distributed dead or live load, or of a live load distributed
symmetrically as regards the center of the span, the shears in each panel on
the left of the center of the span are positive, while those in the panels on the
right are negative,' and the stresses througnout the truss are such that the
ties sustain tension, and the struts, compression; the tendency, In each
panel, being to elongate that diagonal occupied by a tie, and to aKorten that
diagonal occupied by a strut.

But the tendency of an eccentric load, such as those shown in Fig. 31,
is to reverse the shears in the panels between it and the center; and, u this
effort, relatively to the other forces, is of sufficient magnitude to reverse the

ROOF TRUSSES. 713

final shear in any panel, the tendency will be to afiorten the diagonal occupied
by & tie, see Fig. 1 (a), and to lengthen any diagonal occupied by a 9trvt.

As explained in 1^ 14 and 15, this condition is met, in the Warren
or triangular truss, by making each web member capable of resisting both
^nsion and compression ; and, in trusses with both vertical and diagonal
web members, by inserting counters.

In a drawbridge or swing bridge, not only the web stresses, but also
the chord stresses, are reversed when the draw is opened or closed.

To provide against p>08sible further increase in live loads, over those
now in use, specifications sometimes require that, wherever the live and dead
load stresses are of opposite character, only 70 per cent, of the dead load
stress shall be considered as effective in counteracting the live load stress.
For other methods of making similar provision, see Digest of Specifications
for Steel Railroad Bridges.

Boot Trusses.

12S, In roof trusses, the dead load, t. e., the weight of the truss itself
and that of the purlins, roof covering, etc., and the snow load, are usually
taken as uniformly distributed. In many oases the sum of the dead and
snow loads is divided equally between the two supports. .In other words,
the end reactions are equal.

If9. The weights of steel trusses, in pounds per square foot of
building space covered, may be taken, for preliminary estimate, at (0.05 to
"i.OS) X span in feet, according to desi^ and loading. Those of wooden
Tusses, with wooden, iron or steel tension members, may be taken at from
tne-tenth to one-fifth lees.

If it is found that the weight of a truss, as designed, considerably^ exceeds
he weight assumed for it in advance, it should be redesi^ed, assuming a new
'eight slightly greater than that obtained from the design.

130. The weights of purlins, of steel or wood, ntiay be taken at from
to 3 lbs. per square foot of building space eovered.

131. The weight of roof covering may be taken approximately as
Uows:

Corrugated iron 2to3 lbs. per sq. ft. of roof surfaee.

Slate 7to9 "

Shingles, on laths, 2to3 "

If on boards, add 3 " " "

If plastered below the rafters, add, 6 " " " "

39. The snow load* in States north of lat. 35^ may be taken as vary^
(chiefly with latitude) from 10 to 30 lbs. per sq. it. of horizontal projec-
i of roof surface.

33. The purlins, stringers, etCy should be so arranged as to carry
weight of roof covering and of snow directly to the panel points, and thus
d transverse stresses in the rafters.

(4. flach truss bears, besides its own weight, half the weight of roof and
'- between the two trusses (or truss and wall), adjacent to it, and each
I point bears half the load between two panel points (or panel point and
nipport) adjacent to it.

M

Jf

n

~1'

n»

B

ligr. 41.

s, Kig. 41, truss TT carries a weight •=» that on the surface between
o dotted lines, DD and £E; and panel point p carries a weight = that
rectangrle mn.

Tlie 'wind is regarded as blowing horizontally upon one side of the
d as exierting a uniformly distributed normal pressure upon that side.
fbllowrin^ table of assumed normal pressures against sloping sur-
nder horisontal wind pressures of 40 lbs. per sq. ft., the values m the
luznn are based upon Hutton's experiments. Here a is the angle
K tlie sloping roof surface and a horizontal plane.

714

TBUB8E&.

Assumed normal wind pres8ure»

wind pressure = 40 lbs. per sq. ft. a =
horiiontal plane.

P

P, in lbs. pmr sq. ft. Horiaontal
angle between roof surface and

a.

6"»
10*
15«
20*
25*
30*

sin 8. 40.sina. Hutton.

0.087
0.174
0.269
0.342
0.423
0.500

3.5
7.0
10.4
13.7
16.9
20.0

6.1
9.6
14.2
18.4
22.U
26.6

a.

35*
40*
45*
60*
66*
60»

sio a. 40 .sin a. Hutt<HL

0.574
0.643
0.707
0.766
0.819
0.866

22.9
25.7
28.3
30.6
32.8
34.6

30.1
33.3
36.0
38.1
39.4
40.0

136. The directions and amounts of the end reactions and of the stresses
in the members, due to wind, depend upon whether one or both supports
are fixed. If both ends are fixed, their reactions are parallel to the normal
wind pressure — «'. e., they are at right angles to that side of the roof upon
which the wind is blowing ; but, if one end is free to slide longitudinally of ths
truss, its reaction is taken as vertical and that of the other is more nearly
horizontal than the normal wind pressure. When one end is free, the stresses
must be determined for wind blowine on the fixed side (in which case it tends
to flatten the roof) and also for wind blowing on the free side, in which ease
its horizontal component tends to shorten the tie-rod and to raise the apex.
The stresses in the members of roof trusses are conveniently found by means
of the method by sections, 1ft 67, etc., or cpraphically, as below.

137 • Fig. 42 (a) illustrates tho graphic treatment of wind stresses for Fig.
42 (6), under the three conditions named, viz. : — case 1, with both ends fixed ;
case 2, wind blowing against the fixed side; and cased, wind blowing
against the free side.*

In Fig. 42 (a), the segments o5, be, ed and de rqpreesnt the normal wind
pressures at the panel points AB, BC, CD and D£ respectively, and ae
therefore represents the total nonnfd wind pressure on the roof* all being
exerted against the left side.*

138. In case 1 (both ends fixed) the segments fa and «/ of the solid line ea

represent the left and right reactions respectively.

139. In case 2 (wind blowing ajsilinst fixed side) the reactions are repre-
sented by the dash line e/'a; and in case 3 (wind blowing against free ade)
by the dash-and-dot line ef'^a.

140. The segments f'f and ff represent the horizontal components of the
right and left wind reactions respectively in case 1 ; and Z'/* that of the wind
reaction of the fixed end in case 2 or case 3, or that of the total wind reaction.

141. Having found, by moments, the end reactions ef and fa for case 1 ;
where, Fig. 42 (6).

. A «

the vertical reactions, ef and af% for cases 2 and 3 respectively, are found by
dropping perpendiculars from e and from a. Fig. 42 (a), npon gf produced.
The reactions of the fixed ends are then given by the closing line, /^ct, in case
2, and by ef* in case 3.

* To avoid the necessity of showing two skeleton figures and two diagrams,
we have supposed the wind to blow always in one direction (vis., agaiyst the
left side) and first one end of the truss and then the other end to be fixed.
In practice, of course, the reverse of this is the case ; t. e., one end or the
other of the truss (if not both) is fixed, and remains so: and the wind may
blow against either side. The figure and diagram will, however, answer for
this latter condition also. Thus, if the wind blow against the left side, as
shown, and if, as in case 2, that side is fixed, then tne diagram, using the
broken lines, e /' a, gives the stresses in the members, as they are lettered.
But now (the left end remaining fixed) suppose the wind to blow against the
free side; i. e., from the right. We may nevertheless suppose the right end
fixed, and the wind blowing against the left side, as in Fig. (6), and find the
stresses in the members from Fig. (a) as it stands, us.ng the dash-and-dot
diagram, « f a; but we must then remember that the stresses thus found for
BG, GF, etc., on the left of the truss, Fig. (b), really apply to the correspond*
mg members, QE, QF, etc., on the right, and viee versa

ROOF TRTOBE8.

715

142. The stresses In th« web members, GH, MN, etc., Fig. (6), and thoee
in the several members, BG, EM, etc., of the rafters, are pven by the oorre-
sponding lines, ght mn, bg^ em, etc., in the diagram, rig. (a).

143. In the leeward rafter, in this case, the stress in the three segmenta
ME, PE, QE, is uniform throughout, and moreover it is the same in each of
tfa« three cases, being -> me >>- p« » q«.

In the four web members. LM, MN, NP, PQ, to the leeward of the center,
the stress, in this case, is wso, being repres^ited by the point, Imnpq, Fig.
(a).

144. The stresses in the several segments, GF, JF, LF, NF, and QF, of the
horisontal tie rod. Fig. (b), are represented, in Fig. (a),

in case 1 (both ends fixed) by gf, if, If, nf and qf;
in case 2 (wind against fixed side) by gf, if, etc. ;
in case 3 (wind against free side) by gf* if', etc.
In each of the three cases there is uniform tension in the three leeward

FlfT. 42.

flegments, LF, NF and QF, of the horizontal tie rod. This uniform
tension is

in case 1 (both ends fixed) ^ If -^ mf — nf;

in ease 2 (wind against fixed side) -= If — mf « nf;

in case 3 (wind against free side) ■- If -• mf ■= n/*.

145. It is thus seen that, in our Fig., with horizontsl tie rod, the differ-
ence in the manner of supporting the ends affects only the horizontal stresses
in the members of that rod, and, through them, the manner in which the
horisontal component f*f of the wind stress is distributed between the two
flupports.

146* If the lower chord were not straight, however, the stresses in- the
rafter and web members wovdd be affected by the difference between the
three eases.

147* The final or resultant stress, in any member, is the algebraio sum pf
tlie dead, snow and wind loads for that member. In some oases, the wind
jmad may diminish or even reverse the stresses due to dead and snow loads.

716

TRUSSES.

148. In timber roof trusses of short span, Figs. 43 to 47, for roofs of dwell«
ings and other smaH buildings, we may, with sufficient accuracy, make a
liberal assumption for load, to include wind pressure. In discussing these
figures, we investigate the stresses by means of the force parallelogram.
For dimensions of such trusses, see ^ 266.

149. In the wooden roof truss. Fig. 43, uniformly loaded along each
rafter, let H I — the weight of one rafter and its load. Then E I =» the hori-
zontal pressure of the head and of the foot of that rafter (the latter being the
tension in the chord), and H E *» the inclined pressure at its foot.

V

Fig. 43.

150. In Fig. 44, make G R = HI. Then G L is the transverse pressure
of the load against the rafter as a beam, and L R is a longitudinal pressure
along the rafter, forming a part of the total longitudinal pressure. .

151. If G R were concentrated at G, L R would be uniform from G to a.
and would not be exerted above G ; but, as G R represents a load uniformly
distributed along the rafter, from top to foot, the pressure represented by
L R increases uniformly, from nothing, at the top, o, to L R, at the foot, a.

152. Of the transverse pressure, G L, one half, — o p, is sustained at the
top, o, of the rafter, and tne other half, =- a g, at the foot, a. At the top,
o p is resolved into the horizontal pressure, o b "» EI, against the head of
the other rafter, and a uniform thrust, o z, along o a.

Fl«. 44

153. It is immaterial whether we thus resolve o p directly into o b and o m
(as though the head of the rafter rested against a verticcd toaZi at o), or whether
we first resolve it between the two rafters, into o e and o r. For in the latter
ease we must add to o c a thrust ( =» o r >" c «) produced in o a by the trans-
verse pressure (similar to o p) of the head of the other rafter ; and the sum of
these two (o e and o r) is » oz.

154. The total longitudinal thrust in the rafter increases uniformly from
0 2, at the top, to o z + L R » a A;, at the foot, where it combines with a q
( a> half the transverse load) to form a v — H E.

155. Tension in chord — IE — iv — «A — n q. Vertical pressure od
support — HI '» at ••a» + on ■■ aa -^ et.

BOOF TBtraSES.

717

156. In Fig. 45, having found, as for Fig. 43. the stresses, etc., due to
the rafters and their loads, remember that the king rod, o n, supports its
own weight plus the portion y yoi the chord *- ^ the chord. Making o t ^
this oonibined weight, we have o m =* o d =* an additional pressure, uniform
throughout each rafter, and c m = c d » an additional tension on the chord.

157. In Fig. 46, assuming, for safety, that the rafter. / h, is divided at its
center, U, make 00 — the weight of 2 r and its load (« r <- half the rafter) .

158. Then e i -■ an additional pressure on V b, e k " pressure on U c;
« t -» « A; =- additional tension on half chord, c b, and « o — 2 ea « load of
and on « r, — additional tension on king post due to both stmts. Then
make ag ^ eo -\- weight of king rod + weight of two struts + wei^t of and
on y y, and proceed as m Fig. 45.

Tig. 46.

159. Each strut will thus bear half of the weight of and on 2 r, or x u, only
when, as in Fig. 46, the inclination of the strut is the same as that of the
rafter. If the strut is steeper than the rafter, it will bear more than half;
but if it is less steep than the rafter, it will bear less than half ; the remainder
being in every case borne by the rafter.

180. The introduction of the struts converts each rafter, considered as a
beam, into two beams of shorter span and bearing less loads.

ri«. 47.

161. In the queen truss, Fig. 47, make og =- total tension in queen rod +
half weight of and on the "straining beam," 2 w. Longitudinal pressure on
9 to *= tension in chord, 6 o, — I E + o c.

718

TBtTSS£8.

Deflections.

1&2» The total deflection of a truss * comprises (1) the elastic or
temporary deflection, due to the stretch t of its several members under the
loading applied to the truss, and (2) the non-elastic or permanent deflec-
tion, due to looseness of its joints. In good construction the latter is
relatively negligible in moderate spans.

The total elastic deflection, D, of a truss, at any point, c, is made up of
partial elastic deflections, d, d, etc., at c, each due to the stretch, k,j in
some member.

Let it be required to find the deflection at a panel point, e (usually the cen-
ter of a span or the end of the arm of a swing bridge or other cantilever) ;
and, for any load or system of loads, let

D =

d -

I -

k -

W -

E -

thd total elastic deflection at c;

the partial elastic deflection, at c, due to the stretch, A;,t in any

member ;
the unit stress in that member ;
the total stress in that member;
the length of that member ;

the stretch t in that member — —irr*

that load which, applied at c, would produce the stiess, P, in that

member;
P

k pi

the modulus of elasticity of the material, •" P "*■ f "" ~jl •

Tig, 48.

163. K^utTalence of Work. In Fig. 48. let any load, W. be applied
at any point, e, of a truss or bar. Then, for a small deflection, t such as may

♦ See "The Application of the Principle of Virtual Velocities to the Deter-
mination of the Deflection and Stresses of Frames,'' by Geo. F. Swain,
Jour, of the Franklin Institute, vol. UKXXV, 1883; "Trusses with Super-
fluous Members,'' by Wm. Cain, Van Noetrand's Magazine, vol. xxvii.
No. 4, October, 1882; "The Graphical Solution of the Distortion of a Framed
Structure," by David Molitor, Jour. Ass'n Eng'ng Societies, vol. xiii,
No. 6, June, 1894; and "The Theory and Practice of Modem Framed Struc-
tures," by J. B. Johnson, C. W. Bryan and F. E. Turneaure, New York,
John Wiley A Sons.

t For brevity we here use the word "stretch" to signify any change of
length, including the shortening due to compression, as well as the elonga-
tion due to tension.

t For the sake of clearness, the stretches and deflections, in our Figs., are

p
exaggerated beyond the limit within which the ratio, s^v, would remain eVen

approximately constant.

DEFLECTIONS. 719

be permitted in trusses, the external work, W d,* of a partial defl^tioii, <f»
due to the stretch, /;, in any member, is practically — the internal work,
P k, of overcoming the reeistins stress, P, in that member, through the dis-
tance, k; or

W d - P A.

Hence, .

^ P I. I. * w

<^ = W* " ^ ' °^ d" P*

In words, the stretch, k, in any one member, is, to the resulting partial
deflection, d, at c, inversely as is any stress, P, in that member, to a load, W,
which, if applied at c, would cause that stress.

Thus, in Fig. 48 (a), where A; is in the same direction as D,t P ^ W, and
D = Jfc.

In Figs. (&) and (c),t suppose the strut incompressible. Then D is due

p
solely to the elongation of the tie. And D =» ^ A; — « A;.

P
In Fig. (c), «. is greater, and (for a given stretch, k, in the, tie) D is there-
fore greater, than in Fig. (&) .

164. Deflection Independent of Nature of Cause of Stretch.

Now it is evident that the deflection, d, at c, depends solely upon the amount
and character of the stretch, k, in the member, and is independent of the
nature of the cause of that stretch. That is to say, any change, k (however
caused), in the length of the member, necessarily contributes its fixed quota,

p
d >» 7K. A;, to the total deflection, D, at c. In other words, since d and k are

mere distances, and since u is simply a ratio, the relation between d and Jc is
a purely geometrical one, and is therefore not confined to deformations pro-
duced by applied loads, but is applicable also to those produced by changes
of temperature, to intentional lengthening or shortening of members, or to
any other cause.

Hence, if a member be in any way lengthened or shortened, by a length, h,

a oorrasponding change, d, » =,, . A « u ^, takes place in the deflection at c,

w

For instance, if we place any system of loads upon a truss, and, by the

principles of statics, determine the resulting total stress, P, and unit stress

p, in any member; we have, for the partial deflection, at c, due to the stretch,

Kt in that member, under the given system of loads,

(For w, see t 165.)

d ^ u k;

and, since k = -^,

rf « P « ^

E •

p
165. To obtain the ratio, u, « ^, for each member, we suppose a

concentrated load applied at c; and, by the principles of statics, find the
resulting total stresses, P, P, etc., in the several members. If the supposed
load, at c, be taken — unity, the stresses, P, P, etc., so found, are the desired
ratios, u, u, etc.

♦ Strictly speaking, with a loafl increasing gradually from 0 to W, and
with resulting stress increasing gradually from 0 to P, we should deal with
the mean load, =* i W, and with the mean stress, = i- P, in each member;
but it will be seen that this would not aflfect the equations derived.

t Where, as in Figs, (a), (6) and (c), only one member is supposed to
change its length, D »> d.

X See note (t) on preceding page.

720

TRUSSES.

166* Summation of Deflections. The total deflection. D, at e,
under the siven system of loads, being » the sum of the partial deflections,
d, d, due (out not necessarily equal) respectively to the stretches, k, k, ip
the several members, we have

D - 2d - 2

p u I
E •

Thus, in Fi@. (d) and (e), we assume the tie extensible and the strut com-
pressible. In Fig. (d), W = Pi + Pa; and W D = Pi A; + P2 A - (Pi +PJ *.

Pi + Pa n.W

Pi , . P«

Hence D — 2tt* — =^.A; + ^.A;

In Fig.

(«), D - SttA?

P P

ui A?i + ua A?j.

167. PositlTe and Negative Stretches. In some oases it may
happen that the change of length of a member diminishes, instead of in-
creasing, the total deflection at the point, c, in question, and must therefore

be taken as negative in summing up the values ofu k — • -p ■; but when c is

the middle point of a span, or the end of a cantilever, all the changes in length
of the members ordinarily contribute to the deflection, and must therefore
be taken as positive.

Theoretically, the formula, D « 2 _ , applies also to the deflections

of arches, dams and other structures composed of blocks ; but, owing to
the uncertainty of the values of E, and to the relative inaccuracy of finbh
in masonry work, it is of but little practical utility in such cases.

168. Redundant or Statically Indeterminate Members. TVusses
frequently contain members whose stresses cannot be found by the principles
of statics. Thus, in Fig. 11 (c), the two diagonal tension members meeting
at the top of either end post are said to be redundant, or statically indeter-
minate, because the principles of statics do not enable us to determine what
proportion of the total load goes to the supports through each of the two
systems. Figs. 11 (a) and (6), composing Fig. 11 (c). But the deflection
formula, just given, enables us to determine the stresses in such members;
for, by means of it,'we may find, separately, the deflection in each of the two
svstems. Figs. 1 1 (a) and (6) ; and the part load, transmitted to the supports
through each of these two systems, is inversely proportional to their deflec-
tions.

BRIDGE DETAILS AND CONSTRUCTION.

General Principles.

169. In general, a truss bridge consists essentially of two or more verti-
cal trusses, AB, CD, Fig. 49, placed side by side, and connected by the
floor system, which, in turn, they support; and bracing (formin^^ a
•* lateral system") is supplied between opposite chords, where practica-
ble, in order to maintain the trusses parallel.

E

G

C

H

■~*~*

nil

^

F

J>

FiffT. 49.

170. The floor system consists ordinarily of floor beams andstrlns*
ers. The floor beams, AC, EF, etc., Fig, 49, are placed transversely to the
bridge, and are attached to the trusses at opposite panel points. Connected
with these and p>erpendicular to them or parallel to the trusses, are the
stringers, GH, IJ, etc. In railroad bridges, there are usually two or more
stringers placed side by side and running the length of the bridge, to support
the ties. In city highway bridges, these stringers are usually spaced at smaller
intervals, and support buckled plates or other form of flooring, on which thf

r

DESIGN. 721

paving is laid. For country highway bridses, the Btringers are frequently
of wood, placed quite near together, and the planks of the floor are nailed
or spiked directly to them.

Solid floors (see Pencoyd floor sections) add to the rigidity and per^
manence of a bridge, and give increased protection to trafiic below, against
injury from falling bodies or in case of derailment. Their shallowness is an
advantage where head-room is an object.

171* Any load, then, is carried first from the ties or floor, etc., to
the stringers, then by the stringers to the floor beams, and finally bv the
floor beams to the panel points of the bridge, where it is carried througn the
trusses to the supports.

172. Pedestals, shoes or bed plates. Fig. 62, bolted to the piers, support
the ends of the trusses. When the bridge is of long span, so that the expan-
sion and contraction due to heat and cold are considerable, expansion
bearings. Figs. 60, 62, must be provided at one end. See ^ \ 205, etc.
For cross-bracmg, see t^ 19, etc.

General Character of Desiflrn.

173. Flexible and Bigrld Tension Members. Adjustable
Counters. Until recently, eye-bars have generally been used for the
tension members of trusses. These are long flat bars, liable to jdeld laterally
under compression, and furnished, at their ends, with eyes or openings,
through which pass pins connecting them with the other members of the
bridge; but rigid built members, capable of sustaining some compression, as
well as tension, are now much used for tension members. Counters were
usually made in two lengths, and were adjustablcy the two lengths being
connected by tumbuckles ; but these rendered it possible to bring undue and
dangerous stresses in the panels, and they are now giving place to counters
made each in one length.

174. Compression members are ordinarily "built up" of angles and
plates, or of channels and plates with latticing, in hollow shapes, bringing
most of the material as far as possible from the neutral axes of the cross-
tection and thus increasing its resisting moment.

175. Pin and Riveted Connections.^ The web members are con-
nected with the chords either by pins or by rivets. In the former case the
truss is said to be pin-connected; in the latter case, riveted.* Until re-
cently, pin-connected trusses have been typical of American practice; but
the Americans are now largely using riveted trusses, for spans up to from
150 to 175 ft., while the Europeans are in some cases using pins. The prin-
cipal advantage claimed for the riveted joint is that it makes a stiffer bridge
and one that will not rattle, and that a riveted truss, computed as if pin-con-
nected, will have an additional margin of safety on account of added stiff-
ness. In the pin-connected bridge, on the other hand, the stresses can be
much more accurately determined, and deflection may take place without
producing twisting or bending stresses in the connections themselves.

176* Tendency to Greater Rigidity. There is a growing tendency
to use stiffer bracing, to design at least all short braces for compression^ and
to make even the longer tension members of channels or angles, forming a
rigid member. ^ Unless pin-connected eye-bars are of exactly equal length,
some of them will receive more than their share of the total stress.

177* Floor-beam Connections. In the United States, floor-beam
eonnections were formerly made by hanging the floor beams from the pins
by means of hangers ; but now, where possible, the ends of the floor beams
are riveted directly to the inner sides of the posts.

178. In tension members, rivets are so arranged as to reduce the net
effective section as little as possible.

179. Compression members are so designed as to place most of the material
as far as possible from their neutral «xes, and they are sometimes strength-
ened by auxiliary ties or posts supporting them at their middle points, in
cases where the resulting saving in material for the member will be consid-
erably greater than the expenditure of material in the auxiliary member.

* Riveted trusses are unfortunately called, also, "lattice girders," "lattice
trusses," ** riveted lattice girders" and "riveted lattice trusses." The term
" lattice " is often appliea to shallow trusses with numerous panels.

46

722

TRUSSES.

180. So far as possible, compression members are made equally strong
against bending about either of the two principal axes, AB and JCY, Fig. 52,
of their cross-sections.

181* Where the same member occurs many times in a bridge, and where,
therefore, an excess of material in the design of such member would involve
a large total 'vaste, the computation of the member is repeated many timea
until the most economical section is found.

182* In metal trusses the shorter members are usually made to withstand
compression, and the longer ones tension, this bein^more eoonomieal <^
material. Thus, the Pratt truss, with diagonal teii£!ir>n members, is used
for steel bridges, while the Howe truss is now built only with wooden diago-
nals.

Tension Members.

^o„ . , - .. maximimi tensUe stress

183. In eye-bars, area of cross-section — -^n in .^ ^ ■ . — .

allowable unit tension

184. The dimensions of the heads of eye-bars are usually determined bv
the manufacturers, and are so deswned as to give ample excess of strength
at the pin-holes; so that, if tested to destruction, ftuly two-thirds of the
number of bars tested shall break in the body of the bar, this being usually
required by specifications. It is important that the proportions oi eve-biur
heads should be such as to ensure thorough working of the metal in the
upsetting process.

4

o-

z

\l/ \l/ \/ \l/

J'ee^

O id 20 30 2o Sd go 70 80 90 tOO ISO 120

Figr- so.

185. Fie. 50 shows, to two different scales, the "packing'' (arrangement
of pins and eye-bars) in the left half of the lower chord of a 150 ft. tnroui^
(skew) span built by the Phoenix Bridge Ck>. in 1900 for the Philadelphia and
Reading Railway (5o. near Reynolds, Fa.

186. BuUt Sections. Hip vertical hangers, non-adjustable counters
and their corresponding mains, are usually built up of rolled steel shapes.
A section in common use, shown in Fig. 51, consists of four angles, connected,
at intervals, by small narrow flat bars, riveted to the angles and running
across zigsag from one to the other. When single, as in Fig. 51, this is ealled
"lacing" ; when double, as in Fig. 52 (6), "latticing." The shaded area of the
angles. Fig. 51, minus that of the rivet holes, is taken as the effective section.

187. Minimum Sections. Specifications (see Di^pests^ usually require
the use of some minimum section. Thus, in a counter in which the stress is
58,000 tbs., 3.5 square inches of cross-section would suffice; but specifioations
frequently forbid the use, in such sections, of any angle smaller than 3^ X 3i
X t. which gives 9.20 square inches gross; or, deducting one rivet hole from
eaen angle, 7.^ square mohes net section.

Compression Members.

188. The computation of a compression member consists of a series of
approximations ; for the unit stress depends upon the radius of gyration, the
radius of gyration on the area of section and disposal of materialwith regard
to the axes, and this, in turn, on the unit stress. See Pillars, under Strength
of Materials.

189. Fig. 52 (a) shows a form often used for posts, and consisting of two
channels, placed with their backs outward and riveted together bj}r lacing.
In Fig. 52 (b) the channels are placed with their backs inward. For econ«
omy, the channels should be so spaced as to make the radius of gyration the

COUPBESeiOH UEUBEBa.

723

■UH about either axis. A-B or X-Y. The radius of gyntion is given in
tlK luDd-books at mtrB uf structural sbapes. See up. 802, etc.

180. The nppeF chnrd eection ia fntquently bnllt upot twochaonelB
and 1 piste, or in some suoli f orra u shown in Fig. 93. conaiating of two verti-
1*1 plain or "webs," n horiaDatsI top piate or "cover." four "angles." and
flatplNSB or bars on each sideof the bottom. lattice bruin k. orlscloi-, ia
pmvided along the bottom, except at panel points, where it is omitted in
order that the poBt and the ties may enter the ohord from iLWlow. In pin-
euaaecled trusses the axis of the pin lies in the line AB.

191. The interior width, «, depends chiefly on the space remiired by the

inaide rivet heads. Usually, Ior^oon™n'ien'ee of oonB^truclior^ the g™iest
width, v/. required is kept constant IhrouBhout the upper chord. TheTieight.
H. depends chiefly on the size of eye-bar head, and is kept constant. The
thicknessea of the web plates, and sometimes also those of the bars and
angles, are varied, along the chiwd, in order to provide, at each point, aoffi-

(«)

lOtt. The end past ia

ily aa a eolumn, but also as a

198. iheend p<Mt latobeoonsidered not only aaaeolumn. but alaoasa
beam subject to shear. OD account of the wind blowing against the top of tfar
aide of the truaa. ThedeaiAaof this built-up form is much the same In princi-
ple AS that given above for a post. Cortain sections are tried, and then

104: Th.

ivjtli » load =

J __j I — ^iad, at the other end (which ia at the middle of the post)

724

TRUSSES.

195* The maximum stresses, due to compression, of course occur about
the m.idclle of the post, while those due to the wind occur near the ends.
Hence it would be unreasonable to require the post to resist all of both
effects simultaneously throughout its lengthy and specifications therefore
usually allow the unit stress, due to dead, live; impact and wind loading com-
bined, to be increased to 21,000 lbs. per sq. in., properly reduced by formula
for compression.

196* * The formula for the strength of long compression members is as fol-
lows, where _ .

P °" average permissible unit load on column, in lbs. per sq. inch;
I ■» length of column;
r = radius of gyration of its cross-section :

21,000

1 +

/a

9000 r«

197« The formula for extreme fiber stress due to combined compression
And bending, is

Q P 4_ MbT

A "^ , P Z«

I —

Ec

Where

P
A

I

I

E

c =

longitudinal compressive force ;

area of section ;

bending moment due to transverse load ;

distance from neutral axis to extreme fibers;

moment of inertia ;

length of beam ;

modulus of elasticity;

coefficient. See Transverse Strength, % 103.

Joints.

198. Pin Plates. Where a pin passes through one or more shapes of
some member, it often happens that tne combined surfaces of the truss mem-
bers alone, in contact with the pin, are insufficient to transmit, by bearing, all
of the stresses to be delivered to the member. There is then danger of
crushing the material which presses against the pin. To obviate this, other
shapes, usually flats and called pin plates or reinforcing plates, are riveted
to the member; giving, in all, sufficient bearing surface for the pin. See
Fig. 55; where the letters denote:

AA, angles,
C, cover,
B, bar.

W, web,
P, pin,
J, jaw.

F, filler,

O, outside pin plate,

T; batten plates.

Tig. 55.

, 199. In Fig. 66, the two channels form the whole member (exoept the lat-
ticing, which cannot be included to resist compression) and the pin passes
through both channels. In the case of a built-up chord section, or of an
end post* Fig. 53, however, the webs form only a part of the section ;

JOINTS AND PINS. 725

while the cover, the angles and the fiats oan receive no stress directly from
the pin, but must receive it indirectly from the web and from the pin plates
attached to it.

200. Where a pin plate is placed on each side of the web, the outside one
must, according to most specifications, cover the angles; and there must, in
addition, be a **flller" between it and the web.

ISOl* Engineers differ as to the manner in which the stresses are actually
transferred through the several parts of a pin connection. We may assume
that the stresses in the pin plates are delivered almost (tirectly to the shapes
of the member. Thus, the outer reinforcing plate probably delivers most or
all of its stress to the anglc3, and little or none to the web.

20i2» In eaeh angle, those rivets which pass through the inner pin plate
must transmit, by means of thoir bearing against the angle, the sum oi the
stresses which they take by shear from inside and from outside. In other
words, these rivets are in double shear.

Pins.

J903* The pin must be designed to resist bending stresses from the mem-
bers through which it passes. It is also subject to Aeax, but this is seldom
A critical point.

204. The pin requiring t)ie greatest crosensection is usually either the one
at the middle of the span and in the lower chord, where the chord stresses
are greatest, or the one at the joint between the end post and the top chord ;
but, as the pins are relatively small members, all the other pins are, for the
sake of uniformity, usually made of the same sise with it.

Fir- ^'7*

Expansion Be»rinss.

205. Expansion bearinss usually consist of a nest of carefully turned
rollers placed between two planed surfaces, shown in principle in Fig. 57.

206* The rollers are steel cylinders, from 3 to 6 ins. diam. ; and 1 to 4
ft. long; planed smooth. From 4 to 8 or more of these are oonnected to'
getherc>y a frame, and one such frame is placed under at least one end of the
truss. The rollers xreet UF>on a strong planed oast bed-plate; bolted to the
masonry below. Under the end of the truss is a similar plate by which it
rests on the rollers. Since a truss of even 200 ft. span will scarcely change its
length as much as 3 ins. by extremes of temperature, the play of tne rollers is
but small. They are kept in line by flanges cast along the side of the bed-
plate. Flanges should also project downward from the upper bed-plate,
8o as completely to protect the rollers from dust, rain, etc.

207. The total displacement, allowed for the free end of the truss, is
usually specified (see Digests) ; otherwise it may be taken as

r> (T — 0 span
^ " 145,000 •

-where T and t — the max. and min. temps, respectively, in degrees F. The
znin. temp, to be expected may be obtained from Weather Bureau records
of temps, in the shade, but the max. should be taken 20° or 80^ higher than
that of the Weather Bureau ; because, in bright 'sunshine, the bridge will
become much hotter than the air.

208. Rockers. In order to restrict the length of the bearing, where the
displacement is moderate, rockers. Fig. 62, are often used instead of rollers.

209. For other regulations and suggestions regarding design of roller
bearings, see Digests of Specifications, and Figs. 60 and 61.

726 TRUSSES.

Loads » Etc.

210. Loads^ Clearance, etc., for Hlgfawasr Bridges. See ako

Digests of Specifications for Bridges.

Weisrhts of crowds. At the Chelsea bridge, London, picked men,
packed upon the platform of a weigh-bridge, gave a load of only 84 lbs. per
sq. ft. At Buckingham Palace, men, wedged as closely as possible upon a
space 20 ft. in diameter, the last man lowered from above, among the others,
gave 130 lbs. per sq. ft. But modern experimenters have easily obtained
loads of from 140 to 160 lbs. per sq. ft. With picked men, averaging 103.2
lbs. each, all facing one way, carefully packed, and confined within an
enclosure 6 ft. square (0.9 sq. ft. per man). Prof. L. J. Johnson, at Harvard
University, obtained a maximum of 181.3 lbs. per eq. ft.* See also pages
766, etc.

Where the enclosure of the space is such that portions of the persona,
standing against the enclosure, may project beyond it, the load, per unit ol
space, is ot course increased ; and, with small sureas, this increase may be
relatively important.

Camber.

311. Amount of Camber. If we divide the span in feet, by 50, the
quot. will ordinarily be a sufficient camber, in inches. This amounts to 1 ia
600. The camber to be used is, however, usually stipulated in the specifica-
tions. See Digests. A well-built bridge, of good design, should not, under
its greatest lo^, deflect more than about one inch for each 100 feet of its
span, or 1 } 1 1200. Indeed, the deflection is frequently much less than this.

313. The excess of length of the upper chord over that of the lower one,
given the span, the depth of truss and the camber, will be —

8 X depth X camber

span
Hiis rule applies closely with any camber not exceeding 0.02 of
the span.

213. Length of diagonal c b, Fig. 58, c 6 = 4ac^ + a b^;
where a c — depth of truss, and o 6 «= c n H z: . _. __

214. Sometimes the elongation or shortening, produced by

the loading, is computed for each member, and the length of each member
affected is correspondingly changed. See Deflections, tH 162, etc.

Examples.

215. Figs. 59 (a) to (u), to a uniform scale of 1 inch «« 60 feet, serve to
indicate current practice respecting the types selected for different spans,
the relation between span, panel length and depth, the spacing of stiffeners
in plate girders, the arrangement of chord and web members, t»s use of rigid
and flexible members, counters and tumbuckles, in trusses, and, approxt-
mately, the dimensions of rigid members and of gusset plates, as seen in
elevation.

216. In each case the left half of the span is shown, and the center line of
the span is indicated by a dot-and-dash line. Through spans and deck spans
are distinguished by the elevation of the roadway, as approximatel: ' indi-
cated at the left support.

217. In Figs, hto g, representing trusses, rigid members are indicated by
double lines, flexible members in verticals and main diagonals by single
lines, and counters by dotted lines. In pin spans, to avoid confusion, the
rigid members are shown cut off near the pins.

218. Figs, (a) to (o) represent standard designs, from 25 -o 200 ft. span, by
Mr. Ralph Modjeski, C.E., for the Northern Pacific Railway Co.: Fig. (p)
a 250 ft. railroad span designed by the Pencoyd Works of the American
Bridge Co. ; Fig. (q) a 3C8 ft. railroad span by the Phcsnix Bridge Co. ; Figs.
(r) to (0 designs for riveted trusses by the Elmira Works of the American
Bridge Co. ; and Fig. (u) a riveted railroad bridge, of 102 ft. span, designed
by he Pencoyd Works.

* Journal. Ass'n Engng Socs, Jan., 1905.

EXAMPLES.

727

219. Fig. (a) represents a beam girder : Figs, (b) to (g) plate girders; and
Figs, (h) to (o) riveted and pin trufeses; Figs. (A) and (i) being riveted, and
Figs, (f) to (q) pin.

](d)

(f)

(fe) #^ ^"^ <^^lllllllll («) (fl') «(1

ID

_

_

[

ri
4

w %V>XIL/^^

TWP€l

0)1

Fiff. 59 (a to 9).

728

TRUSSES.

t320. Fie. M represents a 128 ft. span for the New York Central and Hud-
son lUver R. K., and Figs. («) and (t) Bp&ha of 143 ft. and 160 ft. respectively
for the Delaware, Lackawanna and Western R. R. Fi^. (r) and («) are modi-
fications of the Baltimore truss. Fig. 15 (6), and Fig.(0 is a quadniplex Warren
truss. Fig. (a) is a skew pan. Fig. (u) is a "pony" span.

Figr. 59 (r).

Plan of
Top Bracing

Plan of JFloor and
SoHom Bracing

Tig. 59 («).

Fi«. 59 (<).

921. Details. Figs. 60 to 65 show a few details of trusses and of plate
girders.

222. Figs. 60 and 61 show left end connections of two through tmsi
bridges (with roller bearings) designed by the Pencoyd Works; Fi^. 61
representing the 250 ft. through pin span shown in Fig. 69 (p), and Fuf. 60
a 124 riveted through span, showing the portal bracing.

_f SeaU for Ftga. «», «S,
883. |%.B2 shows the

Fig. 63 xtowB a floor Uam
connection ot the 160 ft.
througb pin eova of tha
same railway, tig. 69 (m)

■WEIGHTB. 781

SM. Fim. M Uld 69 npresent respMtiTsly s £0 ft. d«lc pliite rirder of
the N. Pao. Ry. Bnd wi 86 tt. through plat* girder by the Penooyd Worka.

ll

, !
__ %

" I

"S

byR&lphModjeekl.C.E., in "Journal of the Western Society

■ See paper by R&lpb ModjeekL C.E., in "J
of EsBineen," Oiicago, F^., 1901, vol. vi. Ni

732 TRUSSES.

weights of steel railroad bridges designed for two locomotives, of
140 tons each, and a uniform train load of 4000 lbs. per foot of traclc. The
weights include the two beams, girders or trusses of one single-tracic span,
with their bracing, metal floor system and end bearings. For wooden noor
system, add 400 lbs. per lineal foot. For pin-connected spans (130 to 200
ft.), the tluee dash diagrams show, respectively, the weights of two trusses
alone, of two trusses and bracing, and of two trusses, bracing and metal
floor. The solid curve includes weights of end bearings.

For weights of combination (wood and iron) railroad bridges, see 1 240.

Highway bridges differ so widely, as to service and design, that it is
searcely practicable to give here useful data as to their weights.

litst of Large Bridges.

Each bridge here given is believed to be (1002) the largest of its type la
the world.

Type Spanning At Span, ft. Built

Truss, Ohio River Louisville 553 1803

Swing, Missouri River Omaha 520 1804

Suspension, ..... East River

("Brooklyn") New York 1595 1883

Suspension, East River

("New"; New York 1600 ♦

Arch (metal) Niagara River Niagara Falls 840 1808

Arch (stone),.... -.Petruff Valley Luxembourg 277 ♦

CantUever, Firth of Forth Queensferry 1700 1890

The highest viaduct is the Gokteik Viaduct, in Burmah, with a maximum
height of 820 ft., and a total length (composed of short spans) of 2260 ft.,
buut in 1901.

Timber Trusses.

226* Timber is now becoming so expensive, except in unsettled regions,
and the labor of designing zo cheap^ that it is no longer found to be good
practice to use unnecessarily heavy timbers, simply for the sake of beinp: '*on
the safe side" and avoiding computations. Henoe, in important bridges,
evBTY part of ea,ch member under stress is usually computed. On the other
lumd, the strengcth of wood is so uncertain an element that, when in doubt,
it is best to adopt that assumption which will require the lai^ger section; and
ample facte rs of safety should be used.

227. Compression members are designed as columns (see PiUars,
imder Strength of Materials) ■ and, if subjected to transverse stresses as well,
these also should be carefully taken into account. All holes and other reduc-
tions of section must of course be deducted from the gross section.

228. In the tension members also, all reductions of section must be
considered; but iron or steel rods are now generally used, in place of wood,
in tension members.

229. In addition, care should be taken ^Jiat the timber can withstand
any crusliing or shearing stresses that may come upon it or be set up in
it. ^ Thus, the ends of posts should be investigated, to see that they are safe
against crushing. Where a post meets another member at an inclined angle
and is to be notched into it, it is economy to compute the depth of notch
required ; as, the deeper the notch, the greater the gross section required for
the notched member. Where bolts are fastened to timbers by nuts, washers
should invariably be placed under the nuts, and the size of washer, necessary
to prevent crushing the wood, computed.

230. Where the wood is subjected to shearing, as where a bolt, passed
through a timber, transmits stress by the bearing of its side against the inside
of the nole. or where there is a step or table which may be sheared off by the
pressure of another piece against it. it should always be seen that there is
sufficient surface along the grain of the wood to take the shear, and some
allowance should be made for the possibility that the grain mav run out to
the surface or to some hole before all the stress can be transferred.

♦ Under construction, 1902,

TIHBBB TBtTSSES.

788

S81« CrcMS-seetlon of Upper Chord. Since it would be inoonve-
nient, in practical eonstniction, to change the section of a timber upper chord
at different points, it is designed throughout to withstand the maximum
stress occurring between any two panel points. Assume width of chord

member. Findr* = (least radius of gyration)' — — — — . Find allowable

unit stress according to column formula given in spei^cations or adopted,
UBLQg the given maximimi stress. Find area required for this unit stress.
Find the resulting depth, which for a horizontal or inclined member is pref-
erably somewhat greater than the width, to allow for bending moment due
to its own weight. If this does not give a good commercial size, it may be
well to revise, m order to obtain a better section.

232« Stmts are-preferably made as wide as the upper chord. Each strut
must be designed separately. Obtain r^, allowable unit stress, etc., as for the
upper chord. For economy, the struts should average nearly square, even
though it should be necessary to alter the section of the upper chord in order
to prevent wide deviation from a square section.

23S» The vertical ties (of iron) may now be designed. Area of cross-
section "- -Ti r-i rr— 7 . But SCO Minimum Section, If 187. The

allowable unit stress

sise of a nut is usually fixed by the diameter of the rod, but the washers
should be so designed as not to crush the wood.

234. The bearings or indentations, required in the upper and lower
chords to hold the inclined members in place, may now be computed. The
component (in the strut) perpendicular to the face or faces agamst which it
presses, is computed, and the necessarv depth obtained, assuming the width
of the lower chord the same as that of the upp^r chord and the struts.

Z35* The section of the lower chord may now be decided upon, since

the reduction of section, due to indentations, is known.

. » . .. maximum stress

Area of net crossHsection ■- ^r-. ri r:^ — : •

allowable unit stress

Ttgo 67.

986. End Joint. Fig. 67. Many different designs for end joints have
been made, proposed and discussed. The ends of the straps should enter
notches in the lower chord, to such a depth that the total stress, taken by the
end fibers of the sides of the notches, is equal to the stress that the ends of
the straps can resist by bending. This depth can be found by successive
trials, or by means of two algebraic equations, in which the maximum allow-
able pressure and the depth of notch are the two unknown quantities.

Determine the shearing surface required to transmit this stress to the body
of the lower chord. This will also determine the space between the notches
and the end of the lower chord. Compute the stress (if any) that remains to

734 TRUSSES.

be transferred, and design the long inclined bolts and thdr Washers accords
ingly. Compute also the compressive area and depth of vertical face ol
lower end of upper chord, required to transmit the horizontal ooaaponent of
its thrust. See also that the lower bearing presents sufficient surface to
resist the vertical component. The kejrs, between the bolster and the lower
chord, must be designed to carry the horizontal component of the wind. For
safety, friction between the two parts should be neglected.

!337. Fif^s. 68 show Joints adapted to most of the cases that occur in
practice with wooden beams, etc. They need but little explanation. Tig,
(a) is a good mode of splicing a post ; in aoing which the line o o should never
be inclined or sloped, but be made parallel to the axis of the i>ost; otherwise^*
in case of shrinkage, or of great pressure, the parts on each side of it tend to
slide along each other, and thus bring a great strain upon the bolts. When
greater strength is required, iron hoops may be used, as at 6, A. and /, instead
of bolts. Fig. (6) shows a post spliced ^y 4 fishing pieces; which may be
fastened either bv bolts, as m the upper part ; or by hoops, as in the lower.
The hoops may be tightened by flanges and E(»-ew8 as at «; or thin iron
wedges may be driven between them and the timbers, if necessary. Fig. C
shows a good, strong arran^ment for uniting a straining-beam k, or rafter ^
and a queen-post u; by lettmg k and I abut against each other, and confining
them between a double queen-post t t; n n are two blocks through which
the bolts pass. A similar arrangement is equally good for uniting the tie-
beam w, with the foot v, of the queens ; with the addition of a strap, as in the
figure. Fig. (e) is a method of framing one beam into another, at right sn^es
to it. An Iron stirrup, as at /, may be used for the same purpose; and is
stronger. Figs. gh,ij are built beams. When a beam or girder of great
depth is required, if we obtain it by merely lasdng one beam flat upon an-
other, we secure only as much strength as the two oeams would have il sep-
arate. But if we prevent them from aliding on one another, by inserting
transverse blocks or keys, as at g; or by indenting them into one another, as
at i j; and then bolt or strap them firmly together to create friction ; we ob-
tain nearly the strength of a solid beam of the total depth ; which strength is
as the aauare of the depth. See Horizontal Shear, % 51, under Transverse
Strength.

238. The strength of a built beam is increased by increasing its depth at
its center, where it is most strained ; as in the upper chords of a bridge. This
may be done by adding the triangular strip y y between the two beams.

239. A piece of plate-iron may be placed at the joints of timbers when
there is a great pressure; which is thus more equally distributed over the
entire area of the joint; or cast iron may be used.

240. Frequently a simple strap will not suffice, when it is necessary to draw
the two timbers very tightly together. In such cases, one end of each strap
may, as at x, terminate as a screw ; and after passing through a cross-bar 2^
all may be tightened up by a nut at x. Or the principle of the dovhU ke^,
shown at K, may be applied. Sometimes, as at A, the hole for the bolt is
first bored ; then a hole is cut in one side of the timber, and reaching to the
bolt-hole, large enough to allow the screw nut to be inserted. This being
done, the hole is refilled by a wooden plug, which holds the nut in place.
Then the screw-bolt is inserted, passing through the nut. By turning the
screw the timbers may then be tightened together.

241. When the ends of beams, joists, etc., are inserted into walls in the
usual square manner, there is danger that, in case of being burnt in two, they
may, in falling, overturn the wall. This may be avoided by cutting the endB
into the shape shown at m.

242. When a strap o, Fig. R, has to bear a strain so great as to endanger
its crushing the timber p, on which it rests, a casting like v may be used under
it. The strap will pass around the back r of the casting. The small projec-
tions in the bottom, beins notched into the timber, will prevent the ftft«^''yg
from sliding under the oblique strain of the strap. The same may be used
for oblique Dolts, and below a timber as well as above it. When below, it
may become necessary to bolt or spike the casting to the under side of the
timber. When the pull on a strap is at right angTes to the timber, if there
is much strain, a piece of plate-iron, instead of a casting, may be inserted
between the strap and the timber, to prevent the latter from being crushed
or crippled; see I and I.

TIMBER DETAILS.

736

TRUSSES.

JS43. Lower Chord Splice. Owing to the length of the lower chord,
it may be necessary to splice it, as in Fig. 69. where the splice is a tabled fish-
plate joint. The number of tables to be used is largely a matter of trial. The
use of too many tables involves too much carpentry, and consequent uncer-
tainty as to distribution of pressures, while the use of too few requires deep
notches, which may too greatly reduce the section. These tables must be
designed to resist bearing against their ends, and to resist being sheared off.
Bolts should be passed through, especially 'at the ends of the fiish-plates, to
prevent them from bending outward ; and the washers should be so designed
as to transmit safely to the wood all the stress that can come on the bolt.

■*-

Jm^

m^

1 1

1 1
1 1

1 1
1 1

1 1

I !

-LL

"ff

I!

w

^

Figr. 69.

S44. Fig. 70 shows a lower chord splice used in connection with the stand-
ard combination (timber and iron) Howe truss bridges of the Chicago, Mil-
waukee & St. Paul Railway See tiF 249-251, Figs. 73. Four of the clamp-
blocks shown are required for each joint, a block being placed upon each, side
of each stick to be spliced. The two opposite blocks forming a pair are held
together, and against the stick, by four through bolts; and the cylindrical
lugs, cast on the surface of each block, enter corresponding holes, bored in
the face of the stick. The two blocks on the same side of a stick are held
together by the hooked clamp-bar. The clamp-key is driven between the
left hook and the beveled key-seat on the left block.

m

fO' O

iO o o o

-ij-

Ciamp BlodcXeft

damp Bloek—ltiffht

Clamp

Inches

''■■■'■

Feet

la

o

Fly. 70.

245. Figs. 71 illustrate a small wooden Howe truM bridge. The top

and bottom chords are each made up of three or more parallel timbers c ee^
placed a small distance apart) to let the vertical tie-rods r r pass between
them. The main braces, o o, are in pairs or in threes. The pieces compos-
ing them abut, at top and at bottom, against triangular angle blockSf •;
which, if of hard wood, are solid ; and, if of cast iron, hollow. Fig. (d), ana
strengthened by inner ribs. These blocks extend across the three or more
chord pieces. Against their centers abut also the counterbraces e. These
are single pieces in small bridges ; or in pairs, in large ones ; and pass between
the pieces which compose a main brace. Where the wooden braces and

TIMBER TBUSSES.

engthenii

"As

kinplaae. Thevtntie^

aod oounlera abut square ngBinat the angle blocks: and are often k""' in
place only by tighteninK thB BOrewa of ths vertical I'

tbe upper chord, form no part of the truss proper.
I O n «■ »

IS, s i aQi!Tt%

W=.

Fig. 72.

247. A

truaa laid f

>rm of lalerftt bracing, Fis. 72. resembles a Horn

le. In it the diaKOoals octhe cross aie slrutBof timber:

pieces rr are round rods. One of the struts is whole, with the excsep-

At the sides of the chords, the ends of the diagonals rest upon a ledEe^omi
by the dotted line i i). about 11 inches wide, cast at the bottom of the caat-

bein^ tiehtened by meaOH of the nut aTnolda the diaaonalB firmly in place';

byt'he same means.

Tbe cast anxle block is as deep a^ a brace; its thicknes nped not PTceed
half an izush. in a large bridge. It has holes for the passage of the rod r r.

7S8

? a sinBlfl track How*

„.. _, _.. _,. with earn weLglung20to

lu i^uK cnuii. Timber not to be stnined mare than gOO lbs. per eq. ineh;

rior quality, requirmg f""" 25 to 27 tons (60j4gOlf6a!)persq. mch to bieaJc it.
Tbe rode to be upnt at their aciev-eada. To each of Cbe two sides of each
lower chord is Buppoeed to be added, and firmly conaocted. a piece at liaaX
half ag thick as one of the chord pieces, aod as loos aa three panels, at th«

t

II

1

vj-

tsx

'.

Ji-

S"

.=».,

»...,.

•ST

r

li

1

51

1

a

1

Jl

1

Jl

1

li

1

i

1

i

i

as

i

I.I.

1

tn..

li

i

1

M9. Standard "comblnE

load brIdieB, CbiCBCo. Milwi

The bridiea are desigoed fi:

train. Compreedve strcKses i

D load of 4000 H

I rail-

inch of net acea nt root of thread. Lateral rods. mai. 15,000 tba. per Bit

Umber. Cross-cieaand suardraila, whiteoak; top chord Paekinc blooi[&
white pine: all other timber, white or Norway pino or Douglas fir.

Combination BHdKeB, Chicago, Milwaukee & St. Paul Rulway:

Total length,

At center.t

t'A

J„^*

At ends,...

3.2*

3, 21

3. n

\i'^*

ii

W«ght, lba„[

130,300

155,S00

182,000

233,100

2M,900

10 ft. nVins'

The panel lenctb ia
""widTd* ft,

IS of web members at

|Wei(ht includes thet

COKBINATIOir TRUSSES.

w-

track span. Of the total weight of the bridge, the trusees have about 63
per cent., lateral Eystems 26, Boor ayiXeai 18, and wall plates 3 per cent.

740 TRUBSES.

250. In all spans, lateral bracing, 6 X 6 ins. at center, 8X8 ins. at ends,
of span; lateral rods, li ins. at center, li to If ins. at ends; collision struts,
S (ond at each end of each truss), 6 X 14 ins.; transverse portal brace, B.
between ends of upper chords (one at each end of brid^), 10 X 12 ins. ;
diagonal portal braces, D (two at each end of bridge), 6X8 ins.

The floor beams are 10 X 16 ins., 20 ft. long, 14 ft. clear span, and 2
ft. 6 ins. apart center to center. The stringers are 5 ins. wide, 12 ins. deep,
placed as shown in the end view. Fig. (&). Ties, 8 ins. wide, 6 ins. deep, 1 ft.
apart, center to center. Guard rail, 8 ins. wide, 5 ins. deep. Under each
end of the lower chord are two timber wall plates, 12 X 12 Ins., 20 ft. long.

251. Figs. 73 show the 99 ft. span. Fig. (a) is a side elevation of half
the span with top and bottom lateral bracing and floor system; Fip:. (6) a
half end elevation, showing portal bracing; Fig. (c) a panel point con-
nection (the same for upper and for lower chord) ; Fig. (d) a cast-iron
angle block for same ; and Fig. (e) a cast-iron angle block for lateral bracing.

Metal Roof Trusses.

252. Among the types commonly used for metal roof trusses are the trian<
gular truss, Fig. 26, and the arch truss; the triangular truss being used for
short spans, and the arch t;^iiss for Ions spans. See tif 255, etc., and
Fig. 76.

253. In roof trusses of short span, carnring light loads, the minimum
sections prescribed in bridge specifications will often suffice for all the mem-
bers. The. connections are made by means of rivets and flat plates, some-
what as shown in Fig. 74.

(a)

Tig. 74.

- 254. In designing such trusses, no matter how tight the stresses, care
should be taken to avoid eccentric loadings, which are apt to occur where the
members are not repeated symmetrically on both sides of the flat plates.

255. Train-shed Roof, Broad Street Station, Pennsylvania
Railroad, Philadelphia. Figs. 75 (aito (/). Built 1893 by Pencoyd
Bridge and CJonstruction Co., erected by Railroad Co. Span, 300' 8*; rise.
108' 6^^; length, 689' 2i*. Twenty trusses, arranged in 10 pairs, 3 pairs
shown in Fig. (c) .

256. Each truss consists of two arched rafters, A B and B C, and a hori-
sontal chord, A C, with three pin joints, A, B and (J, Fig. (a).

Each rafter is composed of two chords, 14 radial braces and 26 diagonals.
For the sake of appearance, the chords are extended across the top panels,
occupied by the two triangular apex members, and ure there connected by
a sliding joint.

257. The horisontal chord AC, Fig. (a), lies below the floor of the
train-shed, and is suspended, at intervals, from girders which support that
floor and which are carried by iron columns in the lower story.

258. End bearings. One foot of each truss rests upon a fixed shoe, bolted
down to the pier ; the other on a nest of 1 1 steel rollers.

259. At each end of the roof, a horisontal wind truss, WW, Fig. (a), is
suspended from the rafters, its ends resting upon brackets riveted to their
iower chords.

METAL RQOF TKUBSE8.

SO. Between ihese horiiontal wind trueaea snd the
I the kJbss curtaJn closing each end of tbe roof,
d inisses (not shown), witli hotiionlal and diagooi

'9 of eye-bars) IS.OIXI. tutal 154,oSo lbs. One
nmewoik of entire train-slied, i Deluding lt\xsiti.
icing, about 7,000.000 Iba.

363.

topermi

Thetr
ttheL

Bveler. Fig!,

siir."l?d

^wa.

joftimb.
ile the d<

5W™1

WBSsodesi

St"

ae*. .

K.'d

>P| section, fr
top of the ti

SSi

XI

at 8. Fig:

,,(a)a
olsott
lainder

nd(<:)
.e foot

f;,.",T.

srs,

througli

' 1? i

erect one pai
last erected.

aboi

lis

i3

'iftijjB

tbel

S's

742

TRUSSES.

265. Dimensions of Arched Roofs of Ijarge Span.

Arches.

Roof.

Span.

Rise.

Length.

Area
Covered.

Train-sheds.

ft. ins.

ft. ins.

ft. ins.

eq. ft.

Pennsylvania Railroad,

Philadelphia. Broad

^

Street Station,

300 8

108 6

689 2

177.160

Pennsylvania Railroad,

Jersey City

Phila. & Reading Railroad,

252 8

90 0

662 6

164,900

Philadelphia. "Reading
Terminal," Market St., . .

259 0

88 3

606 8

131,260

New York Central & H. R.

R. R., New York. Grand

Central Station, ^

199 ?,

94 0

662 0

129,866

Midland Railwayr London.

St. Pancras Station, ....

240 0

107 0

706 0

169,400

Cologne, Germany. Nave, .

209 7

78 8

836 0

175,200

Exposition Buildings.

Machinery Hall, Paris,

1889. Nave

362 9

149 0

1.380 0

500,600

Manufactures and Liberal

>

Arts Building, Chicago,

1893,

368 0

20)3 0

1,268 0

466,600

Timber Roof Trusses.

206. Dimensions for small timber roof trusses. Figs. 43 to 47,
5^ 148, etc., of white pine. Span — 4 X rise. Combined weight of trusBes,
roiof and load, including snow and an allowance for wind, 40 lbs. per square
foot of roof surface. Trusses 12 feet apart, center to center. Safety factor
«■ 3; b = breadth; d =» depth. For Fig. 46, struts 4.5 X 4.6 ins. would
suffice ; but, for practical reasons, the struts are general^ made as wide as
the rafters. For Fig. 47, the straining beam is 12X12 ins. For Figs. 45,
46 and 47, two sets of dimensions are given ; the first for ohord unloaded ; the
second for chord loaded with 100 lbs. per square foot.

Span

h
30

40
60

Rise.
Ft.

Rafters.
Ins.

Chord.

King or

Queen

Rod.

Ins.

Diam.

Fig.

Timber.
Ins.

Iron.

Ins.

Diam.

Chord.

b.

d.

10
9
11
8
10
12
14

b.

d.

10
9
11
8
16
12
12

43
46

46
47

7.5
7.5I

10 1

16 j

6

6.5

8.6

6

8
10
12

5

6.5

8.5

6

8
10
12

1
1

• •

1.6

• •

1.6

. .

1

\*

2

2*

unloaded

unloaded

loaded

unloaded

loaded

unloaded

loaded

TRANSPORTATION AND ERECTION. 743

TRANSPORTATION AND ERECTION.

267* Girders should be loaded for transportation on flat cars, with web
vertical, and with bearings at the points distant i span from the ends. Where
too long for two cars, one or more idle spacing cars are used and the points
of support are pivoted.

268. Girders may be erected by means of gin poles, derricks, gallows,
etc., or they may be skidded from the cars and lowered to place by jacks and
blocking. Ginpoles should have at least four guys, with tacklee, for easy
adjustment. Efoisting may often be done by means of a locomotive running
on the tracks of that part of the structure which is already completed.
Ropes are used at about i their ultimate strength.

•

269. Viaducts are usually built from above, by means of an overhead
projecting traveler. Sometimes by means of a cableway ; but this method is
slow. In some cases the traveling tower is on the ground, and reaches to the
top of the viaduct. Or, the viaduct may be erected by means of false-work,
or from an existing structure.

270. liOng span bridges are ui^ually built upon a' platform of false-
work or OB a row of trestle bents, well braced.

271* Erection. Upon the false-works the lower chords are first laid,
as nearly level as may be. The upper chords are then raised upon temporary
supports which foot upon the one that carries the lower chord. The upper
chords are first placed a few inches higher than their intended positions, in
order that the web members may readily be dipped into place. When the
web members are in place, the upper chords are gradually lowered until all
rests upon the lower chords. The screws are then gradually tightened, tp
bring all the surfaces of the joints into their proper contact; and by this
operation (the upper chord members having the necessary excess of lezigth),
the camber is formed, and the lower chords are lifted clear of the nUse-
works ; the truss now resting only upon its permanent supports.

272. False-work is ordinarily constructed of hemlock or pine, costing
about $20 per 1000 ft. board measure. Allow about $15 per 1000 ft. B. M. for
framing, etc. $5 to $15 per 1000 ft. B. M. may usually be obtained for old
material ("salvage '0*

The main members are usually 12 X 12 ins., and the diagonals 3 X 12.
Bolts, i inch. Owing to the temporal^ nature of false-work and to the sal-
vage which may be obtained for it if it is not too badly cut up, it is advisable
to use plenty^ of material of good standard sizes, especially m longitudinal
bracing, which may be placed between alternate pairs of bents, forming
towers.

273. In soft bottoms, the false-work may rest upon piles, to which
the uprights of the false-work mav be notched and bolted, or banded. Not
less than 4 piles per bent should be used. As manv as 24 have been used.
Piles should be braced below water-mark. Bents shoi^d be built in stories
of from 12 to 30 feet each. Connections should be made, by means of side
pieces or fish-plates.

274. With rock bottom, in a strong current, it may be ezpedi«it to
sink cribs filled with stone, as a foundation for the false-work.

275. The erection of cantilevers and suspension bridges requires
much time ; but their use is often necessitated by the impossibility of erecting
false-work.

276* Renewal of bridges may be accomplished by displacement; either
by protrusion, where the new span is skidded longitudinally along the track;
by transverse displacement, where both old and new spans are placed on
tracks running normally to the bridge; by vertical displacement, either ris-
ing or descendihg; or by pivotal displacement, the old span swinging out
about a pivot, and the new span swinging into place about another pivot.

277. Cautions. In erection and renewal, consider dead weight of
bridge, effect of impact of current of stream, impact of boats, ice, drift, etc.,
especially when floods are to be apprehended, and strains of hoisting tackle.

744 TRUSSES.

For rigidity, a liberal safety factor must be used. Drift may pile up and
form a dam. Trestle bents, in the water, increase the velocity and scour
of the current, and may thus cause undermining. False-work may be pro-
tected against drift by fender piles. Provide against eccentricities of wind
stress. Numerous accidents have shown the expediency of guarding the
unfinished truss itself against high winds. All lateral and other wind
bracing should be in place, and secured, before the false-works ore re-
moved and the trusses allowed to rest upon their final bearings.

Avoid dropping tools, etc. Even very small pieces, falling from a great
height, are dangerous to life and even to the bridge. Hooks in tackles are
liable to break or to pull out. Travelers should be well guyed and clamped,
and carefully watched.

TRUSS SPECIFICATIONS. 745

DIGESTS OF SPECIFICATIONS FOR BRIDGES

AND BUILDINGS.

(1) Steel railroad, highway and electric railroad bridges.

(2) Combination (wood and steel) railroad bridges.

(3) Steel roof trusses, framework and buildings.

The following Digests of Specifications for Bridges and Buildings are in-
tended primarily to give a general view of the essential features of current
practice in such matters, and only secondarily to indicate the practice of
any particular company.

(1) DIGEST OF SPECIFICATIONS FOR STEEL. RAILROAD

AND HIGHWAY BRIDGES.

List of Specifications TJsed.

A, American Bridge Company,

Genteral Specifications for Steel Railroad Bridges, 1900.
Aa» American Bridge Company,

General Specifications for Steel Highway Bridges, 1901.

B, Baltimore & Ohio Railroad Company,

General Specifications for Railroad and Highway Bridges, Roofs,

and Steel Buildings, 190L
Cf Cooper, Theodore — ,

General Specifications for Steel Railroad Bridges and Viaducts,

1901.
Cc, Cooper, Theodore — ,

General Specifications for Steel Highway and Electric Railway

Bridges and Viaducts, 1901.
D, Delaware, Lackawanna & Western R. R. Company,

Specifications for Steel Railroad Bridges, October, 1899; revised

to July, 1900.
£, Erie Railroad Company,

General Specifications for Bridges, 1900.
G, General practice.
Oo, Oaborn Engineering Company,

General Specifications for Highway Bridge Superstructures, 1901.
P, Pennsylvania Railroad Company,

Standard Specifications for Steel Bridges, January 1, 1901.
B, Philadelphia & Reading Railway Company,

Specifications for Steel Bridges, 1898; revised February, 1901.
T, New York Central & Hudson River R. R. Leased and Operated Lines,

General Specifications for Steel Bridges, 1900.

I. GENERAL DESIGN.

Limiting Spans for Different Types.

Beams AND Girders. A Aa B C Cc D T

ft ft ft ft ft ft ft
Rolled beams, solid

floor, etc., up to up to up to up to up to up to up to

20 40 20 20 40 20 25

Plate girders, 20 to 25 to 20 to 20 to 20 to 20 to 25 to

100 80 100 120 80 100 100

Riveted trusses,* 100 to 40 & 100 to 75 to 40 & 90 to 100 to

140 over 120 150 over 160 200

Pin trus.ses over over over over over 150 & 200 &

140 140 120 120 120 over over

Riveted trusses,* under 100 ft: pin trusses, over 100 ft, Oo.
Depth of truss, min, - one-eighth of span, Oo.

♦Unfortunately called ** lattice girders." A; "lattice trusses'* and
•• riveted lattice girders," C; "riveted lattice trusses," D, Y.

746 TRUSSES.

A, Aa, Am B Co; B, B <fe O; C, Cc, Gooperi D, D L & W; £, Erie;

Classification of Highway and Electric Railroad Bildses.

Aa Cc

TAl* City bridges having buckle-plate floors, and paving on concrete
A < base.

(,A2* City bridges having plank flooring.
B B* Suburban or interurban bridges for heavy electric ears.
C C Town or country bridges for light eleotnc cars or heavy loada>
D D Country bridges for ordinary highway traffic.
£1 El Bridges for heavy electric or motor cars only.
E2 '£2 " ** light " " ••

Camber.

Top chord panels longer than lower chord panels by one-ei^th of an indi
in 10 ft - 1 in 960, A. B, C, E, B, T.

Highway bridges, threensixteenths of an inch to every 10 ft, Cc«
About three-fourths of an inch in 100 ft ■- 1 in 1600, D.
Sufficient to bring joints of compression chord to a square bearing when
truss is fully loaded. Each member built longer or shorter in proportion to
the stress to which it is subject under a fuU dead and a full live load, so that
under full loading it will have its normal length, Oo.

Cross Section of Bridge.

Gage, usually, 4 ft 81 ins. Distance, een to cen of tracks, 12 to 13 ft.

Width between trusses or girders in deck spans. Pin spans, min,
0.05 span. Riveted trusses (D), 10 ft. Plate girders (6), 5 to 7 ft. Spans
not over 60 ft, 7 ft: 60 to 100 ft, 8 it; over 100 ft, about one-twelfth of
span, D. Plate girders (Y), over 60 ft, in proportion to hei^^t.

Clearance in through spans, on tangents, G*

a 3 to 3f ft

b 7 ft

c 5 to 51 ft I ;

d 4 to 6 ft AU-6-*

e 10 to 14 ft '

f 1 to 6 ft '

h 20 to 22 ft

Minimum Clearapce on Curves. ^^^

Same min clearance as on tangents, A ; Ditto for car 74 ft long, 48 ft oen
to cen of trucks, 10 ft wide, B ; Ditto for ear 75 ft long, 54 ft cen to cen of
trucks. Additional clearance '^ 0.8 d ins on each side; « 1.6d ins between
tracks ; where d — degree of curvatiu^ » central angle subtended by a chord
of 100 ft, C ; increase lateral clearance at top of car 2.5 ins for each inch at
superelevation of outer rail, C. Cen line of bridge bisects middle ordinate
and is parallel to chord, T.

Highway Bridges. Headway. 14 ft. Oo. For classes A, B, C, and
E, min — 15 ft, Aa, Cc; for a width of o ft over each track, Aa. For
Class D, 12.5 ft, Aa, Cc. Horizontal clearance. Min 14 ins greater
than width of roadway between wheel guards, Aa. For electric cars, 6.5 ft
from cen of track, Aa ; 7 ft, Cc. On curves, provide for a oar of 45 ft extreme
length, 8 ft wide, 20 ft between truck centers, Aa. Width between centers
of trusses, min — 0.05 span, G.

Tension Members.

In general, hip verticals and one or two panels of lower chord at each end
of span are required to be of rigid section, so as to resist both tension and
compression.

Angles, used as tension members, must be fastened by both l^;s, C» D |
or the section of one leg only will be considered effective, C.

Adjustment.

Avoid adjustable members, A, Aa, D (P except in counters). Avoid

* Cc, Classes A and B shall be designed to carry, at any future time, a
double track electric railway.

TRUBS BPECIFICATIONS. 747

G, gen'l; Oo^Osb'o; P, Pa; B, R'd'g; Y, N YC; Aa, Cc, Oo, H'way.

adjustable counters. C» Counter rods and ties in pin spans adjustable, Y;
screw ends upset, C» E» Y, screw threads U. S. Standard, C, Cc, D* E ; diam-
eter, at base of thread greater than in body of bar by onensixteenth of an inch,
D; about 10 per cent, Y: 17 per cent, Oo.

Rods with welded heads must be of wrought iron, Oo« Loops must de-
velop full strength of bar, Oo.

Compresfllon Members.

End posts and upper chords have 2 webs, a cover plate on top flange;
batten or tie plate, and lacing on bottom flange, G

Not more than one plate, and that not thicker than one-half of an inch
(in highway bridges tnree-eighths of an inch), shall in general be used as a
cover plate, C, Cc. Cover plate must not extend more than 4 ins beyond
outer row of rivets, D.

Joints between sections spliced on all sides with at least 2 rows of
closely pitched rivets on each side of joint, C. Abutting surfaces faced A
(D, P, except in top flanges of girders), E, R, Y. No reliance on abutting
surfaces, E.

Lattice Bars.

Width, from 1.5 to 2.5 ins. Thickness, in sinj^le lattice, one-fortieth,
distance between rivets, in double lattice, one-sixtieth, G. If over seven-
sixteenths of an inch, use angles, Oo. An^e with axis of member; in single
lattice 60^, double 45^ G.

Pitch, width of channel + 9 ins, Ct Cc ; 8 X least width of segment, P, B*

Double lattice bars riveted together at their intersections, C» D.

Batten Plates. (Tie Plates, Stay Plates.) . Min length generally — from
0.75 to 1.5 X its own width. Min width, 0 ins, or 0.66 X own length, or ->
least width of member, Oo. Min thickness — three-e^hths of an inch or one-
fiftieth to one-sixtieth of distance between centers of nvets, G. Rivet spac-
ing (D) max 4 ins cens.

Pin Joints.

Eye Bars. Thickness, min, 0.625 inch, or 0.2 X width of bar, Oo.
Heads upset, rolled or forged. A, B. No welds allowed A* D» except (B) to
form loops of laterals, counters or sway rods, B. Upset and die-forged, Y.
Beads not more than one-sixteenth of an inch thicker than body, Df P*
Bars annealed, G | before boring, D. No forge work after boring, B. Bars
to be placed side by side must be bored at the same temperature. A, Aa, Y.
Pins must pass through without driving. Eye bars working together must
be clamped together, and bored at one operation, Oo.

Distance between pin-holes max variation, one sixty-fourth to one thirty-
second of an inch ; or one sixty-fourth of an inch in from 20 to 25 ft, G.

Built Tension Ilf embers. Net section through pin hole «- 1.25 to 1.50
X net section through body of member. Net section back of pin hole =■
0.76 X net section through pin hole, or — 0.80 to 1.00 X net section through
body of member, G ; proi>ortion for double shear on section from back of pin
to end of plate, Oo | length of plate back of pin, min. 2.5 ins, Oo. Distance,
back of eye to back of member, greater than radius of pin, Y.

Pin Holes. Clearance between pin and hole, from one-fiftieth to one
thirty-second of -an inch, G.

Pin Plates or BeinforcingT Plates. At least one plate on each side
must extend not less than 6 ins beyond edge of batten plate, G.

Pins. Up to 7 ins diam, rolled, P ; over 7 ins, forged, C.
Diameter, min, from 0.66 X to 0.85 X largest dimension of any of its eye
bars, G.

Plate Girders.

Min depth; about one-ninth to one-twelfth of span, G.

Proportions of l¥eb and Flange. Bending moments resisted entirely
by the flanges, shear resisted entirely by the web plate, C, Cc, B; except
when the web is made in one length or is fully spliced to resist the bending
stresses, in which case one-sixth of area of cross section of web plate may be
oonsidered effective as flange area, Oo.

748 TRUSSES.

A, Aa, Am B Co; B, B & O; C, Cc, (Dooper; D, D L & W; E, Erie;

One-eighth of gross area of web included in flange, A, Aa, B; if length =
90 ft or over, E: if length is less than 50 ftj only the cover plate and the
horizontal legs of the flange angles are to be included in the flange area, E;
no part of web included in flange, C, Cc, D, P, R, Y.

Web. Thickness, min, three-eighths of an inch, G ; in highway bridges
(Cc, Oo), five-sixteenths of an inch.

Total shear, acting on side next to abutment, to be taken as transferred
into flange angles within distance *= depth of girder. A, Aa, B, E, Oo, P.

Web Splices. A plate on each side of web, G; at least three-eighths
inch thick, A, B; at least five-sixteenths inch, or three-fourths as thick aa
web, and wide enough for 2 rows of rivets on each side of splice, Oo.

Stiffeners. Generally required at ends and at points of concentrated
load. Intermediate stiffeners required usually when unsupported distance
between flange angles exceeds 50 to 60 X web thickness; when shear ex-
ceeds 10,000 — 75 X -, . W- — t C : in highway bridges when shear exceeds

thickness

12,500 — 90 X -^^^^ , Cc ; when shear exceeds ^P'OOO X J jj^, ^here t =
thickness . , d^;

"^ 3,000 t2
web thickness, d = distance between flanges, Oo.
Spacing usually = depth, or = 5 or 6 ft.

Unit stress, max, 10,000 — 45 —, C ; in highway bridges, 12,000 — 65 —,

r r

where 1 = length of stiffener, r — its least radius of gyration, Cc.

Dimensions of angles usually 3i- X 3 X A to 5 X 3i X i.

Flanges.

Unbraced length of flange (compression flange, C, P) max «= 12 X width,

B, P, B, Y; 16 X width. A, C, Cc^: 20 X width, D; in highway bridges.
= 20 X width, Aa, = 25 X width, Oo.

Comp. flange has same gross area as tension flange. A, Aa, B, C, Cc,Oo. P.
Cover plates must not extend more than 5 ins, or 8 X thickness of first

Elate, beyond outer line of rivets. A, Aa, C, Cc. If of unequal thickness, the
eaviest {jHates are next the angles, and the lightest outside. A, Aa. B, C,
Cc, Y. One must extend full length of girder, B, E. Others must be lon§;
enough to take 2 extra rows of rivets at each end, C, P.

Bracing, Riveting, Bearings. See Bracing, Riveted Joints, and Beai^
ings. below.

Beam Girders.

Beams in groups of 2, 3 or 4 for each rail, 10-inch channel separators, about
3 ft apart, riveted to webs, B.

Bracing.

Composed of rigid members, riveted. A, Aa, C, Cc, Y; members inters
sect each other, and other members to which they are connected, on common
center lines, passing through all centers of gravity. Attachments riveted
symmetrically in all directions, Y.

For Beam Girders. Ten-inch channel strut at each end; with 2 or 3
beams, angle bracing between girders ; with 4 beams, angle struts about 6 ft
apart. Connections to have at least 3 rivets, B.

Lateral.

B, in through spans. Top bracing; portal struts at ends; intermediate
struts as deep as the chords; single angles. 3^ X 3^ X f, intersecting in each
panel, for single track; double angles, latticed, for double track.

B, in through spans, bottom bracing; end bottom strut and intermediate
angles, riveted to each other and to stringers at each intersection. Not less
than 4 rivets at each intersection and at each end connection.

B, in deck bridges, complete upper and lower systems at each panel.

Oo, bottom end struts in all spans, whether deck or through.

Y, top and bottom lateral bracing in all deck bridges and in all through
fridges having sufficient head room. Lower lateral bracing in all through

TRUSS SPECIFICATIONS. 749

G, gen'l; Oo, Osb'n; P, Pa; B, R'd'g; Y, N Y C; Aa, Cc, Oo, H'way.

, — . 1

bridges. Upper system in all stringers framed between and riveted to floor
beams where length of stringers exceeds 15 times width of stringer flange.

Y, in deck bridges without metal floor system, upper bracing of cross-struts
at each panel point, composed of 4 angles latticed, with same depth as upper
chord; stiff diagonals intersecting in each panel and riveted to each other
at each intersection.

For Plate Girders.

Deck. Between upper flanges, angles with at least 4 rivets at connections.
Through ; lower lateral bracing of angles, intersecting in each panel, riveted
to each other and to stringers at each intersection, B.

Lateral bracing angles generally of same size as those in stiffeners, R.

Cc, 4n highway bridges, a buckle plate floor may be considered as the re-
quired system of lateralbracing at the floor level.

Sway (diagonal, cross, vibration or wind) and Portal.

Proportioned to resist unequal loading of trusses in double traek spans, "E,
B; end sway bracing to transmit all horizontal forces to abutment, G; to
earry half the max stress increment due to wind <Sc centrif . force. A, Aa, B, P«

In deck spans, at each panel point, A. Aa, D, £, P, B> Y.

Overhead bracing in through spans whose depth exceeds 25 ft, C» D* P,
Y; in highway bridges, 20 ft, Cc} 25 ft, Oo.

in ix>ny trusses and through plate girders, at ends and at each floor beam
or cross strut, A, Aa. K; at every panel point, D.

In through and half through plate girders, at each floor beam and at each
end, or, if there is a solid floor, not over 8 ft apart, Y.

In deck plate girders, rigid cross frames at ends and max 20 ft apart, Y;
sway frames of at least 4 angles at ends and at points 12 to 14 ft apart, B;
through, not more than 12 X flange width along top flange, B.

Riveted Joints.

Rivet Holes, In I beams, must be drilled, B.

May be punched; in steel not over i to f inch thick, 6.

Sub-punched one-eighth of an inch smaller, and reamed to one-sixteenth
of an inch larger, than rivet, in steel ovier five-eighths to three-fourths of an
inch thick; in connections for floor beams and stringers to main trusses or
girders, E.

No drifting allowed, A, B, C, D, E, R.

No interchange of pieces after reaming, D, P.

Hole larger than rivet by one-sixteenth of an inch, G.

Die larger than punch by max one-sixteenth of an inch, G,

Distance from edge of plate to center of rivet. Min, 1.25 to 1.6 ins, or 1.6
to 2 diams of rivet. Max, 4 to 5 ins, or 8 X thickness of plate, O.

Pitch Min = 3 X diam of rivet, general; preferably 4 X diam, Oo.

Max pitch in line of stress, 5 to 6 ins, or 16 X thickness of thinnest outside
plate connected; normal to stress, 30 to 50 X thickness of thinnest outside
plate connected. At ends of compression members (or of built members in
tension, B) ; for a length of 1.5 to 2 X width or depth of member, 3.5 to 4 X
diam of rivet, G.

In plate girders, for rivets connecting web to a top flange supporting the
track, max — 3 ins, R.

Rivets. Diam generally three-fourths or seven-eighths inch. Heads
hemispherical, G. Height of head, min » 0.6 diam, R.

Driving. Avoid hand riveting. Machines, direct-acting,^ worked by
steam, hydraulic pressure or compressed air, capable of maintaining applied
pressure after upsetting, G.

Floor.

Floor Beams. Depth, min, ==» i X length, Y. In railroad bridges and
important highway bridges, riveted to posts of trusses or to webs of plate
girders, G. Given also a bearing on lower flange of girder or on a bracket,
G. In default of such bearing, increase number of rivets by 25 per cent, R.

Hangers, when permitted, not adjustable, C, Cc. Hangers made of plates
or shapes, Oo.

Stringers. Depth, min, == i X length, Y. In highway bridges. Classes

750 TRUSSES.

A» Aa, Am B Co; B, B & O; C, Cc, Cooper; D, D L <fc W; E, Erie;

Al and A2, of steel; Classes B, C, and E, track strineers of steel; Class D, of
wood or steel, Cc. In railroad bridges, and preferably in highway bridges,
riyeted to webs of floor beams and supported by their flanges or by brackets,
B, R. Value of this bearing neglected in determining niunber of rivets
required, B.

Spacing, cen to cen, 6 ft, 6 ins, A, B, C, D, T; 5 ft, E| double track,
through, generally 6 f t, B ; single track, 8 ft, R.

Trough Floors. Troughs rectangular, built of plates and angles, and
riveted to main girders or trusses by angles, and, when practicable, by bracket
angles under the lower horizontal plates. Gusset plates riveted to girders
and troughs at distances of not over 8 ft, T.

Bottom filled with a binder composed of 1 cu ft of clean, sharp gravel,
screened to one-fourth of an inch, to 1^ gallons of No. 4 asphal^ paving com-
position or enough to fill voids. Gravel first heated to 300° F and the whole
mixed at that temperature, T.

Wooden Floor. Continued over abutments. A, B, C, B.

Ties or Floor Beams. Long leaf yellow pine or white oak, G.
Width, 8 ins, A, B; 9 ins. B.

Depth, 8 ins, for 7 ft span of tie, to 14 ins for 12 ft, B ; 12 ins, R ; 10 ins, T.
Notched down one-half of an inbh ; max 1§- ins, B.

Spacing. Usually 6 ins clear; 16 ins oen to cen, R. Every 3d, 4th, or
6th tie fastened to stringer by i inch bolt or lag screw, 6.

Wooden Joists In Highway Bridges. Width, min 3 ins or 0.25 X
depth; spacing, max, 2 to 2.5 ft. Ends of joists lap past each other at bear-
ings on noor beams, with 0.5 inch space between them for circulation of air.

Wooden Floor Beams for Electric Railroad Bridges, Classes El
and E2 Min 6X6 ins, spacing max 6 ins, notched down one-half of an inch
and secured to girders by three-fourths-inch bolts not more than 6 ft apart.
From center of span toward end, so notched (Cc) as to reduce camber.

Guard Rails. 6X8 ins, yellow or white pine, 6. Inner face not less than
3 ft 3 ins from center of track. A; 3 ft 7i ms, B; 5 ft 4 ins, T; 7 ft li ins
apart, clear, R. Notched ^ to li ins over ties, G. Fastened to every
3d or 4th tie (to each tie, R) and at splices by three-fourth-inch bolt or lag
screw, G • Splices over floor timbers, with half-and-half joints of 6 ins lap, G •

Wheel Guards and Curbs in Highway Bridges. Wheel guards
6X4 ins, blocked up from floor plank by blocks 2X6 ins, 12 ins lon8^ not
more than 5 ft apart, bolted to stringeis through blocking pieces, tnree-
fourths-inch bolts, G. In electric railroad bridges (Cc, Claefs E) guard
timbers min 5X7 ins, notched 1 inch over floor timber and secured by three-
fourths-inch bolt to every third floor timber and at each splice.

Buckle Plates. Min five-sixteenths of an inch thick for roadway, one-
fourth of an inch for footwalk, crown 2 ins, for widths of 4 ft under roadway,
5 ft under footwalks. Preferably in continuous sheets of panel lengtJis.
May be pressed or formed without heating.

Bearings on Abutments and Piers.

Permissible load on masonry foundations, max, poimds per sq inch.
400, A, Aa, P; 300, B; 250, C, Cc, D, E, R; dead load, 500; live load,
260, T.

Bed Plates. Of medium steel, C, Cc. Min thickness, three-fourths to
1 inch; in highway bridges, one-half of an inch. Max fiber stress 12,000 lbs
per sq inch, E.

Where ends of two spans rest on one pier, spans are tied together, or have
bed plate, three-eighths to three-fourths of an inch thick, continuous under
both, G.

Sheet lead, one-eighth to one-fourth of an inch thick, between bed plate
and masonry, G.

Anchor bolts, 1 to 1.25 ins diam, 9 to 12 ins in masonry, G; fastened with
sulphur, R; with cement, C, Cc, Y.

Pedestals. Of riveted plates and angles, C; or cast steel, T. Base
plate and connecting angles, min three-fourths to seven-eighths of an inch
thick, B, C, Y. 2 rows of rivets in vertical legs, C, Y.

TRUSS SPECIFICATIONS. 751

G, een'l; Oo» Osb'n: P, Pa; B, K'd'g; T, N Y C; Aa, Cc, Oo, H'way.

Expansion Bearings. Provide for temperature range of 150^ F, A, Cy
E, P» K| for expansion of 1 inch in 100 ft, D, T.

One end sliding, usually in spans less than 60 to 00 ft, G.

One end on friction roUers. usually in longer spans, G; in all trusses, T«

Hinged bolster at each end, in spans from 80 to 100 ft, G.

Rollers rest on bars 3X1 inch, spaced 2 ins and riveted to bed plate, B*
Free ends anchored against lifting and against moving sidewasns, ۥ T.

Rollers. Of machinery steel, C, Cc. Min diam, 3 to 4 ins. A, B, D, E, P»
B} 3 ine up to 100 ft span, 1 inch for each additionfd 100 ft, C, T. Max
pressure on rollers, in lbs per Unear inch, 700 yd, B| 1200 )/d. A, B, P|
300 d, C, D, E; 600 d. Go; d -- roller diam, ins. Length, ins, — 000 yd, T.

n. MATEBIAL.

Boiled and Cast Steel and Iron.

Boiled steel in superstructures in general.

Cast steel in bed plates in special oases, in machinery of movable bridges.

BoUed Iron in loop-welded rods, P| in laterals and unimportant mem*
bers, B«

Cast Iron in bed plates in special cases and in machinery of movable
bridges.

Boiled Steel, Grades.
Soft* In general, in all principal parts.

Medium. In pins, friction rollers, lateral bolts, bearing plates, eye-bars,
ftliHing plates and bed plates; permissibly (C) in compression m chords,
posts, and pedestals.

Btvet. In rivets.

Machinery. In expansion rollers, C.

Boiled Steel. Manufacture.

^ee also Digest of Specifications of Intemat'l Ass'n for Testing Materials.)

All to be made by open-hearth process.

Slabs for rolling plates are hammered or rolled from ingots of at least twice
their cross-section, A, B.

Plates up to 36 ins wide rolled in universal mill, I>, B| or have edges
planed, D.

Boiled Steel. Manipulation.

Annealing. Eye-bars heated to uniform dark red and allowed to coo)
slowly, P; members worked at blue heat are heated to a uniform bright red
(not exposed to direct flame) and allowed to cool slowly, B.

Steel must not be welded, B. No reliance upon welded steel, C.

No .work put upon steel at or near blue heat, or between boiling-point and
point of ignition of hardwood sawdust, C«

Boiled SteeU ^Shop Work.

Sheared edges of steel thicker than five-eiehths inch shall be planed, B.

All sheareaedges (in medium steel, D) snail be planed off to a depth of
0.26 inch, D, T | except web plates of girders over 36 ins deep when covered
by flange plates, and nllers where sheared edges are not seen, D. Grinding
not accepted as equivalent of planing, except for lattice bars, T.

No sharp or unfiUeted re-entrant corners permitted, D. T. Where a

Slate, angle or shape has been cut into, the fillet, as well as the cut, must be
nished with sharp cutting tool, or with chisel and file, so that no sign of the
punched or sheared edge remains, D.

Angles or bent plates, used as end connections on girders, floor beams or
strineers, must be accurately fitted, so that when the member is milled to
lengftn not more than one-sixteenth of an inch will be taken off these connec-
tions at their roots, D.

Material bent bv punching must be straightened before bolting up, B.
Web plates, if buckled, must be cold-rolled to remove the buckles, D.

Spbced chord sections must be assembled and strung out in shop in lengths
of not less than three sections, and, after being drawn up into contact at

T52

TRUSSES.

A» Aa» Am B Co; B, B & O; C, Cc, Cooper; D» D L & W; E, £ri«|

ioints and lined up with splice plates in place, the field rivet holes shall be
reamed to a tit before taking apart, and tne assembled parts, with their splice
plates, match-marked, D.

Riveted members must have all parts pinned up and drawn together before
riveting up, D.

In cases of skew work, or of complicated connections, or of a large number
of pieces of one and the same kind, the work sh&U be set up ana fitted to-
getner in the shop, sufficiently to insure against any misfit, D.

Abutting surfaces a|> ends of sections of compression members, and ends
of members to be framed together, are usually required to be faced.

Boiled Steel. Requirements. •

See also Digest of Specifications of International Association for Testing
Materials.

TEN81L.E TESTS.

Specimens of Medium, Soft and Rivet Steel. For tests of full site eye-bars

see below.

Ultimate Strength, u, and Elastic Limit, el, in thousands of lbs per sq
inch. Elongation, s (stretch), and reduction of area, a, in peroentases ci
original dimensions. Elongation measured in a length of 8 ins.

Medium or
••Pin" Steel.

Soft or "Bridge"
Steel.

Rivet Steel.

u

el

s

a

u

el

8

a

u

el

8

a

A, Aa . .

60-70

0.5 u

22

52-62

0.5 u

25

• •

48-^8

0.5 u

26

B ....

• •

• •

• •

58-63 30

25

• •

61-56

27

26

C, Cc ...

60-68

0.5 u

22

54-62 0.5 u

25

a •

50-58

0.5 u

26

D ....

62-70

0.5 u

22

54-62, 0.5 u

26

• •

48-56

0.5 u

28

E ...

• •

• •

• •

56-64 0.58 u

27

45

• •

• •

• ■

Oo ..

60-70

35

22

52-62 32

25

• •

50-60

30

26

P ....

62-70

33 17

46

52-62! 28

25

50

48-56

28

28 56

R ...

60-68 0.5 u I20

• •

52-60' {«|8^}

56-64' 0.6 u

i

25

• •

48-^56

28

28 ..

Y....

62-70 0.6 u 25 45

1

26

50

48-56

• •

28 55

1

4t
ft

• •

Specimens from metal over five-eighths of an inch thick, el -> 0.66 u, T*
eye-bars, same requirements as for medium steel, D*
u = 63, B.

over 1 .5 ins thick, deduct from el 1 for each one-
eighth of an inch ; el, min » 20, C.
and pins u — 62-70, 1 — 0.6 u, s — 26, a —

45. Y.
15, C, Cc.
(medium, soft or rivet) s, 5 per cent less, A.
(soft) s - 20, B.
and rollers, s = 10, D.
rollers and bearing plates, u =■ 70-78, s ■- 22, Y.

<(

4(
(f

CI

f<
«c
cc

M
M

«
(i

••

C(

(•
41
44
4t

pms, s
it

<<

BENDING TESTS.

In medium steel, specimen to bend through an angle of 180* around a bar
of diam = 1 to H X thickne.«is of "specimen, without showing fracture on
outside of bend ; in soft and rivet steel, to bend fiat upon itself.

NirKmQ TEST.

When nicked and bent around a bar of diam » thickness of rod. rivet
steel shall show a gradual break and a fine, silky, homogeneous fracture, D.

DUIFTINQ TEST.

Center of hole as in ordinary practice, or 1.5 to 1.87 ins or 2 diams froia
%lge of plate; enlarge to 1.25 to 1.50 diam, G.

TRUSS SPECIFICATIONS. 763

G,gen*l; Oo, Oab'n; P,Pa; B,R'd'«; T, N YC; Aa» Cc, Oo, H'way.

- — ■ - _ ■ -

ANQLE TEST.

Angles of all thicknesses must open flat. Aneles not over one-half of an
inch thick must bend shut, cold, under hanuner dIows without sign of frac-
ture, B.

TEBT PIECES.

Minimum section, usually one-half sq inch. Length, min, 8 to 12 ins.
Tests are usually required for each melt or blow.

TESTS OF FULL SIZE ETE-BARS.

Ultimate lbs per Elastic limit lbs per

sq inch sq inch Elongation per cent

min min min

A.„ (5,000 less than) m u * t

, Aa . . . . I gjjj^ specimen / ^^ between necks

B — •••• «■«» »•«»•' {l^fef^SSS^I

C, Cc 56,000 10 between necks

D 58,000 30,000 12inl0ft

Oo * *. ! *. *. '. '. *. ' *65,bbb* .* * .' .' .' .* .* * .* .' * * .' * * .* .' * .* .' .* .* .' .' .* .* .* .* 12.5 in 15 ft
P 48,000 27,000 14f in 10 ft

("58,000 * ) ft K „u / 13 between necks

B '}56,000t> u.&mt 1 10 between necks

(.48000 t 27,000 15 between necks

Y 68,0p0 33,000 10 in 20 ft

In general not over 4 per cent of total number of bars in bridge will be
tested, R; at least 4 per cent, and not less than 3 bars, B.

75 p>er cent of fracture must be silky, the remainder fine granular, B.

Break in head shall not be cause for rejection —

(a) if bar develops 10 per cent elongation (12.5 per cent in 15 ft, Oo) and
the reaulred ultimate strength (ultimate 56,000, C, 55,000, Oo) and if not
more than one-third of all the bars tested break in the head, A» C, Oo*

(b) if bar stretches 14 per cent and if a second bar breaks in body and the
'average stretch of the two bars is not less than 16 per cent, P.

Company pays for bars which meet requirements, less scrap value, 6.

TESTS OF COMPLETED STRUCTURE.

Specified loads, or their eauivalent, passed over structure (in railroad
bridges at a speed of not over 60 miles per hour> and brought to a stop at any
point by means of air or other brakes) or maximum loaa rested upon struo*
ture for 12 hours. After test, structure must return to its original position
and must show no permanent change in any part, C*

COMPOSITION.

Phosphorus, max percentage.

In acid steel, 0.06 to 0.08; in basic steel, 0.04 to 0.06; in castings 0.08.

Sulphur, max percentage, 0.04 to 0.06.

BCAXIMUM PERMISSIBLE VARIATION FROM SPECIFIED CROSS-SECTION

OR WEIGHT.

2.5 per cent, G, except in extra wide plates, D. Oo, P.

In plates over 40 ins wide, in proportion to width, up to 5 per cent in plates
OO ins or wider, D.

1.6 per cent; where plates 36 ins and wider form 40 per cent of total, 2 per
cent in excess, Y.

Long plates, ^ inch out of line in 20 ft, \ inch in 40 ft, R.
Shapes or plates, 3 per cent short in thickness; plates 80 ins wide,^ per
cent, B.

* t Medium steel. * Bars not over 10 sq ins. 1 20 sq ins. Proportional
values for intermediate areas, t Soft steel. ^ In bars not longer than 20
ft bet'ween necks. || In bars longer than 20 ft between necks, f Max, 16.

48

754 TRUSSES.

A, Aa, Am B Co; B, B & O; C, Cc, Cooper; B, D L & W; E, Erie?

Steel Castings.

ManufaGture. Open hearth, A, Aa, D> P» T; acid, T; annealed, P,
R, T. Carbon, per cent, 0.25 to 0.40, 6.
Phosphorus, per cent, max, 0.08, B, T.

TENSILE TESTS.

C, D,E,P, B.

Size of

test piece,

ins.

% square

or
i round

{

i round
about 6
long

Ultimate

stren^h,

lbs per sq

inch, min.

(a)

(b)

65,000

to
70,000
55,000'

to
65,000
72,000'

to
80,000

Elastic

limit,

lbs per sq

inch, min.

33,000)
or y
0.5 ult )

0.5 ult

Elonga-
tion
per cent,
in 2 ins
min.

10 to 15

20

15

Reduo-

tion
of area
percent.

20^ P

25

BSNDIira TEST.

T (a), for general purposes, bed plates, pedestals, etc., to bend 90°, to a
radius = diameter of test piece.
T (b), for drawbridge rollers, etc.

Boiled Iron.

Requirements in Osbom's specification for highway bridges, Oo. Made
from puddled iron or rolled from fagots or piles of No. 1 wrought iron scrap,*
alone or with muck bar. Tensile strength, min, 48,000 lbs per sq inch (50,-
000, B) ; yield point, 25,000 lbs per sq mch (26,000, B) ; elongation, 20 per
cent in 8 ins; in sections weighing less than 0.654 lb per lineal ft, 15 per cent.
Specimens cut from bar as rolled must '^end through an angle of 180 under a
Succession of light blows , when nicked and bent, fracture shall be generally
fibrous and free from coarse crystalline spots: not over 10 per cent of the
fractured surface chall be granular ; specimens heated bright red shall bend
through an angle of 180° under a succession of licht blows not delivered
directly on the bend; must not show red-shortness. In flat and square
bars, one-thirty-second of an inch, in roimd iron 0.01 inch, variation either
way in size will be allowed, Oo.

Cast Iron.

Tough gray iron. A, D, E, B ; unless otherwise specified, A. B.

Transverse strength. Bar 1 inch square, 12 ins span, to near 2,500 lbs,
center load. Must deflect 0.15 inch before rupture, G. Bar 1 inch square,
4.5 ft span, to bear 500 lbs center load, £, B.

Phosphor Bronze.

1 inch cube, under compression, elastic limit, 20,0(X) lbs.
lbs, permanent set max onc'^ixteenth of an inch, B.

Under 100,000

Timber.

Sap wood not allowed in more than 10 per cent of the pieces of one kind,
and no piece will be accepted showing sap covering more than 0.25 X the
width of the piece on any face at any point, or more than half the thiokn4 —
of any plank at its edge, at any point, Oo.

TRUSS SPECIFICATIONS. 756

e, gen'l; Oo, Osb'n; P, Pa; R, R'd'g; Y, N YC; Aa, Cc, Oo, H'way.

lU. LOADS.

1. Vertical Loads.

(Dead and Live Loads and Impact.)

Dead Loads^in Steam Railroad Bridges.

Dead load => weight of metal + n lbs per lineal ft of track, C, D» £» R» Y«
n = 400, C^D, E; n = 500, R; n - 620, Y.

Timber taken at 4.5 lbs per ft, B.M., G. Ballast, 110 lbs per cu ft, C.
Rails, splices, and joints taken at 100 lbs per lineal ft of track. A, B, C.
Rails, splices, guard rails, etc., at 160 lbs per lineal foot of track, P.
Two-thirds of dead load assumed to be carried by loaded chord, Y ; in
spans less than 300 ft, B ; in longer spans calculate distribution, B.

Dead Loads in Highway and Electric Railroad Bridges.

Iron, 3.33 lbs per lineal ft of bar! sq inch area, Oo.

Steel, 3.40 " " " " " " 1 sq inch area, Oo.

Timber per ft board measure. "L Aa; creosoted, 5, Oo; oak, 4.5, Cc, Oo|
other hard woods, 4.6, Cc; yellow pine, 4, Oo; spruce and white pine, 3.5,
Cc; white pine and cedar, 3, Oo.

Concrete, etc., lbs per cu ft, 130, Aa; stone concrete, 125, Oo; cinder
concrete, 100, Oo. Stone, 150, Oo» granite, 160, Aa.

Brick, 150, Aa; 125, Oo; sand, 100, Oo. Asphalt, 130, Aa; 90, Oo.

Rails, fastenings, splices and guard timbers, 100 lbs per lin ft of track, Aa.

Live Loads for Steam Railroad Bridges.

THEODORE COOPER'S STANDARD LOADING.*

^n C )( X JC J o o o Q ^O tA.A.AJ — o 9 o o ^^

lbs on one

pair

of wheels

Train

for each track.

load,

lbs per

Driver

Tender

lin ft.

d

t

U

27,000

17,550

2,700

,30,000

19,500

3,000

35,000

22,750

3,500

40,000

26,000

4,000

50,000

32,500

5,000

Fig. 1.

TWO CONSOLIDATION LOCOMOTIVES, WITH THEIR TENDERS AND TRAINS.

Load in

Truck
(bogie)
Class b

E 27 13,500

E 30 15,000

E 35 17,500

E 40 20,000

E 50 25,000

A, adopts Cooper's loading. See C, above.

B, Cooper's Class E 50, unless otherwise specified.

I>, E» P, H, Loads and spaces differing slightly from Cooper's.
X, Cooper's Class E 40.

* In Mr. Cooper's system of standard loading, the No. (27 to 50) follow-
imS the letter £ in the class designation gives the load, d, on one pair of

40
irivers, in thousands of pounds. In each class, d »= 2 b == - ^ t — 10 TJ.

Since these ratios are constant for all classes, the stresses due to any class are
proportional to the number of the class. The cost and weight of metal, in
^rid^es of all kinds, built under C specifications, will be, in each class, about
LO per cent greater than in- the class next lighter.

756 , TRUSSES.

A, Aa, Am B Co; B, B & O; C, Cc, Cooper; D, D L &W; E, Er»;

ALTERNATIVE LOADINGS.

Use Fig. 1 or the alternative, whichever gives the greater stresses.
WW WW

Jj—d— ,1

■^_^. 9 Q Q 0

if — <^— J J'e.i* — 9 + 7 — ^ 7 ^

Figs. 2 and 3.

Load on one pair of wheels.

Id -6ft;W = W- 50,000 lbs, Above E 40, 60,000 lbs. C.
d = 7 ft; W = W = 65,000 lbs, D.
d « 7 ft; W = W = 60,000 lbs; L - 4,500, Y.
di = 9 ft; W = W = 66,000 lbs 1 «
da - ds = 7 ft; w = w - 30,000 lbs/ ""
Add 30 per cent in figuring floor beams, stringers, hangers, suspenders,
and other floor connections. Add 0 to 30 per cent for spaiis from 100 ft
down to 25 ft, D.

ON CURVES.

Distribution of live load between the two trusses.

W = P — - V- — ; where W == proportion of live load borne by the outer

truss; P = live load at panel considered; m — middle ordinate of entire
curve on span ; b =» dist betw cena of trusses. Make both trusses alike, B, T.

SPECIAL LOADINGS.

For rivets connecting upper flange angles with web in deck girders carrying
the floor directly on the top flanges, and in deck spans with wooden floor
beams, when distance between trusses exceeds 6 ft, 60,000 lbs on one pair of
drivers, distributed equally over three ties or floor beams, P.

For floors, the load on a single pair of engine wheels distributed over 4
ties, B ; over 3 ties, C. For trough floors, 60,000 lbs on one pair of wheels,
distributed over two troughs, Y.

THREE-TRUSS BRIDGES.

In double-track deck spans, all three trusses of equal stren^h, C.
In plate girder bridges of more than one track, center girder figured for
0.75 X the live load, E.

FUTURE INCREASE OP LIVE LOADING.

Only 70 per cent (50 per cent, R) of the dead load shall be oonsidered
effective in counteracting live load stress. A, B. Use 1.5 X live load, E.

*' That the heavier of these engines (see •€,• under 'Standard^ Loading,'
above) is close to the possible maximum, considering the limitations of the
permissible cross-section of existing railroads and the mechanical details of
design and i)roportions, is not improbable. That the economical tendency
toward heavier and heavier engines will in the near futiu-e reach the heavier
class E 50 upon the most important roads is to be expected. The cars will
also follow the same tendency for many kinds of traffic, as experience jiisti-
fies the advance. There are now in use self-dtlmping coal cars of a nominal
capacity of 100,000 i)ounds, which have, on four axles, a total load of 146,000
pounds (10 per cent increase over nominal capacity) on a wheel base, for two
adjacent cars, of 17 ft, 2 ins. These cars on all ordinary bridfcs produce
strains equivalent to those of E 33." — Theodore CJooper.

■

Members subject to reversal of stress must be so designed that a live load
n per cent greater than that specified shall not increase their unit sti
more than n per cent, n » 25, C; 50, B| 100, P«

' '.

G, geni

TBUBS SPECIFICATIONS.

; Oo,Osb'n; P, Pa; B, R'd'g; Y, NYC; Aa, Cc, Oo,

re Loads for Hiehway and Mectric Railroad Bridg

757

H'way.

Aa. For the Floor and its

(Am. Bridge Co.) Supports.

and Uni-

Cc, Concentrated, form

CTheo. Cooper.) (c).

Wagon Car

(S. (b) ^

on Per

Class.* each sq ft,

track. Ids
tons tons

A 24 100

B 12 or 24 100

C 12 or 18 100

D 6 . . 80

El 24

i!i2. •....•• •• lo • ■

For the Trusses.

Per lin ft of Per sq ft of
single track, remaining floor.

(Proportionally for inter-
mediate spans.)

Spans

Spans

Spans

Spans

up to

200 ft

up to

200 ft

100 ft

and

100 ft

and

over

over

lbs

lbs

lbs

lbs

1,800

1,200

100

80

1,800

1,200

80

60

1,200

1.000

80
Up to
76 ft

60

• ■

• •

80

55

1,800

1,200

• •

• •

1,200

1,000

• ■

• •

(a) On two axles, 10 ft cens (and, Aa, 5 ft «ige) ; in classes A, B, and C,
assiuned to occupy a width of 12 ft in single line (or, Cc, 22 ft in double
line) on any part of the roadway.

(b) On two axles, 10 ft centers.

(c) In classes A, B, and C, on remainder of floor, including footwalks. In
class D, on total floor surface.

Oo. Osbom Engineering Co. Highway. May specify any combination
of the following loadings, according to character of bridge and of load.

Uniform loads, lbs per sq ft. For spans up to 150 ft, 100 on roadway and
SO on sidewalks, or 80 on both. For spans over 150 ft, 80 or 60 on both.

A steam road roller; axles 11 ft apart, forward roll 4 ft face, two rear rolls
5 ft oens and each 20 ins face. 15,000 or 9,000 lbs on forward roll and 10,000
o 6,0(X) lbs on each rear roll;

A horse roller, 12,000 lbs on roll, 5 ft face;

A wagon load, 10,000 lbs on two axles, 8 ft apart, 5 ft gage;

Two electric cars on each track; Fig. a.

A train of electric cars on each track; Fig. b.

A train of coal cars of 60.000 lbs capacity; Fig. c.

tbs,\ SOjOOO

Fig, a

-J L-

SOfiOO

ft.

■ao-

■T-^

ibm.i_

SOJOOO or SOjMO \ ^' |

n a

SOfiOO or SOfiOO

jQ Q.

_n o

J

-19 H«-«-^»|* 1*-

>t< o- >kjg

»».!_

O O

02J0OO

Fig.c

_J L

92/)O0

o o

ft. J<j->|< —^O i^if->\fr-8-

n n

Q o

^5-^8

♦Class A, city bridges. Class D, ordinary country highway.

B, suburban or interurban. " El, heavy electric railway only.

C, heavy country highway. " E2, light electric railway onlv.

it

758 TBUSSES.

A, Aa, Am B Ck); B, B & O; C, Cc, Cooper; D, D L & W; E, Erie;

FUTURE INCREASB OF LIVE LOADING. HIGHWAY.

In electric railroad bridges, Class E, only 70 per cent of dead load stress to
be considered as effective in counteracting the live load stress, Aa. For
bridges carrying electric or motor cars, counters so proportioned that a,
future increase of 25 per cent in the specified live load shall not increase the
unit stress more than 25 per cent, Cc*

Impact.

I — S 1 , onn'' ^^®^ I "■ impact stress to be added to the live load stress;

S — calculated max live load stress; 1 — length in feet of loaded distance
which produces the maximum stress in the member, A.

I«= S (o.l + f^^X Mr. G. Bouscaren. C. E.

In Highway Bridges; I ^ 25 per cent of live load stresses Aaj I — L> -♦.
(L -f D), where L and D = live and dead load stresses. Oo«

2. Horizontal Forces.

(Drag, Centrifugal and Wind.)

(a) Ijongitudtnal.

,Drag. In bridges for steam and electric railroads, provide for a longitu-
dinal force, at the rails, = 0.2 of the max live load. In double track (Y),
provide for trains moving either way.

(b) Transverse.

(1) Centrifugal Force*

F = centrifugal force; W = weight of train on bridge; d ■- degree of
curvature = central angle subtended by a chord of 100 ft; v ■» velocity in
miles per hour; o = a coefficient.

A, F =» c d W. For d up to 5°, c «- 0.03. Deduct from c 0.001 for each
degree over 5°. Train on each track.

B, K, F = 0.02 of the live load for each deg of curvature. B, up to 6°. De-
duct 0.001 for each degree over 5°.

C, F, computed for v = 60 — 3 d on steam railroads, == 40 on electric rail-

roads; force acting 5 ft above base of rail.

D, v = 60.

E, F = force due to that uniform load which would produce the max speci-
fied live load bending moment on span; v •" dO.

Y, F = W v2 d H- 85,666. Up to d = 4°, v - 60. For d over 4*. v —
60 — 2d. Max train load on each track.

(») Wind.

(a) ON BAILROAD BRIDGES.

Wind pressure, in lbs per sq ft, = w; in lbs per lin ft — W.
w = either 30 lbs per sq ft on exposed surface of trusses and floor and on
that of a train of 10 ft average height, beginning 30 ins above rail base;
or 50 lbs per sq ft on exposed surface of trusses and floor; whichever
gives the greater stresses, A, P.
In truss spans over 200 ft & in plate girders, w — 30 lbs per sq ft of exposed
Burf of 1 girder and floor, + W on train for lower chord, as oelow, B.
W — L + U. L — pressure in lbs per lin ft on loaded chord, U on un-
loaded chord. W ircludes both wind on bridge and wind on train.
Wind on bridge. L - U = 150, B, C,* D, B; - 200, E.
L — 200, on double track 300, acting 8 ft above rail top;) ^
U — 150, on double track 225, acting at cen of chord, j
Wind on train. L - 300, B, D, K,t Y; =» 450, C,t - 400. E,

lu*

*In spans over 300 ft, add to U 10 lbs to each additional 30 ft. C«

fActmg 7.5 ft above rail, B.

iActing 6 ft above rail base. Includes lateral vibrationa of trains.

TRUSS SPECIFICATIONS. 769

«, gen'l; Oo, Osb'n; P, Pa; B, R'd'g; Y, N Y C; Aa, Cc, Oo, H'way.

A* Wind stress, Sw, in any truss member, C (main trues member, D )
chord or end post, B). need be considered only (I) wben Sw exceeds 30 per
cent, G (25 per cent, D, B), of max stress, S, due to dead and live loads.
Then increase section to bring Sw within limit, C» Pt K. (2) When Sw,
alone or in combination with temperature stress, can balance or reverse S, C*

Anchorage. In determining the requisite anchorage for the loaded struc-
ture, the train is assumed to weigh 800 lbs per lineal foot. A, B, C, P; 600
lbs per lineal loot, &•

(b) ON HIGHWAY AND SLBCTRIC RAIUtOAD BRIDGES.

Either 30 lbs per sci ft on the exposed surface of all trusses and floor, +
150 lbs (180, Oo) per lineal foot of a train covering the span; or 50 lbs per
sq ft on ihe exposed surface of all trusses and floor; wnicfaever gives the
greater stresses, A^ Oo* ^

On each chord. 150 lbs per Im ft, of span, due to bridge, and on the loaded
chord 150 lbs per lin ft of snan additioxtal due to train. For spans exceeding
300 ft, add 10 lbs on each cnord for eadi additional 30 ft, Cc*

Wind stresses (in truss members, Cc; in chords and end posts, Oo) to be
provided for only when the wind stress exceeds 25 per cent of the max dead
a.nd live load stresses (of the sum of all other stresses, Oo), or when the wind
stress (alone or in oombination with temperature stress, Cc) can (neutralize
or, Cc) reverse the stress in the meniibw, Cc» Oo.

IT* STB£SSES AND DIMENSIONS.

Effective Span and Depth.

^ In pin spans, span and depth are measured between centers of pins. In
riveted trusses the span is measured between centers of end bearings and
tbe depth between centers of gravity of chord sections. In plate girders the
span is measured between centers of end bearings, and the deptn between
centers of gravity of flange areas or over backs of flange angles, whichever is
the less. In floor beams the span is measured between centers of trusses,
and in stringers between centers of floor beams, G.

Limiting: Unit Stresses.

Tension.

Net section. The net section of any tension member or flange is deter-
mined bv a plane cutting the member square across at any point. The great-
est number of rivet holes which can be cut by the plane, or come withm an
inch of the plane, is deducted from the gross section, B. The rupture of a
riveted tension member is considered equally probable, either through a
-transverse line of rivet holes, or through a diagonal line ci rivet holes where
trbe net section does not exceed by 30 per cent the net section along the
^xansverse line, C, Cc.

In deducting rivet holes for net section, their diameter is taken at one-
eighth of an inch greater than that of the cold rivet, G; for countersimk
irivets (Oo), one-fourth of an inch greater.

Maximum permissible tensile stresses, in lbs per sq inch.

Medium Soft
Steel Steel
^^9 Aa, Under vertical forces only or horisontal forces only 17,000 15,000
Under vertical and horizontal forces combined 21,000 19,000
^, For "Bridge" (soft) and Rivet Steel, same as Medium Steel under A.

!>• For soft steel : For dead load ; live load.

Eye-bars 14,000 9,000

Built sections 12,500 8,500

CJonnters 8,500

For dead and live load.
Hip suspenders, floor beam hangers, members sub-
ject to sudden loading 7,500

Tension flanges of plate girders and rolled beams 9,000

Bracing 12.000

760 TBUSSES.

A, Aa, Am B Co; B, B & O; C, Cc, Cooper; D, D L AW; E, Erie}

Main members of trusses, flanges and webs of girders and floor beams
for double track, floors and girder flanges with ballast floor, add 10
per cent

For medium steel, add 10 per cent.

^ «^««/- ™iii stress \
E. 8.000(l + — ^^-^S;^)-

Oo. (Highway Bridges.) Medium steel, 22,000 ; soft steel, 20,000; wrouefal

iron, 18,000.
P. M = max calculated stress in member
m = mm

Let r = S ; let k = t—t-* Then M (1 + k) shall not exceed 15,00a
M 1 -r r

Long hip verticals must have 25 per cent excess strength; short floor

beam hangers 50 per cent excess, P.

T. Soft steel. Chords and web members of trusses, and flanges of

plate girders, floor beams and stringers.

Dead load and drag 16.000

Live load and centrifugal force 8,000

MAXIMUM STRESSES IN TIMBEH, LBS PER SQ INCH.

Trans- Bear-
verse End ing Shear
For Highway Bridges, Oo. load- bear- Short across along

ing ing column* fibre fibre

White oak 1,400 1,300 1,000 550 300

Long leaf pine 1,600 1,300 1,000 350 200

White pine 1,100 900 700 200 150

Hemlock 950 850 650 200 100

Extreme fibre stress, in floor beams, max, yellow pine and white oak,
1,200 lbs per sq inch; white pine and spruce, 1,000, Aa, Cc.

Compression.

p = permissible working stress in compression member, in lbs per sq inch.

f = generally the permissible stress in tension member, in lbs per sq inch.

a = a coefficient.

1 = length of piece, in ins, between cens of connection.

r » least radius of gyration of cross-section of member, ins.

p — V

1 +

r2 a

f a

*„ (In medium steel 17,000 11,000

'^' \ In soft steel 15.000 13,500

B. In soft steel 17.000 11.000

C. See below.

Dead load
f a

,000 18.000

D ■{ to to

,600 24,000

ri2,(
ll2,i

Live load
f a

8,000 18.000

to to

8,500 24,000

E. f = 8.000 (l + ?51?_i!!^\ . a = 36,000 with both ends fixed; a

\ max stress/ •

24,000 with one end fixed; a = 18,000 with both ends hinged.
Oo. (Highway Bridges.) f » 22,000 for medium steel, 20,000 for soft steeU

18,000 for wrought iron ; a as in £, above.
P. f = 15,000; a = 13,600.

B. f - 6.600 (l + — ®*''®^^ t) ; f max - 8.000; a - 40,000 with fla*

\ max stress / ^^

ends; a «= 20,000 with pin ends.
When one end is pinned, p » mean of values derived as above.
For angle iron struts, see below.

^Length not over 12 X least side. tMin stress = dead -^ live load streea

TRUSS SPECIFICATIONS. 761

G.gen'l- Oo, Osb'n; P, Pa; R, R'd'g; Y, N YC; Aa, Cc, Oo, H'way.

Y« Soft steel in chords and web members:

i, fa

For dead load and drag 16,000 18,000

For live load and centrifugal force 8,000 18,000

C, Cc. p = M — c — . •

r

For medium steel in stationary structures:

Dead load Live load

M c M

Chord segments, stiffeners. 20,000 90 10,000

For highway bridges 24,000 110 12,000

End and other posts I to^^'^^

8,500
to 9,000

18.000 \ to 80 \
t? w u X. '^ i 20,000 J 90 J 10,000

For highway bridges {1022,000 jto 80 ito llloOO

Lateral struts, rigid bracing 1

for railroad and highway > 13,000 60 8,666 40

bridges I

For soft steel, deduct 15 per cent; for movable structures, deduct 25 per
cent.

R« Angle iron struts.

With flat ends, p - 9,000 — 30 - ; with pin ends, p — 9,000 — 34 -=-. In

r r

lateral and cross struts, add 30 per cent.

Length of compression members, max, — 40 to 45 diameters, or 100 to
120. In highway bridges. 120 to 140 r, Aa ; 100 to 120 r, Cc s 125 to 150 r, Oo»
where r ■« least radius of gyration.

Unsupported width (distance between rivets) of plates subject to com-
pression, max -> 45 X thickness, Oo; 30 X thickness, C» Cc, D; in cover
plates of top chords and end ixysts, 40 X thickness, C. Cc, D ; or, if a greater
width is used, effective section shall be taken as 40 X thickness, C, Cc»
Distance between supports in line of stress, max = 16 X thickness, Oo.

Timber columns, whose length exceeds 12 X their least sides, in highway

bridges, Oo.

C
Max unit stress — 15 —

^ + Toeod*

where C » 1,000 lbs per sq inch for white oak and long leaf pine, 700^ for
white pine, 650 for hendocK; 1 » length of column, between supports, ins;
d *-" least side, ins, Oo.

Alternating Stresses.

Total sectional area of member to be made -' su'm of areas required for
both stresses. A, B.

Area sufficient to resist either stress plus 0.8 (0.6, B; 1.0, Y) X the lesser
■tress, C, Cc, D, B, Y. ^

Permissible working stress, in lbs per sq inch :

=» o r)fvn / 1 I max stress of lesser kind \ _
'^ ' \ 2 X max stress of greater kind/*
M » max calculated stress of greater kind, ^

m -» max calculated stress of lesser kind, I

Let r =« S. Let k - ^— -. Then M (1 + k) shall not exceed i ^'
M. 2 — r 1

15,(X)0 lbs per sq in, J

IN BRIDGES FOR HIGHWAYS AND FOB EL&CTRIC RAILROADS.

In Classes A, B, C, and D, members proportioned for that stress which re-
quires the lar^r section. In Classes E 1 and £ 2, make sectional area <» sum
of areas required for the two stresses, Aa. Members designed to resist either
stress and given 25 per cent excess of strength in their joints and connec*
tions. Oo.

762 TRUSSES.

A, Aa, Am B Co; B, B & O; C, Cc, Cooper; D, D L & W; £» Erie;

Shear and Bearing Stresses.

Shear in web plates, max, lbs per sq inch. 10,000, B ; 4,000, E ; 5,000,
R; 13,000, Pt in medium steel, 10,000. A, Aa; in soft steel, 9,000, A. Aa;
across grain, o,000, D; with grain, 5.000 (net section), D; dead load, 10,-
000, Y; live load. 5,000 (gross section), Y.

Shear and Bearing on Biirets, Bolts and Pins. Maximum, in lbs
per sq inch.

Shear Bearing

Medium Soft Medium Soft

A, Aa, B 12,000 11,000 24,000 22.000

C 9,000 9.000 16,000 16,000

Cc 10,000 10.000 18,000 18.000

Oo 10,000 10,000 22,000 20,000

P, B 7,500 7.500 12,000 12.000

Y, Shear — 0.75 S ; bearing — 1.60 S. S •" permissible unit stress in tension.

In field riveting, increase number of rivets 25 per cent, A. A a. B, Oo« P;
if machine driven, 10 per cent. A, Aa, P; in stringers and noor beams, one-
third, P. Take 0.66 to 0.80 X stress as above, C, Cc, D, B, Y.

In floor connections, use 0.8 X stresses as above, C, Cc; add 20 per cent
to number of rivets, Y.

In wind and sway bracing, use 1 .25 to 1.5 X stresses as above, C, Cc^ D, R.

Rivets with countersunk heads taken at 0.75 X value of rivets with full
heads, P.

Bearing, on phosphor bronse disks, 5,000 lbs per sq inch, B»

Bending Stresses*

Stress in extreme fibres, under bending moments^ max, lbs per sq
inch.

In pins and bolts, 25,000, B: 18,000, C; 20,000, Cc| 15,000, D.R; 16,000.
Y; in pins, closely packed, 25.000, Oo; in medium steel, 25,000; in soft
steel, 22,000, A, Aa, P. Centers of bearings of strained members taken as
points of application of the stresses. A, Aa, R. Applied forces considered
a.s uniformly distributed over the middle hau of the bearing of each member,
C, Cc. Bending calculated \r< m distances between centers of bearing, Oo.

In rolled beams and channels, 14,000, P.

In wooden floor beams, 1,000, A, B, C, P.

Compound Stresses.

Compound (axial and bending), maximiun, lbs i>waq inch.

In end posts of through spans, dead + live + wind + bending, max «

15,000, R.

Proportion the member to resist sura of direct stress, plus 0.75 bending

8 000
stress, A, Aa, B, P, R. Max — '- — ^ , where I — length, ins; r =>

^^ 40,000 r«
least radius of gyration, ins.

If pins are out of neutral axis of section, max must include the additional
stress due to the eccentricity, R.

Bending moment at panel points assumed equal and opposite to tliat at
the center, A, Aa. If fibre stress due to weight of member alone exceeds 10
per cent of the allowed unit stress on such member, the excess must be con-
sidered in proportioning the areas, C, Cc, R.

Minimum Dimensions.

Minimum thickness of plates, in railroad bridges, three-eighths of an
inch for main members, five-sixteenthp of an Inch lor laterals; m lii|^wmy
and electric railroad bridges, five-sixteenths to one-fourth of an inch.

diam of rod. three-fourths of an inch, Oo« Rods and bars, min section, 1 sq
inch, D, R; counters 1.5 sq ins, D, P. Posts, in pin spans, min width 10
ins, A. In posts of through spans, channels min 10 ins, B. Angle, min, 3.1
X 3 X five-sixteenths, B.

TRUSS SPECIFICATIONS. 763

G, gen'l; Oo, Osb'n; P, Pa; B, R'd'g; T, N Y C; Aa, Cc, Oo, H'way.

V. PBOTECTION.

At Shop. After removing loose scale and rust ; 1 coat pure boiled linseed
oil A, Aa, B, D, E» P, R; raw linseed oil, C, Cc; with 10 per cent in
weight of lampolack, D ; standard red lead paint,* T.

Inaeeessible Parts. 2 coats iron ore paint in pure linseed oil, A, Aa, B.
C, Cc, G, R: standard red lead paint,* Y; 1 coat, D ; 1 heavy coat red lead
in raw linseed oil, P; 2 ooata, 18 lbs red lead in 1 gal boiled linseed oil, Oo*

Finished Surfaces. Coated with white lead and tallow. General.

Surfaces in Contact. Painted before joining, A, Aa, B, C, Cc, R, Yj

with 2 heavy coats red lead in raw linseed oil on each surface, Y.

After Erection. 2 additional coats of paint in pure linseed oil. A, Aa,
B, C, Cc; 2 coats of paint, of different colors, R; 2 heavy coats asphaltum
varnish, Y.

At least 48 hours allowed for drying of each coat, Y*

Columns, etc., for 6 ft above surface of street, etc., 2 heavy coats as-
phaltum varnish: under sides of bridges, rest of columns, etc., 2 heavy
coats standard white paint;* ballast side of trough floors, 1 part by weight
refined Trinidad asphalt and 3 parts straight run coal tar pitch at 300" F, Y.

Wherever there i() a tendency for water to collect, the spaces must be filled
with a waterproof material, C, Cc.

First coat paint uf graphite or carbon primer. Oo.

In highway bridges, upper surfaces of metal floor plates thoroughly coated
with asphalt, Oo.

VI. ERECTION.

The Contractor is usually required

(1) to unload materials after delivery, to furnish falseworks and appli>
ances, to remove the old bridge, to alter existing bridge seats;

(2) to drill and set anchor oolts, to erect and adjust the suijerstructure,
and scr.ietimes to furnish and place the wooden floor beams ;

(3) to remove falseworks and appliances ;

(4) to keep the road open for traffic and to avoid interference with any
other thoroughfare by land or water and interference with other contractors ;
to furnish and p>ay watchmen; to keep material clean and in good order; and
to assume all risks of damage to persons or property by reason of storms,
floods or other casualties ;

(5) to furnish pilot nuts for the protection of the ends of pins in driving.

<2) DIGEST OF SPECIFICATION- FOR COMBINATION RAII>

ROAD BRIDGES.t

By Baltimore and Ohio RrJlroad Co., 1901.

I. GENERAL DESIGN.

Type, Howe.

Rods of steel, with upset ends; standard nut and lock nut on each end.
Cast Iron joint boxes and packing spools.

Steel gib plates from 1.25 ins thick for 1.25 inch rod, to 1.75 ins thick for
2.5 inch rod.

Splices in lower chord generally of steel construction.

n. MATERIAL.

Lumber, Georgia yellow pine, white oak or white pine.

Rolled steel. Open hearth. Ultimate strength 60,000 lbs per sq Inch,

* Standard red lead paint. 5 gals contain 100 lbs pure red lead, 4 gals pure
raw linseed oil, one-half pint Japan, free from benzine, Y.

Standard white paint. 5 gals contain 42 lbs pure white lead in oil, 21 lbs
-white zinc in oil, 3 gals pure raw linseed oil, Y.

At least 48 hours between coats, and between final shop coat and load-
ing Y.

T To be used only for temporary purposes.

764 TRUSSES.

permissible variation, 5,000 lbs; elastic limit, 80,000 lbs; elongation 25 pef
o|nt in 8 ins; to bend ISO*' flat upon itself.

III. IX)ADS.

Dead Load.
Timber taken at 4.5 lbs per foot board measure. Track 100 lbs per lin ft.

Live Load.
Ilax intended load + 25 per cent, to provide for increase and impact.

IT. STRESSES AND DIMENSIONS.

lyimitins TTnit Stresses.

Timber, lbs per sq inch, max Yellow pine White pine White oak

Bending or direct tension 1,200 800 1,000

Ck>lumns under 17 diams in length 900 600 750

Columns over 17 diams in length . . 1,200-18 n 800-12 n 1,000-15 n

where n = length •+- least thickness;
n max — 40.

Shearing, along ptan 150 100 200

Bearing, in direction of grain 1,500 1,000 1,250

Bearing, perpendicular to grain . . . 350 200 500

In columns made up of several sticks placed side by side, and bolted
together at intervals, each stick treated as an independent column.

Steel rods, max unit stress "> 12,000 lbs per sq inch.

Floor beams designed to carry the dead load and the heaviest engines in
service without impact allowance. Reinforce for future increase of Toads.

For loadings in excess of that used in designing, reduce speed from 60 to
15 miles per hour, as loads increase to limit of 25 per cent increase of load.

V. PROTECTION.

Steel rods, i^bs, etc., 1 coat of paint in shop ; 2 after erection.
Wood, at joints and at points of contact, to be painted.
Bolt and rod holes to be saturated with paint.

(3) DIGEST OF SPECIFICATION FOB ROOF TRUSSES*
STEEIi FRAMEWORK AND BUILDINGS.

By Baltimore and Ohio Railroad Co., 1901.

I. GENERAL DESIGN.

Made principally of shapes. No adjustable members, except in lateral
bracing. Lateral bracing proportioned for a full wind pressure of 30 lbs per
sq ft of exposed surface, acting in an;^ direction. Tension members in brac-
ing must in all cases pull directly against a stiff strut. If building is enclosed
and the work is exposed to the action of gases, no open spaces less than 1
inch wide left between members, or open pockets inaccessible for painting:.

n. MATERIAL.

Min thickness, 0.25 inch. When subject to the action of gases, five-six-
teenths inch if building is open; 0.375 inch if enclosed.

m. LOADS.

Snow, 20 lbs per sq ft of horizontal projection of roof surface. Wind, 30
lbs per SCI ft, horizontal, in any direction. Min total, 40 lbs per sq ft.

Covering. For roofs, and for sides unless otherwise ordered, corrugated
sheets. No. 22 gage, 26 ins wide; corrugations, 2.5 ins; 3 ins for slope of 1
on 2; 6 ins for less slope. Purlins not more than 4 ft apMirt between centers.

IV. STRESSES.

Columns sustaining roof are considered as hinged at base, unless so an-
chored as to be absolutely fixed.

Unit stresses, if subject to no moving load other than wind, see B, in
Digest of Specifications for Steel Bridges, and Digest (2) of B ft O R R Speci-
fication for Combination Bridges. Stresses given in the latter to be in-
creased 25 per cent.

V. PROTECTION.

Three coats of paint. If exposed to gases, use bridge i)aint (see B in
Steel Bridge Specifications) ; if not, use standard building paints.

SUSPENSION BBIDGES.

765

SUSPENSION BEIDGES.

Art. 1. Table of data required for ealealatlnfr tbe mai]
•halns or cables ofsnspeiiMon bridir^s. Original.

Defleotlon

in parts

of the

Clierd.

Defleotlon
in Deci-
mals of

tbdChetd.

Length of

Main Chains

between Soa-

ponsion Piers,

la parts of the

Chord.

Tension on all
the Main
Chains at

either Sospen-
sion Pier, in
parts of the
entire Sus-
pended Wt.

of the Bridge,

and its Load.

Tension at the
Center of all
the Main
Gbaina ; in
parts of the
entire Sus-
pended Wt.

of the Bridge,

Angle of
Direc-
tion of

tbeChains
at tbe
Piers.

Natural Natural
Slneofthe Cosine of
Angle of the Angle
Direction of Diree-
of the tionofthe
Chains, at Chains at
the Piers, the Piers.

Des. Min.

1-40

.026

1.002

6.06

5.00

.0906

.9960

l-»

.0286

1.002

4.40

4.ST

6 81

.1186

.9936

1-90

.0383

1.008

S.78

3.76

7 86

.1321

.9913

I-S

.04

1.004

8.16

3.12

9 6

.1660

.9874

1-10

.05

1.006

S.55

2.61

U 19

.1961

.9806

1-19

.0526

1.007

2.48

1.88

11 63

.3060

.9786

1-18

.0565

1.008

1.30

2.25

12 82

.2169

.9762

1-lT

.0588

1.009

S.18

3.11

13 14

.2290

.9734

1-16

.0625

1.010

S.06

1.00

14 2

.3426

.9701

1-15

.0667

1.012

^94

1.87

14 55

.2678

.9663

1-M

.0714

1.013

..82

1.74

15 67

.2747

.9616

MS

.0769

1.016

1.70

1.62

17 6

.2941

.9658

1-lS

.0833

1.018

1.57

1.49

18 83

.3180

.9480

1-11

.0919

1.022

1.46

1.37

19 69

.3418

.9396

1-10

.1

1.026

1.36

1.25

31 48

.3714

.9286

1-9

.1111

1.033

L2S

1.12

IS 58

.4062

.9138

.?f

.125

1.041

1.11

1.00

26 S3

.4471

.8945

.1429

1.063

1.01

.881

89 45

.4961

.8736

S-90

.16

1.068

.972

.838

80 68

.6145

.8574

i^

.1667

1.070

.901

.760

83 41

.5547

.8320

.3

1.008

.800

.625

88 40

.6247

.7808

•225

1.1X2

.747

.655

42 0

.6690

.7433

!f

.86

1.149

.707

UMO

46 00

.7071

.7071

.3

1.205

.651

.417

60 13

.7682

.6401

H

.8833

1.247

.625

.876

68 8

.8000

.6009

A

.4

1.332

.689

.312

68 3

.8483

.6294

9-90

.45

1.406

An

.178

60 67

.8742

.4855

yi

.5

1.480

.669

.260

63 26

.8944

.4472

These ealoulatioQs are based on tbe assumption that tbe curve formed by tbe main chains is a
parabola ; wblob is not strictly correct. In a finished bridge, tbe carve in between a parabola and a

catenary ; and is not attsoeptible of a rigoroos determination. It lOay Save SOiUe tiH>a«
ble in mai&ingf the drawins^ of a suspension bridge, to remember that when the
deflection does not exceed about -j^ of the span, a segment of a circle may be used instead of th«
true earre ; inasmuch as the two then coincide very eloaely ; and the oMtre so as the deflection be.
I lase than <]^. The dlaaiMioBs taken fhmi tha dr»wtnt of a Mgnoii viU wuver all the pui^
t of estimating the qnantitiee of materials.

Tbe delleetion usaally tuiopted by enfrinMini for greAt apuu b

Abont ^ to ^ tlie span. As much ai ^ is generally eenflned to small apaas. The bridge wlU

be stronger, or will require less area of oable, if the defleotlon Is greater ; bnt it tben undulates more
readily ; and as undnlations tend to destroy tbe bridge by loosening tbe Joints, and bv increasing tbe
momentum, they must be specially guarded against as much as possible. The usual mode of doing
thte la hj trussing the hand-railing; which with this view may be made higher, and of stouter tim-
bera than would otherwise be necessary. In large spans, indeed, it may be supplanted by regular
Inldge- trusses, sutBoiently high to be braced together overhead, as in tbe Niagara Railroad bridge,
where the trusses are 18 ft high ; supporting a single-track railroad on top ; and a common roadway
of 19 ft clear width, below.*

• The writer believes himself to have been tbe first person to suggest the addition of very deep
trusses braced together transversely, for large sospensloa bridges. Earlv In 1851, be designed such
• bridge^ with four spans of 1000 ft each ; and two of SOO; with wire cables ; and trunses 20 ft high.
It was intended tbr crossing tbe Delaware at Market Street. Fbilada. It was publicly exhibited for
•everal months at tbe Franklin Institute, and at tbe Merchants' Exchange; and was finally stolen
firom tbe ball of the latter. Mr Roebling's Niagara bridge, of 800 ft span, with trusses 18 ft high, waa
not aommeneed until the latter part of 1862 ; or about 18 months after mine bad been publicly ex
MMtad.

766

StTBFENSIOir BRIDGES.

Another very Important aid Is found in deep longitudinal floor timbera. firmly united nhere their
ends meet each other. These asaiiit by distribuUng among several suspeuder-rods, and by that
means along a considerable length of main cable, the weight of heavy pasaiug loads ; and thus pre-
vent the undae undulation that would take place if the load were oonoentrated upon only two opposite
BUBpoMlers. With this Tiev, the wooden stringers under the rails on the Niasmra bridge ue made
virtually 4 ft deep. The same principle is evidently good for ordinary trussed bridges.

Another mode of relierlng the main cables is by means of cabte'ttaya ; which are bars of Iron, or
wire ropes, extending like e y, Fig 1, from the saddles at the points of suspension e, d, obliquely down
to the floor, or to some part of the truss. In the Niagara bridge are 64 such stays, of wire ropes of
1% inch diam ; the longest of which reach more than quarter way across the span from each tower.
TlMj transfer much of the strain of the wt of the bridge and its load direotij to the saddles m the tiv
of the towers : thereby relieving every part of the main eable, and dimiafehins ondvlatton. They
end at e and d, where they are attaohed, not to the cables, but to the saddles. They of eonrse do not
relieve the hack slays.

THe greatest daiiffer arises Urom tbe aetloii of sf r«ms wtiitfs
Strltcluir below the floor, and either lifting the whole platform, and letting
it fall suddenlv ; or imparting to it Tiolent wavelike undulations. The bridge of 1010 ft span aeroaa
the Ohio at wheeling, by Charles Ellet, Jr, was destroyed in this manner. It is said to hare andn-
lated 20 ft vertically before giving way. It had no effective guards against undulation ; for although
its hand-railing was trussed, it was too low and slight to be of much serrioe in so great ai «»«.
Many other bridges have been either destroyed or ii^urwl in the same way. When the height or the
roadway above the water admits of it, the precaution mav be adopted of tie-rods, or anchor rods,
under the floor at different points along the span, and earned from thence, inclining downward, to
the abutments, to which they should be very strongly oonflned. In the Niagara Railroad brldgo 5C
suoh ties, made of wire ropes 1^ inch dlam, extend diagonally from the bottom of the bridge, to tho
rocks below. They, however, detract greatly fh>m the dignity of a structure.

Ifr Brunei, In Mo'rae cases, for checking undulations tnm violent winds striking beneath the plat
Ibrm, used also inverted or up-cttrving eables noder the floor. Their ends were strongly confined te
the abuts several ft below the platform; and the cablea were oonneeted at intervals, with the jpiaA-
torn, so as to hold it down.

Art. 2* The angle adffy or act', Fig 1, which a tang dg or ei to the cnnre at
either point of suspension c or d, forms with the hor line ed or ohord, Is called the ail§^l9 Of
direetion of the Hiain chains* or eables, at those poluts. FrequenUy the eo^e
eA, and dr, of the chains, called the haehstayS, are carried away f^om the suspension p|«a
in straight lines ; in which case the angles Idr, eeh, formed between the hor line e I and the ohatf «
Uaelf, become the angles of direction of the backstays.

P i

Twioe the defleotlon a 6

Sine of anuria of direction aAg^ ,,

|/ (twice the det5ection;s -f- (Half the ehord)i

RovB 1. The direction of the tang dg or ci, can be laid down on a drawing, thus : Continue the
Hue a h, making it twioe as long as a b ; then lines drawn from d and e to its lower end, will be teaga
to the parabolio curve at the points of suspension.

NOTK 2. If the Chord e «!• be nothor, as M>nietimes is the case, the ancle
must be measured from a hor line drawn for the purpoMe at each point of snspenaian} aa the two
angles will in that ease be unequal, the piers being of unequal heights.

TenMton on all the main Hair the eatfre suspended weight of the olep

chains or cahles^toi^ther, __ span and its load

at either one of the piers* Sine of angle of direction a d g

c or €if Fly 1«

^v

(H Span)« + (2 Defl)«

8 Deflection

Half the entire suspended
weifrht of the dear span X
and its load

Half the entire ses-
ponded weight of
the clear span aad
its load.

Cosine of angle of
direction adg

or

ffeiiM&on on all the main
cha ins or cables* toicether*
at the middle* b, of the
N|»an, Tiff 1.

or _^__

Twice the deflection
The diff between the tensions at the middle, and at the points of suspension, is so trifling with the
proportion of chord and deflection commonly adopted in praettoe, vis, from about -^ to ^, that U
is usually neglected : iiiRRmueh as the saving in the weight of metal would be fUlly compensated tbr
by the increased labor of manufacture In gradually reducing the dimensions of the ebalns (kt>m th%
points of suspension toward the middle; and in preparing flttlngs for parts of many diflbrent aliea.
The reduction has, however, been made in some large bridges with wrought-lron main ehainsj bal
<m none with wire cables. «

Sine of angle of direction adg

Half the entire suspended weight of ^ Half tkt
the clear spHu and its load ^ >"^»

gtrSFENSTON BBIDGES.

767

Art. 8 A* As It i8 sometimes convenient to form a roagh idea at the moment, of
llie site of cables reonired for a bridfe, we saggMt the Ibllowing rale for finding approzimacelj the
area in sq ins of tolia iron in the wire reqaired to sostaln, with a safetj of 3,* the weight of the bridfe
itself, together with an extraneous load of 1.205 tons per foot ran of span ; whioh oorresponds to 100
ft« per sq ft of platform of 27 ft dear awtUabte width. This solBeee for a double oarriage-way. and

vwo footwajs. The deflection is assumed at -^ of the span ; and the wire to have an nltimate
■tieagtih of 86 tons per eolid square Incl),

For spans of 100 tt or more,

RuLB. M nit the span in Ibet. by the sqnare lool of tbe span. Divide tlie prod by 100. To tlie
qnot add the sq rt of the span. Or, as a formula,

Area of solid nutal of aU tpan X tqrtof ap0n

theea6leM;in»quare{n*; = •}• tqrtoftpan.

for spotM ovor 100 feet 100

For ipaiui less than 100 feet, proportion the area to that at 100 ft.

If a deft of -ji^ij- is adopted instead of -^t the area of the cables may be rednoed very nearly ^ pari^

The followlnir table Is drawn np from this rnle. The 3d col

glTce the united areas of all the actual wire cables, when made up, including Toida. (Original.)

Feet.

Solid Iron

in all the

Cables.

Areas of
all the

Finished
Cables.

Feet.

Solid Iron

in all the

Cables.

Areas of

all tbe

Finished

Cables.

8j.an
Feet.

Solid Iron

in all the

Cables.

Areas of
all the

Finished
Cables.

Sq. Ins.

Sq. Ins.

Sq. Ins.

Sq. Ins.

Sq. Ins.

Sq. Ins.

1000

848

446

400

100

128

150

80.6

S9.S

900

800

S86

860

84

106

126

26.2

38.8

800

854

826

800

60

89

100

30

25.6

700

212

272

950

• 55

71

76

16

19.2

000

171

219

800

42

64

60

10

12.8

600

1S4

--72

176

86.4

4A.1

25

5

6.4

Having the areas of all the aotnal cables, we oen readily find their dlam. Tlraa, snppose with a

172
apan of 500 ft, we intend to use four cables. Then the area of each of them will be —- = 43 sq ins«

and from tbe table of circles. we see that the corresponding diam is ftill 7fi inn.

The above areas are supposed to allow for the increased wt of a depth of truss, and other additions
necessary to secure the bridge f^m violent winds, and from undue vibrations from passing loads.

When these considerations are neglected, and a less maximum load assumed, the following descrip*
tlons of the Wheeling and Freyburg bridges show what rednotiona are preotloable. Weight, ivS>
elently provideA for, is of great aerviee in reducing nndalatitm.

We do not think that diagonal horizontal bracing should, as is nsnal, be omitted under the floor.
It may readily be eflteted }ay Iron rods.

AU the eabiea need nut be at the aides of the bridge. One or more of them may be over its axis^
especially in a wide bridge. One wide footpath in the center may be used, instead of two narrow
ones at the sides.

The platform or roadway should be slightly cambered, or curved upward, to the extent say of about

*-J^ of the span.

* The writer miiat uot be understood to advocate a safety of 8 uffitiaac lUO lbs per sq ft, in addition
to the weight of tbe bridge, in all cases. He believes that limit to be about a suflBcient one for a pro-
perlj designed wire suspension bridge for ordiuftry travel ; but for an important railroad bridge, he
would (according to position, exposure, Sto) adopt a safety of at least from 4 to 6 against the greatest
poasible load, added to the wt of the bridfre. A train of ears opposes a great surface to the aotlon of
elde winds : and trains must run during violent storms, as well as daring oalms ; but a large epea
Vldce for common travel is not likely to l>e densely crowded with people during a severe storm.

768

_. J (he lMtoh4toys- eft Anil <lr. Flu 1. ud

■trklna an th« ulera, nr tuwsn, or pUIbts. If tbe ""*'* ^|.™^';;^^^|J^2^

1, S nnd 4, tbn plan

aUSP£K6[OIT BRtUOEK

on the pier i rnnn •

Id Wtt^- Uk oUlqvItT flf tin pmnnmuUtwi*! cqiUdvlhebuflor Uih plat tutmmrj ai iba**
br IbpurAw; bub !■ Fig 4, AlHVd. Tfab UBdnieMapn>ag«d bj tbr k oriMOfOal itxpnatM or t^

barbrt^vlirttn Ibli koriiaalB] rbnii u7tf « wll'l flTt lb* nrilosl comuoDHui or iElh prauiin Jv.

Rorlaontnl pnll litwnnl hrthe mitln cbnln ^Tniloii x C'^mtaimtt

" " •lltivnni ny (lie bnck-hlAy ^Teailan X CnilHOf I'h

-VerllenI preaanre by mnlN ohniM ^Ticiimi x sik^di >■'(.

" ** '* bnck-Hlny = tuAb x atur >> i d ■:

SUSPENSION BEIDGEH.

Art. If. lrth.ail,l«pMifrMlyoT6rBloo»p(n,d,Fig

,(,™npporwdb

■ llnkL

lUalTii rriiDi lb. P<«1 pin .. .od ..i.bl.or m«.lD( rml7 ntnu. bolb

Ha pli» ; (b. t«<>l« Id UU bxil^ ollt. h EitCn, b> e^i»l t.

PI

Art. «. But tr ttia end! or tli« ublx Hud bacli-iuy.

rtl. 1 B. 1 C .bd 1 D. » Uie up «f lbs plu, M Had! /ut U ■ l.uct

.c .^.D (BlsD !• •ocporLsl bi »II>no<> kiDucb pliitbrm DQ lop
n1I.c T»li| l»a a Lib iniol ^ U»D lOb .irUb ok Ut bu.k..lv '>ll

rf rf'

Sis!;=."r"-ss:i.'s» jj::

;:^-^-

— T-

"H V • -^^

-F-^ '1

^^

r^

L^..*B

rf (f'

■Ion on the Ineb-atMy, iiDd of ti» prpH.
■Dra on the pferi on rfu fn eiihcr Fig 4 ffl

_ol

^ Ol

%=^4^

770

SUSPENSION BRIDGES.

d 9 and 4' r are equal, an ai^ also their boriiontal campomnU pd and d'o • and Che preaenrw ea tbi
pier are vertical ; and if channel of temperature or of loading produce sHght ehangea in (be angtef
ddg andt (i'm the truek will (by reason of the inequality thus brought about between the hori-
zoutal eompeneats) mere fkr enough to restore the equality between the angles, and between the
horitontal oomponenta, and consequently the pressure upon the pier will at all times be vertteal.

Art. 9, To find, approximately, the length of a main cliaiii

ehdi Ptg. 1 ; having the span o d, and the middle deS a b. See preoeding table. Art 1.

Half length of main chain = f/i^ (detlS) + ()j chord)*.

In Menai bridge (he chord ed is 579.874 ft : and the defl is 43 ft.

According to the above formula, the entire length is 588.3 feet. By aotoal measoremeBt Um chain
U precisely 590 feet. The approximate rule below gives 589.764 ft.

NoTK. The lengths obtained by this rule are only approximate, because the calculation la baseC
opon the sttpposltion that the chains form a parabolic curve : whereas, in fact, the curve of a BnislMd
bridge is neither precisely a parabola, nor a catenary, but intermediate of the two.

The following simple rule by the writer is quite as approximate as the foregoing tedious one,
when, as is generally the case, the defl is not greater than ^^ of the ehord, or span.

Length of main chain when defi does not exceed one-twelfth of the span = chord •\- .23 del.

Art. 10. To Hud, approximately, the length of the vert
snspendlnfp rodii x y, 4^c, Fif^ 1 ; assaminy the curve to
he a parabola.

Let X, Fig 1, be any point whatever in the curve ; and let x w be drawn perp t» tfa« chord e d ; aa4
c/perp to a&; then In any parabola, ^aaifl i aw^ : : ab : bf. And 5 /thus found, added t»ht,
(which is supposed to be already known, being the length decided on for the middle suspending rod,l
gives X g, the length of red reqd at the point x; and so at ally otliMr point.

Ifhf thn» fonnd he tAken IVom the middle deflection a 6.
it IfWvt'N fr or ; and thns auy deflection w x of the main cLaifi or cable, may Iw
found wh«n wt> know its hor dist, aw, trom the center, a, of the spaa.

In the foreKoiug rule, the floor of the bridge is supposed to be straight : bnt generally it is raiMd
toward the center; and in that case, the rods must Arst be caleulated as if the Hoor were straight*
and the requisite deductions be made afterward. When it risHis in twe straigiK lines maeting in tte
center, the method of doing this is obvious. When an arc of a circle is used, its ordioatea nuiy ba
calculated and deducted from the langtlis obtained by this rule.

Or, having drawn the curve by the rule for drawing a paraliola, the aimensiona can be appioz*

tmated to by a scale. The adjustments to the precise lengtlM must be made during the actaat oen*
striKtion of the bridge, by means of nnts on their lower sorew'-enda. The rods require, thoretw%
only to be made long titough at first.

The towers, piers, or pillars, which imhold the ehatits op
cables, admit of an enmess Tariety In mmtgn* According to dr-

cumstanees, they may consist each of a single vertical piece of timber, or a pillar of east or wrought
iron ; or of two or more suah, placed obliquely, either with or withont oonneotiag pieces ; Uke titm
bents of a trestle. Or they may be made (with any degree of or-

namentation) of cast-iron plates ; as in iron house-fronts. Or thej may be of maaonry, brisk, a»
ooncrete; or of any of these combined.

Each of the snspendinf^rods, through which the floor of the bridge is

upheld by the main chains, requires merely strength suflScient to support safely the giea<eat loaA
that can come upon the Interval between It and half-way to the nearest rod on each nae of It ; fn>
dnding the wt of the platform, Ac, along the same intervaL

In anchoring the bachstays into the frronnd, it ta nee^tnary to

seonre for them a sufiOciently safe resistance against a pull equal to the strain, upoit the backstay.

As to the anchoraire of the cables t)e1ow the surftice of the gronnd,
natural rook of firm character Is the most favorable material that can present Itself. When It Is not
present, serious expense in masonry must be incurred in large spans, in order to seour« the necessary
weight to resist the pull of the cables. Our Figs 4^ give ideas of the modes most frequently adoptad.
For a very small bridge, such as a short foot-bridge, for instance, the backstays may simply ba an*
chored to large stones, (, Fig A, buried to a sufficient depth. Or, if the pull is too great for so simpla
a precaution, the block of masonry, mm, may be added, enclosing the backstay. A close ooverlnf
of the mortar or cement of the masonry has a protecting effect upon the iron.

To avoid the necessity for extending the backstays to so great a dist under gnnmd, thsy are osvallT
curved near where they descend below the surface, as shown at B, D, and E ; so as sooner to reaon
the reqd depth. This curving, however, gives rise to a new strain, in tbe dlraotion shown by tha
arrows in Figs B and D. The nature of this strain, and the mode of finding Its amonnt, (knowing
the pull on the backstay,) are very simple ; and fully explained under the head of Funicular Ma>
chine. The masonry must be disposed with rererenoe to resisting this strain, as well aa

that of the direct pull of the backstay. With this view, the blocks of stone on which the bend rcMt
should be laid in the position shown in Fig D ; or by the single block in Fig B. Sometimes the bend
is made over a cast-Iron chair or standard, as at x, Fig F, firmly bolted to the masonry.

Fig B shows the arrangement at the Niagara railway bridge of 8'21 H ft span. The wirt baetltays
end at ce; and from there down to their anchors, they consist of heavy chains; each link of which
is composed of (alternately) 7 or 8 parallel bars of flat iron, with eye ends, through which pass bolts

Each of the 7 bars of each link is 1.4 ins thick, by fins wide, near Um

8TIBPEKSIOK BKIDOE8.

77i

lovest part of the ehatn ; bat they grodnalty Inereaae tram tbenoe upward, until at c, e, wher« thej
QDite vlth the wire cable, the aeetienal area of each Unk to 93 sq ina. These chain backstays pass In
a curve through the massive approach walls. (28 ft high,) and descend vertically down shafts «, «, 25
f( deepen the solid rook. Here they pass through the cast-iron anchor-plates, to which they are oon-
Ciicd below br a bolt 3H ins diam. The anchor-plates are 6J>j fbet square, and 2}^ ins thick ; except
for a space of about 20 ins by 26 tna, at the center where the chains pass through, where they are 1

flMAIhlek. Tbrongh this thick part la a separate opening for eaeh bar eomposing the lowest link.
Fron this part also radiate to the outer edges of the lower face of the plate, eight ribs, 2}4 ioo thick.
The shafts «, «, have rough sides, as they were blasted ; and average 3 ft by 7 ft across ; except at the
botcom, where tbey are 8 ft square. They are eompletely fUled with cement masonry, with dressed
beds, well in oontact with the sides of the sh^s; and thoroughly grouted, thus tightly enveloping
(hciehains at every point; as does also the masonry of the approach wall tow; which extends 28 ft
above ground ; and is 6 ft thick at top, and 10}^ ft thick at its base on the natural rock.

tif Figs i^, shows a mode that may be used in most eases, for brMges of any span. The depth
and the area of transverse section of the shaft, and consequently the quantity of masonry in it, will
depend chiefly npon whether it is sank through rock, or through earth. If through firm rock, then
If Ha sides be nade irregular, and the masonry made to fit securely into the irregularities, much re-
Ummm may be placed upon it to assist the weight of the masonry in resisting the pull on the baok<
■tays. larth also assists materially in this respect.

F is the arrangement in the Chelsea bridge of 383 feet s(MUi, across the Thames, at London ; Thos.
Pafle, eng. The space from one wall & 5, to the opposite one, is 45 feet; and is built up solid with
brlekwork and concrete; except a passage-way 4, ft wide, and 5 ft high, along the backstay ; aad a
•mall chamber behind the anchor-plates. It rests chiefly on piles.

The arrangement by Mr Brunei, in the Charing Cross bridge,LondoB,*laTery similar. In it also
the entire abutment rests on piles ; and is 40 ft high, 30 ft thick, and solid, except a narrow paseage-
wt^ along the chains. The backstays extend inle it 80 ft. Span 676 feet. Defl 5^ feet.

6 la intended merely as A general hint, which, yariously modified, may find its application In the
ease of u small tem(iorary, or even permanent bridge ; for the number of pieces, i, <, to, may be in>
eraaaed to any necessary extent ; and they may be made of iron or stone, instead of wood.

In erAer thMt tlie iMiefestAys may be aec«MiiMey thej are fre-

qoenUy oarried tlirough openinga left in the masonry for the purposa Thus, the maxses, mm,
of maeoary, at A and B, Flga 4^, instead of being made solid, may consist o f two parallel walls,
l>etween which the backstay maj pass; and the anchor-stones, or anchor-plates, will extend
•oroae tbe ipaee between the walls, and have tbetr bearings against the end* of the walls. In D,
■, and F, the eable aaay be Mipposed either te be tightly surrounded by the masonry and gronted to
it» or ela« to be surnwndsd hry a cylindrical passage-way like a culvert, so as to be at all times accea*
•Ible.

Soft Itiable stone must be carefully excluded fh>m such parts of the anchorage as are most
dlreetly opposed to the pull of the baetistays.

If blocka of stone large enengta for seeniing good bend are not proonrable, heavy T-niiaf haraef
Iron, at I-beams, may be advantageously introduced for that purpose.

The masses most be founded at such a depth as not to slide by the yielding of the earth in firont
of them.

For safbty, it Is well to disregard the effect of Motion in dtminishing the tension on the backstay,
and to regard that tension as eontinning nniferm throughout the backstay to its end, even when th»
baelutaj la curved and imbedded in the masonry, as at E, Figs 4}4.

The side parapets should be high and stout, so as to act as stiffening
ladoa, and should not be f estrieted to service as mere hand-rails or guards. As a rule of thumb

^ V' span, provided the depth be not less than that required for a
The' parapets should be stoutly constructed, with special attention to the strength of

truadoa,

their dapth maybe made

bandtraU. . , ,

their Joints, for Uiese are exposed, by the nndulatlohs and lateral motions of the bridge, to violent
Aoranging forces in all directions.

* Removed to Clifton, England, in 1863, and replaced by an iron truss railway sud foot bridge*

772

BIVETS AKD RIVETING.

BIVETS AIID ETVETING.

The welffhtfl in the foUowlns table of ooane Inolnde the head; bat the leasfliA* ainraalf
•re taJcea " auder the head ; " or are those of the ihaiilcs only. In practice, discrepancies of 5 or •
per ct in wt may be expected.

Length

of Shank.

Ins.

8

3.0

3.8

4.6

6.4

6.2

6.9

7.7

8.5

9.2

10.0

10.8

11.5

12.3

13.1

13.8

14.6

15.4

16.2

16.9

17.7

18.4

19.2

20.0

21.5

23.0

24.6

26.1

29.2

32.2

35.3

38.4

K

IMsmeton of Blveto In Inehea.

tVa I V4,

Welfht of 100 BlYetih In ponda.

8.5

••••••■

9.9

17.3

•••«•••.

11.2

19.4

26.6

88.9

•••••••

12.6

21.5

28.7

48.1

65.3

91.5

13.9

23.7

81.8

47.8

70.7

98.4

15.3

25.8

S4.9

61.4

76.2

105

16.6

27.9

87.9

65.6

81.6

112

18.0

30.0

41.0

69.8

87.1

119

19.4

32.2

44.1

64.0

92.5

126

20.7

34.3

47.1

68.1

98.0

138

22.1

36.4

60.2

72.3

103

140

23.5

38.6

63.8

76.5

109

147

24.8

40.7

66.4

80.7.

114

164

26.2

42.8

69.4

84.8

120

161

27.5

46.0

62.6

89.0

126

167

28.9

47.1

65.6

98.2

181

174

30.3

49.2

68.6

97.4

186

181

31.6

61.4

71.7

102

142

188

33.0

68.6

74.8

106

147

196

34.4

55.6

77.8

110

168

202

35.7

57.7

80.9

114

158

209

37.1

69.9

84.0

118

168

216

38.6

62.0

87.0

122

169

223

41.2

66.3

93.2

181

180

286

43.9

70.5

99.8

139

191

260

46.6

74.8

106

147

202

264

49.4

79.0

112

166

213

278

54.8

87.6

124

178

234

806

60.3

96.1

136

189

256

838

65.7

105

148

206

278

861

71.2

113

161

223

300

888

123
183
142
150
169
167
176
184
198
201
210
218
227
286
244
258
261
270
278
287
904
821
838
855
889
428
467
491

The dinm of rivets for bridge work is from H ^ ^ inch: usually % to
^; and for plates more than .5 inch thick, it is about 1.5 times ihe thicicneas;
and for thinner ones about twice ; but these proportions are not closely adhered
to. The common form of rtfrets as sold is shown at R, Fies 8, a head
and the shank in one piece : and S shows the same when after being heated
white hot it is inserted into its nole, and a second head (conical) formed on it by
rapid hand-riveting as it cools. When lonicer than about 6 ins they
are cooled near the middle before being inserted, lest their contraction in cooling
should split off their heads. The hemispherical heads often seen, called smAp
heads, are formed by a machine. The two heads alone require abont
aa much iron as 3 diams length of shank. Ijenyth of a head ■■ about 1
diam of shank ; and its width about 2 diams of shank.

Rlvetinff of Steam and Water Tiffht Joints.

Joints for boilers and water-tight cisterns are usually proportioned about
as per the following table by Fairbairn ; and are made as shown either by Fig 1,
or Fig 2. Fig 1 is called a si n|ple- riveted, and Fig 2 a double-riveted

lap-joint. The dist a a, or c c, is the lap.

Mr Fairbairn considers the strength of the single-riveted lap-joint to be about
.56 ; and that of the double-riveted, about .7 that of one of the full unholed

RIVETS AND RIVETING.

773

plates, when both joints are proportioned as in his following table. But some

later experimenter^ consider about

^-^=^-^ . a. ^ ^ t

^. 1 /

1

J

v>^

I

o

0

o

J

-cr

a

o
o
o

o
o

IC

Fig 1. Fig 2.

proportions include friction (Art 4), without which they would 6e abofd A and .5.

.6 and .6 as nearer the correct aver-
age. Experiments on the subject
are quite conflicting; and it is
plain that no one set of propor-
tions can precisely suit all the dif-
ferent qualities of plate and rivet
iron. . With fiedr qualities of both,
there is every reason to rely upon
.5 and .6 (or about one-seventh

8 art less than Fairbairn's assump-
ion) as safe for practice. These

Fairbairn's table for proportlonlngr the riwetin§s for steam

and water-tifplit lap-joints.

Thiokness of
«Mb plaie.

-16

Diameter of
riveu.

Length of BbaQk
before driving.

From center to
oenter of rivets.

Lap in single
riveting.

Lap in doable
riveting.

BiTetinsT of iron yirders, brldipes, Ae.

N

KC

r^^ r^

e'-^^ — c^«

a

iff

[

(H>' 'O

w

ooo O^O

} o
OOO'iOD^ o

^ o

OQOO O

Figs a

ooo
ooo
ooo

■^=^0

M

py. ■

5^

Art. 1. Tbe snbjeet of riTetinip is abstruse, and involved in
much uncertainty ; and experimental results are very discrepant. We here pro-
pose merely to confine ourselves to what is considered the best joint ; and for
fsafety we shall omit friction; see Art 4. In girder and bridge work the lap-
joints above described are seldom used. Instead of them, the plates p, Figs 3, to
be joined, are butted up square against each other, thus forming a batt«Jolnt,
i «, Fig D; and are united by either a single coTering-Plate, eover,
^rrapper, fish-plate, or welt e e, Fig K ; or the best of all by two of them,
88 at A, or 0 0, 0 o, Fig B. In what follows, the term plate never includes the
covers. The single cover, like the lap-joint, allows both plates and cover to bend
under a strong pull, somewhat as at W, thus weakening them materially ; whereas
the double cover oo,oo, Fig B, keeps the pull directly along the axis of the plates,
thus avoiding this bending tendency. It also brings the rivets into double shear.
thus doubling their strength. When there Is but one cover, it should be at least
as thick as a plate ; and when there are two, experience shows that each had bet-
ter be about two-tMrds as thick as a plate, although theory requires each to be
but kt^faa thick as a plate.

774 EIVBTS AND RIVETING.

Tlie length w w of covers across the joint is equal to that of the joint

Butts require twice as many rivets as laps, because in the lap each
rivet passes through both the joined plates; and in the butt through only one.

Tbe rivets an4l plate on one side only (right or left) of the joint-
line i i of an^ properly proportioned baft-Joint D, represent the full strength
of the joint, inasmuch as those on one side pull in one direction, i^ainst those on
the other side, which pull In the opposite direction. Therefore in designing sudi
joints we need keep in mind only those on one side, as is done in wluit followB.
Thus a single, double, or triple>riveted butt«j4»int D implies one, two, or three
rows of rivets on each sine of the joint-line i i, and parallel to it. In a prop-
erly proportioned lap the strength is as ail the rivets, because one-half of them
do not pull against the other half^ but one end of every rivet pulls in on« direc-
tion, and its other end in the opposite direction.

Tiie net iron, net plate, or net Joint-, \& that which is left l»etf:woeB
the rivet holes, and outside of the two outer ones, all on a straight line drawn
through the centers of the boles of one row. Its width and area are called the net
ones of the joint. That between oUuar rows does not increase the«tt«ngtk.

In Figs 3, N, and K, the rivets are in singple sliear, while those in A and B
are in double sliear.

Art. 2. Bridfpe-Joints are not re%nired to be steam or irater-
tig^bt like those of ooilers or cisterns ; and, therefore, by increasing the hreadth
of the overlap, or the length of the covers, the rivets may be placed in several
rows behind each other, as the 3 rows of 3 rivets each in M and D, instead of only
one row of 9 rivets, as in L. By this means, without losing any of the strength of
the 9 rivets, or of the net iron, we may narrow the width of the plate to too. ex-
tent equal to the combing diams (6 in this case) of the holes thus dispensed with
in the one row. Moreover, by using more than one row we lessen the weaJcening
eflfect shown at W. This mode of placing the rivets directly behind each other in
several rows, as at M, and at the left-hand half of Fig D, constitutes Mr Fair-
bairn's chain rivetinip; but the joint will be somewhat stronger if the rivets
are placed in sigrzafpine order, as in the right-hand half of Fig D.

Tne dist apart ofthe rows from cen to cen should not be leu
than 2 diams. It is questionable to what extent this increase in the number of
rows may be carried without an appreciable loss of atroB^di in the rivets conae-

auent upon the impossibility of quite equalizing the strains on the separate rows.
lut it is probable that if we do not exceed 2 or 3 rows in laps, or the same num-
ber on each Hde of the joint-line in butts, we may in practice assume that each
row, and each rivet, is nearly equally strained.

Rivet-boles are nsually of about one<4ixteenth inch greater diam than the
original rivet, so as to allow the hot rivet to be easily inserted. The 8ub6e<^uent
hammering swells the diam of the rivet until it fills the hole. We may either
take this increased diam of rivet into consideration, as we have done, in calcul*-
ting its shearing and crippling strength, as explained farther on, or with reference
to increased safety we may om'it it. Orilled rivet-boles are said to be better
than punched ones, as the drilling does not injure the iron around them; but on
the other hand their sharper edges are said to shear the rivets more readily.
Hence, such edges are sometimes reamed off. Both these points are, howerer,
disputed ; and both modes are in common use.

Tbe dist from tbe edipe of a bole to the end of a plate or cover should
not be less than about 1.2 diams, to prevent the rivets from tearing out the end
of the plate ; nor nearer the side edge of a plate than half the clear dist between
two holes as given by the Rule in Art 5. Tne first is rather more than Fairbaim
directs.

Rivet boles weaken tbe net iron left between them, not only by the
loss of the part cut out, but either by disturbing the iron around them, or perhaps
by changing the shape of the net line of fracture, which may not then leaist
tension as well as while it was a continuous straight line. Some deny both cauiie
and effect entirely, each party^ basing its opinion on experiments. But the man
of evidence seems to the writer to show that the net iron loses on an average
about one-seventh of the strength due to the net width. With a view to safety,
which we consider to be of paramount importance, we shall in what follows
assume (until the (^^uestion is definitely settled) that there is such a loss of
strength in the net iron.

Rivete€l Joints for resistingr compression should depend, not as
might be supposed upon their butting ends, but upon either tbe shearing or the
crippling strength of the rivets; for contraction or bad work may t<hrow: the

BIVETS AND BIYETING.

775

Sreasare on the riyets. JHaeblne riveting is somewhat stronger than that
one (as is assumed in our examples) by hand. Ttie tblclLness of plates

used in girders, tubular bridges, 4&c, is uauaUr .25 to .5 inch ; with thicker ones
up to 1 inch sparingly in large ones. A pacKinnf pleee, as the shaded piece
In P, is one inserted between two plates to prevent their being bent or drawn
together by the rivets.

Art. ^ A riweted Joint miiy yield In (liree w»n after being
pvopedy pro{>ortioned. namely, by the cdnearing of its rivets; or oy the pulliKg
apart of the net plate between the rivet holes ; or by the eripplinff (a kind^
compeesion, mashing, or criimpUng) of the »l»tes by the rivets when the two are
too xorclblv palled against each otl^. It also compresses the rivets themselves
transverselv. at a less strain tban tbe sbearlnff ones and this parUftl
yielding of both plates and rivets allows the joint to stretctiv and may thus
produce injurious unlooked-for strains in other parts of a structure, considerably
before th^re is any danger of actual fracture. Or in steam and water joints it may
eanae leaks, without farther inconvenience, or danger. For a long time this
crippling had entirely escaped notice, and it was supposed that the only important
pout ij| designing a riveted joint was that the tensile strei^h of the net plate,
and the shearing strength of the rivets should be equal to each other.

Tbe erlppflny strensrth of a Joint Is as the number of rivets, In a lap,
or the number on one side of The joint-line in a butt X dlam x thickness of joinM

Slate. This product gives the crippled area of the joint. We shall here call the
lam X thickness of plate, the eripplinn: area of a rivet. If there are 2 or
more plates (not eovers) on top of eacn other at one joint, their united thiel^ness
is used for flading tbe crippUng area. Tbe nltlmate erliMpMnif nit,
by which tbe abonre product is to be multiplied for the actusd ultimate crippling
strength of the joint, may be safely taken at about 60000 fl>s, or 26.8 tons, per sq
inch.

Tbe dlani of a riwet in ins to resist safely a given single-shearing
ioroe is found thus: Mult the aheving foioe by the ooei of safety, that is by the
number, 3, 4, or 6, (&c, denoting the required degree of safety. Call the product p.
Mnlt the ultimate shearing strength per sq inch of the rlv9et4ron, by the decimal
.7854. Call the product d. Divide p hy b. Take the sq rt of the quotient. The
diearing force ^sul the aheartng strength must beth he in either 8>s or tons.

Or by a formula,

Dlam in ins

V

Shearing force X coef of safety
TTlt shearing strength per aq inch X .7854

If tbe riwet is to be donl»le«sbeared, first mult only half the shearing

force by the coef of safety. Then proceed as before.
On near enough for practice, mult the diam in single shear by the decimal .7,
Ive ultimate stiearluip unit for average rivet-iron may be taken at

About 45000 S^s, ox 20.1 tons per sq inch of circular sheared section.

Table of altiiiiate sinirle sbeariaff strenytb of rlwets.

(market sizes), Ia single shear ; at 45000 9>s or 20.1 tons per sq inch.

Tbis table is not to be used when as in our " Example," Art 5, the
efl*ipplinir strength of the livet governs the strength of the joint.

Ifilie rivet is in double sbear it will have twice the strength in the
table.

For tbe4i»m in double sbear to equal the strength in the table, mult
the diam in the ishle by the decimal .7 ; near enough for practice ; strictly, .707.

Diam.
Ins.

Diam.
Ins.

Iba.

Tons.

Dlam.
las.

%
H
H

Diam.
Ins.

lbs.

Tons.

Dlam.
Ins.

Diam.
Ins.

Sw.

Tons.

%

.125
.187
.290
.812
.375
.487
.500

662
1242
2209
3452
4970
6766
8836

.246
.554
.986
1.54
2.22
3.02
394

.562
.625
.687
.750
.812
.875
.937

11183

13806
16706
19880
23332

31064

4.99
6.16
7.46
8.88
10.4
12.1
13.9

1
1%

1.000
1.062
1.125
1.187
1.250
1.812
1.875

85343
39899
44731
49838
55224
60686
66820

15.8
17.8
20.0
22.2
24.6
27.2
29.8

776 RIVETS AND RIVETING.

The tensile strenirtla of a properly proportioned Joint fa

equally as either the sectional area of the net ^late (not covers) across the cen-
ters of only one row of rivets : or as the shearing or the crippling (as the case
may be) areas of all the rivets in a lap, or of all the rivets on one side of the
joint4ine in a butt. The tensile strength of fair qualitv of plate iron, before the
rivet holes are made, averages about 45000 fts, or 20.1 ions per sq inch ; but we
shall for safety assume, as stated in Art 2, that the makine of the holes reduces
the strength of the net iron that is left about one-seventh part, or to 38^ fte,
or 17.2 tons per sq Inch.

Rem. Ewen tbis is considerably too grreat for laps, or for butts
with one cover, owing to the weakening of the iron in such by the bending eAiown
at W, Figs 3. But we are not speaking of such.

Art. 4. Tbe friction between tbe plates in a lap, or between the
plates and the covers in a butt, produced by their being pressea tightly together
by the contraction of the rivets in cooling, adds much to the strength of a joint
whUe new, perhaps as much as 1.5 to 3 tons per sq inch of circ section of all the
rivets in a lap, or of all on one side of a single-cover butt ; or 3 to 6 tons of all on
one side of a double-cover butt. In quiet structures, this friction might dontinue
to exist, either wholly or in part, for an indefinite period ; but in bridges, <&c. sub-
ject to incessant and violeni jarring and tremor, it is probably soon diminished,
or entirely dissipated. Hence good authorities recommend not to rely on it, ana
it is, therefore, omitted in what follows.

Art. 5. We now give rules for finding the number of rivets required for a
double cover butt-joint (the only kind of which we shall treat), and their
clear or net distance apart. This dist + one diam is the plteb of the rivets, or
their dist from center to center. The principle of the rule will be explained
further on, at Art 7.

First, select a diam of rivet either equal to or greater than .85 times the
thickness of the plate. In practice they are generally 1.6 times for plates ^ inch
or more thick ; and 2 for thinner than }^ in.

Second, mult the greatest total pull in pounds that can come upon Hie entire
joint by the ooef (3, 4, or 6, &c) of safety, and call the product p.

Tblrd, multiply the crippling area of the rivet (that is, its diam X the thick-
ness of plate) by 60000. The prod is the ult cripplinsr strength of a riv«t. Call it tn^

Fonrtb, divide p by m. The quotient will be the number of rivets to sustain
the given pull with the reqd degree of safety.

Then, tbe clear distance apart will be

Number of rows X Piam X gOOOO
38500

Flftb. The clear dist from either end hole of a row to the side edge of the plate^
should be not less than half the clear dist between two rltets in a row.
Example. A double-cover butt-joint in .5 inch thick plate is to bear an actual

J nil of 33750 fi>s, with a safety of 4; or not to break with less than 83750 X ^ -"
85000 fi». How many rivets must it have; and how far apart most they be? -
First, Here .85 times the thickness of the plate is .6 X .85 =■ .425 inch ; there*
fore, our rivets must not be less than .425 inch in diam ; but we will take .76 indi
diam. _^

Necond, The greatest pull X coef of safety « 83750 X * — 135000 lbs -. p.
Tblrd, The crippling area of a rivet X 60000 » .76 X .6 X fiOOpo » 22500 — m,

Fonrtb. ^ *■ ^,^ »* 6 rivets required on each side of the Join t-Une.
' m 22500 ♦

And tbe dear spa0e or net width between them will be, IT tbo • rlTetS
are in one row s

Diam X 60000 4500Q

38500 88500 *«'«~»™«

1 9188
And the pitcb » net space + diam — 1.1688 H- .76 -• 1.9188 Ins, — ' "
— 2.56 diams. •'**

In practice, to avoid troublesome decimals, we might make the net space 1.2 Insi
and the pitch 1.96; but to show farther on the working of the rule, we adhere III
the more exact ones.

Flftb. The clear dist from each end hole to the side edge of the plate Is half of
1.1688 — .6844 ins.

Tbe entire wldtb of net iron Is equal to one clear space X number of
rivets a. 1.1668 X 6 » 7.0128 ins; and the entire width of plate is equal to
pitch X number of rivets, » 1.9188 X 6 » llJil28 ins.

BiTBiB A2n> BnrEnna 777

The ftrea of eroMSfletton of nnboled plate toll J128X •B — S.76048q Ins; Hsteii*
rile strength beflire the lioles are made to MM4 x 46000 — 259088 1Mb

The strength of our Joint, omitting fHction, is therefore a^i^p^^ ■■ .62 of that of tht
original nnholed pUte. • 269038

If the 6 riTets are In 2 rows of 8 riTets each, the elear dlst be*

tween two rlirets in one row will be twice as great as before, or twice 1.1688

= 2.3376 ins. Plteli » 2.8876 + .76 — 3.0676 ins » 8.0876 -t- .75 >«- 4.12 dtoBM.

Clear dlst from end hole to side edge of plate =* half of 2.8876 a 1.168a

Entire widtli of net Iron — 2.8376 X 3 <-> 7.0128 ins. Sntire width

of plate » 8.0876 X 3 >. 9.2628 ins. Area of eroMi seetlon of nnboled

plate»9.2628x.5a4.63i4 sqins. Ultimate tensile strenrtb, nnholed

^ 4.6314 X 46000 »■ 208418 lbs. Vlt strenffth of riveted Joint, omitting

135000
friction » _.-.,- — .66 of that of the nnholed plate.
206413

Thus we see that the anraiigement with two K)wt ghres the same strength as one

row, with a less total width and area of plate. It of coarse requires lonffer cown.

If the 6 rivets are In 8 rows of 2 riyets each, the area of eross
section of the nnholed nlate is 4.2565 sq ins. Its tensile strenstfa«

191542 lbs. Strength of riveted Joint ■■ » .7 of that of the nnholed plate.

The entire width of net iron (7.0128 Ins); its area (7.0128 X .5 — 8.6064 sq ins);
and its ultimate tensile strength (8.6064 X 88500 ^ 135000 lbs), are the same in each
case. The last is the required breaking strength of the joint, as in the beginning
of our examine; and is equal to the oomUned crippling strength of the six rlTeta.

Art. 6. The dlntance apart of the rows, from center to center of
rivets, should not be less than two diameters of a rivet-hole.

Rem. 1. With our constants for tension, shearing, and compression, the
rivets will not vleld first by shearing in a double-cover butt (and
of course in double shear), except when the diam is either equal to or less than
.86 of the thickness of the plate, which will rarely happen. At .85 the crippling
and shearing strength of a rivet are equal when using our assumed coeffls of crip-
pling, shearing, and tension.

Hem. 2. Our example was choeen to illustrate the rule. It will rarely hap-
pen in practice that the rule will give a number of rivets without a fraction ; or
that may be divided by 2 and by 8 without a remainder. In case of a fraction, it
l0 plainly best to call it a whole rivet ; although the joint Uiereby becomes a trifle
stronger than necessary. Or rivets of a slightly diff diam mav be used. If the
nnmMor of rivets comes out sav 7 or 9. we may make 2 rows of 8 and 4, or of 4 and
6, Ao. Moreover, the width of the plate is frequentlv fixed beforehand by some
requirement of tne structure, and we must arrange the rivets to suit, taking care
ki all cases to maintain the calculated ares of net iron in one row, ac.

We have (as we ftt first said we should do) confined ourselves to the

simple buttpjoint with 2 covers, and with the
p , rivets in either 1, or in 2 or more parallel rows

^ I I C on each side of the joint-line ; this being the

^ I I < strongest and the one in most common use in
^ i \ y engineering structures. Necessity at timee

^ * I K cafls for less simple arrangements, for which
It \ ^® cannot afford space, and the strength of
1 ^ which is not so readily calculated. These
sometimes yield results which appear strange
to the uninitiated ; thus, this lap-joint breaks
across the net iron of one plate, along either c c or o o, wfiere there is most of it, and ,
where, therefore, it might oe supposed to be the strongest.

Rem. 4. The followlni^ table shows approximately the comparative
strengths of the common forms of joints when propwly proportioned : varying
with quality of sheets, and of rivets :

■

With
___^ Motion.

The original unholed plate 1.00

Double-riveted butt with two covers. 80

Double-riveted butt with one cover 65

Single-riveted butt with one cover .50

Double-riveted lap 65

Single-riveted lap 50

Without

Motion.

1.00

.64

.52

.40

.52

.40

778

BIYSIS A^D BJVBTnn}.

0. The above tabiilur rtrencths for tbe l»|i^«»anta will be «ipn>z«

iniBtely attained bv adopting the ^Uewijig proporttona, aceofdiaff aa tbe jo&nt is
double- or sin|;le-riTeted.

• ••• ••a*^* t**«9«

QalliBg tbickneas of plate..
Then make dlam of riTet...
" brudth of lap.

Diteb teom. cea to een

olat from md of plate to

«dfff of hoi«B ,

dlst apart of rows from
can to can

<t

(t

u

Doable viv, ■!«■•«.

Made vftT.

lo thiokoeues. Ib Aiaou.

1.
1.67

e.o

7.0

IjO
M
4.2

•

1.

1.67
5£7
4.5

£
IjO
&4
2.7

2Ji

L2

Z»

1.2

ZM

2.0

Hem. 6. If two or more plates on top of eacti otber* as tbs

four in A B or M H, are to be lointed together so as to act as one plate of the
<%lcknea8 c «, the diams of the rfyets, and the thioknetn of the cefrers ce^ee will
depend tipon whether the iunctions of the plates are an In one Ikie with each
other as at 0 c, in A B« or wnather they break joint with each other as at 0, 1, 2, 3
in M H.

a

^

It is plain that the two oovers c <; by means of their connecting livets oonTey
from A to B, across the Joint c c, all the strength that partly compensates for Um
severance of the four piates at that joint ; whereas me two covers e e, « e. and
their rivets in like manner oonvey from « of one single plate, to o of the a^iclniog
one, across the joint between those two letters, only the rtrenath that partiy ooni*

rDsates for the eeveranoe of that single plate ; and so with the joints at 1, 2, and
Therefore the covers c «, and their rivets, most be four times aa strong as thoai
at any one of the four joints 0, 1, 2, 3. The first, c Cy are to be regarded as Mining
two solid plates A and B, eacn of the fourfold thlekness e e ; and the cAhefs as
joining two of the sUigle thickness. The covers e c will, therelbre. each be about
two-thirds of the thickness o e ; and the others eaeh about two>thirds as thick m
a single plate. Thus, sui^xtse each of the i plates inABorMHtobe^ tech thick ;
making ceZ ins. Then each cover, c, is ^ of 3 ins, or 2 ins thick ; or the two covers^
<;«, together 4 ins, which is thus the effective thickness of the joint, ec. But each
cover, e e, is only %of% inch, or % inch thick ; and the effective thickness of Joint
at either 0, 1, 2, or 3, is that of the 3 unbroken plates plus that of the 2 covers, or
<3X^+(2Xi^) = 3^in8.

Art. 7. Principle of tbe Rule in Art 5. With oar constanta for
shearing (45000 lbs per square inch) and for crippling (60000 lbs per squase inchX and
with diameter of rivet equal to, or greater than, .85 times the thickness of the plate,
as by our rule, the crippling strength of a double oov«r butt joint will be «q<iai ie, er
less than, its shearing strength. Therefore, to avoid waste of material, eiUter in the
plate or in the rivets, we must make

'^'SSl' »°ll''of1?«''ir^ - CrtPPH-g ^-e«. of •« th, rl«t* Or.

Crippling area ^ Crippling _ Total irambtr
of one rivet ^ nnit ^ of rivets.

plate of plate ^ umt

Now, by Art 3, the crippling area of a rivet is ■» diam of rivet X thickness of
plate. We take the crippling unit at 60000 Kts; and the tension uiit at 88600 Iba
Therefore (transposing) we must make

fP«foi ««f ^iA^u ^***" o' ^ Thickness ^ mww| v» T<>*»' number
^ Thickness of plate X 88800

RIVETS ANJ} RIVJSTtNO. 779

By making the clear dutance between each end rivet of a row and the side
edge of the plate = half the clear distance between two rivets in a row ; and
calling the sum of the two end distAuces one epace, we have

Number of tpaea _ Number of riveis

iu a row in a row. •

So that

Tlie clear distance between two riTets'ln a row*

wliich is

_ Total net width of plate

~ Number of spacaa in a row

«« Total net width of plate
Number of riwU iu a row

Diam of ^ Thickness v, tst^anti sy Total number
rivet ^ of plate ^ "^"^ ^ of rivets

*^ Thickness v> aornn w Number of rivets
of plate ^ ****~ ^ In a row.
But

Total number of rivets

is also

Number of rivets in a row

= Number of rows.

Therefore, omitting "thickness of plate," common to both numerator and
denominator, we have, as in rule in Art 5,

Clear distance Diam of rivet X flOOOO X Number of rows

"" 38600

Bnt If tbe dtemeter of the rlwets Is less than 0*85 times the
thickness of the plates, the shearing strength of a double-cover butt joint
(with our assumed constants for shearing and crippling) is less than its crippling
strength. Itt such cases, for the clear dbtance betweeu two rivets in a row, say

^_ J* ^ Circular area of a rivet X Shearing unit

Clear distance = Thid^ness of plate X Tension unit ^ ^

Bern. 1* Butt Joints in donble shear, or with 2 covers, being the
only ones here considered, and inasmuch as rivets m^ always be used with a diam
^%ater than .86 of the thickness of the plate, we may in practice always use the
Rule in Art 5 forsnch Joints; and, therefore, we gave it alone.

Rem. 2. When nslngr these mies fi^r other kinds of Joint,
ench as laps, or bntts with single covers, remember that the rivets in snch are in
slnarle shear; and, therefore, we can use Knle In Art 6 (for crippling) only when
the diam is either 1.7 or more times tbe thickness of plate. If less, nse
Rule above for shearlniT* <^11 oi^ ^* assumption that our foregoing coefs of
crippling and shearing are nmd.

•

Bnt the e<»ef for tension niast be changred for each kind of these
other joints, to allow for the weakening effects of the oen^ng shown at W, Figs
3. as deduced approximately from experiment. The writer believes that the fol-
lowing tension units will give safe approximate results without friction. For
doable-coTcr butts, double-riveted, 88500 S>s per sqinch, as adopted above.
For donble-rlveted laps, or one-cover butts, 28000. For slngrle-rl veted
laps, or one-cover butts, 24000. But, as before remarked, no great certainty is
attainable in riveting.

Bern. 3. A Joint may fall by cripplinKr without the facts being
known or even suspected, for it does not implv that anything breaks, but.
merely that the joint has stretMted ; and thfs might not be detected even on
a ^ight inspection of it. Still it might, and probably often has sufficed to endanger,
and even destroy both bridges and roofs by generating strains where none were
provided for.

780 RAILROAD CURVES.

RAIIiROAD CUKTES.

Definitions.

A circular railroad curve abed, Fig. 1, is an arc of a circle joining two straight
litfes, or tangrento, e t and i 2, in tne survey.

Tlie point of enrTe is the beginning a of the curre, or that end of it
first reached by the survey in its progress. ,

Tbe point of tang^ent is the other end d of the curve.

Tbe point of intersection or apex is the point i where the two tan-
gents 6< and iz intersect.

P. C, P. T. and P. I. The stakes driven at the point of curve, point of
tangent and point of intersection are marked P. C, P. T. and P. I. respectively,
and the points and stakes are commonlv referred to by those letters. The point
of intersection, however, is not always located.

The apex distance * is the distance aiordi measured along a tangeut,
from either end a or d of the curve, to the apex i or intersection of the two
tangents.

t

A curve may be located by settinjj up the transit at the point (as a) where the
curve is to join either tangent, laying off equal angles iab^baOfCad, and meas-
uring off the equal chords (usually 100 feet) ab,0Cf cd.f Inasmuch as these
equal chords are usually laid off with the full length of a chain or steel tape, we
shall call them cliains, ta distinguish them from other chords, such as a cor
a d, etc., which may be drawn to the curve.

Tlie total ann^le of the curve is the angle iiz between the two tangents.
It is equal to the central angle aod subtended by the curve.

The deg^ree of cnrvatnre is the angle aob,boc, etc. subtended at the
center by a chain. It is equal to the deflection ang^le }m6c formed be-

*The apex distance is often, but unfortunately, called the " tangent," and
sometimes the "tangent distance."
t But see Sub-chains.
t Many writers call iab,bac, etc. the d^fleetUm angle.

RAILROAD CURYES. 781

»

tween any chain be and the extension dm of an acUoining chain ab^or to the
augle tsn formed between the tangents at and b n which touch the curve at the
twu ends a and 6 of a chain. It is therefore the angle through which the direC'
tion of the line deflects in that portion ot the curve subtended by one chain.
The sharper the curve, the greater the deflection angle and the shorter the
radius oa^ ob^ etc.

A one-deiri^®^9 two-degree, three-degree, etc. curve is one whose deflection
angle or degree of curvature is 1*^, 2°, 8*^, etc.

Tlie tangential anffle is the angle t a ft, 5 a c, etc. used in laying off the
curve. It is equal to one-half the deflection angle.

Tlie deflection distance. Let any chain, as aft. Fig. 2, be extended to
m, and b m made = ab. Then the distance m c from m to the end c of the next
chain 6 c is called the direction dittanoe of the curve.

Tbe tangential distance. Let any chain, as & c, be extended to v, and
let cv be mac^ = bc. Also let c 2 = cv be laid off upon a tangent to the curve at
c Then vz is called the tangential distatice of the curve.

By means of the deflection and tangential distances given in the tables, pp.
784-786, a curve may be located without a transit by means of a chain (or 100
feet tape) for measuring bm^cOf etc., and a rod or tape, etc. for measuring m c,
vs, etc.

An ordinate is any line drawn from a chord to the onrve, at right angles
to the chord, whether the chord be a " chain " or not. *

Tbe middle ordinate is the ordinate en. Fig. 2, at the middle point «
of a chord b c.

Snb-chains, etc. For facility of explanation we have hitherto treated of
curves as being composed entirely of full chains ; but such curves seldom occur
in practice. Usually, after dividing a curve into as many full chains as possi-
ble, there is a fraction of a chain, or «t^-chain, left over. Besides, the chances
are that a curve will not begin or end at a full 100-feet station of tbe survey,
but at a point between two such stations, as in Fig. 3 ; and, inasmuch as it is
desirable to carry, throughout the curve, the same numbering of the stations as
we have on the tangents, the P. C. and the P. T. are in these cases treated as
fractional stations.

Thus, in Fig. 3. the P. C, at a, is supposed to be 41 feet beyond st^ition 122.
We therefore call the P. C, in this case, station 122+41, and station 123 thus
falls in it-s proper place, at A, 100 feet in advance of station 122, as measured, first
along the tangent za 41 ffeet to the P. C, and then along the sub-chain a 6 of 59
feet.

.Similarly, the P. T., in Fig. 3, happens at a point, d, on the tangent d y, 20.8
feet back from station 125, or 79.2 reet (the length of the sub-chain cd) beyond
station 124 or c. The P. T. therefore becomes station 124 + 79.2.

782

RAJLROAB CURVES.

Owing to the oocurreDce of the sign + {phu) in the nnmber of the P. C. or
P. T. of a curve beginning or ending with a sub-chain, such a station is com-
monly called a ** plus."

Figr.3

The snb-tanjcentlal anirle. Fig. 8, is the angle bat(=adb) or deh
{= dae) sabtended by a sub-chain ; the vertex, a, d or c. of the angle lying in
the curve itself. We shall eive the names initial and final sub-tangential
angles to the angles bat anddch subtended by the initial and final sub-chains
a b and c d respectively. If a < be made = a 6, and eh = cd. the chords tb and
h d are called the initial and final snb-tanirential distances respec
tively.

Snb-defieetion angrl^^* ^''^g- ^- T^^ initial sub-deflection angle is
the angle sbc formed between the first full chain be and the extension ft « of the
initial sub-chain ab. The final sub-deflection angle is the angle kcd between
the final sub-chain cd ana the extension ck of the preceding mil chain be. It
bshe made = bc, and ck = cd, the chords s c and k a are called the Initial and
final snb-defiection distanc€Mi respectively.

A long: ebord is a chord, ae or ad. Figs. 1 and 3, subtending two or more
chains or sub-chains.

A simple enrve. Figs. 1 to 3, is one in which the radius remains of con-
stant length throughout and in which the curvature is all in one direction.
Such a curve is therefore an arc of but one circle.

Fifir.4

A compound enrve, as a & cd, Fig. 4, is one in which the curvature is all
in one direction, but which consists of cGcular arcs described with two or more

BAILBOAD CURVES.

783

dli&rent radii, m ob, (/b, o^^, etc. The pertions, as ab^bc^edf deacribed witk
the different radii, are called the bvaiicii«s of the curye.

Fig. 5

A refrerse cvrre, abc. Fig. 5, consists of two curves immediately adjoin-
Inf one another (i. 0., trithoat any straight track between them) and curving in
opposite directions. The radii o&, o'b^ of the two portions, or branches, of a
reverse curve may be of equal or of unequal length, and the total angles, aob,
bo'Cy of the two branches, may be equal or unequ^.

784

BAILBOAD&

Table of Radii, Middle Oirdliiates, Ae, of Carres. Ohoid lOO feet.
Contains no error as great as 1 in the last fi^^uro.

Aug. of
Den.

Bad.
inft.

DeiL
Dl«t.
inft.

Tang.

Dlst.

inft..

Mid.
Ord.

▲ng.oT
Defl.

Bad.
inft.

DelL

DiBt.

inft.

Tans.

Di«t.

Inft.

Mid.
Old.

o •

1

848775

.029

.014

.004

0 »

1 86

8681

8.798

1.896

.848

8

171887

.068

.029

.007

88

8608

8.861

1.485

.866

8

114692

.067

.048

.011

40

8438

8.909

1.464

.884

4,

86944

.116

.068

.014

42

3370

8.967

1.488

.871

6

68766

.146

.072

.018

44

8806

8.026

1.512

.878

«

67296

.176

.067

.022

46

8248

8.084

1.543

.886

7

49111

.904

.102

.026

48

8188

8.14S

1.571

.896

3

42972

.888

.116

.029

60

8126

8.860

1.60O

.400

9

88197

.962

.181

.038

62

8070

8.867

1.629

.407

10

84877

.891

.146

.036

54

8016

8.816

1.668

.414

u

31262

.880

.160

.040

56

2964

8.874

1.687

.431

IS

28648

.849

.174

.048

68

2918

8.488

1.716

.429

IS

26444

.878

.189

.047

a

2866

8.490

1.746

.486

14

24666

.407

.208

.061

8

2818

8.648

:.774

.448

16

22918

.486

.218

.064

4

2778

8.606

1.808

.461

16

21486

.466

.282

.068

6

2729

8.664

1.888

.466

17

20222

.494

.247

.062

8

2686

8.728

1.861

.466

18

19096

.624

.262

.065

10

8646

8.781

1.880

.478

19

18094

.668

.276

.069

18

2606

8.888

1.919

.480

to

17189

.682

• .291

.073

14

2566

8.887

1.948

.487

21

16370

.611

.306

.076

16

2628

8.956

1.978

.486

32

16626

.640

.820

.080

18

2491

4.014

8.007

.602

23

14947

.669

.884

.063

80

2466

4.078

8.066

M»

«4

14324

.696

.849

.087

38

2421

4.181

3.066

.616

^

13751

.727

.368

.091

24

2387

4.180

8.094

.588

36

13222

.766

.878

.096

86

2366

4.S46

8.188

.681

27

12732

. .786

.892

.098

88

2828

4.306

2.162

.638

28

U278

.814

.407

. .102

80

2292

4.368

2.182

.646

29

11864

.844

.422

.106

8S

2262

4.481

2.210

.562

80

11469

.878

.436

.109

84

2832

4.480

2.240

.660

<1

11000

.90S

.451

.118

86

8804

4.687

3.268

.667

32

10743

.931

.466

.116

88

2176

4.686

2.298

.674

88

10417

.960

.480

.120

40

2140

4.654

2.327

.562

84

10111

.989

.494

.128

42

2122

4.718

2.866

.589

86

9822

1.018

.509

.127

44

2096

4.771

2.886

.606

36

9549

1.047

.523

.131

46

2071

4.829

2.414

.60S

87

9291

1.076

.538

.134

48

2046

4.888

2.444

.611

88

9047

1.106

.652

.138

50

2022

4.946

2.478

.618

89

8815

1.134

.667

.142

62

1999

6.003

2.501

.686

40

8694

1.164

.682

.146

64

1976

5.061

2.680

.688

41

8385

1.193

.696

.149

66

1953

6.120

2.660

.646

42

8186

1.222

.611

.168

68

1832

6.176

2.58R

.647

43

7995

1.261

.625

.166

8

1910

6.286

2.618

.664

44

7813

1.280

.640

.160

2

1889

6.294

2.647

.663

46

7689

1.309

.664

.164

4

1869

5.360

2.676

.688

46

7478

1.388

.669

.167

6

1848

6.411

2.706

.676

47

7814

1.867

.683

.171

8

1829

6.467

3.784

.688

48

7162

1.396

.696

.174

10

1810

6.626

8.768

.691

49

7016

1.425

.712

.178

12

1791

6.688

3.798

.686

60

6876

1.464

.727

.182

14

1772

6.648

3.831

.706

61

6741

1.488

.741

.186

16

1764

6.701

2.850

.718

62

6611

1.518

.767

.189

18

1786

6.760

. 2.880

.790

63

6486

1.542

.771

.193

20

1719

6.817

2.906

.787

64

6366

1.571

.786

.197

22

1702

6.876

2.987

.784

66

6251

1.600

.800

.200

24

1686

6.986

2.967

.748

66

8139

1.629

.816

.204

26

1669

6.998

2.906

.748

67

6031

1.668

.829

.207

28

1668

6.060

8.095

.756

. 68

5927

1.687

.844

.211

80

1637

6.106

8.064

.794

69

6827

1.716

• .858

.214

32

1622

6.166

8.068

.771

1

5730

1.745

.872

.218

34

1607

6.828
6.881

8.118

.778

2

5545

l.SW)

.902

.226

36

1592

8.140

.785

4

5372

1.862

.931

.233

38

1677

6.841

8.170

.798

6

5209

1.920

.960

.240

40

1668

6.806

8.U9

.800

8

.5056

1.978

.969

.247

42

1649

6.466

8.828

.807

10

•4911

2.036

1.018

.256

44

1686

6.615

8.867

.814

12

4775

2.094

1.047

.262

46

1621

6.575

8.887

.889

14

4646

2.162

1.076

.269

48

1606

6.681

8.816

.888

16

4523

2.211

1.106

.276

50

1486

6.689

8.845

.88$

18

4407

2.269

1.134

.284

52

1482

6.748

8.874

.8a

90

4297

2.327

1.163

.291

64

1460

6.807

8.408

j&a

22

4192

2.386

1.192

.298

66

1467

6.868

8.488

.868

24

4093

2.448

1.221

.306

68

1446

6.980

8.460

.885

96

3997

2.602

1.251

.813

4

1488

6.960

8.480

.818

28

3907

2.560

1.280

.320

6

1408

7.185

8.6CS

.891

80

3820

2.618

1.309

.327

10

1876

7.IT1

SJ85

.998

88

8737

2.676

1.338

.834

16

1848

7.416

8.IQ6

Ml

84

3667

2.734

1.367

..H42

20

1889

7.561

8.781

jm

Fable of KwUI, Hlddle Ordlaalea, Ae, of Oarreo. Obord IDC

(CoDllDUsd.)

Ths TugentUI Auile 1* iJwi^a ans-hclT of th« Angl< of DallKtloii.

To fflnd lane

lan»ntlBl nnd deflection aniclea

„^„.-. ^.■i^ftafflljochord bjths r»d. Th» qqot will
ug. Find .hli I«.«l »ng )n the ULlB - - ■
To aud tbe der dint for «

ebords lOO f[ lonK- Di' 10000 by tb»

To ■nd the def dtot for eqaal ehorda of say rive" lenstb.

DlTChaid bfrad, MuU qaot br cbord. Ordlr sq of chord b; rad.
To And th« taiigl disl for eanHl rhordaof any K'ven lenrtb.

Ftnt Hod Cb« laoEl sog u above. Divide it bv ^. Find iu tlie Ijible of unt alnea
thanatBlneofthequot, Mult this mt sine by Ihagirem^hord. Mult prod bj 2.
To Bod ih« rKd Ui any Kl*^'! d«f anc. for equal cbords of aar
length. Ulrlde the def aag br 2. Find oat line oribe quoUent. DlTide b*U
the cliotil bf tbia mt line.

* The middle ordinate far a nd of 600 ft

{cboid 100 rt,) ma;

7o6

CIRCULAR CURVES.

llAdllt Ae, of Curires; In metres. Chord, 20 metres ^ %

d ek ametres.

The stakes, at the ends of the 2-dekametre chords, should be numbered 2, 4, 6, Ac;
meaning '2, 4, 6, Ac, dekametres. The tangential angle in the table will then giTS
the amount of deflection per unit (dekametre) of measurement.

•

s

-a

ii

II

11

■ 9>

i
s

It

•

ii

^2

S9

^a

Sx

S!4

^

S3

143.36

»a

S5<

O

H

M

o
.058

.029

99

o

H

2.790

1.396

a

0° icy

0° 6'

6875.50

.007

8°

0'

4°

0'

.349

20

10

3437.75

.116

.058

.015

10

6

140.44

2.848

1.426

.356

30

16

2291.84

.175

.087

.022

20

10

187.63

2.906

1.454

.364

40

20

1718.88

.233

.116

.029

SO

16

134.94

2.964

1.483

.371

60

26

1375.11

.291

.145

.036

40

20

132.35

3.022

1.512

.378

lo 0'

30

1146.93

.349

.175

.044

50

26

129.85

3.080

1.541

.386

10

35

982.23

.407

.204

.051

HO

C

30

127.45

3.138

1.570

.39S

20

40

859.46

.466

.233

.058

10

36

126.14

3.196

1.599

.400

30

46

763.97

.524

.262

.066

20

40

122.91

3.264

1.629

.407

40

.60

687.57

.582

.291

.073

30

46

120.76

3.312

1.668

.416

60

66

626.07

.640

.820

.080

40

60

118.68

3.370

1.687

.422

go (y

1° 0'

572.99

.698

.349

.087

60

66

116.68

3.428

1.716

.42»

10

6

528.92

.766

.878

.096

10°

0'

50

O'

114.74

8.486

1.745

.487

20

10

491.14

.814

.407

.102

20

10

111.06

3.602

1.803

.461

30

16

468.40

.873

.430

.109

■

40

20

107.68

3.718

1.861

,4M

40

20

429.76

.981

.466

.116

11©

0'

80

104.83

3.834

1.919

.480

60

25

404.48

.989

.494

.124

20

40

101.28

3.950

1.977

.4M

ap (y

30

382.02

1.047

.524

.131

40

60

98.39

4.065

2.036

.609

10

36

361.91

1.105

.553

.138

12P

C

6°

fy

96.67

4.181

2.093

.524

20<

40

343.82

1.163

.582

.145

20

10

93.09

4.297

SL162

.639

30'

45

327.46

1.222

.611

.153

40

20

90.65

4.418

2.210

.653

40

50

312.68

1.280

.640

.160

199

(y

.

30

88.:«

4.628

2.268

.668

•60

56

298.99

1.338

.669

.167

20

40

86.14

4.644

^.326

.682

4<» 0'

2p iy

286.54

1.396

.698

.175

40

60

84.05

4.760

2.384

.607

10

6

275.08

1.454

.727

.182

14°

0'

70

O'

82.06

4.875

2.442

.912

20

10

264.51

1.512

.756

.189

20

10

80.16

4.900

2.500

.686

30

16

254.71

1.570

.786

.196

40

20

78.34

6.106

2.558

.641

40

20

246.62

1.629

.814

.204

15°

0'

SO

76.61

5.221

2.616

.666

60

25

237.16

1.687

.844

.211

20

40

74.96

5.336

2.674

.670

5° (y

30

229.26

1.746

.873

.218

4(J

60

73.37

6.452

2.732

.686

10

35

221.87

1.803

.902

.225

16°

0'

8°

(y

71.86

6.667

2,790

.9»

20

40

214.94

1.861

.931

.233

20

10

70.40

6.682

2.848

.714

80

46

208.43

1.919

.960

.240

40

20

69.00

6.797

2.006

.729

40

60

202.30

1.977

.989

.247

17°

O'

80

67.65

5.912 2.9641

.748

60

66

196.53

2.036

1.018

.256

20

40

66.36

6.027

3.022

.758

6° (y

39 0'

191.07

2.093

1.047

.262

40

50

66.12

6.142

3.080

.ni

10

5

185.91

2.152

1.076

.269

18°

O'

0°

<y

63.92

6.267

3.138

.787

20

10

181.03

2.210

1.105

.276

20

«

10

62.77

6.372

3.196

.802

30

16

176.39

2.268

1.134

.284

40

20

61.66

6.487

3.-254

.816

40

20

171.98

2.326

1.163

.291

10°

O'

80

60.59 6.602

3.312

.831

60

26

167.79

2.384

1.192

.298

20

40

59.55

6.717

3.370

.846

r> (y

30

163.80

2.442

1.222

.306

40

60

58.66

6.831

3.428

.860

10

36

160.00

2.500

1.251

.313

20°

C

10°

0'

57.59

6.946

3.486

.875

20

40

156.37

2.558

1.280

.320

21°

(/

30

64.87

7.289

3.660

.919

30

45

152.90

2.616

1.309

.327

22°

0'

11°

0'

62.41

7.632

3.834

.963

40

50

149.58

2.674

1.338

.335

2S°

0'

30

60.16

7.976

4.008

1.007

60

65

146.40

2.732

1.367

.342

24°

<y

12°

O'

48.10

8.816

4.181

1.061

25°

(y

30

46.20

8.658

4.355

1.0M

Radios

Half the chord

- = Half the chord y «°8ecaiit of tangvntial

Sine of tangential angle "*" *"* *^^"™ ^ angle.

Deflection dist = ^^"'^^;J.^^^^"^ « Twice the chord X •»'"* ^Li^f*"^

Tanarentlal dIst =» Twice the chord X sine of half the tangential angle.

Midale ord =» Radius X (1 — cosine of tangential angle) => Half the chord X
tangent of half the tangential angle.

For cnrves of 60 metres, or greater, radius, tbe ordinate at 0 metres trom
the end of the 20-metre chord, or midway between the end of the chord and the mid*
41e ordinate, may be taken at three-fourths of the middle ordinate.

TABLE OF liONG CHORDS.

787

Table of Iiony ClioMls.

Len^^lis of Chord in ft| required to subtend from 1 to 4 stations of 100 ft eaeh.

2 8ta.

20O.O
200.0
200.0
200.0
200.0
200.0
200.0
200.0
200.0
200.0
200.0
200.0
199.9
199.9
199.9
199 9
199.9
199.8
199.8

3 8ta.

800.0
800.0
800.0
300.0
299.9
299.9
299.8
289.8
299.7
299.7
299.6
299.6
299.6
299.5
299.4
299.3
299.2
29».l
299.0

4 8ta.

Ang.

of

Defl.

ISta.

2St8L

3Sta.

400.0

%

100

199.7

298.9

899.9

&>

100

199.7

298.8

399.9

/4

100

199.7

298.7

399.8

Iz

100

199.7

298.6

399.7

7«

100

199.6

298.5

899.6

7<^

100

199.6

298.4

399.5

1^

100

199.6

•^98.3

399.4

iz

100

199.6

298.2

899.3

yi

100

199.6

298.1

399.2

9P

100

199.6

298.0

399.1

^

100

199.6

297.9

399.0

za

100

199.6

297.8

398.9

74

100

199.4

297.7

898.7

9°

100

199.4

297.8

398.6

^

100

199.4

297.4

398.3

k

100

199.3

297.8

398.0

100

199.2

297.2

397.8

\(fi

100

199.2

297.0

897.6

4StA.

397.5
897.3
897.0
396.7
396.6
396.2
896.0
396.7
395.4
896.1
394.8
894.5
394.3
394.1
393.7
393.2
392.8
392.4

£levatl4»ii of outer rail in curyes theoretically is equal in ins to (square
of Tel in I't per sec X §auee in ins) -*- (Rad of curve in ft X 32.2). Experience
has shown that half an incn for each degree of def angle (100 ft chords) does yery
well for 4 ft 8.6 ins gauge up to 40 miles per hour. At 60 miles use 1 inch per deg.
In dangerous places this may be increased for safety against high winds. Ap-
proaching the curye raise the outer rail at the rate of 1 inch in about 60 or 80 ft.

When the ends of a curve are tapered off* by transition curves, the rise is made
upon the latter.

Relation of radius to lengrth of wbeel-base.

Mr. A. M. Wellington * found by experiments with models that a rigid truck
passing around a curve, whether alone or
coupl^ with another truck, assumes
the position shown in this Fig.,t i.e., the
flange of the outer front wheel presses
against the outer rail, and the rear axle
coincides with a radius to the curve.
Then, for the angle A, between that
radius and the radius K which passes
through the center of the front axle :

wheeKbaseB

sine of A = rr — r^— ;

radius R

and the space d between the flange of the outer hind wheel and the outer rail is,

d = radius R X versed sine of angle A, very nearly.

For a given wheel-base B, we have, approximately,

d =■ (d for a 1° curve) X degrefe of curvature ;

and the inner hind wheel will touch the inner rail when d becomes equal to the

total room for play left between the wheel-flanges and the rails, i. e., when

_ - ^ total play

degree of curvature = t-, ri-—

° ofor a 1" curve

This commonly occurs on European railways, where the cars have rigid wheel-
bases much longer than our pivoted trucks, and where d for a given radius is
therefore much greater than with us. Hence the inner rails on curves are more
generally worn there than here.

For a wheel-base 5 ft. long, " d for a 1° curve " is 0.0022 ft. It varies (nearly) as
the square of the length of the wheel-base, d is independent of the gauge.

♦ The Economic Theory of Railway Location, New York, John Wiley <l Sons, 1887.

f In our figure, necessarily much exaggerated, we omit, for simplicity, the treads
of the wheels, all of which are supposed to rest on the rails, and show only so much
of their flanges as extends below the top of the raiL

788

TABLE OF ORDINATEB.

Table of Ordlnatea 5 fU apart. Cliord lOO fL

For Bailroad Ourreit

Ordinatofl for angles intermediate of those in the table can at once be found hj
•imple proportion.

DisUiUMs of the OrdlnatM from tb« end of th« 100 f»«t Ohord.

ADff. of I Mid.
Defl. 50 ft.

S

8

4
8
13
16
30
34
38
13
86
40
44
48
53
56

4
8
13
16
30
34
38
33
86
40
44
48
53
56

4
8
13
16
30
34
38
83
36
40
44
48
53
56

4
8
13
16
30
34
38
83
86
40
44
48
53
56

10
30
30
40
50

10
SO

so

.014
.030
.043
.068
.073
.067
.103
.116
.131
.145
.160
.174
.189
.304
.318
.333
.347
.363
.876
.391
.306
.330
.334
.HA9
.364
.378
.393
.407
.422
.436
.461
.465
.480
.495
.509
.523
.538
.^2
.567
.583
.596
.611
.635
.640
.654

.683

.713

.727

.743

.756

.771

.786

.800

.814

.829

.843

.858

.873

.909

.945

.981

1.017

1.054

1.091

1.137

1.164

1.200

.014
.029
.043
.058
.073
.086
.101
.115
.130
.144
.158
.172
.187
.203
.316
.331
.346
.360
.374
.288
.803
.317
.331
.345
.360
.374
.389
.403
.418
.433
.446
.461
.475
.490
.504
.618
.538
.547
.663
.676
.590
.606
.619
.684
.648
.663
.677
.601
.706
.730
.784
.749
■768
.777
.799
.806
.831
.836
.850
.864
.900
.986
.973
1.008
1.044
1.080
1.116
1.168
1.188

.014
.038
.041
.066
.070
.068
.096
.113
.136
.140
.168
.167
.181
.195
.309
.333
.387
.363
.365
.279
.398
.807
.331
.886
.349
.363
.877
.891
.406
.419
.488
.447
.461
.476
.489
.608
.617
.681
.646
.659
.673
.687
.001
.616
.639
.648
.657
.671
.686
.689
.718
.727
.741
.766
.769
.783
.797
.811
.836
.889
.874
.909
.944
.979
1.014
1.048
1.088
1.118
1.168

86rt. {

sort.

36 ft.

.018 !

.013

.010

.036 1

.034

.033

.038 :

.087

.088

.063 i

.049

.044

.066 !

.061

.066

.077

.074

.066

.003

.066

.077

.106

.098

.088

.119

.110

.099

.138

.133

.110

.146

.185

.131

.168

.147

.183

.171

.159

.148

.186

.171

.154

.198

.188

.164

.311

.196

.176

.334

.906

.186

.387

.330

.196

.361

.383

.307

.364

.344

.318

.377

.356

.339

.391

.369

.340

.304

.381

.361

.317

.398

.368

.880

.806

.378

.348

.818

.384

.866

.880

.396

.870

.343

.806

..388

.864

.S16

.897

.866

.837

.409

.879

.886

.436

.891

M»

.487

.403

.860

.460

.416

.871

.468

.438

.883

.476

.440

.808

.489

.463

.404

.508

.466

.416

.516

.477

.436

.529

.489

.486

.642

.601

.447

.656

.618

.468

.660

.536

.409

.583

.588

.480

.696

.560

.491

.608

.563

.603

.631

.574

.613

.686

.687

.638

.649

.699

.684

.663

.611

.646

.676

.628

.666

.688

.686

.667

.703

.648

.678

.715

.600

.689

.738

.678

.600

.741

.686

.611

.764

.607

.631

.768

.709

.683

.781

.731

.648

,794

.784

.666

.837

.764

.683

.860

.796

.TOO

.898

.836

.786

.936

.866

.764

.959

«{K9D

.791

.998

.917

.618

1.096

.947

.846

1.068

.978

.879

1.093

1.009

.900

I

.008
.018
.038
.087
.047
.066
.066
.076
.064
.008
.108
.113
.133
.181
.140
.160
.169
.168
.177
.187
.197
.306
.316
.334
.383
.343
.361
.361
.370
.380

.386
.808
.317
.836
.884
•S4o
.365
.364
.378
.383
JI91
.401
.410
.419
.488
.486
.448
.467
.466
.476
.486
.494
.506
.618
•531
.681
.641
.660
.669
.682
.606
.639
.668
.676

.788
.746
.769

.008
.015
.033
.090
.087
.046
.063
.058
.066
.074
.061
.068
.096
.103
.111
.118
.136
.133
.140
.148
.166
.168
.171
.178
.186
.193
.900
.306
.316
.233
.380
.337
.346
.353
.360
.267
.375
.383

.397
.804
.313
.319
.326
.834
.841
.849
.857

.871
.378
.886
.894
.401
.408
.416
.438
.481
.488
.446
.464
.483
.601
.619
.688
.667
.676

.618

.006
.010
.016
.030
.036
.061
.036
.043
.047
.063
.067
.063
.068
.078
.078
.0H3
.088
.094
.099
.104
.109
.114
.130
.126
.130
.136
.141
.147
.152
.167
.163
.167
.178
.178
.188
.188
.194
.199
.304
.309
.314
.319
.336
.380
.386
.340
.346
.361
.367
.363
.367
.378
.378
.383
.388
.298
.398
.304
.800
.814
.837
.840
.864
.867
.880

.406
.419
.483

6fl.

.009
.006
.006
.011
.014
.017
.019
.033
.014
.OST
.000

.on

.006
.068
.041
.043
.046
.049
.063
.066
.067
.060
.068
.066
.060
.073
.075

.on

.060
.088
.086
.068
.090
.008

.103
.104
.lOT
.110
.118
.110
.118
.131
.134
.137
.180
.183
.186
.188
.141
.144
.146
.149
.163
.166
.166
.160
.168
.1«
.178
.179

.im

.1«

.m

.814

TABLE OF ORDINATE8.
Tteble of Ordiiiai«s 5 tt apart. — (Oontlnaed.)

789

DlBtano*! of (he OrdiuafeM from the end of the 100 feet Ohord.

Aug. of
Ded.

Mid.
5Cft.

46 n.

40 fL

stn.

80 ft.

86 ft

80 fi.

16 ft.

ion.

6fl.

0 '

640

1.386

1.334

1.188

1.124

1.089

.937

.793

.631

.446

.236

60

1.373

1.260

1.228

1.157

1.070

.964

.816

.649

.458

.241

6

1.300

1.396

1.268

1.191

1.100

.982

.Kt9

.668

.472

.248

19

1.S46

1.338

1.393

1.234

1.130

1.009

.863

.666

.485

.256

»

1.383

1.S68

1.338

1.256

1.161

1.036

.886

.706

.498

.362

SO

1.419

1.404

1.362

1.290

1.192

1.064

.909

.724

.511

.368

40

1.465

1440

1.397

1.32S

1.322

1.091

.932

.742

.624

.376

50

1.491

1.476

1.433

1.356

1.253

1.118

.956

.761

.537

.283

7

1.638

1.513

1.467

1.389

1.284

1.146

.979

.779

.561

.290

10

1.664

1.548

1.503

1.422

1.314

1.173

1.002

.798

.564

.397

ao

1.600

1.584

1. 637

1.464

1.345

1.300

1.026

.816

.676

.804

90

1.637

1.630

1.673

1.488

1.375

1.338

1.048

.886

.590

.811

40

1.678

h656

1.607

1.521

1.406

1.366

1.071

.864

.603

.318

60

1.710

1.693

1.641

1.553

1.436

1.383

1.096

.872

.616

.324

8

1.746

1.738

1.677

1.587

1.467

1.310

1.118

.891

.629

.888

SO

1.665

1.836

1.783

1.687

1.569

1.392

1.188

.946

.669

.353

9

1.966

1.944

1.886

1.787 •

1.661

1.474

1.268

1.003

.708

.373

SO

3.074

2.063

1.991

1.887

1.742

1.566

1.328

1.057

.748

.394

10

3.188

3.161

3.096

1.987

1.834

1.637

1.396

1.114

.787

.416

«>

3.388

3.269

3.301

2.087

1.996

1.719

1.468

1.1T0

.827

.436

11

3.401

3.377

3.306

2.186

3.018

1.802

1.638

1.336

•OOo

^467
.478

so

2.511

3.486

3.411

2.286

2.110

1.884

1.609

1.382

.906

»

3.630

3.694

3.616

2.886

3.303

1.967

1.681

1.339

.946

.499

so

3.7S0

3.708

3.621

2.486

3.386

2.049

1.760

1.396

.966

.680

IS

2.889

2.811

2.726

2.585

3.887

3.132

1.820

1.461

1.025

.641

so

3.949

2.920

3.833

2.685

3.479.

8.214

1.891

1.607

1.065

.663

li

3.068

3.028

3.937

2.786

3.571

2.387

1.961

1.664

M06

.588

so

8.168

3.136

3.043

2.884

3664

2.379

2.031

1.620

1.144

.604

16

3.377

3.246

3.147

2.984

3.756

8.463

3.103

1.676

1.184

.636

so

3.387

3.364

3.353

3.084

3.848

3.644

3.173

1.732

1.224

.646

16

3.496

3.462

3.358

3.184

3.941

3.637

3.343

1.789

1.264

.667

17

3.716

3.680

3.669

3.884

8.136

3.793

3.384

1.902

1.844

.709

18

3.936

3.897

3.779

3.584

8.310

8.968

3.526

8.014

1.424

.761

19

4.166

4.115

3.990

3.784

3.495

•S.13S

2.666

3.137

1.504

.793

so

4.376

4.332

4.301

3.984

3.680

3.388

2.808

3.240

1.583

.836

32

4.816

4.768

4.624

4.386

4.050

S.630

3.093

2.467

1.744

.933

U

6.866

6.204

6.048

4.789

4.423

3.963

3.379

2.696

1.905

1.006

36

6.697

6.642

6.473

5.193

4.798

4.386

3.665

3.934

2.068

1.094

88

6.189

6.079

5.898

5.695

6.171
6.514

4.632

3.963

8.164

2.232

1.181

SO

6.683

6.517

6.323

5.999

4.968

4.239

S.3AA

2..H86

1.26S

S3

7-037

6.967

6.751

6.406

5.932

6.397

4.630

3.619

2.565

1.36»

34

7.473

7.396

7.179

6.813

6.300

5.637

4.832

3.864

2.733

1.446

S6

7.918

7.841

7.609

7.222

6.679

6.978

6.116

4.090

i.901

1.536

S8

8.867

8.286

8.041

7.633

7.060

6.390

5.410

4.337

3.U69

1.686

40

6.816

8.731

8.474

8.044

7.442

6.668

6.706

4.666

3.238

1.718

Gaag^e on cnrTes. Let B = radius of wheel from center

to tread, F — depth of flange of wheel, and L = the
length of that portion of the wheel-flange which ex-
tends below the top of the rail, = 2 |/(R + l7» — R*,
all in inches. Then if, on the curve, we widen the
gauge by a quantity, in inehe*^ =

_ . L,/g<<-f length of rigid wheel-base, ft.

Q — L., \ns. X gnQge^ ft. + 2 X rad. of the curve, ft. '

the wheels will have approximately the saine play
on the curve as on the tangent. For a rigid wheel-
base 14 ft. long and drivers 4 ft. diam., with Vyi inch
flanges, Q is about 0.02 inch (= one-fiftieth of an io.) for each degree of curvature.
Many roads use the same gau^e on curves as on tangents. Others widen the
gauge on curves by from one thirty-second to one-eighth inch for each degree of
curvature, seldom, however, exceediug 1 inch as a maximum. In Philadelphia
the Pennsylvania Railroad has freight-car sidings of 60 feet radius ; track gauge
(same on curves as on tangents), 4 It. 9 ins. ; standard wheel-gauge, 4 ft. 8^ ins.

*■ Except for very sharp curves and for very short wheel-bases with large wheels
and deep flanges, it is amply approximate to say

_ . , - T 1 . v w wheel-base in feet

Q in inches = L in mches X oTT — :; — :; 1— =■•

2 X rad. of curve in ft.

790

LEVEL CUTTIKGS.

To prepare a Table, T, of Iievel (TnitlnffS) tor every ^
foot of helfflil;, or deptli.

of a

4-^

Let the fig repreeent the catting ; or, If InTeirtid,
the flIUng ; in whioh the horisontal lines are evp-

poMd to be JL- foot apart. Firit oalcalate tbt

area in square feet, of the layer a & co, adjoinini

the roadway a b. Then llnd how many oaUe

yards that area gi^es in a distance of 100 flseC

J) O CI. These eabic yards we will call T; they form ths

first amount to be put into the Table T.
Kezt caleolate the area in square feet of the triangle a no. Moluply this area by 4. Find hev
many cubic yards this increased area gives in a distance of 100 fteet. Or they will be found read;
calculated below. We will call them y. This is all the preparation that is needed before

oommenciag the table.

fixam^-Let the roadbed a b be 18 feet, and the stde'Slopee 1^ to 1. Then for the area of a I e •:
sinoe the side-slopes are 1^ to 1 ; and • t Is .1 foot; e o must be 18..S feet; and the mean length d
a 6 CO must be 18.16 feet. Consequently, the area is 18.15 X .1= 1-816 square feet; whioh, in a

181 ft

distanoe of 100 feet, glres 181.6 cubic feet ; whioh is equal to -—^ =6.7828 cnbid yards ; or T.

Next, as to the triangle ano: its height a n being .1 foot, and its base n o .15 feet ; its ares
= ' — r-^ — = ' — =r.0076 square ft. This multiplied by 4, gives .08 square feet ; which, in a distanoe «f
100 fiset, gives .OS X 100 = 9 cubic feet; which is equal to -- = .1111 cubic yard; or y.

Having thus found Y and y, proceed to make out the table in the manner following, which Is st
plaifl as to require no explanation. The work should be tested about every 6 fbet, by ealoulating tfai
area of the full depth arrived at ; multiply it by 100, and divide the product by 87 for the oable yards
The cubic yards thus found should agree with the table.

X eeee

y

... V. I^A^

... .1111

y

6.8333
... .1111

y

6.9444
... .1111

y

7.0555
... .1111

y

7.1666
... .1111

.. Y. 6.722 .1

6.8333

13.5555 .2

6.9444

•

20 5000 .3

7.0555

27.5555 .4
7.1666

7.2777

34.7222 .5

7.2777

42.0000 .6

Tabli T

Height.
Fe«t.

.1
.2
.3
.4
.5
.6

Cub. Yds.

Ao.

6.72 Y.
13.6
20.6
27.6
34.7
42.0

Th« following table containB y, ready calculated for different side-slopes. It plain]|
remains the same for all widths of roadbed.

Side-slope.

y

Side-slope.

y

i^ to 1

0185

0870

.0741

.0926

nil

l^tol

2 tol

jft^tol

1296

x| to 1

1482

5i to 1

iflOT

1 to 1

2j2 to 1

8 tol

4 tol

.lgB2

IW to 1

2222

1)1 to 1

296S

RAILBOADB.

791

Table 1. liCTel Cnttinss.*

Boadway 14 feet wide, side-slopes 1^ to 1.
For liiigle-traok embankment.

Hfllfht

lAft.

.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

Oa.Tds.

Oa.TdA.

Oa.Tdi.

Ga.Yda.

Oa.Td«.

Ott.Yds.

Ou.Yds.

Ou.Tdfc

Oa.Tili.

Oa.TdB.

0

6.24

10.6

16.1

21.6

27.3

33.1

390

45.0

51.2

1

67.4

63.8

70.2

76.8

83^

90.3

97.2

104.2

111.3

118.6

2

1259

133.4

141.0

148.6

156.4

164.4

172.4

180.5

188.7

197a

3

205.6

214.1

222.8

231.6

240.5

249.5

258.7

267.9

277.3

286.7

i

296.3

306.0

315.8

325.7

336.7

345.8

366.1

866.4

376.9

387.5

5

398.1

408.9*

419.9

430.9

442.0

453.2

464.6

476.1

487.6

. 499.3

6

511.1

523.0

535.0

547.2

559.4

571.8

584.2

596.8

609.5

622J)

7

635.2

648.2

661.3

674.6

687.9

701.4

714.9

728.6

742.4

756^

8

770.3

784.5

798.7

813.1

827.5

842.1

856.8

871.6

886.5

901.6

9

916.7

931.9

947.3

962.7

978.3

994.0

1010

1026

1042

1058

10

1074

1090

1107

1123

1140

1157

1174

1191

1208

1225

11

1243

1260

1278

1295

1313

1331

1349

1367

1385

1404

12

1422

1441

1459

1478

1497

1516

1635

1564

1574

1593

13

1613

1638

1652

1672

1692

1712

1733

1753

1773

1794

14

1815

1885

1856

1877

1898

1920

1941

1962

1984

2006

15

2028

2050

2072

2094

2116

2138

2161

2183

2206

2229

16

2262

2275

2298

2321

2344

2368

2391

2415

2439

2463

17

2487

2511

2535

2559

2584

2608

2633

2658

2683

2708

18

2733

2759

2784

2809

2835

2861

2886

2912

2938

2964

19

2991

3017

3044

3070

3097

3124 •

3151

3178

3205

3232

20

8259

3287

3314

3342

3370

3398

3426

3454

8482

3510

21

3539

3567

3596

3625

3654

3683

3712

3741

3771

3800

22

3830

3859

3889

3919

3949

3979

4009

4040

4070

4101

23

4132

4162

4193

4224

4255

4287

4318

4349

4381

4413

24

4444

4476

4508

4541

4573

4605

4638

4670

4703

4736

25

4769

4802

4835

4868

4901

4935

4968

6002

5036

6070

26

5104

5138

5172

5206

5241

5275

5310

5345

5380

5415

27

5450

5485

5521

5556

5592

5627

5663

5699

5735

5771

28

6807

5844

5880

6917

5953

5990

6027

6064

6101

6139

29

6176

6213

6251

6289

6326

6364

6402

6440

6479

6517

80

6556

6594

6633

6672

6711

6750

6789

6828

6867

6907

81

6946

6986

7026

7066

7106

7146

7186

7226

7267

7307

82

7348

7389

7430

7471

7512

7553

7595

7636

7678

7719

88

7761

7803

7846

7887

7929

7972

8014

8057

8099

8142

84

8185

8228

8271

8315

8.358

8401

8445

»89

8532

8576

85

8620

8664

8709

8753

8798

8842

8887

8932

8976

9022

86

9067

9112

9157

9203

9248

9294

9340

9386

9432

9478

87

9524

9570

9617

9663

9710

9757

9804

9851

9898

9945

88

9993

10040

10088

10135

10183

10231

10279 •

10327

10375

10424

80

10472

10521

10509

10618

10667

10716

10765

10816

10864

10913

40

10963

11013

11062

11112

11162

11212

11263

11313

11364

11414

41

11466

11516

11567

11618

11669

11720

11771

11823

11874

11926

42

11978

12020

12081

12134

12186

12238

12291

12343

12396

12449

48

12502

12555

12608 12661

12715

12768

12822

12875

12929

12983

44

13037

13091

1.3145

13200

13254

13309

13363

13418

13473

13528

45

13583

13639

13694

13749

13806

13861

13916

13972

14028

14084

46

14141

14197

14254

14310

14367

14424

14480

14537

14595

14652

47

14709

14767

14S24

14882

14940

14998

l.iOSO

15114

15172

15230

48

15289

15847

15400

15465

155'24

15583

15642

15701

15761

15821

40

15880

15030

15999

16059

16119

16179

16239

16300

16360

16421

50

16481

16542

16603

16664

16725

16787

16848

16909

16971

17038

51

17094

17156

17218

17*280

17343

17405

17467

17530

17593

17656

52

17719

17782

17846

17908

17971

18035

18098

18162

18226

18290

68

18854

18418

18482

18546

186U

18675

18740

18805

18S70

18935

54

19000

19065

19131

19196

19262

19327

19393

19459

19525

19591

56

19657

19724

19790

19857

19923

19990

20057

20124

20191

20259

56

20326

20393

20461

20529

20596

20664

20732

20800

20869

20937

67

21005

21074

21143

21212

21280

21349

21419

21488

21557

21627

58

21696

21766

21836

21906

2l97fi

22046

22116

22186

22267

22327

50

2S808

22469

22540

22611

22682

22753

22825

22896

22968

23039

60

23111

28183

23255

23327 123399

23472

23544

23617

23689

'23762

4 From the Authoin " Measuretuent and Cost of Earthwork."

792

RAILROADB.

Table 3. I^evel Caittncs.

Roadway 24 feet wide, side-slopea 1^ to 1.
For donble-traok embankment.

Heifht
teFt.

.0

.1

.2

.8

.4

.6

.6

.7

.8

.9

Cn.Yda.

Cn.Y<U.

Oa.Tda.

Gu.Tda.

Go.Yda.

Oa.Tda.

Oii.Td*.

Ca.Tds.

Co.Tda.

Ca.Tds.

0

8.94

18.0

27.2

36.4

46.8

66.3

64.9

74.7

84.6

1

94.4

104.6

114.7

124.9

135.3

' 145.8

166.4

167.2

178.0

188.0

2

200.0

211.2

222.4

233.8

245.3

256.9

268.6

280.5

292.4

804.4

8

316.6

328.9

841.2

863.7

366.3

879.0

391.9

4048

417.8

431.0

4

444.4

467.8

471.3

484.9

498.6

612.4

626.4

640.4

554.0

608.8

6

683.3

697.8

612.4

627.1

642.0

656.9

671.9

687.1

702.3

717.7

6

733.3

748.9

764.7

780.6

796.4

812.5

828.7

844.9

861.3

877.8

7

894.4

911.2

928.0

044.9

962.0

979.2

996.4

1014

1031

1049

8

1067

1085

1102

1121

1139

1157

1175

1194

1212

1231

9

1250

1269

1288

1807

1326

1346

1365

1385

1405

1426

10

1444

1465

1485

1605

1525

1546

1666

1687

1608

1020

11

1650

1671

1692

1714

1735

1767

1779

1800

1822

1846

12

1867

1889

1911

1934

1956

1979

2002

2026

2048

20T1

18

2094

2118

2141

2165

2189

2213

22S6

2261

2286

2309

14

2333

2368

2382

2407

2432

2467

2482

2607

2682

2568

16

2688

2609

2635

2661

2686

2718

2739

2766

2791

2818

10

2844

2871

2898

2025

2962

2979

8006

8034

8061

aosk

17

3117

3145

3172

3201

8229

8267

3286

3814

8342

3871

18

3400

3429

8468

3487

8616

8646

8676

8605

8636

3065

»

3694

3726

3756 •

3785

8816

3846

3876

3907

8988

3900

20

4000

4031

4062

4094

4126

4167

4189

4221

4262

4286

21

4317

4349

4881

4414

4446

4479

4512

4646

4678

4011

22

4644

4678

4711

4745

4779

4813

4846

4881

4915

4049

23

4983

6018

5052

6087

6122

6167

6192

6227

5202

6296

24

5333

5369

5405

6441

6476

6613

6649

6686

6621

6068

25

5694

6731

6768

5806

5842

6879

6916

6064

6991

0029

26

6067

6105

6142

6181

6219

6257

6295

6384

0872

6411

27

6450

6489

6628

6667

6606

6646

6686

0726

0766

0806

28

6844

6886

6925

6965

7005

7046

7086

7127

7108

7»9

20

7250

7291

7332

7374

7416

7467

7499

7641

7682

7626

SO

7667

7709

7761

7794

7836

7879

7922

7965

8008

8061

31

8094

8138

8181

8225

8269

8313

8366

8401

8446

8480

82

8533

8578

8622

8667

8712

8767

8802

8847

8892

8938

33

8083

9029

9075

9121

9166

9212

9269

9805

9361

9398

34

9444

9491

9538

9585

9632

9679

9726

9774

9821

9860

35

9917

9965

10012

10061

10109

10167

10205

10254

10802

10351

86

10400

10449

10498

10647

10596

10646

10696

10746

10796

10846

37

10894

10945

10995

11045

11096

11146

11196

11247

11298

11349

38

11400

11461

11502

11554

11606

11667

11709

11761

11812

11806

30

11917

11960

12021

12074

12126

12179

12232

12286

12838

12391

40

12444

12498

12551

12605

12659

12713

12766

12821

12876

12029

41

12983

13038

13092

13147

13202

13267

13312

13367

13422

1347P

42

13633

13589

13645

13701

13756

13818

13869

13925

13981

1403^

43

14094

14151

14208

14265

14322

14879

14436

14494

14551

14009

44

14667

14725

14782

14840

14899

14957

16016

15074

16132

16191

45

15250

15309

15368

15427

15486

15646

16605

15666

16726

16786

46

15844

15905

15965

16025

16086

16146

16206

16207

103-28

10380

47

16450

16511

16572

16634

16695

16757

16819

16881

10942

17006

48

17067

17129

17191

17264

17316

17379

17442

17506

17668

17081

40

17694

17758

17821

17885

17949

18013

18076

18141

18206

18200

60

18333

18398

18462

185-27

18592

18667

18722

18787

18852

18018

61

18983

19049

19115

19181

19246

19313

19379

19446

19611

19678

62

196U

19711

19778

19845

19912

19979

20046

20114

■20181

20249

63

20317

20385

20452

20521

20589

•20657

20725

20794

20802

•20081

64

21000

21069

21138

21207

21276

21346

21415

21485

21565

'21026

65

21694

21765

21835

21906

21975

22046

22116

22187

22268

22829

66

22400

22471

22542

22614 22685

22757

22829

22001

22972

•28046

57

23117

23189

23261

'23334 2MQ6

23479

23552

23625

23098

23771

58

23844

23918

23991

24065 ; 241 39

24213

24286

'24861

24436

24509

50

24583

24658

24732

24807 24882

24957

25032

26107

26183

26868

60

25333

26409

25485

25561 25636

25713

25789

26866

26041

20»I8

For ooDtinuatlon M 100 foeb m« Tasli 7.

BAILBOAJ>S.

793

Table 3. I^evel CnUlnffS.

Boadway 18 feet wide, side-elopes 1 to 1.

For single-traok ezoavatioa.

DeiSth
in Ft.

.0

.1

.2

.3

.4

On.Tdi.

GD.Tdi.

CB.Ydi.

0«..Yds.

Oa.Tda.

0

6.70

13.5

203

27.3

1

70.4

77.8

85.3

92 9

100.6

2

148.1

166.3

164.6

172.9

181.3

8

233.3

242.3

261.3

260.3

269.5

4

325.9

335.6

345.3

355.1

366.0

6

425.9

436.3

446.8

467.4

468.0

6

533.3

544.5

655.7

667.0

578.4

7

648.1

660.0

672.0

684.0

696.1

8

770.4

783.0

795.7

808.6

821.3

9

900.0

013.4

926.8

940.3

953.9

10

1037

1051

1065

1080

1094

11

1181

1196

1211

1226

1241

12

1333

1349

1365

1380

1396

13

1493

1509

1625

1642

1568

14

1650

1676

1693

1711

1728

15

1833

1861

1869

1887

1906

16

2015

2033

2052

2071

2089

17

2204

2223

2242

2262

2281

18

2400

2420

2440

2460

2481

1»

2604

2624

2646

2666

2687

20

2815

2836

2858

2880

2901

21

30:J3

3056

3078

3100

3123

22

3259

3282

3306

3328

3352

23

3493

3616

3540

3564

3588

24

3733

3768

3782

3807

3832

25

3981

4007

4032

4057

4083

26

4237

4263

4289

4315

4341

27

4600

4627

4653

4680

4607

28

4770

4798

4825

4853

4881

29

5048

6076

5105

5133

6161

30

5333

6362

5391

6420

5449

31

5626

6666

6685

5715

5745

32

5926

5956

6987

6017

6048

33

6233

6264

6296

6327

6368

34

6648

6580

6612

6644

6676

85

6870

6903

6936

6968

7001

36

7200

7233

7267

7300

7334

37

7537

7671

7605

7640

7674

38

7881

7916

7951

7986

8021

39

8233

8269

8305

8340

8376

40

8593

8629

8665

8702

8738

41

8959

8996

9033

9071

9108

42

9333

9371

9409

9447

9485

.43

9715

9753

9792

9831

9869

44

10104

10143

10182

10222

10261

46

10600

10640

10580

10620

10661

46

10904

10944

10985

11026

11067

47

11315

11356

li:i98

11440

11481

48

11733

11776

11818

11860

11903

49

12169

12202

12246

12288

12332

60

12593

12636

12680

12724

12768

51

13033

13078

13122

13167

13212

52

13481

13527

13572

13617

13663

63

13937

13983

14029

14075

14121

54

14400

14447

14493

14540

14587

65

14870

14918

14965

15013

15061

66

16348

15396

15445

l.'>493

16541

67.

15833

15882

15931

15980

16029

58

16326

16376

16425

16475

lfi526

59

16826

16876

16927

16977

17028

60

17333

17384

17436

17487

17538 1

.6

Ca.Ydi.

34.

108.

189.

278.

376.

478.

589.

708,

834.

967.

1108

1266

1412

1575

1745

1923

2108

2301

2601

2708

2923

3146

3376

3612

3856

4108

4368

4634

4908

6190

5479

5775

6079

6390

6708

7034

7368

7708

8056

8412

8775

9145

9523

9908

10301

10701

11108

11623

11945

12376

12812

13266

13708

14168

14634

15108

15690

16079

16576

17079

17590

.6

Ott.YdB.

41.3

116.1

198.4

288.0

386.0

489.6

601.3

720.6

847.3

981.3

1123

1272

1428

1692

1763

1941

2127

2321

2521

2729

2945

3168

3398

3636

3881

4134

4394

4661

4936

5218

6608

5805

6109

6421

6741

7067

7401

7743

8092

8448

8812

9183

9561

9947

10341

10741

11149

11666

11988

12418

12866

13301

13754

14214

14681

15156

16638

16128

16625

17129

17641

.7

.8

Ctt.YdB. Ga.Yds.

48.0

55.7

124.0

132.0

207.0

216.7

297.4

306.8

396.1

405.3

600.3

611.3

612.9

624.6

732.9

745.3

860.3

873.5

996.1

1009

1137

1152

1287

1302

1444

1460

1608

1625

1780

1798

1960

1978

2146

2165

2340

2360

2542

2562

2761

2772

2967

2989

3191

3213

3422

3445

3660

3685

3906

3931

4160

4185

4420

4447

4688

4716

4964

4992

6247

5276

5637

6667

6835

5866

6140

6171

6463

6485

6773

6805

7100

7133

7436

7469

7777

7812

8127

8162

8484

8520

■8848

8885

9220

9268

9600

9638

9986

10025

10380

10420

10782

10822

11191

11232'

11607

11649

12031

12073

12462

12505

12900

12945

13346

13391

13800

13845

14260

14307

147-28 •

14776

15204

15252

15687 15736

17177

16227

16675

16725

17180

17231

17693

17745

.9

Oa.Yd&

63.0

140.0

224.5

316.3

416.6

522.3

636.3

757.8

886.7

1023

1167

1318

1476

1642

1816

1996

2184

2380

2583

2793

3011

3236

3469

3709

8966

4211

4473

4743

5020

6304

5696

5S96

6202

6516

6838

7167

7503

7847

8198

8556

8922

9296

9676

10064

10460

10863

11273

11691

12116

1-2549

12989

13436

13S91

14353

14823

15300

15784

16276

16776

17282

17796

For oontiaaatioD to Mv lect deep, see Table 7.

794

BAILBOADB.

Table 4. I^evel CnttlnffS.

Roadway 18 feet, side-slopes 1^ to 1.

For aingle-traok excayation*

Depth
in Ft.

.0

.1

.2

.8

.4

.5

.6

.7

.8

.9

Ov-TdB.

Oa.TdB.

Oa.Tda.

Ca.Yd«.

Oa.Yd«.

Gu.Tda.

Oa.TdB.

Ca.Tdi.

Go.Ydji.

Cn.Tdi.

0

6.72

13.6

20.6

27.6

34.7

42.0

49.4

56.9

64i

1

72.2

80.1

88.0

96.1

104.2

112.5

120.9

129.4

138.0

146.7

2

156.6

164.5

173.5

182.7

191.9

201.3

210.8

220.4

23U.1

240i)

8

249.9

260.0

270.1

280.4

290.8

801.3

811.9

822.6

333.4

344.5

4

355.5

366.7

378.0

389.4

400.9

412.5

424.2

.436.0

448.0

460.0

5

472.2

484.5

496.9

509.4

522.0

534.7

647.6

560.6

673.6

686.7

6

600.0

613.4

626.9

640.5

654.2

668.1

682.0

696.1

710.2

724.§

7

738.9

753.4

768.0

782.7

797.6

812.5

827.6

842.7

868.0

873.4

8

888.9

904.5

920.2

936.1

962.0

968.1

984.2

1001

1017

1038

9

1050

1067

1084

1101

1118

1136

1152

1169

1187

1206

10

1222

1240

1258

1276

1294

1313

1881

1349

1368

1387

11

1406

1425

1444

1463

1482

1501

1521

1541

1660

1680

12

1600

1620

1640

1661

1681

1701

1722

1743

1764

1785

13

1806

1827

1848

1869

1891

1913

1934

1956

1978

2000

14

2022

2045

2067

2089

2112

2135

2158

2181

2204

2227

15

2250

2273

2297

2321

2344

2368

2392

2416

2440

2465

16

2489

2613

2638

2663

2688

2613

2638

2663

2688

2713

17

2739

2765

2790

2816

2842

2868

2894

2921

2947

2973

18

3000

3027

3064

3081

3108

3135

3162

3189

3217

3245

19

3272

3300

3328

3366

8384

3413

3441

8469

3498

3627

20

3566

3685

3614

3643

3672

3701

3731

8761

8790

3820

21

3850

3880

3910

3941

3971

4001

4032

4063

4094

4125

22

4166

4187

4218

4249

4281

4313

4344

4376

4408

4440

23

4472

4605

4637

4569

4602

4635

4068

4701

4734

4767

24

4800

4833

4867

4901

4934

4968

6002

5036

5070

6105

25

5139

5173

5208

5243

5278

5313

53.18

5383

5418

6453

26

6489

5626

5560

5696

5632

5668

5704

5741

5777

5813

27

6850

5887

5924

5961

5998

6035

6072

6109

6147

6185

28

6222

6260

6298

6336

6374

6413

6451

6489

6528

6667

29

6606

6645

6684

6723

6762

6801

6841

6881

6920

0960

30

7000

7040

7080

7121

7161

7201

7242

7283

7324

7865

81

7406

7447

7488

7529

7671

7613

7654

7696

7738

7780

82

7822

7866

7907

7949

7992

8035

8078

8121

8164

8207

83

8250

8293

8337

8381

8424

8468

8512

8556

8600

8645

84

8689

8783

8778

8828

8868

8913

8958

9003

9048

9093

85

9139 .

9185

9230

9276

9322

9368

9414

9461

9507

9553

80

9600

9647

9694

9741

9788

9835

9882

9929

9977

10025

87

10072

10120

10168

10216

10264

10318

10861

10409

10458

10607

88

10556

10605

10654

10703

10752

10801

10851

10901

10950

11000

89

11050

11100

11150

11200

11251

11801

11352

11403

11454

11505

40

11556

11607

11658

11709

11761

11818

11864

11916

11968

12020

41

12072

12125

12177

12229

12282

12335

12888

12441

12494

12547

42

12600

12653

12707

12761

12814

12868

12922

12976

13030

13065

43

13139

13193

13248

13303

18358

13413

18468

13523

13578

13633

44

13689

13745

13800

13856

13912

13968

14024

14081

14137

14198

46

14250

14307

14364

14421

14478

14535

14592

14649

14707

14765

46

14822

14880

14938

14996

15054

15118

15171

16229

15288

15347

47

16406

15465

15524

15583

15642

15701

15761

15821

15880

15940

48

160OO

16060

16120

16181

16241

16801

16362

16423

16484

16546

49

16606

16667

16728

16789

16851

16018

16974

17086

17098

17160

50

17222

17285

17347

17409

17472

17535

17598

17661

17724

17787

51

17850

17918

17977

18041

18104

18168

18232

18206

18860

18425

62

18489

18553

18618

18683

18748

18813

18878

18943

19008

19078

58

19139

19205

19270

19336

19402

19468

19584

19601

19667

19788

54

19800

19867

19934

20000

20068

20135

20202

20269

20887

20406

55

20472

20540

20608

20676

20744

20818

20881

20949

21018

21087

66

21156

21225

21294

21363

21432

21501

21571

21641

21710

21780

67

21850

21920

21990

22061

22131

22201

22272

22848

22414

82485

58

22566

22627

22698

22769

22841

22918

22984

28066

28128

88S00

59

23272

23345

28417

23489

23562

28635

23708

28781

28864

28M7

60

24000

24078

24147

24221

24294

24368

24442

24616

24600

24685

For oontiDaatitQ to 100 ftfM OMfft aee Tabl« 7.

RAILROADS.

795

Table 5. I<eTel Cattlncs.

Roadway 28 feet wide, side-Alopee 1 to 1.
For donble-traok ezoaTationi

Deoth
mFt.

.0

.1

J2

.3

.4

.6

.6

.7

.8

.9

OiuTds.

CD.Tda.

Cu.Ydi.

Ca.Yda.

ClLTdK.

Oa.Tda.

Ca.T<lB.

Ca.Tdt.

Ca.TdB.

Ca.Tda.

0

10.4

20.9

31.4

42.1

62.8

63.6

74.4

85.3

96.3

1

107.4

118.6

129.8

141.1

152.4

163.9

175.4

187.0

198.7

210.4

2

222.2

234.1

246.1

268.1

270.2

282.4

294.7

807.0

319.4

331.9

3

344.4

357.1

369.8

382.6

395.4

408.3

421.3

434.4

447.6

460.8

4

474.1

487.4

500.9

514.4

528.0

541.7

555.4

569.2

583.1

597.1

5

611.1

625.2

639.4

653.7

668.0

682.4

696.9

711.4

726.1

740.8

6

765.6

770.4

7854

800.4

815.5

830.6

845.8

861.1

876.5

891.9

7

907.6

923.0

938.7

054.5

970.3

986.2

1002

1018

1034

1060

8

1067

1083

1099

1116

1132

1149

1166

1182

1199

1216

9

1233

1250

1267

1285

1302

1319

1337

1354

1372

1300

10

1407

1425

1443

1461

1479

1497

1515

1534

1552

1570

n

1589

1607

1626

1645

1664

1682

1701

1720

1739

1759

12

1778

1797

1816

1836

1855

1876

1895

1914

1934

1964

18

1974

1994

2014

2034

2055

2076

2095

2116

2136

2157

14

2178

2199

2219

2240

2261

2282

2304

2825

2346

2367

16

2389

2410

2432

2454

2475

2497

2519

2541

2563

2586

16

2607

2630

2652

2674

2697

2719

2742

2765

2788

2810

V

2833

2856

2879

2903

2926

2949

2972

2996

3019

3043

18

3067

3090

3114

3138

3162

3186

8210

3234

3269

3288

19

3307

3332

3356

3381

3406

3431

8455

3480

3605

3630

ao

3556

3581

3606

3631

3657

3682

8708

3734

3759

3785

81

3811

3837

3863

3889

3915

3942

3968

3994

4021

4047

22

4074

4101

4128

4154

4181

4208

4235

4263

4290

4317

23

4344

4372

4399

4427

4455

4482

4610

4538

4566

4694

fl4

4622

4650

4679

4707

4735

4764

4792

4821

4850

4879

25

4907

4936

4966

4994

5024

5053

5082

5111

5141

5170

26

5200

5230

5259

6289

'5319

5349

6379

6409

5439

5470

27

5500

5530

5561

5591

5622

5653

5684

6714

5745

5776

38

5807

5839

5870

5901

5932

5964

5995

6027

6059

6090

99

6122

6154

6186

6218

6250

6282

6315

6347

6379

6412

80

6444

6477

6510

6543

6575

6608

6641

6674

6708

6741

81

6774

6807

6841

6874

6008

6042

6975

7009

7043

7077

82

7111

7145

7179

7214'

7248

7282

7317

7351

7386

7421

88

7466

7490

7525

7560

7595

7631.

7666

7701

7736

7772

84

7807

7843

7879

7914

7950

7980

8022

8058

8094

8130

86

8167

8203

8239

8276

8312

8349

8386

8423

8459

8496

86

8533

8570

8608

8645

8682

8719

8757

8794

8832

8870

87

8907

8945

8983

9021

9059

9097

0135

9174

9212

9250

88

9289

9327

9366

9405

9444

9482

9521

9560

9599

9639

89

9678

9717

9756

9796

9835

9875

9915

9954

9994

10034

40

10074

10114

10154

10194

10235

10275

10315

10356

10396

10437

41

10478

10519

10559

10600

10641

10682 110724

10765

10806

10847

43

10889

10030

10972

11014

11055

11097

11139

11181

11223

11265

43

11307

11350

11392

11434

11477

11519

11562

11605

11648

11690

44

11733

11776

11819

11863

11906

11949

11992

12036

12079

12123

46

12167

12210

12254

12298

12342

12386

12430

12474

12519

12563

46

12607

12652

12696

12741

12786

12831

12875

12920

12966

13010

47

13056

13101

13146

13191

13237

13282

13328

13374

13419

13465

48

13511

13587

13603

13649

13695

13742

13788

13834

13881

13927

49

13974

14021

14068

14114

14161

14208

14255

14303

14360

14397

60

14444

14492

14539

14687

14635

14682

14730

14778

14826

14874

61

14022

14970

15019

16067

15115

15164

15212

15261

16310

15359

62

15407

15456

15605

16064

15604

15653

15702

16751

15801

15850

68

1590O

15950

15999

16049

16099

16149

16199

16249

16299

16350

64

16400

16450

16501
17010

16551

16602

16663

16704

16754

16805

16856

66

16907

16959

17061

17112

17164

17215

17267

17319

17370

66

17422

17474

17526

17578

17630

17682

17735

17787

17839

17892

67

17944

17997

18050

18103

18156

18208

18261

18314

18368

18421

68

18474

18527

18581

18634

18688

18742

18795

18849

18903

18957

69

iHtlll

19065

19119

19174

19228

19282

19337

19391

19446

19501

60

19656

19610

19665

19720

19775

19831

19886

19941

19996

20062

For contin nation to 100 feet, see Table 7.

796

RAU.BOAD&

Table 6. I<eTel CnUinirs.

Boadway 28 ft wide, sidenslopee 1^ toX
For donble-traok exoaTation.

Depth

inVt.

.0

.1

.2

.8

.4

.6

.6

.7

.8

J9

Cu.Yds.

OcYdB.

Cii.Yd«.

GlLTdB.

Cu.Yd,.

Cu.Yda.

CB.Yd<.

Ca.Y<lB.

Cu.Yds.

0O.TdB.

0

10.4

21.0

81.6

42.4

63.2

64.2

763

86.5

07.0

1

109.8

120.8

132.6

144.3

166.1

168.1

180.2

192.4

204.8

217.2

2

229.6

242.3

266.0

267.9

280.9

294.0

807.2

320.5

334.0

847^

8

361.2

374.9

388.8

402.8

416.9

431.1

446.4

469.9

474.4

480.1

4

503.7

618.6

633.6

648.6

663.9

679.3

694.7

610.2

626.8

641.6

5

667.6

673.4

689.6

706.7

722.1

738.6

766 0

771.7

788.4

8Wit

6

822.2

839.3

866.6

873.8

891.2

908.8

926.4

944.2

962.0

98OJ0

7

998.1

1016

1036

1063

1072

1090

1109

1128

1147

1106

8

1186

1204

1224

1243

1263

1283

1308

1322

1343

1368

9

1383

1408

1424

1446

1466

1486

1607

1628

1649

1671

10

1602

1614

1636

1667

1679

1701

1723

1746

1767

1700

11

1812

1836

1858

1881

1904

1927

1960

1978

1997

2020

12

2044

2068

2092

2116

2140

2164

2180

2218

2238

2282

18

2287

2312

2337

2362

2387

2413

2488

2464

2489

2616

14

2641

2667

2693

2619

2646

2672

2608 .

2726

2762

2770

15

2806

2838

2860

2887

2916

2942

2970

2997

8026

8068

16

3081

3109

3138

3166

8195

3223

3262

3281

3310

8880

17

8368

3397

3427

8456

3486

3616

3646

3676

3606

8686

18

3667

3607

8728

3768

8789

3820

3861

3882

3913

3044

19

3976

4007

4039

4070

4102

4134

4166

4198

4231

4268

20

4296

4328

4361

4894

4427

4460

4493

4627

4660

4604

21

4627

4661

4695

4729

4763

4797

4832

4866

4900

4985

22

4070

6006

6040

6076

6111

6146
>606

6181

6217

6263

.6288

28

6324

6360

6396

6432

6469

6642

6678

6616

6662

24

6689

5726

6763

5800

6838

5876

6913

6951

6089

6027

26

6066

6103

6141

6179

6218

6267

6296

6334

6373

6412

26

6461

6491

6630

6670

6609

6649

6689

6729

6769

6809

27

6860

6890

6931

6971

7012

7053

7094

7186

7176

7217

28

7269

7300

7342

7384

7426

7468

7610

7662

7694

7637

29

7680

7722

7766

7808

7861

7894

7937

7981

8024

8007

80

8111

8166

8199

8243

8287

8331

8376

8420

8464

8600

81

8564

8698

8643

8688

8734

8779

8824

8870

8916

8961

82

9007

9068

9099

9146

9191

9238

9284

9331

9378

9426

83

9472

9619

9566

9618

9661

9708

9756

9804

9851

9900

84

9948

9997

10046

10093

10142

10190

10239

10288

10387

10386

86

10136

10484

10634

10683

10638

10683

10732

10782

10832

10882

86

10933

10988

11034

11084

11136

11186

11237

11288

11339

11391

87

11443

11494

11646

11698

11649

11701

11753

11806

11868

11910

88

11968

12016

12068

12121

12174

12227

12281

12334

12387

12441

89

12494

12648

12602

12656

12710

12764

12819

12873

12928

12982

40

13037

13092

13147

13202

13267

13312

13368

13423

13479

13586

41

13591

13647

13703

13769

13816

13872

13928

13986

14042

14099

42

14156

14213

14270

14327

14386

14442

14600

14668

14616

14678

43

14731

14790

14848

14906

14966

15024

15082

15141

15200

16260

44

15318

15378

15437

16497

15556

15616

15676

16736

15796

16866

46

15917

15977

16038

16098

16159

16220

16281

16342

16403

16466

46

16626

16587

16649

16711

16773

16836

16897

16959

17021

17064

47

17146

17209

17272

17336

17398

17461

17624

17587

17651

17714

48

17778

17842

17906

17969

18033

18098

18162

18226

18291

18856

49

18420

18486

18560

18616

18680

18746

18811

18877

18942

19008

60

19074

19140

19206

19272

19339

19405

19472

19638

19605

19672

61

19739

19806

19873

19940

20008

20075

20143

20211

20279

20347

62

20416

20483

20651

20620

20688

20757

20826

20894

20963

21082

63

21102

21171

21241

21310

213«0

21450

21519

21689

21659

21780

64

21800

21870

21941

22012

22082

22153

22224

22296

22366

22488

66

22509

22581

22662

22724

22796

22868

22940

23012

23085

23167

66

23230

23302

23376

23448

23621

23694

23667

23741

23814

28888

67

23961

24036

24109

24183

24257

24331

24406

24480

24554

24629

68

24704

•24779

24854

24929

25004

25079

25165

25230

26306

26381

69

25457

25533

25609

25686

25762

26838

25915

25992

26068

26146

60

26*222

26299

26376

26464

20.'>31

26009

26686

26764

26842

20096

Fur contintiation to lUU ftet, see Table 7-

BAILROADS.

797

Table 7. IieTel Onttlnys.

OoaOnvatlMi of th« six feragoing Tablw of Cubto Contents, to 100 llMt of height or depth.

Height

Table

Table

Table

Table

TabU

• Table

or Depth

in Feet.

1

2

8

4

•

5

6

Ca. TdB.

Ca. Ydi.

Ca.T<la.

Oa. Ydi.

On. Yds.

Ca. Yda.

61

238,T5

26094

17848

24739

20107

26998

.5

24201

26479

18108-

25113

203S6

27390

62

24570

26867

18370

26480

20667

277'85

.6

24942

27257

18634

25868

20949

28183

63

25317

27650

18900

26260

21233

28583

.5

25694

28046

19168

26635

21519

28986

64

26074

28444

19437

27022

21807

29393

.6

26457

28846

19708

27413

22097

29801

65

26843

29260

19981

27806

22389

80213

.6

27-231

29657

20266

28201

22682

80627

66

2762-2

30067

20533

28600

22978

81044

.6

28016

30479

20812

29001

23276

81464

67

28413

30894

21093

29406

23674

81887

.5

28812

31313

21376

29813

23876

82312

68

29215

31733

21669

30222

24178

82741

.5

29620

32167

21945

30636

24482

83172

60

30028

82683

22233

81050

24789

83605

.6

30438

83013

22523

31468

26097

34042

70

30852

33444

22S14

31889

25407

34481

.5

31268

33879

23108

83313

25719

84924

71

31687

34317

23404

82739

26033

85369

.5

32108

34757

23701

83168

26349

85816

72

32533

35200

24000

83600

26667

86267

.6

3-2960

36646

24301

34036

26986

86720

78

33390

36094

24604

84472

27307

87176

.5

33823

36646

24907

84913

27631

87636

74

34250

87000

26214

86366

27966

88096

.5

34697

37467

26622

35801

28282

88561

76

35139

37917

25832

36250

28611

39028

.6

35582

38379

26144

36701

28942

89498

76

36029

38844

26468

37166

29174

89970

.5

36479

39313

26774

87613

29608

40446

77

36931

39783

27092

88072

29944

40924

J>

373S6

40257 .

27411

88536

30282

41405

78

37844

40733

27733

89000

80622

41889

.6

38305

41213

28066

89468

30964

42376

7»

38768

41694

28381

89939

31307

42865

.6

392:^5

42179

28708

40413

31653

43357

80

39704

42667

290;{7

40889

32000

43852

81

40650

43660

29700

41860

32700

44850

82

41607

44644

30370

42822 .

83407

45859

83

42576

46660

31048

43S06

34122

46880

84

43555

46667

31733

44800

34844

47911

86

44546

47694

32126

46806

35674

48954

86

45548

48733

33126

468-22

86311

60008

87

46561

49783

83833

47850

87056

61072

88

47585

60844

34548

48889

37807

52148

80

48620

61917

35270

49939

38567

63236

00

49667

53000

86000

51000

39333

64333

91

60724

54094

86737

52072

40107

65443

02

51793

55-200

37481

53156

40889

66563

93

62872

66n7

38233

64260

41678

67694

94

53983

67444

38993

65356

42474

58837

96

65065

68683

39759

66472

43278

59:>90

96

66178

69733

40533

57600

44089

61156

97

67302

60894

41315

58739

44907

62331

96

68437

62067

42104

59889

45733

63618

99

69583

63250

42900

61050

46567

64716

100

60741

64444

43704

62222

47407

65926

798

RAILBOAJDS.

Table S,

Of Cable Tarda in a 100-foot station of level cutting or filling, to be added to, or Biib>
tracted from, the quantities in the preceding seren tables, in case the excar*'
tiooA or embankments should be increased or diminished 2 feet in width.

Coblo Yardiinal

ength of 100 feet ; breadth 2 feet; and of difierent depths.

Height or

Depth

in Feet.

Cabto
Yards.

Height or
Depth

la Feet.

•

Oabie
Yards.

Height or

Depth

la Feet.

Cable
Yards.

Height or

Depth

la Feet.

Cnbio
Yards.

Height or

Depth

in Feet.

Ovbie
Yarda.

.6

3.70

.5

152

.5

800

.5

448

.6

606

1

7.41

21

156

41

804

61

452

81

eoo

.5

11.1

.5

159

.6

307

.6

466

.6

604

2

14.8

22

163

42

811

62

459

82

607

.5

18.5

J5

167

.6

315

.6

463

.6

611

3

22.2

23

170

43

319

63

467

83

616

.5

26.9

.5

174

.6

322

.5

470

.5

619

4

29.6

24

178

44

826

64

474

84

622

JS

33.3

.5

181

.5

330

.6

478

.6

626

&

37.0

25

186

45

833

65

481

86

630

.s

40.7

.5

189

.5

837

.6

485

.6

633

6

44.4

26

193

46

841

66

489

86

637

.6

48.1

.5

196

.6

844

.5

493

.6

641

7

61.9

27

200

47

848

67

496

87

644

.5

56.6

.6

204

.6

362

.5

600

.6

648

8

69.3

28

207

48

866

68

604

88

662

.5

63.0

.5

211

.5

869

.5

607

.6

656

9

66.7

29

216

49

863

60

611

89

669

.5

70.4

.5

219

.5

367

.5

616

.6

663

10

741

80

222

50

370

70

619

90

667

.6

77.8

.6

226

.5

374

A

622

.6

670

11

81.5

SI

230

51

378

71

626

91

674

.5

86 2

.5

233

.5

381

.6

630

.6

678

12

88.9

32

237

62

885

72

633

92

681

.5

92.6

.6

241

.5

889

.5

637

.6

686

13

96.3

33

244

63

893

73

641

93

689

.5

100

.5

248

.6

896

.6

644

.6

608

14

104

34

252

54

400

74

648

94

696

^

107

.6

256

.5

404

.6

662

.6

700

15

111

35

269

66

407

75

666

96

704

.5

115

.6

263

.5

411

.6

669

.5

707

16

119

36

267

66

416

76

663

96

711

.6

122

.6

270

.5

419

.6

667

.6

716

17

126

37

274

67

422

77

670

97

710

.5

130

.6

278

.6

426

.5

674

.6

722

18

133

38

281

68

430

78

678

98

726

.5

137

.6

285

.5

433

.5

681

.5

730

19

141

80

289

69

437

79

686

99

733

.5

144

.5

293

.5

441

.5

589

.6

787

20

148

40

296

60

444

80

593

100

741

BxMARK. Tbe forearolnfr tables of level cntttnga mmj also be
used for widths of roadway ipreater than tiioee at the heads
of the tables. Thus, suppose we wish to use Table 1, for a roadbed m n, 16 ft
wide, instead of o 6, which is only 14 ft, and for which the table was calculated. It
is only necessary first to find the vert dist s a, between these two roadbeds ; and to
add It mentallj/ to each height £ s, of the given embkt, when taking out firom ths

a

I
1

BAILBOAD6. 79d

table the numbers of cub ydi corresponding to the heights. By this means we obtain
the contents of the embkt c & op, for any required dist. Next, from these contents
subtract that corresponding to the height 8 a, for the same dist. The remainder, will
plainly be the embkt mn op.

In prsctlce it will be sufficiently correct to take tato the nearest tenth of a
foot, which will save trouble In adding it mentally to the heights in the tablet.

If tlie roadbed is narrower than the table, as, for instance, if mn be
the width in the table, but we wish to find the contents for the width cb, then
first find sOf and calculate the cubic yards in 100 feet length of cbmn. Then,
in taking out the cubic yards from the table, first subtract « a mentally from
«ach height; and to the cubic yards taken out for each 100 feet, opposite this
reduced height, add the cubic yards in 100 feet of c6fiin.

To avoid trouble with contractors about the measurement of rock
cuts, stipulate in the oontraet, either that it shall conform with the theoretical
cross section ; or that an extra allowance of say about 2 feet of width of cut
will be mad^ to coyer the unavoidable Irregularities of the sides.

SbriBkaire of Smbankment. Although earth, when first dug, and
loosely thrown out, tvkUs about \ part, so that a cubic Tard in place averages
about 1| or 1.2 cubic yards when dug: or 1 cubic yard dug is equal to f, or to
.8333 of a cubic yard in place ; yet when made into embankment it gradually
subsides, settles, or shrinks, into a less bulk than it occupied before being dug.

The following are approximate averages of the shrinkage ; or, in other words,
the earth measured in place in a cut, will, when made into embankment, occupy
a bulk less than before by about the following proportions :

Gravel or sand about 8 per ct ; or 1 in 12)^ less.

Clay ** 10 per ct; or 1 in 10 less.

Loam " 12 per ct ; or 1 in SUless.

Loose vegetable surface soil " 15 per ct ; or 1 in ^ less.

Puddled clay. ** 25 per ct ; or 1 in 4 less.

The writer thinks, from some trials of his own, that 1 cubic yard of any hard
rock is place, will make from IH to IV cubic yards of embankment; say on an
average 1.7 cubic yards. Or that 1 cubic yard of rock embankment requires
.5882 of a cubic yard in place. He found that a solid cubic yard when broken
into fragments, made about as follows

Cubic
yards.

In loose heap 1.9

Carelessly piled 1.75

Carefully piled 1.6

Bubble, very carelessly scabbled 1.5

Bubble, somewhat carefully scabbled...- 1.25

Of which there were

Solid

62.6 per cent.
67 "

63 . "
67 «

80 *'

Voids

47.4 percent*
43 «

37 "
33 "

20 **

800

COST OP EARTHWORK.

COST OF EARTHWOEK.

Art« 1* It i« sdvlsable to psy for tbts kind of work hy the oubic yard of aaeeaeoMon only j li^
stead of allowing separate prioes for excavatioa and embankment. By thii means we get rid or tb«
dlfDoulty of measurements, as well as the controTersies and lawsnitR which often attend the deter-
nlnation of the allowance to be made for the settlement or subsidenee of the embankmenta.

It is, moreover, oar opinion that Jostice to the contractor should lead to the EnflrUsb pi*aCa

tlce of pnylnsr the laborers by tbe cubic yard, instead or by the day.

experience fully proves that when laborers are scarce and wages high, men can scarcely be depended
apon to do three-fourths of the work which they readily accomplish when wages are low, and when
fresh hands are waiting to be hired in case any are discharged. The contractor Is thns plaoed at the
mercy of his men. The writer has known the most satisfactory results to attend a system of task-
work, aooompanied by liberal premiums for all overwork. By this means the Interests of the laborers
are identified with that of the oontractor ; and every man takes care that the others shall do their
teir share of the task.

Ellwood Morris, G E, of Philadelphia, was, we Y>eUeTe, the first person who properly Investicmted
ttie elements of cost of earthwork, and reduced them to such' a form as to enable us to caJonlato the
total with a considerable degree of accuracy. He published his results in the Journal of the Pranklla
Institute in 1841. His paper forms the basis on which, with some variations, we sdall consider the
matter : and on which we shall extend it to wheelbarrows, as well as to carts. Thronghout this paper
we speak of a cubic yard considered only as solid in its place, or before it is loosened for removal. It
is scarcely neoessary to add that the various items can of course only be regarded as tolerably clbse
approximations, or averages. As before stated, the men do less work when wages are high ; and nsors
when they are low. A great deal besides depends on the skill, observation, and energy of the oon-
tractor and his superintendents. It is no unusual thing to see two oontrfotors working at the saine
prices, in precisely similar material, where one is making money, and the other losing it, fH>m a want
of tact in the proper distribution of his forces, keeping his roads in order, having his earts and bar-
rows well fliled, Ao, tto. Uncnmmonly long spells of wet weather may seriously affect the cost of exe-
cuting earthwork, by making it more diffloult to loosen, load, or empty ; besides keeping the roads la
bad order for hauling.

The aggregate cost of excavating and removing earth is made up by the foliowin|; items, namely:
1st. Lootening the tarth ready for the ahoveUera.
2d. Loading it by ahovela into the carte or harrifwe,
8d. Hauling, or toheeling it away, ineluding emptying and retttming.
4th. Spreading it out into eueceative layera on the embankment.

5th; Keeping the hauling-road for carte, or the plank gangtaaye for ftorrMW, <n g90d order.
6th. Wear, tharpening, depreciation, and inter eet on coat of tome.
7th. Superintendence, and footer- carriere.
8th. Ftqfit to the contractor.

We will eonsider these Items a little in detail, basing our calculations on the assumption that (
Uon labor costs $1 per day, of 10 working hours. The resulto in our tables must therefore be ln>
•jreased or diminished in about the same proportion as common labor oosts more or less than this.

Art. 2. lioonenlni? the eartb ready for tbe sboTellera. Thiais

generally done either by ploughs or by picks ; more cheaply by the first. A plough with two horsea,
and two men to manage them, at $1 per day for labor, 75 cents per day for each horse, and S7 cents
per day for plough, including harness, wear, repairs. Ac. or a total of tS.87, will loosen, of strong
Vavy soils, f)-om 200 to 300 oubic yards a day, at ftvm 1.S3 to 1.29 ceatt per yard; or of ordinary
loam, from 40u to 600 oubio yards a day, at from .97 to .64 of a cent per yard. Therefore, as an ordl-
nary average, we may assume the actual cost to the contractor for loosening by the plough, as M-
lows: strong heavy soils, 1.6 cents ; common losm, .8 cent; light sandy soils, .4 cent. Very stiff pure
clay, or obstinate cemented gravel, may be set down at 2.6 cente ; they require three or four horsea.
By the ^ick, a fair day's work is about 14 yards of stiff pure day, or of cemented gravel ; 25 yards
of strong heavy soils; 40 yards of common loam; 60 yards of light sandy soils — all measurad in
place ; which, at $1 per day for labor, gives, for stiff clay, 7 cents ; heavy soils, 4 cente : loam, 2.6
oentn; light xandy soil, 1.666 ceuto. Pure sand requires but very little labor for looeenlng; .5 of a
■ent will cover it.

Art. 3. ShoTelllniir ^l>o loosened -earib Into carts. The amouat

shovelled per day depends partly upon the weight of the material, but more upon so proportlonlag
the number of pickers and of carte to that of shovellers, as not to keep the latter waiting for either
material or carte. In fairly regulated gangs, the shovellers into earts are not actually engaged ta'
■hovelling for more than six- tenths of their time, thus being unoccupied but four-tenths of it; while,
under bad management, they lose considerably more than one-half of it. A shoveller can readily
load into a cart one-third of a cubic yard measured in place (and which Is an average working oart-
load), of sandy soil, in five minutes ; of loam, in six minutes : and of any of the heavv soils. In seven
minutes. This would give, for a day of 10 working hours, 120 loads, or 40 cubic yards of light sandy
soil ; 100 loads, or S3^ cubic yards of loam ; or 86 loads, or 28.7 yards of the heavy soils. But from
these amonnte we must deduct four-tenths for time necessarily lost; thus reducing the actual work-
ing quantities to 24 yards of light sandy soil, 20 yards of loam, 17.2 yards of the heavy soils. When
the shovellers do less than this, there is some mismanagement.

Assuming these as fair quantities, then, at $1 per day for labor, the actual cost to the eontraelsr
for shovelling per oubic yard measured in place, will be, for sandy soils, 4.167 ocnte; loam, 6 cento;
heavy soils, clays, Ac, 6.81 cente.

In prsctiee, the carte are not usually loaded to any less extent with the heavier soils than with the
lighter ones. Nor. indeed, is there any necessity for so doing. Inasmuch as the dlfferenee of wslght
of a cart and one-third of a cubic yard of the various soils is too slight to need any attentlen : c«p^
uially when the cart-road Is kept in good order, as it will be by any contractor who underatands hir

OOBT OF EABTHWORK. 801

own Intarwt. Neither ii it bjoeeeary to modify the load on aoooant of any tUgkt ineUnattcnt wbiok
m»7 ooenr in the gradinc of roada. An earth-cart weighs by itself aboat H * (on.

Art;. 4. IIanlln§r away (be earth i damping, or emptying ;

and retlirnlni^ to reload. The arerage speed of horses in haoling is about S^ miles
per boor, or 900 feet per minate ; whioh is eaual to 100 feet of trip eaoh way ; or to 100 feet of Uad,
as the distanoe to which the tarth i* tkauUd is teohnioally oalled.* Besides this, there is a loss of
aboat fbnr minutes in every trip, whether long or short, in waiting to load, dumping, taming, 4c.
Hence, every trip will occupy as many minutes as there are lengths of 100 feet eaoh in the lead ; and
four minntes besides. Therefore, to find the number of tripe per day over any giren average lead, we
divide the namber of minutes in a working day by the sum of 4 added to the number of 100 feet
lOBgthe oontained in the distanoe to whioh the earth has to be removed ; that is,

The number (900) o/minuUt ^ a working day __ the number o/ tripe, or loade
A-\-the number of 100/«el lenffthe in the lead " removed per day, per eart.

•

And sinoe M of a cubic yard measured before being loosened, makes an average cart-load, the num-
ber of loads, divided by 8, will give the number of cubic yards removed per day by each cart ; and
the onbio yards dlTided Into the total expense of a eart per day, will give the eoet per oublo yard for
hauling.
RuASK. When remoTing koae reek, which requires more time for loading, say,

No. of mimimiee (600) in a worMng day _ jg^ ^f j,,,^ remoeed,
• + So. «/ lOO-ZM Umqthe o/ Uad P«r day, per eart.

In leads of ordinary length one driver ean attend to 4 carts ; which, at f 1 per dd^, Is SB eenti per
mn. When labor ii $1 per day. the expense of a horse is usually about 75 cents ; and that of the
eavt, inelnding harness, tar, repairs. Ac, 36 cenu, making the total daily cost per eart $1.36. The
expense of the horse is the same on Sundays and on rainy days, as when at work : and tliis oonsid-
erallon is included in the 75 cents. Some oontraotors employ a greater number of drivers, who also
help to load tbe carts, so that the expense is about the same in either case.

BxAMPLS. How many cubic yards of loam, measured in the cut. can be hauled by a horse and cart
In a day of 10 working hours. (000 minutes,; the lead, or length of haul of earth being 1000 feet, (or
10 lengths of 100 feet,) and what will be tbe expense to the contractor for hauling, per cubic yard,
asenming the total cost of eart, horse, and driver, at $1,367

^•^' 4-H0ton<n*e.o/100/sst. = u" = " '^•*- ^"^ ^F" = "'» "^ ^'''^^
In this manner the 3d and Sd columns of the fbllowing tables have been ealonlated.

Art. 5. Spreadlnir« or leTelllnff off tlie earth into reffular
thin layers on the embankment, a bankman win spread rrom so to loo cubic

yards of either common loam, or any of the heavier soils, clays, Ac, depending on their dryness.
This, at $1 per day, is 1 to 2 cenU per cubic yard ; and we may assume l^i cents as a fair average
fbr each soils : while 1 cent will snSlce for light sandy soils.

This expense for spreading is saved when the earth is either dumped over the end of the embank*
0Mnt, or is wasted ; still, about yi oent per yard should be allowed in either case for keeping the
damping- places olear and in order.

Art. 6. Keeplnfc the cart-road in fpooti order for banlins.

No ruts or puddles should be allowed to remain unfilled : rain should at once be led off by shallow
lltobes : and the rosd be carefully kept in good order; otherwise the labor of the horses, and the wear
of carts, will be very greatly increased. It is usual to allow so much per cubic yard for road repairs ;

but w^suggest so much per cubic yard, per 100 feet of toad ; say -|^^ of a cent.

Art. 7. Wear, sharpeninfp, and depreciation of picks and

shovels. Bxperlenoe shows that about Hot a, cent per cubic yard will cover this item.

Superintendence and water-carriers. These expenses win vary with

loeal circumstances ; but we agree with Mr. Morris, that 1 ^ cents per cubic yard will, under ordinary
dreumstaoces, cover both of them. An allowance of about }4, c^^t may in justice be added for extra
trouble in digging the side-ditehes ; levelling off the bottom of tbe cut to grade ; and general trimming
up. In verv Mght cuttings this may be increased to H mnt per cable yard.
At ^ cant, all tbe items in this article amount to 2 cents per eobio yard of cut.

Art. 8. Profit to the contractor. This may generally be set down at f^m 8 to
15 per cent, according to the magnitude of the work, the risks incurred, and various incidental cir-
cumstances. Out of this item the contractor generally has to pay clerks, storekeepers, and other
agMts, as well as tbe expennes of nhanties. kc ; although these are in most oases repaid by the proSte
of tne stores; and by the rates of boarding and lodging paid to the contractors by tbe laborers.

Art. 9. A knowlednre of the foreirolnip items enables ns to
ealenlate with tolerable accuracy the cost of removing: earth.

For example, let it be required to ascertain the cost per cubic yard nf excavating common loam, meas-
ured in place; and of removing it into embankment, with an average haul or lead of 1000 feet; the
wages of laborers being $1 per day of 10 working hours ; a horse 75 cu a day ; and a cart 25 cts. One
driver to four carts.

J)t When an entire cut is made into an embankment, the Mean kanl is the dist between centers
of gravity of the cat and embkt.

51

802

008T OF EARTHWORK.

JTere we have eoMt of loosening, laif by pick, Art 2, per cubic yard, eof.
Loading into carte. Art. 8. " "

SatUin^ luOO feet, oM calculated previouelff in example. Art. 4, "
Spreading into layers. Art. 6, "

ICeeiping cart-road in repair. Art. 6, 10 lengths of 100 ft.

Various Uems in Art. 7,

Cent*.

2.50
6.00
8.74
1.50
1.00
2.0U

Total cost to contractor.

Add eontrador^e pn^, say 10 per cent.

Total cost per cubic yard to the company, S2.814

It is ea«7 to ooDstmot a table like the foUoving, of costs per cubic yard, for different lengths of lead.
Oolamns 2 and S are first obtained bj the Rule In Article 4 ; then to each amonnt in column 3 is added

the variable quantity of y^ of a cent for erery 100 feet length of lead, for keeping the road in order;
and the constant quantity (for any given kind of soil) ooinposed of the prices per eubie yard, for
loosening, loading, spreading, or wasting, ko, either taken (fom the preceding articles ; or modiflsd
to salt particular clrcunutanoes. In tbie manner the tables have been prepared.

By Carta. I^abor $1 per day, of 10 working: boars.

1

L

1

Oommon Loam.

Strong HoATj Soils.

-!

If

ts

. 2

*2.

%l

£■2

if

TOTAL OOST PER CUBIC
YARD, BXCLUSIVE OF

TOTAL COST PER CUBIC
YARD, EXCLUSIVE OF

It

&§

PROFIT TO CONTRACTOR.

PROFIT TO COHTRACTOR.

•oA

■si

1

H

hi

pi

w

hi

Feet.

Gu.Tds.

Cts.

Cts.

cu.

cu.

Cts.

Cts.

Cts.

Cu.

Ote.

25

47.0

2.66

18.69

12.44

11.99

10.74

16.00

14.75

18.50

13.35

50

44.4

2.81

18.86

12.61

12.16

10.91

16.17

14.92

13.67

13.43

76

42.1

2.97

14.06

12.80

12.36

11.10

16.86

15.11

18.86

13.61

100

40.0

3.12

14.22

12.97

12.52

11.27

16.53

15.28

14.03

13.18

150

86.4

848

14.68

13.83

12.88

11.68

16.89

15.64

14.30

18.14

900

83.3

8.76

14.96

13.70

13.26

12.00

17.36

16.01

14.76

13.61

300

28.6

4.87

16.67

14.42

13.97

12.72

17.96

16.73

16.48

14.83

400

26.0

6.00

16.40

15.15

14.70

18.46

18.71

17.46

16.21

14.96

600

22.2

6.68

17.18

15.88

15.43

14.18

19.44

18.19

16.94

U.«

eoo

20.0

6.26

17.86

16.60

16.16

14.90

20.16

18.91

17.66

16.41

TOO

18.3

6.87

18.67

17.32

16.87

16.62

20.88

19.63

18.38

17.13

800

16.7

7.48

19.28

18.03

17.58

16.33

81.69

20.34

19.09

17.M

900

16.4

8.12

19.92

18.67

18.22

16.97

22.28

30.98

19.73

18.48

1000

14.3

8.74

30.74

19.49

19.04

17.79

23.06

21.80

30.56

19J0

1100

13.8

9.40

21.60

20.25

19.80

18.66

88.81

28.66

21.31

20.06

1200

12.6

10.0

2^20

20.95

30.50

19 25

24.51

23.26

22.01

20.76

1800

11.8

10.6

32.90

21.66

21.20

19.95

35.21

38.96

23.71

S1.4t

1400

11.1

11.2

23.60

22.36

21.90

20.66

86.91

24.66

28.41

SS.M

1600

10.6

11.9

24.40

23.15

23.70

21.46

26.71

25.46

24.31

33.96

leoo

10.0

12.6

26.10

23.86

23.40

23.16

27.41

26.16

34.91

S3.06

1700

9.52

18.1

26.80

24.66

24.10

23.85

28.11

36.86

25 61

34.16

1800

9.09

13.7

26.50

26.26

24.80

23.66

28.81

27.66

36.31

3ft.66

1900

8.70

14.4

27.80

26.05

25.60

24.36

29.61

28Ji6

27.11

35.86

2000

8.3S

16.0

38.00

26.75

26.30

26.06

80.31

29.06

27.81

36.56

2250

7.54

16.6

29.86

38.60

28.16

26.90

83.16

80.91

29.66

28.41

2600

690

18.1

31.60

80.35

29.90

38.66

83.91

83.66

81,41

30.16

Mmlle

6.58

19.0

32.64

31.39

30.94

29.69

34.95

88.70

82 45

9\m

3000

5.88

21.2

36.20

83.96

33.50

82.t&

37.61

86.26

35.01

83.76

8260

5.48

22.8

87.06

36.80

86.36

34.10

39.36

88.11

86.86

85.61

3600

5.18

24.8

88.80

87.56

37.10

35.85

41.11

39.86

38.61

87.96

8750

4.82

26.9

40.66

89.40

38.95

37.70

42.96

41.71

40.46

89.91

4000

4.54

27.6

42.60

41.26

40.80

39.55

44.81

43.56

42.81

41.06

4260

4.3)0

29.1

44.86

43.10

42.65

41.40

46.66

46.41

44.16

42.91

4500

4.0S

80.6

46.10

44.86

44.40

43.16

48.41

47.16

45.91

44.66

4750

8.88

32.2

47.95

46.70

46.25

45.00

50.26

49.01

47.76

46.61

6000

3.70

83.8

49.80

48.65

48.10

46.86

52.11

60 86

49.61

48.36

1 mile

8.62

86.5

51.78

60.63

50.08

48.88

64.09

53.84

51.59

80.84

iiim.

2.86

43.8

61.40

60.15

59.70

58.46

68.71

63.46

61.31

66.96

IH m-

2.40

62.1

71.02

69.77

69.32

68.07

73.33

72.06

70.63

69.86

IH >"•

2.07

60.4

80.64

79.39

7894

77.69

82.96

81.70

80.4ft

1t.30

3 m.

1.82

68.7

90.26

89.01

88.56

87.81

92.5T

91.SS

90.07

88.13

COST OF EABTHWORK.

803

By C^rts. I^abor ^1 per day, of 10 workluir hoars.

H

la

I

*o 8

ti

a

reet.

SO

76

IQO

ISO

300

- .WO

400

SOO

MO

TOO

800

MO

lOM

IIW

ISW

IMO

14M

15M

IMO

17M

1800

IMO

MOO

SSSO

xoo

3000

8360
8600
8760
4000
tiSO
4SM
4780
SOW
1 mile

\U

3 m.

lace, for
only.

— jS

O.M

■ 3

M O

•2»

Zl

5 ?

t»t«

^«

»8.

oi

*-_7

•a"

b^

jl

51

9

a

Ca.YdB.

Gts.

47.0

2.M

44.4

3.81

4:i.l

2.97

40.0

3.13

38.4

. 3.43

SS.S

8.75

38.6

4.37

25.0

5.00

33.3

5.63

M.0

6.25

18.3

6.87

18.7

7.48

15.4

8.13

14.8

8.74

13.8

9.40

12.5

10.0

11.8

10.6

11.1

11.3

10.5

11.9

10.0

13.5

9.53

13.1

9.00

13.7

8.70

14.4

8.38

15.0

7.54

16.6

6.M

18.1

8.58

19.0

5.88

31.3

5.48

23.8

5.13

34.3

4.83

36.9

4.54

27.5

4.M

29.1

4.08

M.6

8.88

32.2

8.70

38.8

3.52

85.5

8.88

a.8

3.40

53.1

3.07

M.4

1.83

68.7

Pve stiff day, or cemented
GraTel.

TOTAL COST PER CUBIC,
TAttD, BXCLUSIYB OF
PROFIT TO CONTRACTOR.

Picked

and
Spread.

Picked

and
Wasted.

Ploughed

and
Spread.

Cu.

Ota.

cu.

19.00

17.75

14.50

19.17

17.93

14.67

19.36

18.11

14.86

19.53

18.38

15.03

19.M

18.64

15.89

M.26

19.01

15.76

30.98

19.73

15.48

31.71

30.46

17.31

33.44

21.19

17.94

83.16

81.91

18.M

33.88

83.68

19.38

24.59

83.34

M.W

35.33

33.98

M.73

36.05

24.80

31.55

36.81

35.56

33.81

37.51

36.26

33.01

38.81

26.M

38.71

28.91

27.66

34.41

29.71

38.46

35.81

W.41

29.16

36.91

81.11

29.M

26.61

31.81

30.56

27.81

38.61

81.36

28.11

83.31

33.06

38.81

86.16

88.91

M.M

86.91

35.M

88.41

87.95

36.70

88.45

40.51

39.26

36.01

48.36

41.11

87.86

44.11

43.86

39.61

45.96

44.71

41.46

47.81

46.56

48.31

40.66

48.41

46.16

51.41

50.16

46.91

58.36

53.01

48.76

55.11

58.86

50.61

57.09

55.84

53.59

M.91

65.46

63.21

7&S3

75.08

71.88

86.95

84.70

81.45

96.57

94.33

91.07

1 ■«•
gss

Cts.

13.35
18.43
13.61
13.78
14.14
14.51
15.23
15.96
16.69
17.41
18.13
18.84
19.48
M.SO
31.06
31.76
33.46
28.16
3S.W
84.M
25.86
36.06
36.86
37.56
29.41
81.16
82.20
34.76
36.61
88.36
40.21
42.06
48.91
45.M
47.51
49.36
11.84
60.M
70.58
80.20
89.83

Light Sandy Soils.

TOTAL COST PER CUBIC
YARD, EXCLUSIYB OF
PROFIT TO CONTRACTOR.

cu.

11.53
11.69
11.88
13.05
12.41
12.78
13.60
14.23
14.96
15.68
16.40
17.11
17.75
18.67
19.33
M.OS
30.73
31.48
3333
23.93
23.63
24.33
35.13
35.88
37.68
39.48
30.47
33.08
34.88
36.63
88.48
40.83
42.18
43.93
45.78
47.63
49.61
58.23
68.85
78.47
88.09

l^J

u

rS3

gS&

(^ ^

s «

Cu.

CU.

10.77

10.25

10.94

10.43

11.13

10.61

11.80

10.78

ll.M

11.14

12.03

11.51

12.75

12.23

13.48

12.46

14.21

13.69

14.93

14.41

15.65

16.13

16.36

16.84

17.00

16.48

17.82

17.30

18.58

18.06

19.28

18.76

19.98

19.46

20.68

20.16

31.48

30.96

33.18

31.66

33.88

32.36

33.58

23.06

24.36

28.86

35.08

24.66

36.93

36.41

38.68

28.16

29.73

29.20

83.38

31.76

34.13

33.61

35.88

35.36

37.78

37.21

39.58

39.06

41.45

40.93

43.18

42.66

45.03

4451

46.88

46.36

48.86

48.34

58.48

57.96

68.10

67.58

77.72

77.20

87.34

86.82

IS
a. I»

3

Cu.

9.50
9.6T
9.86
10.08
10.39
10.76
11.46
13.21
12.94
13.66
14.38
15.09
15.78
16.55
17.31
18.01
18.71
19.41
20.21
30.91
21.61
22.31
23.11
38.81
36.66
37.41
38.46
81.01
83.86
34.61
M.46
88.31
40.18
41.91
48.76
45.61
47.69
57.21
M.83
76.45
86.07

Art. 10. By wlie^lbairrOWS. The oo«t by barrowt may be estimated in the same
maaner aa by earU. *8ee Articles 1, Ao. Men in wheeling move at about the same average raU aa
boraes do in hanllng, that is, 2H mites an boor, or 200 feet per minnto, or 1 minute per erery lOO-feet
length of lead. The time occupied in loading, emptying, Ao (when, as is usual, the wheeler loads hla
Dim barrow,) ia about 1.85 minutes, without regard to length of lead ; besides which, the time lost ia
ee— aional alu>it reau, in adjoating the wheeling- plank, and in other incidental causes, amounu to
about JL pari of hU whole time; so that we must in practice consider him as actually working but
f hours out of his 10 working ones. Therefore

Th* number of in<nitte« in a worMng d«y X .9_ __ (]k« wwaJbtr of trip$ or of loadt
1.26 + the number of lOOfeet lengiht'of lead ~ removed per day per harrow.

See Resaark, next page. . , , ,. _^ *

The namber of loads divided by 14 will pive the number of cub yards, since a cub yard, measured

in plaoe, sTcrages about 14 loads. And the cost of a wheeler and barrow per day. (say 91 per man.

and 5 oenu per barrow,) divldad by the number of cub yards, will {Ito the oost per yaj-d for loading

wheeling, and emptying.

loTHdhatnt ](W fML (or lOlnjrLba ur lOQ

008T OF SASTHWOBK.

lb jia Ear luiliii. >lH>)IPi

Zi^'li

l.t+Ko9fVja-J»tHmglkt

h

1

8

1

3

c„.„l^

8to«»H«Ty&>U«.

i

;z'v"™"i™'

YiMl, EICI.DBIVB Of

H

111

111

M

u

1^1

n

111

]

1

1

0«.

is
as

1

1

ou.

1

on.

iiis
II

cu.

Of.

s

IS

3

a

OOBT OF EARTHWORK*

805

*By Whe«llMirrows. lAbor $1 per day, of 10 working lionnk

3

a

oe, for
)tjing.

Pure Stiff Clay, or Oe-
mented Ontvel.

Light Sandy Soils.

ai

if

«►

it

3
II

TOTAL COST PBR OUBIC

TOTAL OOBT PER CUBIC

Sf

YARD, BXOLnSIVE OF

YARD, EXCLU8IYB OF

2 V

PROFIT TO COKTRICTOR.

PROFIT TO CONTRACTOR.

Z^

•

9^

e g

as

?

u

hi

•8 .«

hi

9 d c
S «

u

B 93

Feet.

Ga.Tda.

Ota.

Ota.

Ota.

CU.

eta.

Cta.

Ota.

CU.

Cu.

»

96.7

i.00

14.69

13.87

10.19

8.87

8.79

8.04

7.52

6.77

60

99.1

4.75

16.80

14.06

10.80

9.55

9.47

8.72

8.20

7.45

76

19.8

6.44

16.09

14.77

11.59

10.27

10.19

9.44

8.92

8.17

100

17.1

e.14

16.74

15.48

12.24

10.99

10.91

10.16

9.64

8.88

UO

14.0

7.50

18.16

16.90

13.65

12.40

12.39

11.57

11.05

:o.80

soo

11.9

8.89

lf.69

18.97

16.09

1.S.77

13.69

12.94

12.42

11.67

160

10.8

10.9

90.96

19.70

16.45

15.20

15.12

14.37

13.85

1.<).10

900

9.07

11.0

99.40

91.15

17.90

16.65

16.57

15.82

15.30

14.55

860

8.14

19.0

98.76

99.50

19.25

18.00

17.92

17.17

16.65

15.90

40O

7.80

14.8

96.90

98.95

W70

19.45

19.37

18.62

18.10

17.35

tfO

0.71

16.0

96.66

96.80

22.06

90.80

20.72

19.97

19.45

18.70

600

0.17

17.0

98.00

96.76

23.50

92.25

92.17

21.42

20.90

20.15

000

6.89

10.7

80.80

99.66

96.30

25.05

24.97

24.29

93.70

92.96

TOO

4.67

99.6

83.70

89.45

99.20

27.95

27.87

27.12

26.60

25.85

8Q0

4.17

96.9

86.90

35.25

82.00

80.75

30.67

29.92

29.40

28.66

SOO

8.76

97.9

89.30

38.05

34.80

83.55

33.47

32.79

32.20

81.45

1000

8.48

80.0

49.10

40.85

37.60

86.35

36.27

35.52

85.00

84.26

laoo

9.91

30.1

47.80

46.55

43.80

42.05

41.97

41.22

40.70

39.90

1400

9.68

41.6

63.40

52.15

48.90

47.65

47.67

46.82

46.30

45.55

1000

9.34

4«.»

68.00

57.75

54.50

53.25

53.17

52.42

51.90

£1.15

1800

9.00

59.5

64 80

63.55

(».30

59.05

58.97

58.29

67.70

66.95

9000

1.81

68.0

70.50

69.25

66.00

64.75

64.67

63.92

63.40

62.65

S900

1.06

03.8

76.00

74.75

71.50

70.25

7017

60.42

68.90

68.15

.9400

1.63

68.6

81.50

80.95

77.00

75.75

75.67

74.92

74.40

73.66

HmtH.

IJO

76.6

88.64

87.39

84.14

82.80

82.81

82.06

81.54

80.W

Art* 12. By wheeled serapei^ and drag werapegw. The body
of the wheeled scraper is a box of smooth sheet-steel about SK^ ft sqiiare by 15 ins
deep, ooDtaining aboat ^ cubic yard of earth when " even fiiin'* The box is open
ill ftt>nt (in some machines it is closed by an " end gate ** when full), and can be raised
and lowered, and revolved on a horizontal axis. To fill the box, it is lowered into,
and held down in, the earth, while the team draws the machine forward. When full,
it is raised to about a foot above ground ; and, on reaching the dump, is unloaded by
being overturned on its axis. All the movements of the box are made by means of
levers, and without stopping the team, which thus travels constantly. The wheels
hare broad tires, to prevent them from cutting Into the ground.

In the drag scraper the box, owisg to the greater resistance to traction, is made
mnch smaller. It contains about .15 to .25 cubic yard in place, and is always open in
front. The operation of the drag scraper is similar to that of the wheeled scraper,
except that the box, when filled, rests upon the gnx>und and is dragged over it by tiie
tnim

Bach scraper ("wheeled" or "drag^') requires the constant use of* a team of two
horses with a driver. Besides, a number of men, depending on the shortness of the
lead and the number of scrapers, are required in the pit and at the dump to load the
scrapers (by holding the box down into the earth) and unload them (by tipping the
box). Except in sand, or in very soft, soil, it is economical to use a plow before
scraping.

The serereet work for the team is the filling of the box ; and this occurs oftenest
where the lead is shortest. Hence smaller scrapers are used on short than on long
hauls. We base oar calculations on the following loads : '

For drag scrapers (used only on short hauls) 2 cubic yard

Tor wheeled scrapers

lead le« than 100 feet 33 «»

" 100 16 300 feet '. 4 '«

*« 400to600feet .5 "

** oTtr 600 feet - m- .6 **

806

OOBT OF EARTHWORK.

The daily ezpenM per scraper, for driver^B wages and the ua»of a 2-hoFBe team, ii'
about 93.50. For leads of 400 feet and oyer, we add 60 cts per day for use of " snatch
team " to help load the larger scrapers then used. One snatch team generally aerres
a number of scrapers.

Owing to the fact that the teams are constantly in motion without rest, they trard
somewhat more slowly than with carts. We take 160 ft per minnte (or 76 ft of lead
per minute; as an average.

In loading and unloading, the teams not only go out of th^ir way in order to tun
around, but travel more slowly than when simply hauling. To cover this we make
an addition of 100 ft to each length of lead, whether long or short, for wheeled
scrapers and for drag scrapers.

We add 1 cent per cubic yard for the cost of loading and dumping the scrapers ; and
estimate the approximate cost of the other items as follows*.

Repairs of cart-road ^ ct per cub yd in place for each 100 ft of lead

light Boils Heavy Soils

Loosening cts per cub yd in place ctspercubydinplaos

by pick * 5.

by plow * 2.

Spreading 1 1.6

Superintendence, wear and tear etc 1 1.

We repeat that our figures are to be regarded merely as tolerable approximations,
and subject to great variations according to skill of contractor and superintendent,
strength of teams, character of material moved, state of weather etc etc.

No. of trips per day ^ No. (600) of mins in a working day
per scraper - No. of 76 ft lengths in (lead -flOOft)

No. of cub yds in place moved __ No. of tripe per ^ No. of cub yds in place,
per day by each scraper day per scraper ^ per scraper per ^p

'^r?SLSng,^^uuS^' = Daily expense of one scraper i et for loading
dumping and returning No. of cub yds in place, moved and dumping

per day by each scraper

Total cost per Cost per cub yd .1 ct per cub yd p„^ • ., „». _^ . ..^

cubic yard in in ^, for inplice foreaih S?^^JSSL?"5 ^t^S-SS

place eiclusive - loading, haul- -f 100 ft of lead, + ^ ^^*'JS^^

of «>nteactor's mg, dumping, for repairs of JJ^^S^A^T ^

profit and returning road »ut«i«««»«ww «-u.

By Wheeled Scrapers. Labor SI per day of 10 working hours.

(a)

(b)

(c)

(d)

•o Si

in

Total cost per eubio yard, Id plaoe, ezeloalve of oontraetor'i profit

b ^

■5.5'

Hh

*^

•ssl
til

nantlty In
hauled per
each scraper

5^ e

Light Soils

HeaTj

Soils

Picked

Picked

Plowed

Plowed

§55

S o..a S

and

and

and

-.*"*

J

9

o

Spread

Wasted

Spread

Wasted

Spread

Wastsl

Feet

cub yds

cts

cts

cts

CU

cts

cts

cto

60

100

4.5

6.6

5.6

12.1

10.6

9.1

7.6

100

90

4.9

7.0

6.0.

12.5

11.0

9.5

8.0

150

70

6.0

8.2

7.2

13.7

12.2

10.7

9.2

200

60

6.9

9.1

8.1

14.6

13.1

11.6

10.1

800

45

8.8

11.1

10.1

16.6

15.1

18.6

12.1

400

45

9.9

12.3

11.3

17.8

16.8

14.8

18.8

600

38

11.5

14.1

13.1

19.6

18.1

16.6

15.1

800

80

14.3

17.1

16.1

22.6

21.1

19.6

18.1

1000

24

17.7

20.7

19.7

26.2

24.7

28.3

11.7

* Li^ht soils can generally be advantageously loosened by the scrapors them-
selves in the act of loading.

COST OF EARTHWORK. 807

By l>i'a8r fiksrapers. Labor $1 per day of 10 working hours.

(a)

(b)

(c)

(cl)

Quantity In place,
hauled per day by
each scraper.

Cost per cubic yard in

Elaoe for loading,
auUng, dumping,
and retujming.

Total coat

per cubic yard in place, exclaslTe of contractor's proflL

Length of lead, c
tance to which
l8 hauled.

Light Soils

Heavy Soils

Spread

Wasted

Picked and
Spread "Wasted

Plowed and
Spread "Wasted

Feet

cub yds

cts

cts

Cts

Cts

eta

cts

cts

50

60

6.9

9.0

8.0

14.5 13.0

11.5

10.0

75

50

8.0

10.1

9.1

15.6

14.1

12.6

U.l

100

45

8.8

10.9

9.9

1614

14.9

13.'4

11.9

150

36

10.8

13.0

12.0

18.5

17.0

15.5

14.0

200

30

12.7

14.9

13.9

20.4

18.9

17.4

15.9

Art. 13. By ears and locomotlTe, on level track. We have based
our calculations upon the following assumptions: Trains of 10 cars, each car con-
taining 1% cubic yards of earth measured in place. Average speed of trains,
inchiding starting and stopping, but not standing, 10 miles per hour, = 5 miles
of lead per hour. Labor $1 per day of 10 working hours. Loosening, loading
(by shovelers), spreading, wear Ac of tools, superintendence, Ac, the same as
with carts, Arts 2, 3, 5, and 7. Loss of time in each trip for loading, unloading,
Ac, 9 minutes, = 0.15 hour. Therefore

Number of tripe per ) ^ The number (10) of hours in a working day
day, per train / .15 + the number of 6-mile lengths in the lead

Number of cubici Number of Number (10) Number (1.5) of cubio
yards in place, per > s: trips per day X of cars in a X yards in place in each
day per train J per train train car

Cost per cubic yard, in place,| q^^ ^y^ t„,n expense* + 1 day's cost of track

for hauling, dumping, and V a - — r- ^— ^ — zi — . , , ^—3 r— r

.retqming j Number of cubic yards in place per day per traia

One day's train expenses :

Cost of 10 cars ® |100.....>.......... ...................m $1000

** looomotiTe ^ 8000

-S4000

One day's interest at 6 per cent, on cost of tndn......................^ |0.6T

Wages of engine driver (who fires his own engine)................;... 2.00

** foreman at dump. 2.00

** 3 men at dump at $1 S.OO

Puel 2.00

Water 1.00

ftepairs of focomotive and cars » 2.33

Total daily expense of one train $13.00

Depreciation (life of rolling stock taken as 10 years)
say SlOO per annum per $1000,
= S400 " " *• train,
= $ 4 " day (assuming 100 working days per year) 4.00

Daily expense and depreciation, one train, $17.00

Taking cost of track, laid, at $2500 per mile, and its life at 5 years, the dally
expense of track, for interest, depreciation, handling and repairs, may be
taken at $6.00 for each mile of lead.

Therefore,

808

COST OP EARTHWORK.

Oont Der cable yard tn place
for hauling, damping, and
retarning

'}

817 -^ (16 for each mile of lead)

Number of Number (10) Number (1.5) oi
trips per day X of cars in a X cubic yards la

per train

train

each car

Ti)tal cost per cubic
yard in place, ex-
clusive of contrac-
tor's profit

Cost per cub yd in Cost per cubic yard, in place, for
place for hauling, , looseuing, loading, spreading or
dumping, and re- "'' wasting, and snperiutendenc*, tc
turning (Arts 2, 3, 5, and 7.)

By Cars and liOcomotlve. Labor SI per day of 10 working hours.

(a)

(b)

(c)

(d)

'4^

•a

biejard, in
>r hauling.
, and re-

Total cost per cubic yard, in place, exclusive of contractor's profit

Light Soils.

Hearj Boils.

ll.

•o

•a

•o

"2

•0

«

•o

«

Length of 1
tanoe to i
is hauled

sua o
9

Gost per en

Slttce. f(
umping
turning.

Si-
ll

WOQ

It

04

a .
eOQ

^

1*

a .

a .
oOQ

a .

ij

op
CU

Miles

Cu. yds.

Cts.

Cl8. Cts.

Cts.

cts.

Ct8.

Cts.

Cts.

Cts.

H

750

2.47

11.30

10.80

10.04

9.04

16.77

14.27

13.27

11.77

yi

600

3.38

12.16

11.16

10.90

9.90

16.63

16.13

14.18

12.68

495

4.84

13.17

12.17

11.91

10.91

17.64

16.14

15.14

13.64

1

420

5.48

14.31

13.31

13.05

12.06

18.78

17.28

16.28

14.78

2

270

10.74

19.67

18.57

18.31

17.31

24.04

22.54

21.54

20.04

3

195

17.95

26.78

25.78

25.52

24.52

31.25

29.75

28.75

27.26

4

150

27.33

36.16

35.16

34.90

»3 90

40.63

39.13

38.13

86.68

Where large amonnts of work are to be done, the steam excairator, lmm4
dredg-e or steain shovel generally economizes time and money. Where tht
depth of cutting is leas than 10 ft, so much time is lost in moving from place to placs
that the excavators do not work to advantage. In sti£f soils, cuttings may be made
about from 17 to 20 ft deep without changing the level of the machine. For greater
depths in such soils the work is done in two levels, since the bucket or dipper cannot
reach so high. But in sand and looae gravel, much deeper cuts may be made from a
single leveL

The excavator resembles a dredging machine in its appearance and operatioii. A
large plate^teel bucket, like a dredging bucket, with a flat hinged bottom, and pro-
vided with steel catting teeth, is forced into and dragged through the eartli by
■team power. It dumps its load, by means of the hinged bottom, either into can
for transportation, or upon the waste bank, as desired.

Each machine is moanted on a car of standard gauge, which can be coupled in an
ordinary freight train. The car is made of wood or iron, as desired, and is provided
with a locomotive attachment, by which it can be moved from point to point as the
work proceeds. The machines can be used as wreeklnijf or derrick ears*
Each machine has a water tank, holding from 300 to 560 gallons, for the aopply of
its boiler.

Before beginning to excavate, the end of the car nearest the Avnrk Is lifted froa
the track by hydraulic or screw jacks, upon which It rests while M-orking.

In stiff soils the excavator leaves the sides of the cut nearly vertical ; and the de-
sired slope is afterwards given by pick and shovel. VHien the soil is hard or much
frozen, it may be loosened by blasting in advance of the excavator.

COST OF EARTHWORK. 809

The czcftTator baa to be mored forward (as tb» work adTaacee) abt 8 ft at a tlmcL
As regularly made, it can dig at a distaQce of 17 ft, borizoatally, from the center of
the car in any direction, and can damp 12 ft above the track. In saud or grarel it
takes ont, while actnally digging, 3 dipperftils (»■ 4^ to tf cub yards in the dipper.
M 8.76 to 6 cubic yards in place) per minnte; in stiff clay, 2 dipperfuls per minute
(a* 3 to 4 cub yards in tho oipper, m 2^ to 3.33 cubic yards in place). An aTerage
oay's work (10 hours) for a ** No 1 ** machine, including time lost in moving the ma-
chine, Ac, is about 500 cubic yards in ** bard-pan," and from 1200 to 1500 in sand and
grayel. Thia allows for the osual and generally ooatoidable delays in having cars
ready for the excavator.

The excavators carry about 80 to 90 lbs of steam. They bum from 100 to 160 lbs
of good hard or soft coal per hour; and require ono engineer, one fireman, one
crauesman, and 5 to 10 pitmen, including a bose. The pitmen are laborers, who
attend to the jack^ lay track for the excavator and for the dump cars, assist iu
moving the latter, bring or pump water, Ac, Ac.

After reaching the site of the work, about 30 minutes are required for getting the
excavator into working condition ; and an equal length of time, after completioB
of the work, in getting it ready for transportation.

The following figures are taken fkom the records of work done by a No 1 machins^
from May to Nov, 188S. The material was liard clay with pockets of sand. Tha
•xpenses per day of 12 working hours, at $1.50 per such day for labor, were

Water (a very high allowance) m.. $ 6.00

Goal, 1^ tons bituminous » 10.00

WMCe of engineer 4.00

** ** cranesman or dipper-tender «...M.M....M....M....... 2.50

** ** pit boes ........................MMM...M.. M.... ......... 3.00

** ** 8 pitmen at $1.60 .•...^.... 12.00

Oil, waste, repairs, Sto (estimated) ....MM.......................... 5.00

Interest on coet ($7600) of machiue.«....M................. 1.25

$44.26

Reduced to our standard of $1 for labor per day of 10 working hours, this would
be say $30.00 per day. Reduced to the name standard, and allowing for the greater
proportional Iocs of time in stopping at eyeoing and starting in the morning: the
average daily quantity excavated, measured in place, was, in shallow cutting, 630
cubic yards; in deep cutting, 1200 cubic yards; average of whole operation, 80O
cubic yards. This would make the cost, per cubic yiurd measured in place, for
loosening and loading into ears, 5.07 cts, 2.6 cts, and 3.75 eta respectively ; while the
cost by ploughing and sboyeling, in strong heafy soils, by Arts 8 and 3, is 7.4 cts ( and
by picking and shoyeling, say 10 cts.

810

008T OP EARTHWORK.

Art. 14. BemoTlnv roek excavation by wh«ell»arpai

A cubic yard of hard rock, in place, or before being blasted, will weigh aboat
1.8 tons, if saDdstone or conglomerate, (150 Sbe per cubic foot :) or 2 tons if good
compact granite, gneiss, limestone, or marble, (168 9>s per cuoic foot ) So that,
near enough for practice in the case before us, we may assume the weight of any
of them to be about 1.9 tons, or 4256 fi>s per cubic yard, in place ; or 158 lbs per
cubic foot.

Now. a solid enbie yard, wlirn brokea ap by blasting for remorai bj whesl>
barrows or earU, will oecupy a spaoe of aboot 1.8, or 1| eablo yards; whereas average earth, whea
loosened, swells to but about 1.2, or 1^ of its original bulk in place; although, after being made late
embankment, it erentaally shrinks into less than Its original bulk. In estimating for earth, tt b
assumed that ^ onbio yard, in place, is a fair load for a wheelbarrow. Such a cobio yard wlli weigh

2430
on an average 2430 lbs, or 1.09 tons; therefore, -r^ =» 174 lbs, is the weight of a barrew>load, tf

2.31 enblo feet of loose earth. Assuming that a barrow of loose roek should weigh aboat the i

42M
as one of eatth, we may take It at ^ of a cubic yard ; which gives -rr- = 177 lbs per load of

rock, occupying X cubic feet of space.

In the following table, columns 2 and 8 are prepared on the same principle as fbr earth, ma directed
In Article 10. Column 4 Is made up by adding to each amount in column 3, .2 of a cent for each la
fiMt length of lead, for keeping the wheellnc-plankB In order ; and 45 cents per cnbic yard, In plaeo,
as the actual cost for loosening, including tools, drilling, powder, Ac ; as well as moderate drainagtb
and every ordinary contingency not embraced In column 3. Contractor's profits, of conrae, are not
here included.

Ample experience shows that when labor is at tl per day, the foregoing 45 cents per eebio yard, la

{lace, is a sufliciently liberal allowance for loosening hiird rock under all ordinarr cireamatancss.
n practice it will generally range between 80 and 60 cents ; depending on the poettlon of the strats,
hardness, toughness, water, and other oonsIderationB. Soft shales, and other allied rooks, may fr»
quently be loosened by pick and plough, as low as 16 to 20 cents; while, on the other hand, slmllow
enttings of very tough rook, with an unfavorable position of strata, especially in the bottoma ef «•
oavations, may cost el* or even considerably more. These, however, are exceptional eases, of oom*
paratively rare occurrence. The quarrying of average hard rock requires about }^ to HJbot powder
per oubio yard, in place; but the nature of the rock, the position of the strata. Mo, may Inereaaeit
to ^ lb, or more. Soft rook frequently requires more powder than hard. A good ehurn-dtiller wHl
drill 8 to 10 feet in depth, of holes about 2H feet deep, and 2 inches diameter, per dav, in averagt
hard rock, at from 12 to 18 cents per foot. Drillers receive higher wages than oonunon laborefa.

Hard Rock, by Wbeelbarrows.

Labor f I per day, of 10 working hours.

Length of

Lead,ordis<

tance to

which the

rock is

wheeled.

Number of
cubic yards,

in place,

wheeled per

day by each

barrow.

Coetper

cubic yard,
in place,

for loading,

wheeling,

and

emptying.

Total cost
per cubic

yard, in
place, ex-
clusive of

profit to
eontraetor.

Length of
Lead, or dis-
tance to

which the

roek is

wheeled.

Number of
cubic yards,

in place,
wheeled per
day by each

barrow.

Cost per
cable yard,

in place,

for loading,

wheeling,

and
emptying.

Total eoil
peroabii

yard, la
plaoe, ex*
elusive ef

profit to
contractor

Feet.

Cubic Yds.

Cents.

CenU.

Feet.

Cubic Yds

Cents.

Cents.

26

12.3

8.64

63.7

600

2.96

85.6

81.T

60

10.7

9.81

64.9

700

2.62

• 40.1

86.6

76

9.68

11.0

66.2

800

2.84

44.8

91.4

100

8.66

12.1

67.8

900

2.12

49.6

•64

160

7.26

14.6

69.8

1000

1.94

64.1

101.1

200

6.26

16.8

62.2

1200

1.66

68.6

UM

2S0

6.49

19.1

64.6

1400

1.44

72.9

129.7

800

4.89

21.6

67.1

1600

1.28

82.2

180.4

350

4.41

28.8

69.6

1800

1.16

91.6

1401

400

4.02

26.1

71.9

2000

1.04

100.8

ia.8

450

3.69

28.6

74.4

2200

.963

110.2

169.6

600

3.41

80.8

76.8

2400

.879

119.6

1«J

Art. 15. Remowlng^ rock excawatlon by carta. A cart-load of
rock may be taken at ^ of a cubic yard, in place. This will weigh, on an averageL
851 ftn ; or but 41Ibs more than a cart-load of average soil. Since the cart itself will
weigh about ^ a ton, the total loads are very nearly equal in both cases. Columns
2 and 8 of the following table are prepared on the same principle as for earth, as
directed in .Art. 4. Clolumn 4 is made up by adding to each amount in column 8;
the following items: For blasting, (and for everything except those in column 8;
loading, and repairs of cart-road,) 45 cents per cubic yard, in place; for loadiniL
S cents, per cubic yard, in place ; and for repairs of road. .2, or f of a cent for ea(£
100-feet length of lead. Ck>ntractor'8 profit not indudea.

eoer of eabthwork.

811

Hard Boek, by Carts.

Labor $1 per day, of 10 working hours.

Lmgibor

Namberof

Ooitper

Total ooit
per oobie

yard, in
plBoe,ex-
claalTe of

profit to
oontraetor.

Length of

Kumber of

Cost per

Total eosi
per cable
Tard, in

plaoe, ex«

eluslTC of
profit to

oontraetor

UMMl.ordl«-

Ottbio yards,

cable yard,

Lead, or die-

cubic yards,

cubic yard,

taoee to

in place,

In place.

tance to

In place,

in place, for

whioh tbe

battled per

for hauling,

which the

hauled per

hauling,'

roekia

day, breaeb

•ad

rookie

day, by each

and

kMdod.

oart.

anptylng.

hauled.

oart.

emptying.

WttH,

OvbteYda.

Cent*.

Oenti.

Feet*

OaUeTda.

Gents.

Cents

1ft

lO.J

0.61

69.6

1800

6.00

36.0

81.6

SO

18.6

0.77

60.9

1900

4.80

96.0

83.6

T6

17.8

7.0S

00.3

3000

4.63

37.1

84.1

100

17.1

7.«

00.6

3:00

4.31

39.7

87.8

160

16^

7.81

61.1

3500

8.87

88.8

90.8

MO

15.0

8.88

61.7

Mmile

8.70

88.7

98.0

SOO

18.8

9.87

68.0

8000

8.88

87.6

96.6

MO

13.0

10.4

04.3

8350

8.13

4Q1

99.6

800

10.9

11.6

06.6

8600

3.93

48.8

108.8

•00

10.0

13.6

06.7

8750

8.76

46.8

106.8

no

9.38

18.6

68.0

4000

8.61

47.9

108.9

AM

8.67

14.6

00.3

4360

8.47

60.6

113.1

too

8.00

15.0

70.4

4600

3.86

68.8

116.8

1000

7.60

10.7

71.7

4760

3.34

66.8

118.8

1100

7.00

17.7

73.9

6000

3.14

58.4

181.4

1900

6.07

18.7

74.1

ImUe

8.04

61.3

11A.8

laoo

0.83

19.8

75.4

1«"

1.67

76.0

141.8

1400

6.00

30.8

76.6

IH"

1.41

88.8

167.6

1600

6.71

31.9

77.9

IH'*

1.33

103.5

174.0

1000

6.46

33.9

79.1

8 "

1.06

116.8

190.4

1700

5.33

34.0

80.4

«M"

.963

ISu.O

306J

*' JLoose roek *^ will cost abont 30 cts per jd leas ; and even s<aid rock will

age about 10 ots leas than the tablee.

Art. 16. Removlnic roek exeawatton by ears- and locomo-
tlwe, on level track. Our calculations are based upon the following assump-
tioiu : Trains of 10 cars, each car containing 1 cubic yard of rock measur^
In place. Average speed of trains, including starting and stopping, but not
■tanding, 10 miles per hour = 5 miles of lead per hour. Labor 91 per day of
10 working hours. Loosening, 45 cts per cubic yard In place. Loading, 8
ctB per cubic yard in place. Cost of track, for interest and repairs, tS per
day per mile of lead. The calculations are the same, in principle, as those
in Art. 13.

Hard Koek, bjr Cars and I^eomotlTe.

Labor 91 per day of 10 working hours.

liength c(f lead, or distance to which the rock

ifl hauled miles 1 8 6 7 10

Number of cubic yards, in place, hauled per

day by each train 2900 1300 800 600 400

Cost, per cubic yard in plaoe, for hauling,

. dumping, and returning cents .6 1.7 8.5 5.7 10.8

Total cost, per cubic yard in place, exclusive

of contractor's profit cents 58.6 54.7 06.5 58.7 63.8

812 TCKNEI&

TUNNELS.

Taniiels for railroads sbould, if possible, be straig^lity espe*
eially when there is but a single track ; inasmuch as collisions or oth«r accidents
in a tunnel would be peculiarly disastrous. A tunnel will rarely be expedient
before the depth of cutting exceeds 60 feet. Firm rock of moderate hardneaSi
and of a durable nature, is tbe most feTOrable material for a tunnel;
especially if free from springs, and lying in horizontal strata. In soft rock, or
in shales (even if bard and firm at first), or in earth, a lining of hard brick or
masonry in cement, is necessary. A tunnel should have a grade or Ineli-
nation in one direction, for ease of future drainage and ventilation. No
special arrangement is essential for ventilation either during conatructioo,
or after, if the length does not exceed about 1000 feet; but beyond that, gen<i
erally during construction either shafts are resorted to, or means provided for
forcing air into the tuouel through pipes from its ends. But after the work it
finisbra, except under peculiar circumstances, nothing of the kind is necessary*
Shafts often xTraw air downwards; and frequently, even when aided by a flteepb
uniform grade, do not secure ventilation. The Mont Cenis tunnel under the
Alps, completed in 1871, Is 7U miles long, and has no shafts, although it grades
up from each end, which is tne most unfavorable of all conditions for ventila*
tion without shafts. It was made so for facilitating drainage. Its ▼entilatioa
is maintained by air forced in from the ends. The Hoosac tunnel. Mass, 4^
miles long, has shafts ; one of them 1030 feet deep: but they were for expediting
the work. iStaalts i^enerally cost from 1)4 to 3 times as much per cubic
yard as the main tunnel, owing to the greater aifficulty of excavating and rs*
moving the material, and getting rid of the water, all of which must be don*
by hoisting. When through^ earth, they must be lined as well as the tunnel;
and the lining must usually be an under-pinning process. Or the lining may
first be built w>eT the intended shaft, and then sunk by undermining it grad-
ually. Their sectional area commonly varies from about 40 to
100 square feet. They have the great advantage of expediting the work by in-
creasing the number of points at which it can be carried on ; but if placed too
close together, their cost more than compensates for this. The air in some
tunnels, while being constructed, is much more foul than in others; so that
after the work has been commenced, shafts with forced air may be expedient
where they were not anticipated. In excavating the tunnel itself, a beadinc
or pa^<4age.way, 5 or 8 feet high, and 3 to 12 feet wide, is driven and maintained
a short aistance (10 to 100 feet, or more, according to the firmness of the ma-
terial) in advance of the main work. In rock, the heading is^ust below the
top of the tunnel, so that the men can convenif^ntly drill holes in its floor for
blasting; but in earth, the heading is driven along the bottom of the tnnneL
that being the most convenient for enlarging the aperture to the to\\ tunnel
size, by undermining the earth, and letting it fall. In earth, the top and sides
of the heading, as well as of the tunnel, must be carefully prevented froa
cavini; in before the lining is built; and this is done by mean's of rows of vertt*
cal rough timber props, and horizontal oaps or overhead pieces, between whieh
and the earth rough boards are placed to form temporary supjMrting sides and
ceiling to the excavation. The props and caps are placed first ; and the boards
are then driven in between them and the earthen sides of the excavation.
These are gradually removed as the lining is carried forward. Tbe UniaVt
when of brick, is usuallv from 2 to 8 bricks thick (17 to 26 inches) at bottom,
and from 1^ to 2^/^ bricks thick at top; and when of rough rabble in cement,
about half again as thick. It is important that the bricks or stone should bii
of excellent hard quality, and laia in good cement. The bricks should be
moulded to the shape of the arch. As the lining is finished in short lengthy
and before the centers are removed, any cavities or woids between it ana
the earth should be carefully and compactly filled up. Even in rock, if mnch
fissured, or if not of durable character, as common shale, lining is necensry.
Tbe cross-section of a single-track railroad tunnel, in the clear of event*
thing, and for cars of 11 feet extreme width, should not be feM than about If
feet wide, by 18 feet high ; nor a double-track one, less than 27 feet wide, by M
feet high ; unless in the last case the materia] Is firm rock, in which t blgn aroh
is not necessary for lining. The roof may then be much flatter, so that a balglil
of 20 feet may answer. With cars of 10 feet extreme width, the width of^«
tunnel may be reduced to 25 feet ; or with 9 feet cars, to 28 feet. Many have
been made 22 feet. The Mont Cenis is 26 feet wide, by 25 high. The rato of
daily progress from exuh fsce of a tunnel varies from 18 inches to 9 feet 01
length per 24^hours, with three relays of workmen. On the Mont Cenis the

TBE8TLES. 818'

tramM were *bovt I to » tMt Atilj tor ■ wbote reir, rrom rich lace. Drills
wo4«d bf compreflBfld air wen em|>Joysd Id Iha headlufEi, which war? 12 feet
wide bi » feat bigb. Ordinarily, from IJi to a ffiM raaj bO Uken sa lisngn.
TbB dilTereDiie of rate of proKreu between a tingle »nd a dnuble tracli tunnel
is not so vreat u mleht he ftuppoiod ; inumucb ai a larger rorca can tie em-
plDied on tbe wider one. If the tunnel i> in eartb.the construction of tbe
llnlog ttboul makes up for tbe slDwai eicaratloo or DOS In nxk. Iq rock. *iUi
■■bar at tl per dar, tbe coat will unuallr Tar^ witb Ibe cbaracter of the rock.
ItamK tots per cubic yard fl>r ttas main LuDuel; and fromts intlO (or tbe
heading: while ihafta will average about 50 per ceot. more than beading. Tits
' eoaor* hlngie-tracli^tmmel, wjien common labor 1b (Jperdaj-.wiil g€n«^\T

itiB uBiiallr met with;
ips or bf balling. Tbe
.g rtd of tbe Biuoke in

.hould be >u

d«T of 24 boura wlib two alill^ uf 12 boun ewfa. wa> a> roliana; bi h>ad
dTJlUnii 2.8 Teet and 2.« feet retpoctlir'lf from each end; bj macblne drllli
(two riTal drills in competilloot G.6 feel ind 7.6 feet. Tbe niitEritl was iiard
g»7 aandilone. For (be wbnle tunnel the rate was about 2 feet per dai.

For farther information rcapecting lunnpia, Ibc reader is referred to Mr.
B. S. DrlDker'a Tsr^ full treatise on (be subject, published bf tbe MSHn Wiley,

TSESTLES.

w I. & [t. 9. <^ T, are

814 TRESTLES.

and 8, to beigtats ftom 20 to 30 ft ; Fig 5, from 80 to 40 ft: Fig 6, fVora 40 to 60 II; ai
rough approxiniations merely. A single framework, such as that shown in eaeh o(
these six figures, is called a **bent/* These bents of course admit of many modifi-
cations. They are nsnally supported by bases of masonry, as in the figures. These
preserre the lower timbera from contact with the earth, which would hasten their
decay. It is udrisable to make these bases high enough to prerent injury from cattle^
orpassing vehicles, Ac. Up to heights of about 40 or 50 ft, a single row of po8t.s or np-
rights, a, a, a, Figs 1 to 9, as shown at « « under Figs 1 and 0, will answer. But as the
height becomes greater, more posts should he-introduced, as shown at 3 a; under Fig
i; or two entire rows of them ; or three rows, as under Fig 7^ and as also in Fig 8,
which is an end Tiew of Fig 7. Figs 7 and 8 bear much resemblance to the trestlst
190 ft high, with masonry bases 30 ft high (8. Seymour, C.B.), wliich carried the
Srie Rway (now the N Y, Lake £rle k West*n R R) over the Ci^oiiesee River at
Portafpe, If IT. There each bent had 21 posts 14 ins square, at its base ; and li
posts of 12 X 12, at its top. The other timbers were 6X 12; many of them were la
pairs, embracing the posts. This single-track viaduct was begun July 1, 1851, and
completed Aug. 14, 1852. It contained l,e02,000 ft (B M) of timber, and 108,868 Iba
of iron. In the foundations were 9200 cub yds of masonry. The entire cost waa
about $140,000. It waa burned down in 1875, and was replaced, in less than 3 mos,
with a single-track viaduct of ivrouffbt-lron trestles, containing, in
all, 1,340,000 lbs of iron, and 130,600 ft (B M) of timber; and casting, complete,
above the masonrv, about 895,000. Frequently the posts of trestles are in pairs;
and the other timbers pass between ; all bolted together.

In Fig 4, the posts n^OyOy are end views of three trestles or bents, snch as Fig 8;
and<< are diag braces extending from trestle to trestle ; the two outer ones inclining
in one direction; and the central one crossing them. These may be placad either
intermediate of the posts, as in Fig 3; with the heads of the two outer ones confined
to the cap c c of one trestle ; and their feet to the sill yyot the next one ; or thej
may all be spiked or bolted to the posts themselves, as in Fig 4. The last is the bes^
as it serves also directly to stiffen the posts: as do also the braces oOyftn^ Fig 2.
Such bracing is too frequently omitted. During the passage of trains, the backward
pressure of the steam, exerted through the driving wheels against the track, pro-
duces a serious strain lengthwise of the road, and tending to npeet the trestles; and
the sudden application of brakes to a moving train, produces a similar strain in the
opposite direction. These strains become moredangeruuflasthe ht increases. Hence
the need for such braces. Usually the outer posts may lean 1.6 to 2.5 ins to a ft.

The posts should not be less than about 12 ins square, except in quite low trestles;
and even then not less than about 10 X 10. The diag bracing may generally be abont
as wide as the posts ; and Iialf as thick. The disit apart of the bedts, when the road*
way is supported by simple longitudinal beams, should not exceed 10 or 12 ft, for
railroads. But if these beams receive support from braces beneath, like ss. Fig 8 : or
fiom iron truss rods, the dist may be extended to 16 or 20 or

more ft. But when the trestles become very high, and contain a great deal of tim«
ber, it becomes cheaper to place them farther apart, say 30 to 60 ft; and to.earnf
the railway upon regular framed trusses, as at t<u. Figs 7 and 8; as in a bridge witli
•tone piers. In the Genesee viaduct, the trestles were 50 ft apart, center to center.

When such a trestle as Fig 8 becomes very narrow in proportion to its height, we
may add to its stability by introducing beams w, extending from trestle to trestle)
and still further by inserting diag braces v v, as in the old Qenesee viaduct.

As tar as practienble, arrange the pieces so that any one may be removed if it
becomes decayed ; and another put iu its place.

On carves, additional strength should be given on the convex side; as suy*
gested by the dotted lines in Fig 5. On very hi^h trestles especially (as well ai
on bridges), wheel-guards, g g, Fig 10, either inside or outside of the lails,
should never be omitted.

In marshy groundy piles may be driven to support the trestlee; or may be left so
far above ground, as themselves to constitute the posts. Snch treaties may often be
used advantageously, even when to be afterward filled in by embkt. They then sus*
tain tlie lails at their proper level until the embkt has reached it final settlement.

They are generally used to avoid the expense of embkt; especially when earth caa
only be obtained from a great dist. Even when earth and timber are equally con-
venient, they will rarely much exceed about half the cost of embkt; even when bvt
about 30 ft high ; but owing to their liability to decay, they should be resorted tt
only in case of necessity ; or as a temporary expedient.

RAILROAD CONSTRUCTION.

815

BAIil^AST.

T»ble of cubic yards of ballRsi per mile of road.

Side^lope of the ballast 1 to 1. Width in clear between 2 tracks 6 ft. The ties
and rails may be laid first, for carrying tho ballast along the line; then raised a
few ft of length at a time, and the ballast placed under them. Deduct for tieSt
as below.

Depth

in

Ids.

Top width,
SnoLa Track.

Top width,
DovBLs Traok.

10 Ft.

11 Pt.

12 Ft.

21 Ft.

22 Ft.

23 Pt.

12
18
M
SO

Cub. Y.

2152
3874
4694
6111

Cub. Y.

2347
3667
5085
6600

Cub. Y.

2543
S960
5474
7087

Cub. Y.

4303

6600

8996

11490

Cub. Y.

4499

6894

9388

11980

Cub. Y.

4695

7188

9780

12470

A BUOft can brealc 8 to 4 cable yards per day, of hard quarried stone to a siise
suitable for ballast; say areraging cubes of 3 Inches on an edge. Where other
iMtllast cannot be had, hard-burnt clay is a good substitute. The slag from iron
Aunaces is excellent. The ties decay more rapidly when gravel or sand is used
instead of broken stone, because these do not drain off the rain, but keep the ties
damp longer.

TIES.

In the Unitsd States the life of a tie is about as follow*:

Average,

ATerage,

Yean.

Yean.

Yean.

Yean.

Ohestnut,

6 to 12

7

White Oak,

5 to 12

7

Oedar,

6tdl5

9

Spruce Pine,

4to 7

. 6

Hemlock,

3 to 8

5

It will often, especially in the case of the softer and more perishable woods, b«
true economy to preserre ties by the injection of creosote. ' Creosote

preserveR the spikeg.

The writer believes that most of the fault usually ascribed to cross-ties, as well as
to rail-Joints, is in reality due to imperfect drainage of the roadbed. Hence, he does
not agree with those who advocate vert long ties ; but considers that with good
ballast, on a well-drained roadbed, S}/^ ft is as good as more; and that 8)^ ft, by 9
ins, by 7 ins; and 2V^ ft apart from center to center, is sufficient for the neaviest
traffic. On many important roads they are but 8 ft ; and on some only 1}4 ft long ;
track 4 ft 8)^. On narrow-gauge roads the ties are generally from 6 to 7 ft long.
The actual cost of cutting down the trees, lopping o£f the branches, and hewing
the ties ready for hauling away to be laid, is about 6 to 9 cts per tie, at $1.75 per
day per hewer.

The narrow bases of rails, resting immediately on the cross-ties, without chairs,
frequently produce in time such an amount of crushing in the ties as to ii^'ure them
materially even before decay begins. Bumetised tiee n)st the spikes away rapidly.
Greosoted ones preserve them.

Cross-tiMi of 8^ feet, by 9 inches, by 7 inches, contain 3.719 cubic feet each;
and if placed 2% feet apart from center to center, there will be 2112 of them per
mile, amounting to 291 cubic yards. Therefore, if they are completely embedded
in the ballast, they will diminish its quantity by that amount. At 2 fpet apart there
will be 2640 of them, occupying 364 cubic yards; and at 3 feet apart, 1760 of them;
t2S43 cubic yards.

816

RAILROAD TIES.

Cubic feet eoDtoined in eross-tles of diflTerent sixes.

Dimensions.

•

Ck>ntenta.

Dimensions.

Contents.

Ft. Ins. Ins.

Cub. Ft.

Ft. Ins. Ins.

Cub. Ft.

8 by 8 by 6

2.667

S}4 by 10 by 7

4.132

8 9 6

3.000

8>| 10 8

4.722

8 9 7

3.500

8>| 12 8

5.667

8 10 6

3.333

9 8 6

3.000

8 10 7

3.889

9 9 6

3.376

8 10 8

4.444

9 9 7

3.938

8 12 8

5.333

9 10 6

3.750

8)4 8 6
S% 9 6

2.833

9 10 7

4.375

3.188

9 10 8

5.000

8>2 9 7
8^ 10 6

3.719

9 12 8

6.000

3.542

TIE PliATES.

Where the rails bear directly upon the ties, the great unit pressure of the
narrow rail base, the churning action of the rail under passing wheels, and the
hastening of decay by the bruising of the wood fibres, cause rapid wear of the
tie immediately under the rail.

Among prominent forms of tie plates are the Servis, the Goldie, the Church-
ward, and the Wolhaupter.

Servis.

GOLDIK,

All consist essentially of a flat iron or steel plate, laid on the tie immediately
under the rail. Spikes, holding the plate and the rail in place, are driven into
the tie through holes in the plate. Nearly all successful forms have two or more
ribs on the lower side. These ribs stiffen the plate, but their principal use is,
by cutting into the upper face of the tie, to prevent motion of the plate and con-
sequent aorasion of the tie. In some forms the ribs run across the fibres of the
tie ; but ribs running mitk the fibres, as in the Servis plate, are usually pre-
ferred, as being more easily imbedded, more difficult to displace, and less in-
jurious to the fibres. Most forms have also a shoulder on the upper side, to assist
the spikes in preventing spreading of the rails, and, in some cases, to act as a
rail brace; but this shoulder is seldom considered essential.

Tie plates erciitly lengthen the life of the tie. On curves and bridges the sav-
ing in a numr)er of cases hfts been estimated at 50 per cent, in cost, and 60 to 75
per cent, in labor. The tie plate has often displaced small gangs of men whose
sole duty it was to replace ties.

Tie plates cost from 5 to 15 cents each ; and placing them costs f^om ^ to IK
cents each.

Most roads use tie plates only under rail Joints; on curves, heavy grades,
bridges, and trestles; in tunnels where there is much dampness; at switches, at
stations, and at street and road crossings ; in yards ; and where sand is moch
used by the locomotives.

KAILROAOe.

Brocy sq liich <tf lectio , . .

rtil; or 10 lfr'714fl tooH p«r miU of sEogLe-tn

Wlinl

Wt ill Iba per yi of rail, of »i

Thna. a nil of 1M lona per mile of alngle Irock. will bare a eeetton of 6MI ag
Ine ; and will weigb MM BM per }d of elDgl* rail. Add foe turoonta, aldians. road-

■ onrmtleo^TOin'plelejIngle-'irKksiiiierslrocluwperweek. " 'J'"™

Steel mlU laat fcom 9 lo -jfi j-ean ; averago 15 .vaan.

818

RAILROAD SPIKISL

RAII<BOAD SPIKES.

The hook-lieacled spikes t, commonly ased for confining rails to
the cros»-ties, vary within the limits of the following table : the lightest
ones for light rails on short local branches ; and the heaviest ones for
heavy rails on first-class roads. The spikes are sold in k^s usually of
160 1m. For the weight of Spikes of larger dimensions, we may near
enough take that of a squsre bar of thi same length. What is saved at
the point suffices for the addition at the head.

Slseinins.
Length. Side.

No. per keg
of 150 lbs.

526
400
705
488
890
296
267

No. per
100 lbs.

350
266
470
826
260
197
171

Sise in ins.
Length.) Side.

No. per keg

i.pe:
160

of 160 lbs.

360
289
218
310
262
196

No. per
100 llM.

288
198
146
207
175
180

A mile of sine le«traelL road, with 2640 cross-ties, 2 feet apart f^om
center to center ; ami with rails of the ordinary length of 30 feet, or fifteen ties
to a rail ; will have 352 rail-Joints per mile ; and,, with 4 spikes to each tie, will

require 10560 spikes, or nearly 37 kegs (5500 lbs.) of 5}4 X Ai & siM in very com-
Butan allowance "must be made for rail-guards at road-crossings, which we

mon use, which weighs a trifle more than V^ lb. per spike.

rail-ir

may assume to be 30 feet wide, or the length of a rail. A guard will usually con-
sist of 4 extra rails for protecting the track -rails, and spiked to the 15 ties by
which said track-rails are sustained. Ck>nsequently such a crossing requires
15 X 8'= 120 spikes. For turnouts, sidings, loss, etc., we may roughw average
700 * spikes more per mile: thus making in all (if we assume one road-crossing
per mile) 10560 -|- 120 -{-700 = 11380 spikes per mile ; or say 6000 lbs. or 40 kegs of
160 lbs.

Adliet«ion of Spikes. Professor W. R. Johnson found that a plain spike
.876, or % inch square, driven 8^ ins. into seasoned Jersey yellow pine or un-
seasoned cbestnut, required about 2000 lbs. force to extract it; flrom seasoned
white oak, about 4000; and from well-seasoned locust, about 6000 lbs. Bevan
found that a6-penny nail, driven one inch, required the following forces to ex-
tract it : Seasoned beech, 667 lbs; oak, 507 ; elm, 327 ; pine, 187.

Very careful experiments in Hanover, Germany, by Engineer Funk
give from 2466 to 8940 fi>s. (mean of many experiments, about 3000 lbs.),
as the force necessary to extract a plain ^ inch souare iron spike, 6
inches long, wedge-pointed for 1 incn (twice the thickness of the spike),
and driven A}4 inches into white or yellow pine. When driven 5 inches,
the force required was about t^ part greater. Similar spikes, A io<^^
square, 7 inches lone, driven 6 inches deep, required flrom 8700 to 6745
lbs. to extract them nrom pine ; the mean of the results being 4873 fi>B.
In lUl cases about twiee as much force wa* required to extract tnemjrom wMk. The

3;>ikes were all driven across the grain of the wood. Experience shows that when
riven vfUh the grain, spikes or nails do not hold with much more than half as
much force.

Jagged spikes, or twisted ones (like an au^er), or those which were either
sweliMl or diminished near the middle of their length, all proved inferior to
plain, square ones. When the length of the wedge point was increased to 4
times the thickness of the spike, the resistance to drawing out was a trifle less.
But see *Mag-spike" in Glossary.

When the length of the spike is fixed, there is probably no better shape than
the plain square cross-section, with a wedge-point twice as long as the width of
the spike, as per this fig.

* This allows that turnouts and sidings amount to about 1 mile of extra track oa
16 miles of road.

BAIL-JOINTS. 819

BAII^-JOINTS.

Art* 1. A track, beiog weakest at the joints between the rails, where they
<^ deprived of their yertical strength, has of course a greater tendency to bend at
those points ; and this bending produces an irregularity in the morement of the
train, which is detrimental to both rolling-stock and track. Moreover, that end of a
imil upon which a loaded wheel is moving, bends more than the adjacent unloaded
end of the next rail ; so that when the wheel arrives at said second rail, it imparts to
its end a severe blow, which injures it. Thus, the ends of the rails are exposed to
fiur more ii:\jury than its other portions. Numerous devices have been resorted to for
strengthening th6 Joints of the rails, with a view of preventing this bending entirely ;
or, at least, of causing the two adjacent rail-ends to bend equally, and together ; so
as to avoid the blows alluded to. None of these Joint-fiststenings, known as chair^
flalk-plates, wooden blocks, Ac, have proved entirely satisfactory.

Much of the deficiency ascribed to the fiurtenings, is, however, really due to wan*
of stability in the cro8S:ttes at the Joints, and more attention must be directed to
this latter consideration, bofore an efficient fastening can be obtained. Observation
shows that when the Joint-ties are very firmly bedded, almost any of the ordinary
fikstenings will (if the Joint is placed between two ties, instead of resting upon a tie),*
answer very well; whereas, when the cross-ties are so insecurely bedded as to play
up and down for half an inch or more under the driving-wheels of the engines, the
■Wrongest and most effective fastenings soon become comparatiTely inoperative. All
the parts of the best of them will in that case becbme gradually loosened, warped,
beni or broken.

Experience has established the superiority of suspended Joints over supported
ones. Long Ikstenings, perhaps, possess but little superiority over short ones, where
the track is not kept in good repair ; for the great bearing of the former, although
imparting increased firmness on a good track, bec9mes converted into a powerful
leverage, by which it accelerates its own destruction, in a bad one. An element in
the iqjury of joints, is the omission of proper fastenings at the center of the rails.
Each rail nhould be so firmly attached to the cross-ties at and near its center, as te
compel the contraction and expansion to take place equally from that point, toward
each end. It would probably be somewhat difiicult to accomplish this perfectly.
The attempts hitherto made have failed.

Under the eixtremes of temperature in the United States, b^r iron expandfli
•r eontracto about 1 part in 916 ; or 1 inch in 76^ feet ; consequently, a rail
80 ft long will vary /» inch ; and one 20 ft long Ailly ^incti.

Beside tliis, the rails are Terjr liable to move or creep
bodily in tbe direction of tbe beamiest trade, especially when the
gmd^ descends in the same direction ; and by this process also the Joint-fastenings
sre exposed to additional strain and derangement.*

All rails appear to become elongated very slightly at their ends by use ; and this
renders a full allowance for contnustion and expansion the more necessary.

Art. 2. ETen Joints and broken Joints. If, in the two lines of
rails forming a track, the Joints are placed opposite to each other, they are called
**even Joints;" while "staggered" or '* broken" Joints are those where each Join!
Id one of the lines of rails is opposite to the middle of a rail in the other line. Iv
the latter case, the Jar of passing from rail to rail is less severe, but of course more
frequent, than where both wheels make that passage at the same time.

Art. 8. Beveled, or mitred Joints. To lessen this Jar, Mr. Sayre sug*
geets cutting the rails so that the vertical plane forming the rail end shall make an
angle of 45° to fXP with the longitudinal vert plane of the web of the rail, instead of
the usual right angle. This would permit the use of longer rails than are now laid,
am the great space (^ inch or more) between the ends of such long rails in cold
weather, would not be so serious an objection when the ends were thus cut obliquely.
This method of cutting the rails has been tried, with good results, but has not yet
come into general use. It is claimed that a comparatively inexpensive change in the
arrangement of the saws at the rolling mill, would permit the rails to be cut with
ends at any angle, as readily as with square ends, and without further increase in
tbe cost of sawing.

• In the first case the joint is called a suspended one; in the last a supported

820 BAiL-Jonrra.

Art. 4. lisli-plBtca, Fig. 1, and Atutlc-mlnUn, Fin 2,

ntmrlv suppUDted h11 olbei forma of Joint od the prin-
cipal railroads of the V. S. Thej are rolled in long
bire, and cut off in aof deeired Ifiogtb, generally about
S tt; HDd are bolted logetbar, and lo tlie nils, bj 4

Art. S. Tke >lati-plBl«>olntwaaoDeDf tb«

urilast goggested. It waa iDtroduoed upon tbs New-
castle sad Frenehtown K E, iu DelBware, by Kobl.
H. Bacr, in IMS. Tba weigbCof aconiplele fiah-plUs
joint, Includlni bolle and nuU, is abaut 20 tta.

Art. 6. Tbe princ

■bicb are drii

ayar.
pial*

ipal adTanlageoC tbe ansle-plBl

igh sjots in^lbelj flanaea looonTinBl

of iKe rails to "ereen."* (See Art L(
I of ilie mill bSTS (o be slatted for this

somewhat loose, provided tKe spites boJiTflrra. Moi
tbeauglejolnt addsgreally 10 ibe Jalera/stceugtbol

antage of tbe angle plate ie tbat It tranefersal least a part of tbe

continue to give aome eupport even If the bolla sboutd becmu
le, provided the spites bo] a flnu. Moreotor, tbe spreading base of

""hS.'"

P'8-2- Pig. 2 A.

Art. 7. Tbe rollowing are usual dimensions of angle plates :

Fop raiU.

HeighL

™.t,T..";

l-Si?'

60B>6 4 7oas
SStbs

U'"-

li'-

S^

•OatbeSt. LoiiiH bridge (steel arcbes) and its eastern approac-b (plate girders

dav, both up and down a grade'or ») feet per mile, aud witb luch force thai,
althoogh various fasteniDgs were used, In order lo preveut tbe creeping, none
proiedr effectual, and tbe track was adjusted daily lo accommodate tbe creeping.

n BaUToad biaeila, Jna. 2d! 1

BAIL-JOINTB.

821

Art. 8. The wheel-tread and 76 9> steel rail, shown (one-fifth of real size) in

Fig 3, are those designed by Robt. H. Sajrre,
€. £., and used by the liehi^b Valley K R,
under very heavy traffia Tlie aDg^le-jplate
lolnt was designed by Mr. Jobn Fritz,
3upt Bethlehem Iron Ck>. Bethlehem, Pa, ana
Sayre.

i:

3 4
INCHES

Fig.3.

These forms of wheel-tread, rail, and splice,
are the result of careful study, and each detail
has been modified from time to time as experi-
ence dictated. The stems of the two plates are
f)laoed wide apart, thus giving the joint greater
ateral strength ; at the same time adding to its
vertical strength by the support given to the lower
side of the rail-head by the upper enlargement c ;
while the lower one a secures a full bKearing on
the flange of the rail. The joint, for 76 S> rail,
complete, 2 ft long, with 4 bolts J^ inch diam,
weighs 40 to 48 &>s, depending upon the thick-
ness of the angle plate. The drilled bolt-holes
in the stem of the rat/, are 1 inch dlam, to allow
the rails to contract aud expand.

Art. 9.

rnrr

0"

Fig.

Figs 4 and 5 (one-fifth of actual size) show an ang-le-plate

Joint made by Cambria Steel Cu, and
furnishedwith their |»ateiit nut-lock,
which consists of a small piece, or " key,"
p, of Bessemer steel, semi-circular in cross-
section at one end, and tapered to a hori-
zontal edge at the other. After the nut
has been screwed to its place, the key is
driven close up to it, and then the pointed
end of the key is bent up (as shown in
Fig 5) by a special tool with a lever
attached. The key is prevented from
falling out sideways by the edge of the
longitudinal groove, Fig 4, in the angle-
plate, into which it fits.

2
4.

INCHES

Fi^,5.

Art. 10. Both fish- and angle-plates are apt to crack vertically about the
middle of their length, or opposite to the joint in the rail. To obviate this, the
** Samson- bar '' (made either of fish or of angle form) was rolled about
half inch thicker at the middle than at its ends. The thickened portion was
about 8 ins long, extending say 4 ins each way from the joint; but the upper
edge of the bar, upon which the head of the rail rested, and in ^^bars the
lower edge also, were made of this increased thickness throughout theii leugth.

Art. 11. Fish- and angle-plates, of all the patterns shown, and others, are
rolled to suit diflTerent sizes and sbapes of rails. The bolt
beads are usually round, and the shoulders of the bolts, immediately
under the heads, are therefore made of owal cross-section, fitting into corre-
sponding oval holes in the fish- or angle-plaf e. The bolt is thus prevented from
turning when the nut is screwed on, and afterwards. Many devices have been
tried, with a view to preventing: the nuts ft*oni ivearingr loose

The plates are fiequently rolled with a longi'itndinal grroove, as wide as
the head or nut of the bolt, and about "% inch deep, running their entire length.
This groove receives either the head of the bolt, which in such cases is made
square or oblong and inserted first, and the nut afterwards screwed on ; or else
the nut is first placed in the groove, and the bolt afterwards screwed into it.
This is intended to prevent the unscrewing of the nut, but cannot be relied upon
to do so.

It is well to have the slots in the flanges of rails or of angle-bars so spaced that

822

KAIL-JOINTS.

tlie two spikeii of a Joint, driven Into the same crosfi-tlO|
■liall not be directly opposite to each other, but '' stHggered/' so aa to
diminish the danger of splitting the tie.
Joints are frequently laid with one fish- and one ang^le-plate.

Art. 12. It will be noticed that both fish- and angle-plates act by plaoing a
support under the head of the rail. The Fisher brldg^e-Joint, Figs 6'to 9,
made by Mr. Clark Fisher, Trenton, N J, applies the support under the bate of th«
rail.

The principal feature of this joint is a flanged beam, Fig 6, about 6 ins wide and
22 ins long, which extends across, and is spiked to, the two joint-ties, as in Fig 7.
The holes for the spikes are placed so that the two spikes in the same tie are not
opposite to each other; and the flanges F F also are staggered, so as not to interfere
with the driying of the spikes. The joint-ties T T are placed 7 inches apart in the
clear. The beam has an upward camber of abont one-eighth of an inch. The two

Fi^.9.

mg,e.

Fig. 7.

Fig. 8.

rail-ends, forming the joint, rest upon the beam, and meet at the middle of iti
length. They are held down to it by a single U-shaped bolt B, of 1 inch diam,
with a nut on each leg. These nuts bear directly upon the borlEontal upper sides
of the " fore-locks " L L, one of which is shown separately in Fig 9. The fore-locks
are rolled to fit accurately to the rail-flanges. The legs of the U-bolt pass first
through the circular holes h A, in the beam. Fig 6 ; next through rounded notches
cut in the corners of the rail-flanges ; then through the holes in the fore-locks ; and
lastly through the nuts. Between the U-bolt and the bottom of the beam is placed
a small piece «, of spring steel, slightly cambered downward, and having two semi-oir-
cular notches for the legs of the U-bolt, which hold it in place. This is intended to keep
the joint elastic, to take up any loose space produced by the wear of the sarfaces in
contact, to render less abrupt the strains on the bolt, and, by keeping the threads
of the nut pressed against those of the bolt, to prevent the nuts from becoming
loose. The joints are shipped from the factory complete, and with all the parts
bolted together; the nuts being screwed down to within about two threads of their
flnal places, so that the ends of the rail-flanges can be easily slid into t)lace under
the fore*locks.

As an additional precaution against creeping of the rails, the rajl-flanges may bo
slotted near their ends, as in cases where flsh-plates are used, and spikes drivon
through these slots. For such cases the beams are punched, at the mill, with four
additional square holes a little further from the edges 6f the beam than the others.
Unlike the angle- and fish-plate joints, the Fisher may be used with any section of
T-rail; and the head of the rail may be made stronger by being rolled pear-shaped,
vhich is inadmissible with fish- and angle-joints, because these require a noarlj

BAIL-JOINTS.

823

boitaontal bearing on the under side of the head. The '* Fisher " requires no drill*
lug or punching of the stem of the rail. It eoste about 25 per cent more than a
fish- or angle-joint for the,wme rail. Its wel^rlKt? complete, for 65-fi> rail, is about
S2t>s.

Mr. Fisher makes also an extra stronar Joint with three U-bolti^
for heavy curves and for places liable to wash-outs. It is intended to support the
Joint, even if the ballast is removed from under the joint-ties. Either of the Fisher
joints can be made of any desired weight. The "Fisher" is largely used on some
of the principal eastern roads, and with very satisfactory results.

Art* 13. The Bonsano rail-Joint, Figs. 10. 11, and 12, invented and
patented by Mr. Adolphus Bonzano, C. £., of Fhiladelphia, is essentially an
angle-plate joint (Art. 6), but, in the Bonzano joint, the horizontal flange of the
angle oar is rolled about 3 inches wider than usual, and, after rolling, and cut-
ting to the proper lengths, its middle portion is pressed downward by dies, form-
ing a girder, 6, which projects downward between the two joint ties.

The broad horizontal flange, being level with that of the rail, afibrdis greatly
increased bearing upon the ties, and adds to the lateral stififness of the joint ; the
girder, G, between the joint ties increases the vertical strength of the joint ; and
the triangular gussets, b, Figs. 10 and 11, securely hold the horizontal flange and
the downward-projecting girder in their relative positions.

The . twd splice-bars, in the Bonzano joint, have a combined cross-section
about i.2 times that of the rail, and, as shown by tests made by Prof. Henry T.
Bovey, at McGill University, Montreal, eoual strength with the rail, while the
ordinary angle bar joint has but one-third that strength.

The joints are made either 30 Inches long for 6 bolts, or from 24 to 26 inches
long for 4 bolts. For 80 fi>. rails, the splice-bars for the 30 inch and 24 inch joints
weigh together about 69 tt>s. and 73 ft>s. respectively.

The price is about 0.3 cent per pound higher than that of the ordinary angle
bar joint.

824

TURNOUTS.

TURNOUTS.

Art. 1. To enable an engine and train to pass from one track, A B, Tig 1, M
another, A D, a turnout is introduced. This conslste esMenflally of •

-JBfA

■witcb, qmp 8,0, trog^f, and two fixed ffaard-ralls, g and ^. If a switch
is made to serve for two turnouts, A D nud A 1>\ Fig 2, one on each aide of the main
track, A B, it is called a tbree-throw swlten.

Fi^. 2.

Fig. 3.

Art. 2. When a train approaches a switch in the direction of either arrow, Fis
1 ; or so that it passes the frog before reaching the noitcii, it is said to ^' trail"
the switch. When it approaches in the opposite direction, passing the twitch befors
reaching Xh^frog, it is said to ^* face" the switch. Fig 3 represents a portion of
a double-track road in which the trains keep to the right, as shown by the arrows.
In this fig, V and W are *^ trailing'*'* switches; and X and Y are ^* facing''* switches.
In order to leave the main track by a trailing switch, a train must move in a direo*
tion contrary to the proper one on said track.

Art. 3. Misplaced switches. A moving train, /octnp any switch, must
plainly go as the switch is set, whether right or wrong. If wrong, serious accident
may result. For instance, the train may run upon, and over the end of, a short
trestle siding, or may collide witli a train standing or moving upon the turnout.
Safety switches, such as the Lorenz, Arts 13, Ac, and Wharton, Arts 18, Ac,
are so arranged that trains trailing them can pass them safely, even if the switch
is misplaced. But in the case of the plain 8tab-swltch« Art 4, when mis-
placed, a trailing train will leave the rails at h and r, or t and u. Fig 4, and run
upon the ties.

Stub-switches are frequently provided with ^^ safety -eastings '* of iron,
bolted to their sides, and reaching from their toes m and «, Fig 4, several feet toward
p and q. These, in case of misplacement of the switch, receive the flanges of tlM
wheels of a trailing train, and guide the wheels safely on to the switoh-raiTs gm IB^
p a. The *" Tyler" switch is arranged in this way-

Art. 4. Tbe

blnnt-ended or atnb-swlteli eonalaU

■joft<r<.niil>,fnu>d pa, Fills I naH. Tbe tniii,gi<.adp,M Sum
thej tri, Axti In liD» wUh lh» mBmtrufa,[onD Ibe ••heel » of tta(

n(vhBTB tb«T>n Id Hoe with (ha maln-tnek nlto.^* und ri)U m nd (

,n »»y ara In liDS with the luroonl-raili, c» Hud uy).

imBlu llnpB or roftd, tb« ewhch-nlla »rsnaiiallT from IS to 36 feel loof rrom

to tA«- Fonnsrl^ their h*e]B,7 Uld p, wen flied b^balngcoDflved la IhoakmA

1 it^d raili by thorl flsh-pUlM. In •liber cue tbej rem^t

sada' b/'the WelT Fn^ Co. Theie bead pit

bewi-pUlte» muat at course be longer, to give room for tbe lltra rall-CDdB side by

FreqiHoIlT ■ pl>lD~>trip of Iron. aboDi B Incbee wids by lialMiKh thick, la ttttat*

>i^rii>uaKa;e; ^oeratlju abawD in^gS. The clamp-bira Hliould be placed, if

■Gauge-

Fig. o. ."cl"

oaar IbsIoea,»«id>,Flg4. It prrjjei'ta beyond the tracl
at Fig> 7 and S. and coDoacted nllh tbs leier. L, by »]
rha lia. T, Fig 4, to wbicb tb* bsad-plalta are foalcued,
Mhers, In order to gi« room for Ihe .wHch-etand.M,Figa
to its upper aarface. Tbis tie sbould also be of larger crc

826

TURNOTJTB,

Art. 5. The Bwlteh-leTeni, and the swltch*stands to which they ut
ftttachedf are made in a great variety of forms. See Figs 7, 8, 9, 14, 15, and 17. That
shown in Figs 7 and 8 is the '* Tamblint^-le^er stand" or *^ Ground-leTer
stand," and, in its nnmerons modifications, is very largely used. It is so arranged,
that, whichever way the switch is set, the crank, 0, is on the dead center, so thai
the lateral strains of passing cars or engines can exert no tendency to tarn it.

Fig. 7.

Fig. a

Tumbling switches are convenient becaase they occupy but little space. By meant
•f a target or lantern, connected with the switch, they may be maao to indicate to
the engine driver the position of the switch.

When the switch is set either way, the lever is padlocked to a staple driren Inte
the tie and passing up through the slot in the handle of the lever. The lever is fre*
quently made with a weight of say 20 Ibe on its free end, to aid in bringing it down
to its proper position.

Art. 6. Fig 9 represent* a common form of the nprifflit lewer and stand.

The switch-rod, B' Figs 4, 7 and 8, is generally attached at ttie lower end, A, of the lever.
The cast-iron frame, F, is fastened to the long tie, T, Fig 4, by large screws or
•pikes, which pass through its broad feet or flanges, B B. The top of the frmme is
provided with two notches, NN, and staples, to which the lever is secured by a
padlock. When this stand Is to be used for a ttre«-throw switch, the frame has
thr^e notches and three staples. The upright stand may be used wlierever it will
not be in the way of passing trains. The target, T, at the top of the lever, by
showing the position of the latter, indicates to the driver of an approaching engine
which way the switch is set.

Fig.O.

MONKEY SWITCH

Tig, 10.

Art. 7. In the ^ Honkey-swltcta,** Fig 10, the crank, o <, is moved hori*
Eontally through an arc of a circle by means of the lerer, h A, about 3 ft long, which
fits upon the square head, s, of the vertical spindle or pin, s o. The switch-rod, B' Figl
4, 7 and 8, is attached tu the pin, iv.

Many modiflcHtlons of the monkey-switch are in use. The spindle, so, is Are*

Siently made long enough to bring the lever to about the level of the hand; and
e lever is permanently attached to the stand, and hinged near the spindle so m
to liang down, out of the way, when not in use. To the top of the spindle is fre-
'■'nently attached a vertical rod of any desired length, and cari7iug at its top a target
oh turns as the spindle does, and thus indicates the f/osition of the switch

TITBITOUTB.

827

Art* 8« All parts of the iwitch-stand, and the tie npon which ft rests, should
bs psrfectly rigid, because it is very important that they shoald hold the ends of
the switch-rails exactly in line with those of the main line and turnout. They
therefore, in view of the great strains to which they are subjeoted, must be strongly
constructed, and frequently looked after.

^

-w

K

Fiff. 11.

AFt. 9. In Figs 1 and 4, do «» is called the switch-ansle. The dist, dm,
Figs 1, 4, and 11, required for the motion of the toes, is callea the tlirow of the
switch. It must be equal at least to.tbe width, d to, Fig 11, of the top of the rail,
in addition to a width, to m, sufBcieut to allow the flanges of the wheels to pass
along readily between b and «, Fig 1, and between r and «. The tops of the rails
are generally between 2 and 2^ ins wide; and about 1^ to 2^ ins suffice for the
flanges. The throw, d m, however, is commonly about 5 ins.

The graafTOf ^iS 6> of a railroad track, is the distance between the wmmt

sides GQ' of the heads of its two rails. Hence these inner sides are called the gwokge
sides of the rails.

Art. 10. The stubHTwitoh is cheaper in first cost than the improved safety
switches. Arts 13, 18, etc, but is less economical in the long run.

As it is Tery essential that the toes of the switch-rails should never come into
contact with the adjoining rail-ends, a space of about an inch must be allowed
at the toes for expansion, and for "creeping" This renders the blows

of passing trains very severe, and injurious to rolling stock, and to the rail-ends.
The latter are worn away rapidly and must be frequently renewed. From the sam«
cause the tie under the head-plate is apt to become loose in its bed.

^t^lj,^p^w^.^(.

by B liimbllng-leTer, L, tie M.
To thB bonzontal siis. A, of

pinion, A Tliis engages In the
le«th or tbe quadrsnt, Q, md
iDOVn It boriionlsllr llirougU a
tbe lever L in thmon from It!
hwn by (he dotted lines. The rod,
icbed »t X, or at X'. to tble qiiai)-

, .... ...... .u ^ ji,g B„j,cl|

to the lertical

be nialn trnck, Ss In Fig 13, the fl:
■ila X' and S' V', irlll push the .wite
hell.™™*; St Ihe same lime necewi
he ntTerse position, and turning the

ikes plsce it a trBlllDg traio on t)ie

Art. IS. In (he Eioren> MttMy-swItcb. the ooDneclIng-bBr, R', ne
BSt to the loes, is provided "Ith a ■prlnK, S, Fig 16. placed snmetimes belwi
the rails, as there Bhown ; Mmetlmes oiiUTde of the track. Tbla Bprlngperm
th<^ moving of the swilcb-rnlla by the wheels of s trailing train, as does 1
■utomallc Bffi(eh.eta>id, Figs 14 and IK; hut after the passage of each xhi
the spring returns the switch rails lo their original position. The blow of i

880 TURNOUTS;

. .lif^|Qrf<yaftoboai,tiid1bM»ti

hnak fh« Ibraier. On ttie other band, IIm oomprenloii «f the epriiM; dnrte|f tlw
paaiaM of the train throash the awftoh, aomedmee Impatra its ehiatiel^. ao that tt
then ndla to retnm the nwltchnll to Ita proper poaltion in contact with the atodt*
rail, and allows it to- remain half an inch or more away from it, and in dani^er ef
beii^ strnpk by the wheel-flancea ef approeohlnsp trains ** faxing ** the switch. A
■imuar amdent may happen during the ordinary working of the switch, if aa
obstacle, as a small utone, becomes lodged Iv^tween the switch-rail and the stock-
rail ; fbr the spring may permit the switchman to Ibroe the swltch-leTsr hooM to JM
place withoat Drlngiitf tfia two raUs properly into eontaet See Art. 14.

Art* 14. De Vonfa safety awiteli-Btand, Tig 17, made by Peana
Steel Od, is designed to mmedy this. In this stand, the spring is placed in, and se-
cured to. a semi-cylindrical iron spring-case or box, B; to the opposite sides of which
ve fixed two hor axles. One of these is shown at A« This axle passes through the

mg.i7.

switch-leTer, L, near its fhlcmm, F. Tt also passes through the inverted T-shaped
slot, H, in the rigid bar, S, which, together with the bar, W, at tiiched to the spring-
case, is Jointed, at J, to the switch-rod, R'. When the switch is proiierly set, either
for the main line or for the turnout, the axle. A, is iu the hoi* part of the slot, H,
and immediately under the vert part, so that there is no obstruction to the more-
ment of the switch, and a trailing train will open a misplaced switch as explained
in Arts 12 and IS. But when the lever is raised, for the purpose of setting the switch
in the other position, the axle, A, rises into the vert part of the slot, as in the fig,
lifting the spring-case with it. If now any obstruction prevents the switch-rul
from l)eing pressed home, the rigid bar, 8, by means of the axle. A, prevents the
lever, L, from moving farther.

Art. 19« Theory would require that tbe lenvtlia off the swltoli-railB,

in split-switches, should vary with the radius of the turnout curve, and formeriy
they were so made. Where this radius is such that a No 10 frog fsee Art 26) is xe>
quired, the switch-rails should, theoretically, l)e 28 f t long. But in practice a onl-
form length of 15 ft (just half the usual length of the steel rail from which the
switch-rails are cut) for all turnouts, gives the best results, combining economy of
manufacture with greater strength, and greater ease of handling, than are possiUe
with much longer rails.

Art. 16. It will be noticed that in point-switches (as also in the Wliarton switeh.
Arts 18, Ac) there can be no such jar as that occasioned in the stub-switch by the
long space between the toes of the switch-rails and the ends of the adjoining rails.

Art. 17* It is important that the thin portions of each switch-rail should Im
carefully shaped so as to receive throughout a firm lateral support from the stock-
rail when in contact with it. Otherwise the switch-rails are in danger of bending
under the lateral pressure of passing trains. This might throw the point out from
the stock-rail, endangering the train.

Art. IT a. Fig. 17 a shows a tlirac«t]iro-«r point avrtteli msde by The
Weir Frog Co., Ciucinnati, Oliio. It has the usual stock rails, 0 and Z, and font
switch rails, A, B, X and Y. The switch rails all slide upon the same set of iron
" friction plates,*' which are spiked to the ties under the rails, but are not shewn in
the llgure. Rails A and B, are held rigidly together by four connecting bars
% a, a, a, while X and Y are similarly connected bj the other four connecting ban

Euh p*lr of ivllch njls. thni foimed, n

aa of ■ ilngle iCuid, pluied an one ilds of the awTtctu

fieiire uowb tbs ewitch u a

(^ nils DUfrT be op«n»d by
The flgore ihom the iwtl<

(Dtes between nlli X and A for Ibe p— ra (/ Iht ftmi* of wheel I^ wUeb wfeari
tbea rnne apon roll A.
~ ■« tr** 0 B, (he rod R a !■ then pn>hMl,mnd ibrowe nlU

ringing 1

m rHlli (

ndlaCiu

xbee, tt ts ImpDflHlbls to flpike the Inner flanges of the etc^
ee minng tbat portlan of their length (Hme 12 l»at rroia tb*

r , thej tome in conMcl with the iwltih-relli. Ae e nitiMitnte

Ihsj on provided with ipeclal eupportlng blocki B. S. on the oater side. In the
Weir ewlteb, each of Iheee li made uf one piece of (liit ber Iron, bent orer utd
twiatwl, nnd eerrlng: ftl» ae n iUdlng pUla, u ihown more cleu-lj in Ihe eecltoo ■ o,

Toennerof IWeteniu tbeend el
mailBAble CBadn^ AL tbTOBgh

ttiet »hlct
J(flg.n0)

.wn in Hg, V

tbauTtngeiDeiithenldlinlUrtoauitiitoo, eicepi thatln Fig. ITcths n
MrtlngHbMtaltolheiMliaf tbe rail eaibown, while Id Fig. 176 II li of
•hspe, end 1> Tl<nUd to thefiatgi of lb» rail.

r

832

III

i!iy

^111 i

Isfs I
alls I

||f = 5

834

TUBMODTB.

Art. 29« FrOffl* The frog is a contrivance for allowtne the flange ot the
wheel on the rail e Xy Fig 1, to cross the rail r s ; and that of the wheel on rir, to croM
ex. The flrst contrlTance for ttils purpose was a l»ar, approxi-
mately of the shape of the rail, pivoted at the point where the center lines of the
rails « ST and r« cross each other, and free to move horizontally about this pivot, so
that it could fomi a portion of e z when the train was passing to or from the turnout^
or a portion of r « when the train was using the main track. Sometimes the pivol

Fiff.20.

through one end of the bar, as In Fig 20, and sometimes through Its e«nUr.
as in FIsr 21. Such bars were generally moved by a rod
(attached at n) and lever, similar to those used for switches;
and they then, of course, required an attendant; bat manjr
attempts have been made to use such firogs by conBectu\,
them with the switch by means of rods, Ac, so that the bar
should move automatically when the switch wa« tamed
Owing to the considerable distance (80 ft, more or less) be>
tween the frog and switch, it has been found diflScalt to
aecure simultaneous movements of the switch and frog, and the contrivances referred
to have not come into extensive use. Such bars, while they avoid the jar produced
by wheels passing across the throat of the frog (Art 36), labor under the same dio*
advantage as the stub-switch. Art 10, In requiring a liberal allowance of space be-
tween their ends and those of the adyoining rails, to avoid any possibility of their
coming into contact.

Flff. 28.

Art* 23. These bars were soon superseded by rfgfd east-lron Arosm» FItf

98 and 23. Theeo were hardened by chilling, so m better to resist the action of

Fisr.28.

passing wheels; but even with this precaution they wore oat so much more rapidly
than the rails, that the wiiig^wm and ie, and the ton§f ae« P, were capped
with steel from ^ inch to 1 inch thick, bolted or riveted to their upper sarfiice«

TUBNOUTS. 835

fbe triangle, P, called the tongue of the frog, is the meeting-point of the two ralli,
fz and /», Fig 1 ; while the wings, w m and i c, are continuations of the rails 0 and r.
The wings give support to the treads of the wheels in passing over the spaces between
the point and w and i, which spaces are left for the passage of the flanges.

The channel is called the montll of the frog at a, Figs 22 and 23; and iti
tliroat at the narrowest part, w i. That part of the tongue back of m, Fig 23, or
between u and p, is called its beel.

The channel is made about 2 ins deep to prevent the flanges from touching its bottom.

The projections, t <, Fig 22, are for bolting ihe frog to the wooden cross ties.

Although one side of the frog forms a part of the turnout curve, its shortness war-
rants us in making both sides,^ o, s t. Fig 23, straight.

Art. 24. Onlde-ralls, or guard-rails, g gr'. Fig 1. Suppose wheels to bo
rolling from A toward B, Fig 1, on the main track ; the switch-rails being in the
dotted positions. On arriving opposite the frog, some Irregularity of motion might
cause the flanges of the wheels running along the rail, r «, to press laterally against
■aid rail. Consequently, after passing the throat, to «, Fig 22, they would press against
the wing, tc; and passing between cand P, they would leave the track; or strike
the sharp end of P, breaking it, and endangering the train. To prevent this, the
guard-rail, fl. Fig 1, is placed so near the rail, 6 A (say 1% to 2 ins from it), that th«
flanges at b A, while passing between it and g, prevent those at the opposite nUl
from pressing against the wing, t c. Fig 22, and from striking the point ; and guida
them safely along their proper channel, t m. Similarly, if wheels be rolling from
A toward D, Fig 1 (the switch-rails being in the positions, 9 m, p <), the centrifugal
force due to the curve would cause the flanges to press against the rail, « a;, and
against the wing, vf m. Fig 22, thus rendering the train liable to the same kind of
accident as in the preceding case. This is prevented, in the same manner as before
by the guard-rail, p', Fig 1, which keeps the flanges in their proper channel
Fig 22. y*

The oarrow flaoce-way between the guard-rail, gr. Fig 1, and th^il, 6 A,
slioald extend at least a foot each way from a point directly opposite the
point,/. Fig 23, of the frog. In a distance of at least about 2 ft more at each of its
ends the guard-rail should flare out to about 3 ins Ax>m the rail, b A, so as to guide
the flanges into the narrow channel. The same with g^.

Onard-rails liave to resist a strongr side pressure, and should
be very firmly secured to the wooden cross-ties. This is usually done by bolting
against them two or more stoat blooke of steel, or of wrought iron, which, in tora*
are bolted to the tlee.

Art. 25. Tlie east-iron firoff, as first made, had no proTlsion for
fisstenlngr it to the rails; but was simply bolted to the cross-ties. It was
afterwards provided with a recess at each end, of the exact shape and size of the end
of the rail. The rail ends were inserted into these recesses, and the frog was thus
kept in line with the rail. In frogs made of rails, the same purpose is served by fish-
er angle-plates, by which tibe ends of the frog are secured to those of the rails.

Art. 26. The leuflftll, a g^ Fig 23, of a cnst-iron frog, usually varies from
4 to 8 ft; and depends upon the angle, qft^ at which the rails, e x and r ;, Figs 1 and
28, cross each other. This is called tiie frog^-ang^Ie. This angle may be expressed
either in degs and mins, or in the number of times the width of the tongue on any
line, as o t, Fig 23, is contained in the distance, gf^ from the point,/, to the center,
^, of that line. This nnmlSer is called the tr^g number. Thus, if the angle,
o/«. Fig 23, is such that the length, gf, is 3, 4, or 10, Ac, times the width, o t, the
frog is called a No 3, 4, or 10, Ac, frog. Fig 28 is a No 3 ; Fig 22, No 5. Frogs are
usnally made of Nos 4 to 12 ; sometimes with half numbers, as 7^^, 8^, Ac.

Art. 27. Draw two parallel lines, b h\dd\ for the top of rail, ea;. Fig 23, and
A h\ kk'y for that of rail, r «; crossing each other at the required angle. Then the
intentection,/, of lines dd' and h k' is the theoretical point of the troK, As
this point would be too narrow and weak for service, it is in practice rounded off
where the tongue is about W inch wide, as shown. If the frog is to be simply abutted
to the rail-ends, sa;. Fig 23, as in some cast-iron frogs, the length, /p, need be only
great enough to give a width, to, suflBcient to accommodate the rail-ends, z and x,
and the hesuls of the two spikes at v which confine them to the ties. If desired, a
portion of the flange of each rail-end may b^ cut away so that the rail-heads come
together; thus diminishing the width necessary for t o, and, of course, the distance,
/p. In the case of cast-iron frogs provided with recesses for holding the rail-ends,
as in Art 25, the width, ot, and length,/^, must of course be greater.

In tk'ogn made of rails, the leng^th must be such that the rail-ends,
« and Xf Fig 23, are far enough apart to give room for fitting to their inner sides the
splice-plates by which they are connected with the frog. Where anp2e-plates
used, this distance must be greater than in the case of jSsV-plates.

TUKNOUTS.

837

into recesses let into the sides of the throat-pieces and blocks. Tbe clamM
are prevented fUrom sliding^ by clips riveted on the Hanges of the rafk
Great care is taken to have all the a^Joininip sarfaces in full contact

with eacli other, throughout, so as to diminish the liability to wear.

Art. 32. In stiflf frogs the parts are held together by bolts passing through
the rails and the throat-pieces. In some there are no throat-pieces, and the
four rails forming the frog are riveted to a wrought-iron plate. All of these
frogs can be made of any desired length and to any desireoangle.

Art. 33. Tbe standard lengrtb of stiff frogs from be tod z, for Nos
4 to 8, inclusive, is 8 ft ; Nos 9 and 10, 9 ft ; Nos 11 and 12, 10 ft ; No 15. 12 ft.

Art. 34^ In the IVelr trog (W«ir Frog CJo.. Cincinnati, OhioV Kgs. 26 A,
26 B and 26 0, the notching of the main or long point s, to receive the short point
d, is ayoided by forging the latter, under hydraulic pressure, to fit the former which

b used tm tbe upper die In the forging process. Hie rail fanahtf; the long point • is
tiras preserved intact throughout the space where the two points are in contact, am
shown In the section at n o^ Vig. 26 B. The short poiat, a, is held tightly against^

B!iK..2e:B

B?ig.-SeC

8ECTI0N.AT«»O.

INCHES.
1-C 1 28 4 "5 0

w/ z w

SECTION AT p q.

and under the bead at, the long point s, by tiie tigfatenlnc of the nuts on the bolts
(whieh are provided with nut>Iocks), and tiius gives it additional support.

The long filling-blocks, t (, are forged In dies ttom wrought iron or steel. They
are then planed to fit the point rails, drilled, and bolted as shown.

Art. 35. Tbe object in redncinar the wldtb of tbe cbannela

for some distance each way from the point, /, Fig 23, so as barely to admit the
flanges freely, is to allow tbe treads of the wheels to have as mach bearing as pos-
sible upon the wings and tongue while moving over the broadest part of the chan-
nel near /. In frogs shorter than about No 4, it is difficult to secure sufficient
bearing fer the treads, even with the utmost allowable contraction of the channel,
when the width of the tires is, as usual, about 6 ins. lu the earliest frogs this diffi-
culty was partially overcome by gradually raising: tbe bottom of tbe
cbannel between the point and wings, so that the wheels, in traversing that part,
ran upon their fiangtt instead of upon their treadg. The Jar occasioned by the
tread. In striking against wing and point, was thus avoided. This arrangement is
still used in eroarings, where the tracks cross at a very obtuse angle, and where,
consequently, tbe wings can give little or no support to the treads. The flanges,
howerer, soon cut gutters in the bottoms of the channels, and thus increase thaif
depths, so that the treads strike the wings and point, as in the ordinary frog.

TURN0UI8.

TOKNOUT8.

If the bearlDg betwc
naf, ia too Bhort, 8 ridgB -uiu uu >»•

Ing from BtD«srd A, n]«y drop betwt r— = — r

il reaches a place, near/, vliere ths polDt is Dsnower tban the Eutter worn Id
the nheel-tread. The ridge ta then Uahle to wedge the aprine-ml i away from
the point/. To avoid this, the wlDe v of the spring-rail, and lU bearing against
the poiDt-rail, have been leagthaned in the preeeot patieru. thug giving a wider
bearing and keeping the ridgeotaworD wheel-tread on (op of the rails nntU the
frog haa been passed. The littiiidnrd lenstli for either style of Bprlnv
rail ft'OK, and for anj angle, is IS feet.

Since Bpring-raU frogs do not preBentatuUbearingtfl wheels entering or leav.
Ing the tumavi, but only to those passing to and fro on the main track, they are
most useful where the greater pan of tbe trsSe mores on the main line, and

Art. 37. In ordering frov>> gWe the frog angle or number, and the

Arl. SS. For Hprlng-rBlI troK», BpecifV Mam whether the turn-
out is to the right or left hind. In the ease of s'iirfrogs, this is not neceesarr.
Art.S9. Tbe Isylns-oat of Taraonla-

tfl>taiIGe,p/, li astralghHlnedrmwn from the tbBorollcal point ^./roffi/i to llie
hsel,i>, of that swllcb-railwhlch. whan opened, forms tbs unn- nil of the OummL
Formerly, when tfaa [arDDut cnrrK wu taken IS BUnlng at tbalssof the switch, ths
tng diat was a stralgbt line fmin the tbeorttlcal point of frag (o the toe, m. Fig 4,
of the outer switch-rail, <7 m. when opened.

tol^sor tbe tonnulas, will be almost Inappreciable. So loo. If afrog nqmberefaanld
bauaed. Inlsmndlateoflhossinlheflnlcnlomn of Ihe table, the olhwdlmeutlons

A rail almost always haa to be cut fn 'two in order to All up the frogdist; and the

Rem. When the lumont leeree a straight track, as In Fig Sft,the ftwiF ancle
Is eqnal lo (he central ingle./c o. When lbs main (rsck ii curTeit, snd the tumont
cnrrei In the opp(Mll« direction (Ptg 3i)), it li eqnal tn the aum (c/n) of the
eentral angles, /c o. /n r^ ; snd when the two enr.e In the iiiinie direcdou (FIf
II), It is «tual to Ibe dlff (n/c) of the csutnl anglee./eo,/aa.

nl-'' Cbm 2

840

TURNoirra.

Art. 40. To lay out a f nrnonti p as. Flar 29, firom » stratiirlit
ira€l£«l>4i« From the culuiun of radii in the table below, select one, co^ suit-

able for the turnout; together with the correspond'

fy ■ »-.....,^^ iug frog number, frog di8t,p/, aBd switch length.

^ I - ^ , Z*^?^ Place the frog so that the main-line side uf its

^j ^'^'^^ ^vi> tongue- shall be at /«, precwe^y in line with the

■^#>" r I ■* ^ ^v /£ ^ inner edge of the rail, tor, and its theoretical

I ^Nv'^^N \ point, /, at the tabular frog dist, jpf^ from the

I /s. \ Vj* starting-point, z>. Stretch a string from q (oppo-

I / \ \ **** P) ^f't *na from it lay oflF the three ordinates

{ / \ from the table; thus finding three points (in addi-

I y tion to a and f\ in the outer curve. Do not, bow-

I / B^ff. SO ever, dme staKes at these points ; but as each o?

{ / them is fonnd, measure off from it, inward, hall

^y the gauge of the track; and there drive stakes.

Do the same from q and/. The five stakes will all
then be In the dotted center line of the turnout. Fig 29; and will serve as guides to
the work, without being liable tn be displaced. The dimensions in the table below
are found by the following f^rmalfui* the main track being straight:

FrOfT ^^ •* *" V Radias co -i- Twice the gadge.

Or, Froc No « *« Half the cotangent of half the frog angle.

BfUlIas e O «■ Twice the gauge X Square of frog number.

Or, Badlnii e o » (Frog dist pf -^ Sine of frog angle; -^ half the gauge.

Or, Radius e o •» (Gauge -»- Versed sine of frog angle) — half the g»uge.

Fro§^ dist pf. a> Frog number X Twice the gauge.

Or, Frog" dist pf. => Qauge j>9 -k- Tangent of half the frog angle.

Or, Frof[ dist pf. »■ (Rad co + half the gauge) X Sine of frog angle.

Middle ord «> ^ gauge, approx enough.

Eaeb side ord... » ^ mid ord <=> ^ (or .188 of the) gauge, approx enoaght

Switch I^ncth - . / Throw in ftxToOOO

a,\>prox enou^i \j Tangential dist for chorda of 100 ft, for rad e o of

^ turnout curve.

TABLE OF TURNOUTS FBOH A 8TBAIGHT TBACK. Fig 20.

Gauge 4 ft V^/^ lAs. Throw of switch 5 ins.

For any other fraagre, the/ro^ angle for any given frog number remains
the same as in the table. The other items mtiy be taken, approx enough, to vary di-
rectly as the gauge.

FrojK
Mumber

Froft
AnjEle

Turnout
JUdliia

DeflAnirof

Tumont

Carve

Fros

Dist

pf

Middle
Ordinate

Side
Orda

Stnb
Bwltak
I^ncth

o r

Feet.

o r

Feet.

Feet.

Feet.

Feet.

12

4 46

1356

4 14

113.0

1.177

.883

84

\\\i

4 58

1245

4 36

108.3

1.177

.883

32

11

6 12

1139

5 2

103.6

1.177

.883

31

101^

5 28

1038

5 31

98.9

1.177

.883

29

10

544

942

6 5

94.2

1.177

.883

28

9}4

6 2

850

6 45

89.5

1.177

.883

27

9

6 22

763

7 31

84.7

1.177

.883

25

8H

6 44

680

8 26

80.0

1.177

.888

24

8

7 10

603

9 31

75.3

1.177

.883

22

7K

7 38

530

10 50

70.6

1.177

.883

21

7

8 10

461

12 27

65.9

1.177

.888

20

«H

8 48

398

14 26

61.2

1.177

.88u

18

6

9 32

339

16 .'>8

66.5

1.177

.883

17

6H

10 24

2S5

20 13

51.8

1.177

.883

15

6

11 26

235

24 32

47.1

1.177

.888

14

i>^

12 40

191

80 24

42.4

1.177

.883

18

4

14 14

151

38 46

87.7

1.177

.888

U

* Shortest length of stub switch that will at the same time form part of the
turnout curve, and give 6 ins throw. Point switches require only half this
throw. In practice, switches are frequently made much shorter than the table

iquires, thereby sharpening the beginning of the curve.

TnRXOUTB.

841

An 41. Tnraaat fnum m cmmd nwla tnuili. Tbe r'>llowlDgcOD-

CMe'l, Vl(' 30; whsn tha iwo curVei dcBett in uppoall'v directions.
Cases. Fig 31; wlMn tlia IwocurTvi Qaflect In ihti BBine dirwiiuu.
llmvlDg delanulMd .pprol udud > nilhis for Ibfl lurnoul cur.t. lak« fi-om lb<
l•M^ p 784, lU eoneapoDdlng <!r«»^um oaxle. ind ibot lor tbe nmlo ciino. i.

,(;, fiod lh»lr dtf-mnc^ In tbe taljlo, p MO, Ood the

I > d^flraion angla {not thp frog mele) pesrtfl to tb«

I \ sum rjr ditf Just found. Ttac frog nnmbpr,

I \ HWitch lenKlh Hod froKdlstKiice pf. in

[ \ the table, oppoaile the defleciion »ii|ile thuu se-

\ lecled, are the propel oi

\ Tlirnrrlii-allu «e should. In (

DeOectlon niis-le of

(Id Case 11 = Tabulai denectic
(in Cue 2) ^ Tsbultur dellectl(

Ban tha dell anglei ara, respactlTaly. V and B°.

In Cms I|1r°.]-2°— 10°. N»»st defl Knileg In talile, p S40, IDCKCande^Sl'.

ir weielaci Wbff.-nbaw Frog Ho, 7>^; Snitch L'gtb, 21 ft; Fr«Dlil~TO.«
IH-dncjiid — B.7 70.7fl;IHflAnglB ior turnout - lOO W — 20 — 8= Wi iUd

In CnaeSi ffi — a' " V: Tabular dad MiglM,pB«0,B"6'and IfiZV. tP fJ

f yet Frog KOyiO: Switch Vstb. Wit; rnigHiit'-SMft — tw]»Uln>'H>»4.1
! DeO Aiigl» = B=V + a° — S^Vi Had, B.J !(• ft, 6=Bi'glwa tfrog No, lOUj
=._,._,..._? .„^.____^_. ^»^~ — ■-» Jiln — •aj&Bfll DcflAngla-

ft! DeO Aiigl»=B=V+a°— S^Vi Ead,B.jIi»ft, 6=Bi'
e-itch L'nh, 29 K; Frog IHit — BS.S ft — twice ^ Id — «« I
6°»l'-f :^ — 7°S1'; iUd.HjTSSn. ,
TUe frog: 4iat p f nakf alao be fonad tbu i

Tsnrent of half/no — —

FrO^nist p^-njlX twlc»the>iIworha]f/Ha.

nvathe ^og with the ni.in-IioB side of 1 1> tongue at/(. Id Una wllh the InnT

diic, p/(fooud u .bivejl from the hee[, p, of the (unw switch-ialL. BtreMh a airing

Kidila ord = (SqnatB of ha
Thew three ordi. and the p
irjlilji alakea. m la Art «0.

If 9/) + t>lce IhE rad of tnnioDt curre. Bach aid

glsU?aiid/. itnlu tpolDU of the enter rail of tb
a w« BHMnra, iDwart, biir tba gance, and diin E cm

842

TUBNOUTB.

Art. 49. To find froff disto, Ae^ by meaiiii of a drawing tm
scale* The frog dist cao generally be found near enough for practice, from a
drawing on a scale of about ^ or Vg inch to a foot. And so in the many cases when
turnonts cross tracks in Tanoos directions, in and about stations, depots, Ac.

Figs 32 and 33 are intODded merely to furnish a few general tiints in regard tt

FifiT. 32.

^-HUL

raob drawings. For Instance, the curves of a main track, as well as those of a tonii
out, generally have radii too large to admit of being drawn on a scale of ^ inch to
a foot, bv a pair of dividers or compasses. But they may be managed thus : Draw
any straight line, a 6, Fig 32, to represent by scale a lOO-ft chord of the curve, divide
it into twenty 6-ft parts, a 1, 1 2, 28, &c, and lay off by scale the 19 corre8i>onding
ordinates, 1 1, 2 2, 33, Ac, taken from the table on page 730. By Joining the ends
of these, we obtain the reqd curve-, a o6, of the main track ; <md of course can draw

the inner line, y 2, distant firom It by scale the width of track, say 4 ft 8^ Ins. Now
let a e 6 and y ^ Fig 33, be a curved main track so drawn ; and let any point m be
taken as the starting-point of the turnout, m v, Ac. On each side of m measure olT
any two equidistant points, n and n, in the same curve ; and through m draw sg^
parallel to n n. Then is m^ a tang to the curve, y m t, at «n. Having determined
on the rad of the turnout curve, m v «, draw that curve by the same process as before;
first laying off the angle, ^ mi, equal to the tangential angle of the curve, taken
from the table, p 788. Then, beginning at m, lay off 5-feet dists along m i ; and fh>m
them, as in Fig 32, draw the ords corresponding to the turnout curve. Through the
ends of these ords draw the curve, mve, itself. Then the frog dist will be the
straight dist from e to v, and can be measured by the scale, within a few inches ; or
near enough for practice. The middle ord of the arc, m v, cannot be found correctly
by so small a scale as ^ inch to a foot, but should be ciUcuIated thus : From the
square of the rad take the square of half the chord, m v. Take the sq rt of the rem.
Subtract this sq rt from the rad. If two other ords should be desired, half way be-
tween 11V and « and the center one, they may each be taken as ^ of the center one.
Make the switch-rail long enough to leave 2)^ ins at its toe between m n and m m.

The ftrog angle at v will be equal to the angle, red, formed between the tang, vr,
to the carve, aeb; and the tang, « d, to th| curve, mve. These tangs are found in
the same way as m^; namely, for the tang,« r, lay off from v two equidistant points,
h and hy on the onrve, aeb; and through v draw v r parallel to kh. Also, for « d,
lay off from v any equidistant points, u and u, on the carve, mve, and through #
draw V d parallel to them. This angle may be measured by a protractor. Or, if on
the two tangs we make v 4 and v 4 eqnal to each other, and draw the dotted line 44;
and tnm Its center at 0 draw 6 v ; then 6 v divided by 4 4 will give the No of the tng,
With-cars, and a little ingenuity, the young student will be able, by similar proo-
asses, to solve graphically any turnout case that may present Itself. The method
by a drawing has great advantages over the tedious and complicated calcalatioas
which otherwise become necessary in cases whe)^e curved and straight tracks istai^
•sot each other in various directions. The drawing serves as a check asainst sertoos
errors, which would be detected at once by eye. None of the graphical measure*
■seats will be strictly accurate ; but with care, none of the errors need be of prae>
Heal importance.

TUBNOUTS,

843

Art* 4S« An experienced track-layer, with a good eye, can place hu own gnide-
flakes by trial on the groand; and by them lay his turnouts with an accuracy as
practically useful as the most scrapulous calculations of the engineer can secure.

The following example. Fig 84, of a turnout Arom a straight track, Y Z, exhibits a
dommon case, iQ which all the work may be performed on the ground, without pre*

vioiu caleolation. Let <vo be the tongne of a frog, with which the assistant has
bean directed to make a tnmoat from T Z; and that he has received no instructions
more than that the tnmont most start at d, and terminate in a track, W, to be laid
parallel to Y Z, and distant ftom it r « or r«, equal to 6 ft.

Place the tongue of the frog by guess near where it must come, having its edge,
V i, jfrecisd^ in line with the inner or flange edge of the rail, b r. Then stretch
a piece of twine along the edge, o v, of the frog, and extending to dg. Try
by measure whether v e is then equal to ed; and if it is not, move the frog along
the line, h r, until those two dists become equal. Then is v the proper place for the
point of the frog ; 6 v is the frog dist ; one-half of e « is the length of the middle ord
of the turnout curve, d v ; and if two intermediate ords are needed i^ s and «, each
of them will be ^ of said middle one.

The frog being now placed, proceed thus : Place two stakes and tacks, x and x, at
the reqd inter-track dist, rx and r«, of 6 ft from the rails, br. Then range by
pieces of twine 0 a; and v/, to find the point, n, of intersection. Then measure nv^
and make n m equal to it. Then is m the end of the reverse curve, v nt, of the turn-
out. The ords of this curve may be found as before ; one-half of n ft being the
middle one, Ac,

RxM. It may frequently be-of nse to remember that in any arc, as v m, of a circle ;
vn and mn being tangs from the ends of the arc; one-half of the diet, A;n, is the
middle ord, k 2, ot the curve ; near enough for most practical purposes, tohenever th«
lenffth of the chords v m, of tht are is not greater than one'half the rod of the circU
^ which the arc it a part Or, within the same limit, vice versa, if we make Ac n equal
to twice kMy then will n be very approximately the point at which two tangs from
the ends of the arc will meet. Also, the middle ord of the half arc, vs or «m, may
be taken ae ^ of the middle ord, ik 4^ of the whole arc

J I

TtTBNTXBLHS.

TUSSTABLEB. o40

TUENTABLES.

ArCl. A tnmtebl« iMaplatfopin, luully rrom 40 in «a ft lime ud
moot a to 10 ft wide (HB Fig IJupoD whidi > laaomnlie und in tender mu,
beinu, und Ihtu b« turned Bmund bor itaraugh ui; poctlen of a circle ; aad thae be
tnuifemd from one Inck to uatber tonnlng Mi/ uigle with it. The table 1> lop-
ported by ■ pliot nndcr lu cenler; and bj wheels or rollore under lie two ende.
FreqnFDIIjr other rollen are added between the ctntBt and enda. BensLtb the plal-

a wall of iraeoiir; or brick abcnt S ft tbick, cappsd with eHhar OQt etone or wood.
Thodlam of the pit in clear of Ihia lining li aboul 2 loe greater than Ihe length of
tlie inrnlable. The lining li generRlly bnllt with a alep, aa aeen In Fig 1. for aup>

^ma La frvqni

^tly le

:ulu' rall.aaatmFlgU. J
I Btep roT the end rolleta, eh

^nlarpilil

t, wiibuut ilepplng down into it; eapeclally when i

tbla Increaeed Dot onJ:r their coet but their wt, »> ai lo make Ihem i

bceldei earning mnch eipenee tor repairs l with greatar troulilo In making them.

For Ibe mlnimnm lenKtli oF ■ lprBt«bl«, add ftomVAio^tlat

engine a

ieDgtbet

span Ibe central anpport ; anil thee telJeTa the end rollera By this mdlna the fric-
Ubn while tarDJDglBConfloedaa much aepoaalbl* to the center of moliuai and la tbrav

846

TUBXTTABLES.

fore more readily oyercome than if it were allowed to act at the circnmf. Th%
•Dgine-men soon learn, by feeling, the proper spot for stopping the engine so aa thni
to balance the platform.

Art. S. Figs 1 to 6 represent the Sellers enst-iron turntable of Wm
Sellers & Co., Phila. It consists of two cast-iron girders of about 1^ ins avera^
thickness, perforated by circular openings to save metal. One of these girders ia
shown in Fig 1 ; and parts of one in Figs 3 and 4. Each girder is in two separate
pieces, which are fastened, as shown in Figs 1, 3, and 4, to a hollow cast-iron ** cen-
ter-box,** A B, Figs 2, 3, 4, and 6, by means of 2^ inch screw-bolts, at/, Fig 3; and
by hor bars, o o, of rolled iron about S% ins square, fitting into sunk recesses on
top of the boxing, and tightened in place by wedges, i i, screw-bolted beneath.

Art. 8. The sides of the center-box are about 1% ins thick. It is sus-
pended from the steel cap, C, by 8 screw-bolts 2 ins diam. On its lower side
this cap has a semi-cylindrical groove extending across it, transversely of the
track, as shown in Fig 6. This groove fits over a corresponding semi-cylindrical
ridge on the top of the cast-iron ^' socket,** t (so called), on which the cap thus
rests. The socket, in turn, rests upon the upper one, u, of two annular steel plates,
u and V, which form a circular box containing 16 steel conical anfl-
ft*lctlou rollers, d d, Figs 2, 6, and 6. These are. about 8 ins in length, and
in greatest diani. They have no axles, but merely lie loosely in the lower part, «,
of the circular box ; filling its circumf with the exception of about l^ inch left for
play. In the direction of their axes they have V^ inch play. The lia, u, of t)*? cir-
cular box, resttf upon the tops of the rollers, wnich separate it from v by aU>ut ^
Inch. V rests upon the top of the hollow cast-iron post. P, which, by means
of its flanges, is bolted to the cap-stone, M, of the foundation pier.

In order to insure a perfect bearing of the revolving surfaces upon eacn other,
and thus diminish the liability to abrasion, the rollers, d d, and the insidea of the
box in contact with them, are accurately finished, as are also the top and bottom of
the roller-box, and the surfaces of the socket, «, and post, P, in contact with thena.
The rollers are oiled by means of the spaces shown by the arrows in Fig 2.

Art; 4. • AcUnstment of the beigtat of tbe table. By turning tha
nuts, N N, of the 8 screw-bolts which support the table, the latter may be raised or
lowered 1 or 2 ins; the cap, G, socket, «, und roller-box, u v, remaining stationary
on top of the post, P. All turntables should have the means of making such
adjustment. Before the nuts, N N, are finally tightened up, the blocks, -w w«
Of bard wood, cut to the proper thickness for the desired ht of the table, are
inserted between the cap, G, and the top of the center-box, A B, as shown.

Tbe bt of tbe table sbonld be such that each of the wheels at its cater
ends shall be ^ inch, in the clear, from the circular rail on which they travel.

Art. 5. At each of the outer ends of the table, the two girders are connected
transversely by heavy cast-iron beams, called ^^cross-fflrts.*^ These project
beyond the girders, and carry the cast-iron end-wbeefs, 20 ins diam, 2 at each
end of the platform. The treads of these wheels are but about 3 or 4 ins below the
bottoms of the girders, and the wheels therefore do not require any considerable
depth of pit for their accommodation. In order that they may roll freely, their
treads are coned, and their axles are made radial to the circular turntable pit. In*
termediate transverse connection between the main girders, is secured by tha
wooden cross-ties notched upon them to snpport the rails, and frequently 10
or 12 ft long, for giving a wide footway across the pit. A lever, 8 or 10 ft long,
fitting into a staple, is used for turning the table, not on account of friction, but aa
a handle for the workmen.

Art. 6. The semi-cylindrical shape of the Joint between the cap, C, and tha
socket, «, permits the slight longitudinal rocking motion of the turntable which
takes place when a locomotive comes upon it or leaves it ; but prevents it from ti|h
ping sideways, as it was apt to do, when, as formerly, the cap rested directly upon
the lid u of the roller-box, and the top of the post P was hemispherical, forming a
ball-and-socket Joint with a casting upon which the roller-box rested.

Art. 7. Ten slses of tbese turntables are furnished for locomotive use.
The diameters are aa follows : 70 feet, 65 feet, 60 feet, 56 feet, 54 feet, 50 feet (two
patterns), 46 feet, 40 feet, 30 feet and 12 feet. For prices, weights, dimensions of pit
and foundations, timber required, etc ; address William Sellers A Co, 1600 Hamilton
St, Philadelphia. Machinery for turning, being considered unnecessary, is not at-
tached, unless specially ordered. Its cost is extra.

The entire cost of excavating and lining the pit; foundation for pivot; circnltf
rail for end rollers, &c, complete for a 56-ft turntable will vary from $1200 to $2600
in addition, depending on the class of materials and workmanship ; and whether the
bottom of the pit is paved or not.

Art. 8. Tbe ylrders of turntables are now very generally made of

TVUmi.'BLBB.
r«Ued-lron plnt«a, Hoh glrdar bstDg Id oaa pfa
of r or chunnol bauns, or or bIkM-I™, ona on BMlnW

Art. ». In the plaleLron InrntablM mad* bj Ihs Ed«« Moor Bildge Worki
WLLiDwigtoo. D»l., thfl croBa-girda™. Q O, Fig. g, are of plate-ipon ; md it thofr «ddi
they bnvB flangw, F F, of anglB-Lton, by whfch they »ro Hvoted to ths naJn-ginl*™. a

}7 whloh tha tabls 1i iniHoded from the cap, C, w
f«d In two lf>HB of thrao bolta a&obi ona niw buJbg om

-. .Vim tlrt"3boIU,H,of ouo'row! Tha hor flauga, bearing HI th«

Buga rroni irhlding nndat lu load. Tare itntls of atiKle-lron are riTat«d to tha web
oT tha cniw«lnlar batwaeo thaboIlL Two of theu atruta are ihonnat K K, Flir7.
Tbair ends abnl agalnat the upper Hangs of Ihe crosa-ginler, and agalnaiJ J.

'Ria S bolts, II H, pug up Ihroo^b Iha tlaDgae of tha ap. C (Ihroa bolls tbrongh
each flanee), and Ihalr nuu real njmn Lta lop.

Art. 11. The enp, 0, la held In plaeo on Iha eocket, >, bj mnaa of
langaa whiob attend down flrani It en both al^ aa ihown la Fig T.

Tha roller^ <i(f,aDd tba iDltar-boi. ua,ara thoaa mada b; Wm. Ballen < Oil
Pb(l»,Arl3. ^

The tat of the table mar be»4|aa(ed, within * range of lor 3 Ina, by
nuaa Qf iba nati, N N, ai tn the Sallan labia, Art 4.

The lower part, t>, of the roller-box, Initud of regtlng dlrectlj upon the potl, P,
aaln the aellere teble, An S.reeUnpon an Iron caBtloK.L.Bhich.ln turn.reela upon

The poet' It built np d( plate- and anglB-lrona rl.fled logelher, and msj be a hol-
low tmnoaled Bqiiare pTnoild, aa abown, or of other abapea. Thoae pmentlng tha

Art. 1B« The tnAln girders are braced by hordiejif rode, whoeeendt

which thejr may be tiKhlened or looaenad.
Tha ramatha In Art 9 on the end wheels of the gellen table apply alio to

top, lo aa to Ineure a iinlfotm bearing, no matter whet Inclination maj be ElreD M
the axla hj tha Tert adjuitment of tha lournal-boxea.

^ TUBNTABLEB.

WOAdeii taratAbles. wltb dobs bnt two cammon vheel mllen mt ucb

end or tbe plitniriD. are somi'tlmes lesorted to nom motiiei or original cost.
ThBjr sre, however, mucli harder to turn, geOBrBili requiring two iiieii, aided

or g to' 12 imil] loUen Iraiefling on a cireu'lar ri^l or e to 12 Imt dinmetei;
■ rouDd the pliot as a center. Tbeae are lotended to lualAiu tlie whole weight;
the EDd rollen iHing go adjusted aa tu touch their rail odIt when tbe plsC^rm
nKkioTtliuu the eouine enters or leaves it. Therefore, thereia)(« realetaDce
rrom Triction than wiien, aa iu Fljts 11, there are odI; the end ivllen r. In thle
lust case, the engine and lender cannot be balincod to preeiaelj upon the
■Leader central piTot, u to prerent a great part or the weight troni tHing
thrown upon the end rollen; thus maiMialJj locieulag (be frlotlonil re>

ling on tie eirciiiar rail. Thm, in Figs 11, (which show one or the manr modM
or framlug a Uole which has only a central pi'ol/, and end rollers r.) the mala
platTorm resia en the girders c, which are strengthened below by braces a; while

across the pit. One-hair°or one arm. "r Vl'sTranaiene platrorm Is intended to

"rV«Sng! '^Ii'ls lidporUui " connect th"e"/ou' ends of.ibeTwrplStftJme b*
rour beams, as tbe whole structure is therebj msterisU; stiffened. In the
figures tbe wheel work Bx x 1b for convenience Impropcrlj ahown aa If It Mood

The figures need hut IlUle explariBtlon. They represent ar iclual U-fool
platform which has been In use for some years. The coDTex root f at the

rest on a slPel lUp ii. This should tp kept well ollf"; and protected' Tnm
dust by a leather collar around;), and resting on pu. Its upper part, about »
Inches diameter. Is cut into a screw with equarc tlireads abbil \i inch Ihlek.

TURNTABLES. 849

its head n,) so as to revolve with it. Strong screw-bolta ii oonnect the aeTeral
timhers at the center of the platform.

B is a light cast-iron stand supporting two bevel wheels about 1 foot diameter,
which give motion by means of an axle dfl% inches diameter to two similar
ones below, shown more plainly at W and x. These last give motion by the
axle 2 to the pinion e, (6 inches diameter, and ^ inches face.) which turns the
platform by working iuto a circular rack <, (teeth horicontaL 1 inch pitch ; 3^
inches face,) which surrounds the entire pit. This rack is spiked to the nnder
side of a continuous wooden curb H, wnich is upheld by pieces F, a few feet
apart, which are let into the wall J J, which lines the pit. The short beam
M N, (about 6 feet,) which carries the lower wheelwork, is suspended strongly
firom the beams of the transverse platform above it. Instead of the two lower
bevel wheels W Y, and the horizontal axle «, a more simple arrangement is to
place the pinion e at the lower end of the vertical axle a; and let it work into
a rack with verticat teeth at u, on the inner face of the stone foundation of the
circular rail. For this purpose the stand B should be directly over u. There
are two cast-iron rollers r, 2 feet diameter, 3 inch face, under each end of the
main platform ; and one under each end of the secondary one.

Although this kind of platform necessarily has much fHction, yet 6ne man
can generally turn a 45-foot one by means of the wheelwork, when loaded
with a heavy engine and tender. Indeed, he may do it with some difficulty
bv hand only, while all is new and in perfect order; but when old, and the
circular railway uneven and dirty, it requires two men at the winches to do it
with entire ease.

As before remarked, the resistance to turning is diminished by employing a
•et of from 8 to 12 rollers or wheels r,
Figs 12, about a foot to 15 inches in diam-
eter, so arranged as to form a circle 8 to 12
feet diameter around the pivot. When this
is done, the main girders of the platform
are placed 8 to 12 feet apart; and long
cross-ties are used for supporting the
railway track. Also, the main girders
are sometimes trussed by Iron rods. FigSi t2i

Fig 12 shows the arrangement of these rollers r, which revolve upon a cir*
celar track «; while the platform rests on their tops by the trace «. The
rollers r are held between two wrought-iron rings o, o, about 8 inches deep,
\4 inch thick, which also are carried by the rollers. From each roller a radial
tMkrod L 1 inch diameter, extends to a rlngnn, which surrounds the pivot p,
closely, out not tightly, so as to revolve independently of it. These tie-roos
keep the rings o o at their proper distance from the uvot, so that the rollers
cannot leave the rails s and n. Between each two rollers, the rings oo should
be strengthened by some arrangement like a, to prevent change of shape. The
pivot p may be as in Figs 11. There must, of course, be the usual two rollers
under each end of the platform, for sustaining the engine as it goes on or off;
but during the act of turning the platform, the whole weight should rest on
the central rollers. Such a platform of 50 feet length can, if carefully made,
be turned, together with an engine and tender, bv one man, by means of a
wooden lever 12 to 16 feet long, inserted in a staple for that purpose; and there-
fore may dispense with the transverse platform for sustaining wheelwork.

Such rollers as have Just been described, in connection with friction rollers.
Fig 0. form perhaps tne best arranf^ement for a laive tnmiiii;

bndse. At least one end of a platform must be provided with a eateli ot
StopTor arresting its motion at the moment it has reached the proper spot.
54

850

ENGINE-HOUSES, ETC.

A common mode Is shown at Fifr 18. It consists of a wrought-iron bar mn, 4
feet long, 8 inches wide, and ^ thick ; hinged at its end m, which is confined to

the top of the platform. Its outer end n is
formeci into a ring V for lifting it. A strone
casting ee (or in longitudinal section at itA
about 15 Inches long, 8 inches wide, and 1
inch thick, is also firmly bolted to the top of
the platform ; and the stop-bar mn rests in its
recess r, while the platform is beingr turned.
A similar casting a a is well boltcn to the
wooden or stone coping e e, which surrounds
the top of the lining wall of the pit. When
the stop-bar reaches this last casting, as the
platform revolyes, it rises up one of iis little
inclined planes tt^ and falls into the recess of
a a, bringing the platform to a stand. When
the platform is to he started again, the bar Is
lifted out of Its recess by the ring F, until it passes the casting; when it is again
laid upon the coping e c^ and moves with the platform ; or, if required, the
hinge at m allows it to be turned entirely over on its back. When there is a
transyerse platform, the proper place for the stop is at that end which carries
the turning gear; as it is there handy to the men who do the tuminff. If there
is only a main platform, the stop may be placed mid war of the rails. Some-
times a eprtnip eateik is plaoea at each end of the platform ; and each catdi
is loosened from its hold at the same instant by a long doable-acting lever. All
the details of a platform admit of much yariety.

TV

Instead of the Motion ft»0en. Fig 5, fHctioi
bali* 5 or 6 inches diameter, of polished steel, are
sometimes used. The piyots also are made in
many shapes.

Platforms lllLe oei. Fi^ t'i* rewolT-
Ukg ar«and one end o as a center of mo-
tion, are sometimes useful. The shaded space is
the pit. If an engine approaching along the track W, is intended to pass on to
any one of the tracks 1, 2, 3, 4, the platform is first put into the required posi-
tion, and the engine passes at once without detention. If the platform is lon&
it will be necessary to have roller-wheels not only under the moving end a, bot
at one or two other points, as indicated by the roller rails e c.

Engrine bonnes, of brick, cost from $1000 to $1900 per engine stall, exdo-
sive of the foundations.

The cost of a complete set of shops of brick, for the thorough re-
pair of about 20 locomotives, and of the corresponding number of passenger
and other cars: together with suitable smith shop, foundry, car shop, boiler
shop, copper and brass shop, paint shop, store rooms, lumber shed, offices, Ac:
completely Aimished with steam power, lathes, planing machines, scales, and
all other necessary tools and appliances, win be about from $75000 to $100000 ex-
clusive of ground. A large yarcl, of at least an acre, should adjoin the buildings.
A moderate establishment, for the repairs of a few engines only, may be bout
and furnished for $26000.

WATER STATIONS.

851

WATER STATIONS.

UTater stations are points along a railroad, at which the engines stop to
take in water. Tbeir distance apart varies (like that of the fuel sta*
tions, which accompany them,) from about 6 miles, on roads doin)( a very large
business : to 15 or 20 miles on those which run but few trains. Much depends,
however, upon where water can be had. It has at times to be conducted io
pipes for 2 or 3 miles or more. The object in having them near together is to
prevent delay from many engines being obliged to use the same station. To
prevent interruption to travel, they are frequently placed upon a side track.
A supply of water is kept on hand at the station, usually in large wooden tubs
or tanks, enclosed in frame tank-houses. The tank-house stands near the track,
leaving only about 2 to 4 feet clearance for the cars. It is two stories high ; the
tank being in the upper one ; and having its bottom about 10 or 12 feet above the
rails. In the lower story is usually the puiup for pumjiing up the water into the
tank; and a stove for preventing the water from freezing in winter.*

The t&nks are usually circular; and a few inches greater in diameter at the
bottom than at the top, so that the iron hoops may drive light. Their
capacity- generally varies from 6000 to 40000 gallons, (rarely 80000 or more,)
depending on the number of engines to be supplied. A tender-tanlL holds
from 1000 to 3000 gallons; and an enelne evaporates from 20 to 150 gal-
lons per mile, depending on the class of engine ; weight of train ; steepness of
grade, Ac. Perhaps 40 gallons will be a tolerably full average for passenger, and
80 for freight engines. The followlngr are the contents of tanhs
of different inner diameters, and depths of water. IJ. S. gallons of 231 cubio
inches ; or 7.4805 gallons to a cubic foot.

Diam.

Depth.

Ck)ntent8.

Diam.

Depth.

Contents.

Ft.

Ft.

Gallons.

Cub. Ft.

Ft.

Ft.

Gallons.

Cub. Ft.

12

8

6767

905

24

12

40607 ■

5429

14

9

10863

1385

26

13

51628

6902

16

9

18535

1810

28

14

64481

8621

18

10

19034

2545

SO

15

79310

10603

20

10

23499

3142

32

16

96253

12868

22

11

31277

4181

34

17

115451

15435

Cypress or any of the pines answer very well for tanks. The staves
may be about 2}4 inches thick for the smaller ones; to 4 or 5 inches for the
largest. The bottoms may be the same. The staves should be planed by ma-
chinery to suit the curve precisely. Nothing is then needed between the staves
to produce tightness. A single wooden dowel is inserted between each two near
the top, merely to.hold them in place while being put together. The bottom is
dowelled together; and simply inserted into a groove veVy accurately cut, about
an inch deep, around the inner circumference of the tub, at a few inches above
the bottoms of the staves.

One of 20 feet diameter, and 12 feet deep, may have 9 hoops of good iron ; placed
several inches nearer tosether at the bottom of the tank than at the top. Their
width 3 inches; the thickness of the lower two,^inch ; thence gradually dimin-
ishing until the top one is but half as thick. The lower two are driven close
together. These dimensions will allow for the rivet-holes for riveting together
the overlapping ends; and for a moderate strain in driving the hoops firmly
Into place.f Three rivets of }4 ^^^^ diameter, and 3 inches apart, in line, are
sufficient for a Joint of a lower hoop. One of 84 feet diameter, 17 deep, may
have 12 hoops; the lower ones 4 inches by ^; with three ^-inch rivets to a
lower hoop-joint.

The bottom planks of the tank must bear firmly upon their supporting Joists,
or bearers.

A tank musthave an Inlet-plpe by which the water may enter it ; a waste*
pipe for preventing overflow ; and a dischargee or feed-pipe 7 or 8
incbes diameter, in or near the bottom; tlirough which the water flows out to
the tender. The inner end of the discharge-pipe is covered by a valve, to be
opened at will by the engine man, by means of an outside cord and lever. To

* A frame tank-house, 18 feet square, with stoue foundations for both house
tnd tank, will by itself cost $400 to $600. A brick or stone one somewhat more.
fSuch a tank, put up in its place, will cost about $400.

852

WATER STATIONS.

Re outer end is generally attached a flexible canvas and gum-elastio hose about
7 or 8 inches diameter, and 8 or 10 feet long, through which the water enters the
tender-tank. Or, instead of a hose, the feed-pipe may be prolonged by a inetal*
lie pipe, or nozzle, sufficiently long to reach the tender; and so Juintod as, wheli
not in use, to swing to one side, or to be raised to a vertical position, (in the last
case it iat called a drop^) so as not to be in the way of passing trains.

The same tank may supply two engines on difl^rent tracks, at once. Tiia
tanks are very durable.

Tlie patent flrostHproof tank of Jobn Bai>nliam, Batavlii,
Illinois, 18 simply an ordinary tank, in which the water is prevented from
fceezing by means, 1st, of a circular roof which protects a ceiling of joists, be>
tween which Is a layer of mortar ; 2d, by an air-space obtained by a similar cefl*
ing beneath the timbers on which the talnk rests. Although the sides are en-
tirely unprotected, no house is necessary ; but merely strong posts and beanie
on a stone foundation, for the support of the tank.* The supply pipes are is
boxes made of boards and tar-paper.

Tanks are Creqnently made reetansmlar. with vertical sides of
posts lined with piank. and braced across in both directions by iron rods. They
are more apt to leak than circular ones. They have been made of iron ; bat
wood seems to be preferred;

The water for sapplytnflr the tanks, may be pumped by hand, steam,
horse, wind, hydraulic ram, or otherwise, from a running stream; from a pons
made by damming the stream if very small or irregular; from a cistern below
the tank ; or from a common well. Manjr roads doing a business of 10 or tS
engines daily In each direction, depend entirely upon wells; and pump by hand;
generally two men to a pump. Those doing a very large business, when the
supply cannot be obtained by gravity, mostly use steam. The windmill is
the most^ecouomical power; and when well made, is very little liable to get out
of order. Of course it will not work during a calm; but this objection may be
obviated in most cases by having the tanks large enough to hold a supply for
several days. Steam, however, is most reliable.

The folloniugr table will give some idea of the power required In
a steam enfi^dne for the pamplnsr. In oidering an engine, specify not
its number of horse-powers, but the number of gallons It must raise In a glvea
number of hours, to a given height; with a given steam pressure faay about 60
to 80 fi>s per square incn.) The pump should besufficientlv powernil not to have
to work at nigiit ; and should be capable of performing at least 26 per cent, more
than its required duty.

A fair aweragr^ horse should pump In S hours the quantitim
contained in the first 3 columns; to the height in the 4th column; or snflicient
to supply the number of locomotives in the 5th column, with about 2000 galloos
each. Two men should do about one-third as much.

OBb.Ft.

Lbs.

f
Gals.

HuFt.

9s. of
Loeos.

Col».Ft.

tiM.

Gals.

Hi. Ft.

ii«.«r

1000

100000

11968

100

•

4571

286714

$4194

86

17

SX)00

125000

14960

80

7H

6883

8838S8

89898

80

90

im

106066

I994n

60

10

6400

400000

47872

20

94

8200

200000

2S986

00

12

8000

WOOOO

09840

90

89

8S55

222222

26596

45

19^4

10667

6o6o67

79787

16

40

4000

250000

29920

40

16

16000

1000000

119680

10

60

A reserToir, with a stand-pipe, or water eolnmn, is preferable
to the ordinary tank, when the locality admits of it; being less liable than the
pump to get out of order; and being cheaper in the end. The reservoir is sup*
posea to be -filled by water flowing into it by gravity ; and to have its bottom si

* The cost of wipdraill alone, for railway stations, varies from about $450 lor
18 feet diameter ; to $1500 for 36 feet diauieter, at factory.

WATEB BTATI0H8. SoS

groDDd SDd the belgbt of th/irater laaj rrqulre. It ms; be euavDied In Ihe
or It m'nj be butll above gniaud. iccordtpg to the IochIIIj. It tuii)' be rootei

bSaa tHa 1a™n d^melfr. Is M^w'^'en evilly uSdergrouDd,) lo wIlhTn »%^

Bboie ib« track, forming • ataiid-ptpe, op wster-Htlnmii i rrom tbs
upper FDd of irhkbtn« valei flovs (tbtouijb sltlier ii bOM or ■ Joip(«d noEile,)
u Id the cue of h l«nk. Sflyer»l lucb pipes, or une larger one, mir be laid, for

Ibe pipe maliFS lis hcnd,«nd b«oinP«>erlle»l,l>a''«l'e foroMnluRand cloalDg

euU; reacbed br |be eoglDS man.

DP waMr. wblle rBUMlaa;, frooi a lone trvoal
u!d between Ibe nITi. The UalTan aboat ^ mile long.

OurflgqieihowsatrBck tanbof Al°<!>> »lled plate-livn, UteibeeUof which
nreejIniloDB. TbelSDgtb*DTeilap«iicbolter2ln9: lesTingG ft u their lAtwina
lODgth. The eheeta are cut slIghtlT Uperlug. ao tbal at one end of eaeh length
tbelniaEblaAlndeeperifaaagttfaeolbeT,aDd thetopiare tbuBkeptfluih witb
«Mb olhsr throngbont. TtaeJolnU are double rlTeted with % Incb rtyeU, aboDt
IH Ih" r™ «nwi lo «nWr, and^ ',^^^'^- *^ *""'' ^^'Ji^^r^l^"^^- ""?
tbe Bides. Theen»s-tl« are nntcbia, r° ahow'nT^TcceKe'thriri^gh, wblch^
loosety held to Ihem bj^lwo ipikea, S anas, m each tie. The hcsda or tbo »p(ke»

Idtngsof I'^x «lnch Uar-iron. 'Jhe anelesanrt tf

._ _engtha

tbrougfaoul Its length.

The seovp on the lender li lowered ialD ibe trough, nnd ntleed
by meani o( a l^ver on tbe flreman'a platform, and la not permitted lo to
bottom of tbe trough.

' Ibf trongli Is Happlled with water b; meam at plpn leiding ftoc
Jarent lank. Tbe eupplj i> regulated bj a man In charge.

Topreient the wat«r from b«eEliiK In nintrr, ateim luled tu the

>] Hangca of tbe t^ X i% io=h angle

noiiidtnpof 11^ X^ inch bar-iron. Theanglea... ... „

;tha of 15 ft. ajid are rireted to tbe sides of the trough conlinuouslj

854

FENCES, ETC.

Evaporation from liOComotives. The evaporatioa is
usually from 6 to 7 lbs of water to 1 lb of fair coal. Hence if we take the average
at 6^ n>Sf or say .8 of a gallon of water to 1 lb of coal, and assume, as on page
800, that a passenger engine eyaporates an average of 40 gallons per mile, and
a freight engine 80 gallons, we shall have very nearly ^^ tons of coal consumed
per 100 miles by the former ; and A% tons by the latter. The evaporation from
a heavily task^ powerful engine may amoudt to 150 gallons or more per mile;
but such is an exceptional case.

Theoretical thickness near bottom of sheet-Iron water
tanks, single riveted ; safety 4 ; ultimate strength of the iron 40000 lbs per
square inch, out reduced say one- half by punching the rivet holes. Although
safe against the pressure qf the ioater^ some are plainly far too thin for handling.

Depth In
Feet.

INNBB DIAMBTER IN FEET.

10 I 15 i SO I S5 I SO I SS

40

THICKNESS IN INCHES.

1

.0026

.0052

.0078

.0104

.0130

.0166

.0182

.0208

5

.0130

.0260

.0391

.0520

.0661

.0781

.0911

.1042

10

.0260

.0521

.0781

.1042

.1302

.1662

.1823

.2083

15

.0391

.0781

.1172

.1562

.1958

.2344

.3734

.3125

20

.0521

.1042

.1562

.2084

.2604

.3125

.3645

.4166

25

.0651

.1302

.1953

.2604

.3255

.3906

.4557

.5208

80

.0781

.1562

.2344

.8124

.8906

.4687

.5470

.6250

Railroad track scales. The capacities are In tons of 2000 lbs or 2240
fts, as may be desired.

Capac-

. Length.

Capac-

Length.

ity.

ft.

ity.

ft

10

12

66

40to 66

15

12 to 15

75

40to 86

20

12 to 16

100

60 to 112

30

20 to 82

160

60 to 128

40

30 to 40

•150

100 to 160

50

40 to 50

Post-and.rall fences, in panels ^ ft long; 5 rails; nsaally co«t between
40 to 100 cents per panel, including the putting up ; or firom $512 to 91280 per mile
of road fenced on both sides, with 1280 panels.

Fence-posts are usually of chestnut, cedar, or white oak. They last about 10 yean
on an average. The usual size is 2 to 3 ins thick X 0 to 7 ins wide, 8 ft long, 6 ft
above ground. Their cost varies greatly ; say from 6^ to 26 cts each ; average, 10
to 15.

Worm fences seven rails high, with two rails on end at each angle, cost about
^th less. Labor $1.76 per day. The scarcity or abundance of timber chiefly in-
fluences the price ; as is also the case with ties.

Barbed Steel irlre Anoe costs, per mile of single row ef fence, pat up,
including the wooden posts and all labor, from $150 to $260, depending on the
height of fence, the varying market price of wire, labor, See.

A^nraj'-statlon house, 80 X 6f) feet, surrounded by a platform 12 feet wide,
protected by projecting roof; for passengers and freight; will cost finom $0000 to
|10,000, according to finish and completeness, at eastern city prices.

^

COST OF BAILROADS. 855

Approximate averaire estimate for a mile of 8liii:le-track

railway. Labor $1.75 per day.

Chrvbhing and dearing, (average of entire road^) 3 acres at $&0 $ 150

€hrading; 20000 cub yds of earth excavation, at 86 cts 7000

*« 20OO cub yds of rock excavation, at $1.00 2000

Mcuonry of culverts ^ drains^ ahutments of small bridges, retaining-waUs, dk;

400 cub yds, at $8, avertige ' • 3200

BaUast; SOOO cub yds broken stone, at $1.00 3000

€^ss4ies; 2640, at HO cts, delivered 1584

Bails; (00 lbs to a yard;) 96 tons, at $80, deKvered 2880

Spikes 150

RaUrwints 300

Sub^divery of materials along the line. 300

Laying track 600

^kncing ; (average of entire road,) supposing only j/^ of its length to be fenced.. 450

Small wooden bridges, trestles, sidings, road-^srossings, cattle guards, ilk, dc 1000

Land damages 1000

Engineering, superintendence, officers of Oo, stationery, instruments, rents,

printing, law expenses, and other incidentcds 2386

Total $20000

Add for depots, shops. IngfQe-boaaes. Pasaenger and Fr«if ht Stfttioiii. PUtforaii,
Wood Sheds, Water Statioiu with their tanks and pamps. Telegraph, Bnginei, Can, Wdfh SflalMi
^mM, 4o, ito. Also for large briagc«. luunelt, Tumouu. Iw.

856

LOCOHOnVES.

BOLLING STOOK.

liOCOMOTIVES.
Dimensions, Weights, Ac

Lists of some of the principal locomotives made by the Baldwin Looomotly«
Works, Philadelphia, liOfL

In the designation of the class, the first number (8, 10, Ac) is the total num-
ber of wheels of the locomotive. Tne second (80, 82, Ac) eives the diameter of the
cylinders, thus : diam. in inches = n/2 + 3, where n » this second number. The
letter (C, D, or E) indicates the number (4, 6, or 8, respectively) of drtvinshwheeia.

The wneel-base is the distance from center to center of the front and buck
wheels. For minimum lensrtli of turntable, add IV^ to 2 ft to the total
wheel-base of locomotive and tender, which is = wheel-oase of locomotive -4-
wheel-base of tender + dist. C to G of front tender wheel and hind engine wheel.

Under ** SerTtee,'* ^m^ms oauenger: V^ freight; H, mixed; H^twUchimg.

Since 1890. the eomponnd loeomoiive has come into extensive use m
many brancnes of service. The Baldwin (Vaudain) type has a high-pressuie
and a low-pressure cylinder on each side. Those of other makers usually have
bat two cylinders, the high-pressure cylinder being on one side and the 16w-
pressure cylinder on the other. The dimensions and weights of compound
engines do not differ materially from those of the corresponding classes of simple
engines, as given below.

For ffauffe of 4 ft 8 1-9 Ins.

1

CjUn-
dtn.

DrivlBg
•Whftli.

WkMl-tan.

Extreme ]|th
loco, and
tender.

Extreme

Eelfkl

onop

Stack

Locomotive.

Ten-
der.

Loco,
and
ten-
der.

GhuBs.

•

§

s

Ina.

1

GO
Ina.

1

1

Driv's

Total.

aboTe
raU.

Ina.

Ft. Ina.

Ft. Ina.

Ft. Ina.

Ft. Ina.

Ft. Ina.

Ft. Ina.

Ft. Ina.

8.80 G

P

18

24

4

66-72

7 6

21 8

16 0

49 3

66 3

9 0

14 6

8.34 G

II

20

24

4

72-78

7 6

21 11

16 6

50 0

66 0

10 0

14 ft

10.84 D

P
F

20

24

6

66-68

11 9

22 11

15 0

60 6

66 6

10 0

14 e

10.88 D

<*

22

26

6

62-72

12 6 ,24 2

16 6

62 3

68 8

10 0

15 0

8.34 D

F

20

26

6

50-66

14 0 22 6

15 0

60 0

66 0

10 0

14 •

8.88 D

X

22

30

6

66-«2

14 0

22 11

16 6

61 0

67 0

10 0

16 0

10.84 E

(1

20

26

8

60-66

14 10

22 8

16 0

50 3

66 3

9 6

14 6

10.88 E

II

22

30

8

60-56

16 0

23 2

16 6

61 3

67 8

10 0

15 0

4.32 G

S

19

24

4

60

7 6

7 6

15 1

38 0

63 10

9 0

14 6

6.86 D

<i

21

26

6

50

11 0

11 0

15 7

40 1

66 9

10 0

14 6

•

Wsigkt in worUag ordtr,.in pormdi.

OtMotty of tiBtor.

Locomotive.

a

Glass.

Service.

Type.

Greatest on

1 pair of

driv. wh'ls

On all
drivers.

1

Total of
looo. and
tender.

Goal,
tons.

Water.
Gals. tM^

8.30 G

Pass'ger

Amer.

88000

74000

106000

66000

172000

^

8300 2700»

8.84 G

««*

<«

44000

86000

122000

80000

202000

7

4000 8883S

10.34 D

P. AF.

lOWh'l

36000

100000

136000

72000

208000

6

8600 80000

10.88 D

«(

<<

44000

124000

164000

80000

244000

7

4000 888SS

8.84 D

Freight

Mogul

42000

120000

139000

72000

211000

6

8600 80000

8.88 D

«t®

62000

150000

171000

90000

261000

8 4600 V75Q9

10.34 E

i«

Gonsol.

34000

126000

140000

68000

208000

6 8400 28889

10.88 E

«

14

45000

170000

188000

90000

278000

8

4600 87W0

4.82 G

Switch

63000
48000

104000
140000

104000
140000

seooo

76000

160000

216000

6
6H

2800 28888

6.36 D

8800 81888

1

LOCOMOTITE8.
For yuKC of 8 ft.

1

telto-

M

¥S..l-ll»M.

4.

a .

:,r;

CllH.

1

1

i

■|

LocomotiTe.

;r

I-Ka

of

Drtv.

™»i.

dw"

(nun
»1L

8^C

10.MD ;

lO-ME

A
H

1*

IS
16

IS
IS

42

8 a"

i 1

ii

40 9

BB 4
40 8

.7 <

8 8

ii

8 0

18 e
It 1

18 S

e«rrlc«.

Tn*

wtiju Is *wHs« Mm. b poini.

CpilJljSftaBdW.

•s

1

1

II

dui.

K

^1
51

Tot»L

Omlg. tm.

lOME

SS&

17000
ITOOO

7000

1

1

86000
40000

U300

"^
^

ISOQ 13Sn

1800 isooo
isoo uwoo

1

a.B.

IMl

«■<!(« 4 ft » ln>.

ci™.

Wkiil-UM.

ii

n.

10

1

ii

n

Drirt

rouL

....

1

4

s

8

lu.
«S

S2
K

FLIu
7 9

Pklu

lis

17 6

FLIu

H)8H

BOSK

Til^ In w«Ab{ndM.

868 LOCOMOTIVES.

Sclecl«d Erie B«IIroad SUuidards, 1903. »•■■«« 4 ft 8 1-2 Ibb.

.on n-

OC^

»0 0 n

c n OOo oc^

.o o o

oc^

nOOn-n^„

noon ni^ «

QQQD_Q_Q^i

nonnn nb.^

i -

M

aboTs

vW

860

LOCOMOnVES.

Perfo:

ee of ItoeoinotlTeB.

Pttnenger enginee usuall j carry Iteel and water snfflelent for 4MI or

50 miles; some, 80 to 70. Freight trains, enough for 20 to 26 miles. Boads,
or dlTislons, with steep grades require the fuel and water stations to be nearer
together than where tne grades are easy.

The following gives the loads (exclusive of locomotive and tender) whidi
tile above described Baldwin enyin®* vUl haul, at their usual speeds, on a
straight track and on dlflterent vrades varying from a level to 8 ft. per 100
ft., or 158.4 ft. per mile. The loads are based upon the assumption that the
MHoalled ^ adhesion " of the locomotive is nine-fortieths of the weight on all
the drivers, and that the condition of road and cars is such that the ftrlctlonal
reslstMiee of the cars does not exceed 7 9E>s. per ton of 2240 its. of their
weight. These are ordinarily favorable conditions. The adhesion is seldom
less than one-fifth, or more than one-third, of the weight on the drivers.

lioads la tons of 2240 IImu (exclusive of locomotive and tender).

Gan^e 4 fU 8 1-2 Ins.

Caass.*

8-ao-c

8-84 C

ia««4-p

10-88-D
8-M-D
a-«8-D

10 84 E

4-3-C
e-86— D

Ser-
Tice.*

P
•*

PF
«<

F
«

«

s

M

Type.

Amer.
((

10Wh»l

<t

1

Mogn

ConsoL
It

On a grade of

Oper
ot. -i
Oft.
per
mile.

1980
2800
2675
8825
8225
4060
8375
4600
2826
8776

cent.=

26.4 ft.

per

mile.

790
920
1086
1860
1816
1660
1876
1876
1160
1660

Iper

cent =3

62.8 ft.

per

mite.

470
646

645
810
796
996
880
1136
TOO
940

IJiper

oent.=3

79.2 ft

per

mile.

820
876
446
660
666
606
680
796
490
#60

2 per

cent.s

106.6 ft

per

mile.

285
276
880
416
416
620
436
600
870
496

cent.—

182 ft

per

mile.

180
210
256
826
826
410
846
470
806
896

Sper
oent.aB
164.4 R
per^'

mile.

140
166
205
260
260

31

286
820

Ctenso 8 fft»

8eiw
▼ice.*

Type.

On a grade of

OtM*

Oper
ct. —

Oft.

per
mile.

HPer

cent.s=

26.4 ft.

per

mile.

1 per

cent.»

62.8 ft.

per

mil&

iHper

cent. =3

79.2 ft.

per

mile.

2per

cent —

106.6 ft

per

mile.

2Kper
cent.—

182 fL
per

mile.

8per

cent.««

154.4 ft

per

mile.

8-22-C
10-24— D

8— 24— D
ia_-26— E

4— 20— G

P
PF

F

<i

S

u

Amer.
lOWh'l
Mogul
Consol.

610
900
916

1226
800

1080

266
406
410
656

866
496

168
246
255
845
280
810

106
170
176
240
186
226

76
126
186
186
126
170

66

100
106
146
100
140

60
76
80
116
80

6— 24— D

m

tlwe foree

•f a locomotive,
in pounds

Square of diam. ^ Single length of ^ J^\yi!^i^!;L
S?l piston in ins. X stro\e in ins. ^ "^^JS. ^! ^iS^^

Diameter of drivin|^wheel in inches.

•Seep. 868.

LOCOMOTIVES.

861

tYom the tractiye force must be deducted 20 to 80 per cent, for internal fric-
tion, eta The effective tractive force cannot exceed the adhesion, or say one-
fourth the weight on the drivers.

The iniiuU steam prewiure tn tlie eylinders is a]ways less than the
boiler pressure ; and the disproportion increases with the speed. Thus, with the
bo/ler pressure about 200 lbs. per squwra ineb, the cylinder preesure, at 8 to 10
mileB per hour, may be from 180 to 190 Ids., while at & speed of SO or 40 miles, it
may be only 160 to 170 Iw. The average cylinder pressure is ascertained by
n^DS of an indicator applied to the cylindw; and its proportion to the initial
pressure depends utou how early in the stroke the supply of steam from boiler
tq cylinder is cut off; or, in other worda, upon the extent to which the steam is
used e^ansively,

.The power and speed of loeomotlires. and (heir coiummp-
tion of fhel and water, vary greatly with circumstances, such as grades
and curvature ; condition of track and rolling stock ; number of cars in train ;
diameters, number and distance apart, of car wheels ; manner of coupling the
cars ; skill of locomotive runner and fireman, Ac, do.

On the Phlla A Readings Ry (Shamokin Division), between Catawissa
and Lofty, 84 miles, freight loeomotiwes are assigned train loads as below :
(according to conditions of the engine and weather) in tons,* including weight
of oars and loading, and 8.26 tons * for caboose, but exclusive of locomotive
and tender. Each locomotive is assisted by a helper of Class 1-5, or 1-7. The
grades and curves are as follows :

Grades.

Feet per mUe.

Miiftn Percent.
^"^■- of total.

than 24.5

Betw 80.62 and 35.05
" 89.60 and 45.40

Total

** 82.94 and 83.26

2.86

80.72

0.45

8.41

90.27

1.32

84.03 100.00

22.72 66.76

Curvature.

Degree of. No. of p^x
curve. curves '***•

8,203
29,694
26,218
10,479

Deflection.

Less than 8<> 24

" ♦* &> 59

«* «« 9P ^

13° 14

«

<t

376
1440
1343
1099

5
52
56
61

(14.13 miles) 183 74,594 4260 44

g

M

i

4
4
2
2
2
2

h

2

Train load, tons.*

1

Drivers.

Cylind's

&p*
Is

120
145
120
145
145
180

Weight, tons.*

Slow
freight.!

Train
No. 82.t

1

6
6
8
8
8
8

Diam.
ins.

18
20
20
20
22
22

24
24
24

28
28

Total
loco.

Teti-
der.

Max.

Min.

Max.

Min.

H— 1
li— 1
I— 1
1—2
1-5
1-7

64

61

50.5

50.5

50.5

56

21.60
47.00
48.26
62.67
63.00
72.00

44.60
60.50
66.75
60.72
70.50
82.00

26.26
34.50
27.50

28.00
32.75
46.00

1653
1638
1793
1898

2193

1028
1128
1198
1268

1463

1533
1633
1743

2038

1023
1093
1163

1368

Special test rans.

Catawissa to Lofty.

Class.

No.

of

cars.

Wt.of
train.

*

Time.
H. M.

Speed

miles

per

hour.

W'ter
used
gals.

Coal

Loco.

Help.

used

Date.

tons.*

Jane 6, 1(

" 7,
March S.

)98

I-l
1-2
L-?

1-6

nnnn

45
45

1826
1968

3 25
8 21
2 %

10.2
10.4
17.0

14500

13675

7000

g

<(

7

1901

3.5

— »

J

* Tons of 2000 pounds.

fRun from Newberry Junction to Tamaqua, 105 miles. Slow freights, 11 to 12
hours ; train No. 82, 10 to 11 hours.

A

862 LOCOMOTIVES.

Hie greater the ratio of live or net load, to dead load or tare, the greater is th«
total tonnage (cars and load) that can be hauled by a given locomotive. In order
to take account of this, a diBcrimination is made, on the Shamokin Diviaion, In
favor of cars having a capacity of not less than 80,000 Ibe. Trains in which from
10 to 19 of the cars. are of such capacity, are given 100 tons additional ; with 20
or more such cars. 200 tons.

On the Canaaian Paclfle Railway, t a ratio of 2 tons of net load, or
*' contents," to 1 ton tare, is taken as standard ; and it is found, on the eastern
lines, where the controlling grades are generally about 1 per cent., that a train of
empty cars offers about 30 per cent, greater resistance than a train of equal weight
with net/tare =2/1. A chart has therefore been prepared, by which the engines
are loaded upon this basis. Heavier grades do not increase the Jriedonal reaiat-
ances. Hence the total resistance increases less rapidly than the grade, so that
on grades steeper than about 1 percent., an addition of 30 per cent, to the resist-
ance of full trains, would give too high a resistance for empty trains of equal
weight, and vice versa.

On the Canadian Pacific, the loading of freight trains is graded as follows,
according to speed, weather conditions, &c.:

Ordinary trains..
Fast trains

Conditions.

Ordinary.

Temp. + 10° to —20® F.
or bad rail.

Temp, colder than — 20**
Fahr.

Reduce schedule by

0 per cent.
10 " "

7 per cent.
12 " "

12 per cent.
15 " "

In making deductions under this table, the probable conditions on the rulina
grade f not those at the starting point, are considered. During snow or wind
storms, loadings are determined by the conditions.

On the Rletamond, Fredericksbnrar A Potomac Radlroad,

freight locomotives draw loads as follows : — exclusive of engine and tender.
Cylinders 18 X 26, 62 in. drivers, 90,000 fbs. on drivers, 630 tons.
" 19X26, " " 102,000 '• " " 700 "

Maximum (and limiting) grade 1 per cent, on tangents.

W<Kkl fnel. A ton (2240 fi»s) of good anthracite or bituminous coal is about
equal to 1^ cords of good dry, hard, mixed woods (chiefly white oak) ; or to 2
cords of sucn soft ones as hemlock^ white, and common yellow pine. Much of the
inferior bituminous coal of Illinois is hardly equal (per ton) to a cord of average
wood.

A cord is 4 X 4 X 8 ft, or 128 cub ft A cord of good dry, white oak (next to
hickory, the best wood for fuel) weighs 3500 fi>s or 1.563 tons. Dry hemlock,
white, or common yellow pine (all of them inferior for fuel) , about .9 ton. Per-
fectly green woods generally weigh about 4^3^ more than when partially dried
for locomotive use ; in other words, a cord of wood, in its partial drying, loses
from ^ to ^ ton of water, and still contains a large quantity of it. Since this
water causes a great waste of heat, green wood should never be used as fuel. The
values of woods as fuel are in nearly the same proportion as their weights per
cord when perfectly dry.

When wood is used, about .2 cord *, or when coal, about 3^ cord of wood, most
be used for kindling, and setting up steam ready for running ; and this item is
the same for a long run as iror a short one ; so that long roads have in this respect
an advantage over short ones, in economy of fuel. Wood has the dlsadvantsn
of emitting more sparks ; and is, moreover, nearly twice as heavy as coal, for tfis
performance of equal duty ; and is, therefore, more expensive to handle. It also
occupies 4 or 5 times as much space as coal.

Up iTi'ades greatly increase the consumption of fuel. Thus, on a road M

miles long, with grades mostly of less than 6 ft per mile, and with very few ex-

• ceeding 14 ft per mile, with coal trains of 784 tons descending, and 291 tons

(empty) ascending, at about 10 miles per hour each way, the coal consumption

t Paper by Mr. Thos. Tait. Manager, Canadian Pacific Railway Co. , read before
the New York Railroad Club, January 17, 1901. Proceedings, vol XI, No. 8.

LOOOMOTTVES.

863

per 100 miles for each ton of total train (indading engine and tender) was 14.5
lbs descending, and 86.6 Bw ascending.

On flrst-olass roads a p— enyer engine will aTerave about 8500O
miles per year, or say 100 miles per day ; a freight engine 25000 miles per
year, or say 70 miles per day.

Goal

burned lbs

per mile.

Arer. net
sp'd miles
per hour.

^1

a

8

&

PS4

^ s

Drivers
diam.ins.

a

Stroke.
Inches.

s

s

I*

3 ^g'g S

CO

oo

dec

S

S8

CO

C4

U3

I

s

i

e

8

Q

a

flS

9

2

s
o

"So

a

kl

s

s

o
a

QQ

I

rf

o <s

i3

a

a

O

I

■s

M
u

o

9

5ZJ

o

co> COS

c«»

2

a

^ <0 00 ^ CO ^ ^

OP t^ !■» 00 t^ go oO

8 as** 8

CO Ol CO ^ o> U) CO

<

0 2 .2 hS loo

T:^ rt 3od . S tf TT-S

mO? »^ •*'« ^g '^ ^g wO'

|3S "SS 2a gS 5 25 2*2

9

g

O)

o»

s

&^

a

Burlin
Quinc

rk Ce
91.

©"•«

ooo

S (3 A

»?

.2 ©OO

J3 ♦ai-i

©^

O

SSi

48

l!

n

« o
S w

1^

3|

i|

00

ft

864 LOCOMOTIVES.

IfOComotlTe expenses per 100 miles ran, will ayerage about as follows:

Paasenger. Freight.

Fuel S8.00 $6.00

Water. 1.00 2.00

Oil, waste, Ac 70 .90

Repairs 4.00 8.00

Engineer and fireman 6.00 6.00

Putting away, cleaning, and getting out.... 1.50 2.00

Locomotiye superintendence 30 .50

S15.60 $25.40

An additional allowance of, say f2 per 100 miles, should be made annually for
depreciation of each en|pLne. An engine in active service, even under
a Judicious system of repairs, generally becomes worthless (except as old iron),
in say 16 years on an average.

CABS.

865

CABS.
Csaal dimensions, wei^tato and capacities. Gan^e 4 fit 8^ ins.

Passenger

Parlor

Sleeper 4..

Baggage, mail,

and express ...

Box and cattle ..

Gondola

Platform

CoaL

Damp

Leagth

of body.

ft.

50 to 60
60 to 70

50 to 60
28 to 36

32 to 36
29 to 36
10 to 18

Width.

ft.

10

li
^tolO

8>^to 9
8 to 9

7Kto sy^

Height

above ralL

ft.

14

4f

It

11 to 123
6 to 73
4 to 43
8 to 9>2
6to 7*

Weight,

empty.

lbs.

SOOOOto 80000
80000 to 100000

<(

40000to 60000
23000 to 36500
20000 to 30000

«i

25000 to
9000 to

83000
11000

Nominal ca-
pacity, in
passengers or

60 to 64

80 to 40

aboat 80

60000 to 80000
t(

II

11

15000 to 20000

* Add 6 inches for brake shaft,
t Add 2}^ to 3 feet for brake shaft

On narrow ffanye (3 ft and 3% ft) roads there is but little uniformity in
car building. Freight cars are usually from 25 to 32 ft long, 63^^ to 8 ft wide ;
capacity, 30000 to 40000 lbs.

Steel Cars, built by the Pressed Steel Car Co., Pittsburg, Pa.

Hopper

Flat

Box»

Furniture ♦ ..
Gondola

Length

oyer end

sills.

Ft. Ins.

31 6

40 0

35 115^
51

43 3

Width

over all.

Ft. Ins.

10 0

10 0

10 2

9 1034

10 0

Height

over alL

Weight

in
pounds.

86,000
28,800
32,300
43,800
35,100

CaiMcity

in
pounds.

100,000
80,000
70,000
60,000

100,000

Height

over
Brake Mast.

Ft Ins^

10 6Ti

6
13
14

8

9

8

s

* Wooden bodies with steel underframing : others all steel.

The aweriwe life of a passenger car is about 16 vears. Average annual
repairs, including painting, $300 to 8700; for mail and express cars, 3150 to
11300; freight cars, $75 to $150.

Allowina; 125 fi>s per passenger, &/ull car-load of passengers (50 to 60 in num-
ber) woula weigh but from 62^ to 7500 B>s, or say 3 tons ; while the cars them-
Belves weigh say 30 tons, or nearly 10 tons of dead loiul to 1 payings ton
of passengers. But, as a general rule, passenger trains are not more than
halir^fiUed ; making the proportion about 20 to 1. The foregoing table shows that
when freisrlit ears are loaded to their nominal capacity, there is less than
about 1-3 ton of dead load per ton of paying load ; or, with cars
half loaded, 2 to 3.

The average cost, in the United States, of moving a passenger one mile is about
double that of moving a ton of freight one mile, while the receipts per paa-
senger-mile are nearly three times those per freight-ton-mile.

Tbe resistance of cars to motion, on a level track, and with cars and
track in fair order, is usually taken at about from 6 to 8 B>s per ton of 2240 lbs.
With everything in perfect order, it may fall as low as 5, or even 4, fbe per ton.
On the other hand, if the wheels are not truly round, and if the Journals are not
well lubricated, it may 'greatly exceed 10 or 12 fi>s.

55

r

8^6 CARS.

To estimate rongrlily tbe speed of a train in which one is riding; ;
if, as usual, tbe rails are 30 feet long. By means of the sound of the trucks m
padsing the joints, count the number of rail-leogths passed in 20 seconds. Thia
number is a very little less than the speed of the train in miles per hour.

For, let n as the number of 3(V>foot radl-lengths passed over in 20 seconds. Then :

Speed in mUes p« hour = ^^^^Jl^ = ^-^^ «•

If the wheels are 28 inches in diameter, as is common in trolley cars and
bicycles, the number of revolutions in 5 seconds (which may sometimes be
counted by means of irregularities in a wheel) will give rerf closely the speed
in miles per hour.

For, let n = the number of revolutions in 6 seconds. Then :

a ji II w 60X60X28Xir« „^^

Speed in miles per hour =* 5 ^ 12 x S280 " ^'^^ ^

BAILBOAD STATIBTtCa.

8tJ7

EAILROAD STATISTICS.

Table 1. IN THE VNITEB STATES.*

Plimt.

Miles built in one year

Miles in mperation ^..

BollinK sioek in operation.
Namberof locomotives

tt

K

passenger cars

baggage, mall, and express cars

freight and other cars

Coot of road and equipment,

per mile, in dollars

total, in millions of dollars

Operation.

For one year.

Passengers carried one mile, per mile of road

Tons of freight carried one mile, per mile of road
C^ross earnings,

per mile of road, from passengers, dollars

" " " freight *'

" " ** maib, Ac " ......

" " total "

per passenger-mile, from passengers, "

" ton-mile, from freight "

passenger earnings -s- total earnings

freight *' -j- *' "

mail, Ac, " -^ " "

gross earnings -i- total investment

Expenses, (For details, see Table 8. )

per mile of road dollars

expenses + gross earnings

Vet earnings.

Net earnings -i- total investment

7174

6498

87801

166817

17412

83241

12330

22958

4475

7253

455450

1061970

51561

53783

4530

8789

65392

75062

368514

474728

1641

1732

4740

4686

230

528

6611

6946

0.0251

0.0218

0.0129

0.0093

0.2483

0.2519

0.7169

0.6817

0.0348

0.0664

0.1186

0.1015

6174

6792

0.6078

0.0854

0.0504

0.0340

1809.

8981
190833

87245

26184

8121

1328084

54607
10254

79182
675748

1597

4952

612

7161

0.0200

0.0078

0.2220

0.6900

0.0880

0.1072

4769
0.6659

0.0359

Table 2. UNITED STATES BT DITISIONS, 1890.*

Plant.

Miles in operation

Cost of road and equipment, per
mile, dollars... ,

•

Operation.
For one year.

Gross earnings per mile, 9

Expenses per mile, $.

Expenses -^ Gross earnings

Eastern
Statm.

Central
States.

Western
States.

Pacific
States.

52735

64818

69562

10666

68911

45648

48746

61696

10131

6932

0.6842

6911

4803

0.6950

4697

3020

0.6430

6948

4575

0.6589

Total,
U.S.

187781
54607

7161

476»

0.6659

* Tables 1 and 2 are baaed chiefly upon Poor's Manual

RAILBOAD STATISTICS.

Tables. IlenBS or total «■!!.- ._

operatioQ o( all the mllroada of the United B»ta. Y«mr ei
From B^wrt of Intentato Gonuuerce C

T0U1»

road.

^'tor

87 40

10 «1

aa 3s

■! J!

1 m

407
67

101

'■1

20

10.72

«^,. ..a „.,. A J jai^aSSO;*;;;

SS

lalesr,[A.„...!r;.

a.ti

l«>,82e,0M

an

SOS

M

Si

i4a,m,Mfi

756

sa as

S) «2

1

IBS

177

'S'S

t:^

Bklances. Swilching chugss, car mileage, &c

2.T4
2.W

2450

36,819,917

194

4297

Each of these

«mhkl4. ^, to
which baa but 1
the neit. Fiie
rlallf Bff^ti tl

aaother

The total annual ezpeiueB on raUr«ad> Ib the UntHiil
Stales ueuallf range between 69 and ISO cents per train mils ; that la, per mils
actually run b; trains. When a road doei a Ter; large baalaesB, and of such a
■ 3 may be heavy, and the oars full {u in ooal-tarrTlng

oad8t,ll

, althougli 01

roada half tl

BAIUtOAD STATISIIOa.

869

Table 4. Oroas ammal eainiliigs per mllie, per _

■Bile, mmd per ton mile, of some or tlie principal tJ. S. rail*
roads In IslNK. From Poor's Hannal,

PennsTlTanU.

New York Central A Hudson BiTor,

Baltimore A Ohio

Chioago. Barlinffton A Qnincy

Philadelphia A Beading

Union Pacific.......

Wabash ,

Atchison, Topcka A Santa F6

Total and averages. United States....

From passengers

FiTom freight

Length,
miles.

Per mile

Per pass
mile.

Permile

Per ton

of road.

of road.

mile.

2780

94690

80.0194

$18877

1K).0047

2395

5730

0.0182

11495

0.0059

2024

2791

0.0174

9909

0.0089

7419

1276

0.0211

3946

0.0086

915

4386

0.0162

19600

0.0078

2421

1865

0.0279

5842

0.0164

2278

1753

0.0189

4044

0.0056

7029

1164

0.0228

4196

0.0102

187781

1684

0.0200

4912

0.0073

Table 5. Annual earnings and expenses of some of tbe
principal railroads In tbe United states In 1899.
Poor's Manual.

PennsWyaaia.

V«w York Central A Hudson BiT«r,

Baltimore A Ohio. .......m....

Chicago, Burlington A Qnincy........

Fhilaaeiphia A Beading.......

Wabash

Atchison. Topeka A Santa T6^

Total and averages, United States....

Gross

LJ^

earnings
permit

of road.

2780

S25787

2896

19286

2024

14160

7419

5942

915

24640

2421

8181

2278

6820

7029

6768

187781

7161

Expenses
permile
of road.

S17670

12163

10858

8662

18422

4713

4671

8927

4760

Expenses

■i- gross

earnings.

0.6904
0.6307
0.7667
0.6118
0.6468
0.5760
0.7288
0.6814
0.6669

Operatlngr expenses. Brio BaMrsad Companjr, IIKMI*

Entire system, comprising Ekie and Ohio Divisions.

Obal used per mile,*
per passenger looomotiTe. lbe.«
▼ork •• *•

switching •• "

pusher •• •*

freight " "

passenger oar,
freight "
100 tons t

00 and wastes

Cylinder oiL

Loco, mileage per qaart .
Iriibiicatinff oil.

Looo. mileage, per qoart...^...
Waste, pounds per 100 miles .....

«

M
M
M
44
41

41
U
41

86.9

93.7

78.4

147.7

162.2

18.8

6.4

20.6

.122.5

.. 49.0
1.0

per per

Cott per mile loco, car

mile, mile.

Fuel cents 7.29 0.59

Bepairs and renewals, ** 7.46 0.67

Oil and waste " 0.38 0.02

Water supply ^ ** 0.42 0.03

Other supplies ** 0.14 0.01

Engineers and firemen ** 6.82 0.52

Boundhouse men ** 1.54 0.12

Total «* 28.99 Tffl

Cost per 100 tons f per mile, 8.88 cents.
Cost of coal, per tonf, arer-

age of anthracite and

bitaminoas 11.25

* 1.6 cords wood taken as eqniyalent to 1 ton (2000 lbs) ooaL
t Tons of 2000 lbs.

870

IBOV AND BT££U

BE4IUIBEMEHTS FOB IBOUT AND

(See alio Bridge Spedfloatioiu.)

Dlffest of Specifleations adopted, subfect to letter ballot, at 4th Annual
Meeting of the American Seetion of the International Asaoelation

for Teatinff JHateriaLi, Jane 29, 1901. Adopted by letter ballot^ 'August,
1901, except wrougbt iron , on which action was deferred.

Process of Mannfkctare.

Wrougbt iron; puddled, charcoal hearth, or rolled from fagots or piles made
from wrought iron scrap, alone or with muok bar added.

Steel castings. Open-hearth, crucible or Bessemer prooess.

Steel forgings. Open-hearth, crucible or Bessemer process.

Steel Rails. Bessemer or open-hearth. Insots shall be kept Teriical lii pit*
heating furnaces. No bled ingots shall beuseoT Sufficient material shall be dis-
carded from the tops of the ingots to insure sound rails.

Steel Splice Bars. Bessemer or open-hearth.

Boiler Plate and Rivet SteeL Open-hearth.

Structural steel for bridees and ships. Open-hearth.

Structural steel for buildings. Opeu-hearth or Bessemer. .

Test Pieces.

For flat plates, the specimen shown in Fig. J shall be used.
For large rounds, test specimen as shown in Fig. K. The center of the speci-
men shall be half way between the center and the outside of the round.
Whenever possible, iron shall be tested in full size, as rolled.
Test specimens shall be cut from bar as rolled.

^^/ ParaUei SeeU(m

t9.70

M*f |0M <Jb«M»-

SI.

-// // //

• I %

//

7

nSAOvim to 79JfQ m

I

\

-4'ji''^107MSntn^

Wfn

tnni

<-^/g~^ii — •-■• ffO^Otntn — >

It

<-Va

n^m

Flflr. K.

IRON AND BTEEi;. 871

Teste.

Nleklnir tests* The specimen shall be slightly and evenly nicked on one
flide, and bent back at this point through an angle of 180*^ by a succession of
light blows.

uot bendlngr tests. Specimens shall be heated to a bright red, and bent
by pressure or by a succession of light blows and without hammering on head.

Cold bending' tests. Specimen to be bent by pressure or by a succession
of light blows.

Tield point. The yield point shall be determined by careful obserration
of the drop of the beam or halt in the gase of the testing machine.

Drop teste. The drop testing macnine for rails shall have a tup of 2000
pounds, the striking face of which shall have a radius of not more than 5 ins ;
and the t^t rail, not more than 6 feet lone, shall be placed head upwards on
solid supports 3 ft apart. The anvil block shall weigh at least 20,000 ens, and the
supports shall be » part of, or firmly secured to, the anvil. Height of drop from
15 n for 45 fi> rail to 19 ft for 85 fi> and over. One test piece shall be selected
from every fifth blow.

Homoireneity teste Ibv fire box steel. A portion of the broken
tensile test specimen is either nicked or grooved A inch deep, in three places
about 2 ins apart and on o);>posite sides. It is then clamped in a vise and broken
off, by light hammer blows, bending away from each groove in succession. The
specimen must not show any single seam or cavity more than 3>^ ineh long.

SfQtes to table, pp. $72 and 878.

!a) To be bent flat,
b) Specimen to be bent about a bar of diameter equal to its cwp diameter or
ckncss.

(e) Specimen to be bent about a bar twice its diameter.

(d) Elongation, min, per cent, in sections less than 0.664 B>p. per linear ft,
grade Aj 19 ; B, 16; €,12/ r -^

le) Nicking test. Max per cent granular surface, gradQ A, 10 : B, 10: C, 16.

(fi Hot bending test. "Bax to be cent without cracking on ouiside of bend.
To be bent flat in each grade; 180^ in grades A and B, and sharply to 9QP
in C. Grade A, heated ydlow and suddenly quenched in water between 90P
and 90*^ F, to bend flat 180°. Also, heated brieht red, split at end, and each part
bent back 180°. Punched and drifted toholeat least 0.9 diam of rod or width of oar.

(ft) Phospliorus, pieces for physical test, 0.06 for each grade.

( n) Bulphuri pieces for phvsi^ test, 0.06 for each graded

(i) Bending. Specimen 1 inch X % inch to bend cold around a diam of 1 inch
without fracture on outside of bent portion.

(j) Bending. ^HBOimen 1 inch X H ^^^ ^ ^°^ <^1^» withoi^t firactnre on
outside of bent portion, around a diameter of 5^ inch.

(k) Same, around diam of 1% ins.

(I) Same, around diam of 1>| ins if not leas than 20 ins diam ; around a diam
pf 1 inch if less than 20 ins diam.

(m) Same, about a diameter of 1 inch.

(n) " " " " H "

(p) Deduct 1 per cent for each % inch in thickness tUbowe % inch, and ^ per

cent for each ^^ inch below t^ inch,
lo lo

(q) Bending. Rivet rounds to be tested of full size, as rolled. PUte speci-
mens shall be 1^ ins wide. For plates not over ^ inch thiek, the thickness
.shall be the same as that of the plate, and the specimen shall, where possible,
have the natural rolled surface on two opposite sides. For plates thicker than
f^ inch, the specimen may be ^ inch thick. Shall be subjected to both cold and
quenched bendine tests. For the auenched test, the material is to be heated to
a light cherry red (as seen in the dark) and quenched in water of temperature
between 80° and 90° Fahr. Samples shall bend flat without fracture on the
outside of bent portion. Bending may be done by pressure or by blows.

(r) For pins, the elongation shidl be 6 per cent less. Center of test specimen
1 inch from surface.

(s) Eye-bars shall be of medium steel. Fullnsized tests shall show 12^ ^
cent elongation in 16 ft of body. Min tensile strength, 66,000 Smb. per sq in.
At least % of eve-bars tested shall break in the body.

(t) Same as (q) but omitting quenching test.

(a) See Homogeneity Test, in text, above.

IRON AND STEEL.

} notes, p. 871.

Reqairemeiits for

Metel.

Wbouqht Iron.

Grade A

Grade B .'.

Grade C

Stbbl Castings.

Hard

Medium

Soft

Steel Forginqs.

Soft or low carbon

Carbon, not annealed.
Carbon, annealed

Carbon, oil tempered.

Nickel, annealed

Nickel, oil tempered..,

Steel Rails.

60 to 69 n>s. per jard.
60 to 69 " " " .
70 to 79 " " " .
80 to 89 " " " .
90 to 100 " " " .

Steel Splice Bars

Open Hearth Boiler

Plate and Rivet Steel.

Flange or Boiler Steel ...

Fire Box Steel

Extra Soft Steel for
Boiler Rivets

Structural Steel for
Bridges and Ships.

Rivet Steel

Soft Steel

Medium Steel

Structural Steel
Buildings.

Rivet Steel

Medium Steel

VOR

Allowable percentage of

Carbon.

Max.

0.40
0.40
0.40

Max. Min,
0.45 0.35

0.48
0.60
0.53
0.55

0.38
0.40
0.48
0.45

Max.
0.15

Phos-
phorus.

Max.

0.08 g
0.08 g
0.08 g

0.10
0.06

ao4

0.04

0.10
0.10
0.10
O.tO
0.10

0.10

acid basic
0.06 0.04
0.04 0.08

0.04 0.04

acid basic
0.08 0.06
0.08 0.06
0.08 0.06

Max.

0.10
0.10

Sul-
phur.

Max.

0.05 h
0.05 h
0.06 h

0.10
0.06
0.04

0.04

0.20
0.20
0.20
0.20
0.20

0.05
0.04

0.04

0.06
a06
0.06

Mangan-
ese.

Max. Min.

1.00
1.00
1.05
1.10
1.10

0.70
0.70
0.76
0.80
0.80

0.60 0.80

0.60 0.30
0.60 0.80

0.60 0.30

NiokeL

Max. Min.

4.0O
4.00

3.00
8.00

IRON AND 8TEBL.

Iron and Steel.

873

See notes, p. 871.

Tensile Testd.

Strength. fi«.persq. in.

Max.

Min.
50,000
48,000
48,000

85,000
70,000
60,000

Average.
68,000
75,000
76,000

85,000
80,000
90,000

Blastic limit

and

Yield Point

fi)s. per sq. in.

Yield Point

Min.

25,000

25,000

25,000

88,250
81,500
^,000

Average.

29,000

87,500

37,500

Elastic Limit

47,600
60,000

Elonga-
tion.
Per-
centage
in 8 ins.

Con-
traction
of Area.

Per-
centage.

Min.
25d
20 d
20 d

15
18
22

ATge.
28
18
23

21.5
24.6
22.5

Min.

20
25
30

Avge.
36
30
32.5

42.6
42.6
47.6

Cold Bending
Tests.

Angle
of Bend.

180
180
180

90
120

180
180
180

180
180
180

How
Bent

a
b
c

1
i
i.

i

m
n
o

See Drop Test, in text, above.

64,000

66,000
62,000

56,000

60,000
62,000
70,000

60,000
70,000

Min.
54,000

66,000
62,000

46,000

60,000
52,000
60,000

50,000
60,000

Min.
82,000

Max. and Min.

I K Tensile
Strength.

1 3^ Tensile f
f Strength. |

) ]^ Tensile f
j Strength. {

Min.
26

180

a

25 p

26 p

28 p

180
180

180

a,q
a,q

a,q

rs.
26 p
25 p
22p

180
180
180

a,t
a, t
b,t

r.
26 p
22 p

180
180

b,t

874 IRON AND STfiBL.

Iron to weakened by extreme cold.

The belief (origioating with Styff of Sweden) is gaining ground that iron and
steel are not rendered more brittle by intense cold^ but that the great number oi
breakages of rails, wheels, axles, ftc, in winter, is owing to the more severe blows
incident to the frozen and unyielding nature of the earth at that period of the year.
But Sandberg's experiments show conclusively tbat although these metals may per-
haps bear as much tt«ady force, graduaUy applied, in winter as in summer, yet their
renstance to imp^dse^ or sudden force^ is not more than ^ or ^ as great in severe
sold ; which renders them less flexible and less stretchy. It is probable that this
&ct does not receive as much attention as it should, in proportioning iron bridges, kc

Some experiments with good wrought iron showed that even at 23P Fi^ or only
0° colder than freezing point, there was a loss of strength of fixtm 2)^ to 4 pex
eent.

Malleable Cast Iron. Experiments by Mr. D! L. Barnes, of Chicago, on
a large number of samples of a single make of "malleable'* cast iron, gave in
most cases tensile strengths ranging from 24000 to 32000 lbs. per square inch,
with an average of about 28000 lbs. The higher figures were obtained generally
with the smallest bars (about 3 X ^ inch) and the lower with the largest tMus
(about 3X1 inch). Pieces planed on all four sides averaged only about 24000 lbs.
per squareHnch. This may explain the difference in favor of the smaller sections,
in which the original '* shell " ferms a larger portion of the whole cross sectioiL

CAST lapir.

Tensile btrength 14,000 to 20,000 lbs ♦ per sq Indi

Compressive strength (average about 100,000)... 90,000 to 180,000 " " "
Transverse strength, bar 1 in sq, 1 ft span,

center load 2600 lbs. Deflection, minimum,

0.15 inch.

Elastic limit about 6,000 Ifaa per sq inch

Modulus of Elasticity " 16,OOP,000 •* " ««

Speei0cat|ons.

Tensile strength.

Bureau of Water, Philadelphia 16,000 to 20,000 lbs per sq Inch

Water Department, St. Louis, Mo 18,000 " " "

Transverse strength.
Bureau of Water, Philadelphia.
1 in sq, 56 ins span, center load 500 9)s.

1 in sq, 86 ins span, " " 750 lbs. Deflection, minimum, 0.4 to 0.6 In.
Water Department, St Louis, Mo.
8 in X 3^ in (laid flat) 18 ins span, center
load 1000 to 1250 9>8. Minimum deflection 0.8 to % inch.

Weiffbt of Cast Iron.

Assnmlns 450 lbs per enb ft, specific gravity 7.2, a cub inch w^gfas

0.2604+ fi>s ; and a pound contains 3.83995+ cub ins.

Table, pag^e 875 : D = thickness or diameter, in inches.

Wt. of plate, 1 ft square, in 9>8 = 37.5 D (Exact) Log W = 1.574 0813 + Log D
" " sq bar, 1 ft loDg, in lbs = 8.125 D^ (Exact) Log W » 0.4JM 8600 + 2LogD
** " rd bar, 1 ft long, in fits == 2.45437 £>8 LogW*e0.889 9400 +2 Log D

" "ball, iuft8= 0.136354D8 Log W = 1.184 6651 + 8 Log D

Weigrbt of a spberieal sbell = weight of ball having outer diam of
shell minus weight of ball having its inner diam.

Weiarbt of pattern. A casting weighs 20 X weight of pattern of per-
fectly dry white pine. If not perfectly dry, although well seasoned, for 20,
substitute 19 or 18.

For lead, at 700 9>s per cub ft. multiply weight of cast iron by 1.555-—;

For eopper, at 550 lbs, multiplv by 1.222 ;

For brass, at 500 lbs, multiplv by 1.111 ;

For wrougfbt iron, at 485 n>s, multiply by 1.0777 ;

For tin, at 460 fi)s, multiply by 1.022 ;

Zinc, at 450 fibs = cast iron.

* High grade irons may reach 80,000 to 40,000 B>s per sq inch, tensile.

WEIGHT OF CAST IRON.

875

TABIiE OF WEIOHT OF CAST IROlf .

At 450 &» per cubic foot ; specific gravity, 7.2.

D = Thiekaess or diameter, in inches. For equivalents in feet, see p 221.

1/32
1/16
3/32

1/8

5/32

8/16

7/32

1/4

9/32

6/16

11/32
3/8

13/32
7/16

16/32
1/2
9/16
6/8

11/16
8/4

18/16
7/8

16/16
U
1/16
1/8
3/16
1/4
5/16
3/8
7/16
1/2
9/16
6/8

11/16
3/4

13/16
7/8

15/16
2.
1/8
1/4
8/8
1/2
5/8
3/4
7/8

Weights, in pounds.

Plate
1 ft sq.

1.172
2.344
3.516
4.688
5.359
7.031
8.203
9.375
10.55
11.72
12.89
14.06
15.23
16.41
17.58
18.75
21.09
23.44
25.78
28.12
30.47
32.81
85.16
37.50
39.84
42.19
44.53
46.88
49.22
51.56
63.91
66.25
58.59
60.94
63.28
65.62
67.97
70.31
72.66
75.00
79.69
84.38
89.06
93.75
98.44
103.1
107.8
112.5

Square

bar
1 ft long

0.0031

0.0122

0.0275

0.0488

0.0763

0.1099

0.1495

0.1953

0.2472

0.3052

0.3693

0.4394

0.5157

0.5982

0.6866

0.7812

0.9888

1.221

1.477

1.758

2.063

2.393

2.747

3.125

3.528

3.955

4.407

4.883

5.3^

5.908

6.458

7.031

7.629

8.252

8.899

9.570

10.27

10.99

11.73

12.50

14.11

15.82

17.63

19.63

21.63

23.63

25.83

28.12

Round

bar
1 ft long

0.0024

0.0096

0.0216

0.0383

0.0599

0.0863

0.1174

0.1534

0.1941

0.2397

0.2900

0.3451

0.4051

0.4698

0.5393

0.6136

0.7766

0.9587

i.160

1.381

1.620

1.879

2.157

2.454

2.771

3.106

3.461

3.835

4.228

4.640

6.072

6.522

5.992

6.481

6.989

7.517

8.063

8.629

9.213

9.818

11.08

12.43

13.84

15.34

16.91

18.56

20.29

22.09

Ball.

0.0001

0.0003

0.0U05

0.0009

0.0014

0.0021

0.U030

0.0042

0.U055

0.0072

0.0091

0.0114

0.0140

0 0170

0.024:i

0.0333

0.0443

0.0675

0.0731

0.0913

0.1124

0.1363

0.1636

0.1941

0.2283

0.2663

0.3083

0.3545

0.4050

0.4602

0.5202

0.5851

0.6652

0.7308

0.8119

0.8988

0.9917

1.091

1.308

1.553

1.827

2.131

2.466

2.836

3.240

3.682

1^

Welg^lits, in pounds.

Plate
Iftsq.

117.2
121.9
126.6
131.2
135.9
140.6
145.3
150.0
154.7
159.4
164.1
168.8
173.4
178.1
182.8
187.5
192.2
196.9
201.6
206.2
210.9
215.6
220.3
225.0
234.4
243.8
263.1
262.5
271.9
281.2
290.6
300.0
309.4
318.8
328.1
337.6
346.9
356.2
365.6
3750
3844
393.8
403.1
412.5
421.9
4^1.2
440.6
450.0

Square

bar
1 ft long

80.62
$3.01
36.60
38.28
41.06
43.94
46.92
50.00
63.17
56.45
59.81
63.28
66.84
70.51
74.27
78.12
82.08
86.13
90.28
94.63
98.88
103.3
107.9
112.6
422.1
132 0
142.4
158.1
164.3
176.8
187.7
200.0
212.7
225.8
239.3
253.1
267.4
282.0
297.1
312.5
328.3
344.6
361.1
378.1
395.5
413.3
431.6
450.0

Hound

bar
1 ft long

23.97
26.92
27.96
30.07
32.26
84.61
36.86
39.27
41.76
44.33
46.98
49.70
52.50
66.38
58.33
61.36
64.47
67.65
70.91
74.24
77.66
81.15
84.71
88.36
95.87
103.7
U1.8
120.3
129.0
138.1
147.4
157.1
167.0
177.3
187.9
198.8
210.0
221.5
233.3
246.4
267.9
270.6
283.6
297.0
310.6
324.6
338.9
353.4

Ball.

4.161
4.681
5.242
6.846
6.496
7.191
7.934
8.727
9.571
10.47
11.42
12.43
13.49
14.61
15.80
17.04
18.36
19.73
21.17
22.69
2427
25.92
27.66
29.46
33.29
37.45
41.94
46.77
51.96
57.62
63.47
69.81
76.57
83.74
91.36
99.40

107.9

116.9

126.4

136.3

146.8

157.9

169.4

181.6

194.1

207.4

221.2

235.6

876 TBIOHT OP CAar-IROIf PIPES.

WEieHT OF CA8T<IBOir PIPES pw niBDlD|r Rmt.

imimlDg Iht weight of cut-lroD it 4MlbgjHr onb It. or .SAOilb iHrcDblDch. St

Ihua AFB DOM cnmmonLj nud«. thej »dd to tha ir«l|ht of

«Kb In^lh or HBtlnn et plixi of ht ill*, (boot u moch u thit of e locheg io
laDitD dT tbe plain pipg h glTan In tbe tibia.

PorlHid-pipvmiilt br1.0:goppgp.nalt brl.t;bra—^adiH^th;

welded Iron. mull b/ l.ueei.oriuld sua ilfHBDtb part.

otMit or wnutbtpfpa aurface Tur »ch 120 cub 11 ol
ft g( boilar (Or web awu cub [t of luch apau.

WEIGHT OF WBOUGHT IRON AHD STEEL.

877

Table of Welgrlit of WROITOHT IROIT And SllBEIi.

At 485 fi)8 per cubic foot ; specific gravity, 7.76. See page 879.

]> »- Thickness or diameter, in inches. For equivalents in feet, see p 221.

1/32

1/16

8/32

1/8

5/32

8/16

7/32

1/4

9/32

6/16

11/32
3/8

13/32
7/16

15/32
1/2
9/16
5/8

11/16
8/4

13/16
7/8

15/16

1.

1/16

1/8

8/16

1/4

5/16

3/8

7/16

1/2

9/16

5/8

11/16
3/4

13/16
7/8

15/16

2.

3.

1/8
1/4
3/8
1/2

5/8
3/4
7/8

Weiffhti

^, in pounds.

Plate
Iftsq.

Square
bar

Bound
bar

Ball

1 ft long

1 ft long

1.263

0.0033

0.0026

2.526

0.0132

0.0103

3.789

0.0296

0.0232

0.0001

5.052

0.0526

0.0413

0.0003

6.315

0.0822

0.0646

0.0006

7.578

0.1184

0.0930

0.0010

8.841

0.1612

0.1266

0.0016

10.10

0.2105

0.1653

0.0023

11.37

0.2664

0.2092

0.0033

12.63

0.3289

0.2583

0.0045

13.89

0.3980

0.3126

0.0060

15.16

0.4736

0.3720

0.0077

16.42

0.5558

0.4366

0.0099

17.68

0.6447

0.6063

0.0123

18.95

0.7400

0.5812

0.0161

20 21

0.8420

0.6613

0.0184

22.73

• 1.066

0.8370

0.0261

25.26

1.316

1.033

0.0359

27.79

1.592

1.250

0.0478

30.31

1.895

1.488

0.0620

32.84

2.223

1.746.

0.0788

35.36

2.579

2.025

0.0985

87.89

2.960

2.325

0.1211

40.42

3.368

2.645

0 1470

42.94

3.802

2.986

0.1763

45.47

4.263

3.348

0.2092

47.99

4.7^

3.730

0.2461

50.52

5.263

4.133

0.2870

53.05

5.802

4 557

0.3323

65.67

6.363

5.001

, 0.3820

58.10

6.960

5.466

' 0.4:^65

60.63

7.578

5.952

0.4960

63.15

8.223

6 458

0.5606

65.68

8.894

6.1)85

0.6306

68.20

9.591

7.533

0.7062

70.73

10 31

8.101

0 7876

73.26

11.06

8.690

0.8750

75.78

11.84

9.300

0.9687

78.31

12.64

9.930

1.069

80.83

13.47

10.58

1.176

85.89

15.21

11.95

1.410

90.94

17.05

13.39

1.674

95.99

19.00

14.92

1.969

101.0

21.05

16.63

2.296

106.1

23.21

18.23

2.668

111.1

25.47

20.00

3.056

116.2

27.84

21.86

3.492

121.3

30.31

23.81

3.968

Weights, in pounds.

Plate
1 ft sq.

126.3
131.4
136.4
141.6
146.5
151.6
166.6
161.7
166.7
171.8
176.8
181.9
186.9
192.0
197.0
202.1
207.1
212.2
217.2
222.3
227.3
232.4
237.5
242.6
252.6
262.7
272.8
282.9
293.0
303.1
313.2
323.3
333.4
343.5
363.6
363.7
373.9
384.0
394.1
404.2
414.3
424.4
434.6
444.6
454.7
464.8
474.9
485.0

Square

oar
1 ft long

32.89
35.57
38.36
41.26
44.26
47.36
50.67
53.89
57.31
60.84
64.47
68.20
72.04
75.99
80.04
84.20
88.46
92.83
97.31
101.9
106.6
111.4
116.3
121.3
131.6
142.3
153.5
165.0
177.0
189.5
202.3
'215.6
229.2
243.3
257.9
272.8
288.2
304.0
320.2
336.8
353.9
371.3
389.2
407.5
426.3
44.5.4
465.0
485.0

Bound

bar
1 ft long

25.83
27.94
80.13
32.40
3476
37.20
39.72
42.32
45.01
47.78
60.63
53.57
56.68
59.68
62.87
66.13
69.48
72.91
76.42
80.02
83.70
87.46
91.30
95.23
103.8
111.8
120.5
129.6
139.0
148.8
158.9
169.3
180.0
191.1
202.5
214.3
226.3
238.7
251.5
264.5
277.
291,
305.
320.
334.8
349.8
365.2
380.9

.9
.6
.7
.1

Ball.

4.485
6.G45
6.660
6.801
7.000
7.750
8.561
9.406
10.82
11.28
12.81
18.39
14.54
15.76
17.08
18.87
19.78
21.27
22.82
24.46
26.16
27.94
29.80
31.74
36.88
40.36
45.20
60.41
56.00
62.00
68.41
75.24
82.52
90.25
98.45

107.1

116.3

126.0

136.2

147.0

168.3

170.1

182.6

195.6 ,

209.2

223.6

288.4

253.9

Welsh* «ri n IM lenarth of FI.AT BOI.I.ED IROS . At 480 llw per
ooMe fbol* Fornat Iron, d«duct J* pwl; far fll««t, add J.; TorcoppAr, add
|l iaruitbnM,Hld^; (Or Iwl, add H 1 for £!>><. d'daot ^.

IBOK AND 8TBSL.

87»

I'

Welarbt of 1 lU in lenfftb of FI«AT BOI<I<ED IRON, at 490 Ibr

per cable foot — (GontiDued.)

Ij

THI0KKB8S TS IKCHBS.

^£

1-16

H

' 316

H

516

H

7-16

H

H

H
35.94

%

1

109i

S.163

4.883

6.486

8.646

10.81

12.97

15.13

17.29

21.62

30.28

34.5S

H

2.188

4.875

6.564

8.750

10.94

13.18

15.31

17.50

81.88

26.26

30.62

35.09

H

2.214

4.427

6.642

8.854

11.07

13.28

15.'50

17.T1

22.14

26.56

31.00

S5.4t

H

t.239

4.479

6.717

8.958

11.20

13.43

15.67

17.92

22.40

26.86

31.34

35.8S

H

a.366

4.531

6.798

9.062

11.33

13.69

15.86

18.12

28.66

27.18

31.72

36.36

11.

3.S91

4.S8S

6.873

9.166

11.46

13.75

16.04

18.33

22.90

27.50

32.08

M.66

S7.Q8

H

2.S18

4.686

6.954

9.871

11.59

13.91

16.28

18.54

83.18

87.62

33.44

H

2.3i4

4.686

7.0S2

9.375

11.72

14.06

16.40

18.75

23.44

28.12

32.80

ST.6»

H

8.870

4.740

7.110

9.479

11.85

14.22

16.59

18.96

28.70

28.44

83.18

8T.tt

H

2.3»6

4.791

7.185

9.582

11.97

14.37

16.76

19.16

23.94

88.74

33.52

S8.88

H

8.422

4.844.

7.206

9.688

12.11

14.53

16.95

19.87

84.38

89.06

33.90

88.tft

H

2.448

4.896

7.344

9.792

12.24

14.68

17.13

19.58

24.48

89.36

34.86

S9.U

H

3.474

4.948

7.482

9.886

12.37

14.84

17.32

19.79

24.74

89.68

34.64

99.68

11.

2.500

5.000

7.500

10.00

12.50

15.00

17.50

ao.oo

36.00

80.00

86.00

40.0t

Weigrl^t of Wrongrlit Iron and Steel.

AmnmlnK 485 lbs. per cab ft,* specific gravity, 7.76; a cubic ineh
weighs 0.28067 BM ; and a pound contains 8.6629 cubic inches.

Table, pagr® ^75 : 1> = thickness or diameter, in inches.

Wt. of plate, 1 ft square, in S>8, = 40.4167 D ; Log W == 1.606 6605 + Log I>
sq bar, 1 ft long, in lbs, = 3.86806 D^ ; Log W = 0.627 3792 + 2 Log J>
rd bar, 1 ft long, In fts, = 2.64527 D2 ; Log W = 0^422 4698 + 2 Log I>

ball, in lbs, « 0.146959 DS; Log W=s 1.167 1966 + 8 Log I>

ti

4(

Weiarlit of a _ f weight of ball having) _ (weight of ball having
flpbencal staell ~~ \ outer diameter of shell j \ inner diameter of shelL

Wel^lits of eqaal maflses.

For lead, at 700 lbs per cub ft ; weight
For copper, " 550 ♦' *• •• '^

For brass, ♦• 500 " " •• '*

For tin, " 460 " •• " ♦•

For sine or .« ^^ „ »
cast Iron,

450

<(

i<

1.44 X
1.13 X
1.03 X
0.948 X

0.928 X

weight

of

wrought

iron

♦Very pure soft wrought Iron weighs firom 488 to 492 !bs per cubic foot ; average
roUed iron about 480. At 480 fi>s, a bar 1 inch square weighs exactly 10 fi>8 per
yard » Z% Smb per foot

880

SH£ET-IRON«

Welclito per Bquare foot of nlTanised sheet iron. Standard
adopted Dj the Amerioan QalTaniaed Iron Aas^n, at Fittsbnrgb, April, 1884.

Urt

*•.

OnnoM

Sqft

He.

Oaoqai

Sqft

Ho.

OnnoM

Bqft

aTotr

38Mlbi.

avoir

nSit.

aTolr

per

periqft

{Mraqft.

per sqft.

2340 lbs.

29

12

2987

24

17

2108

10

38

1086

S8

13

2767

28

10

1886

18

38

048

27

14

2660

22

21

1706

17

43

838

26

16

2380

21

24

1493

16

48

746

25

16

2240

20

28

1280

14

60

697

The iralTaniBliiir ^ simply a thiii film of sine on both sides of tht

•heet, •■ Id what i« known as " tinned plates," or " tin ; " whieh are in reality sheet iron elmllarly
•eaied with tin. Zino, like tin, resists corrosion from ordinary atmospheric inflaenoes, maeh better
than iron ; and henoe the nse of these metals as a protection to the iron. A well galranised roo(
of a good pitch, will saflfer but little from 6 to 6 years' exposure without being painted. It will thaa
take paint readily, and shoald be painted. It is better, however, always to paint tin ones at onoe.

Paint does not adhere well to new sine, and this is the principal

reason why new galvanised roofs are not painted ; bat this may be remedied by first bmshtnc the
sine over with the following : One part of chloride of copper, 1 part nitrate of oopper, 1 psu-t of sal-
ammoniac. Dissolve in 64 parta of water. Then add 1 part of commercial hydroohlorie acid. When
brushed with this solution, the line turns black ; dries within 13 to 24 hours, and may then be painted.
Paint of some mineral oxide of a brown color is generally used; one ooat being applied to both
■ides in the shop ; and the other after being put on the roof. Repainting every S or 4 years will saJBoe
afterward. Ungalvanised iron (called black ibom, for distinction) is also very enduring for roofs, if
veil painted every 1 or 2 years. The chief advantage of galvanized roofing is that it does not require
painting so often as the black. The galvanising adds about ^ of a lb per square foot of snrftoe, or
about H ^ per sq ft of sheet as coated on both sides ; without regard to the thieknese of tho elieeti
Paint for roofs should not have much dryer. See Painting.

The sulphurous fumes fk'om coal are very corrosive of

hthbr oalvanixco cm black mox ; as may be seen in shops, railroad bridges, or engine booses,
roofed with either ; if efficient means are not provided for carrying off the smoke : and the same with
other metals. Thk aoid or oak thibkr is said to destroy the sine of galvanised iron.

Flat iron is usually nailed upon a sheeting of boards; but the strength of oormgated iron
obviates the necessity for this, and enables it to stretch 5 or 8 ft from purlin to purlin, withont inter-
mediate support. The corrugated sheets are riveted together on the roof, by rivets of galvanind
wire about one-eighth inch thick, 800 to a pound, well driven (so as to exclude rain) S or 4 inches
apart, all around the edges. The rivet-holes are first punched by machinery, so as to insnre ooinei-
denoe in the several sheets ; and the rivets are driven by two men, one above, and one beneatti the
roof. For black iron, ungalvanised nails, boiled in linseed oil as a partial preservative fk«m met, are
commonly used ; as also in shingling or slating. Galvanised ones, however, wonld be better in aD
these cases ; or even oopper ones for slating because good slate endures much longer than either
shingles or iron, and therefore it becomes true economy to use durable metals for fastening it. In
none of these cases, however, are the nails fully exposed to the weather.

The sheets of flat iron are put tofpether by overlapping' and

voLD»e THE KDOK8, muoh the same as shown by the fig page 916, head Tin ; the joints whioh run
up and down the roof being the same as at s a, and the horisontal ones as at t (;
except that inasmuch as these are nv,t soldered in the iron sheets, the Joint is made
about ^ to 1 inch wide, instead of ^ inch, the better to provide agmnst leaking.
Cleats are used as in tin, with 2 nails to a cleat. The iron plates are best laid on
sheeting boards ; but in sheds, &c, are sometimes laid directly on rafters, not mors
than about 18 ins apart in the clear ; the plates being allowed to sag a little between

the rafters, so as to form shallow gutters. In such cases it is well to bevel off the tops of the rafters

slightly, as in this fig.

A serious objection to iron as a r€»of coverinfp, is its rapid con*

densation of atmospheric moisture; which falls from the iron in drops like rain, and may do injuiy
to ceilings, floors, or articles in the apartments immediately beneath the roof. Painting does net
appreciably diminish this ; it may, however, be obviated by plastering.

CorrugTAted sheet iron. The size .of sheets generally used for corrugating,
is so inches wide by 96 inches long. Corrugation reduces the width to 27^ inches. When tiie cor*
rugated sheets are laid upon the roof, the overlapping of about 2H inches along the sides, and of 4
Inches along their ends, diminishes the area of roof covered by a sheet, to about seven-eighths of that
of the entire corrugated sheet itself; or, the weight per square foot of roof covered, will be about
one-seventh greater than that per square foot of the corrugated sheet ; or, the weight of oormgated
iron per square foot of roof covered is about one-fifth greater than that of the flat sheets fhmi which
it is made.

About 6 inches are usually allowed for the extension over the eaves.

The weights per square foot corresponding to the diflbrent numbers of the Birmingham wire gaaga,
vary somewhat with the different makers. The two styles of corrugation given in the table Mow,
ix'l}i and 2yi X Ht are those most frequently used.

COBRUGATED SHEET IROK.

881

No.
Bins^m
wire ga.

Thick-
ness

in ins.

Wt in lbs per
sq ft of sheets.

Ift in fta per
sq ft of roof.

Black

20
22
24
26

BUck

.085
.028
.022
.018

Black

1.84
1.50
1.20
1.00

Lead oot'd
or galv'd

2.
1.6
1.25
1.12

Black

2.12
1.73
1.38
1.15

Leadcot'd
or galv'd

2.3
1.84
1.44
1.29

mmm'miw//dm

3.9
X 27ilL« ^

Streniptb of Corrugated Iron. Experiments by the antbor.

First. A. sbeet d <f, of ITo. 16 Iron,

(about -^ inch thick,) 27 ins wide, by 4 ft long,
with five complete corragations of 5 ins by 1 inch,
waa laid on supports 3 ft 9 ins apart. A block of
wood c, 9 ins wide, by 7 ins thick, and 30 ins long,
was placed across the center, and gradually load^
with castings weighing 1600 fba.

Thia eaaaed a d«fleotion st tbe oenter of prvcisely H an
iaeh. Od the remoTal of tbe load after an hour, no perma-
Mnt set waa appreciable. The seTeritj of tbe teat waa par-
poaely increaaed bj applying the aereral eaatlngs very
waghlv, joltiac the whole aa aiMh aa poaaibU.* The ana-
pended area of the aheet waa 8.44 aq ft ; and sinoe the actual eeiiler load of lOOO Iba fa aI>ont eqalf a-

3000
lent to 9000 Iha eg«a% dUtribtOed, it amounta to . .=355 Iba per aq ft diatributed. But SOOO lbs

0.44-

distributed would prodnoe a deflection of but about fUll ^ of an ineb. Again, 855 lbs perleq ft
la about 4 times tlie weisht of the greateat crowd that oould well congregate upon a floor. Gonae*
qvently thia iron, at 8' 9'' apan, is safe in praotioe for any ordiaaiy crowd. Uoreever, anoh a crowd

would prodooe a oenter deflection of only the ^th part of ){ of an inch ; or -Jw- of an inch; or y4^
«f tbe clear span ; which is bat two*thirds of Tredgold'a limit of -t^ of the apaa.

In one experiment the ends «f the sheets rested upon supports dressed so as to present undulations
eerrespondlng tolerabW closely with the shape of the corrugation* | but in the other the supports
were flat, and each end of the aheet reated only upon the lower points of tlie oormcations. "So ap-
preolabto diffierenoe waa obserred in the results.

Seeond. An areb of No. 18 (^
Inch) Iron, corrugated like the foregoing,
bnt the depth of corrugation increased to
IV^ins by the processor arching the sheet;
^esr ensan 6 ft 1 inch ; rise 10 ins ; breadth 27
Ins, (of which, however, only 25 ins boie
i^ainst the abutments.)

Each fbot o of the arch abutted upon a easting /,
tbe Inner portion t of which waa aadalated on top. to
eorrespona with the corrugations of the arch, which
rested upon it. At y, (one-fourth of the span.) two
weeden blocks were placed, oooupylng a width of 9
laobes, and extending Aoross the areb ; on them was
pHed A load, {, of castings, to tbe extent of 4480 lbs,
V t tons. Under this load the arch descended about
Half an inch at y, becoming flatter on that side and

tflghtly more cunred upward along the unloaded sMe n. Two stiQllar blocks w«re then placed at n,
•ad two tons of load, «, were pil^d upon them, in addition to the i tons at I; making a total of 8900
lbs, or 4 tons. This brought the arch more nearly back to its original shape; but still slightly
■ttughtened at both n and y, and a little more curred in the center. The load was then increased to
10000 lbs, and left standing fbr several days. Two iron ties, each ^ by IHt which were used for pre*
Viating the abutment castings / from spreading, were found to have stretched nearly }i of an iaeh.
Additional ones were inserted, and the load increased to a total of 6 tons, or 18440 lbs : parts of It on
9 and I, and part in the shape of long broad bars of iron at tbe center of the arob, below the loads a
and I, and between n and y. Bo far aa could be Judged by eye, the shape of the arch was now almost
perlbct. T%e lemdt a and 1 did not touch each other. After standing more than a week, the load
was accidentally overturned, crippling the arch. The load waa equal to about 1000 lbs per sq ft of

the arch. Such arches have since come into common use instead of brick, for
fireproof 0oors,

CnrTed roofli of 25 to 30 ft span, rising about 14 Bpan. may be made
of ordinary corrugated iron of Nos 16 to 13, riveted as usual ; and having no acces-
•ories except tie-rods a few feet apart ; continuous angle-iron skewbaeks ; and thin
▼•rtlcal rods to prevent the ties from sagging.

• Without letting the deflection exceed H inch ; which was prevented by a stop nnder the

66

S

IHON PIPES, TUBfS AND FrTTINGS.

Welded wronsbt-lroa plp«>

InnOT DUm.

i

il

1°

i

Inner Diw..

^1

Is

^1

1

1

1
1

i

: as

^

lD>.

¥

I

era

0S7
0117

o!osB

0.M1
ft}"
U'.140

ji.a

•

0.M8
0,259

1

ss.n

I

l.M

1.3»

i.a

z»

2^
3.M
«.]>

4.7S

. rrninga n>r Wroactat-lpoa Plpea.

bolter

ti.be.. In

1

ngttasu

ftoVtX

li

'Si'-

?

li

Thlek.

Nom

8?
^1

li

Tbick-

Norn

If

1

Ins.

a.vf

B

WO
IS

IbB.

lise

t.01

O.Ei
ft84

1

6^

B.IM

WG
9

1

51

1

■

1

i

i.'w

■ plpa glra tba '
' diiaaanU, «m |

BOLTS, NUTS, WASHEKS.

883

Screw Tlipeads, Bolto, Nnts, and Washers.

fiferew threads, a = angle between two sides of a
thread ; P =s pitch ; w — width of tiat top or bottom of
each thread; all measared in a plane containing the
axis of the screw ; N = number of threads per inch, =
1/V. In the Sellers or FrankUn Institnte
Standard, proposed by Mr. William Sellers and
adopted by the Institute in 1864, a = 60° ; 8=P ; u? = c
= P/8 ; F = 0.76 P ; M = P cos a/2 = 0.8660 P ; D (diam-
eter) = d + 2 X 0,866 X 0.75 P = <!+ 1.299 P. Under the
name of Tnlted States Standard, the U. S. Navy
Department in 1868 adopted the Sellers system, except
for finished heads and nuts, which it made the same as
for rough heads and nuts.

d

ins

Ti86
.240
.294
.844
.400
.454
.607
.620
.731

ins

^2
.0074
.0078
.0089
.0096
.0104

.oiia

.0126
.0138

N

20
18
16
14
13
12
11
10
9

ins

d

ins

.837
.940
1.066
1.160
1.284
1.389
1.491
1.616

ins

7oi66
.0178
.0178
.0208
.0208
.0227
.0250
.0250

8~

7

7

6

6

6H

6

6

ins

d

ins

1.712
1.962
2.176
2.426
2.629
2.879
3.100
8.317

ins

.0277
.0277
.0312
.0312
.0357
.0357
.0884
.0413

jr

4

3

ins

d

ins

8.567
3.798
4.028
4.256
4.480
4.730
4.953
5.203
5.423

ins

.0418
.0485
.0454
.0476
.0500
.0500
.0526
.0526
.0555

N

Dimensions of Heads and ITnts.

Finished.

X

H
H

H (\n head)
(in nui)

Rongrh.

^;r^.

l^D + 1-16 inch.
D — 1-16 inch.

Figs. 2

In the Whitworth (English) standard thread, the angle a, Fig 1, is 55P.
The tops and bottoms of the threads are rounded, instead of flat as in tne Ameri-
can standards. The number (N) of threads per inch is the same as above for
diams of bolt up to three ins, except for D = >^ inch ; where N ^ 12.

In the International metric screw thread, adopted at Zurich,
October, 1898, the Sellers thread profile is used. The dimensions are as follows,
all in millimeters :

Diam.

6

78

9

10

11

12

14

16

18

20

2224

27

30 83

36

39

42

46

4852

66

60

64

68

72

76

80

Pitch

1.0)1.26 1.6

1.751 2.0

2.5

3.0

3.6

4.0

45

6.0

5.5

6.0

6.6

7.0

Intermediate diameters are to be of an integral number of millimeters, and
of the same pitch as the .next smaller diameter in the table. Thus, for diam 66
or 69 mm ; pitch = 6.0 mm.

Plate-iron washers. Standard sizes. Diameters of washers and bolt-
holes in inches. Approximate thickness by Birmingham wire gauge. Approxi-
mate number in oue fi>.

Diams.

5-16
5-16

7-r«

Ths.

18
16
16
16
14

No.

450
210
139
112
68

Diams.

9-16
13-16

Ths.

14
12
12
10
10

No.

43

26

22.6

13.1

10.1

Diams.

15-16
11-16

in

Ths.

9
9
9
9
9

No.

8.6

6.2

5.2

4.

2.8

884

BOLTB, NUTS, WASHEB8.

TlQ

A tquart boa4 «nd oat logetiwr, w«igh abovt ms mook M a losglh of the 1h>U «qml to 7 «r 8 f

D. Baaagon, 6 or 7.

With tiw above dimensioDB % bolt will tM«rmllj fUI bj brMkinfl
off betwMD th« hemd and the not, where the diameter it deereanf
by outting the thread, rather than by utripping off its thradi.

Tbe diani I> of tlie thread must of course be greater

than that required |o bear aafely the proposed tenaile strain, by an amoaiii
equal to Iwies the depth of the thread. The waste of iron, whioh would
result from making tbe mnUr* hoU of this greater diam, la freqnantiy
avoided by making the Iwlt firom a bar of only sufficient dimeoalons to bear
tbe strain safely, and npsettilii^ ItJi ends as in Fig 3,
thus Inoreasing their diam snfDcientiy to allow for the ontting of th*
threads.

In carpentry, as well as in ties for masonry, vosJkers, mt », of eltkcr aas»
or wrought iron, are placed between the timber, or stone, and tlw iMsd
and nut; in order to distribute tbe pressure over a greater earfaoe, and
thus prevent ornshlne ; especially in timber.

When mncn strained agralnst wood, the side

of a square wrought-iron washer; or the diam ww of a oiroular one, should not be less than 4 diains
of the screw, as in the flg ; and its thickness, (10, H diam ot UcM.

I'wo such square washers will together weigh as much as 18 diama in
length of a round rod of the same diam as the screw. Two round
washers will weigh together as much as 14 diams of rod of same diam
as sorew. In either case, a square head and nut will weigh as much
as 6 diameters. Cast-iron washers, being more apt to split under
heavy strains, may be made about twioe as thick as wrooght ones.
When the strain is very great, the diam of the washer may be 6 or
4 times that of tbe screw ; and its thickness equal to diam ; but 4
diams will suffice for most practical purpoaes, or even 2.5 when there
is bat little strain, and tbe thickness may then be but .1 or .2 diam of
bolt.

Table of machine and car bolts, with

•quare and hexagon heads and nuts. Figs 4 and 5 ; made by Hoopes
A Townsend, 1330 Buttonwood St, Phila. AH tbeir bolts
are cut with U. S. Standard threads, as

Fifir.4. Fi|f.5.

per first table on p 883, unless otherwise ordered. Discounts, see price llBt.

Length, ins
exclusive of head.

Weight, fl)8 of
100 bolts.

List price, $ per loa

. 1

Min.

Max.

Min.

Max.

Min.

MaT,

1

2

u

8
<<

12

((

20

t(

24
«t

(1

3.9
6.2
9.7

14.7

20.4

26

87

68

97.7
145.0

13.2

20.3

43.5

68.3

122.0

151.0

224.0

880.0

470.0

625.0

1.70
2.00
2.40
2.80
8.60

6.20

««

7.20
11.20
16.00

2.74

8.56

6.76

7.00

13.22

19.26

22.30

29.70

42.00

fi6.60

ExiNftnslon bolts, for fastening plates, timbers,
etc., to walls of brick or masonrj. Toe wedge-shaped
nut, traveling up the bolt, as the latter is turned,
presses the wings against tbe sides of the hole, which,
in practice, is drilled just large enough to admit the
nut and wings, so as to prevent the former fW>m turn-
ing with the bolt. If the hole is made larger, aa
shown, the nut must be held by a small wedge.

BOLTS, NUTS, WASHEBS. 885

■

Ijoelc-iiiit WMdieiv. When bolts are snbjeeted to much
rough jolting, as at rail-joiDts, &c, the nuts are licu>le to wear loose^
and unscrew themselves. On railroads this is a souree of great
annoyance, and innumerable deyices for preventing it have neen
tried. The Terona lock-nut washer * is a simple circular washer
made of steel ; with a slit s s cut through it, tearing sharp edges.
On one side, a, of the slit, the metal is pressed upward about ^
inch ; and that on the other side, e, downward, the same distance ;
JBO that a perspectiTO view would be somewhat as at t. Now, when
the nut is screwed down over the washer, in the direction of the
arrow, the slit offers no obstruction ; but if the nut afterward
tends to unscrew itself the sharp upper edge of the slit, along a, presents ft-iction
against the bottom of the nut, which tends to hold it in place. Besides,* the
washer, by its elasticity, tends to resume its original shape, and thus presses tho
threads of the nut against those of the bolt ; and the additional friction thus
produced also aids in holding the nut.

Another lod^-nut washer consists of a long strip of steel, with tvfo holes, each
of which has its edges formed like those of a Verona washer, and through each
of which passes one of the bolts of the rail-joint.

Another device is to cut, at the end of the screw, a few threads of a screw of
less diameter than the main one, and in the opposite direction. The nut is then
•crewed upon the larger diameter: and after it the lock-nut is screwed in the
other direction upon the smaller diam, until it comes into contact with the main
nut. In the SnutH lock-nut bolt, this second nut is only about % inch thick ;
and after being driven home, one of its corners is bent over the edge of the
main nut.

The Atwood lock-nuts take advantage of elasticity in the nut itself, which
is obtained either by slitting the nut, or by reducing its thickness near the
bolt hole.

It Is claimed that if the threads of an ordinary bolt and nut are carefully cut,
80 as to be in contact with each other throughout, no look-nut contrivance is
necessary, because the friction between the two tnreads is distributed over a
larger surface, and abrasion does not take place so readily as if the threads
touched each other at only a few points. The nuts are therefore less apt to wear
loose under repeated jarring.

Owing to the difficulty of obtaining such perfect fitting bolts and nuts, due to
the wear of the cutting tools used in Uieir manufacture, bolts and nuts have been
made in which the thread on the bolt difl^ slightly in shape from that in the
nut. l^ey also furnish nuts in which the thread, instead of being of uniform
shape throughout, gradually becomes deeper and thicker, by having its side angle
made moi« acute, and its top truncated. These nuts are used with bolts having
the usual uniform thread. The bolt enters the nut upon the side where the
thread is of the same shape as its own ; but its thread encounters, and is forced
into, the gradually narrowing and deepening path between the threads of the
nut. In £K>th devices, the enforced conformity between the two threads is relied
npon to give the desired completeness of contact between them. The greater force
required In screwing on the nut also increases the friction between the threads.

BUCKIiEB PI.ATES.

Buckled plates are usually of steel, 3^ to ^ in thick and 8 to 4 ft sq ; some-
times in long plates having several buckles each. Buckle 2 to 8 ins. Flat rim
or fillet, 2 to 4 ins. They are used for the floors of buildings and of highway
bridges.

Total permissible load, lbs. on a single square buckled plate of any size and
thickness.f Load » 4 A; / A; where k = permissible unit stress in metal, &s per
sq in, say 6000 ; t = thickness of metal, ins, and h » depth of buckle, ins.

BucklAd plates are stronger, and require less concrete, etc, for filling, when laid
with convex side down. They weigh but little more than flat plates, or about 10
lbs per sq ft per ^ in of thickness.

* Invented by Mr. Thomas Shaw, M. E., of Philadelphia.

t*' Steel in Construction," by Ptocoyd Iron Works, Philadelphia, 1900, p 147.

8»b WEIGHT OF HETALS.

WEIOBT AVD STBENOTH OF IBON BOI.TS. (Orlgitikl.)
Slameten, weights, and approilmmte braking itniiiB, Rir round bolts)

hreaklng strnlii per squBrs inch uiumed a» (ollonii: VputI IticL square, otl

A longupf<«t rod Is naBtronger tbnn one not upset, ugHlDHtjtw^l/anpftHfloidfl
orstrmlnL Butb wlil then brrak at about midlength^ under equal pulls. Insucli

Square bmru. Strength ar wt ^ 1.2T3 X atrengtb w weight of nwnd bu.

t^wpmrimn.

{^vf^

_r

~1:M Xwei

gj"?

■imlU

iroobai.

Kia>«i]a(sed.«BpHt.

"^

^^2

Wellb.' Br«t-

r

Bl™.

WMiW

«'r°.1n!

;

ot

P^IM

""■ 1 •■"'■■

"■

\

JUB

.MS

FU.

1439

.85

.821

III.

B.IO
8.6»

Nl

3.22

14,1

jn*

siai

a"

2.45

16.1

.897

u

i4aii2

^

A.U

m|

:73

2

Itifl

ii'i

fi

£14
23>

■

11

\0A

IMSi

l§

2fl3

2
2

B

iS

s

3:1a

38.t

8ia

?

3*

aJI

127.2

2B4(r2g

s&a

A

SM

lis.

Im 'lis

37BM

1*36

liil

i

W2

18t"?

43W«

HO

S.28

42!3

«8464

67i

44

4.W

ets

d

41

383.6

euoM

wu

2S,a

sell

T08B08

8S.1

H

S.OS

il

fi«

M4

S1536

IM

988

1

are

3WJ

7B833

LSS

10&

8S2ae

05.2

421.1

12&

WIRE OAUOE8.

887

The Birmingham wire naM is
new British w g went into effect March

the one In most general use for Iroa. The
1st 1884. In the " American ** w g of Dar-
ling, Brown & Sharpe, Providenoe It. I., each diam or Uiick is » the next smaller
one X 1.122932. We take the wt of wrot iron per cub ft at 485 lbs in the first two ;
and at 486 in the last. For the wt of steel, mult that of iron by l.Ol. For
lead, mult iron by 1.46. For ■!■«, mnlt iron by .9. For bram (approz), mult
iion by 1.06. For eopper , mult iron bgr 1.18C

BtrmiiiflrJiaiii W. Oa.

lfewBrltlflliW.Ga.|

American W. Qa.

IMamof

«■*. A

Wtof

Diam of

Wtof

Diam of

Wtof

He.

wire, or
thleknoH

Wtof
Iron wire.

iron
ikeets,

wlM, or
thloknes*

"Wtof
Iron wire.

iron
sbeeU,

wire, or
thiclcness

Wtof
iron wire,

iron
sheets,

ofiheot,
ina.

IB lbs per
UnfU

inlboper
■qft.

of sheet,
Ins.

in 08 per
Un ftT

in lbs per
sqft.

of sheet,
ins.

ln!bsper
Un ft.

in lbs
per sqft

7-0

•••••• ••••••

H.....««».J

•••<•• ••»— »

.600

.661

20.21

6-0

••••••••••••

•••••• ••••••

.464

.569

18.76

6-0

• ••••• ••••••

„„„,„,„

.482

.494

17.46

4-0

.464

.646

18JI6

w400

.423

16.17

.460000

.661

18.68

S-0

•426

.479

17 J8

.873

.366

15.03

.409642

.446

16.68

S-0

J80

JI88

16.86

.848

.820

14.06

.364796

.363

14.77

0

.840

.906

18.74

.824

.278

13.09

.824861

.280

13.16

1

JOO

JB»

12.18

.800

.288

1243

.289297

.222

11.70

s

.284

.214

11.48

.276

.202

11.16

.257627

.176

10.43

8

.269

J78

10.47

.262

468

1049

.229423

.139

9.291

4

.288

.160

9.619

.282

.143

9.377

.204307

411

8.273

6

.220

428

8.892

.212

.119

8.668

481940

.0877

7.866

6

^

409

8.206

.192

.0976

7.760

.162023

.0686

6.661

7

j0869

7.276

.176

.0820

7418

.144285

.0552

6.842

8

a66

J0721

6.669

.160

.0677

6.466

.128490

.0438

5.203

9

.148

.0680

6.981

.144

J0648

6.820

.114423

.0847

4.633

10

.134

.0476

6^16

.128

.0434

6.173

.101897

.0276

4425

11

J20

.0382

4.850

ai6

.0367

4.688

.090742

.0218

3.674

12

.109

.0316

4.406

.104

.0286

4.208

.080808

.0178

3.272

18

.096

.0289

8.840

.002

.0224

8.718

.071962

.0187

2.914

14

.083

.0188

8.356

X)60

.0169

8.288

.064084

.0109

2.595

16

J072

.0107

2.910

.072

X>1S7

2.910

.067068

.00863

2.310

16

.066

.0112

2.627

.064

.0106

2.587

.050821

.00684

2.053

17

.058

.00891

2.344

.066

i)0632

2.263

/>46257

.00648

1.882

18

.040

.00636

1.980

.048

.00610

1.940

.040903

.00430

1.681

19

.042

.00467

1.697

.040

.00423

1.617

.085890

.00341

1.452

20

.086

.00826

1.416

.066

.00344

1.456

.081961

.00271

1.293

21

.082

.00271

1.293

.062

.00269

1.293

.028462

.00215

1.152

22

.028

.00206

1.132

.028

i)0207

1432

.026346

.00170

1.026

23

.026

.00166

1.010

.024

.00152

.0700

.022572

.00185

.913

24

.022

.00128

.8892

.022

.00128

.8801

.020101

.00107

.814

25

J02Q

.00106

.8088

.020

.00106

.8088

.017900

.000849

.724

26

J018

J000859

.7226

.018

.000667

.7276

.015941

.000673

.644

27

i)16

.000678

.6467

X>164

.000712

.6628

.014195

.000584

.574

28

J014

.000619

.6668

.0148

.000579

.5982

.012641

.000423

.611

29

j018

.000448

.6254

U>136

.000489

.6497

.011257

.000886

.455

80

J012

.000382

.4850

.0124

.000408

.6012

.010025

.000266

.405

81

JOIO

.000266

.4042

.0116

.000857

.4688

.008928

.000211

.360

S3

JOOO

.000216

.8688

J0106

.000309

.4866

.007950

.000167

.821

83

J0O8

XXX)170

.3288

.0100

.000266

.4042

.007080

.000188

.286

84

MI

/)00180

.2829

.0093

.000224

.3718

.006805

.000106

.254

86

J006

U)000662

.2021

.0084

.000187

.8395

.006615

.0000887

.226

86

.004

/X)00424

.1617

.0076

.000153

.8072

.005000

X)000662

.202

87

AAA^^Aa ^Aa a

•

.0068
.0060
.0052
.0048
/)044
.0040
.0086

.000122

.0000952

.0000714

.0000608

.0000618

.0000428

.0000344

.2748
.2426
.2102
.1940
.1778
4617
.1466

.004453
.008965
.003631
.00314^

.0000626
.0000417
.0000380
XI000262

480

88

■ #oew* ws •

A ■ ^^k^ A A A ^ A A A

469

80

• ■ OWV ••••••

0 O ■ wmm V V V O • •

;;;;;;;;;;;;

442

40

AAA ^^^ A A B A ^ a

&^^^^^ A^^^^A

.137

41

■■••W •••••■

0OA### v#W*0

42

43

......M....

•••••••••••«

•••••• eeeoeo

44

■AAAMA AMMBMM

X082
.0028
.0024

iNno

X016

.0000271
.0000207
.0000152
.0000106
.0000068

4298
4182
.0970
.0608

46

W^^F^r^^ WW WW^W

46

••0*0 #••#•••

47

•*«*. ••••oee

48

ooo •■••oa^Stf
•••••• oooo**

•••••« o«««««

888

WIRE QAUOEB.

Amerieau icanfr® for sheet and plate iron and steel (19MI). We omit
the columns of weight in kilograms per square foot and in poands per square
meter, and simplify the headings of the remaining columns.
An Act estabnshing a standani gauge for sheet and plate iron and steeL
£e it enacted by the Senate and Mouse (^ RepreaeiUatives of the VnUed Slate* (^
America in Congress assembled. That for the purpose of securing uniformity tki
following is established as the only standara gauge for sheet and plate iron and
steel in the United States of America, namely :

Approximate thickn«u.

Weight.

No.

Inohes.

MilUmeten.

Per «q. foot.

in AToIrdnpoi*

Periq.
meUr, in

oonces.

pounds.

kllosr«infc

7-0

1-2 =.5

12.7

120

20.00

97.65

6-0

15-32 =.46875

11.90625

18.75

91JX

5-0

7-16 =.4375

11.1125

380

17.60

as.44

4-0

13-32 =.40625

10.31875

$60

16.25

79.38

»^

3-8 =.875

9.525

240

15.

78.24

2-0

11-32 =.34875

8.73125

)20

13.75

67.18

6i.oe

0

5-16 =.ai2S

7.9376

12.50

1

9-32 =.28125

7.14376

|80

11.25

64.98

2

17-64 =.285625

6.746875

170

10.625

61.88

8

1-4 =.26

6.35

160

10.

48.82

4

15-64 =.234875

5.958125

150

9.375

46.77

5

7-82 =JJ1875

5.55625

140

8.75

42.72

6

13-64 =.203125

5.159875

130

8.125

89.67

7

3-16 =.1875

4.7626

120

7.6

86.6e

8

11-64 -=.171875

4.365625

110

6.875

^.67

S

5-82 =.15625

3.96875

100

6.26

^.52

10

9-64 =.140625

3.571875

90

5.635

t7.4B

11

1-8 =.125

3.176

80

6.

:t4.4a

12

7-64 =.109875

2.778125

70

4.375

ti.ae

13

3-32 =.09375

2.38125

60

3.76

18.81

14

5-64 =.078125

1.984375

50

3.125

15.26

15

9-128 =.0703125

1.7859375

45

2.8125

18.78

16

1-16 =.0625

1.5876

40

2.5

12.21

17

9-160 =.06625

1.42875

36

2.25

10.99

18

1-20 =.65

1.27

32

2.

9.765

19

7-160 =.04375

1.11125

28

1.75

8.544

20

$-80 =.0375

.9525

24

1.50

7.324

21

11-320 =.084375

.878125

22

1.875

6.713

22

1-32 =.03125

.798750

20

1.26

6.103

28

9-320 =.028125

.714875

18

1.126

6.498

24

1-40 =.025

.685

16

1.

4.882

25

7-320 =.021875

.555626

14

.875

4.272

26

3-160 =.01875

.47625

12

.75

8.662

27

11-640 =.0171876

.4365625

11

.6876

8.367

28

1-64 =.015625

.396875

10

.625

8.062

29

9-640 =.0140625

.8571875

9

.5685

2.746

SO

1-80 =.0125

.8175

8

.5

2.441

31

7-640 =.0109375

.2778125

7

.4875

2. 186

82

13-1280=.01015625

.25796875

H

.40625

1.989

88

3-320 =.009375

.2:^125

6

.876

1.8)1

84

11-1280 =.00859375

.21828125

H

.34375

1.67»

85

5-640 =.007812$

.1984975

or

.8126

1.526

86

9-1280=.00708l25

.17859875

n

.28126

1.373

87

17-2660=.006640625

.168671875

.265625

1.297

38

1-160 =.00626

.15875

?

.25

1.221

And ou and after July first, eighteen hundred and ninety-three, the same and
no other shall ift used in determining duties and taxes levied by the United
States of America on sheet and plate iron aad steel. But this act shall not be
construed to increase duties upon any articles which may be imported.

Sec. 2. That the Secretary of the Treasury is authorized and required to pv»*
pare suitable standards in accordance herewith.

Sec. 8. That in the practical use and application of the standard gauge herebv
established a variation of two and one-half per oent. either way may be alloweoL

Approved March 3, 1898.

GIRGUIiAB HEAaUBE.

889

CIRCUIiAR MEASVBE.

Uaed in oomparing cross sections of wires, etc.

A circular unit is the area of a circle whose diameter is one linear unit.
Thus, a circular inch is the area (=0.7854 square inch) of a circle whoee
diameter is one inch.

The following table is adapted, by permission, from Mr. Carl Hering's valu-
able Tables of Equivalents of Units of Measurement, New York, 1888. Inas-
much as we take 1 meter —- 39.37 inches, instead of 39.37079 inches, our yalues
differ slightly from his.

Logarithm.

1 0 mil ♦ = 0.78640 Q mil* - 1.895 0899

» 0.00064516 O miUimeter ~4.809 6692

= 0.00050671 D millimeter 1.704 7691

1 D mil * = 1.2732 O mils* j0.104 9101

^ a00082145 O millimeter... 1.914 5798

1 0 Millimeter ^ iseao Qmils* 8.190 8896

= 1217.4 D mils* 3.085 4807

• a* 0.78M0 D millimeter ~ ^.1.896 0899

1 D millimeter = 1973.5 Qmils* 3.296 2409

= 1^32 O millimeters 0.104 9101

SI^IftON STAJTBABD WIB£ OAVOK.

Adopted by the Associated £dlson Illuminating Companies.

In this table the gauge number is approximately equal to
Y^ X area of cross section in cLrcnUtr mils*
= T^ X square of diameter in mils. *

No

Diameter,

No.

Diameter,

No.

Diameter,

in mile.

in mile.

in mils.

3

54.78

65

254.96

160

400.00

5

70.72

70

264.58

170

412.32

8

89.45

75

273.87

180

434.27

1?

109.55

80

282.85

190

486.89

15

122.48

85

291.65

200

447.22

20

141.43

90

800.00

220

469.05

23

168.12

96

308.23

240

489.90

80

173.21

100

316.28

260

509.91

85

187.09

110

331.67

280

529.16

40

200.00

120

346.42

300

547.73

45

212.14

130

360.56

820

565.69

50

223.61

140

374.17

840

583.10

65

234.53

150

387.30

360

600.00

00

244.95

• 1 mU » T^^ inch.

890

WIBB OAUOBB.

Ho trade Stapidlty is more thoroughly senseless than the sdherenoe ti
the Tarious Birmingham, Lancashire, Ac, gauges; instead of at once denoting flie
thickness and diameter of sheets, wire, Ac, by the parts of an inch ; as has 1od|
been suggested. Thus, No. ^, or No. ^ vrirO) or sheet-metal of any kind, shoold
be understood to mean 14^^ vj ^^ ^u ^^<^^ diam, or thickness. To avoid mistakes,
which are very apt to occur from the number of gauges in use; and from the absurd
practice of applying the same No. to different thicknesses of different metals, in dif<
fisrent towns, it is best to ignore them all ; and in giving orders, to define the diam<
eter of wire, and the thickness of sheet-metal, by parts of an inch. Or the weight

er hundred ft for wire ; or per sq ft for sheets, may be employed. We believe that
e foregoing Birmingham gauge applies to sine, copper, brass, and lead; althou^
it is generally stated to be for iron and steel only. Another Birmingham gauge it
used for sheet -brass, gold, silver, and some other metals; but we have never seen it
stated what those others are. There are different gauges even for wire to be used
for different purposes; and various firms have gauges of their own ; not OTen aocoid'
ing among themselves.

As Mr. Stubs makes various Bnglish gauges, the term ''StalMi flTAHS^" ^
iUeJf means nothing. Generally, however, in our machine shops, it applies to the
Birmingham gauge of the preceding table.

BlnDlnirtaain gauge for sheet BnuiB, SilTer* Ckild, and all m«ta]|

except iron and steel ?

Ifo.

Thkkii's.

No.

Tttlskn's.

No.

Thiokn's.

Vo.

Thkkn't.

No.

ThiekD*M.

No.

ThlohDla

•

Ineh

Inoh

Ineh

iBOb

Inch

Insfc

1

.004

7

.015

13

.086

19

.064

25

.095

81

J3S

2

.005

8

.016

14

.041

20

.067

26

.103

82

.143

8

.008

9

.019

15

.047

21

.072

27

ai3

83

.145

4

.010

10

.024

16

.051

22

.074

28

.120

34

a48

6.

.012

11

.029

17

.057

23

.077

29

a24

85

.158

6

.013

12

.034

18

.001

24

.082

80

.126

86

.167

Tbe mills roUlnip sheet Iron In the United States generally
use the following, which varies slightly from the Birmingham gauge :

No.
1

lbs per
sqft
12.50

No.
8

lus per
sqft
6.86

No.
15

lbs per
sqft
2.81

No.
22

lbs per

2

12.00 .

9

6.24

16

2.50

23

1.12

3

11.00

10

5.62

17

2.18

24

1.00

4

10.00

11

6.00

18

1.86

25

.90

5

8.75

12

4.38

19

1.70

26

.80

6

8.12

13

3.75

20

1.54

27

.72

7

7.50

14

3.12

21

1.40

28

.64

When 'Wire, Bheet-metalf dc«.y are ordered by gauge number, and it if
sot specified what gauge i» intended ; dealers in the United States fill the older ••
follows :

Brass, bronze or German Silver in sheets, German Silver wire, brazed braw, bronM^
line or copper tubing, by Brown A Sharpens (or " American '*) gauge.

and

Oopper in sheets : brass and copper wire ; seamless braas, bronze or oopper tabingj
id imaU brass rods; by Stubs* (or Birmingham) gauge.

Unannealed or hard

has about ^ths tbe strengtha oi tht table p. 891,

Unannealed or bard nnMSunre naa about %tm the strengtna <n
and about X more weight If annealed, only full half the atrength.

Hard copper ivlre maj be taken at ^ of the tsbolsr ftreogtha. and ftai
I more weiirht

IRON WIRE.

891

Tftbl« of Cbareoal Iron Wire made by Trenton Iron €o.,

Trenton, N. J. The numbers in the first column are thoee of l^e Trenton Iron
Oo's gUVkge* The corresponding diameters in the second column will be seen to
be sconewhat less than those of the Binningham gauge.

Ko.

Diam.
las.

Lineal
feet to the

POODd.

Tenille

Str'gth

Apprax

Iba.

No.

Dlam.
Ins.

Lineal

feet tc the

Pound.

Tensile
Str'gth
Ayprox

No.

Dlam.
Ids.

Lineal

feet to the

Poand.

00000

.450

1.863

12598

11

.1175

27.840

1010

26

.018

1164.689

0000

.400

2.358

0965

12

.105

34.219

810

27

.017

1305.670

000

.360

2.911

8124

13

.0925

44.092

631

28

.016

1476.869

00

.330

3.465

6880

14

.080

58.916

474

29

.015

1676.989

0

.305

4.067

5926

15

.070

76.984

372

30

.014

1926.321

1

.385

4.645

5226

16

.061

101.488

292

31

.013

2282.658

2

.265

6.374

4670

17

.0525

137.174

222

32

.012

2620.607

8

.245

6.286

3948

18

.045

186.335

169

83

.011

3119.092

4

.225

7.454

3374

19

.040

235.084

137

34

.010

3778.684

5

.205

•8.976

2839

20

.035

808.079

107

85

.0095

4182.508

6

.190

10.453

2476

21

.031

392.772

••• •••

36

.009

4657.728

7;

.175

12.322

2136

22

.028

481.234

•«• •■•

37

.0086

5222.086,

8

.160

14.736

1813

23

.026

603.863

»«» •••

38

.008

5896.147

9

.145

17.950

lo07

24

.0226

745.710

••• ••■

39

.0075

6724.291

10

.130

22.838

1238

26

.020

948.896

•«• •••

40

.007

7698.268

Tbe wire in this table Ib supposed to be bard, bright,
The figures in the column at tensile strength are based upon tests

charooal iron wire fi*om Trenton blooms.

The tensile strengtti of wire made of is about

Good refined iron 15 percent. less

Swedish charooal iron 10

Mild' Bessemer steel 10

Ordinary crucible steel 25

Special crucible steel 30 to 120

Annealing renders wire more pliable and ductile, but less elastic ; and reduces the
tensile strength by from 20 to 26. per cent

u

more

u

or unannealed.
made with good

than that of

bright charcoal

wire, given in

the above table.

To And approximatel7 the nnmber of stralfrlit wires that
can be i^ot Into a cable of i^lven diameter.

Divide the diameter of the cable in inches, by the diameter of a wire in inches.
Square the quotient Multiply said square bv the decimal .77. The reeult will be
correct within about 4 or 5 per cent at most, in a cylindricai cable.

Tbe solidity, or metal area of all the wires in a cable, will be
to the area of the cable itself, about as 1 to 1.3. In other words, the area of the
Toids is nearly ^ that of the cable ; while that of the wires is fully % that of the
esUe. All approximate.

892

I BEAMS.

I< = span in ft

Q

W = uniformly distribated safe load In Bm « =-

XJ

M =a moment of load, in ft-fiw = -^ °^ "a ~ T5"

M

S = stress in extreme fibres, in lbs per sq in =3 -=.

I, 1 = moment of inertia ; I, about XY ; i, about A B
B, r =» radius of gyration ; B, " " r, '

X

=s ** section modulus

>»

<«

i(

4( 4(

12 M

S

C = coefficient for uniformly distributed safe load = WL « 8 If =

88X
12

Cg, for static loads; S = 16,000 lbs. Cjq, for moving loads ; S = 12,600 fts.
0 E= distance required to make r = B

Section
* index.

H.

Depth

ins.

Weight

per ft

Iba.

Area of

section

sq in.

Web

thickness

ins.

Flange

width

ina.

B 1

it

24

(1

100.00
80.00

29.41

0.7$4
0.500

7.254
7.000

B 2

20

100.00
80.00

29.41
23.73

0.884
0.600

7.284
7.000

B 8

20

75.00
65.00

22.06
19.08

0.649
0.600

6.399
6.250

B80

18

70.00
55.00

20.59
15.93

0.719
0.460

6.290
6.000

B 4

15

100.00
80.00

29.41
23.81

1.184
0.810

6.774
6.^00

B 5

15

75.00
60.00

22.06
17.67

0.882
0.590

6.^2
6.000

B 7

15

55.00
42.00

16.18
12.48

0.6M
0.410

6.746
6.600

B 8

12

55.00
40.00

16.18
11.84

0.822
0.460

6.612
fi.250

B 9
If

12

35.00
31.50

10.29
9.26

0.436
0.360

5.086
5.000

BU

u

10

40.00
25.00

11.76
7.37

0.748
0.810

6.099
4.660

B13

9

(1

35.00
21.00

10.29
6.31

0.732
0.290

4.772
4.330

B15

" 8

<<

25.50
18.00

7.50
5.33

0.541
0.270

4.271
4.000

B17

7
(t

20.00
15.00

5.88
4.42

0.458
0.250

3.868
3.660

B19

6

17.25
12.25

5.07
3.61

0.475
0.280

3.575
3.330

6 21

5

14.75
9.75

4.34
2.87

0.504
0.210

3.294
3.000

B23

4

10.50
7.50

3.09
2.21

0.410
0.1.90

2.880
2.660

B77

it

3

7.50
5.60

2.21
1.63

0.361
0.170

2.521
2.880

I*BEA1CS.

893

I-BEAMS.

A

'^^

X-

f-Y

B

D-

B

H

Tbe table siT«i the maximum
and the minimum weight of each section.
The minimum weights are standard.
Others are special.

Caution. — With very short spans,
the loads found hy means of columns
Cg and Cm. although safe against bend-
ing, may be so great as to endanger a
crushing of the ends of the beam, or of
the walls, etc., under them, unless the
beam has, at its ends, a greater length of
bearing than would otherwise be needed.

2380.3
S087.9

1655.8
1466.5

1268.9
1169.6

921.3
795.6

900.5
795.5

691.2
€09.0

511.0
441.7

821.0
268.9

228.3
215.8

158.7
122.1

111.8
84.9

68.4
66.9

42.2
86.2

26.2
21.8

15.2
12.1

7.1
6.0

2.9
2.5

i

R

ins.

r
ins.

X

48.56
42.86

9.00
9.46

1.28
1.36

198.4
174.0

52.65
45.81

7.50
7.86

1.34
1.39

165.6
146.7

30.25
27.86

7.58
7.83

1.17
1.21

126.9
117.0

24.62
21.19

6.69
7.07

1.09
1.15

102.4
88.4

50.98
41.76

5.53

5.78

1.31
1.32

120.1
106.1

30.68
25.96

5.60
5.87

1.18
1.21

92.2
81.2

17.06
14.62

5.62
5.95

1.02
1.08

68.1
58.9

17.46
13.81

4.45
4.77

1.04
1.08

53.5
44.8

10.07
9.50

4.71
•4.83

0.99
1.01

88.0
36.0

9.50
6.89

3.67
4.07

0.90
0.97

31.7
24.4

7.31
5.16

8.29
3.67

0.84
0.90

24.8
18.9

4.75
3.78

3.02
3.27

0.80
0.84

17.1
14.2

3.24
2.67

2.68
2.86

0.74
0.78

12.1
10.4

.2.36
1.85

2.27
2.46

0.68
0.72

8.7
7.3

1.70
1.23

1.87
2.05

0.63
0.65

6.1

4.8

1.01
0.77

1.52
1.64

0.57
0.59

3.6
3.0

0.60
0.46

1.15
1.23

0.52
0.53

1.9

1.7

^8

lbs.

2,115,800
1,855,900

1,766,100
1,564,300

1,353,500
1,247,600

1,091,900
943,000

1,280,700
1,131,300

983,000
866,100

726,800
628,300

570,600
478,100

405.800
383,700

333,500
. 260,500

265,000
201,300

182,500
161,700

128,600
110,400

93,100
77,500

64,600
51,600

38,100
31,800

20,700
17,600

^m
lbs.

1,653,000
1,449,900

1,379,800
1,222,1G!Q

1,057,400
974,700

853,000
736,700

1,000,600
883,900

768,000
676,600

667,800
490,800

445,800
373,500

817,000
299,700

264,500
203,500

207,000
157,300

142,600
118,500

100,400
86,300

72,800
60,600

50,500
40,300

29,800
24,900

16.200
13,800

D

ins.

17.82
18.72

14.76
15.47

14.98
15.47

13.20
13.95

10.75
11.26

10.95
11.49

11.05
11.70

8.66
9.29

9.21
9.46

7.12
7.91

6.36
7.12

5.82
6.32

5.15
6.60

4.33
4.70

Section
index.

B 1

((

B 2

i(

B S

((

B80

B 4

<<

B 5

B 7

i(

B 8

(<

B 9
•»

Bll

(i

B13

<i

B15

({

B17

B19

II

B21

B23

<<

B77

«

894

CHANlfElJS.

W

■pan in ft

- uniformly distribated nfo load in Bm » =-

Li

moment of load, in ft-ttM

WL

8

C

8

SX
12

S =3 stress in extreme fibres, in Bw per sq in = ^

1, 1 = moment of inertia ; I, about XY ; I, about A B

r = radius of gyration ; H,
"section modulus"

u

t$

u

u

X:

12 M

S

coefficient for uniformly distributed safe load » WL = S'Mi

8SX
13

C^, for static loads; S — 16,000 lbs. Cm, for moring loads ; S >» 12,500 lis.

D :

» distance required to make r = R

flection
index.

H.

Depth

ins.

Weight

per ft

lbs.

Area of

section

sq in.

Web

thickness

ins.

Flange

width

ins.

C 1

15

55.00
33.00

16.18
9.90

0.818
0.400

3.818
S.400

C 2

12

40.00
20.50

11.76
6.03

0.758
0.280

3.418
2.940

C 3

10

t4

35.00
15.00

10.29
4.46

0.823
0.240

3.183
2.600

C 4

9

4<

25.00
13.25

7.36
8.89

0.615
0.230

2.815
2.430

C 6

14

8

21.25
11.25

6.25
3.35

0.582
0.220

2.622
2.260

C 6

7

19.75
9.75

6.81
2.86

0.633
0.210

2.613
2.090

C 7

6

K

15.50
8.00

4.56
2.38

0.563
0.200

2.283
1.920

C 8
«

6

11.50
6.50

3.38
1.95

0.477
0.190

2.037
1.750

C 9

4

n

7.25
5.25

2.13
1.55

0.325
0.180

. 1.725
1.580

C72

<<

3

4t

600
4.00

1.76
1.19

0.362
0.170

1.602
1.410

Fire-proof floors of I beams and briek-arcbes.

The arches are usually •• four-inch" — or " half a brick" deep; spad« *i fron> 4
to 6 feet ; rise about one-twelfth to one-sixteenth of the span, lie-rods, 2*, %C
inch to 1 inch diameter, from 4 to 6 or 8 feet apart, and anchored into each wall

with a stout washer, W, At each wall an angle iron, a, or a tee iron is generally
used instead of a beam. The spandrels are leveled up with oonorete, enclosing
wooden strips, m m, about 1 inch X 2 inches, two OTer each arch. To tnese stript
the flooring is nailed.

CHANNliXS.

^

Tbe table ylvea t

CADtloD,— With ver

bearing tfi an vouJd othuvlse be needed.

1

■

,S.

L.

X

£ 1 fe

»

Seetioo
index.

430.2

12.1S
8.28

5.19 (

Ma

57.4

4441500

*^iZ

8,58

CI

lail

S.91

isi <

80S

S?;J

3N),20I>
227,800

178,000

?:l?

C,2

"^l

t^

a?? S

?"

w:i

24fi.400

I92.M0
111,600

Im

C_3

70.7

l.!7

t'i 2

637

16.7

167,600
112,200

";S

4.84

CJ

sals

125

"I s

530

ail

127,4*0
86,100

99.600
67;300

4:94

^..^

2i:?

S:^

l?2 0

^

M

101,100
66,800

62^200

3.48

c^a

1b:S

1.28

IM S

m

ts

69,SO0
46,200

54,800

1^

CJ

":1

":«

!:S S

483
498

4.1
8.0

44,400
31,600

24;7I»

ifa

^J

aJ

S:S

1.48 {

4K

20;2O0

16^800

kw

c 2

i,i

0.31

I.OS (

«1

}:J

14.700

slioo

]ft

C^72

Thew
"a del

le crowd

4-lacb a

.wi

'70 lbs.
1 bardly

~:^Zi

g BDd wood
01 or flour.

naoo

iDg, bul

ere foot.

896

ANGLES AND T SHAPES.

CABHEOIB ANr€»IiBS

I,*
X,

r

distance between center of gravity and back of flange W
i( <( «( (t <( (< It ti ^

moment of inertia ; I» about XY ; i, about A B

= least " section modulus" ;

It

(t

i(

i(

<i

12 M.

S ""

radius of gyration ; B, " " R', "

least radius of gyration, about neutral axis forming aoate angle a
with each flange. In angles with equal legs, a =» 45°

For T shapes only.

Section
index.

Size

Thick-

Weight

Area of

d

8

H \V

nesss

per ft

section

ins.

ins.

I

IDS. ins.

ins.

lbs.

sq ins.

Anf^les ^rith Uneqaal

♦A 150
•A 159

A 89
A 168

A 92
A 177

•A 178
•A 186

A 187
A 96

A 196
A280

♦A204
*A 97

•A 212
•A 98

A220
A 228

A 229
A 237

A 238
A 245

•A246
•A 251

A 262
A 257

•A 258
*A262

A 264
A 269

•A 270
•A 275

♦ A 276
•A 277

♦ A 278
•A 279

I 7

17

I

6
6

6
6

^

d
5

5
5

^1

X4
X4

X31
X3)

X4

X4

X3>|

5
5

X3
X3

I 4^X3
4^|X3

X3^

X3
X3

33^X3
3>|X3

4

14

I 4
,4

33
31

1X2

3KX2
3>iX2

3X2
8X2;

3 X2
3 X2

2KX2
, 2>|X2

I 2^4 X 1
i 2H X 1

I 2 XI?

I 2 XI'

! iH X 1

Vb

I

Vb

v.

8

I

H
\%
\i

IB

g
^
^

32.8
15.0

9.50
4.40

2.71
2.50

0.96
0.75

46.87
22.56

30.6
12.3

9.00
3.61

2.17
1.94

1.17
0.94

80.75
13.47

28.9
11.7

8.50
3.42

2.26
2.04

1.01
0.79

29.24
12.86

24.2
11.0

7.11
3.23

1.71
1.53

1.21
1.03

16.42
8.14

22.7
8.7

6.67
2.56

1.79
1.59

1.04
0.84

16.67
6.60

19.9
8.2

5.84
2.40

1.86
1.68

0.86
0.68

13.98
6.26

18.5

7.7

5.43
2.25

1.65
1.47

0.90
0.72

10.33
4.69

18.5
7.7

5.43
2.25

1.36
1.18

1.11
0.93

7.77
8.66

17.1
7.1

6.03
2.09

1.44
1.26

0.94
0.76

7.34
8.88

15.7
6.6

4.62
1.93

1.23
1.06

0.98
0.81

4.98
2.88

12.4
4.9

8.65
1.44

1.27
1.11

0.77
0.61

4.13
1.80

9.0
4.3

2.64
1.25

1.21
1.09

0.59
0.48

2.64
1.36

9.5
4.5

2.78
1.31

1.02
0.91

0.77
0.66

2.28
1.17

7.7
4.0

2.25
1.19

1.08
0.99

0.58
0.49

1.92
1.09

6.8
2.8

2.00
0.81

0.88
0.76

0.68
0.51

1.14
0.51

5.5
2.3

1.63
0.67

0.86
0.75

0.48
0.37

0.82
0.34

2.7
2.1

0.78
0.60

0.69
0.66

0.87
0.85

0.87
0.24

1.8
1.0

0.53
0.28

0.48
0.44

0.29
0.26

ao9

0.09

7.53
3.95

10.75
4.90

7.21
8.34

9.28
4.67

6.21
2.72

3.71
1.75

3.60
1.7S

5.48
2.5B

1.66

8.3S
1.58

1.72
0.78

0.75
0.40

1.42
0.74

0.67
0.89 •

0.64
0.29

0.28
0.12

0.12
0.09

ao4

0.02

* Special sections, f For M and S see p. 892 or p. 894

anox.es and t shapes.

897

AVD T SHAPES.

I

-X----Y-

i

-w-

-W-

I

I

I

-X--YJ--

i

-w-

S^

H

^

-X---I---Y-

H

-W-dt

B

'm

Section
Index.

Majdmum and Minimum 'Weight of each Section.

10.58

2.96
1.47

8.79
1.60

2.90
1.23

8.31
1.57

2 52
1.02

1.74
0.75

1.71
0.76

2.30
1.01

1.68
0.74

1.65
0.72

0.99
0.41

0.53
0.26

0.82
0.40

0.47
0.25

0.46
0.20

0.26
0.11

0.12
0.09

0.05
0.03

2.19
2.26

1.85
1.93

1.85
1.94

1.62
1.69

1.53
1.61

1.55
1.61

1.38
1.44

1.19
1.26

1.21
1.27

1.04
1.10

1.06
1.12

1.00
. 1.04

0.91 ,
0.95

0.92
0.95

0.75
0.79

0.71
0.72

0.63
0.63

0.41
0.44

0.89
0.95

1.09
1.17

0.92
0.99

1.14
1.20

0.96
1.03

0.80
0.85

0.81
0.88

1.01
1.07

0.83
0.89

0.85
0.90

0.67
0.74

0.53
0.57

0.72
0.75

0.55
0.57

0.56
0,60

0.40
0.43

0.39
0.40

0.27
0.29

0.88
0.89

0.85
0.88

0.74
0.77

0.84
0.86

0.75
0.76

0.64
0.66

0.64
0.66

0.72
0.73

0.64
0.65

0.62
0.68

0.53
0.54

0.44
0.46

0.52
0.53

0.43
0.43

0.42
0.43

A 150*

5.01

A 159*

8.02

A 89

8.82

A 168

7.88

A 92

3.25

A 177

A99

A 178*

2.34

::::::::::::::::::::::::::::::::::::

A 186*

4.88

A 187

1.94

A 96

4.45

A 196

1.89

A 280

3.62

A 204*

1.54

A 97*

2.92

A 212*

1.26

A 98*

2.87

1

A 220

1.23

A 228

2.20

A229

0.96

A 237

1.85

A 238

0.75

A 245

1.80

1

A 246*

0.63

'

A 251*

1.15

1

A 252

0.56

1

A 257

1.00

1

A258*

0.54

A 262*

0.70

1

A 264

0.29

A 269

0.59

0.39
0.40

0.30
0.31

0.22
0.22

'

A 270*

0.23

A 275*

0 23

A 276*

0 18

1

A 277*

0.09

A 278*

0.06

, 1

A 279*

1

* Special sections.

57

898

ANGLES AND T SHAPES.

CAIUTECIIE AJrOUSS

4 «■ distance between center of grayitr and back of flange W

^ _^ (i << 4( It « i( <t (k T7

1, 1 »■ moment of inertia ; I» about XY ; i, aboat A B

Xf X =a least " section moduliis" ; 3

It

i<

tr -^

It

ti

II

II

12 M,

S

«i

II

radius of gyration ; By «• 9

least radius of gjration, about neutral axis forming acute angle •
with each mtnge. In angles with equal legs, a = 45*^

coefficient for uniformly distributed safe load : "j

Cg for static loads ; fibre stress = 16,000 fts. >-For T shapes only.

Cj^, for moving loads ; fibre stress » 12,000 S»s. J

Section
index.

Size

H W

ins. ins.

Thick-

nesss

ins.

Weight

per ft

Ibe.

Area of
section
sqins.

ins.

m
ins.

I

Angrles witb-EqnAl

A 118
A 103
A 86
A 88
•A 94

•A
A
A
A
A
A
A

17
18
90
26
99
84
40

•A 41

•A 45

A 46

A 100

•A 51

•A 101

A 66

A
A
A
A

60
61
65
66

A 102

A
A
A
A

70
73
78
80

•A 81

A

A
A

82
83
84

56.9

26.4

87.4

14.8

80.6

12.3

19.9

8.2

17.1

7.1

11.4

4.9

8.5

4.5

7.7

8.1

6.8

2.8

5.8

2.5

4.6

2.1

8.4

1.2

2.4

1.0

1.5

0.8

1.0

0.7

0.8

0.6

16.78
7.75

11.00
4.36
9.00
3.61
5.84
2.40
6.06
2.09
8.86
1.44
2.50
1.31
9.25
0.90
2.00
0.81
1.56
0.72
1.80
0.62
0.99
0.36
0.69
0.80
0.44
0.24
0.29
0.21
0.26
0.17

2.41
2.19
1.86
1.64
1.61
1.89
1.29
1.12
1.17
0.99
0.98
0.84
0.87
0.78
0.81
0.69
0.74
0.68
0.66
0.67
0.59
0.51
0.51
0.42
0.42
0.85
0.84
0.30
029
0.26
0.26
0.23

97.97
48.63
38.46
16.39
19.64
8.74
8.14
8.71
5.26
2.45
2.62
1.24
1.67
0.98
1.23
0.55
0.87
0.89
0.64
0.28
0.36
0.18
0.19
0.06
0.09
0.044
0.067
0.082
0.019
0014
0.012
0.009

TSliw

T
T
T
T
T
T
T
T
T

50

57
61
3
72
77
82
12
16

5 X3
4 X6
4 X3
8MX8H
8 X4

8 X8K
2J^X8

2KX
1%X

13.6

15.6

9.8

11.7

11.8

8.5

7.2

4.9

3.1

8.99
4.56
2.78
8.45
8.48
2.49
2.10
1.44
0.90

0.76
1.56
0.78
1.06
1.32
1.09
0.97
0.99
0.54

2.6

10.7
2.0
8.7
5.2
2.9
1.8
0.66
0.28

5.6

2.8

2.1

1.8»

1.21

0.98

a64

O186

0.1t

* Special sections, f For M and S see p. 892 or p. 804.

aiiql.es and t shapes.

AHV T SHAPES.— Ooatlnnad.

,J^

Maximum >nd

Minimu

IB Wdcht ot e

ch Seeilon

2.U

l.H
1.02

1

L

J

•.

0.

e.

0.

r

0.

e.

a.

•.
ft

f%

n 1

D.M
(I.BI

0.8S
O.BS

i

o:23

0.033
O-OM
ftOlT

A S4

Bcleclcd Sactloa*.

1.41

i

i

1.19

IS

oiei
o!«8

».410
lE^lBO

US!
!5!

1

480

i

iSO

s

i"

900

SEPARATORS FOR I BEAMS.

CARWEOIE STANDARD CAST IRON SEPARATORS FOR

I REAMS.

Separators for W^ 2(/' and 24^' beams are made of %'' metaL
O'' to W beams are made of >^" metal,
y beams and ander are made of ^" metal

«

Designation
OF Beam.

r^

ST

Ins. Lbs.

Distances.

o
02

Ins.

Ins.

Bolts.

N

GQ

In.

■♦a

« s

§s

a

Ins.

t

a

Ins.

Weights.

a
a

I

--0

Lbs. Lbs.

hi

2

Lbs

LbSL

Sepaiwtors witb Two Rolts.

B 1

B 2

B 3

B 80

B 4

B 5

B 7

B 8

B 9

24
20

80.
80.

1^

13>J

^

20

65.

7

18
15

55.
80.

6%
6^

15

60.

5f|

ll'l

15

42.

12

40.

6

12

81.5

10%

6%

8.41
8.41
3.23
3.16
3.35
3.23
2.98
2.98
2.92

.250

<(

t<
It
(I

K
l(
If
((

82.
28.
25.
16.
15.
16.
15.
11.
11.

Separators with One Bolt.

6.60
8.10
8.10
2.75
1.75
1.76
1.75
1.50
1.50

B
B

8
9

B 11
B 13
B 15

B 17
B 19
B 21
B 23

B 77

12

12

10

9

8

7
6
5
4
8

40.0
31.5
25.0
21.0
18.0

15.0

12.25

9.75

7.50

5.50

5

f

3

3

1.49
1.46
1.40
1.34
1.28

1.25
1.22
1.16
l.lt
0.70

.125

(t
<(
((
«<

((
(I
<i
«<

.09

10.
10.

8.

7.

6.

4.
4.
3.
3.
2.

1.50
1.50
1.25
1.20
1.00

.75
.60
.60
.40
.25

-

Z BAR COLUMNS.

For ana of iHtlon,
w^bt vet jtri, lext
TBdfiii ot gjrMion ind
nfelosd, se« t)>bJ«,pp.

ly-

ISA

Ji

SA

•1

Ji.

900

SEPARATORS FOR I BEAMS.

CARNEGIE STAHTDABB CAST IBOUT SEPARATORS FOE

I BEAMS.

Separators for 18^', 2(K' and 24^' beams are made of %'* metal.
" •' G'' to W beams are made of 3^" metal.

" " V beams and under are made of %^^ metal

Dbsionation
OF Beam.

a

M

§

N

4

Ins. Lbs.

Distances.

Ins.

u

-2

V 9
^ a

Ins.

Bolts.

45

QQ

In.

I

»4

0)

8

Ins.

a

Ins.

s
a

Lbs.

Weights.

a

9> ^ o

5 •§

Lbs.

S
2

•8

Lbs

§

Lb&

Separators wltb Two Bolts.

B 1

B 2

B 3

B 80

B 4

B 5

3 7

B 8

B 9

24
20
20
18
16
15
16
12
12

80.
80.
65.
55.
80.
60.
42,
40.
81.6

8.41
8.41
3.23
8.16

ass

3.23
2.98
2.98
2.92

.250

u
u
<(
l(
«
l<
It

32.
28w
25.
1&
16.
16.
15.
11.
11.

6.B0
S.10
8.10
2.76
1.75
1.75
1.75
1.50
1.50

Separators with One Bolt.

B 8
B 9
Bll
B 13
B 15

B
B

17
19

B 21
B 23

B 77

12

12

10

9

8

7
6
5
4
3

40.0
31.6
25.0
21.0
18.0

15.0

12.25

9.75

7.50

5.50

IIH
9

7%

i

6

f

5

3
3

.49
.46
.40
.34
.28

.25
.22
.16
.18
0.70

.125

((

iC

((
*(

((
<t
it
«

.09

10.
10.

8.

7.

6.

4.
4.
3.
3.
2.

1.50
1.50
1.25

i.ao

1.00

.75
.60
.60
.40
.26

Z BAB COLUUNB.

For area of bwIIob,
■eight per yard, leui
Tsdius of gyration and
safe load, see tablet, pp-

Thicknw.

see

ngure a

K.VB.

i

Met^

A

B

"

D

E

r

«

H

'

1

A

12A

31

^

2

2

2H

8

3

1

12|

A

2

2

2

8
6
8

3

s

J^

laf

3A

6A

2
2

2
2

I»

3
3

a
«

ft

12

3|

BA

2

2

2

2

?'

8
7

3

i

"IF
if

t"

1"

3 "
3

3
3

1

9 "
9

4

i

s

^

V

3

3*

3
3

P

*A

i

i

4

4

1

I

5
6
5

I

3
3
3
3
3

3ft

9

1

f

JL

16

6

■s

^A

3

3

sj

101

6ft

1

16A

5

3

3

sft

10

5A

16
16

6

s

i

ef

3
3

3
3|

3
3

9
9

6

N

!

«

ISA
16

16

16A

5H

1

i

3
3
3

3
3

3

3ft
3?

9
9
9
9
9

6
5
5
5
5

Is—

18«

;

4
4

2

3

3A

:»

6
6

i

19

7

4

2

3

11

6

18H

18* i
ISA

1

6

a*

6
6

4
i
4

4

2
2
2
2

3
3

lOJ

lo!

6-
6
6.
6

\

1

?

181
18H

6

a

4

4

2
2

if

lot

t

902

ZBAR COLUMNS.

CARNEOIS STEEIi Z-BAR COIiVlUrS.

Table of Safe lioads as given by Carnegie Steel Co., for colamns with
square ends. Safety fisctor ■« 4. The loads giv«n are based upon the followtng
afiowed stresses in pounds per square lucb :

For lengths of 90 radii or less, 12,000.
" " over 90 radii, 17,100 — 67—.

Each Z-bar oolumn is made up of four Z-bars and one web-plate (all of nni*
form thickness) bolted or riveted together, as shown in the figure on page 901.

6«Uicli Steel Z«bar Colamiie.

Composed of four Z-bars about 3 inches deep and one web-plate 5^ inches wide.

Thickness Qf metal, \
inch. j

i

A

i

A

: i

A

Area of section, sq. )
ins. j

9.81

11.7

18.6

16.0

17.6

ao.o

Weight per yard,)
pounds. j

95.1

119.4

138.6

162.9

179.7

208.7

Least rad of gyr, )
inches. j

1.86

1.90

1.88

1.93

1.90

1.96

Length of column.
Feet.

Safe load of column, in pounds.

12 or less.
14
16
18
20
22
24
26
28
30

111800
111400
104600
07600
90800
84000
7720O
70460
6340O
56600

140600

140600

133000

124600

116200

107800

99400

91000

82600

74200

168200
168200
153200
143400
133400
123600
113800
103800
94000
84000

191600
191600
182600
171200
159800
148600
137200
126000
114600
103400

211400
2U400
199800
1^^
171400
161800
149200
186400
128800
111000

289600
289600
229600
215600
201600
187600
173600
169600
145600
181600

8-iiieli Z«bar Colmnns.

Composed of four Z-bars about 4 inches deep and one web>plate 6^ inches wide.

Thickness of metal, )
inch. j

i

A

♦

A

i

A

i

«

f

Area of section, sq. )
ins. J

11.3

14.1

17.1

19.0

21.9

24.8

Ctf> n

29.0

81.9

Weijtht per yard,)
poapds. /

114.9

144.8

174.0

194.1

221.1

262.8

267.6

296.4

836.2

Least rad of gyr,)
inches.

2.47

2.52

2.67

2.49

2.65

2.60

2JB

2.58

2.68

Length of column.
Feet.

Safe load of oolumn, in pounds.

18 or less. -
20
22
24
20
28
30
32
84
86
38
40

135000

180000

123800

117600

111400

105200

98800

92600

86400

80200

74000

•67800

169600
165000
157400
149600
142000
134200
126600
119000
111200
103600
96000
88200

204800
201000
191800
182600
178600
164600
155400
146400
187400
128200
119200
110000

228400
221000
210600
200200
189600
179200
168800
158400
148000
139400
127000
116600

262400
256400
244800
238000
221200
209400
197600
186000
174200
162400
150600
189000

297000
292800
279800
266800
258800
240600
227600
214660
201600
188600
176600
162600

236400
221200
207000
192600
178800
164400

848600
342600
827000
811600
206200
280800
26S400
250000
284600
219200
M880Q
IIMOO

882400
879200
862600
846000

829400
812800
896400
279800

288800

BhBBOO

818M0

Z-BAR COLUMNS.

903

CARHEGIE STEEI. Z-BAB COI^UJIINS.
Table of Safe Ijoads ((xmtiDued).

10-inch Steel Z-bar Columns.

Composed of foar Z-bare( about 5 inches deep and one web-plate 7 inches wide.

Thickness of metal, 1
ineh. i

Atea of section,
ins.

sq.)

Weight per yard,)
po.unds. J

lioast rad of gyr,
inches.

}

A

{

A

i

A

*

H

f

15.8

19.0

22.3

24.5

27.7

80.9

32.7

35.8

161.1

194.1

227.4

249.9

282.6

815.6

830.0

368.4

8.08

3.13

3.18

3.10

8.15

8.21

8.18

3.18

JL

39.0

897.8

3.26

JiCngth of column.
Feet.

22 or less.
24
26
28
30
82

88

40
42
44
46
48
50

Safe load of column, in pounds.

189400228400 267800

185600225200266200

178600 217200 256600

171600208800/247000

164600

157600

1

136&0O
129600
122600
115400
1084001
1014001
94400

200400
192200
183800
176660
167200
158800
150600
142200
134000
125600
U720O

8200285:

«6oom

237400

227600

21

208600

199000

189400

179800

1702p0

160600

151000

141400

294000
289200
278400
267600
256800
246000
200
4400

202800
192000
181200

382400
.329600
317400
305400
298400
281400
269400
257400

213600345400978200

159600

233400
221200
209200

170400 197200

185200

148d00'173200

371200
370600
357400
844200
881000.
317800
304600
291400

392000
387200
373000
858600

429800
427800
412400
897000

265000
251800
238600
225400
212200
199000

8444001^1600
330000.366200

316800
301400
287200
273000
258900
244400
280200
215800
201600

350800

820D0O
SOftOO
2^9200
27S80O
258^
243000
227600

468000
468000
453200
436800
420400
404000
387600
371200
354800
338200
321800
305400
289000
272600
256200

IS-ineb Sieel Z-bar Golumna.

Composed of four Z-bars about* 6 inches deep and ona web-plate 8 inches wide.

Thickness of metal, )
inch. J

t

A

i

A

i

«

i

if

*

Area of section, sq. )
ins. /

21.4

25.0

28.8

81.2

84.8

38.5

40.6

44.1

47.7

Weight per yard,
pounds.

218.1

256.6

293.4

818.6

865.6

892.7

413.4

449.7

486.8

Least rad of gyr,")
inches. /

3.67

8.72

3.77

8.70

3.76

&78

a68

8.66

3.64

Length of column.
Feet.

Safe load of column, in pounds.

26 or less.
28
80
82

86
88
40
42
44
46
48
50

256600
254000
246000
288000
230300
2SK00
214200
206200
198200
190200
182400
174400
166400

300600
299400
290200
281000
271800
262600
258400
244200
235000
225800
216600
207200
198200

345200
345000
385200
324800
314400
304000
293600
283000
272600
262200
252400
241400
231000

874600
872000
S6040O
349000
837400
825800
814200
302800
291000
279600
268000
256400
244800

418200
417800
405000
892200
879600
866800
364000
841400
328800
316000
808200
290600
277800

'462000
460600
44660P
432600
418400
404200
390200
376000
361800
347800
333600
319600
805400

486000
481600
466400
461400
486400
421200
406200
391200
376000
361000
845800
830800
815800

529000
522800
506M)0
496000
478400
456600
440400
423800
407400
391000
374400
858000
341400

572200
564200
546400
528400
510400
492600
474600
456600
438800
420800
402800
384800
367000

For loads greater tban those g^lven in the tables, the Z-bar
columns may be re-enforced by additional plates, riveted to the flanges. The
addition of such plates does not in any case diminish the least radius of gyra-
tion. Hence the same load per square inch qf crosjt'section may be used.

904

PHCENJX COLUMNS.

T»ble of rolled-lx^n Bevmeiit-eolaiiiiui of the

Iron Co, 410 Walnut St, PhiUda.

Pboeatx

Tb« diaioiisions ciTra

are subject to slight Tariations which are unavoidablv
in rolling iron shapes. The weightm of columns gi-ron
are those of the 4, 6, or 8 segments, of which ttiej art
composed. The »hanks of the rirets used in joining tbem
together, of course, merely make up the quantity of metal
punched or drilled out, in making die holes ; but the rire^
fuadi add from 2 to 6 per cent to the weights giren. The
rtveto are spaced 3, 4, or 6 ins apart from oen to
oen.

Any desired thickness between the minimoa
and maximum for any given size, can be furnished.
We give the dimensions, weights, &c, corresponding ts
the principal thicknesses. G columns have 8 segment^
£, 6 s^m^nts. All others, 4 segments.

A

M

tt

M
U
U

Bt

u
u

Mm

Diameters^ ins.

^

u

J«

(I

M

U

!.ft

M

M
M

11

i«

H
t(
M

!>/

One column.

of cross
- sec,
sqins.

8.1

4.8
6.8

e.8

6.4
9.2
12.
14.8
7.4
10.6
18.8
IT.
10.
18.
26.2
88.2
41 Ji
16.8
26.4
37.8
40.8
61.8
24.
36.
62.
68.
92.

Wtper

ft run,

lbs.

12.6
16.
19.8
82.6
21 Jt
30.6
40.
49.8
24.6
86.8
46.
66.6
Ii8.8
00.
94.
110.6
187.8
66.
88.
126w
166.
206.
80.
ISO.
178.8
226.6
a06.6

I<east
radof

ins.

1.46
1.60
L66
1.60
1.92
2.02
2.11
2.20
&84
2.43
2.62
2.61
2J60
2.98
8.16
8.84
3.62
4^18
4.36
4.66
4.78
4.91
6.46
6.69
6.77
6.96
6A

Slseof
BlTeta.

«4
1<

«Kx

THE GRAY COJAnat.

Tbe Onty- Colnma, deilgiied lud Hl«nl«d br Hr. J. H. Qnj, coudBt^
tn Itsorlslniif form, o( BDg1«, oonneoted at IntciTili D (ganenllr of 2 ft fl liuf
tjj truuiene beui Ue-pliil« T. dbhiIIt »X% Im. Tbig ooDitniclloa rendera
tba puta of ths eolnmD eulLr ucesslble, for palntlDg. etc, but under truut.
TOM or bnekUnfi itraiHa the colamD must not umewbat Itke s mtBnguliir
trwMDe wlthont dunnalH. To remedy thin, m later form, Ihe " twelfe-auglo"
oolomn, Flg5, hM Dean d»Lgaed. bsTiua, In Ihe rauwe column, FIgi 1 sod 1,
Imtawl of Uw bent tie-pUte* T, four addltioasl aiiglei, runulng LoDglUidlnallr
Uke tha othan, lud placed oeotrall;, as ebown. Tb«Be angles supply tbe mluioD
witb too iielH, laieneotlug at right angln.

Fig 3 ibowa pUt«a rl

rof bDlldluga. It I*

blocka. The riTela an Dsuallr ^ Inoh

nia Bafa load, Id ponnda per sq inch, of tba ordlnaiy oolainh, la atat^ i
17,100 — ST ~, when L — lengtli of Dolumn and r - Its le«M nd of gTntloii.

Tba OMt of ttaa Ctr«r colamn, at shcm, Is from 1 to l.S oenta per ft> plni tb
•Oltoftheanglca.

906

THE GRAY COLUMN.

Oray Golanui. lilat of Selected Staes.

Size

S

ins.

Angles

ins.

Area

sqin.

Mom
of in.

r

LeicA

rad.

of gyr.

ins.

Safe loads* for

column lengths

of

12 ft.

30 ft.

Square Oolnmns. Figs. 1 akd 2.

9
it

10
<i

12

<(

18
i<

14
<(

4«

it
tt
t(

15
tt

4t
4(
«

16
«

tt

18
tt

tt

20
«

80

8.48

64

2.7

7.8

115

18.00

119

2.6

6.8

250

9.52

95

3.1

9.6

136

20.00

179

3.0

9.0

285

16.88

241

3.8

14.4

250

30.00

327

8.3

10.9

436

16.88

285

4.1

16.8

255

43.52

552

8.6

IS.O

646

16.88

336

4.6

20.2

256

29.36

526

4.8

18.5

446

26.00

444

4.2

17.6

390

30.00

468

4.0

16.0

450

81.92

597

4.4

19.4

485

39.36

624

4.0

16.0

590

36.00

626

3.8

14.4

685

22.00

496

4.8

23.0

335

31.92

6^

4.7

22.1

490

39.86

731

4.4

19.4

600

30.00

653

4.7

22.1

440

41.84

817

4.4

19.4

635

46.88

828

4.2

17.6

710

22.00

570

5.1

26.0

340

36.88

912

5.0

25.0

570

55.62

1134

4.6

21.2

850

22.00

746

5.8

33.6

345

36.88

1182

5.7

32.5

576

46.88

1465

5.6

31.4

730

36.88

1485

6.4

41.0

560

67.62

2588

6.2

88.4

1065

41.80

4147

9.9

98.0

680

264.40

22688

9.2

84.6

4286

80
16S
100
205
196
325
205

210

3U

355
S96
470
430
280
405
^0
300
520
570
285
480
700
300
495
630
510
980
630
3980

Wall Columns. Fig. 3.

12

14

tt

15
<i

16
tt

18
20
30

14.88

94

2.5

6.2

906

18.00

160

3.0

9.0

265

29.52

241

2.9

8.4

420

21.00

217

3.3

10.9

306

31.38

317

8.2

10.2

465

16.50

200

8.5

12.2

2140

85.16

875

3.8

10.9

610

27.66

434

4.0

16.0

415

31.38

562

42

17.6

475

31.38

im

6.8

46.2

405

198.30

8937

6.7

44.9

8160

130
186
295
226
835
185
380
830
380
440
2785

Corner Columns. Fio. 4.

3>^X3>iX %
4 X5 X %

23.48

272

3.4

11.6

845

15.76

288

4.3

13.5

240

24.91

•425

4.2

17.6

875

260
105

805

* In.thousands of B>s, hj formula: Saf« load =* 17100 — 67~in]bapertqin.
t Three 1-in plates riveted to each pair of angles. Fig. 2.

eiKENQTH OF IRON FILLABS.

tSr"

TABLE OF BKSAKING lAADS OF IBOM PIEXABS,

•n*. Dedocsd from Go

in practice. If tbe pllbw !■ reetucB*

Id loDB per <(|tu» Inch of nutal uw. Dedacsd from Gotdan. Tta* eiidi an
nDpoMd M be plued to form perfeollY trua beuriDgi ; end all

. -,[i.™relj thee™ in practice, irtbti pllli._ _

[vara, uaatb* Itutitde for ameiimireof lenitb. (OrigiiulJ

RvoDd.

HoUow
Sqasre.

BoUd
Boi»d.

Boltd

»M

rilf.

Sni^:!^

■£

■ sswtff

OMt.

Wrt.

CHt.

■Wrt.

Caat.

wrt.

emit.

Wrt.

i

i

i
i

1

'

11

■is
i

1
l!

1:
a

1"

1

I

!,

[
>

1

i

[J!

VBlt

910

STBENQTH OF IRON PILLARS.

HOIiliOW CTI^IlTBllICAIi WBOUOHT IBOIT PIl^I^ABS.

Table 4, of breakiniT loads In tons of bollow eylindrleat
wronirht iron pillars, wlt.b flat ends* perfectly trae, and
firmly fixed, and the loads presslniT equally on every |Mrl
of the top. Calculated by Ctordon's formula.

(Original.)

I

.3
8

4

6

•

7

8

9

10

11

13

M

14

15

n

IB
10
16

WBOXrOHT IBOir. THIOKKBSB H INCH.

Outer diameter in inohea.

H i

1 1

IH 1

IH \ IH \

8 1 3^ J

f 3M 1

*H 1

»

BRBAKINQ

LOAD.

TOAS.

TOBSi

Toni.

T0B«.

Tons.

Teiu.

Tom.

Ton*.

Tons.

T«IM

$.64

5.37

6.88

8.50

10.1

ll.T

18.3

14.8

16.4

18.0

S.94

4.64

6.33

8.00

9.6

11.3

13.8

14.5

16.1

17.8

3.80

8.86

5.57

7.88

8.9

10.6

13.2

18.9

15.6

17.3

1.77

8.13

4.74

6.36

8.1

1:!

11.6

18.8

15.0

16.7

1.36

3.51

4.07

5.66

7.8

10.8

13.6

14.3

16.0

1.04

3.08

8.46

4.91

6.6

8.8

9.9

11.6

18.4

15.a

.81

1.65

3.91

4.34

5.7

7.4

9.1

10.8

13.6

14.4

.61

1.86

3.46

8.67

5.1

6.7

8.8

9.9

11.7

18.5

.50

1.06

3.06

8.18

4.5

6.0

7.6

9.1

10.8

i3.C

.41

.96

1.75

3.77

4.0

5.4

6.9

8.4

IQll

11.8

.84

.81

1.53

3.41

8.6

4.8

6.3

7.7

9.8

11.0

.»

.70

1.84

3.14

8.3

4.8

5.6

7.0

8.6

10.3

.84

.60

1.16

1.88

3.8

8.9

6.3

6.6

8.0

9.6

.81

.58

1.08

1.09

3.5

8.5

4.7

6.0

7.4

8.9

.19

.47

.91

1.60

3.8

8.3

4.8

6.6

6JI

8.8

.18

.43

.84

1.88

3.1

3.9

4.0

6.1

6.4

7.7

.14

.88

.67

1.11 1.7

3.4

8.4

4.4

6.6

6.8

.37

.56

.91 1.4

3.0

3.8

8.7

4.7

6.8

1 .9

1.4

3.0

3.6

8.4

4.3

H

1

1

8
4
fi
$

7

8
t

a

IS

It

u

it

Weiffht of one foot of Iraffth of pillar, in pounds.

.830 ( 1.15 i 1.47 I 1.80 | 3.13 | 3.45 { 3.78 | 8.11 | 8.48 1 8.n

Area of ring of aolid metal, in aquare inches.

.346 I J44 I .443 I .540 I .688 I .786 | .886 | .988 | 1.08 | 1.18

a

WBOUQHT IKOXr.

TBICKNESS H XNOK.

5

II

Outer diameter in tnohea.

P

J

3 1

3K 1

SH 1 2^ 1 8

1 SH 1 4 1

4M

1 5

1 6H

1 •

BRIAKINO LOAD.

TODI.

's*.*-

Ton*.

Ton*.

TODI.

Tons.

Toot.

Tons.

T«at.

Toai.

Tom.

1

31.9

36.4

38.8

n.4

84.5

40

47

63

M

66

73

1

3

Sl.l

94.8

S7.«

80.7

83.9

40

47

68

60

66

73

3

8

19.9

98.1

98.4

29.7

83.0

89

46

62

59

66

71

9

4

18.6

n.8

95.8

38.6

81.9

88

«

51

58

64

71

t

5

17.0

30.4

88.6

37.8

80.7

87

44

50

87

68

70

6

16.4

18.6

32.1

95.7

99.3

86

43

49

66

63

8

«

7

18.9

17J

30.6

33.8

27.8

84

41

47

64

61

»

8

13.6

16.6

19.1

33.8

25.9

8i

40

46

68

60

t

9

11.3

14.3

17.5

20.6

24.8

80

88

44

61

56

&

•

10

10.0

18.0

16.1

19.1

23.7

99

87

48

06

57

Q

It

11

9.0

10.7

16.7

17.6

21.1

97

85

41

48

65

61

It

13

8.1

IM

18.6

16.4

19.6

96

88

40

46

64

61

18

18

7.8

9.6

13.4

15.1

18.3

94

81

88

44

63

8

18

14

6.6

8.8

11.8

14.0

17.0

98

80

86

46

61

14

16

6.0

8.0

10.4

13.9

16.6

91

98

84

41

40

IS

11

16

5.6

7.8

9.6

13.0

14.6

90

97

88

40

47

64 .

U

18

4.6

6.0

8.0

10.8

13.7

18

94

80

87

48

80

u

SO
85

8.8

5.1

6.8

8.7

11.0
7.9

16
19

21
Iff

Sft

9r

17
14

84

97
99

18

40
88
37
93

47

8

97

90
96
80
86

80

18
10

86

40

14
11

18
16

U
19

40

46

46

60

8

13

16

60

WelgHt of one foot of lengtH ct pillar, jb pounds.
4.60 I 5.38 I 5.90 | 6.53 | 7.90 | 8.50 | 9.83 | 11.1 | 12.4 | 18.7 | 16.9

Area of ring of solid metal, in square inohes.
1.67 I 1.77 I 1.96 I 2.16 | 2.55 | 2.95 | 3.84 | 8.78 I 4.13 | 4.61

1.38 I

8TREKQTH OP IRON PILLARS.

911

KOIXOW CrCXUTBRICAI. WBOUOST IBOH PII^ULBS.

Table 4,

(Continned.) (Original.)

3

WBOUQHT IBOir. THICKNESS H INCH.

«

%i

Outer diameter in inohea.

u

s

6

1 6H

1 «

1 «»

1 7 1 7H 1 8 1 8X 1 0 1

10 1

11

19

J

BKBAKIlfO LOAD.

Tom.

Tons.

To*!.

Tens.

Tom.

Toni.

Tom.

Tom.

Ton*.

Tons.

Tons.

T«ni.

i

llf

^^

ISO

1S9

166

in

189
186

901

914

988

268

990

2

t

11)

itt

180

140

168

174

199

913

987

181

988

4

6

10 i

110

188

146

158

171

164

197

910

986

161

988

6

8

10

114

In

140

164

167

101

194

$

983

958

984

8

10

9

^

188

180

140

169

170

100

998

964

280

10

la

M

110

190

148

1X7

171

iS

199

994

HO

976

12

14

m

16

100

t99

18T

161

106

194

919

m

972

14

16

n

m

108

117

lli

146

|M

178

187

213

240

968

10

18

n

as

97

110

1S8

153

m

160

907

988

968

10

3

M

3

91,

•8

iil

181

146

169

178

201

927

967

90

58

88

198

188

Ifil

166

193

890

960 •

91

9$

ft

64

70

80

'S

116

19

148

167

188

919

941

96

9f

4

B»

61

74

100

118

19T

141

107

196

934

80

8i

8

m

68

04

76

87

80

IM

196

161

118

107

36

48

i

88

44

88

64

76

86

96

no

186

)«6

190

40

4ft

St

80

98

46

66

66

.76

87

9B

198

I^

174

46

M

j§

84

81

S

60

06

76

87

109

186

168

60

OT

\^

n

94

89

48

61

60

60

88

108

139

00

7^

11

u

10

fl

98

84

40

48

66

IS

91

Ul

70

8t

0

u

14

10

99

87

89

87

44

67

g

93

80

90

T

•

U

14

18

89

91

91

86

48

78

80

100

0

1 T

t

19

U

18

99

96

80

41

68

66

100

Wallrlit of one foot of lenctb of pdllar. in pounds.
98.0 I 90.91 98.01 81.4 1 84.01 96.0 | 89.8 | 42.0 | 44.7 | 49.7 | 66.0

I 00.8

▲veft Of rins of Mild netal, in sauaro inohoa.
iJtn \ 7.861 8.04 I 9.48 1 10.9 1 11.0 | 11.8 | 19.6 | 18.4 | 14.9 16.6 | 10.1

TOible 4« (GontinQed.) (Original.)

s

WBOVGHT XHON. THIOKXniBB X INOB.

a

!i

Outer diameter tn inohea.

V

J

It

1 1*

1 w

1 10 1 17 1 18 1 90 1

91 1

94 1 96 1

98 1 80

BBtADNO LOAD.

Tmi.

Tons.

Tons.

Tom.

Tons.

Tons.

Tons.

Tons.

Tons.

Tons.

Tom.

Tons.

1

000

068

704

788

806

864

966

1066

1167
n40

1257

1867

1468

I

10

688

088

601

749

796

846

949

1040

1248

1854

1457

10

8

MS
479

SI

061
804

!B

769
699

810
ISO

913
060

1010
973

1190
1077

1238

noi

1827
1989

1480

\1&

20
80

40

416

470

698

684

636

•M

600

919

ion

1130

1387

4ft

60

866

406

489

616

670

027

740

848

961

1007

1179

1294

50

«

f&

400

469

606

660

668

781

881

1006

1115

1228

60

3M

848

698

448

480

600

716

834

080

1046

1100

•»

80

su

S6

988

344

892

440

643

649

757

868

978

1092

80

90

186

992

961

308

347

899

488

590

694

800

910

1023

90

100

167

%

226

269

803

346

4)6

532

631

786

843

955

100

110

184

\&

927

384

809

880

474

666

ffi

770

%

110

116

lit

136

192

226

359

330

416

606

607

126

160

88

101

122

146

171

198

968

328

405

486

574

666

150

176

09

78

96

113

183

155

208

266

831

400

478

600

176

100

4»

00

74

89

106

124

168

216

969

828

896

407

900

196

' Weight of one foot of lenirtli of pillar, in pounds.

I 130 I 147 I 157 I 168 I 178 | 190 | 290 | 241 | 262 | 283 | 304

Area of ring of aolid metal, in aquare inohee.
87.7 I 40.01 44.0 \ 47.1 | 60.8 | 58.4 \ 69.7 | 66.0 1 79.8 1 78.6 I 84.8 | 91.1

The breaking IoomIs Ibr lens tblcfcneraes may safely be Msoined i*

Olminish at the same rate as the thiokness.

912

8TBENOTH OF IBON PILLAB&

Table of approximate averaffe ultimate loada In Iba per
•qaare Ineb, as found by experiment with ear^uUy prepared specimens. In

Sractlce, allowance must be made for the loagher ctiaracter of actual work, for
arrings etc etc.

1

Length -»• least ndins of
' gyration.

•

Pencoyd Angles, Tees, I beams and Channels.*

Phoenix
columna.t

ast radius of
.ion.

Steel.

Iron.

Iron.

Hard; M
permt
oarbon

Mild;. 13
p«r eent
earbon

Flat
ends

Flat
ends

Fixed
ends

Flat
end*

Hinsed
ends

Bound
ends

Flat
ends

9

17
20

ao

40

60

60

70

80

90

100

120

140

160

200

800

••••••••

100000
74000
62000
60000
68000
65600
68000
49700
46500
40000
33500
28000
19000
8500

••••••••

70000
61000
46000
44000
42000
40000
38000
86000
34000
80000
26000
22000
14800
7200

46000
43000
40000
88000
36000
34000
32000
31000
30000
28000
25500
23000
17500
9000

•••••••a

••a«»««a

46000
48000
40000
38000
36000
34000
32000
30900
29800
26800
28500
20000
14500
7200

•a ■•••«•
••••••a*

46000
48000
40000
88000
86000
83750
31500
29750
28000
24800
21000
16500
10800
6000

*

44000

40250

86600

83600

80600

27750

25000

22750

20500

16600

12800

9500

6000

2800

67200
60400
4S000
40000
87000
87000
87000
87000
86000
85000
35000
84600

....•«.•
-

s

17

20

SO

40

50

60

70

80

90

100

120

140

160

200

aoo

The following simple formula, bj Mr. D. J. Whittemore, was found tl
Tory cloeely with the results of the experiments on Phoenix coluninatf

Breaking load in lbs

per sq inch of area

of cross section of pillar

..^ -^ , 625000
[(1200-H)X«0]+2^

where H

length of pillar
diam D, flg p 904

both In the same unit.

Mr. Christie* adopts the following formula for obtaining the proper
af safety for pillars of wrought iron or steel:

For flat and Axed ends, Faetor of safety i- 8 + ( .01

length

)

least rad of gyr
For hinged and round ends, Factor of safety — 8 -f- (.015 s — . f ^ 1

It will be noticed that the factor of safety, as found by these formula, in-

. . . - - - " gji

of its cross section ; and is greater for round and hinged ends than for flat and

retY
creases with the ratio of the length of the pillar to the least radius of
of its crosa
fixed ends.

rration

• 8«e " Wrovgbt Iron and Stael In Gontlrnetlon ", b.r Peoeoyd Iron Warka ; pabUshad ^ Joka
WDar a Sons, New York, 1884.
t Saa Traniaetlona, Amarioan Sooiaty of Olrll Bnclnaan ; Jaa, Fab and Maroh 18SI.

STRENQTH OF IRON PILLARS.

913

Ultimate erlpplinir streng^ihs la lbs per sq inch of metal
section of the fonr wroniptat iron pillars below. These formulaa
sure deduced by Ghs. Shaler Smith, IVom many tests b;^ G. Bouscaren, C. E., of
Ijurge pillars of good American iron. The lower Table is an abridgment of the
iUI ones by C. L. Gates, C. E., in the Trans. Am. Soc. C. E., Oct., 1880.

_, length between end bearings ,...., j • * v. a

V. =» —^ — ; — 7-t: : — 5 — ^^ both in the same measure ; and is to be squared.

kasi dtameter d

For safety take from %to%, according to circumstances.

Flat onds.

One pin end-
Two pia ends.

42500

1 +

H2

4500
40000

1 +

H«

2250
36600

1 +

1500

,^V^f^

36500

1 +

H8

3750
36500

1 +

H«

2250
36500

1 +

1750

36500

1 +

Ha

2700
36500

1 +

H«

1500
36500

1 +

1200

mtlmate and safe loads in lbs per sq inch, of the above four pillars, with
flat ends, and equally loaded. Coef of Safet3r=>4 + .05H. By G.L Gates, C.E.

H.

A. Square Col.

B. Phoenix Col.

G. American Col.

D. Common Col.

nt.

Safe.

nt.

Safe.

Ult.

Safe.

Vlt.

Safe.

]«

87067

7822

40476

8521

34434

7249

33693

7093

16

S6876

7683

40212

8377

34167

7118

33339

6946

18

86470

7443

39645

8091

33597

6856

32589

6651

20

86024

7205

39030

7806

32982

6596

31790

6358

S2

85544

6970

38373

7524

32327

6338

30952

6069

26

84767

6622

37317

7110

31285

5959

29639

5646

m

33344

6063

35424

6440

29435

5352

27375

4977

96

31806

5531

33406

5810

27512

4789

25108

4367

40

SOldS

5033

31352

5226

25584

4264

22919

3820

45

28562

4570

29310

4690

23701

3792

20857

3337

BO

26932

4143

27321

4203

21900

3369

18952

2916

K

25333

8728

25415

3765

20203

3004

17214

2550

m

23787

8398

23611

3373

18621

2660

15643

2235

68

914

FLOOR SECTIONS.

PENCOTD FliOOR SECTIONS.

L ^ span, in feet.
C a- coefficient.
W = distributed load, in lbs, per foot of floor width.

w-5

Cormirated floorlnir* for bridges and buildings.

W — load producing fiber stress of 15,000 fba per square inch.

SECTION 210 M.
Dimensions in inches.

Thickness, inches.

Weight,

Web. Flange.

fi>B per sq fL

C

A i

14.8

44,000

if A

18.4

55,000

A 1

21.9

66,000

H A

25.5

77,400

1 i

29.1

88,800

SECTION 260 M.
Dimensions in inches.

Thickness, inches.
Web. Flange.

i i to|

A i tof

I I toi

Weight,
S»s per sq ft.

20.0 to 30.7

26.5 to 37.2

29.4 to 40.1

105,000 to 186,000
143,000 to 224,000
153,000 to 237,000

Z Bar FloorlniT*

W =* safe load.

^

■xr

f^^r^.

r^ ''^^^

■tr

T

Section
No.

Dimensions, Thickness, ins.
in inches. Z bars. Plates.

Weight,
fcs per sq ft.

1

A B C I>( i )

15 6 9 4JAj- itoi

r 25.9 to 36.1

■I 29.1 to 39.3

( 32.3 to 42.5

93,400
104,000
114,400

2

18 8 10 ^\ \\ A to A

( 32.1 to 42.3

^ 35.2 to 45.4

( 38.4 to 48.6

143,000
155,000
166,400

8

21 9 12 6 -1 /a I 1 to 1

1 39.3 to 49.5

^ 42.4 to 52.6

(45.6 to 56.7

203,400
217,400
231,000

to 147,400
to 157.000
to 167,000

to 209,400
to 221,400
to 283,000

to 281,000
to 294,000
to 907,200

CHAINS.

915

W£IUiIT AJTB STRENOTja OF IRON CHAINS.
Table of strength of chains.

Chaing of aaperior iron wiU require ^ to J^ more to break them. (Original.)

Diam of rod
of which
the links
are made.

Ins.

3-16

K
^6

13-16

Weight

of chain

per ft run.

Pds.

.5

.8
1.
1.7
2.
2.5
3.2
4.3
6.
6.8
6.7
8.
9.

Breaking strain
of the chain.

Pds.

1731

3069

47M

6.922

9408

1*2320

15590

19219

23274

27687

32301

37632

43277

Tons.

.773

1.37

2.14

3.09

4.20

5.50

6.96

8.58

10.39

12.36

14.42

16.80

19.32

Diam of rod
of which
the links
are made.

Ins.

Weight

of chain,

per ft run.

Pds.

10.7
12.5
16.
18.3
21.7
26.
28.
32.
38.
54.
71.
88.
105.

Breaking strain
of the chain.

Pds.

49280

59226

73114

88301

105280

123514

143293

164505

187152

224448

277088

aS5328

398944

Tons.

22.00
26.44
32.64
39.42
47.00
55.14
63 97
73.44
83.55
100.2
123.7
149.7
178.1

The links of ordinary iron chains are usually made as short as is consistent
with easy play, in order that they may not become bent when wound around
drums, sheaves, Ac; and that they may be more easily handled in slinging
large blocks of stone, Ac. U. S. Government experiments, 1878, prove that
studs weaken the links.

When so made, their weight per foot run is quite approximately 3>^ times that
of a single bar of the round iron of which they are composed. Since each link
oODSlsts of two thicknesses of bar, it might be supposed that a chain would
possess about double the strength of a single bar : but the strength of the bar
becomes reduced about 30 per cent, by being formea into links ; so that the chain
has but about 70 per cent of the strength of two bars. As a thick bar will not
sustain as heavy a unit stress as a thinner one, so of course, stout chains are
proportionally weaker than slighter ones. In the foregoing table, 20 tons
per square inch. Is assumed as the aveorage breaking strain of a single straight
bar of ordinary rolled iron, 1 inch in diameter or 1 inch square; 19 tons, from
1 to 2 inches : and 18 tons, from 2 to 8 inches. Deducting 30 per cent from each,
we have as the breaking strain of the two bars composing each link, as follows :
14 tons per square inch, up to 1 inch diameter ; 13.3 tons, from 1 to 2 inches: and
12.6 tons, from 2 to 3 inches diameter ; and upon these assumptions the table is
based. The weights are approximate ; depending upon the exactness of diameter
of the iron, and shape of link.

916

TIN AND ZINC.

i

TIJT AITD ZINC.

The pare metal is called block tin. When perfectly pnre, (whicb It
rarely is, being purposely adulterated, frequently to a large proportion, with the
cheaper metals lead or zinc,) its sp grav is < .29 ; and its weight per cub ft is 4S5 lbs.
It is sufficiently malleable to be beaten into tin foil, only y^Vo* ^' '^ inch thick.
Its tensile strength is but about 4600 B>8 per sq inch.; or about 700O fi»fl when made
into wire. It melts at the moderate temperature of 442*^ Fah. Pure block tin ii
not used for common building purposes ; but thin plates of sheet iron, covered wjtii
it on both sides, constitute the tinned plates^ or, as they are called, the tiny used for
covering roofis, rain pipes, and many domestic utenaili. For roofs it is laid on boards.

Tbe sbeeli
of tin «%re ami*
ted as shown in
this fig. First, se?'
eral sheets an
joined together in
the shop, end for
end, as at tt; hj
being first bent
oyer, then hask-
mered flat,and th«B
soldered. These an
then formed into s
roll to be carried
to the roof; a roll

being long enough to reach from the peak to the eaves. Different rolls being spread
up and down the roof, are then united along their sides by simply being bent as at a
and «, by a tool for that purpose. The roofers call the bending at « a cUntble grotnt^
or double lock ; and the more simple ones at f, a single groove^ or lock.

To hold the tin securely to the sheeting boards, pieces of the tin 3 or 4 ins long,
by 2 ins wide, called cleats, are nailed to the boards at about every 18 ins along th*
Joints of the rolls that are to be united, and are bent over with the double g^roove i.
This will be understood from y, where the middle piece is the cleat, before being
bent over. The nails should be 4-penny slating nails, which have broader heads
than common ones. As they are not exposed to the weather, they may be of plain iicOi
Mnch use is made of what is called leaded tin^ or temes« for roofing. It ii
limply sheet-iron coated with lead, instead of the more costly metal tin. It is not
as durable as the tinned sheets, but is somewhat cheaper.

The best plates, both for tinning and for ternes, are made of charcoal iron ; which,
being tough, bears bending better. Coke is used for cheuper plates, but inferior ai
regards bending. In giving orders, it is important to specify whether charcoal
plates or coke ones are required ; also whether tinned plates, or ternes.

Tinned and leaded sheets of Bessemer and other cheap steely are now moch osed.
They are sold at about the price of charcoal tin and terne plates.

There are also in use for roofing, certain compound metals which resist tamisk
better than either lead, tin, or zinc ; but which are so fusible as to be liable to be
melted by large burning cinders falling on the roof from a neighboring conflagration.
A roof covered with tin or other metal should, if possible, slope not much less than
five degrees, or about an inch to a foot ; and at the eaves there shonld be a sudden
fall into the rain-gutter, to prevent rain from backing up so as to overtop the double-
groove Joint «, and thus cause leaks.' Where coal is used for fuel, tin roofs shonld.
receive two coats of paint when first put up, and a coat at every 2 or 3 years after.
Where wood only is used, this is not necessary ; and a tin roof, with a good pitch,
will last 20 or 30 years.

Two good workmen can put on, and paint outside, from 260 to 300 sq ft of tin roof,
per day of 8 hours.

Tinned iron plates are sold by the box. f heee boxes, unlike glass, have noi equal
areas of contents. They may be designated or ordered either bj their names or
sizes. Many makers, however, have their private brands in addition; and some of
these have a much higher reputation than others.

TIN AND ZINC.

Tabic of Tinned and Terne Plat* a.

Cnotlon.— BoiM oftan oDDttln muldcrmblT Ihi wdght of Hn plat* tlwB tta
labia nqnlm; Iba plMs being loUad Uiiu wd plstod tbln, Id oidsr to eubW
Wnntiinin to gat jaj loi mora ■Mtutal Una tfaaj ntimlah.

...

..

w"V

Hirk.

n,lTT

...

„.'S,.

lUI

,•1

1

L

1

!no

i:»

,2=.

KMt.

li!

it

,sa.

....

ail

ai».

.„.

jiia

"ii"

1

h

I

»»

1

1

S5S

HX»

"5"

ItXW

!"

I

1

i

i

1

IK

ilS!?

"

Mi

i

™'f'

Te

f

7

if

.on Df u Uuk Uilu, DF l.n aa p« h ft ; No it = .OU liHk, tut IM Di ; Mo IS ^ ^1 lniD. ud
l.OBi: Nolt^ MilDSIi. udl.tiai par*] ft. Asjot IbHasuBban mil la nxd on non, ta
wbSoi pptpu* H ■bgatd ba nrj tMr«.
Water k«pt In alne TcneU Is •■!£ to b«Dome Injurloiu to health ; sod

uj laUurtoaa tiaoU. TkU la paaalUjowlDC te (be IHb thu aanlB>-plp« belu abort. Lbs wm«r

WEIGHT OF HETALB.

RMf copper la nauallj In ■beets of 2<j feet x B feet ; or r^>^ square teet,

lolnta am feriaed bT only OTerUppiag mid bendlPi; Die abnU. mucb u ihowf
bj the figs on pHus Sift; except Ibst tbs horiuulil joiDln ara bent u locked

Slieet leiMl. List of altuidu^ welgrlllB Id lbs per squiira foot. Thick'
ueeaea tn decinuiU t>t ma iDcb.

WelKbt or MeMI 1

W ^ Weigbt of bill), III pclUlla^ D

= IXameie.

of ball, iu inehe.

Lead =(-O0lbapercubf() W =

.212106 D>

i,«»=T.m

Copper^ (.ioOtbsperoubft) W =

,1S6CMI«

lo»W- 1.221

Brus =(6001baper™bft| W-

,1515MI>»

1«KW = 1.180

fK|\.S!} = !»•-„,,=

.146959 D"

10|:W = U67

SSo } - (460 [b» per cub ft) W -

.139354 D9

i«gir=~i.i34

'4T31«gD
For Hteel, Hronybt Iron and enst Iron ballB, see also isble^ ft

>i L

■".*■ "^".iT'

^F

i

? i-i

^Ffs

;).

-H

- I'-

IS

7s

Si

zi^l

s^

gS

7|i!i-f

t

: ■

■0 ?;

ij..

1

ISI^

!■>.

■ii

:«

'"■

L^! !!

;:

lis:

l^t'r

■>

-f.

*,

'S

.T,

U

'!♦

Ii

::

**»;?

Vtll

:;

A

]|

f.

mISTL'S-

SW^XK-

^^l^^,%7

nt

upow

but

cms

mm

• held by copper cle«ta; i

METALS.

919

^

ROIiI^ED liEAD, COPPER

, and BRASS : Sheets and Bars.

Thieknesa
or

LB AD.

COFPXB.

BRASS.

Tbicknesf
or

JMftmeter,

Diameter.

or lids,

Sheeta,

Square

Bound

Sheets,

Square

Bound

Sheets,

Square

Bound

or side,

- in

per

Bars;

Bars;

per

Bars;

Bars;

per

Bars;

Bars;

in

InehM.

Sqnare

IFoot

1 Foot

Square

IFoot

IFoot

Square

IFoot

IFoot

Inches.

Foot.

long.

long.

Foot.

long.

long.

Foot.

lotig.

long.

Lbs.

Lbs.

Lbs.

Lbs.

Lbs.

Lbs.

Lbs.

Lbs.

Lbs.

1-32

1.86

.005

.004

1.44

j004

.003

1.36

.004

.003

1.82

M6

8.72

.019

.015

2.89

.016

.012

3.71

.014

.Oil

1-16

333

5.38

.044

.034

4.33

.034

.027

4.06

.033

.025

S-33

H

7.44

.078

.061

5.77

.060

.047

6.43

.066

.044

H

633

9.30

.121

.095

7.20

.094

. .074

6.75

.088

.069

5-32

S-16

11.2

.174

.137

8.66

.136

.106

8.18

.127

.100

8-16

7-32

13.0

.237

.187

10.1

.184

.144

9.50

.178

.136

7-82

H

14.9

.810

.244

11.5

.240

.189

10.8

.336

.177

S?16

5-16

18.6

.485

.381

14.4

.376

.295

13.5

Ji53

.277

?.i.

22.3

•698

.548

17.3

.541

.425

16.8

.508

.399

?-l.

26.0

.950

.746

20.3

.736

.578

19.0

.691

.543

H

29.8

1.24

.974

28,1

.962

.755

21.7

.90S

.709

H

916

33.5

1.57

1.23

26.0

1.22

.955

24.3

1.14

.900

9-16

H

87.2

1.94

1.53

28.9

1.50

1.18

27.1

1.41

I.ll

H

11-16

40.9

2.84

1.84

31.7

1.82

1.43

29.8

1.70

1.34

11-16

H

44.6

2.79

3.19

34.6

a.16

1.70

32.5

3.08

1.60

«

13-16

48.8

8.27

8.57

37.5

2.55

1.99

35.3

3.38

1.87

18-16

K

52.1

8.80

2.98

40.4

2.94

2.31

37.9

2.76

2.17

H

15-16

66.0

4.87

3.42

43.3

3.38

2.65

40.6

8.18

2.49

15-16

1.

69.5

4.96

3.90

46.2

3.85

3.02

43.8

8.61

2.84

1.

13<

66.9

6.27

4.92

52.0

4.87

3.82

48.7

4.57

8.60

IH

IH

74.4

7.75

6.09

57.7

6.01

4.72

54.3

6.64

4.48

iH

IM

81.8

9.37

7.37

63.5

7.28

5.72

59.6

6.83

5.37

19i

m

89.3

11.3

8.77

69.3

8.65

6.80

65.0

8.12

6.38

IH

1«

96.7

13.1

10.8

75.1

10.2

7.98

70.4

9.53

7.49

IK

1«

104.

15.2

11.9

80.8

11.8

9.25

75.9

ll.l

8.68

1«

IH

112.

17.5

18.7

86.6

13.5

10.6

81.3

12.7

9.97

IH

t-

119.

19.8

15.6

92.8

15.4

12.1

86.7

14.4

11.8

2.

Seamless brass tabes. Principal sizes. Extras, in cents per pound,
#ver base price. For base price, see price list.
Copper tabes, 8 cents per pound extra.

Thickness.

Outer Diameter, inches.

Stubs
gage.

Ins.

H

12
«

13
15
18
21
28

1

IK

2

1
<(

4

6

9

13

3

4

5

6

7

T%

4
11
16
18
20
22

0.238

0.120

0.065

0.049

0.035

0.0295

0.0230

••••■•••

••■••■• e

40
43
50
65

6

it

8

9

13

16

22

3

K

4

6

10

15

24

1
*(

4

7
11
16

2

i*

7
11
15
20

5

«

11
15
19

9

((

15
19
23

13

u

19
23

27

18
tt

24
28
82

25

920

METAU9.

Awenkge ultimate tensile strengtli of BEetals.

The ultimate tensile or pulling loiid per square inch of any
material is frequently called its constant, coefficient, or modultu of
tensioB, or of tensile strength.

Antimony, cast

Bismuth, cast

Brass, caat 8 to 13 tons, say 18000 to 29000 Ihi

* wirei unannealed or hard, 80000. Annealed..
Broiue, phosphor wire, hard, 160000. Annealed....

Ck)pper, cast 18000 to 80000
'^ sheet.

i(

<i

bolts, 28000 to 88000.

wire (annealed 16 tons); unannealed.

Gold, cast

" wire, 2SO0O to 30000.......

Gun metal of copper and tin, 28000 to 66000

'* '* caat iron, U. S. ordnance, 36000 to 40000

Iron, cast, English ....18400 to 22400.

" " ordinary pig..l3000 to 16000.

American cast iron aTorages one-fourth more than the above.

Ayerage cast iron, when sound, stretches about .00018 ; or 1 part
in 6666 of its length : or }4 inch in 67.9 ft. for erery ton of ten-
sile strain per sq inch, up to its elastic limit, which is at about
^ its break-strain. The extent of stretching, howeyer, varies
much with the quality of the iron ; as in wrought-iron.

CSast, malleable, annealed 18 to 25 ton&

Iron and Steel, rolled.— See Digests of Specifications.

Lead, cast, 1700 to 2400 by author..

" wire, 1200 to 1600. Pipe 1600 to 1700 "

Platinum wire, annealed, 32000. Unannealed

Steel and Iron, rolled. — iSee Digests of Specifications.

Silver, cast..

Tin, English block

'• wire

Zinc, ca8t...3000 to 8700; (the last by author)

Pounds

Tods

per

per

sq. inch.

aq. la

1000

.45

8200

L4

23500

10.5

49000

22

63000

28.1

24000

10.7

30000

lU

83000

147

60000

26£

20000

8.9

27500

12i

39000

17.4

88000

17

17900

8

14600

6.17

u

48160

206O

1660

6600O

4100O
4600
7000
8360

2L5

a74

25

las

2.0
8.1

IJ

lArae bars of metal bear less per sa inch than small ones.
Iron l»ars re-rolled cold have tensile strength Increased 25 to 50pc(
ct, with no increase of density. They are said to lose this strength if reheated.

^

METALS.

921

Sheet lead is sometimes placed In tlie Joints of stone col-
umns, with a view to equalize the pressure, and thus increase the strength of
the column. But experiments have proved that the effect is directly the reverse,
and that the column is materially weakened thereby.

ATcraiT® crnshins load for Metals.

It must be remembered that these are the loads for pieces but two or three
times their least side in height. As the height increases, the crushing load
diminishes. See "Strength of Pillars."

Cast Iron, usually

It is usually assumed at 100000 lbs, or say 45 tons p«r sq inch. Its
eroshiDg strength is usoallv from 6 to 7 times aa great as its tensile.
Within its average elastic limit of about 15 tons per sq inob, average
oast iron shortens about 1 part in 5555 : or % inch in 58 ft under each
ton per sq inch of load ; or about twice as much as average wrought
iron. Hence at 15 tons per sq inch it will shorten about 1 part in 370;
or ftiU % inch in 4 feet. Different oast irons may however vary 10 to
15 per ct either way from this.

v. S. Ordnance, or gun metal : Some

IFrouKbt iron, within elastic limit

Its elas& limit under pressure averages about 18 tons per sq inch.
It begins to shorten perceptibly under 8 to 10 tons, but recovers when
the load is removed. With tnm 18 to 30 tons, it shortens permonsntiy,

about i^th part of its length ; and with from 27 t5 30 tons, about -j^^b
part, as averages. The crushing weights therefore in the table are
not those which absolutely masb wrought iron entirely out of shape,
bat merely those at which it yields too much for most practical build-
ing purposes. About 4 tons per sq inch is considered its average safe
load, in pieces not more than 10 dlams long ; and will shorten it % inch
in 30 ft. average.

Brass, reduced ^^th part in length, by 51000 ; and 3^ by

Copper, (cast,) crumbles

(wrought) reduced M^ pvt in length, by

Tin, (cast,) reduced J^th in length, by 8800; and % by

liCad, (cast.) reduced U of its length, by 7000 to 7700....

" By writer. A piece finch sq, 2 ins high , at 1200 lbs the com-
pression was 1-200 of the ht ^ at 2000, 1-29; at 3000, 1-8; at
6000, 1-3 ; at 7000, 1-2 of the ht.

Spelter or Zinc, (cast.) By writer. A piece 1 inch

square, 4 ins high, at 2000 lbs was oompresned 1-400 of its ht ; at 4000,

1-200 ; at 6000, l-lOO; at 10000, 1-38 ; at 20000, 1-15 ; at 40000 yielded

rapidly, and broke into pieces.

Steel, 224000 lbs or 100 tons shorten it ttom .2 to .4 part.

" American. Black Diamond steel- works, Pittsburg, Penn.

experiments by Lieut W. H. Shock. U. S. N., on pieces ^ in

square; and 3^ ins, or 7*8ides long.

" Untempered. 100100 to 104000

" Heated to light cherrv red, then plunged into oil of 82^ Fab,

173200 to 199200

" Heated to light cherrv red, then plungnA into water of 79°

Fah: then tempered on a heated plate, 325400 to 340800....

'* Heated to light cherry red, then plunged into water of 79°

Fah, 275000 to 400000

Elastic limit, 15 to 27 tons

Compression, within elas limit averages abt

I part in 13300, or . 1 of an inch in 1 11 ft per ton per sq inch ;
or .1 of an inch in 5.3 ft under 21 tons per sq inch.

Best Steel Isnife edgres, of large B R weigh scales

are considered safe with 7000 lbs pres per uneal inch of edge ; and

solid cylindrical steel roJlers under bridges, and

rotttng on tUO, safe with V'diam in ins X 3 100 000, in lbs per lineal
inch of roller parallel to axis. And per the ftaBie« for

Pounds per
sq. inch.

85000 to 125000

175000

22400 to 85840
29120

Solid east iron wheeLi rolling on wrought iron, )/Diam ins X 852 000.

Solid tteel

*' cast iron, y'Diam ins X 222 222.
" steel, V^Diam ins X 1300000.

.165000.
.117000.
.103000.
...15600.
....7350.

.102050.
.186200.
.333100.

.337800.
..47040.

Tons per
sq. inch.

38 to 5«

78.1

10 to IC

18

78.6
52.2
46.0
6.92
3.28

46.6

83.1

148.7

150.8
21

" wronghtiron, |/Diamins X 1024000.

" " " " " cast iron, )/Diam ins X 850 000.

From " Speoifleations for Iron Drawbridge at Milwaukee," by Don J. Wbittemore, 0. K.

922

STONE, ETC.

Average ultimate tensile strenctlis of Stone, cte.

The ttrengtbA in all these
tablMmay readilv be one-third

Pounds

Tons

Pounds

Tons

part more or less than our

per

per

per

per

averages.

sq. inch.

sq. ft.

MarbIe,strong,wh.Ital J .*
" Champlain,Tarie-

sq. inch.

sq. ft

Brick. 40 to 400

220

14.1

1034

665

Caen stone, 100 to 200

150

9.7

gated*

1666

lOTl

« Glenn's ril8,N.Y.

blk * 760tol034.

892

67.4

" Montg'y CO, Pa,

gray ♦

1176

75.6

" " white*...

734

47.2

*• Lee,Ma8s,white.*

875

56^

Cement and concrete,

" Manchester, Tt,*

see articles, Cement
and Concrete.

560 to 800

675

43.4

** Tennessee, varie-

gnted*

1034

66 J^

Oolites, 100 to 200

150
70

9.7

Plaster of Paris, well set.

4.6

Rope, Manilla, best

12000

771

Glass, 2500 to 9000

6750

869.6

" heniD. best

16000
106

966

Glue holds wood together

Sandstone. Ohio*

676

with from 300 to 800...

550

35

Pictoii, N. S.*

434

27.9

Horn, ox

9000

579

" Conn red ♦.

690

37.9
159.1

Ivory

16000

1029

Slate Lehieh *.

2475

Leatier belte, 1500 to

" Peach bot'm,* 3026

50(0. Good

8000

193

to 4600

3812

2461

Mortar, common, 6 mos

Stone, Ransome's artif....

300

19J

old, 10 to 20

15

.96

Whalebone

7600

489

* By tlie autlior'B trials with one of Kiehl6'8 testing machines.
brakenD^sqinokM.

fiectioas

STONE, ETC.

923

Ultimate averaye crnflhliiK loads in tons, per saaare
foot, for stones, Ac The stones are supposed to be on bed, and the neiehta
of all to be from 1.5 to 2 times the least side. Stones generally begin to crack or
split under about one-half of their crushing loads. In practice, neither stone nor
brickwork should be trusted with more than }^ to -j^th of the crushing load, ac-
oording to circumstances. Wben tborouKhly wet some absorbent sand-
stones lose fully half their strength.

Granites and Syenites.

Basalt

limestones and Mar-
bles*

Oolites, good

Brownstone :
Connecticut —

"Building"

"Bridge"

Tons per
sq. ft.

800 to 1200

Brick*

Brickwork, ordinary,
cracks with*

Brickwork, good, in ce-
ment*

Brickwork, first-rate,
in cement

Slate..

Caen Stone

" " to crack

Chalk, hard

Plaster of Paris, 1 day
old

Mean.
Tons.

250 to 1000
100 to 250

570 to 970
400 to 6H0

40 to 300

20 to 30

30 to 40

50 to 70

400 to 800

70 to 200

20 to 30

750

700

625
175

775
535

170

25

35

60

600

135

70

25

40

Tons per
sq. rt.

Mean.
Tons.

Cement, Portland,

neat,U. S. or foreign,

7 days in water 75 to 150 112.5

Common U.S. cements,

neat, 7 days in water 15 to 30 22.5
ConcreteofPort.

cement, sand, and

gravel or brok stone

in theproper propor-

tions,rammealmold 12 to 18 15

6 months old , 48 to 72 60

12monthsold i 74 to 120 97

With good common

hyfl cements,

abt .2 to .25 as much
Coigrnet beton, 3

months old 100 to 150 12C

Rubble masonry,

mortar, rough | 15 to 35 25

Glass, green,crowu and|

flint 11300to2300 180f

or 8 times that of granite
llce,flrmt I 12 to 18 | 15

Crnshinic bei^ht of Brick and Stone.

If we assume the wt of ordinary brickwork at 112 lbs per cnb ft, and that it would
crash under 30 tons per sq ft, then a rert uniform column of it 600 ft high, would
crush at its base, under its own wt. Caen stone, weighing 130 tbs per cnb ft, would
require a column 1376 ft high to crush it. Average sandstoues at 145 lbs per cub ft,
would require one 4158 ft high ; and average granites, at 165 lbs per cub ft, one
of 8145 feet. But stones begin to crack and splinter at about half their ultimate
crushing load; and in practice it is not considered expedient to trust them with more
than V^th to ^th part of it. especially in important works; inasmuch as settlements,
and imperfect workmanship, often cause undue strains to be thrown on certain
parts.

The Merchants' shot-tower at Baltimore is 246 ft high ; and its base sustains 6^4
tons per sq ft. The base of the granite pier of Saltash bridge, (by Brunei,) of solid
masonry to the height of 96 ft, and supporting the ends of two iron spans of 455 ft
each, sustains 9^ tons per sq ft. The base of a brick chimney at Glasgow, Scotland,
468 ft high, bears 9 tons per sq ft ; and Professor Rankine considers that in a high

gale of wind, its leeward side may have to bear 15 tons. The highest pier of Rocque*
kvonr stone aqueduct, Marseilles, is 305 ft, and sustains a pressure at base of IZ^
tons per sq ft.

* Trials at St. liOnis bridire, by order of Capt James B. Eads, G. E.,

Bhowed that some magueslan limestone did not yield under less than 11 00 tons per aq (t. A column
8 ins high, 9 ins diuin, ihortened 0.00*25 innh under pressure; and recovered vben relieved.

Sxperiments made with tbe OoTt testing machine at Water*
town. Mass, 1892-3^ gave 1400 tons per sq ft ultimate crushg load for whits
Mi4 Mae marble from Lee, Mass, 700 for blue marble from Ifontgomery Co, Pa, 960 for limestone froa
Conshohoeken, Pa, 500 for limestone from Indiana, 840 for red sandstone from Hummelatowa, Pa,
S80 to 1000 for yellow Ohio sandstone ; Phila bricks, flatwise ; hard, machine-made, 350 to 700 toiwt
hand-made, 700 to ISOO; pressed, machine-made, 450 to 580; Brickwork eolumns, IS ina iq and 18 IM
kif h ; in lime, 100 tons ; in cement, 150.

T Experiments by Col. Wm. Ludlow; U. 8. A., with Govt teetlng machin«a, in 1381, gave from SI
lo 84 tons per eq ft for pure, hard loe ; and 16 to 50 tons for Interior grades. Tlie speeuMns If asi
IS-iosh evbes) oompreusd H*ol Inoh before eruslilns.

924

BTONE BEAMS.

STOBTE BEAMS.

Table of safe qnlescent extraneous loads for beams of ffoocl
bulldins' frranite one inch broad, supported at both ends, and loaded at the
center; assnming the safe load to be one-tenth of the bi-etikiiig one; and the latter
to be 100 lbs for a beam 1 inch square, and 1 foot clear spun. The half weight of
the beams themselvefl is here already deducted at 170

lbs per cub ft.

•

a

OLEAB

SPANS nr

FEXT

•

1

3

3

4

5

■
6

7

8

10

13

16

90

^

Bate center loads in pounds.

10

5

40

30

18

10

90

45

38

31

17

160

78

63

88

81

36

31

350

134

83

61

48

40

34

seo

178

119

88

70

58

48

43

83

490

844

163

130

86

78

67

66

46

86

37

16

688

818

313

156

136

104

88

76

59

47

86

S3

999

488

831

846

187

163

138

130

94

76

68

88

1439

718

478

357

884

386

301

174

187

111

85

68

1959

878

660

487

888

823

274

288

186

153

118

81

3558

1378

860

686

507

431

858

813

346

301

157

109

8388

1818

1077

806

oa

534

455

886

818

357

900

141

»

8888

1886

1338

885

784

660

563

480

888

318

348

171

n

4888

3417

1608

1305

861

8G0

682

6M

470

887

803

S16

u

5T68

3877

1816

1484

1145

851

813

708

563

468

862

960

it

7388

8643

3425

1815

1450

1205

1030

886

713

588

462

8SS

so

8896

4496

3885

3343

1781

1489

1273

1110

883

728

578

415

38

10688

5441

8634

2714

8168

1803

1543

1345

1068

888

686

605

86

13868

6476

4814

8381

2581

2147

1886

1608

1376

1064

883

606

If nniformly distrlbated over tlie clear si»aii, the safe extranoooa
loads will be twice as ^reat as those in the table.
For g^ood slate on bed the safe loads may be taken at about 3 times; t^jt

Sood sandstone on bed at about one-half; aad for Hpood marble or
Imestone on bed at about the same as those in the table.

MORTAR. 925

MOBTAB, BBIOKS, &o.

I.IME HOBTAR.

Art. 1. MortAr* The proportion of 1 measure of quicklime, either in ir^
regular lumps, or ground, and 5 measures of sand, is about the average used for
common mortar, by good builders in our principal Atlantic cities; and if both
materials are good, and well mixed (or tempered) with clean water, the mortar is
certainly as good as can be desired for such ordinary purposes as require no addi-
tion of hydraulic cement. The bulk of the mixed mortar will usually exceed that
of the dry loose sand alone about ^ part.

Quantity reqntred. 20 cub ft, or 16 struck bushels of sand, and 4 cub ft, or

3.2 struck bushels of qolokliine, the measures slightly shaken in both oases, will make abt 22^ cub ft of
mortar; suflacient to laj 1000 brioks of the ordinary average sise of 8)4 by 4 by 2 ins, with the coarM
mortar joints usual in interior house-walls, rarying say from H to )i inch. With such joints, lOOi
■noh bricks make 2 cubic yards of massive work. Nearly one-third of the mass is mortar. For
outside or showing joints, where a whiter and neater looking mortar is required, house.builders in-
crease the proportion of limv to 1 in 4, or 1 in S. For mortar of fine screened gravel, for oellar-walla
of stone rubble, c coarse brickwork, 1 measure of lime to 6 or 8 of gravel, is usual ; and (he mortar
is good. In STsrage rough massive rubble, as in the foregoing brickwork, about one- third the mass is
mortar: consequently a cubic yard will require about as much as aOO such bricks ; or 10 cubic feet. (8
■truck bushels) of sand ; and 2 cub ft, or 1.6 bushels of quicklime. Superior, well-scabbled rubble,

eareftally laid, will contain but aboat -^ of its bulk of mortar ; or 5H oub ft sand, and 1.1 cub ft lime,
per cub yard.

For public engineering works, especially in maasWe ones, or where exposed to dampness, an addi-
tion should be made in either of the foregoing mortars, of a quautitv of good liyd
cement, equal t^ about % of the lime; or still better, % of the lime shoula be
omitted, and an equal measure of cement be substituted for it. If exposed to water while
qvlte new« QM little or db lime outside.

With bricks of 8^ by 4 by 2 ins, the following are the qaaiitlties of mor-
tar mud of bricks for a cubic yard of massive work.

Thickness Proportion of Mortar No. of Bricks No. of Bricks

of Joints. in the whole mass. per ouS yard. per cub foot.

■i-inch about ^ 638 23.63

1 << <i 1

? ¥

fii II 3
TIF

J " " ^ 475 17.60

I " •• y^ 433 16.04

In estimating for biricks in massive work, allow 2 or 3 per ct for waste ;

and in common buildings, 3 per ct. or more. Much of the waste is incurred in cutting bricks to fit
angles, ko. In Philadelphia a barrel of lump lime is allowed for 1000 bricks ; or for 2 perches (25 cub
ft each) of rough cellar-wall rubble. Somewhat less mortar per 1000 is contained in thin walls, than
in masslTC engineering structures ; because the former have proportionally more outside face, which
does not require to be covered with mortar; but thin walls involve more waste while building; so that
both require about the same quantity of materials to be provided. Careful experiments show that
mortar becomes harder, and more adhesive to brick or stone, if the proportion of lime is increased.
Hence, on our public works the proportion of one measure of quicklime to S of sand, is usually spec-
ified, but probably never used.

liime is usually sold in lump, by the barrel, of about 230 lbs net,
or 250 0>s gross. A heaped bushel of lump lime averages about 75 lbs. Orovnd qnlckltme,

loose, averages about 70 lbs per struck bushel ; and 3 bushels loose just fill a common flour barrel ; but
f^om 8.5 to 3.75 bushels, or 245 to 280 lbs can readily be compacted into a barrel.

Oeneral remarks on mortar and lime. On too great a pro-
portion of our pablio works, the common lime mortar may be seen to be rotten and useless, where it
has been exposed to moisture ; which will be carried by the capillary action of earth to several feet
above the natural surface; or as far below the artificial surface of embankments deposited behind
abutments, retaining-walls, to. The same will frequently be seen in the sofllts of arches under em-
bankments. Common lime mortar, thus exposed to constant moisture, wiil never harden properly.
Even when very old and hard, it absorbs water freely. CevMnt also does so, hut hardens.

Bricfcdust, or burnt clay, improves common mortar ; and makes it hvdraulic.
In localities where sand cannot be obtained, burnt clay, ground, may be substituted ; and will gen*
erally give a better mortar.

Protection of quicklime from moisture, even that of the air, is
absolutely essential, otherwise it undergoes the process of air-slacking:, or

574 21.26

522 19.33

926

MORTAR.

■poDtaneoos alaokiog, by which it beoomes redaoed to powder as wheo slscked bj water aa asa«k
bint without heatiDg, and with but little awelliDg. Aa thta air elaokiog requires from a few moDtha ta
a year or more, depending on quality and ezpoaure, it givea the lime time to absorb anflloient carboaie

acid ft'om the air to injure or destroy its efficacy. But qaicklime irill keep

good for a lonv time if first ground, and then well packed in air-tight
.rrela. The grinding aUo breaks down refractory particles found in all limea, and which Iqjure the
Mortar by not slaoking uniii it baa been made and uaed. For the aame reamo it is better that lime
■hoaUl not be made into mortar aa aoon aa it is aiaoked, bat be allowed to remain alaeked for a day or
two (or even sereral) protected from rain, sun, and dust.

Ume slaeked iu grent bulk may char or eren set fire to wood.

Uine paste and mortitr will keep A»r yearn, and improve, if well
buried in the earth. Alao Ibr months if merely ooTered In heapa under shelter, with a.thiek lajer of
■and. The paste ahrinks and oraoka In drying ; but the aand In mortar prevents this.

As approximate averages varying much according to the character and

degree of burning of the limeatone : ana to the fineneaa or coarseness of tbe sand, one meaasre of
food qniokllme, either in lump, or ground ; if wet with about H a measure of water, will within leaa
than an hour, slack to about 2 measures of di7 powder. Ami if to this powder there be added about
9i mnre measures of water, and 3 measures of ary sand, and the whole thoronghly mixed, the reanU
will be about 3>^ meaaurea of mortar. Or the same slacked drr powder, with about 1 measore of
water, and 5 meaaurea of sand, will make about 5^ measures of 'mortar. In both cases the bulk of
the mortar will be about H P^^rt greater than that of the drv SHud alone. If ^ of a measure of water
be uaed for alaoking, the reault, instead of a dry powder, will be about 1^ measures of atiff paste ; or
with I whole meaaure of water for slacking, the result will be about IH[ measures of thin paste, of
altout the proper consistence for mixing with the sand. Very pure, fat limes, alack qoicklr, and make
about from 2 to 3 meaaurea of powder ; while poor, meagre ones, require more time, and swell less
Slow slacking, and small swellug, in case the lime haa been properly burnt, are not in general bad
propertiea ; but on the contrary, naually indicate that it ia to stmie extent hydraulic. In this case it
makes a better morter : especially for works expoaed to moiatore, or to the weather. Very pare limes
•re the worst of all for such exposares; or are bad weather-Umit i and in important works, shoald
never be used without cement.

Shell lime appears to be about the same as that f^om the purest limestones;

but that from chalk is still more inferior, and wilt not bear more than about 1 H measures of aand;
its mortar never becomes very hard. Madrepores (commonly called ooral) appear to foraish a UsM
intermediate between those of chalk and limestone- They require to be but moderately barnu

Tlie averag^e weig^lit of common hardened mortar is about 105 to 115 Ss
per cub ft. ^

Oront is merely common mortar made so thin as to flow almost like cream.

It Ih intended to (111 interstices left in the mortar-JointM of rough masonry ; but unless it contains a
large amount of cement, it is probably entirely worthless ; since tbe great quantity of water injurea
the propertiea of lime; and moreover, Ita ingredients aeparate from each other; tbe aand settling be-
low the lime. Besides this, it will never harden thoroughly in the interior of thick masses of ma-
sonry ; indeed, the same may probably be said of any common lime mortar. In such positions, it hsa
been found to be perfectly soft, after tbe lapse of many years.

Both the sand and the water for lime mortar, should be firee from elajr and

salt. The clay may be removed by thorough washing; but it is extremely dif-
flcnlt to get rid »f the salt fhim seashore sand, even by repeated washings. Enough will generally
remain to keep the work damp, and to produce efflorescences of nitre on the surface,* whether with
lime, or with cement mortar. Slacking by salt water gives less paste than fkesh.

Mortar should not be mixed upon the surface of clayey ground ; but a rough board, brick, or stone
platform should be interposed. Pit sand sifted ft-om decomposed gneiss, and other allied rooks, is ex.
eellent for mortar ; its sharp angles making with the lime a more coherent mass than the rounded

Eains'of river or sea sand. Mortar shoald be applied wetter in hot than in cold weather; especially
brickwork ; otherwise the water is too much absorbed by the masonry, and the mortar ia thereby
injured.

The tenacity, or eobesi ve strenictii, that is, the resistance to a poll

of good common lime mortar of the usual proportiona of lime and sand, and 6 months old, is aoont
from 15 to 90 0>a per aq inch ; or .96 to 1.9 tona per aq ft. With leaa aand, or with greater age, it will
be atronger.

Ttae crnshlnic strenfctli of good common mortar 6 months old ia from 150
to 800 Iba per sq inch, or 9.7 to 19.S tons per aq root.

Tke fiii<llnfc resistanee, or that which common mortar opposes to any

fbree tending to make one ooume of masonry slide upon another, is stated by Boudeiet, to be but & fts
per aq inch ; or about one third of a ton per sq ft, tn mortar B montba old.

Transwerse strensrtli of good common mortar 6 months old. A bar 1

Inch aquare and 12 ina clear span, breaks with a center load of i to 8 Iba.

Tbe lime in mortar decays wood rapidly, especially In dose,

damp aituationa. Still the aoaking of timber for a week or two in a aolntion of qaicklime in water
appeara to act aa a preservative. Iron* ao completely embedded in mortar as to exclude air and
moiatore. haa been found perfect after liUO years; bat if the mortar admiu moisture the iron deoaya.
So, probably, with other metala.

Tbe adbesion to common bricks, or to ronnrta rnbble at any

age will average about H of tbe cohesive atrength at the aame age; oraay 13 to 2i ftaper sq inch, er
.7S to 1.& too per aq ft at 6 months old. If care be taken to exclude duat entirelv, bv dipping eaeh
brick into water before laying it. or by aprinkling the atone by a hose, Ao, the adhesion will be in*
treaaed. On the other hand, much duat may almoat prevent any adheaion at alL The preeantlon of
wetting is especially neoeaaary in very hot weather, to prevent the warm brioka or atooe fh>m IJI^
imm the Hevtar by the rapid abaorption and evaporailoD of tvs water. The aih isles le veif
— nath ksrSpreiMed krtotek or to amoothiy dressed or sawed stone la eonsiderabty lass.

BRICKS.

927

BRICKS.

Art. 2. Brieks, sise, weiclit, Ae, A brick 8.25 X 4 X 2 ins contains
66 cub ins ; or 26.2 bricks to a cub ft ; or 707 bricks to a cub yard.

In ordering a large number, a minimum limit of dimensiou should be specified,
in order to prevent fraud. A brick % inch less each way than the above, con*
tains but 52.5 cub ius ; thus requiring full 25 per cent more bricks to do the same
work, and 26 per ct more cost lor laying, which is geuerally paid by the 1000.

The weljKlit of a good commou brick, 8.25 X 4 X 2 ins, will average about
4.5 Jbs ; or 118 lbs per cub ft = 3186 tbs or 1.42 tons per cub yard ; or 2.01 tons per
1000. A good pressed brick of the same size will average about 5 tt>s, = 131 fts
per cub ft = 3537 ft>s or 1.58 tons per cub yd; or 2.23 tons per 1000. Since the
weight of hardened mortar averages but little less than that of good common
brick, we may for ordinary calculations assume the ireiirht Of brichwork,
with common bricks, at 1.4 tons pt^r cub yard, or 116 !bs per cub ft ; and, yrith
pressed brick, at 1.56 toQS per cub yd, or 129 lbs per cub ft.

In water, either brick will in a few minutes absorb from ^ to % lb of
water: or 0.1 to one-seventh of the weight of a pressed brick, or ^ to one-third
Of its bulk.

BTamber of bricks 8>^ X 4 X 2, required per sq foot of wall, allow-
ing for the usual waste in cutting bricks to fit corners, jambs, &c.:

Wall 21K ins, or 2^ brick 85 bricks

*4 ♦• or 3 " 42 "

Wall 814 ins, or 1 ^i^ick 14 bricks

** 12% " or 13^ ♦* 21 "

17 " or 2 •' 28

t(

It

Ijayingr* P^r day. A bricklayer, with a laborer to keep him supplied with
materials, will, in common bouse walls, lay on an average about 1500 bricks per
day of 10 working hours. In the neater outer faces of back buildings, from 1000
to 1200; in good ordinary street fronts, 800 to 1000: or of the very finest lower
story faces used in street fronts, from 150 to 800, depending on tbe number of
angles, Slc. In plain massive engineering work, he should average about 2000
per day, or 4 cub yds ; and iu large arches, about 1500, or 8 cub yds.

Since bricks shrink about -^ part of each dimension in drying and burning,
the moulds should be about ^ part larger each way than the burnt brick is
intended to be. Good well-burnt bricks will ring when two are struck together.

At the brick-yards about Philadelphia, a brick-moulder's work is 2333 bricks
per day ; or 14000 per week. He is assisted by two bovs, one of whom supplies
the prepared clay, moulding sand, and water; while the other carries away the
bricks as they are moulded. A fourth person arranges them in rows for drying.
About ^ of a cord, or 96 cub ft of wood, is allowed per 1000 for burning. Where
coal is used, the kilns are fired up with anthracite, and the finishing is done with
bituminous. One ton of t:oa1, in all, makes 4500 bricks.

For paTlngf sidewalks tbe bricks are laid on a 6-inch layer of gravel,
which should be free from clay, and well consolidated. With bricks of 8>4 X 4
X 2 ins, with joints from ^ to M itich wide, a square yard requires, flatwise,
38 bricks; edgewise, 73; endwise, 149. An average workman, with a laborer to
supply the bricks and gravel, will in 10 hours lav about 2000 bricks; or 53 sq yds
flat, 27 edgewise, 13 endwise. When done, sand is brushed into the joints.

Art. 8. Tiie crushiair strength of bricks of course varies greatly.
A rather soft one will crush under from 450 to 600 fba per sq inch ; or about .SO
to 40 tons per sq ft ; while a first-i'ate machine-pressed one- will require about 20O
to 400 tons per sq ft, or about the crushing lim'it of the best sandstone; two-
chlrds that of the best marbles or limestones ; or ^ that of the best granites,
or roofing slates. But masses of brickwork crush under much smaller loads
than single bricks. In some English experiments, small cubical masses, only
9 inches on each edge, laid in cement, crushed under 27 to 40 tons per so ft.
Others, with piers 9 ins square, and 2 ft 3 ins high, in cement, only two days
after being built, required 44 to 62 tons per sq ft to crush them. Another,
of pressed brick, in best Portland cement, is said to have withstood 202 tons
per sq ft; and with common lime mortar only 3^ as much.

It must, however, be remembered, that crackinec and splitting usually com-
mence under about one-half the crushing loads. To lie safe, the load should not
exceed 3^ of the crushing one ; and so with stone. Moreover, these experiments
were made upon low masses ; and the strength decreases with the proportion
of the height to the thickness.

The pressure at the base of a brick shot-tower in Baltimore, 246 feet high, is
estimaied at63>^tons per sq ft; and in a brick chimney at Glasgow, Scotland,
468 feet high, at 9 tons. Professor Rankine calculates that in heavy gales this is
increased to 15 tons, on the leeward side.

928

BRICKS.

with our preseDt Imperfect knowledge on thii ■al|)eot, It cannot be considered iiafe to expose em
nrtt-class pressed brickwork, in etmtu, to more than 12 or 15 tons per sq ft : or sood hand-moulded,
to more than two-thlrda as mneh. ^^

Tenstle strenytb of brick, 40 to 400 ftw per sq inch ; or 2.6 to 26 tons per sq ft

The Enffrllsli rod of brickwork is 806 cub feet, or llUcub yards: and

reqolres about 4&00 bricks of the English standard sise ; with about 75 cub ft of mortar. The Enclish
hundred of lime, is a oub yd.

FrOBen mortar. There is risk in using common mortar in oold weather. If the eoM
should continue long enough to allow the frozen mortar to set well, the work may remain safe ; but U
a warm day should occur between the freesing and the setting of the mortar, the aun shlninc on one
side of the wall may melt the mortar on that side, while that on the other side may remain fTosei
hard. In that case, the wall will be apt to fall ; or if it does not, it will at least always be w«ak ■ for
mortar that has partially set while froien, If then melted, will nerer regain its strenstb. fir the
vriter's own trials hydranlic cements seemed not to be injured by fMeilng.

Ezpertmeiitfi for ronderinff brick masonry Imperwloiui to

water* Abstract of a paper read before the American Society of Civil Engineers, May 4, ISn,
by William L. Dearborn, Civil Engineer, member of the Society.

The face walls of the Bsck Bays of the Oate-houses of the new Oroton reaenrolr. located nortk
•r Eighty-sixth Street, in Central Park, were built of the best quality of hard-bamt brlok ; laid is
mortar composed of hydraulic cement of New York, and sand mixed in the proportion of one meskvn
of cement to two of sand. The space between the walls Is i ft ; and was filled with concrete. The ftaa
valla were laid up with great care, and every precaution was taken to have the Jointe well «Hied ui4
insure good work. They are 13 ins thick, and 40 ft high ; and the Bays when ftall generally have M ft
ef water in them.

When the reservoir was first filled, and the water was let into the Gate-houses, It was foand to filter
through these walls to a considerable amount. As soon as this was discovered, the water waa drawn
out of the Bays, with the Intention of attempting to remedy or prevent this infiltration. After eare-
ftilly considering several modes of accomplishing the ol^ect desired, I came to the oonelaaion to trr
" Sylvester's Process for Repelling Moisture from Bxtomal Walls."

The process consists in using two washes or solutions for covering the surface off brlok wmlla ; see
eomposed of Castile soap and water ; and one of alum and water. The proportions are : three^uar*
ters ef a pound of soap to one gallon of water ; and half a pound of alam to four gnllona of water
both substances to be perfectly dissolved in the water before being used.

The walls should be perfeotlv clean and dry ; and the temperature of the air should not be beloe
50 degrees Fahrenheit, when the compositions are applied.

The first, or soap wash, should be laid on when at boiling heat, with a flat brush, taking oare not
to form a froth on the brickwork. This wash should remain twentv-fonr hours : so as to oeoome dn
and hard before the second or alum wash is applied ; which should be done In the same manner ss
the first. The temperature of this wash when applied may be 60^ or TO^^ ; and It sbonld also remaia
twenty-four hours before a second coat of the soap wash is put on ; and these ooate are to be repealed
alternately until the walls are made impervious to water.

The alum and soap thus combined form an Insoluble oompound, filling the pores of the masonry,
and entirely preventing the water from penetrating the walln.

Before applying these compositions to the walls of the Bays, some ezperlmente were made to tail
the absorption of water by bricks under pressure after being covered with these washes, in order ti
determine how many coate the wall would require to render them impervious to water.

To do this, a strong wooden box wait made, put together with screws, large enough to hold 3 brtdst
and on the top iras inserted an inch pipe fortv feet long.

In this box were placed two bricks after being made perfectly drv, and then covered with a ooekef
each of the washes, as before directed, and weighed. They were then suhjeoted to the pressaie of a
eolumn of water 40 feet high ; and, after remaining a sufficient length of time, they were taken set
and weighed again, to aaoertain the amount of water they bad absorbed.

The bricks were then dried, and again coated with the washes and weighed, and sofejeeted to press-
ure as before : and this operation was repeated until the bricks were found net to absorb any water.
Four coatings rendered the bricks impenetrable under the pressure of 40 ft head.

The mean weight of the bricks (drj) before being coated, was S7i lbs; the mean absorption was
one-half pound of wate& An hydrometer was used in testing the solutions.

As this experiment was made in the fall and winter, (186S,) after the temporary roofs were put en
to the Oate-house, artificial beat had to be resorted to, to dry the walla and keep the air at a proper
temperature. The cost was 10.06 ots per sq ft. As soon as the last coat had become hard, the water
was let into the Bavs, and the walls were found to be perfectly impervious to water, and thej still
remain so in 1870, after about 6^ yeans.

BxicK ABCR (FoorwAv OP FIioH Bridov). The brick arch of the footway of High Bridge la the
arc of a circle 29 ft 6 in radius ; and is 12 in thick ; the width on tep Is 17 ft; and the length ooverad
was 1381 ft.

The first two courses of the brick of the arch are oomposed of the best hard-burnt brick, laid edge-
wine in mortar oomposed of one part, by measure, of bydraalio cement of New York, and two parte
of sand. The top of these bricks, and the inside of the granite coping against which the two tep
Ooiirses of brick rest was, when they were perfectly dry, covered with a coat of asphalt one-half an
inch thick, laid on when the asphalt was heated to a temperature of fl'om 800^ to inSP Fahrenheit.

On top of this was laid a oonrse of brick flatwise, dipped In asphalt, and l^d when the asphalt waa
hot; and the Jointe were run full of hot asphalt.

On top of this a course of pressed brick was laid flatwise In hydraulic eement mortar, forming the
paving and floor of the bridge. This asphalt was the Trinidad variety : and waa mixed with 10 per
eent, by measure, of coal tar ; and 26 per cent of sand. A few experimente for tasting the strength
ef this asphalt, when used to cement bricks together, were made, and two of them sire given below.

Six bricks, pressed together flatwise, with asphalt Jointe, were, after lying ilx months, broken*
The distance between the supporte wss 13 Ins ; breaking weight, WW lbs ; area of single Joint, I8>tf a«
laa. The asphalt adhe: .d so strongly to the brtok as to tear away the snrfaee la many i '

BRICKS. 929

Two brioki pressed together end to end, oemenied with asphalt, were, after lying 6 months, broken.

The disunce between the supports was 10 ins ; area of Joint* 8W sq ins; breaking weight, 150 lbs.

The area of the bridge ooTered with asphaltea brick, was 39065 sq fL There was used M200 lbs ef
asphalt. 33 barrels of coal tar, 10 onb yds of sand, 93800 brioks.

The time oocupied was 100 days of masons, and 148 days of laborers. Two masons and two labiir*
•rs wUl melt and spread, of the first ooat, 1650 sq ft per day. The total cost of this coat was 5.2S
aents per sq (t, exelnsiTe of dnty on asphalt. There were tiiree grooves, 2 ins wide by 4 ins deep,
made entirely aoross the briok areh. and inunediately under the first coat of asphalt, diriding the
aroh into four eqaal parts. These grooves were filled with elastio paint cement.

This arrangement was intended to guard agidnst the evil effeots of the ooniraotion of the areh in
winter; as it was expected to yield slightly at these points, and at ao other point; and then the
dastie eement would prevent any leakage there.

The entire experiment has proved a very sucoessfol one, and the areh has remained perfeotly tight.

In proposing the above plan for working the asphalt wUh the brickwork, the object was to avoid
depending on a large aontlnued surface of asphalt, as is usual in covering arches, which very fre-

Jinently eraoks f^om the greater contraction or the asphalt than that of the masonrv with which it ie
n oontaot ; the extent of the asphalt on this work being only about one-quarter of an inch to each
briok. Tlds is deemed to bo aa essential element in the snooess of the impervious oovering."

A eheap and elfeetive prooees for preventing the pereolation of water through the arches of aque
4aots, and even of bridges, is a great desideratum. Many expensive trials with resinous oompounds
ksTs proved failures. Hydraalio eement appears to merely diminish the evil. Mooh of (he bnmlde
ii probably due to oraoks produced by ehanges of temperatore.

The white eflioreseenee so oommon on walls, eepecially on those of brick,
to due to the presence of solnble salts in the bricks and mortar. These are dissolTed,
and carried to the face of the wall, by rain and other moisture. Sulphate of magne.
da (Bpeom Salt) appears to be the most frequent cause of the disfiguration. In many
places mortar lime is made from dolomite, or magnerian limestone, whidi often oon^
taJns 30 per cent or more of magnesia ; which also occurs frequently in brick clay.
Coal generally contains sulphur, most firequently in combination with iron, forming
the w^-known ** iron pyrites ". The combustion of the coal, as in burning the Itme*
■tone or clay, in mannfSstctures, in cooking etc, conTorts the sulphur into sulphurous
add gas, which, when in contact with magnesia and air, as in the lime or brick kiln,
or in the ilnished wall or chimney, becomes sulphuric acid and unites with the mag*
nesia, forming the solnble suAphais. We are not aware of any remedy that will pre*
vent its appearance under sudi circumstances ; but the formation of the sulphate may
be preTented by the use of limestone and brick-clay free from magnesia.

59

930 CEMENT.

CEMENT.

General Principles.
The elements chiefly concerned in the action of lime and cement

Calcium,

Aluminum,

Carbon, C \ Oxygen, O.

mortars are —

Silicon,
'Hydrogen,

Oxygen combines with each of the others, forming oxides. Thus:
Calcium oxide, CaO, is lime;
Aluminum oxide, AJ^Os,* is alumina;
Carbon dioxide, CO^, is carbonic acid ;
Silicon oxide, SiO«, is silica, or silicic acid;t
Hydrogen oxide, HgO, is water.

Limestone is a calcium carbonate, or combination of lime and carbonic
acid, CaO + COj, or CaCOa.

Clay (including argillaceous minerals in general) is an aluminum silicate^
or comoination of alumina and silicic acid, AI2O8 + SiOs.

Lime. When limestone (without clay) is " burned," its COg is driven off,
and the remaining ("quick") lime has a strong affinity for water, absorb^
ing it with such avidity as to develop heat sufficient to produce steam,
the generation of which disintegrates and swells the mass. Combining thus
with the water, the lime forms calcium hydrate, CaO.H^, or QiH|0|.
This process is called slaking or slacking! and lime which nas satisfied its
affinity for water is called slaked (or slack) lime. When slaked lime is used
as mortar, it gradually absorbs carbonic acid from the air, forming calcium
earbonate, the water being liberated and evaporated. Hardened lime nu>rtar
may thus be regarded as an artificial limestone.

Cement. When aluminum silicate, such as clay, in sufficient Quantities,
is burned with calcium carbonate, such as limestone; the burned product,
called cement, is deficient in, or devoid of, the slacking propertv; but, on
the other hand, when il^is made into mortar, the combinations, formed be-
tween the elements of tne lime, the alumina, the silica and the water, during
the burning, and afterward in the mortar, are such that they readily prooeea
under water. Chemists differ as to the nature of these combinations. If
free lime remains in the mortar, calcium carbonate is formed by reabsorption
of carbonic acid from the air, as in the case of lime mortar.

Setting, or the loss of plasticity, usually occurs within a few hours (some-
times within a few minutes) after mixing cement with water; whereas
hardening (which appears to result from a different set of chemical pro-
cesses) often proceeds for months or even years.

The property of setting and hardening under water is called hydraulic^
ityi and cements which do not slack, but which harden under water, are
called hydraulic cements; or, more briefly, cements.

Hydraulic lime is a name given to cements j(much used in Europe)
whicn, while to some extent hydraulic, do not contain enough of the hydrau-
lic elements to prevent slaking. The slaking, however, is slower, and the
swelling less, than with lime proper.

The ratio of the weight of aluminum silicate to that of the lime, in a ce-
ment, is called its hydraulic index. Other things being equal, it may
be used as an indication of the hydraulicity of the cement.

Natural and Portland Cements. Many natural limestones contain
clay in such proportion that they afford cements when burned. Cements
so made are called natural ; while those which result from the burning of arti-
ficial mixtures of lime carbonates and aluminum silicates are called artifioiaI«
and, when certain refinements (see below) are observed, "Portland" cement.

In making natural cement» the material is burned in lumps; bui for
Portland cement the material is finely ground before burning, and the

* The subscripts indicate the combining ratios of the several elements.
Thus, in alumina, Al2C)3 means a compound of 2 atoms of aluminum with 3
of oxygen.

t Quarts is silica ; and most of the sand used in mortar ib quarts sand.

CEMEKT MOBTAR.

931

burning is done at a hi|^ temperature, producing incipient vitrifaotion. In
both natural and Portland cements, tne burned product is ground to an
impalpable powder.

In natural cement, the hydraulic index usually varies between 0.60 and
1.50; in Portland cements, between 0.40 and 0.60.

The higher cost of Portland cement is due to the more careful selection of
the materials and to the more elaborate and expensive treatment given them,
resulting in the ultimate attainment of much greater strength.

The name Rosendale, originally and properly restricted to natural
cements made in Ulster County, New York, is often applied indiscriminately
to American natural cements in general.

Limestones containing manrnesia are called Dolomitic Limestones or
Dolomites. The presence of more than 3 per cent, of magnesia, in the
finished product, is usually considered objectionable.

Cement Mortar.

Cement mortar consists of cement and some inert granular material,
aa sand, fine gravel or g^und cinder, mixed with water.

Owing to tne cheapness with which cements are now manufactured, and
the superiority of the n!kortars made from them, the latter have to a great
extent superseded lime mortars, even in ordinary building operations.

Amount of Mortar Required for a Cubic Tard of Masonry.*

Mortar.
Description of Masonry. Cu. yd.

Min. Max.

Ashlar, 18' courses and i'^ joints, 0.03 0.04

" 12* " " " " 0.06 0.08

Brickwork (bricks of standard sise, 6i X 4 X 2i ins.) :

i» joints 0.10 0.16

I' to i' joints, 0.25 0.36

f ' to V joints 0.35 0.40

Rubble, of small, rough stones, , 0.33 0.40

" " large stones, rough hammer-dressed 0.20 0.30

Squared-stone masonry, 18' courses and f ' joints, 0.12 0.16

" " 12' " " " " 0.20 0.25

Cement and Sand Required for 1 Cubic Tard of Mortar.*

of Sand to 1
of Cement.

Mortar Proportioned
by Weight.

Mortar Proportioned by Volumes
' of Packed Cement and
Loose Sand.

Portland.

Natural.

Portland.

Natural.

u

Cement.

Sand.

Cement.

Sand.

Cement.

Sand.

Cement.

Sand.

Bbl.

Cu.Yd.

Bbl.

Cu.Yd.

Bbl.

Cu.Yd.

Bbl.

Cu.Yd.

0

7.40

0.00

7.91

0.00

7.40

0.00

7.91

0.00

1

4.05

0.67

4.92

0.61

4.17

0.67

4.58

0.58

2

2.80

0.78

3.43

0.72

2.91

0.78

3.04

0.76

3

2.00

0.85

2.54

0.80

2.08

0.85

2.24

0.81

4

1.60

0.89

2.04

0.84

1.66

0.89

1.70

0.86

5

1.30

0.91

1.64

0.86

1.35

0.91

1.39

0.88

6

1.10

0.93

1.40

0.88

1.14

0.93

1.28

0.89

♦ Taken, by permission, from "A Treatise on Masonry Construction," by
Prof. Ira O. Baker. New York, John Wiley A Sons. 9th edition, 1899.

932 CEMENT.

The effects of cold upon Portland cements, althoiu^ it retards the
setting, do not appear to be serious otherwise. Even if Portland cement
mortar freezes almost as quickly as the masonry is laid with it, it does not seem
to depreciate materially. We have found this to be the case also with lime
mortar ; even when, a few hours after freezing, the temperature became so hi^
as to soften the frozen mortar afifain. But although the mortar of either lime
or cement may not thereby be injured, the work, especially in thin brick walk,
may be ruined and overthrown. Thus, if, soon after the mortar, through the
entire thickness of such a wall, be frozen, the sun shines on one face of it, so as
to soften the mortar of that face, while the mortar behind it reniains hard,
it is plain that the wall will be liable to settle at the heated face, and at least
bena outward if it does not fall. Coatings of cement, applied to the backs
of arches on the approach of winter, and left unprotected, have been found
entirely broken up and worthless on resuming work the next spring.

Alternate freezing and thawing are apt to disintegrate botn natural
and Portland cement mortars.

The heating of sand and cement^ in freezing weather, seems to be a
bad practice, especially if they be placed in cold water. But for use out of
water Mr. Maclay says they may be heated to 60® or 60®. Cold w^ater for
mixing is probably no farther injurious than that it retards the setting.

Strengths.

Factors Affecting Strength. The strength of samples, under test, is
much affected by the temperature of the air and water, as also by the d^ree
of force with which the cement is pressed into the molds; by the extent of
setting before being put into the water, and o^ drying when taken out; and
stUl more by the pressure under which it sets, which increases the strength
materially. On this account, cements in actual masonry may, under ordi-
nary circumstances, give better results than in tests of samples. Tlie
causes named, together with the degree of thoroughness of the mixing or
gaging, the proportion of water used, and other considerations, may easily
affect the results 100 per cent, or even much more. Hence the discrepanciefl
in the reports of different experimenters. Specimens^ of the SEune cement,
tested under apparently similar conditions, may give widely different results.

The Bureau of Surveys, Philadelphia, requires, 1901, the follow-
ing tensile strengths, in lbs. per sq. inch; 1 day in air, remainder in water:

7 days. 28 days.

Portland, neat 500 600

" 3 parts sand, laboratory test, 170 240

•' 3 " '• mortar from mixing box. 125 175

Natural, neat, 200 300

" 2 parts sand, laboratory test, 120 200

" 2 " *' mortar from mixing box, 50 125

See also diagrams, p. 933, and Requirements, p. 942.

Portland and Natural Cements. Effect of Age. The diagram *
opposite illustrates approximately the strengths of average Portland and of
average natural cements, neat and with proper doses of sand, up to an age
of two years. Tests may readily vary 10 per cent, or more either way from
the average.

Cements of the same class differ much in their rapidity of hardening.
At the end of a month one may gain nearly one-half of what it will gain in a
year, and another not more than one-sixth;* yet at the end of the year both
may have about the same strength. Hence, tests for 1 week or 1 month
are by no means conclusive as to the final comparative merits of cements.

Many Tears are required to attain the greatest hardness;
but after aoout a year the mcrease is usually very small and slow, especially
with neat cement. Moreover, any subsequent increase is a matter of little
importance, because generally by that time, and often much sooner, the work
is completed and exposed to its maximum stresses.

There seems to be a period, occurriiig from a few weeks to several months
after laying, during which cement and its mortars for a short time not
only cease from hardening, but actually lose strength. They then recover,
and the hardening goes on as before. This feature is not indicated in our
diagram.

* Bee Richard L. Humphrey, in "Cement," Chicago, May. 1899.

STBENGTH.

ffood qiulily, itiil. with l.£or2m«asar«flofj!Anii, give a mortar strujiKenQiuch
for mosl eDKioeering purposes; but a good Portland will give one equilly
Htronj with 3 or 4 measuna of sand: and wilL, therefore. & equally cheap
at Iwica the prica ; bcflidas requiriBc tua haudliDg. Btorinc, and testinic of only
ball the cumber of paokaeee.

Any addition of sand weakens cement, enpetsially as regards ten-
Al'tbough, with Band, the Btrengtb of the mortar may never attain to that of
i be cut twice aa utronjf as in 7 daya.

ronaaain;

.ich at the end of a year wih be 3,
■ end of a year g "~ "

about S or 6 times the Btterigth of tte others in 7 days, they still averag
about 2.S to 3 times as strong in a year or longer,
Dlacram * eihlbitlnB, approximately, the effect of land. i

different ages from 1 week
to 1 year. The four solid

Portland oementa, and the

t, the I

h kind of

I of 1

week, reapeetively, beiriB-
Ding at the top. The
eurves for aatoral cement
■ra earried oiJy to 6 part*

a "UsHmry Coiwtruotion."

934 CEMENT.

Mr. Wm. W. Maclayt C. E.,* found that, in the testina of cements, the
temperature of the air and water had far more influence than had before been
suspected, but the ultimate effects of temperature, within certain limits, are
fortunately not so important in actual practice as the first experiments
might lead us to infer. Work must go on notwithstanding changes of tem-
perature, but we must take care that our mortar shall at all times be strong
enough, even under their most injurious influences. Cements in open air are
certamly more or less injured by drying instead of aettino, when the tempera-
ture exceeds about 65^ to 70**. But if mixed only in small quantities JEit a
time, and guickljr laid in masonry of dampened stone, so as to be sheltered
from the air, the injury is much reduced. The sand and stone should both
be damp, not wet, in hot weather, and a litUe more water may be used in the
cement paste; also, if possible, 'Uot only the mortar while being mixed, but
the masonry also, should then be shaded.

The compressive strengths of cements and cement mortars, in cubes,
appear to be about 8 to 10 times their tensile stren^hs. The crushing
strength, with sand, increases with age much more rapidly than the tensile
strength, and the more so, the greater the proportion of sand. Cements are
seldom tested in compression.

The shearing strength of neat cements averages about one-fourth of
the tensile strength.

The adhesion of cements to bricks or rough rubble, at differ-
ent ages, and whether neat or with sand, may probably be taken at an aver-
age of about three-fourths of the cohesive or tensile stren^h of the cement or
mortar at the same age. If the bricks and stone are moist and entirely free
from dust when laid, the adhesion is increased; whereas, if very dry and
dustyj especially in hot weather, it may be reduced almost to nothing. The
adhesion to very hard, smooth bricks, or to finely dressed or sawed masonry,
is less than the adhesion to rough and porous surfaces.

Abrasion. See '*Mr. Eliot C. Clarke»*' p. 937.

Weight. See also Weight, p. 939, and (2) Specific Gravity, p. 940.
Weight is an uncertain indication. A coarse-ground cement weieps heavier,
but gives less strength, than the same cement more finely groimd.

Color. See "Variations in Shade," p. 936.

Fineness. See "Mr. Eliot C. Clarke," p. 936, ^'Sieves,'* p. 938,
"Fineness," pp. 938, 940, and "Requirements," p. 942.

Cement, when freshly ground, is not so good as when a few weeks old.

Precautions.

The engineer should reserve the right to take a sample from each
package, and to reject every package of which the sample drawn out does
not satisfy the stipulations. On works using large quantities, one person
should be specially detailed to this duty.

Protection from moisture, even that of the air, is very essential for
the preservation of cements, as well as of quicklime. With this precaution,
the cement, although it may require more time to set, will not other-
wise very appreciably deteriorate in many months.

Setting.

Slow setting does not indicate inferiority; for many of the best cements
are the slowest setting. A layer of very quick-setting cement may partially
set, especially in warm weather, before the masonry is properly lowered and
adjusted upon it, and any disturbance, after setting has oommenoed,
is prejudicial. Such cements are. to be regarded with suspicion, and sub-
mitted to longer tests than slow ones. Still, quick-settingoements are beet
in certain cases, as when exposed to running water, etc. They may be ren-
dered slower by adding a bidk of lime paste equal to 6 or 15 per cent, of the
cement paste, without weakening them seriously.

As a general rule, cements set and harden better in water than in air,
especially in warm weather. If, however, the temperature for the first few
di^s does not exceed 55^ to 65*^ Fahrenheit, there seems to be no appreciable
difference in this respect ; but in warm air, setting eement, in drying, loses
the moisture upon which the operation of hardening depends. It therefore
sets without hardening. In hot weather every precaution should be used
against this.

♦ "Transactions American Society of Civil Engineers," Dec., 1877.

«AND. 935

SflEnd Betards Setting. In our experiments with various hydraulic
oements, of the consistence of mortar, even without sand, we have 'detected
no change of bulk In setting. But Mr. Clarke (see p. 936) found an
expansion of not more than 0.001 part in any dimension.

Sand.

The best sand is that with gjains of very uneven sizes, and sharp. The
more uneven the sixes, the smaller are the voids, and the heavier is the sand.
It is generally considered that the sand should be well.washed if it contains
day or mud. But see "Adding Clay,** p. 936. Mr. Clarke says, "the
finer the sand, the less is the strength."

Proportton of Sand. As a general rule, with cements of good quality,
we shall have mortars fit for most engineering purposes if we do not exceed
from 1 .5 to 2 measures of dry sand to 1 of the common cements ; or from 2 to
3 of sand to 1 of Portland.

Voids in Sand. Since a cubic foot of pure quartz weighs 165 lbs., it
follows that, if we weigh a cubic foot of pure dry sand, either loose or rammed ;
then, as 165 is to the weight found, so is 1 to the solid jpart of the sand. And, if
this solid part be subtracted from 1, the remainder wul be the voids, as below.

Wt., lbs. per cub. ft., dry. 80 85 90 95 100 105 110 115

Proportion of solid, . . 0.485 0.515 0.546 0.576 0.606 0.636 0.667 0.697
Proportion of voids, . . 0.515 0.485 0.454 0.424 0.394 0.364 0.333 0.303

But the sand, when wet in mortar, occupies about from 5 to 7 per cent, less
niace than when dry; the shrinkage averaging say 6 per cent. ; thus making
the voids 0.304 of the 105 tb. sand when wet; and 0.364 of the 95 lb.; the
mean of which is 0.334. But. to allow for imperfect mixing, etc., it is better
to assume the voids at 0.4 oi the dry sand. Moreover, smce the cements,
as before stated, shrink more or less when mixed with water, and worked up
into mortar, it would be as well to assume that, in order to make sufficient
paste to fill the voids thoroughly, we should use, of dry common cement
slightly shaken, not less than half the bulk of the dry sand ; and not less than
46 per cent, if Portland.

To And the percentage of voids, pour into a graduated cylindrical
measuring-glass 100 measures of diy sand. Pour this out, and fill the g^lass
up to 60 measures with water. Into this tprinkle slowly the same 100
measures of dry sand. These will now be found to fill the glass only to say
94 measures, having shrunk say 6 per cent. ; while the water will reach to sav
121 measures; of which 121 — 94 — 27 measures will be above the sand;
ieaving 60 — * 27 -*■ 33 measures filling the voids in 94 measures of wet sand;
showing the voids in the wet sand to be H •» 0.351 of the wet mass. If the
sand is poured into the water hastily, air u carried in with it, the
voids will not be filled, and the result will be quite different.

Compressibility of Sand. Careful experiments of our own, with or*
dinary pure sand from the seashore, both dry and moist (not weO. sa^o the
following results. The dry sands were oomj[>aoted by thorough shaking and
jarring; the moist sands by ramming in thm layers. Sand B was of much
ftner'grain than A. C consisted of the finest sifted grains from B.

Perfectly dry. Moist.

Lbs. per cubic ft. Reduction Lbs. per cubic ft. Reduction

^..— ■>i>^^..«s.,^^<» of bulk, ,„-—**«...,.-->>_— —-^^ of bulk.

Sand. Loose. Shaken, per cent. Loose. Rammed, per cent.

A 97 112 13.4 86 107.5 20

B 88 101.6 13.4 69 107.5 33.3

C 82 98.5 16.8 103.5

None of these sands, when dry and loose, if poured gently into water to a
depth of 15 inches, settled more than about one-fifteenth part; the coarsest
one, A, considerably less.

Water Bequired. See "Mr. Eliot C. Clarke," p. 936.

Cold water for mixing is probably no further injurious than that it
retards setting.

Salt. See "Mr. EUot C. Clarke.'' p. 936.

936 CEMENT.

For pointinsi the best Portland is none too eood, and is best used neat,
but it is dften used with from 1 to 2 parts of sand. Mix under shelter, and in
quantities of only 2 or 3 pints at a time, usinc very little water ; so that the
mortar, when ready for use, shall appear rather incoherent, and quite defi-
cient in plasticity. The joints bein^ previously scraped out to a depth of at
least half an inch, the mortar is put m by trowel; a strai£ht-ed£^ being held
just below the joint, if straight, as an auxiliary. The mortar is then to be
well calked into the joint by a calking-iron and hammer; then nxore mortar
is put in, and calked, until the joint is full. It is then rubbed and polished
under as great pressure as the mason can exert. If the joints are very fine;
they shomd be enlarged by a stonecutter, to about 1-4 inch, to receive the
pointing. The wall should be well wet before the pointing is put in, and kept
m such condition as neither to give water to, nor take it from, the mortar.
In hot weather, the pointing should be kept sheltered for some days from
the sun, so as not to dry too quickly.

Preservation of Metals.

We have found, by ten years' trial, that if, after setting, dampness is abso-
lutely excluded, cements preserve iron, lead, zinc, copper and brass ; and that
plaster of Paris preserves all except iron, which it rusts somewhat unless the
iron ib galvanized. Lime-mortar probably preserves all of them, if kept
free from damp.

Efflorescence.

Natural cemebts, when used as mortar for brickwork, often disfigure it,
especially near sea-coasts, and in damp climates, by white efflorescenoe
which sometimes spreads over the entire exposed face of the work, and also
injures the bricks. This also occurs in stone masonry, but to a muoh less
extent, and is confined to the mortar joints. It injures only porous stone.
It is usually a hvdrous carbonate of soda or of potash, or sulphate of lime
(Epsom salts), often with other salts. As a preventive, General Gilmore re-
commends to add, to every 300 lbs. (1 barrel) of the cement powder, 100 tbs.
of quicklime, and from 8 to 12 tbs. of any cheap animal fat ; the fat to be well
incorporated with the quicklime before slacking it, preparatory to adding
it to the cement. This addition will retard the setting, and somewhat dimin-
ish the strength of the cement. It is said that.linseed oil, at the rate of 2
gallons to 300 lbs. of dry cement, either with or without lime, will in all
exposures prevent efflorescenoe ; but, like the fat, it greatly retards setting,
and weakens the cement. See also Bricks.

Mr. Eliot C. Clarke has published * the results of a series of expoi-
ments made for the Boston Main Drainage Works. From his paper we con-
dense as follows, by permission:

Variations In shade, in a (riven kind of cement, may indicate differenees
in the character of the rock or degree of burning. Thus, with Rosendale, a
light color generally indicates an inferior or underbumed rock. A ooarse-
ground cement, light in color and weight, would be viewed with suspicion.

The highest strength was obtained by the use of Just enough water
to dampen the cement thoroughly. An exoess of water retards setting.
Natural cements need more water than Portland; fine-ground more than
coarse; quick-setting more than slow. Neat Rosendale, a year old, was
strongest with 35 per cent, water. Neat Portland, same age, with 20 per
cent.

The finer the sand, the less the strength.

Salt, either in the water used for mixing, or in that in which the cement
is laid, retards setting somewhat, but has no important effect upon the
strength.

Adding clay gives a much more dense, plastic, water-tight paste, useful
for plaster or for stopping leaks. Half a part of day did not seem to weaken
mortar materially, except in the case of sample blocks exposed to the weather
for 2^ years after a week's hardening in water.

A year's saturation in fresh or salt water, and in contact with oak, hard
pine, white Pine, spruce or ash, did not affect the mortars.

With sand, fine-ground cements make the strongs^ mortar; but when
tested neat, coarse-ground cements are strongest. Hiis is espeoialfy the
ease with Portlands.

♦ "Trans. Am. Soc. C. E.," April, 18S5.

BEGOMMENDATIONS. 937

Qobd results were obtained from mlxlngr different cements. A
mortar of half a part each of Rosendale and Portland, and two parts sand,
was stronger, at 1 week* 1 month, 6 months and 1 year, than the avera^ oi
two mortars, one of 1 part Rosendale and one of 1 part Portland ; each with 2
parts sand. Mixtures of Roman (quickHsetting) and Portland (slow) set
about as quickly as Roman alone, and were much stronger.

Portland resisted abrasion best when mixed with 2 parts sand; Rosen-
dale with 1 part. A little more or less sand rapidly reduced the resistance in
both cases.

, Cements expand in setting; but not more than 1 part in 1000 of any
given dimension.

Sand cement or silica cement is made by mixing cement with
quartz sand (silica) and grinding the mixture. It is claimed that the cement,
in the mixture, becomes much more finely ground, and that a mixture of 1
part cement and 3 parts sand can therefore carry, in mortar, nearly as much
sand as could the pure cenient alone before this treatment.

The fineness of cement and sand is indicated as follows, where the
large numerals represent the siev6 numbers; the small numeral, to the left
of each sieve number, represents the percentage retained upon that sieve ;
and the final small numeral, to the right of the last sieve number, represents
the percentage passed bv tne last sieve. The sum of the small numerals
=- 100. Thus, » 20 ^* 30 * 40 '*^ means that 5 per cent, was retained on a
No. 20 sieve, 15 per cent, on No. 30, and 35 per cent, on a No. 40, while
the remaining 45 per cent, passed the No. 40 sieve.

Properties and tests of cement. Reeommendations of
Amenean SoeietT of Civil Snidlneers. Digest of Fioal Report of
the Committee* on a Uniform System forTestsof Cement, Trans. Am. Soc. C. E.,
Vol. xiv, Noyember, 1885.

The first tests of inexperienced, though intelligent and careful persons, are
usually very contradictory and inaccurate, and no amount of experience can
eliminate the variations introduced by the personal equations of the most con-
Bcientioos observers. Many things, apparently of minor importance, exert so
marked an influence upon the results, that it is only by the' greatest care in
everv particular, aided by experience and intelligence, that trostworthy tests
can be made.

Onlv a series of tests for a oonsiderable period, and with a full dose of sand,
will show the full value of any cement ; and it would be safer to use a trust-
worthy brand without applying any tests whatever, than to accept a new article
which had been tested only as neat cement and for but one day.

It is recommended that tests be confined to methods for determining (1)
fineness, (3) liability to checking or cracking, and (3) tensile strength ; and, for
the latter, for tests of 7 days aud upward, that a mixture of 1 part of cement to
1 part of sand for Natural f cements, and 3 parts of sand for Portland f cements,
be used, in addition to trials of the neat cement. The quantities used in the
mixture should be determined by weight.

The tests should be applied to the cements as offered for sale. If satisfactorv
results are obtained with a full dose of sand, the trials need go no further. If
not, the coarser particles should first be excluded by nsing a No. 100 sieve.
In order to detennine approximately the grade the cement would take if ground
fine, for fineness is always attainable, while inherent merit may not be.

The amount of material needed for making five briquettes of the stand-
ard size recommended is, for the neat cements, about 1.66 pounds, and for those
with sand, in the proportion of 3 parts of sand to 1 of cement, about 1.26 pounds
of sand and 6.66 oances of cement.

♦ Q.A. Gillmore, Chairman, D. J. Whitteraore, J. Herbert Shedd, Eliot C.
Clarke^ Alfred Noble, F. O. Norton, W. W. Maclay, Leonard F. Beckwith,
Thomas G. McColIom.

I Where the word '* natural " is used in this connection, it is to be understood as
. ng applied to the lightly burned natural American or foreign cements, in
•oatradiBtinction to the more heavily burned Portland cement, either natural
or artificial.

938 OEMENT.

Becommendatlons of Am. Soc* CItU Engn. Continued.

SampUnfiT* Usually, where cement has a good reputation, and is used in
large masses, as in heavy concrete foundations, or in the backing or heart"
ing of thick walls, the testing of every fifth barrel seems to be sufficient ; bat in
very important work, where the strength of each barrel may in a great measure
determine the strength of that portion of the work where it is used, or in the
thin walls of sewers, etc., etc., every barrel should be tested, one briquette being
made from each.

In selecting cement for experimental purposes, take the samples from the
interior of the original packages, at sufficient deptn to insure a fair ezponoitof
the quality. Store the samples in tightly closed receptacles, impervious to light
or dampness, until required for manipulation, when each sample of cement
should be so thoroughly mixed, by sifting or otherwise, that it shall be unifoim
in character throughout its mass.

Slewes. For ascertaining the fineness of cement it will be convenient to om
three sieves, viz.: (Sizes of wire by Stubs gage.)

Na 50— 50 meshes per linear inch ( 2,500 meshes per square inch ) , No. 85 wirt
No. 74—74 " " " " (6,476 « " " " ) , No. 87 wire.
No. 100-100 " «« " " (10,000- « " " " ), No. 40 wire.

• For sand, two sieves are recommended, viz.:

No. 20—20 meshes per linear inch (400 meshes per square inch). No. 28 wim
No. 30—80 " " " " (900 " " " " ), No. 81 wiit

Standard sand. Sands looking alike and sifted through the same sievei
may give results varying within rather wide limits. The use of crushed quarts
is recommended, the degree of fineness to be such that the sand will all pass a
No. 20 sieve and be caught qn a No. 80 sieve.

Mixing:, etc The proportions of cement, sand, and water should be ctre-
fuUy determined by weight, the sand and cement mixed dry, and all the water
added at once. The mixing must be rapid and thorough ; and the mortar, whick
should be stiff and plastic, should be firmlv pressed into the molds with the
trowel, without ramming, and struck off level ; the molds in each instance, wfaik
being charged and manipulated, to be laid directly on glass, slate, or some other
non-absorbent material. The molding must be completed before incipient set-
ting begins. As soon as the briquettes are hard enough to bear it, they shoald
be taken from the molds and be kept covered with a damp cloth until they are
immersed. For the sake of uniformity, the briquettes, both of neat cement and
those containing sand, shonld be immersed in water at the end of 24 hoars,
except in the case of 1 day tests.

Fresh, clean water, having a temperature between 60° and 70® F., should he
used for the water of mixture and immersion of samples.

The proportion of water required varies with the fineness, age, or other condi-
tions of the cement, and the temperature of the air, but is approximately ai
follows :

For neat cement : Portland, about 25 per cent.; natural, about 30 per cent.
For 1 cement, 1 sand : about 15 per cent, of total weight of sand and cement
For 1 cement, 3 sand : about 12 per cent, of total weight of sand and cement
The object is to produce the plasticity of rather stiff plasterer's mortar.

(1) Fineness. It is recommended that the tests be made with cement that
has passed through a No. 100 sieve. See p 937.

(2) Cliecfelngr or cracklnfr. Make two cakes of neat cement, 2 or S
inches in diameter, about \^ inch thick, with thin edges. Note the time in
minutes that these cakes, when mixed with water to the oonsistenoy of a stiff
plastic mortar, take to set hard enough to stand the wire test recommended by

Gen. Gillmore, viz., -J^ inch diameter wire loaded with ^ pound for initial set,

and ^ inch loaded with 1 pound for final seti One of these cakes, when hard

enough, shonld be put in water and examined from day to day to see whether It
becomes contorted, and whether cracks show themselves at the edges. Such con-
tortions or cracks indicate that the cement is unfit for use at that time. In
some cases the tendency to crack, if caused by the presence of too rooek
unslaked lime, will disappear with age. The remaining cake shoald be kcqpt !■

TESTING.

939

Becommendations of Am. Soc. Civil Engrs. Continued.

•

the air, and its oolor obmrred, which, for a good cement, should be uniform
throughout : yellowish blotches indicating a poor quality ; tne Portland cements
being of a bluish-ffray, and the natural cements being light or dark, according
to the character of the rook of which they are made. The color of the cements
in air indicates the quality much better than when they are put in water.

(8) Tensile strenfr^l^* An average of 5 briquettes may be made for each
test, only those breaking at the smallest section to be taken. The briquettes
should always be broken immediately after being taken out of the water, and
the temperature of the briquettes and of the testing room should be constant
between 60° and 70° F. .

The stress should be applied at a uniform rate of about 400 pounds per min-
ute, starting at 0. With a weak mixture use half the speed.

The molds fhmished are usually of iron or brass. Wooden molds, if well
oiled to prevent absorption of water, answer a good purpose for temporary use.
but speedily become unfit for accurate work. Our figures show the form of
briquette and of metal mold recommended.

\La

Section through a-h

The elips should be hung on pivots, so as to avoid, as fur as possible, cross
strain upon the briquettes.

Weight. The relation of the weight of cement to its tensile strength is an
uncertain one. In practical work, if used alone, it is of little value as a test,
while in connection with the other tests recommended it is unnecessary, except
when the relative bulk of equal weights of cement is desired.

Settingr. The rapidity of setting furnishes no indication of ultimate
strength. It simply shows the initial hydraulic activity.

^niek-aettiiiir cements are those which set in less than half an hour; and
•low-setting cements are those requiring half an hour or more to set. The
cement must be adapted to the work required, as no one cement is equally
ffood for all purposes. In submarine wort a quick-setting cement is often
unperatively demanded, and no other will answer, while for work above the
water-line less hydraulic activity will usually be preferred. Each individual
case demands special treatment. The slow-setting natural cements should not
become warm while setting, but the quick-setting ones may, to a moderate
extent, within the degree producing cracks. Cracks in Portland cement indi-
cate too much carbonate of lime, ana in the Yicat cements too much lime in the
original mixture.

940 CEMENT.

Properties and Teste of Cement. Benort of Board U. SL A.
Eni^lneer Officers. Properties and tests of rortlaiv), Natural and Pus-
solan* cementa. Digest of a Ueportof Majors W. L. Marshall and Smith S.
JLeach aud Capt Spencer Cosby, Board of Engineer Officers, on test.'^g Hydraulic
Cements. I^ofessional Papers, No. 28» Corps of Engineers, U. S. A., 1901.

Unfortunately, tests for acceptance or rejection must be made on a product
which has not reache<l its iiual stage. A cement, when incorporated iu masonry,
undergoes chemical changes for mouths, whereas it is seldom possible to
continue tests for more than a few weeks at the most.

A few tests, carefully made, are more valuable than many, made with iess cai«.

Cement which has been in storaipe for a long time should be eareftelly
tested before use, in order to detect deterioration.

A cement should be rejected, without regard to the proportion of fnilares
among saniplea tested, if the samples show dangerous variation in quality or
lack of care in manufacture, and resulting lack of uniformity in the product.

The practice of ottering a bonus for cement showing an abnormal strength
Is oiyeetionable, as it leads to the production of cements with defects not
easily detected.

For Portland or Puzzolan cement, make tests for (1) fineness of grinding ; (2)
specific gravity ; (3) soundness, or constancy of volume in setting; (4) time of
setting, and (5) tensile strength. For Natural cements omit tests (2) :ind (3).

(1) Fineness. Cemen tit ions quality resides principally, if not wholly, in
the very Unely ground particles. Use a No. 100 sieve, woven from brass wire
No. 4C Stubs gage; sift until cement ceases to pass through. The percentage
that has passed through is determined by weighing the residue on the sieve.
The screen should be frequently examined to see that no wires have been
displaced. See p 937.

{%) Speeifie sraTity. The specific gravity test is of value in determining
whether a Portland cement is unadulterated. The higher the burnin,<x, short of
vitrification, the better the cement and the higher the specific gravity. If under^
burned, the specific gravity of Portland cement may fall below S ; if overburned.
it may reach 3.5. Natural cement has a specific gravity of about 2.5 to 2.8, and
Puzzolan about 2.7 to 2.8.

The temperature may vary between 60° and 80° F. Any approved form of
volumenometer or specific gravity bottle may be used, graduate to cubic centi-
meters with decimal subdivisions. Fill the instrument to zero of scale with
benzine. Take 100 grams of sifted cement that has been prevlouslv dried by
exposure on a metal plate for 20 minutes to a dry heat of 212° F., a!id allow it to
pass slowly into the benzine, taking care that the powder does not stick to the
sides of the graduated tube above the fluid^nd that the funnel, through which
it is introduced, does not touch the fluid. The approximate specific gravity will
be represented by 100 divided by the displacement in cubic centimeters. The
operation requires care.

(8) Stonndness, and (4) settings qnalltftes. The temperature should
not vary more than 10° from 62° F. For Portland cement use 20, ifor Natural SO,
and for'Puzzolan 18 per cent, of water by weight. Mix thoroughly for 5 minutes.
On glass plates make two cakes about 3 inches in diameter, }^ inch thick at the
middle and drawn to thin edges, and cover them with a damp cloth. At the end

of the minimum time specified for initial set, apply needle -^ inch diameter,

weighted to % pound. If an indentation is made, the cement pafises the require-
ment for initial setting. Otherwise the setting is too rapid. At the end of the

maximum time specified for final set, apply the needle -^ inch diameter, loaded

to one pound. If no indentation is made, the cement passes the reqairemeut fbr
final set. Otherwise the setting is too slow.

(Tonerally speaking, both periods of set are lengthened by increase of moiftture,
and shortened by increase or temperature.

*By Portland cement, in this report, is meant the product obtained by
calcining intimate mixtures, either natural or artificial, of argillaceous and
calcareous substances, up to incipient fusion. By Natural cement is meant
one made by calcining natural rock at a heat below incipient fbsion, and grind-
ing the product to powder. By PnSsolan is meant tne product obtained by
grinding slag and slaked lime, without subsequent calcination.

TfiSTIKG. 941

Becommendafions of Board of IT. S# A* Engineer
r Officers. Continued.

In gaging Portland cement in damp weather, the samples shoald be thoroughly
dried before adding water. This precaution is not deemed necessary witn
Natural cement. Sufficient uniformity of temperature will result if the testing
room be comfortably warmed in winter, and if the specimens be kept out of the
sun in a cool room in summer, and under a damp cloth until set. Temperatures
may vary between 6QP and 9fP F., without atteoting resnlts more than the
probable^error in the observation.

Boiling test. Place the two cakes under a damp cloth for 24 hours. Place
one of them, still attached to its plate, in water 28 days ; immerse the other in
irater at about 7(P F., and let it be in a rack above the bottom of the receptacle;
heat the water gradually to the boiling point, maintain the heat for 6 hours and
then let cool. The boiled cake should not warp or become detached from the
plate, or show expansion cracks. If the cold-water cake shows evidences only
of swelling, the cement may be used in ordinary work in air or fresh water for
lean mixtures, but if distortion or expansion cracks appear in it, the cement
should be rejected*

Aeeeleratecl tests are not generally recommended, but where a test must
be made in a short time, the boiling test is considered about the best. It not
only gives short-time indications, but at once directs attention to the presence
of ingredients which might lead to disintegration. On the other hand, it may
lead to the rejection of a cement which would behave satisfactorily in actual
work and which would stand the test after air-slaking. Sulphate of lime, while
enabling cements to pass the boiling tests, introduces an element of danger.

(6) Tensile tests are preferred to flexural or compressive tests. Sand
tests are the more important and should always be made ; and neat tests should
be made if time permits.

A cement which tests moderately high at 7 days, and shows a substantial
increase in strength in 28 days, is more likely to reach the maximum strength
Biowly and retain it indefinitely with a low modulus of elasticity, than a cement
which tests abnormally high at 7 days with little or no increase at 28 days.

Use briqu^tes or the form recommended by the American Society of Civil
Ekigineers,* measuring 1 inch square in cross-section at place of rupture, and
held bv close-fitting metal clips, without rubber or other yielding contacts. The
teitB should be made immediately after taking the briquettes from the water.

Meat tenndle tests. Use unsifted cements. For Portland cement, use
20; for Natural, 30; and for Puzzolan, 18 per cent, water by weight. Place the
cemen t on a smooth non-absorbent slab ; in the middle make a crater sufiicient to
hold the water; add nearly all th^ water at once, the r^nainder as needed ; mix
thoroughly by turning with the trowel, and vigorously rub or work the cement
for 5 minutes.

Place the briquette mold on a glass or slate slab. Fill the mold with consecu-
tive layers of cement, each to be 3>^ inch thick when rammed. Give each layer
80 taps with a soft brass or copper rammer weighing 1 pound, having a face %
inch diameter or 0.7 inch square, and falling alx)ut |^ inch.

After filling the mold and ramming the last layer, strike smooth with a trowel,
tap mold lightly on side, to free, cement from plate, remove the plate, and leave
for 24 hours, covered with a damp cloth. Then remove the briquette from the
mold and immerse it in fresh water, which should be renewed either continu-
ously or twice ttx each week during the specified time.

Tensile tests with sand. For Portland and Puzzolan cements, use 1
part cement to 8 parts sand ; for Natural or Rosendale, 1 to 1. Use crushed
quartz sand, passing a No. 20 standard sieve, and being retained on a No. 30
standard sieve.

After weighing carefully, mix dry the cement and sand until the mixture is
uniform, add the water as in neat mixtures, and mix for 5 minutes. The con-
stituents should be well rubbed together.

For maximum strength in tested briquettes, Portland cements reanire
water = 11 to 12)^ p^ cent, by weight of constituent cement ; Naturu, 15 to
17; and Puzzolan, 9 to 10.

A machine which applies the stress automatically and at a nnlform rate

* See page 989.

942 CEMEIH!'.

Becominendatloiis of Board of U. 8. A* Enslneer
Officers. Continued.

of inerease is preferable to one oontrolled entirely by hand. The stveas
should be increased at the rate of about 400 flbs. per minute. A rate materiidly
greater or less than this will give different results.

The highest tensile strength from each set of briquettes made at any one time
is to be considered the governing test.

Field teste are recommended, whether or not the more elaborate tests
above described have been made. In connection with tests of weight and fine*
ness, and observations of texture and hardness in the work, field t^ts ofteD
suffice for well-known brands, showing whether the cement is genuine and
whether it is reasonably sound and active. Pats and balls of neat cement from
the storehouse, and of mortar from the mixing^ platform or machine, should be
frequently made. Estimate roughly the setting and hardening qualities by

Eressure of the thumb-nail ; hardness of set and strength by breaking with the
and and by dropping upon a hard surface. The boiling test may also be used.
Should the simple tests give unsatisfactory or suspicious results, then a full series
of tests should DO careftilly made.
A cement may be rejected if it fails to meet any of the following requirementa

t Reqalremento.

Portland.
Slow. Quick.
Fineness. Percentage to pass through a No.

100 sieve as in (1) 87 to 92*

Specific gravity. Between 3.10 3.10

and 8.26 3.25

Time of setting. Initial, not less than 45 m. 20 m.

nor more than 30 m.

Final, not less than 46 m.

nor more than 10 h. 2.6 h.

Tensile strength, neat,

- 7 / 7day8t 460 400

B)s. per sq. in. j 28 daysf 640 480

Tensile strength. With sand, as in (6).

» ^ : / 7 daysf 140 120

lbs. per sq. m. |28 daysf 220 180

*92 per cent, is quite commonly attained by high-grade American Portlands,
but rarely by imported brands. For the latter, use 87.
f B<^ect any cement not showing an increase at 28 days over 7 days.

Natural.

Puzzolan.

80

Not

given

20 m.

97
2.7
2.8

45 m.

'4h!

10 h.

90
200

350
500

60
160

140
290

CONCRETE. 943

CONCRETE.

Cement concrete is an artificial stone, composed of cement mortar
mixed with an "aggregate " consisting generallv of broken stone, but often of
gravel and sometimes of brick-bats, oyster shells, etc., or of all these together.

In concrete, as in mortar, it is advisable, on the score of strength, that all
the voids be filled or overfilled .

Taking the voids, in broken stone, in gravel and in sand, at SO per cent,
of the mass, or equal to the solid portion, we have, for 1 cubic yard of
concrete :

1 cu. yd. broken stone ) (4 parts stone

0.5 cu. yd. sand > •■ -< 2 parts sand

0 .25 ou. yd. cement j i 1 part cement

or

1 cu. yd. broken stone
0.5 cu. yd. gravel
0.25 cu. yd. sand,
0.125 cu. yd. cement

8 parts sandstone
4 parts gravel
2 parts sand
1 part cement.

Caution. When a solid body is reduced to a mass consisting of broken
pieces separated by voids, the increase in bulk is due solely to the voids, and
18 equal to the space occupied by them. Hence the ratio between the in-
crease of bulky or " swelling.** and the original bulk is that of the voids
to the original, and not to the final bulk. Thus, if a solid cubic yard of stone,
after being broken into pieces, occupies twice as much space as before, then
the increase in bulk, or the space occupied by the voids, is » that occupied
by solid pieces = Aa// that occupied by the entire broken mass.

It is doubtful whether hard roclc* when blasted and made into embank-
ment, settles to less than 1.67 cubic ^ards for each original cubic yard. As
the result of certain embankment m hard sandstone, Mr. EHlwood Morris
gives 1.417 yards of embankment for each solid yard; but the rough sides of
rock excavations make it difficult to measure them with accuracy, and this
may have affected his result.

Stone Crushers. Principal sizes. From catalogue of Farrel Foundry
and Machine Co. For prices, see Price-list.

Sixe of Stone, Capacity
Ins. Tons.

7X10 {«g
16X10 {120

per Day,
Sise, Ins.*

?n

Horse-power
Required.

8
15

8,000
16,400

24 X 13 {f^
36 X 24 {|gg

3 I
5

30
65

20,000
50,000

Concrete mixers are of

various tyi

MS. Some have a

fixed tro

with a revolving; worm; others a square^ dox revolving about its diagonal
axis, or a revolvmg drum. The gravity mixer is simply a steel trough, setat
an an^e of 18** to 22^ with the vertical, and armed intemalljr with fixed
steel pins and deflectors. The materials for the concrete are fed in at the top
of the trough, and, passing downward through it, are mixed by their contact
with the pins and deflectors. The following figures, from the catalogue of
the Drake Standard Machine Works, refer to the type first named.

Capacity, Horse-power Weight,

Cu. Yds. per Day. . Required. Lbs.

400 20 5,500

200 25 4.000

100 15 2,700

75 10 1,800

25 6 1,600

Compressive Strength.

The strength of concrete is affected by the quality of the broken stone, as
well as bjr that of the cement, the degree of ramming, etc.
Ramming adds about 50 per cent, to the strength.
Slow-settmg cements are best for concrete, especially when to be rammed.

944

CONCRETE.

Experiments on 12-lnch conerete cubes, rammed in cast iron
molds, by A. W. Dow, U. S. Inspector of Asphalt and Gement.* The
cubes were thoroughly wet twice daily.

The results for one year are means of five cubes ; the rest are means of two
cubes. Deduct from 3 to 8 per cent, for friction of press.

The materials were as follows:

Cement. Portland Natural

Per cent, retained on sieve of 100 meshes per linear inch, 8.5 14

Time for initial set, minutes 190 20

" " hard " " 305 36

Tensile strength as follows, lbs. per square inch:

1 Day. 7 Days. 1 Mo. 3 Mos. 6 Mos. 1 Year.

Portland, neat, 441 839

" 3 parts stan-
dard broken quartz, 248 429 398 428 474
Natural, neat, 96 180

dard broken quartz, 91 188 327 414 48fi

Sand used in concrete.

No residue on a No. 3 sieve; 0.5 per cent, passed No. 100. Voids 44 per
oent., with 4.4 per cent, water.

Broken Stone. Gneiss. Of Nos. 6 and 12 (table below) 3 per cent
retained on 2.5 inch mesh; all on li inch. Others, 0 retained on 2.5 inch;
nearly all on 0.1 inch. For voids, see table, below.

Gravel. Clean quartz, passing a If-inch mesh, 2 per cent, passing a Na
10 mesh. Voids, 29 per cent.

Water. With Portland cement, 0.09 cu. ft. ( = 5.7 lbs.) per cu. ft. of
rammed concrete; with natural cement, 0.12 cu. ft. ( = 7.5 lbs.).

Crushing Strength of 12 in. Concrete Cubes, in lbs. per sq. in.

£bcperiments by A. W. Dow, as above :
Parts by volame ; cement, 1; sand, 2; aggregate, 6.

Aggregate

VoidR in Aggregate.

Crushing Strength,
lbs. per sq. in., after

No.

Stone,
rts.

OQ

1

Per

Cent.

of Vol.

Per Cent,
of Voids

10

45

3

6

1

i

Filled with
Mortar.

Days.

Days.

Mos.

Mos.

Year.

«

PQ

o

a 8

6

45.3

83.9

908

1790

2260

2510

3060

3

3

35.5

107.0

950

1850

• •

2070

2760

i ' 9

4

2

37.8

100.6

• •

• ■

• •

• •

2840

t 10

6

• •

39.5

96.2

• •

* •

• •

• ■

2700

*^ 12

* •

6

29.3

129.1

694

1630

2680

1840

2820

6

• ■

45.7

83.9

• •

• •

1630

1530

1860

1

6

« •

45.3

83.9

228

539

376

795

015

g 2

3

3

35.5

107.0

108

364

593

632

841

9 3

4

2

37.8

100.6

• •

• •

• •

• •

915

"S 4

6

• •

39.5

96.2

• •

• •

• •

• •

800

5^ 5

• «

6

29.3

129.1

87

421

361

344

763

6

6

• •

45.7

83.9

• •

• •

596

• *

829

To test the effect of the quantity of water used, Mr. Dow made four
12-inch cubes, of 1 part ]^ortland cement, 2 parts sand, 6 parts coarse gravel;
two so wet that water ran from the concrete when tamped in the mold; the
other two with just sufficient water to make a plastic paste, as in tlie tests

* Report of the Operations of the Engineering Department of the District
of Columbia for the year ending June 30, 1897.

CONCRETE BEAMS. 945

corded in the table. ^ At the end of a year the two wet cubes gave 2300 and
2500 lbs. per square inch respectively; average. 2400. The drier cubes gave
2513 and 3037 respectively; average, 2775.

In the floors of graving docks at Genoa, Italy, concrete of 1 part Portland
cement, 2 parts sand, 3 parts small gravel, carries safely 107 Ids. per square
inch; safety factor 15. In the concrete bridge over the Danube at Munder-
kingen, Germany, the maximum pressure is said to vary from 500 to 560 lbs.
per square inch. Test pieces broke with from 2000 to 4650 lbs. per sq. inch,
varying with their age.

TransTerse Strength.

Concrete Beams. Tests bv Boston Transit Commission, 1895-96.
Beams 6 ins. square; spans 30 and 60 ins. ; age from 29 to 37 days, 24 hours
in air, remainder in water.

S » modulus of ruptiu^ >» unit stress in extreme fibers at instant of rup^

ture, in lbs. per sq. inch.
a

W '- 7^ "> extraneous center breaking load, in lbs., on a beam 1 inch

lo

square, 1 ft. span.

Mixture.
Portland Sand. Stone,
cement.

1 2 4i 40 t4
1 2i 4 38 t<

Effect of size of stone. Mixture, 1 -
wet ground. 6 tests for each size.

Wt.,Lbs.perCu.Ft.

9StS

ests

-2i — 4;

Voids.
Per Cent

48
45
40
40
37
33i.

S.

Max.

531
441

24 hours

Min.

158
131

in air, :

S.

A

Av'ge.

304
280

29 days Id

Stone. Concrete.

2 to 1.5 95 156
1.5 to 1 95 156
1 to 0.5 99 150
0.5 to 0.25 100 150
0.25 and less 94 142
mUed 108 152

Max.

328
349
305
312
248
349

Min.

263
272
219
229
214
230

S

Av'ge.

298
311
263
261
227
293

When masonry is backed by concrete, the two are liable in time
to crack apart, owing to unequal settlement, especially if the ramming has
not been thorough.

In variable climates, cast-iron cylinders, filled with concrete, are

frequently split horisontally by unequal expansion and contraction. In
nucn structures it is safest to consider the cylinders as mere molds for the
concrete ; and to depend only upon the concrete for sustaining the load.

Molded blocks of Portland concrete, of even 50 tons weight, can
generally be handled and removed to their places in from 1 to 2 weeks.

Ramming of concrete; when properly done, consolidates the mass
about 5 or 6 per cent., rendering it less porous, and very materially stronger.
The rammers, like those used in street paving, are of wood, about 4 ft. long,
6 to 8 ins. diameter at foot, with a lifting handle, and shod with iron ; weight
about 35 tbs. They are let fall 6 or 8 ins. The men using them, if standing
on the concrete, should wear gum boots to preserve their feet from corrosion
by the cement.

Ramming cannot be done under water, except partially, when the con-
crete is inclosed in bags. A rake may, however, be used gently for leveling
concrete imder water.

The size of the broken stone for concrete is generally specified not
to exceed about 2 ins. on any edge ; but if it is well freed from dust by screen-
ing or washing, all sizes, from 0.5 to 4 ins. on any edge, may be used, care
bemg taken that the other ingredients completely fill the voids.

60

946 CONCRETE.

Concrete b used for brlngrlnir up uneven foundations to a level

before starting the masonry. Bv tnis means the number of horisontal
joints in the masonry is equalized, and unequal settlement is thereby pre*
vented.

Concrete may readily be deposited under virater in the usual
way of lowering it, soon after it is mixed, in a V-shaped box of wood or plate
iron, with a lid that may be closed while the box descends. The lid, however,
is often omitted. This box is so arranged that, on reaching bottom, a pin may
be drawn out by a string reaching to the surlaoe, thus permitting one of tbe
sloping sides to swing open below, and allow the concrete to fall out. The
box is then raised to be refilled. In large works the box may contain a cubic
yard or more, and should be suspended from a ti^veling crane, by which it
' can readily be brought over any required spot in the work. The concrete
may if necessary be gently leveled by a rake soon after it leaves the box. Its
consistency and strength will of course be impaired by falling through the
water from the box ; and moreover it cannot be rammed under water with-
out still greater injury. Concrete has been safely deposited in the above-
mentioned manner in depths of 50 ft.

The Tremle, sometimes used for depositing concrete under water, is a
box of wood or of plate iron, round or square, open at top and bottom,
and of a length suited to the depth of water. It may be about IS ins. diam.
Its top, which is always kept above water, is hopper-shaped, for receiving the
concrete more readily. It is moved laterally and vertically by a traveling
crane or other device suited to the case. Its lower end rests on the river
bottom, or on the deposited concrete. In commencing operations, its lower
end resting on the river bottom, it is first entirely filled with concrete, which
(to prevent its being washed to pieces by falling through the water in the
tremie) is lowered in a cylindrical tub, with a bottoin somewhat like the box
before described, which can be opened when it arrives at its proper place.
When filled, the tremie is kept so by fresh concrete, thrown into the hoi)per
to supply the place of that which gradually falls out below, as the tremie is
lifted a little to allow it to do so. The weight of the filled tremie compacts
the concrete as it is deposited. A tremie had better widen out downward
to allow the concrete to fall out more readily.

The area upon which the concrete is deposited must previously be sur-
rounded by some kind of inclosure, to prevent the concrete from spreading
beyond its proper limits ; and to serve as a mold to give it its intended shape.
This inclosure must be so strong that its sides may not be bulged outward by
the weight of the concrete. It is usually a close crib of timber or plate iron
without a bottom ; and will remain after the work is done. If of timber it
may require an outer row of cells, to be filled wjth stone or gravel for sink-
ing it into place. Care must be taken to prevent the escape of the concrete
through open spaces under the sides of the crib or inclosure. To this end
the cru) ma>^ be scribed to suit the inequalities of the bottom when the latter
cannot readily be leveled off. Or inside sheet piles will be better in some
cases ; or an outer or inner broad flap of tarpaulin may be fastened all around
the lower edge of the crib, and be weighted with stone or gravel to keep it in
place on the bottom. Broken stone or g^ravel or even earth (the last two
where there is no current), heaped up outside of a weak crib, will prevent the
bulging outward of its sides by the pressure of the concrete. After the con-
crete has been carried up to within some feet of low water, and leveled off, the
masonry may be started upon it by means of a caisson, or by men in diving
suits. Or, if the concrete reaches very nearly to low water, a first deep
course of stone may be laid, and the work thus brought at once above low
water without any such aids.

The concrete should extend out from 2 to 5 feet (accoriding to the
case) beyond the base of the masonry. All soft mud should be removed
before depositing concrete.

Bags j>artly filled with concrete, and merely thrown into the water,
are used in certain cases. If the texture of the bags is slightly open, a por-
tion of the cement will ooze out, and bind the whole into a tolerably compact
mass. Such bags, by the aid of divers, may be employed for stopping leaks,
underpinning, and various other purposes, that may suggest tnemselves.
Such bags may be rammed to some extent.

Tarpaulin may be spread over deep seams In rock to prevent

the loss of concrete; and, in some cases, to prevent it from being washed
away by springs.

^

LAITANCE. 947

"When concrete is deposited in water, especially in the sea, a pulpy gela-
tinous f9uid exudes from the cement, and rises to the surface. This causes
the water to assume a milky hue; hence the term laitance, which French
engineers apply to this substance. As it sets very imperfectly, and, with
some varieties of cements, scarcely at all, its interposition between the layers
of concrete, even in moderate quantities, will have a tendency to lessen, more
or less sensibly, the continuity and strength of the mass. It is usually re-
moved from the inclosed space by pumps. Its proportion is greatly dimin-
ished by reducing the area of concrete exposed to the water, by usmg largg
boxes, say from 1 to li cu. yds. capacity, for immersing the concrete."

948 MODERN EXPLOSIVES.

MODEEN EXPLOSIVES.

Art. 1. Mest of tlie explosives, which, of late years, have been takine
the place of gunpowder, consist of a powdered substaDce, partly saturated
<rith nitro-glycerine, a fluid produced by mixing glycerine with nitric and sul-
phuric acids.

Art. 2. Pure nltro-srlycerlne, at G0° Fah, has a sp grav of 1.6. It is
odorless, nearly or quite colorless, and has a sweetish, burning taste. It is poison-
ous, even in very small quantities. Handling it is apt to cause headaches. It is
insoluble in water. At about 306° Fah it takes flre, and, if unconfined, burns
narmlessly, unless It is in such quantity that a part of it, before coming in coo-
tact with air, becomes heated to the exploding point, which is about 880° Fah.

N-G, and the powders containing it, are always exploded by meauns at
feharp percussion. See Arts 36, Ac. After N-G is made, great care is required to
wash it completely from the sarplus acids remaining In it from the
process of manufacture. Their presence, either in the liquid N-6, or in the
powders containing it, renders tne N-G liable to spontaneous decomposition,
Which, by raising the temperature, increases the danger of explosion.

Art. 3. N-O freezes at about 45° Fah. It Is then very cliflicalt of
explosion, and must be thawed gradually j as by leaving it for a sufficient
length of time in a comfortably warm room, or by placing the vessel containing it
In a second vessel containing hot water, not over 100*^ Fah ; but never by exposing
It to intense heat, as in placing it before a fire, or setting it on a stove or boiler.
Elxtra strong caps are made for exploding N-G and its powders when frozen.

Art. 4. N-G, owing to its iucompressibility, is liable to explosion
tbrouifb accidental percussion. This, and its liability to leafc-
ag^e, render it inconvenient to transport and handle. Hende it is rarely used in
the liquid state in ordinary quarrying and other blasting. In the oil regions of
Penna, it is largely used in oil w^ells, in order to increase the flow. I<'or this
purpose it is couhned in cylindrical tin casings, from 1 to 5 inches diani, called
torpedo-shells. These are suspended from, and lowered into the well by means
>f, a cord or wire wound on a reel ; and are destroyed when the charge is ex-
ploded. They are about 1 inch less in diam than the well, and contain usually
from one to twenty quarts = 3 lbs, 5^oz to 66 lbs, 63^ oz of N-G. They are
pointed at their lower ends, in order to facilitate their passage through the oil or
water which may be in the well. When a greater charge than about 6S^ lbs is
required, two or more of these shells are placed in the well, one on top of another,
the conical point on the lower end of each one fitting into the top of theon^ next
below. In this case, the N-G is fired by means of a cap or series of caps placed
In the top of the charge before it is lowered. When the charge is in place, the
caps are exploded by electricity led to them by conducting wires, as in Art 37, or
(as in the method more commonly practised) by letting a weight fall on them.

When a well has been repeatedly torpedoed, and a cavity has thus been formed
in it so large that the space sunounding a torpedo would interfere too greatly
with the eflfect of the explosion of the N-G oh the walls of the well, the latter is
placed directly in the well, by lowering a tin cylinder, filled with it, and pro-
vided with an automatic arrangement which allows the N-G to escane when at
the bottom of the well. The N-G is then fired by a torpedo suspendea on a line,
and having caps placed in its top. These caps are exploded by a leaden or iron
weight sliding down the line, or by electricity. When the rock is seamy, the
N-G is confined in short cylindrical tin shells, lowered into the cavity, and fired by
a torpedo. It is also used for increasing the flow of springs of water. It of course
cannot be used in hor or npivard holes, such as often occur in tunneling, Ac.
Art. 5. N-G explodes so suddenly that very little tamplna: Is re-
quired. Moist sand or earth, or even water, is suflScient. This, witli the fact
that N-G is unafiected by immersion in water, and is heavier than water, render
it particularly suitable for sub-aqueous work, or for holes containing
water, provided the rock has no seams which would nermit the N-G to escape.
If the rock is seamy, the N-G must be confinea in a water-tight casing.
Such casings, however, necessarily leave some spaces between the rock and the
explosive, and these diminish considerably the effect of the latter.

Art. 6. The great explosive force of N-© Is due partly to the very
large volume of gas i n to which a small quantity of it is converted by explosion, and

MODEBN EXPLOSIVES. 949

partly to the mddtnneit with which this conTorsion takes place, the gases being
liberated almost instantaneously,* while with gunpowder their liberation requires
a longer time. The suddenness of the explosion increases its effect, not only by
applying all of its force practically at one instant, but also by greatly heating the
gases produced, and thus still further increasing their volume.

Art, 7. The liquid condition of N-G is useful in causing it to fill tbe drlll-
liole comipletely, so that there are no vacant spaces in it to waste the force
of the explosion: On the other hand, the liquid form is a disadvantage, becausei,
when thus used without a containing vessel in seamy rock, portions of the N-0 leak
away and remain unexploded and unsuspected, and may cause accidental explosioi
at a future time.

Art. 8. N-d is stored In tin cnns or earthenware Jars. Iff
properly washed from acid it does not injure tin. For transportation, these cans o%
jars are packed in boxes with sawdust, or in padded boxes, and loadM in wagonf^
The B R companies do not receive it.

Art. 9. When N-G and its compounds are eompUtdy exploded, tlie i^A*^
iriven out are not tronblesome, but those resulting from incomplete explosion
such as generally takes place, or from combustion, are very offensive.

Art. 10. For convenience, we apply the name ^'dynamite '* to any expla
aive which contains nitro-glycerine mixed with a gn^nular absorbent; *^trn«'
^lynamite" to those in which the absorbent of the N-G is **Kieselguhr,"t ot
some other inert powder which takes no part in the explosion; and ^*fals^
dynamite ** to those in which the absorbent itself contains explosive substance*
other than N-G.

Art. 11. Tiie atoorbent, by its granular and compressible condition,
aets as a cushion to the Jf-G^ and protects it from percussion, and from
the consequent danger of accidental explosion.

N-G undeivoee no change in composition by being absorbed ; and it then freezes,
burns, explodes, &c, under the same conditions as to pressure, temperature, Ac, af
when in the liquid form. The cushioning effect of the absorbent merely renders it
more difficult to bring about sufScient percussive pressure to cause explosion. The
absorption of the N-G in dyn enables the latter to be used in hor holes, or in holes
drilled upward.

Art. 12. N-G and dyn explode much more readily when rlfi^idly
eonflncNlf as by a metallic vessel, or by the walls of a hole drilled in rock, than
when confined by a yielding substance, as wood. Therefore the fact that dyn, not
being liquid, can be packed in wooden boxes, renders it safer than N-0 which has to
be kept in stone or metal vessels.

Art. 18. True dynamites must contain at least about 50 per
cent of N-G. Otherwise the latter will be too completely cushioned by the absorbent,
and the powder will be too difficult to explode. False dynamites, on the contrary,
may contain as small a percentage of N-G as may be desired; some containing aa
little as 15 per cent. The added explosive substances in the false dynamites generally
contain large quantities of oxygen, which are liberated upon explosion, and aid in
•ffacting the complete combustion of any noxious gases arising from the N-G.

Art. 14. Dynamites which contain larse percentagres of
K-O explode (like the liquid N-G. Art 6) with great suddenness, tending to thaUer
the rock in their vicinity into small fragments. They are most useful in very hard
rock. In such rock, STo 1 dynamite, or that containing 75 per cent of N-G, is
roughly estimated to have about 6 times the force of an equal wt
ofKunpowder.

For soft or decomposed rochs. sand, and earth, the lower grades
of dynamite, or those containing a smaller percentage of N-G, are more suitable.
They explode with less suddenness, and their tendency is rather to upheave large
masses of rock, Ac, than to splinter small masses of it. They thus more nearly re«
semble gunpowder in their action.

Judgment must be exercised as to the g^rade and quantity of explosive
to be used in any given case. Where it is not objectionable to break the rock into
small pieces, or where it is desired to do so for convenience of removal, th« higher,
ihattering grades are useful. Where it is desired to get the rock out in large masses,
as in quarrying, the lower grades are preferable.

• For very difficult work in hard rock, and for sabmarine blasting, the highest
grades, containing 70 to 75 per cent of N-G, are used. A small charge of these does
the same execution as a larger charge of lower grade, and of course does not require

* Such sudden liberation of gas is called ^* detonation."

t KlMelguhr if an ssrtbj, silieious Hmestooe, composed of the fossil remains of small sheila

■ash shell aeu as a mlaote reeeptaole for aitro-f 1 joerine. Kieaelf ohr is found In Hanover, Oermaaj-

aa4 in New Jersev*

950 MODERN EXPLOBIVES.

the drilling of 00 Urg* a hole, lu rabmarine work tbeir sharp exploaioa to aal
deadened by the water.

For general railroad trork, ordinary tonneling, mining of ores, Ac, the »irer>
ag^ vra^le. containing 40 per cent of N-G, is used ; for quarrjrlmgp, 35 per
cent; Tor blasting^ stumps, trees, piles, Ac, 80 per cent; for s»ita «md
•artb, 16 per cent.

Art. 15. Dynamite, like N-O, can be readily exploded under
water, provided it is so immersed as not to be ieattered ; but lon^p ea^poanre
to water is injurious to it. In the higher grades, the water, by it« greater
affinity fur the absorbent, drives out the N-Q. In the lower grades it is apt to wash
away the salts used as additional explosives.

Art. 10. In dyns containing a large percentage of N-O, the latter is liable
to exude in liquid form, or to *' leak," especially in warm weather, and then to
explode tbroiigh accidental percassion. The same danger exists, even though the
percentage of N-G be small, if the absorbent has but small absorbing power, and ^
consequently, easily saturated.

Art. 17. True dyn resembles moist brown sugar. Its properties an
generally those of the N-G contained in it. Thus, it takes fire at about 360<> F, and
burns freely. It freezes at 46° F, and is then difficult to explode. It is not exploded
by friction, or by ordinary percussion, but requires, for general purposes, a atrong
cap, or exploder, containing fulminating powder, see Arts 36, 88, Ac. It may, how*
ever, be exploded by, a priming of gunpowder, tightly tamped, and fired by an ontt*
nary safetv-fuse.

Art. 18. Tbe cbargre sbonld fill tbe eross section of tbe
bole as completely as possible. If water is not standing in tlie hole, tbe cartridge
should be cut open before insertion, so that the powder may escape ftt>m it and fiU the
hole ; or the powder may be simply emptied from the cartridge into the hole.

Art. 19. For blasting^ ice in place, holes are cut in ft, and a number ofdya
cartridge's (one of which must contain an exploding cap) are tied together and low-
ered from 1 to 6 ft into the water. They are fired as soon as possible after immer*
sion, to avoid the danger of freesing. Electrical exploders (Arts 87, Ac,) are best
for sub-aqueous work.

Art. 20. Dyn is useful for breaking up pieces of metal, such as old
cannon, condemned machinery, ** salamanders " (masses of hardened slag) in bla^t
furnaces, tc. In cannon, the dyn is of course exploded in the bore. In other piecea^
small holes are generally drilled to receive it; but plates, evened considerable thick*
ness, may be broken by merely exploding dyn upon their surface.

Art. 21. For blastinfp trees or stumps, one or more cartridges are
fired in a hole bored in the trunk or roots, or under the latter. This shatters botli
trunk and roots. A tree may be felled neatly by boring a number of
small radial holes into it, at equal short dists in a hor line around its circnmf, and
by means of an electric battery (Arts 37, Ac), exploding simultaneously a smaU
charge of dyn in each. Or a single long cartridge may be tied around the tnmk ai
a small tree, and fired.

Art. 23. Piles may be blasted in the same way aa trees ; or a hole may
be bored for the cartridge in the axis of the pile; or the cartridge may be simply
tied to tbe side of the pile at any desired ht.

Art. 23. The higher grades of dyn, like N-G, require bat little tmntp*
inir* Uiie a wooden tampins^-bar, never a metalMe one, for any explosive, if
a charge of dyn *" banffS fire/* it is dangerous to attempt to remove it. Remove
the tampiruj^ all but a few ins in depth, on top of which insert another cartridge,
containing' an exploder, and try again. See electrical exploders, Arts 87, Ac. Dyn,
like N-G, if frozen, must be thawed graduMy^ by leaving it in a warm room, far
from tbe fire; or by placing it in a metallic vessel, which is then placed in another
vessel containing hot water. The water should not be hotter than can be borne by
the hand. Otherwise the N-0 is liable to separate from the absorbent. The K-0 la
dyn may freeze without cementing together the particles of the absorbent; ia
which case the powder of course is still soft to the touch. An OTereiiar|pe of
N-Q, or of dyn, is liable to be burned, and thus wasted, giving off ofiensive gasea.

Art. 24. Dyn is sold in eyllndrleal, paper>eoTerea cart*
rldgpes, from J^ to 2 ins in diam, and 0 to 8 ins long, or longer. They are taj>
Dished to order of any required size, and are packed in boxes containing 25 lbs or 61
lbs each. The layers of cartridges are separated by sawdust.

Art. 29. Some of the B R companies decline to carry dyn or N-O ia aajr
shape. Others carry dyn under certain restrictions, based upon State laws; pro-
viding that it must be dry (i e, that no N-Q shall be exuding from it); that boxea
and cars containing it shall be plainly marked with some cautionary words, as ** ex-
plosive,** ** dangerous,'* Ac; that the cartridges shall be so packed in the boxes, and
the boxes so loaded in the cars, that both shall lie upon their sidex^ and tbe boxsf

HODEBH EXPLOSIVES.

951

be In no danser of falling to the floor ; that caps, &c, shall not be loaded in the
same car with dyn, Ac. <&c.

Art. 26. A sreai many Tarletles of dyn are made. They differ
(ffenerallj but slightly) In the composition of the absorbent, and in the method
of manuracture. Each maker usually makes a number of grades, containing
different percentages of N-Q, <&c, and 'gives to his powders some fanciful name.

Art. 27. The following table of explosiTes, made by the Repauno
Chemical Co, Wilmington, Del, and known as ** Atlas" powders, gives the
percentage of N-G iu each.

Brand.

Percentage
of N-G.,

Brand.

Percentage
of N-Cf.

A

75

I)+

33

B+

60

B

60

E+

27

C+

45

E

20

c

40

The absorbents contain : in *' A" brand, 18 per cent wood pulp and 7
per cent carbonate of magnesia; in " G" brand (the average grade), 46 per cent
nitrate of soda (soda saltpetre), 11 per cent wood pulp, anaS per cent carbonate
of magnesia ; in " E " brand, 62 per cent nitrate oT soaa, 16 per cent wood pulp.
See, and 2 per cent carbonato of magnesia.

Art. 28. ^ Miner's Friend ^ powder contains nitrate of soda, wood
palp, resin, and carbonate of magnesia. It freezes at 43P, and is then, like other
dyn, difficult to explode. When used under water, the cartridges should not be
broken, because the powder is injured by direct contact with water. Their
•* Hecla" powder is a lower grade. It is in granulated form, like ordinary
blasting powder, but is said to be much stronger. It is intended as a substitute
for it.

Art. 20. ** iidant" powder is dyn proper, containing 75 per cent N-G,
and 25 per cent Kieselguhr obtained near their works In Sexr Jersey. The
lowest grade, branded *'M," contains 20 per cent N-G. The name "giant
powder" was originally applied to dynamite in general.

Art. 30. Other brands are ** Hercnles " powder and ** Jndson R R P
powder," a substitute for ordinary blasting powder. It is put up in water-

Kroof paper bags, of 63^, 12^, and 25 fi>s each, and these are packed in wooden
ozes holding 60 ft>s eacn. *^ Judson F F F dynamite '* is a higher grade,
in cartridges of the usual shape, packed in 50-fi> boxes.

Art. 31. ^ Raclcaroek '' cartridges are said to contain no N-G, and to
Ae entirely inexplosive until immersed, for a few seconds, in an inexplosive
liquid furnished by the same Co. They are then allowed to stand for 15 mins,
alter which they may be used at any time. ' They are fired in the same way as
dyn,- and can be used under water. The mfrs claim that they " approximate
N-G in strength, and are stronger than dyn."

Art. 32. The following^ explosives are made and used in
SiUrope, but have not yet been regularly imported into the U. S.

Ckmipressed gpnn-eotton, is cotton dipped in a mixture of nitric and
sulphuric acids, then r.duced to a fine pulp, and made into discs 1 to 2 ins thick,
ana % to 2 ins diam, or larger. It is generally used wet, for the sake of greater
safety. It then requires extra strong caps or primers. Roughly speaking, it is
about as strong as dyn No 1, but is less shcUtering in its effect. Being lighter
than dyn, it requires larger holes; and, owing to its rigidity, is less easily in-
serted, and does not fit the hole so completely. When dry, it is very inflam-
mable, bat, if not confined, it burns harn^essly. It contains no liquid, to freeze
or to exude; and is safe to handle.

Art. 33. Tonlte consists of finely divided gun-cotton mixed with nitrate
of baryta. It is compressed into candle-shaped cartridges having, at one endL a
recess for the reception of an exploder containing fulminate of mercury. Th«
cartridges weigh about the same as dyn. They are generally made waterproof

952 MODERN EXPLOSIVES.

Art. 34. Fordte, IdtboCraeteiir. and naaUn are foreign makes oi
nitro-elycerine explosives. In Dualln the absorbent is sawdust. It has greater
bulk than dyn for a given wt, and requires larger holes.

Art. S5. ExplofliTe ffelatine is a transparent, pale yellow, elastic
substance, and is composed oi90 per cent N-G and 10 per cent gan-cotton. It is
less sensitive than dyn to percussion, friction, or pressure, and is not affected by
water. Its specific gravity is 1.6. It burns in the op«n air. For comply
detonation a special primer is required. The addition of a small proportioo o(f
camphor renders it still less sensitive, and increases its explosive force. Tb$
camphor evaporates to some extent.

In some experiments on the power of different explosives to increase the contents
of a small cavity in a leaden block, explosive gelatine caused an increase 60 per ceot
greater than that caused by dyn No 1. In hard rock the diff would probably hsTf
been greater. The increase was 10 per cent less than that caused by N-O.

Art. 86. Tbe eap or explCMlery used with ordinary safety fuse for Of
ploding N-O and dyn, is a hollow copper cylinder, about y^ inch diam, and an luck
or two in length. It contains from 15 to 20 per cent, or more, of fulminate of mcr-
cary, mixed with other ingredients into a cement, which fills the closed end of tin
cap. The cap is called ** single-force,*' ** triple-force," Ac, aoeording to the qnaotii|
of explosive it contains.

The end of the fuse, cut oflT square, Is Inserted Into the open end of this cap, br
enough to touch the fulminating mixture in it. In doing this, care must be takn
Bot to roughly scratch the latter. The neck of the cap is then pinched, near ill
open end, so as to hold the fuse secareiy. The cap, with the fuse thus attached, ii
then inserted into the charge of N'O or dyn, care being taken not to let the foM
eome into contact with the explosive, which would then be burned and wasted. If
m dyn cartridge is used, the fuse, wltn cap, is first Inserted into it. The neck of the
•artridge is then tied around the fuse with a string,, and the cartridge is then rea4y
to be placed in the hole and fired.

Art. 87. Tbe Siemens mag^eto-eleetrlc blastin8;> lappa*
ratas, now In general use, consists of a wooden box about as large as a transit
box. Outside it has two metallic binding-posts with screws, for attaching the two
wires leading to the exploder. From the top of the box projects a handle at tbt
end of a vert bar. This bar, which is about as long as the box is high, Ts made so
as to slide up and down in it, and is toothed, and gears with a small pinion inside
the box. When a blast is to be fired, the bar is drawn up, by means of the handls^
as far as it will come. It is then pressed quickly down to the bottom of the box.
In its descent it puts into operation, by means of the pinion, a magneto-electriB
mackine inside the box. This generates a current of electricity, which increases la
force with the downward motion of the bar, but which is confined to a short circnit
of wire toithin the 6ox, until the foot of the bar strikes a spring near the bottom of
the box, breaking the short circuit and forcing the electricity to travel through the
two longer ** leading wires,** which lead it from the two binding-posts on the ontsidi
ff the lx>x to the cap or exploder placed in the charge.

Art. 38. Tbe cap used with this machine is similar to that used with safe^
fhse (Art 36), except that its mouth is closed with a cork of sulphur cement, through
which pass the two wires leading from the electric machine. The ends of these
wires project into the fulminating mixture in the cap. They are ^ inch apart, hot
* are connected by a platinum wire, which is so fine as to be heated to redness by ths
mrrent firom the battery. Its heat ignites the fulminate and thus explodes the cmw

Art. 89. Wbere a namber of boles are to be fired slmiil*
taneonsly (thus increasing their effect), each hole has a platinum cap inserted
into its charge, and one of the stiort wires attached to each cap is joined to one of
those of the next cap, so that at each end of the series of caps there is one fk-ee end
of a short wire. Each of these two ends is fastened to the end of one of the leading
wires, placing the whole series *' in one circuit.*' Where the holes are too far apart
for the caps to be thus Joined by the short wires attached to them, the ends of ths
latter are connected by cotton-covered ^* counectiny wires.**

HODEBK EXFIiOSrVES.

968

Art. 40. The magDeto-electiical maelkine weighm about 16 tt«. It can
fire about 12 caps at once.

Caps for ordinary fuse and for electrical firing, (Uses, wires, electrical machines,
Ac, are sold hj most of the makers of, and dealers in, erplosives, rock-drilling
machines, &c.

Art. 41. Simultaneous firing of a number of holes can be conveniently
accomplished only by electricity. Electric blasting apparatus is specially useful
for blasting under water, where ordinary fuses are apt, especially at great depths,
to become saturated and useless.

If an electrical machine fails to fire a charge, it is known that the charge
cannot explode until the attempt is repeated. Therefore no time need be lost,
and no risks run, on account of " hanging fire."

dUlfPOWDER.

The exploalwe foree of powder is about 40000 lbs. or 18 tons, per squait
inch. Its welflrlit averages about the same as that of water, or 62^^ lbs per
cubic foot : hence, l.fi» = about 28 cubic inches. In ordinary quarrying, a cubie
yard of solid rock in place^ (or about 1.9 cubic yards piled up after being quar-
ried,) requires from ^ to ^ ft. In very refractory rock, lying badly for quarry-
ing, a solid yard may require from 1 to 2 lbs. In some of the most successful
great blasts fbr stone for the Holyhead Breakwater, Wales (where seTeral
tliousands of lbs of powder were usually exploded by electricity at a single
blast,) from 2 to 4 cubic yards solid were loosened per &> ; but in many instances
not more than 1 to 1^ yards. Tunnels and shafts require 2 to 6 fi>s per solid
yard ; usually 8 to 5 wb. Soft, partially decomposed rock frequently requires
more than harder ones. Usually sold in kegs of 25 lbs.*

Welffbt of powder ]

In one

foot deptli of bole.

IMameter of bole

lin

l^ins

IJ^ins

2 ins

2J^lns

Sins

liirelfl^lit of powder
avoirdupois

OlbSoz

OlbSOK

Ofi> llOB

lB)4oz

21b

2lbl3oa

Diameter of hole

8Hln

4 ins

4HinB

Sins

^}4 ins

6 ins

We1«l^t of Powder
avoirdupois

3lbl4oE

69> 9oz

fflb 60B

mi4os

9Ib8oE

UlbSoB

964 TZMBBB.

PBESEBTATION OF TIMBER.

Art. I. (a) The decay of timber is caiised by the srowth and

activities of fungi. The minute spores of one of these fungi, gernunatinf
on a piece of wood, send out fine threads, which enter the wood cells and
aoon give off a complex compound called a ferment or ensyzne, which dis*
solves certain parts of the wood fibre. The dissolved fibre serves as food
for the fungus. The threads throw out brancheB and sub-branches, and
soon the timber is permeated by a mass of such threads, the growing pvtt
of which give off ferment. The action of the ferment changes the chftrniRtl
and physical properties of the wood, rendering it, in some cases, like brown
charcoal, in others white, soft and stringy, and the wood is said to be rotten
or decayed. Eventually some of the threads grow out from the stufaot
of the timber, and form toadstools and other excrescences. Under thesa
are found cavities containing thousands of spores, which, when ripe, are
blown off into the air and settle upon other timbcn» where the process is
repeated. Moisture and heat are favorable to the growth of the fungi, as
are also the starches, sugars and oils found in the cells 9f the sapwoodbiit
wanting in the heartwood.' If protected from the action of these fungL
wood will last indefinitely. Hence the accumulation of deadwood sfaowd
be avoided.* If air is excluded, as when timber is kept constantly and
entirely immersed in salt or fresh water, the fungi cannot thrive. Saft
confined in timber with air, ferments, producing dry rotf as where besms
are enclosed air-tight in brickwork, etc. ; and where green timber is painted
or varnished, or treated with creosote, etc. The sap then not only prevents
the thorough penetration of the oil, etc, but may cause the ^^reater part of
the wood to rot although its firm outer shell tavea it a deceptive appearance
of strength, (b) Sap should therefore be first removed by seasoning!
i e, either by dr^^ng the wood in air at natural or higher temperaturea
or by first steaming the wood under pres so as to vaporise the sap, ana
then removing the latter by means of a vacuum. Thorough seaaoniiog of
large timbers In dry air at ordinary temperatures may require years; and
too rapid kiln-drying cracks and weakens the wood. But it is questionable
whether steaming and vacuum remove sap as thoroughly as do the slower
dry processes, (c) Alternate exposure to water and air is very destructive.
It causes wet rot.

Art. 2. Sea-worms. The limnoria terebrans works from near hidi-
water mark to a little below the surface of mud bottom; the teredo navatit
within somewhat less limits. The teredo is said to be rendered less activl
by the presence of sewage in water.

Art. 3. (a) The best timber-preserving processes are practically nselefl
unless thorouprhly well done. If the gain in durability will not war-
rant the expenditure of time and money read for this, it is more eoononiieal
to use the wood in its natural state, (b) The woods best adapted to
treatment are those of an open or porous texture. They absorb the oU
etc better than the denser woods; and their cheapness renders the use of
the treatment more economical, (e) Most of the processes in common use
seem to render wood less combustible, (d) After treatment by any process,
the wood should be well dried before using.

Art. 4. (a) Creosote oil, or dead oil, is the best known preservative.
Against sea-worms it is effective for 15 to 25 years, and is the only known
protection, (b) As temporary expedients, piles are sometimes covered with
sheet meial, or with broad-headed nails driven dose together. These rust or
wear away in a few years. Oak piles, cut in January, and driven with the bark
on, have resisted the teredo for 4 or 5 years; and cypress ^les, well ebarred,
for 9 yeaxa. (c) For ordinary exposures on land, 8 to 10 lbs of creosote ou

£er cub ft are reqd — say 670 to 830 lbs per 1000 ft board measure — 90
> 40 lbs per cross tie of 4 cub ft. For protection sfuunst sea-worms 10 to
12 lbs per cub ft suffice in climates like those of Great Britain and the
Northern U S; but in warmer waters where the teredo is very active, from
14 to 20 lbs per cub ft are used. Large timbers may not require saturatioa
throughout, and thus may take less per cub ft. But see (i) and end of Art.
1 (a), (d) Creosote oil weighs about 8.8 tbs per IT. S. gallon, (e) The
sticks should be reduced to their intended final dimensions and framed (if
framing is reqd) before treatment ; especially if for exposure to teredo, which

* See paper by Dr. Hermann von Schrenk, read before the American
Railway Engineering and Maintenance of Way Assodatton, March, 1001.

TIMBER. 955

isjBure to attack any spots which (as by subsequent cutting) are left unpro-
tected, (f) Creosoted ties have remained sound after 22 yesjrs* exposure
The creosote protects the spikes from rusting, (g) Spruce and tamarack,
owing to their irregular density, are imsuitable for creosoting. (h) Creosote
renders wood stiffer and slightly more brittle. In hot weather it exudes to
some extent and discolors the wood. Its smell excludes it from dwellings.
(1) It does not wash out from the wood, but often fails to penetrate the
heart-wood. Then, if any sap remains, decay begins at the center. See
end of Art 1 (a). Bumettizing the cen of the stick (see Art 7) and using a
coating of creosote outside, has long been suggested as the best possible
method. It is the principal feature of the AJlardyce process. This is
cheaper than thorough creosoting. In the RHtgers process, which has
been successfully employed for ties in Germany since 1874, the creosote and
a solution of zinc cmonde are injected simultaneously. (J) In the creo*
resinate process * the preservative fluid consists of creosote 38 per cent,
formaldehyde 2 per cent, and melted resin 60 i>er cent.

Art. 5» (a) Mineral solutions are Inferior to creosote, even on land;
.and useless in running water or against sea-worms; but they approximately
double the life of inferior timber under ordinary land exposures; and their
eheapness permits their use where that of creosote is too expensive, (b)
They render wood harder; and brittle if the solution is too strong. They
are liable to be washed out by rain, etc. Hence the outer wood decays first.
See Art 4 (i) Art 8 (b) (c) (d). (c) A committee of the American Soo of Civ
Engrs,t after collating a large number of experiments, recommended Biu>
nettizinflr (Art 7) for damp exposure* as that of cross ties, damp floors,
etc; and f£yani zing (Art 6) for comparatiTely dry situations with
exposure •to air and sun-light, as in bridge timbers, for which it is better
suited than Biunettizing because it seems to weaken wood lees. In such
exposiues it preserves wood sometimes for 20 to 30 years.

Art. 6. (a) Kyanlzingr consists in steeping the wood in a solution of
1 lb of bi-chloride of mercury (corrosive sublimate) in 100 lbs of water, (b)
It is usual to idlow the wood to soak a day for each inch of the thickness or
least dimension of the piece, and one day in addition, whatever the size.
(c) Oen'l Cram found the process very unh ealthy, " salivating all the men **i
but Mr. J. B. Francis, at Lowell, and Mr. H. BisselL of the Eastern R R of
Mass, had little or no trouble in this respect. The sublimate, however,
which is very poisonous, is apt to effloresce, and the use of the timber is thus
rendered djuigerous. (d) The process is valuable for timber placed in
moderately damp situations, but the salt is liable to be washed out by run-
ning water. Kyanized spruce fence posts, planted 4 ft in the ground, at
Lowell, Mass, in 1850, were examined m 1891, and most of them were found
Tery sound both above and below the surface of the ground.

Art* 7* (a) Bumettizingr consists in immersing the wood for several
bours in a solution of 2 lbs chloride of zinc in lOQ lbs of water, under a
pres of from 100 to 300 lbs per sq inch.

Art. 8. Other preyentives. (a) Steeping in a solution of sulphate
of copper (blue yitriol) has been extensively used, but does not seem to
have been permanently successful. The blue vitriol washes out readily.
(b) In the Barschall or Hasselmann process,^ introduced in Germany
m 1887, in the U S in 1899, the wood is boiled, at a temperature from 212^
to 284^ Fahr. and under a pressure of from 15 to 45 lbs per sq inch, in a
solution of iron, copper and aluminum sulphates and "Kainit,''^a sulphate
of magnesia and potash, mined at Stassf urt. Germany. The solution is said
to carry off the sap (timber being more readily treated by this process when
SPsen than when seasoned), while the copper destrojrs the funjd, and the
son forms an insoluble compound with the cellulose or woodv fibre. It is
olaimed that the process g^-eatly hardens the wood, especially the softer
varieties, rendering them suitable for ties, without impairmg their strength,
elasticity or pliabmty. (c) The Wellhouse process injects first a solu-
tion of chloride of zinc with glue, and then one of tannin (both under
pressure), in order to diminish the subsequent washing out of the chloride.
In a later modification^ the zinc, glue and tannin solutions are injected
•eparately. Several millions of ties have been treated In this way. Tbo

* See "A Proposed Method for the Preservation of Thnber," by F. A.
Kummer. Transactions, Am Soc C E, Vol XLIV, December. 1900.
t See Tranmctions, Am Soc C E. July, Auk «nd Sept, 1886.
t Raihroad (3aaette, February 9. 1900.

956 TIHBEB.

proeen is not reoommended for sub-aqueous use. (d) Processes in whidi
the wood is treated by painting or soaklnfif * are: Carbolineum Avenarius
(Tar-oil, chlorine, ete)» Ligni &klvor (Tar-oil, etc), Woodiline and Spirit-
tiDe (chemical solutions) and a distillate of pine used by the Penns^hranis
Bailroad Co. for car work, (e) Fence-posts etc seem to be preserved, to
some extent, by having only their lower ends dipped in tar well boiled to
remove the ammonia* which last b destructive to wood. The upper end
must be left untarred to let the sap evaporate, (f ) Attempts at wood pre*
aervation by means of Taper of creosote etc have proved failures.
(g) While wood . remains thoroughly saturated with petroleum it does
not decay. But unless the supply is kept up the oil evaporates and leaves
the wood unprotected, (h) Cottonwood ties laid upon a soil contain-
ing about 2 per cent carbonate of lime, 1 ^r cent salt and 0.5 per cent
each of potash and oxide of iron, on the Umon Pacific R. R. in 1868. were
found in 1882 "as sound and a good deal harder than when first laid,"
although such ties in other soils lasted but from 2 to 5 years. (1) The ust
of solutions of lime and of salt; and charring the siuiace; are sometimes
found useful in damp situations.

* See Report by O. Chanute to the American Railway Engineering and
Maintenanee of Way Association, March, 1901.

STRENGTH OF MATERIALS.

967

Art. 4. YTltimate nyrerage tensile or cobeslve strenflpth of

Timber,

filing tke least weights in ponnds which, if attached to the lower end of a vert rod
one inch square, firmly npheld at its upper end, would break it by tearing it apart
Tor large timbers we recommend to reduce these constants ^to y^ part.

Tbe streagtba in all then tables maj
readily be one-tbird part more or leas
than ear averages.

Alder

A«h, English

** American (author) abt.
Birch

** Amer'n black

Bay-tree

Beech, English

Bamboo

Box

Cedar, Bermuda.

** Guadaloupe

Chestnut

" horse

Cyprus

Elder

Sim

" Canada

Fir, or Spruce

Hawthorn...

Haael

Holly

Hornbeam

Hickory, Amer'n

Lignum Yitn, Amer'n

Lanoewood

Larch, Scotch

Locust

Maple

Lbs per
sq. inch.

14000
16000
16500
15000

7000
12000
11500

6000
20000

7600

9500
13000
10000

6000
10000

60O0
13000
10000
10000
18000
16000
20000
11000
11000
23000

7000
18000
10000

u
u
it

Mahogany, Honduras

" Spanish

Mangrove, white, Bermuda....

Mulberry

Oak, Amer'n white

« basket

** « red

" Dautsic, seasoned ....

Riga

English

live, Amer'n

Pear

Pine, Amer'n, white, red, )
and Pitch, Memel, Riga^. 3

Plane

Plum

Poplar

Quince

Spruce, or Fir

Sycamore

Teak

Walnut

Yew

Acrcws the n^min* Oak

" «. *» Poplar

♦' " " Larch,900to
« Fir, A Pines

Lbs per
sq. inch.

8000
16000
10000
12000

10000

10000

10000

11000

11000

7000

7000

10000

12000

15000

8000

8000

2300

1800

1700

550

Thkse ark averaqes. The strengths yary much with the age of the tree ; th«
locality of- its growth ; whether the piece is from the center, or from the outer por<
tions of the tree ; the degree of seasoning ; straivihtnessof grain ; knots, Ac, Ac. Also,
inasmuch as tlie constants are deduced from experiments with good specimens of
small sise, whereas large beams are almost invariably more or less defective from
knots, crookedness of fibre, Ac, it is advisable in practice to reduce these constants
as recommended abore.

* EflTect of Tappitijir Trees for Tnrpentine. Preliminary experi-
ments by the Forestry Division of the U. S Department of Agriculture upon
long-leaf pine from Alabama indicate that (contrary to the generally received
impression) " turpentine timber," t. e. the tiniber^of trees that have been
*' boxed " (robbed of their turpentine), while it has slightly less tensile and
shearing strength, is from 20 to 30 per cent, stronger in compression (whether
with or across the grain) and under transverse strain. In the ** turpentine tim-
ber," however, the k'esin collects in spots, gumming ihe tools, and thus rendering
the timber harder to work than that of trees which have not been deprived of
their turpentine. The specimens tested were taken mostly at heights of from 7
to 33 feet above ground. 'Circular No. 8. Issued 1892.) Boxed and unboxed
timber are frequently called " bled " and " unbled " respectively.

958 BTBEKOTH OF UATEBIAIA.

Art. I. €*aqireaalve fllrcBBrtha of AmcrlcMi wood*, mIh

tbmijt dBfl tar^fitUf uatontd. Approakaiftt* ftvengH dadnced from iuildt «Kp*ft
aant* mada wilh the n 3 Qort Msting imohtns •! WklBrtavn, Hub, by Ht. 8. P,

B)wrpL«,f[U-ibscanii»af 1880, Seasoned woods nalaCcraghtng much bsttir

SSi"''' ~l^ ISSJ

pllob »ot Jan

0| ftuuura {battonvDod),

nMt employed In thalji

'SET W

slloir ptnac^ apraca, and onfln«rT im

iI«iBlatMforbrldg8a,roof«,eto,cn

:u^u, in ahort bloclcm; »;»««, h

pruclfre perfecllf aquatJia preHure !■ nn
jmprpBflJoD, «itb ^rly HSBon^d vhlca pi

l^to^lnoh; wWch Ml eqosl to from ^ lo \< Inch pir foot of hetghl; orftanA
loAof th»bBlgbt: tb8 0(«in bemg.bont ^ tnch to a foot, or X «r tba halghL
T[n3erlOOO(ilba taU], or IMOIbt per aq Inoh.tbsj apllt badly; and In Hma ata

iBpK^lmani 4 cBDtlmaIrM (1,67 Jocb) aqnin, »2 nntlmatni (12.« loa) laai.
Wbcn Ihs leiigib eimadi 10 tinici the l«ait etde, tt Woodsn Plllata.
tSpaFlni«iii4eenl<inetrc'B<1.eTlDFb)iquara, 18 c^ntlmtU'ea (»3 Ini) longl kid

ofan Iron punch 4 cnilmalraa iqotn, or Jnil coreriQg'ihe enlln wi'dtZ af'oe

loada prodncfng .n iDdeotatlnn of .01 Inch. Thu MooDd eolqmn (taeadwl ''.I") r™

BTREMQTH OF HATERUU.

k( tbe center i l«cethe>

iHnuderBAld loadh,

Tbs Hfa lotd It ban one-stilh of [fan brsaklDi; Lewd.

For tbe aemt Iwids, dedcct I4 Ifae nt of th* brnn lt>slf. Th« d^flaiUm.

IiOBtiB applied flnddei>I|r tUI donU« tbe deBecttoiu In the table ; tt

Csntloa. Intsniiicb as Ihli tubls HU bued apOD veil mgoaed, stiaiglit
grained pieces. Aw ^m koola. and other defocu. we must nol iu prucCiee taka

more lb»n about til ' ' '

buUdlng timber a'
tbe deSecttoDB.

ObaerrolBO that aurtableliforeafe center loads, but It Is piai a that

tbe load Toald b*ve te be sailsliied by a mere kaire-edEe, at theHrvceDlerof
the beam. Nov, In tbe InsUnce Rem. p. BSO, if ve attempted to austalD tbe center
load of SOTGIba npoa sucb a fenife-edge, It oould at once cut tbe beam In two. U
we even applied It along Sort lua of tbe length, it would cut Into ll. aad w^ should
not bBTs a saTet j of 6 uilnst erusbtng tbe top uf the beam until as In the case ol
tbe ends we distributed tbe load along full 46 Ins of length, ox about aHoalbrt.

Wl*k (he Mft londn In thin taUe * beux in>7 b«itd to
A|[aliiat eraablnr nt the eiidn.

960

STRENGTH OF MATERIALS.

Table,

» eontinned.

(Original.)

D«pth
of

Span 18 ft.

Span 20 ft

Span 26 ft|Span 30 ft|8pan 36 ft|8pan 40 ft.|

\0L

beam.

load

def.

load

def

load

def

load

de£

load

def.

load

def

lUTSOI

beam.

Ins.

fi>8.

ins.

9>B.

ins.

JbB.

ins,

fi>8.

ins.

Bm.

ins,

fbB.

ins.

fits.

6

160

1.4

135

1.8

108

2.9

90

4.6

77

6.6

67

9.2

12

7

204

1.2

184

1.5

147

2.6

122

3.9

105

6.8

92

7.6

14

8

267

1.0

240

1.3

192

2.1

160

3.2

137

4.6

120

6.4

16

9

338

.92

304

1.2

243

1.9

202

2.8

174

4.0

152

5.5

18

10

417

.82

375

1.0

300

1.7

250

2.5

214

3.6

188

4.9

20 1

11

605

.74

454

.93

363

1.6

302

2.2

269

8.2

227

4.3

22 ;

12

600

.68

540

.85

432

1.4

360

2.0

808

2.9

270

S.9

24

14

817

.68

735

.72

688

1.2

490

1.7

420

2.4

367

3.2

28

16

1067

.60

960

.63

768

1.0

640

1.5

548

2.1

480

2.8

32

18

1350

.45

1216

.56

972

.90

810

1.3

694

1.8

607

2.6

36

20

1666

.40

1500

.50

1200

.79

1000

1.2

867

1.6

760

2.2

40

22

2017

.37

1815

.45

1452

.72

1210

1.1

1087

IJi

907

2.0

44

24

2400

.33

2160

.41

1728

.66

1440

.96

1234

1.3

1080

1.8

48

26

2817

.31

2526

.38

2018

.60

1684

.88

1449

1.2

1263

1.6

52

28

3267

.28

2940

.35

2352

.65

1960

.81

1680

1.1

1470

1.5

56

30

3750 .26

3375

.33

2700

.50

2250

.76

1928

1.1

1687

1.4

60

32

4267 1 .26 3840

.30

3072

.45 2660

.71

2194

1.0

1920

1.3

64

34

4817 1 .23 433o

.29

3468

.44 2»90

.67

2477

.92

2167

1.2 68

36

6400 1 .22 4860

.27 3888 1

.43 3240

.63 2777 1

.86

2430

1.1

72

White oak, and best Soatbern pitch pine will bear loads ^
greater.
For cast iron, malt the loads in the table by 4.5 ; and for irroag^ht by

6.3. For these new loads, mult the delk by .4 for cast ; and by .3 for wrought.

If tbe load is equally distributed over the dpan, it may be twice as
great as the center one, and the defs will be IJ^ times those in the table. If the
loads in the table be equally distributed along the whole beam, the defs will
be but five-eigbth8 as great as those in the table. When more

accuracy is reqd, half the wt of the beam itself must be deducted from the ceotet
load; and the witole of it from an equally distributed load. The wt of the beam, in
the last column, supposes the wood to be but moderately seasoned, and therefore to
weigh 28.8 lbs per cub ft.

IJses of the t'orciroinK table. £z. 1. What mast be the breadth
<>f a hor rect beam of wh pine, 18 ins deep, supported at both ends, and of 20 ft elsst
length between its supports, to bear safely a load of 5 tons, or 11200 fi>s at ita center?
Here, opposite the depth of 18 ins in the table, and in the column of 20 feet lengths,

we find that a beam 1 inch thick will bear 1215 &>b ; consequently, il^ := 9.22 in^

the reqd breadth ; for the strength is in the same proportion as the breadth.

Ex. 2. What will be the safe load at the center of a Joist of white pine, 18 ft long,
8 ins broad, and 12 ins deep? Here, in the col for 18 ft, and opposite 12 ins in depth,
we find the safe load for a breadth of 1 inch to be 600 9)8 ; consequently, 600 X 3 »
1800 ft)s, the load reqd.

Rem. Cautions in the use of the above table. For instance, in
placing very heavy loads upon short, but deep and strong beams, we moat tctke care
that the beams rest for a suiflcientdist on their supports to prevent all danger from
crushing at the ends. Thus, if we place a load of 6075 B>s at the center of a beam
of 4 feet span, 18 ins deep, and only 1 inch thick, each end of the beam sustains •

fifk'JK

vert crushing force of -r- = 3087 fi>s, and that sidewlse off the vrain, in

which position average white pine, spruce, and hemlock crush under about 800
ibs per sq inch, and do not have a safety of 6 until the pressure is reduced to aboat
183 fi>s per sq inch. Therefore our beam, in order to have a safety of 6 against
crushing at its ends, must rest on each support 3037 4- 133 = 23 sq ins ; or for a
safety of 4 nearly 16 sq ins. When a pressure is equally distributed side-
wise (that is, at right angles to the general "direction of the fibres) over the entire
pressed surface of a block or beam (to ensure which, the opposite surface must be
supported throughout its entire length) the resulting compression might readily
escape detection unless actually measured. But when a considerable pressure b
applied to only a portion of the surface, as of caps and sills where in contact with
the heads and feet of posts, or at the ends of loaded Joists or girders, the com>
pression becomes evident to the eye, because the pressed parts sink below the
unpressed ones, In consequence of the bending or breaking ot the adjacent tihrm.
What in the first case (especially if slight) would be called compression, would

STRENGTH OF MATERIALS. 961

in the second be called ernslitiiir 9 even when neither might be so great as
to be unsafe.

Owing to the resistance which said adjacent fibres oppose to being bent 01
broken, it is plain that a given pressure per sq ineli, or per 84 foot, &q.,
will cause somewhat less compression or crushing when applied to only a part of
a surface, than when to the whole of it.

Tlie irriter has seen 40 half seasoned hemlock posts, each 12 ins square,
footing at intervals of 5 ft from center to center, upon f^imilar 12 X 12 inch hem-
lock sills, to which they were tenoned, and which rested throughout their entire
length on stone steps. Each post was gradually loaded with 32 tons, or equal to
say 500 lbs per sq inch; and their feet all crushed into the sills from ^ to V^ inch.
Their heads crushed into the caps to the same extent. In practice the pres-
sure at the heads and feet of posts is rarely, if ever, perfectly equable; ana the
same remark applies to the ends of loaded joists, girders, Ac, in which a slight
bending will throw an excess of pressure upon the inner edges of their supports

I

ni

BIBEiraTH OF 1UTEBU.I&

STRENGTH OF WOODEN PILLARS. 963

WOODEN PILLABS.

The strengths of pillars, as well as of beams of timber, depend mnch on their d«-
p^ree ot seiuioilliiir* Hodgkinson foand that perfectly seasoned blocks, 2 diams
long, required, in many cases, twice as great a load to cmsh them as when only
moderately dry. This should be borne in mind when building with green timber.

In Important practice, timber should not be trusted with more than V^ to V^of ks
calculated crushing load ; and for temporary purposes, not more than 34 to ^.

Mr. <%arlea Sbaler Smltb, €. E., of St. I^ohIb, prepared tlie
following^ fonnala for the breaking loads of either sqnare or rectangular
pillars or posts, of moderately seasoned white, and common yellow pine, with flat
ends, firmly fixed, and equally loaded, based upon experiments by himself.

It is Gordon's formula adapted to those woods ; and gives resul ts considerably
•mailer than Hodgkinson's, It is therefore safer.

Call either side of the sqnare, or the least side of the rectangle, the breadth. Then,

5000t

Breakg load in lbs, per

Kale, sq inch of area, of a

pillar of W or T pine

' ~ 1 /^ ^' length in ins

1 +

/sq of length in ins \

\sq of breadth in ins ^ J

Or in words, square the length in ins ; square the breadth in ins ; div the first sqnare
by the second one ; mult the quot by .004 ; to the prod add 1 ; dir 6000 by the sum.

Ex. Breakg load per s<i inch, of a white pine pillar 12 ins square, and 30 ft, or 360
ftks long. Here the sq of length in ins is 300* ~ 129tK)0. The square of the breadth is

,_. -.. .129600 , 5000

12« = 144; and -j^ = 900 ; and 900 X .004 = 8.6; and 3.6 + 1 - 4.6. Finally, — ^

= 1087 lbs, the read breakg load per sq in. As the area of the pillar is 144 sq ins.
the entire breakg load is 1087 X 144 =s 156528 lbs, or 69.9 tons.

Recent experiments on wooden pillars 20 ft long, and 13 ins square, by Mr.
Kirkaldy, of England, confirm the far greater reliability of Mr. Smith's formula.
Hence we present the following new set of original tables based upon it.

For solid pillars of etrnt iron and of pine, whose heights range
firom 6 to 60 times their side or diam, we may say, near enough for practice, that ft
•ast iron one is about 16^ times as strong as a pine one ; but no such approximata
ratio holds good between wrought iron and pine, or between cast and wrought iron«

t Tbe teMkinff toed la lbs per sq Inch in skert biMks. by Mr. Salth.

964 BTBEKOTH OF VOODEV

Table »t bnakliiK load* In toaa af •qoar* plllan mf b»ir
eaBABMl whtM sr «onmoa ;ellaiv pine fflnmly axed and
qaslly loaded. By C.SUaler Snilili-»foniinl». (OrlglMl.)

u

Sid.

ot«

tu

1* pla« pillar, 1> Inehea.

H

1 MU 1 'M 1 >N 1 t 1 IM 1 IM 1 IK 1 > 1 >M 1 )M 1 S)< 1 •

1
1

i

'1
1

1;S
1

;

1

UIAE

1
i

f
1

TUL T

:i
J

1
1

1
I'

t

1

1
i"

f 5

II

side ata^mrt pine pillar, Ib iBclias.

ii

Ik 1 •« M« 1 t 1 )M 1 (M 1 i« 1 1 1 >M 1 «M 1 •« 1 I 1 TM

i
1

J

1:1

1
1

Too.

1

i

B

]

1

II

1
1

toil

1
1

I

i

STKEMOTH OF WOODEN PILUlBS.

Bide orsqaKre plB« pillar.

li

8ld« «raqPBr« pin* plllu-. In In

eh».

fi

1W< 1 11 1 llt« 1 IIM 1 11H 1 11 1

1 lOKl ii' 1 nsi ] I1M 1 UK 1 11

1

1

f

Si

III

1

BBB

J

Tom.

1
1

0 LO
1

AD.

r

1

Kemitrks. Mr Utrhaldv (Onnd tor Klacn lUld nsBtale flr^

a> It long, &Dd ^3 IDH squire, (or 18}^ Sides high.) 14S and ise UDS UOiI ; or .87)

plnni 160 tona loUl ; or .947 Uin, or 2121 lbs, per aq loch. HodgkloHin vould gin
EftcSor MrKlrkkldj'sW-h pillars gboiMned kbout ^ of ma iDCb tots] : or .03

Tbe ordter hu^nDwn 8 uDbrHced pin'sira of taemlsek,ta]erBblrHii»iied,

12 Inn squire, ind 12 ft blRfa. to be graduillr louled escb wttb SI tana, or 71680
lbs toUl; {nr.KS^ton.orlSS Ibg per sq inch) without appreclabis Tielding. As-
■umiDf thslr ilrsDKth and stlmaa to be about as for Hr Smitb'a pins, (as '- -"

our Ublesjlbey sboaM b» him yield at 39.9 tons toUI. V""- "- —

■^ ■■■■ . -■ -«, ihcT should jleld M

idr«ddeAl,*t9I.BIoDB.

966

8TRENOTH OP WOODEN FILLABS.

Table of breaking loads In tons of flqaare pillars of balff-
■eaaoned white or eommon yellow pine, witb flat ends
flrmly- fixed, and equally loaded. By C. Bhaler Smith's formola.
(Continued.)

Ab this table was partly made by interpolation^ the last figure is not always pre>
cisely correct.

Original.

eight
feet.

Side of square pine pillar in inebes.

It

n.s

13 1 14 1 15 1 16 1 17 1 18 1

19 1 20 1

21 1 22 1 23 1 24

txif

BREAKING LOAD.

ToDfl.

Tout.

Tom.

Tons.

Tona.

Tont. Tons. {

Tom.

Ton*.

Toni.

Tons.

Tons.

4

858

418

482

562

625

703

786

872

964

1060

1161

1266

4

6

336

394

456

526

599

676

760

847

938

1033

1134

1236

«

8

806

367

429

500

5V2

649

732

818

910

1006

1106

1208

8

10

281

339

400

466

537

612

694

780

870

964

1064

1166

10

12

252

307

365

432

502

576

656

740

829

922

1022

1124

12

14

225

277

333

397

464

536

614

696

784

876

973

1074

14

1«

201

250

303

363

428

497

573

652

739

829

925

1024

16

18

179

224

274

331

392

458

631

608

692

780

873

972

18

20

160

201

248

301

359

422

492

566

647

732

822

919

20

22

143

182

224

274

329

388

456

526

604

686

773

866

2S

24

127

163

203

249

301

367

421

488

563

642

726

816

24

26

116

148

184

226

275

328

889

453

523

599

680

767

25

28

103

133

167

206

252

302

359

420.

490

560

638

721

28

SO

93

121

152

189

231

278

332

389

453

522

597

677

30

82

84

109

138

173

212

256

307

361

421

487

558

635

3S

34

76

99

126

159

196

237

284

335

392

455

523

697

34

36

60

91

116

146

180

219

264

312

366

426

490

660

38

38

63

84

107

184

166

203

245

290

341

897

458

525

38

40

58

77

99

124

154

188

227

270

318

372

429

494

40

42

54

71

91

115

143

175

212

253

298

349

403

465

4S

44

50

66

84

107

133

163

198

236

280

328

380

488

44

46

46

61

78

99

123

152

185

221

263

308

358

413

46

48

43

57

73

92

115

142

173

207

247

290

337

389

48

50

40

53

68

86

107

133

162

194

231

272

317

867

50

62

37

50

64

81

101

124

152

182

217

256

300

347

62

54

35

47

60

76

95

117

144

172

205

242

283

328

54

66

33

44

56

71

89

110

135

162

198

228

267

810

56

68

31

41

52

67

84

103

127

153

182

215

253

294

58

60

29

38

49

63

79

98

120

144

172

204

240

280

60

66

26

33

43

65

69

86

105

126

151

179

211

246

65

70

22

29

87

48

60

74

92

111

134

159

187

218

70

76

19

25

33

42

53

66

82

98

118

141

166

195

75

80

16

22

29

37

46

58

72

87

106

125

148

174

80

85

14

19

26

33

41

52

66

78

94

112

132

156

85

90

13

17

23

80

37

46

58

70

85

102

120

141

90

95

12

16 21

27

33

42

53

64

77

98

108

127

95

100

11

14 19

24

30

38

48

58

70

84

99

117

100

110

10

12

16

20

26

33

40

48

58

70

82

97

110

120

9

11

14

17

22

28

34

41

40

60

71

83

120

130

7

9

12

14

18

23

29

36

43

62

61

72

130

140

6

8

10

12

16

20

25

31

87

44

53

62

140

150

5

7

9

11

14

18

22

27

32

88

46

64

160

160

5

6

8

10

13

16

20

24

29

34

41

48

160

170

4

6

7

9

11

14

17

21

26

80

86

48

170

180

4

6

6

8

10

12

15

19

22

27

82

88

180

190

8

4

5

7

9

11

14

17

20

24

29

84

190

900

8

4

6

6

8

10

12

15

18

22

216

81

900

STRENQTH OF VOODBN FILLAB8.

967

Breaking loads of half seaaoned aqiiaro pine pillava.

s«

.a 3

s«

BB LOAD rWM ■« DT.

?5

am MAD PKB ■« nr.

M)3

BK LD nm ■« nr.

wo
.IT*""

BR LD rsB nq, n.

Hfl

M a

sd s

Ton*.

Lbs.

Tent.

Lb*.

Tost.

Lbs.

Toni.

Lba.

1

2.2232

4980

26

.6027

1350

61

.1960

489

76

.0924

207

2

2.1969

4921

27

.6697

1276

62

.1888

428

77

.0902

202

8

2.1544

4826

28

.6398

1209

63

.1826

409

78

.0879

197

4

2.0978

4699

29

.6116

1146

64

.1768

896

79

.0662

198

5

2.0290

4646

30

.4868

1087

66

.1706

882

80

.0839

188

6

1.9513

4371

31

.4607

1082

56

.1647

369

81

.0821

184

7

1.8665^

JH81

32

.4379

981

67

.1598

868

82

.0799

179

8

1.7772

3981

33

.4166

933

68

.1645

846

83

.0781

176

9

1.6867

3776

34

.3969

889

69

.1496

836

84

.0763

171

10

1.6942

3571

35

.3781

847

60

.1451

826

85

.0746

167

11

1.6040

3369

36

.3699

809

61

.1406

816

86

.0728

168

12

1.4165

3173

37

.3447

772

62

.1362

806

87

.0714

160

13

1.3317

2983

38

.8296

738

63

.1321

296

88

.0696

166

14

1.2513

2803

39

.8152

706

64

.1286

288

89

.0688

158

16

1.1745

26:31

40

.3018

676

66

.1260

280

90

.0670

160

16

1.10/7

2470

41

.2889

647

66

a2io

271

91

.0656

147

17

1.0353

2319

42

.2772

621

67

.1179

264

92

.0638

143

18

J)723

2178

43

.2661

696

68

.1147

267

93

.0625

140

19

.9134

2046

44

.2564

672

69

.1112

249

94

.0616

138

20

.8585

1923

45

.2466

650

70

.1085

243

95

.0603

136

21

.8076

1809

46

.2657

628

71

.1054

236

96

.0689

132

22

.7603

1703

47

.2268

608

72

.1027

230

97

.0576

129

23

.7166

1605

48

.2183

489

73

.1000

224

98

.0567

127

24

.6755

1513

49

.2100

472

74

.0973

218

99

.0554

124

25

.6380

1429

60

.2031

466

76

.0951

213

100

.0645

122

968

PLASTBBINQ.

PLASTEEING.

Thx plastering of the intiide walls of buiidiugs, whether done on laths, bricks, ai
stone, generally cunsidts of three separate coats of mortar. The first of these is called
by workmen the rough or scratch coat; and consists of about 1 measure of quicklime,
to 4 of sand ; (which latter need not be of the pul-est kind ;) and ^ oaeasure of bol-
lock or horse hair ; the last of which is for making the mortsr more cohesive, and
less liable to split off in spots. This coat is about f>£ to ^ inch thick ; is put ob
roughly ; and should be pressed by the trowel with sufficient force to enter perfectlj
between and behind the laths; which for facilitating this should not be nailed
nearer together than J^ an inch. In rude buildings, or in cellars, Ac, this is oftes
the only coat used. When this first coat has been left for one or more days, accord-
ing to the dryness of the air, to dry slightly, it is roughly scored^ or scrcOched, (henoe
its name,) with a pointed stick, or a lath, nearly through its thickness, by lines mn-
ning diagonally across each other, and about 2 to 4 ins apart. This gives a better
hold to the second coat, which might otherwise peel off. If the first coat has be-
come too dry, it is well also to dampen it slightly as the second one is put on.

The second coat is put on about ^ to ^ inch thick, of the same hair mortar, <»
coarse stuff. Before it becomes hard, it is roughed over by a hickory broom, <h
some substitute, to make the third coat adhere to it better.

The third coat, about l/^ inch thick, contains no hair; and for giving it a still
whiter and neater appearance, more lime is used, say 1 of lime, to 2 of sand ; and
the purest sand is used. This mortar is by plasterers called stucco ; a name
also applied to mortar when used for plasterint; the outsides of buildings. Or in*
stead of stucco, the third coat may be, and usually is, of hardfinisih^ or gauge stuff;
which consists of 1 measure of ground plaster of Paris, to about 2 of quicklime,
without sand. Hard finish works easier ; but is not as good as stucco, for walls in-
tended to be painted in oil. The plaster of Paris is for hastening the hardening.

Bitber of these third ooats is smoothed or polished to a greater or less extent, acoording to whettv
it is to show, or to he papered, painted, Ac. The polishing tools are merely, the trowel ; the hand-
loat, (a kind of wooden trowel ;) and the water>brush, (a short-handled brush for wetting the earfses
part at a time with water, in order to polish more treelj.) For finer polishing, a float made of cost
Is used. The smooth pieoe of board about 10 to 12 ins square, with a handle beneath, on whioh the
plasterer holds his mortar until he pats it on to the wall with his trowel, is oalled a hawk. '

The more thoroughly eaoh eoat is gone over with the water-brush and trowel, (which prooeas is
oalled hand-lloaHag,) the firmer and stronger will it be. Frequently only two coats of plastering are
put on in inferior rooms ; or where great neatness of appearance is not needed. The first is of hsir
mortar, or coarse stuff; this Is scratched with the broom, and then oovered by the finishing ooat sf
finer mortar, (stucco.) If this last is nearly all lime, or with but very little sand, to make It werfc
easier, it is called a slipped ooaX. Without any sand it is called Jbie ttmff. Neither is as gooid ai
stucco, if the wall is to be papered. When this is the case, the third coat also may have a litOe hair,
to give it more strength ; but this is not absolutely necessary.

A very good eBiect may be produced in station- house*, ohnrehes, fto, by only two ooata of piaster la
which fine clean screened gravel is used instead of sand. When lined into regular eoorsea, It tobum -
bles a buff-colored sandstone, very agreeable to the eye.

In purchasing plastering hair, care must be taken that it has not been taken flrom salted hides,
inasmuch as the salt will make the walls damp. For the same oause sea-shore sand should not be
used. It is almost impossible to wash it entirely free fkcm salt.

In briok walls intended to be plastered, the mortar joints should be left very rough, to let the plas-
ter adhere. If it is put on smooth walls, without first raking out the mortar to the depth of nearlj
an inch, it is very apt to fall off; especially from outside walls; as can be seen daily in any of our
cities. As this raking out of briok joints is tedious and expensive, it would generally be better te
use paint rather than plaster. The walls should also be washed olean from all dust ; and ahould hs
slightly dampened as the plaster is put on.

To imitate granite on outer walls : after the second or smooth ooat of plaster is dry, it reoeives a
eoat of lime wash, slightly tinted bv a little umber, or ochre, &c. After this is dry, in case it appears
too dark, or too light, another may be applied with more or less of the ooloring matter in it. PinaUy,
a wash of lime and mineral-black is tprinVLtd on f^m a fiat brush, to imitate the black specks of
granite. Bv this simple means, a skilful workman can produce excellent imitations. The horlfeontal
and vertical Joints of the imitation masonry, may be ruled in by a small brush, asing the same Uaek
wash, and a long straight-edge.

The rough surfaces of all walls are more or less warped, or out of line ; and it is not possible fMr
the plasterer to rectify this perfectly by eye, as may be seen in almost everv house. Even in what
are oalled fint-elass ones, a quick eye oan generally detect onsightly undulations of the plaatend
■orfheee.

To prevent this, the process of sereedinfT ^ resorted to. Screeds are a kind of

gauge or guide, formed by applying to the first rough ooat, when parilv dried, borisontal strips of the
plastering mortar, about 8 ins wide, and f^m -i to 4 ft apart all around the room- These are made Is
project from the first eoat, out to the intended face of the seoond one : and while soft are eareftolly
made perfectly straight, and out of wind with each other, by means of the plumb-line, straight-edga
4c. When they become dry, the second ooat is put on, filling up the broad norisontal spaces between
them ; and is readily brought to a perfectly fiat surface, corresponding with that of the eereeds, hj
means of long straight-edges extenoing over two or more of the latter.

A day's work at plastering.

A plasterer, aided bv one or two laborers to mix his mortar, and to keep his hawk mipplMi, esa
average from 100 to 900 sqpare yards a daj» of first ooat; about ^m Boeh of leooad; aad half as

BLATINO.

96»

■nok «r thtrd, whleh reqniret mora ewe. The amooBt wlU depend vpon the nusber of eaglae, eiat
•f roome, whether on oeiUnge or on walU, ko, ko.

Gen Olllmore's estiniate of eo«t of plasterlngr 100 square yards
with 2 or with 8 coats. Common labor $1 per day.

Materials.

QnJaUline... ....•

** for line itnff..

Plaster of Parle

Lnihs

Hair

Common Sand

White Sand

Kaile

If aeon's labor

Laborer

Cartace

Cost of 100 sqnare yards.

Tbree Goats.

Two Goats.

Hard finished work.

Slipped ooat finish.

4easki.

$4.00

SJieasks.

fs.sa

^ "

.86
.70

aooo

4.00

fOOO.

4.00

4ba8h«is.

.80

8 bashels.

.00

7 loads.

1.00

Oloads.

1.80

SHbnshds.

.25

ISlhe.

.90

ISlbs.

.90

4 days.

7.00

8H days.

8.12

8 days.

8.00

a days.

2.00

2.00

i.ao

$25.50

«f.95

This aiBowits to VH *ts per sq yd for S eoals; and say 20 cts for Seoats.

PlRflferlngr lalbs are usually of split white or yellow pine, in lengths of
•boat 8 to 4 fbet ; and henoe eallod 8 or 4 ft laths. They are about IH ins wide, by H inch thick.
Thej are nailed up horisontally, abont H inoh apart. The upright stads of partitions are spaoed at
•noh distances apart, (generally abont 15 ins from center to center,) that tiie ends of the laths may
be nailed to tbem. Laths are sold by the handle of 1000 each. A square fbot of surfaoe requires IH
four feet laths ; or 1000 such laths will oorer 886 eq fU Sawed laths may be had to order, of any re>
quired length. A carpenter can nail up the laths for from 40 to 80 sq yds of plastering in a day at
10 hours ; depdbding on the number of angles in the rooms, Ao.

*-#-

SLATING.

SooFnra slates are usually fix>m V^ to ^ inch thick ; about -A- being a commoM
areraffe. They may be nailed either to a sheeting of rough boardi (e, ffy in the fig)
from fiU>\\^ inch thick, (which should be, but rarely are, tongued and groovad,)

970

SLATING.

Uid horizontally from rafter to rafter ; or Blopins, from purlin to purlin as the
case may be; or to stout laths 1 1 1 about 2 to 3 ins wide, and from 1 to 1^
thick, nailed to the rafters at distances apart to suit the gauges of the slatoi
Two nails are used to each slate ; one near each upper corner. They may be either
of copper, (which is the most durable, but most expensiye.) of zinc, or of either
galvanized or tinned iron. The last two are generally used ; or in inferior work,
merely plain iron ones, previously boiled in linseed oil, as a partial preserra-
tive (torn rust. Bust, however, sometimes weakens them so much that they
break; and the slates are blown off in high winds, to the danger of pasaexB by.
Since good slate endures for a long series of years, it is true economy to use
nails that are equally durable. In iron roo&, the slates, instead of being nailed
to boards, are sometimes tied directly to the iron purlins, by wire. A SQoareof
slating, shingling, Ao, is 100 sq ft.

In laboratories, chemical factories, Ac, subject to acid ftimes, it is difficult to
provide a meul fMteniog that will not be eaten away. In anoh oaeee it is beet to depend ehleflj upon
a layer of morur between the slatee. Thii will harden before the metal faateninse give way ; and
will hold the slatee in place, while new fastenings are being inserted.

The least pitell oonsidered advisable for a roof, to prevent rain or snow tmm being driven
through the interstices between the slates, is i^jMut 38H° t or 1 vert to S hor : which corresponds to
a rise of ^ the span in a common doable pitched roof. Bat even at steeper pitches, rain, and more
partloalarly snow, will be foroed through the roof by violent winds; especial] v if laths alone be ased
or even boarding alone. To avoid this, a layer of mortar about % inch thick, may be epread ever
the touching surfaces of the slstes if on laths. If on boards, the same prooesa may b« adopted; or
the more common one of first covering the boards with a layer of what is called Hating /M ; but
which in reality is merely thick brown psper, soaked in tar. This Is sold in long continuous rolls,
28 ins wide, and weighing from 40 to 60 lbs. A 60 lb roll will oover about SOO eq ft of roof. With

i roper precautioos against the admission of rain and snow, a pitch as flat as 1 in 2^, or even 1 in
, may be adopted.

The thickness of slate on a roof is doable ; except at the h^t i;i; Ac, where It ia trlpla. The
lap is measured fh>m the nail hole (under i) of the lower slate, to the lower edge or • tail, s, of the
upper one; audis usually about 8 ins. In order that the showing lower edges of the slates shall,
when laid, form regular straight lines along the roof, the nail hole* are made at equal distances fk«m
said lower sdges ; so that any irregularity of length Is ooncealed f^om view at the hidden heads vt
the slates. The slater estimates the length of his slate fkt>m the nail hole to the tail; discarding the
narrow strip between the nail hole and the head. If fh>m this reduced length the lap be dedocted,
then one-half of the remainder will be the gaitge, weathering, or margin, of the slating; or, la
other words, the thawing or expoted width of the courses of slates. The gauge In ina mnltipUel
by the width of a slate hi ins, gives the area in sq ins of finished roof oovered by a single slate ;
and if 14i (the sq ins in a sq foot) be divided by this area, the quotient will bethe number of alalss
required per sq ft of roof. The upper side of a slate is called its hack ; the lower one, Ita bed.

Slating, like shingling, must evidently be commenced at the eaves, and extended upward. Blaos
the beds of the slates are not exactly parallel to the boarding, and oonsequently do not real flat npoa
it, those at the lower edge w would easily be broken. To prevent this, a tiUing ttrip (a
stout wide lath, with its upper side planed a little bevelling, to suit the slope of the slatas) Is Int
nailed around near the eaves, for the tails of the lowest course of slates to rest on. This is shown en
a larger scale at T.

Slate of the best quality has a glistening semi-metallic appearance, somewhat like that of a i
face of paper rubbed with black-lead pencil. That of a dull earthy aspect, is softer, mora i

bent, and consequently more liable to yield to atmospheric influences, rain, f^Mt, fto. Iron pyrltss
frequently occurs in slate; and since it always decomposes and leaves holes, should never be admitted
on a roof. Of two qualities of slate, that which absorbs the least weight of water, whan pieces of
equal sixe are soaked for an hour or two, is generally the best; being least liable to split bj frost,
and become weather-worn. This test is easilv applied.

In England the dlflSsrenft slxea are dlsungnished by absurd names of no meanins- In the
United States they are called 6 bylS's; 16by24's, Ac, according to their measures in Inches. Tbej
may be cut to order, of almost any prescribed dimensions, or shape. Tho^ in common use vary ttom
about 7 by 14, to 12 by 18. The first forms about 6 to 6 inch courses ; and the last about 7 to 8 inch;
depending upon how far fh>m the head the nail holes are pierced. The farther this is, the firmer
will the slating be.

Slate roofs, like iron ones, heat the rooms immediately below them very much. This is somewhat
diminished when the slates are on boards, instead of laths ; and still more by a cost of plaster be-
neath. They are also liable to break when walked on ; less so when bedded in mortar.

Welgpnt of slate roofii. Slate weighs about 175 Bis. per cub foot; therefore,

a sq ft, yi inch thick, weighs about 1.8 lbs; i^, 2.7 lbs; and ^ thick, S.Slbs. But owing to th«
overlapping, asqusre foot of roof requires about 2.^ sq ft of slate of ordinary sises; andtf tbs
slate is laid on boards an inch thick, the weight per sq ft of roof will be increased about SW fts;
or with \% inch boards, 2.8 lbs. Laths will weigh about % lb per sq ft of roof.
Hence,

▲pproz Waicht

of one an ft of

Slating, in lbs.

Slate H inch thick on laths 4.75

" " on 1 inch boards 6.76

" " on IH " " T.SO

•• 8-16" . on laths 7.00

on 1 inch boards. 9.00

" " " onlH" " 9.65

" H " on laths 9.S6

" " " on 1 Inch boards IIJS

" •' " on IH " " 11.80

If slating felt is used, add ^Ib ; or if the slates are bedded in H inch of mortar, add S lbs.

SHINGLES. 971

for the total weight borne by the roof tnuw, thatof the pDrline aim mwt be added. This wiU
xu»t Tary muoh fh>m the limiu of 1 ^ to S lbs per aq ft in roofi of moderate span. Add for wind and
■now, eay 20 B>i per eq ft ; and finally add the weight of the truu itaeif.

For stopplniT ttke Joints between slates (or shingles, ftc) and chimneys,
dormer windows, Ike, a mixture of etifT white- lead paint, a« aold by the keg, with sand eQough to pre-
went it ftx>m running, ii very good ; espeolally if protected by a ooverlng of stripe of lead, or copper,
tin, ko, nailed to the mortar-joinu of the chimneys, after being bent so as to enter said jolnu ; which
should be scraped out for an inch In depth, and afterward refilled. Mortar protected in the same
way, or eren unprotected, is often used for the purpose ; but is not equal to the paint and sand. Mor
tar a few days old, (to allow reflraotory particles of lime to slack,) mixed with blaoksmith's oinder»
and molaasoa, is muoh need for this purpose , and becomes very hard, and drectlTe.

SHINGLES.

Wmra cedar shingles are the best in use ; and when of good quality will last 40 or
SO years in our Northern States. They are usually 27 ins long ; by from 6 to 7 ins
wide ; about ^ inch thick at upper end ; and about % at lower end or butt ; and are
laid in courses about 9% ins wide ; so that not quite ^ of a shingle is exposed to the
weather.

They are Qsoally laid in three thicknesses ; except for an inch or two at the upper ends, where there
are Ibar. They are nailed to sawed shingling-laths of oak or yellow pine; about 16 ft long; 3^ ia«
wide, Mid 1 inch thick ; placed in horisonul rows about 8^ ins apart. These are nailed to the raft<
•rs. or purlins : wbieh. for laths of the foregoing sise, should not be more than 2 ft apart fh>m oenter
to oenter. Two nails are used to each shingle, near its upper end. They should not be of less sise
than 400 to a lb. Wrought nalle being the strongest, are the best; out ones are apt to break

by the warping of the shingles. Two pounds of snob nails will suCBoe for 100 sq ft of roof, ineludlng
waate. An average shingle IVi ins wide, in %)4 inch oourses. expoees 639^ sq ins ; making 2}i shingles
to a sq ft of roof: but to allow for waste, and narrow shingles, it is better in praetice to allow about S
shingles to a sq ft.

Shingling, like slating, mast plainly be begun at the eaves : and extended upward. For closing th^
joints between the shingles, and chimneys, dormer windows, Ac, see at end or Slating.

Qypross and white pine are also muoh used for shingles, being mnch cheaper, but scarcely half as
durable. All shiogles wear quite thin In time by rain and exposure. In warm damp climates they
all deoay within 6 to 12 years.

■ ^

PAINTING.

principal material used in house-painting, is either white lead, or oxide of
zinc, ground in raw (unboiled) linseed oil, by a mill, to the consistency of a thick
paste. In this condition, it is sold by the manufacturers in kegs of 25, 50, and 100
ms. To prepare it for actual use, merely requires the addition of more linseed oil,
iay 3 or 4 pints to 10 lbs of the keg paint, for thiiming it sufficiently to flow readily
uiiuder the brush.

Good painting requires 4 or 5 ooate ; but usually only 4 are used In principal rooms ; and S In inftoloh
ones. Bsoh coat must be allowed to dry perfectlv before the next one is put on. One lb of the keg
paint will, after being thinned, cover about 2 so yds of first coat; 3 yds of second; and 4 yds of each
snbaequent coat ; or 1 sq yd of 8 coats will require in all, 1.06 As ; or 4 coats, 1^ fts ; of 5 coats, 1.58
l>s. The reason why the first coats require so much more than the subsequent ones, la that the bare
snrfaoe of tbe wood absorbs it more.

When, as is usual, raw or unboiled oil is used for thinning, drytira mnst be added to it; otherwise
the paint might require several weeks to harden ; whereas, with drjers, from 1 to 8 days, according
to the weather, soffloe for each coat to become bard enough to receive tbe next one. Tbe dryers most
sommonly used, are powdered litharge, in the proportion of one heaped teaspoonfnl : or Japan var*
aish, 1 table-spoooful, to 10 lbs of the keg paint. Either sugar of lead, or sulphate of zinc, may also
be used instead of litharge ; and in tbe same proportion. Although both litharge and Japan vamtsb
are dark-colored, yet the quantity is so small as not to appreciably affect tbe wbitenesa of the paint.
If the vamiah ia used in exceaa, aa ia often done in the hurry to have work flniiihed, it producea
eracka all over tbe aurface. No drjer ia necessary if paintera' boiled oil be used for thinning. Mere
boiling will not canae oil to harden more rapidly ; but that intended for painters, has litharge added
to it previously to boiling ; in tbe proportion of 1^ 0>s to each 10 gallons of raw oil. In some works
written for the use of house painters, it Is asserted that boiling renders the oil too thick for any but
eoarse outdoor work. But this is entirely a mistake; for if the boiling be properly done, the oil
will be quite thin enough for the best inside work ; and will moreover be olearer than while raw ; and

972

FAINTING.

VUl laipart to Ui« pmioied rarfM* » more •htnlnf appMmaoe. The teat ahoiild be barely
to prodoM botliog ; or about 400° Fata. The boiling thoold oontinue aboat 1^ hoars ; the oil bdag
thoroughly eUrred ai short InterraU, to preyent the litharge from eettUng at the bottom. The fire
may thea be allowed to lubelde; when the operation wiU be oompleted. A aedimeiit will then form
at the bottom ; which muit be left behind when the oil it poured off. Although no dryer ia neeecsaty
with thia oil, ■till a little litharge may be added when great expedition demand* it. Painters rarely
ase this oil. on aocount of lu tri&lng inarea«e of cost.

Another •ubttanee much uaed with the thinning oil, (ezoept for the first ooat,j is splrito of turpen-
tine ; called " turp" by the workmen. The quantitT of oil may be diminished, to the extontof the
added turp. This being more fluid than oil, causes the paint to work more pleasantly under the brash.
It moreover diminishes the tondency of the paint to beoome yellow ; espeoially in rooms kept closed
for some time. It is also much cheaper than oil. It should not be used, or but sparinglr, for exposed
outdoor work ; inasmuch as its tondency is to impair the firmness of the paint ; and althonch its
•flbots are scarcely appreciable indoors, they are qulto apparent when the work has to t«aist the
weather. As the fashions ohange in hoase«paintlng, the surface is at times required to present a
shining or glossy finish ; at other times a deeid one is in vogue. The glossy one is that whieh the
Mint will naturally have, provided that no more turp than oil be used in the thinning. The dead
Inish is obtoined'hy using no oil, bat tarp alone, (or the last ooat; which in that case is oallcda

fi\tHng eoQt. Although turp is not properly a dryer, still, as it eraporatos qolokly, it Caoilitates the
rdening of the paint.

In outdoor work it is usaally advisable to use more dryer than inside, so that the paint may seanar
become hard enough not to be Injured by dust or rain. Otherwise less wonid be better.

When, instoad of a whito finish, one of eome other oolor is required, the ooloring in^redieBt It
mixed with the whito paint to be need in the last ooat only ; although two ooloring ooato are sease-
times found to be necessarv before a satisfactory elEeot is prodaeed. The ooloring ingredients may be
indigo, lampblack, torra sienna, amber, ochre, chrome yellow, Venetian red, red lead, Ac, Ac; whieh
are ground in oil, ready for sale, by the manufacturers of the white-lead and sine painto. They are
■Imply well stirred into the whito paint.

All surfaoes to be painted, should first be thoroughly dry, and tn» from dost. If on wood, att
plane-marks, and other slight irregularities, should first be smoothed oiT by sand-paper, when the
neatest finish is required. Also, aU heads of nidls must be punched to about H inch below the tor-
faee. To prevent knou ftom thowing through the fioished work, (as those in whito or yeUow pine
would do, on aocount of the oontalned turpenUnc,) they must first be killed, as it is tormed. A asosi
and eflbetive way of doing this, is by covering them with two eoato of shellao varnish ; whieh, whea
dry, should be smoothed by sand-paper. Another mode, not qoito so certain, is by one or fcwo eeatt
of whito lead mixed with thin glue-wator, or tiju, at it it ealled.

Aftor thete preparations, the first, or priming coat, is put on ; in which there should be no toxp;
because it would sink at once into the bare wood, leaving the whito lead behind it, in a nesu-ly dry
friable condition. After this the nail holes, eraoks, Ac, must be filled with oommon glasiera' pnt^,
made of whiting (fine clean washed ehalk) and raw Unseed oil ; boiled oil will not answer ; the potty
would be friable. The putty would be apt to fall out, if pat in before priming ; beoaose tbe weed
would absorb the oil, and the putty woold then shrink. After the firit coat is perfecUy dry, the
second one is put on ; and for it about 1 measure of turp may be mixed with S measures of the thin-
ning oil. In the third, and any subsequent coato, equal measures of turp and oil, may be osed tar
thinning, if the work is required to dry wUh a glon ; but if it is to finish dettd, the last eoat most
be a JUUttng one ; or one in which the thinning oil Is •ntirely omitted, and torp aloae sahatitatod
for it.

Painters generally clean their brushes by merely pressing oat most of the paint with a knifle ; aad
then keep them in water until further use. If to be put awav for some time, they may be thoronchty
•leaned by turp ; or by soap and water. To prevent a bard skin tnm forming on the top of thtir
paint when not osed for some days, they pour on a little oil.

The beat paints for preserTinfp iron exposed to the weatlier»

a4>pear to be pulverized oxides of iron, such as yellow and red iron ochres; or brown hematite inm
ores finely ground ; and simply mixed with linseed oil, and a dryer. Whito lead applied directly to
the iron, requires incessant renewal : and indeed probably exerta a oorrosive elfeet. It may, hew-
ever, be applied over the more durable colors, when appearance reqaires it. Bed lead is said to be
Terr durable, when pure. An insUnoe is recorded of pump-rods. In a well 200 ft deep, near London,
which, having first been thus painted, were in use for 45 years : and at the expiration of that time,
their weight was found to be precisely the same as when new; thos showing that nut had nc*
affected them.

When tbe sise of the exposed Iron admits of It, Ito freedom fh>m rast may be very mvcb promoted
by first heating it thoroughly; and then dipping it into, or washing it well with, hot linseed oil;
which will then penetrate Into tbe interior of tbe iron. For tinned iron exposed to the weather, on
rooffe, rain pipes, Ac, Spanish brown is a very durable color. The tin is frequently foaad perfectly
bright and protected, when this color has been osed, after an exposure of iO or 50 years. Whilt
paint washes off In a few years by rain.

Plastered walls should If possible be allowed to dry for at least a year, before being painted in oB
otherwise the paint will be liable to blister. They may, if preierred, be frescoed (water-ooleiL.
mixed with size) to the desired tint during the interval.

The painting of unseasoned wood hastens its decay. If the lorfhce to be painted la greasy, the
grease must first be removed by wator in which is dissolved some lime.

Washes for outside work. Downinf, in his work on country houses^
reoommends the following: For wood'unrk; in a tight bushel, slack half a bashel of fTesh liau. by

Knring over It boiling wator snOoient to cover it 4 or 5 ins deep ; stirring It antil slaeked. Add f
I of sulphate of sine (white vitriol) dissolved in water. Add wator eno«ch to bring all to the eea-

sistenoe of thick whiwwash. Apply with a whitewash brush. This wash is whito; bat it may be
Mlored by adding powdered ochre, Indian rod, umber, Ac. If lampblaek is added to water-oolors, U

QLAS8, AND GLAZINa.

AlMD, Hotbir. hM u lUiid I'S cu » j»ri . WSii baiiblulod; lOasini r» IIdihI dIL; KB
irja: U >• lailT illwd ibur olein hhiI; 1 Ite n> Bnlar. AMiaj UUH. hj Kr>i»>(><ii^

CWment n»r sMppInK Joints, xurh u smoDd chimneys, Ac, Ac While

QLASS, AlTD SLAZINQ.

TABLE OF NVnBERS OF PANES IIT A BOX.

974 GLAflS.

The bflst qnalltiea of Amerioan glase made In the Tldnity of PfaUadelphiay
Boston, Pittsburg, Ao, are for most purely ua^ftU purposes, as good as those from
foreign oountries ; but when the highest degree of beauty is required, as in the
lower front windows of first-class dwellings, fkncy stores, Sui, polie^ed pfavte*
glass of England, France, or Germany, must be used, although the price for
moderate sized panes is from 6 to 8 times as great as that of the best quality
single-thick American. Its perfectly smooth surface, free from distorted refleo*
tions, also makes it the best for covering pictures ; still, if carefully selected
American panes be used for this purpose, few except critics in glass will detect
the difference.

A tblck arlasB is made expressly for floorlngr* up to 1 inch thicks
and up to 50 inches by 9 feet dimensions. Also, for skylights, from ^ to ^ inch
thick. This can be fUrnished to order of any size up to 40 inches by 8 or 10 feet
The smaller sizes can also be had ground. Grinding prevents the entrance of
the Aill i^lare of the sun ; and, moreover, diffuses the fight over a much greater
width of space below.

Strengrtb of ylass. Tensile 2500 to 9000 lbs per square inch. Boston rods
by author, a'^OO to 5200. Crushing strength, 6000 to 10000 lbs per square inch.
Transversely, (by the writer's trials,) flooring glass, 1 inch square, and 1 foot
between the end supports, breaks under a center load of about 170 fts ; con-
sequently, it is considerably stronger than granite, except as regards crushing ;
in which the two are about equal.

Remark. Window and other fflass which contains an excess of potash or of
soda is very liable to become dull in time, owing to the decomposition of those
ingredients by atmospheric influences.

ROPE.

975

ROPE.

Xlie strengrtli of rope varies jgreatly. Pieces from the same coil may vary
25 per cent. The table below supposes an average quality Manila. Grood
Italian hemp is considerably stronger. The tarring^ of roi>es is said to
lessen their strength ; and, when exposed to weather, their durability also. We
believe that its use in standing rigging is partly to diminish contraction and
expansion by alternate wet and drying weather. A few months of exposed
work weakens ropes 20 to 60 per cent.

•

Table of Manilla rope.

Diam.

Circ.
Ins.

Wtper
foot,
lbs.

Break]
Tons.

ing load,
lbs.

Diam.
Ins.

Circ.
Ins.

Wtper
foot,
lbs.

Breaking load.

Ins.

Tons.

lbs.

.239

%

.019

.25

560

1.91

6

1.19

11.4

25536

.318

1

.033

.35

784

2.07

6K

1.39

13.0

29120

.477

IK

.074

.70

1568

2.23

7

1.62

14.6

32704

.636

2

.132

1.21

2733

2.39

7V^

1.86

16.2

36288

.795

2^

.206

1.91

4278

2.55

8

2.11

17.8

39872

.955

3

.297

2.73

6115

2.86

9

2.67

21.0

47040

1.11

Z%

.404

3.81

8534

3.18

10

3.30

24.2

54208

1.27

4

.528

5.16

11558

3.50

11

3.99

27.4

61376

1.43

4K

.668

6.60

14784

3.82

12

4.75

30.6

68544

1.59

6

.825

8.20

18368

4.14

13

5.58

33.8

75712

1.75

5}i

.998

9.80

21952

4.45

14

6.47

37.0

8288G

Working: loads. For manila ropes from 1 to 1% ins diam, running at
different speeds over 'sheaves of the diaras stated, Mr. C. W. Hunt (Trans Am

Speed

Slow

Medium

Bapid

ft per min

50 to 100
150 to 300
400 to 800

as for work on

derrick, erane, quarry
wharf, cargo

C

0.140
0.056
0.028

1" rope 1%" rope
D D

8
12
40

14
18
70

Snch ropes wear out rapidly. A rope 1^ ins diam wears out in lifting from
7,000 to 10,000 tons of coal. On the other hand, 1^ inch transmission ropes,
running 6000 ft per min and carrying 1000 H. P. over sheaves 6 ft and 17 ft in
diam, last for years.

Mr. Hunt's figures for ultimate strength, based upon tests of full-sized speci-
mens of manila rope made by three independent rope-walks and purchased in
open market, are practically identical with those given in our table above, as
are also those of Prof. B. Kirsch, of the Imperial Boyal Technological Industrial
Museum, Vienna, quoted by Mr. Hunt.

976

WEIGHTS AND STRENGTHS OF WIRE ROPES.

WEIGHTS AND STBEITGTHS OF WIRE ROPES.

Wire Rope manufactured by John A. Roebltiig''fl Sons C?o., Tren-
ton, N. J.- The prices and weights giyen are for ropes with hemp centers.
When made with toire centers, the prices per foot are lO per cent, higher, and
the weights 10 per cent, greater.

Trade
No.

Diam. Approx.
in circum.
ins. in ins.

Wt.
perft^
in lbs.

Approx. break-
ing strength * in
tons of 2000 fiE>s.

Iron.

\j, steel.

Minimum diam.
of drum in feet.

Iron.

C. steeL

Price in cents
per foot.t

Iron.

C. SteeL

Standard Holstins Rop«, with 6 strands of 19 wires each.

4

Z%

3

2

1^

8.00
6.30
4.85
4.15
3.55
3.00
2.45
2.00
1.58
1.20
0.89
0.62
050
0.39
0.30
0.22

78

62

48

42

36

81

25

21

17

13
9.7
6.8
5.5
4.4
3.4
2.5

156
124

96

84

72

62

50

42

34

26
.19.4

13.6

11.0
8.8
6.8
5.0

18
12
10
8K

7

%%

6

4

P

1>^

117
92
80
63
57
48
40
33
26
20
16
12
10
8

1^

US,
111

98

74

66

56

46

38

30

28

18

14

12

11

10

9K

Transmission or Hanlag^e Rope, with 6 strands of 7 wires each.

11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

i

4

Z%

3

2

IS
1^

8.66
8.00
2.45
2.00
1.58
1.20
0.89
0.75
0.62
0.50
0.39
0.30
0.22
0.15
0.125

34
29
24
20
16
12

9.

7.

6.6
5.3
4.2
8.3
2.4
1.7
1.4

68

58

48

40

82

24

18.6

15.8

13.2

10.6
8.4
6.6
4.8
3.4
2.8

9H

8

6^
5

r'
I

2

Notes on the Use of Wire Rope, by the Boebling's* Company.

The ropes with 19 wires per strand are the more pliable, and therefore best
adapted lor hoistings and runnings rope. The others are stiffer and better
adapted fofg-uys, Ac. Ropes of iron or steel, up to 8 inches diameter, made to
order. Hemp center rope is more pliable than wire center. Wire rope mnst
not be coiled or uncoiled like hemp rope. When on a reel, the reel
should be mounted on a spindle or flat turn-table in order to pay off the rope.
When forwarded in a small coil without a reel, roll the coil on tne ground like a
wheel, and thus run off the rope. Avoid untwisting and short bends. To
preserve irire rope, apply raw linseed oil (which may be mixed with an

*For the safe worhins load, take one-fifth to one-seventh of the
breaking load, according to speed.

t Discounts, 1901: bright rope, 80 per cent and 7>^ percent.; galvaniaed, 26
per cent and 7j^ per cent

WEIGHTS AND STRENGTHS OF WIRE ROPES. 977

equal quantity of Spanish brown or lamp-black) with a piece of sheepskin,
keeping the wool against the rope. If for use in water or under-

ground, add 1 bushel of fresh-slacked lime and some sawdust to 1 barrel -of
tar. Boil the mixture well aud saturate the rope with it while hot.

Oalvaniaed wire rope for rigging is cheaper and more durable than hemp
rope ; and does not stretch permanently under great strains. Its bulk is one-
sixth and its weight one-half that of hemp rope. Roebllng's wire rope has been
made the standard by the United States Navy Department. Shackles, sockets,
swivel-hooks, and fastenings, dbc., furnished ana put on and splices made.
Pulley-wheels furnished. Also galvanized steel cables for suspension bridges.
Crucible cast-steel wire ropes are much more durable than iron ones. They
should be kept well lubricated.

Patent .Flattened Strand Wire Rope.
ManufactuxQd by A* I<eselien A Sons Rope Oo., St. lioals. Mo. .

Hoisting Ropes.

Haulage and Transmission BopesL-

Breaking • strength
in tons of 2000 fbs.

Minimum diara.of

• List price f per

.■•'

-8

9^

drum in feet.

foot, in cents.

i

■t-f

4^

•

■t-f

-1.3

•

■«-+

tj

.F4

^ >

2

c

.^

GO

g

.^
#«

3

s

••^

'^

»-•

S

&

i

u

2

• S3

%

A vera
per

1

Crucibl

stee

Swedes

3

Crucibl
stee

1

OS

3

w

Crucibl

ste<

OQ ■

S

5

Hoistingr Rope.

8.50

260

176

75

<0 OD

9

10

257

182 •

152

6.50

211

140

66

5|

8

9

202

144

120

5.00

168

109

54

7.25

7.5

173

121

104

3.71

. 124

81

40

> ^ flj

A <T» 2:

5.75

6.5

123

86

74

2.50

84

56

28

5

0

82

59.5

52

2.15

67

47

21

4.5

4.5

68

50

43

1.70

56

38

17

3 ►. M

4

4

56.5

39.5

34 -

1.25

40

29

13

3.5

3.5

45

30

26

0.96

32

21

9

9;s *

3

3

35

24

21

0.67

22

15

6

«8'g

2

2.5

26-

18.25

15.5

0.44

13

9

4

1.5

1.75

19.5

14.5

10.5

Hanlaffe and Transmission Rope.

i

2.40

80

54

2.00

64

45 1

1.64

53

36

1.20

38

27

0.93

30

20

0.68

21

14

0.40

13

9

81

54

•

45

67

45

36.5

53.5

35

29

42

27.5

22

34

20.5

17.5

24

14

12.5

ir.5

10

8.25

1

♦Working load — 0.2 X breaking strength.
+ Discount, 1901, 30 per cent, and T^ per cent.
J "Hercules." ''Made from a specially drawn
which is solely made for this brand of rope."

62

and patent tempered steel,

978 PAFBB.

PAPEE.

M sbeeta 1 qnire. 20 qnirei 1 resm.
Slaes of drawlnir PApers.

Idi. Int.

ADtlqaarlu 31 X 62

Double Blephant 26 X 40

Atlas 26 X 34

Imperial 21 X SO

Ins. I0S.

Super Royal 19 X 27

Royal 19 X 24

Medium 17 X 22

Demy ^.. 15 X 20

Gap IS X 17

The English drawing-papers are slrouger nnd superior to the American. Those
by Whatman bare a high reputation ; they are, however, of different qualities. When

f taper is pasted on musliii, the difference in quality is not so Important. Of paper
n rolls, the German makes are the beet. There is but little of other makes imported.

Botb white and tinted papers, for the use of engineers, are made
in continuous rolls, without seams. Widths 36. 42, 54, 58, and 62 ins; usual
lenffths 40 yds ; but can be had to order to 400 yds or more. These may also be
baa mounted on muslin, in rolls 10 to 40 yds long.

Gartridg^e or pattern paper is furnished in long rolls, of same lengths as
white paper, mounted or not ; widths up to 54 ins. Color, a light buffi

TraeinfT paper. Most of that sold, whether domestic or foreign, tears so
readily as to be of comparatiTcly little serrice. Parchment paper, 87 and 88 ins
wide, rolls of 20 and S3 yds, is better, but does not take ink perfectly.

Traein^ eloth, usually called tracing muslin^ and sometimes vellum etath, is
altogether preferable to tracing paper, on account of its gretA strength. Widths
18, 80, 36, and 42 ins ; lengths to 24 yds.

Profile paper is made in widths of 9 ins and 20 ins, and in single sheets
or in long, continuous rolls.

CroMi section paper, mounted or unmounted, tracing paper and cloth,
are furnished in sheets and in rolls, ruled in quarters, fifths, eighths, tenths,
twelfths, and sixteenths of an inch, or in millimeters.

Colors. Since the introduction of blue printing, tinted drawings are seldom
made, except for architectural effect; but colors may be used to adrantage on
black-line prints from tracings, p 432 d. A good draughtsman needs but few
colors; say India ink, Prussian blue, lake, or carmine, ught red, burnt amber,
burnt sienna, raw sienna, gamboge, Roman ochre, 8u> green. Winsor A Newton's
colors are among the best in use. Purchase none but the Tery best India ink.
Cakes of colors should always be wiped dry on paper, after being rubbed in
water ; and but little water should be used while rubbing ; more being added
afterward.

I^ad pencils. Genuine A. W. Faber's Nos. 2, 3, and 4, are very good. The
hardness increases with the number. Nos. 3 and 4 are good for field-book use : which
to prefer, will depend on the character of the paper; No. 3 for smooth, and No. 4 for
the coarser or more granular papers. His leered pencils are of a higher grade and
better suited for draughting. " H " stands for " hard," " B " for *' soft." The degree
of hardness or of softness is indicated by the number of H's or of B^s. "F** (inteiv
mediate) corresponds with No. 3. Dixon's American pencils are good. The oflBoe
dranghtflman should have a flat file, or a piece of fine emery paper glued to a strtp
of wood, upon which to rub his lead to a fine point readily, after using the knife.

BLCE-PKINT8. 97t»

BLUE-PRINTS, ETC *

Art. 1. (a) In order to obtain the best results, all umneeessary ex-
posure, either of the sensitized paper or of the solations, to sunlight or to
other white light should be avoidea.

(I») CleanllaesB is of the first importance. The vessels in which the solu-
tions are made and mixed must be scrupulously clean, and, if washed with soap,
must be oaref\illy rinsed with clean water. They should be left full of water
when not in use. The presence of free alkali of any kind is fatal to good re-
sults, and immediately destroys the blue cOlor of a finished print. See Art. 19 (b).
The solutions must not be allowed to oome in contact with iron.

Art. S. (a) The Solution used in sensitizing the fiaper for blue-prints
is usually that of /erricvanide of- potassium {red prussiate of potash) f and am-
moniocitrate of iron (citrate of iron and ammonia) in water.

(b) The two salts are usually dissolved seimrately and the two solu-
tions then mixed. The potassium salt should be broken up fine. The iron salt
is usually quite pure and dissolves very rapidly. It may be kept indefinitely iu
a solid state if perfectly dry, but it readily absorbs moisture, and then becomes
sticky and unfit for use; and the solution is apt to become mouldy after a few
days, either alone or when mixed with the potassium solution. Hence, it should
be prepared (in a dark room) in small quantities as required.

Art. 8. (a) The following is an average of several recipes that give ex-
cellent results:

Solution A.. 1 ounce of red prussiate of potash to 6 ounces of water, or 2)4
onnoes of the salt to a pint of water.

Dissolve thoroughly and filter. The solutions may be sufficiently filtered
through raw cotton, and much more rapidly than through paper.

Solution B. IH ounces of aminoniocitrate of iron to 6 ounces of water, or 4
ounces of the salt to 1 pint of water.

Dissolve thoroughly. Filter, unless the solution is perfectly clear.

(b) Keep the two solutions in separate glass-stoppered bottler in a dark place
until they are to be used. Then mix them in equal parts, and filter the mix-
ture. Take care that no undissolved particles of the red prussiate get into the
double solution. It must be rejected when its brown color changes to bluish green.

(c) The combined solution will eost amateurs from 1 to 2 cents per ounce to
make. About 4 ounces will suffice for coating 100 square feet of paper.

(d)' If a lew drops of strong ammonia solution be added to the citrate solu-
tion, B. until the odor is quite perceptible, the addition of a saturated solution
of oxalic add in water to the double solution will basten tbe prlnt-
inirtn cloudy weather. 10 per cent, of the oxalic-acid solution will in-
crease the rapidity of printing about 2^ times ; 20 per cent., 5 times ; 30 per
cent., 10 times r but with more than 20 per cent., it is difficult to get clear white
lines. In sunlight the difference is much less marked. (Engineering News^ Dec.
15, 1892.)

Art. 4. (a) Where fine work is not essentiaLany well-sized paper, suffi-
ciently tough to bear the washing, will answer. For important work use paper
of fine uniform texture and smooth hard surface, free rrom injurious chemical
substances. If the solution penetrates below the surface, a portion of the chem-
icals may remain in the paper in spite of the washing, ana damage the result.
Many papers are made especially for this purpose. The Saxe ((lerraan) and
Bives (French) papers are considered among the best. Johannot and Steinbach
papers give good prints, but are not very strong. Weston's and Scotch linen
papers are stronger, and the latter gives excellent prints. Before sensitizing
a large quantity of paper of a new kind, try a small sheet of it. Ijlnen for sen-
sitizing is also sold by dealers in photographic material and engineers' supplies.

Art. 5. (a) The solution Is appltcMl (in the dark room of course) to
one side only of the paper. This is sometimes done by **floatlni^'* tlie
paper upon the solution, taking care that none gets upon the hack of thie sheet.

♦ See "Modern Heliographic Processes," by Ernst Leitze: D. Van Nostrand
Ck> , New York, $3.00; a work to which we are indebted for many valuable sug-
gestions.

"Modern Reproductive Graphic Processes," bv Lieut. J. S. Pettit, D. Van
Nostrand Co., Science Series, No. 76, 50 cents, deals chiefly with artistic photog-
raphy, lithography, etc.

See also paper by Benj. H. Thwaite, Proc's, Inst'n Civ. Eng'rs, Vol. Ixxxvi, p.
812, reprinted in Engineering News, Nov. 27, 1836.

t Not the /errocyanide or j/e/Zof* prussiate.

9i*0 BLUE-PRINTS.

The paper is held by two diagonally opposite oornen, and the diagonal joiDing
the other two corners is then allowed to touch the surface of the liquid. Then
the two- coiners held in the hand are dropped. first one and then the other. The

Saper should then be lifted, one half at a time, to see whether any air bubbles
ave been formed under it. if so, they may be removed b^ drawine orer the
solution that half of the sheet under which they occur, while the other half is
held up from the liquid. One or two minutes suffice for floating, and the paper
is then drawn out over an edge of the bath, draining off the surplus liquid.
This process requires a tray larger than the sheet, and the inner surface of the
tray must be not only water- proof, but also proof against chemical action firtNoa
the solution. Considerable care is required in the manipulation.

Art. 6. (a) The solution is usually applied bv means of a soft wide brvak
(such, for instance, as those used for wetting the leaves in letter-copying boob)
or a large soft spongpe entirely free from sand or other grit.

Art. 7. (a) In applying the solution, the paper may be laid upon a board oot-
ered with .^ofi smooth oll>€loth« which, after each sheet is sensitized, should
be wiped off, to avoid smearing the back of the next sheet.

(b)Tlie operation must be quickly performed, so that no portion ofs
sheet may become dry before its entire BUTf&qe has been coated. For very large
sheets it may be necessary, for this reason, to employ two persons. First cover
the sheet by' strokes of the wet sponge or brush, moved in the direction of the
length of the paper, and then, immediately, by light strokes at right angles to
these and with the sfK>nge or brush squeezed out, so that the solution may be
uniformly and thinly distributed over the entire surface. Wash out the sponge
immediately in the dark room.

Art. 8. (a) The paper is then hung up to dry in the dark room, bv means
of clips, of any convenient form and free from iron. Small sheets may be hung
by one corner ; larger sheets by two adjacent comers, or by three or more places
{according to size) along one edge, taking care to buckle this edge slighUy, so
that the paper may not he stretched in drying. If the sheets are hung overt
rod or rail the solution will dry unevenly at the bend. In order that the whitei
in the print may be clear, the air should be warm, so that, the paper may dir
quickly and the solution be thus prevented from penetrating it deeply.

Art. 9. (a) Make sure that the paper is perfectly dry before it is used
or put aw:iy, and see that it is kept both dry and (i4frk until it is wanted for uft
If carefully prepared and preserved it will retain sensitiveness for a long tine,
but the best results are obtained with fresh paper, and it is best not to keep it
more than a month or two. . *

Art. 10. (a) The traclnfr paper or tracing cloth should be of a blmdi
cast (a yellow paper delays printing), thin (see Art. 15, ), and as nearly

transparent as possible. It should be pre.served, both before and after drawing,
from long exposure to light, which tends to render it opaque.

(b) Both before and arter drawing, it should l>e kept either flat or rolled, and
not folded, because folds render it difficult to bring the drawing into perfect oon-

^4^act with the sensitive paper in printing.

Art. .11. (a) The drawing: or tracing should be made with the best
India ink, rubbed very black. The addition of a little gamboge or chrome yel-
low increases the opacity. Lines drawn in chrome yellow and in gambose print
well ; but Prussian blue or carmine should be rendered more opaque by the addi-
tion of a little Chinese white or flake white. Hold the tracing up to a strong
light, in order to detect any weak places in the lines.

Art. 12. (a) Printing: consists in exposing the sensitive paper to tbe
action of light, the drawing being placed between the light and the sensitive
surface. Tbe arc electric ll^ht prints more slowly than direct sunlight, bat
has the advantage of constancy in allweathers and at all hours, and of fixedness
of position. Pee Art. 16 (a).

(D) Place the frame with its face perpendicular to the rays of light, as nearlv
as may be, and see that no shadows, as of trees, buildings, etc.. are allowed to faU
upon a portion of the drawing.

(c) All handling of the paper, such as cutting it to size or placing it in the
frame, should be done in a weak light.

Art. 13. (a) To secure close contact between the tracing and tbe sensitive
paper (see Art. 15, ) they are usually placed in a printlngr-fk*aine. Tbe

essential parts of an ordinary frame are : the frame proper, a plate of clear glass
for the passage of the light, and a padded back, which, by mean** of clamps and
springs, presses the two sheets closely together and against the glass.

(b) Tbe tracing is laid in tlie frame, with its drawn side next to the glass
(but see Art. 16 b), and then the sensitive paper, with the sensitive

side next to the tracing. Finally, the padded back is placed in the frame.

BLUE-PRINTS. 981

(e) The back is often made in two halves, hinged together and each provided
with a spring, so that one half may be raised to- permit examination of tlie
progress of the exposure, while the other half, remaining clamped, holds tlie
uracing and the sensitive paper in position.

(d) i^y using a frame left open at both ends long strips of sensitive paper mi>y
be used, a part at a time, the rest being rolled up at tne ends of the frame at d
wrapped for protection from light.

(e) In any frame it is important that the glass be suflSciently thick to with-
stand the pressure reouired in order to secure close contact between the two
papers (see Art. 15, below), of excellent quality, and free from defects which
would obstruct or unequally refract the light. The glass) should be carefully
cleaned before printing.

(f ) Improved forms of print! ng-fnimes have rubber air-cushions in place of
flannel pads. In others the necessary pressure is secured by means of a vacuum
produced between the tracing and the glass by means of a pump.

{fg) Printing-frames are supplied by dealers. The prices, including glass, vary
Arom about 92 for frames lU x 12 inches, to $30 or $45 for frames 36 x 60 inches.
lYames running on rollers, with fittings for exposing them outside of windows,
are also furnished, at prices varying with the dimensions and the requirements.

(li) For large blue-prints, Prof. E. C. Cleaves, of Cornell University, uses,
instead of a frame, a wooden cylinder covered with felt and revolving on its
axis. Upon this cylinder the tracing and sensitive paper are stretched by means
of a suitable clamping device, and the cylinder is then revolved in the sunlight.
This method dispenses with the use of glass. It of course requites a longer ex-
posure than the ordinary method. (Trans. Am. Stjc. Meek. Eng., vol. viii, p. 722.)

(1) For still larger prints. Prof. R. H. Thurston stretches the two papers upon
a thin board, which is then sprung into a curve and held in that shape, keepins
the papers in tension upon the convex side. This method also dispenses with
the use. of glass, and, the curvature of the board and the papers being but
slight, the whole of the pnper is exposed to the light at one and the same time.
{Trans. Am. 8oe. Mfch. Bng., vol. ix, p. 696.)

Art. 14. (a) Tlie time reonlred for exponnre varies with the
color, directness, and intensity of the light, with the thickness and opacity of
the tracing paper, with the blackness of the drawing, with the materials and
the care usea in sensitizing the paper, and with the freshness of the latter, from
two or three loinutes to hours or even days. Roughly, we may say that in full
sunlight, in Philadelphia, about three minutes ordinarily suffice from noon to 2
P. M., and ten minutes at 10 a. h. or 4 p. h. ; in the shade, thirty to forty-five
hiinutes at noon ; but no fixed rules can be given. Experience must decide
in each case. A preliminary experiment may be made with a small frame. If^
the back of the frame in in two or more pieces, the process may be inspected from
time to time.

(b) If perfectly opaque Ihk be properly used, the blue background may be

Srinted very dark without spoiling the lines, but over-exposure in printing reh-
ers the background fii-st blackish and then of a dingy shade. See .Art. 17 (c) and
Jd). DrH winffs in pale ink must be printed very lightly, in order that the

lines may remain white, and it is best to use with them a weak light, or to pro-
tect them by tissue paper or ground glass. See Art. 18 (a).

Art. 15. (») To obtain perfectlv sharp impressions, the side of the tracing
. upon which the drawing is made should oe in IminediRte contact with
, the sensitized surface of the blue-print paper, especially if, as with sunlight, the
direction of the light is variable ; for, if any appreciable distance intervenes
between the two, as in printing through cardboard (see Art. 16, below), the
shadows cast by the lines of the tracing will move over the sensitized surface as
the direction of the light changes, and thus give a blurred impression. In most
cases, however, it is practically out of the question to place the two surfaces in
this way, because that position gives a reversed impre&rion as regards right and
left.* Hence a thin tracing paper or linen is recommended in Art. 10 (a). For
the same reason it is imperative that the two papers be firmly and evenly
pressed aminst the glass.

Art. 16. (a) By using a light which is constant in position, relatively to
the surface of the tracing, such as an arc electric light, it is possible, by prolong-
ing the exposure for hours or even days, to obtain blue-prints f)roin draw-
insti made upon atoat drawing: paper or even upon bristol board.

(d) With sunlight the same object may be accomplished, either by placing the
original with it« back to the glass, and the sensitive paper (which sliould be very

* A print, thus reversed in position, may of course be easily read by means
ef a mirror. This is commonly done with Patent Office drawings.

982 BLUE-FRINTB.

thin) with its back to the sunlight, or by placing the printing firame in the bottom
of a deep and narrow box, so that the light can shine directly upon the frame
only when approximately parallel with the long sides of the box. To print
rapidly, the sunlight must be kept full upon the frame by frequently moving
the t)ox.

Art. 17. (a) The print, when sufficiently exposed, is taken from the frame,
and both its race and back are waslied tnorouybly In clean water until
the characteristic blue color is perfectly developed.

(b) The washing should be done in a. tray with a flat bottom larger than the
largest print to be washed, and care should be taken not to injure the sur&oe
of the prints bv hard rubbing or by sharp bending, or otherwise. It is better
to have a circulation of water in the tray, not only to keep the water clean, bat
also to bring about the necessary agitation of the prints without handling them.

(e) The washing may be hastenedjand dark or ^'over-exposed "prints may be
lightened somewhat, by having the water warm, say at 90^ or 100^ Fahrenheit.

(d) Over-exposed prints may also be lightened by immersing them in water
rendered slightly alkaline by ammonia. In this bath they at once assume t

Kurple tint, which soon becomes weaker. At the proper moment, which most
e learned by experience, the alkaline action must be stopped by drawing the
print rapidly through a solution of 1 part of hydrochloric (" muriatic ") add
(H. Gl.) in 100 parts of water.

(e) Continue washing until the water has for some time come off perfectly
clear. Then hang the prints up smoothly to dry.

Art. 18. (a) After washing, the application of a solution of from 1 to 5
per cent, of hydrochloric acid, or of oxalic acid, in water, intensifies the blue
color, and is therefore useful in bringingout pale or "under-exposed" prints;
but the prints must then be afterward washed again in pure water. Hydro-
chloric acid applied b^me washing, or to imperfectly washed prints, will make
the lines show blue.

Art. 19. (a) To erase a (white) line on a blue-print, go over the line with
the sensitizing solution applied with a clean brush or quiil pen. This should be
done in a weak light. Then expose the entire print and re-wash.

(b) Wblte lines are adfded to blue prints, usually in Chinese white;
but the blue color may be removed, showing the white paper beneath, by apply-
ing a saturated solution of concentrated lye (caustic soda or potash^ of of car-
bonate of soda* or carbonate of potash, with a fine clean pen nearly dry. If
laid on too freely, it spreads rapidly. Even if the pen is perfectly clean, the sur-
face thus produced has a yellowish cast as compared with the white of the
paper. The carbonate solutions act more slowly than the lye, but not lea
surely, and they are not iniurious to the skin, whereas the lye burns badly.
The ordinary lime-water sold by druggists makes little or no impression upon
the blue color. If red, instead of wblte, lines are desired, mix with the
soda or potash solution ordinary carmine writing-ink, in such quantity (to be
ascertained by trial) as will give the desired color.

Art. 20. (a) Blue prints which are to be subjected to much handling shoukl
be mounted upon cloth, or the prints may be made, in the first place, iip<Mi
sensitized tracing linen.

Art. SI. (a) Processes grivini; a wbite arronnd. with either blue
or black lines, are usually so complicated as to be oeyond the reach of most
engineers. Their results, also, are generally uncertain, even when applied by.
experts ; the background often lacking in whiteness.

(b) Tandylie paper (Eugene Bietzgen Co., Chicago) and Madnro
paper give excellent aark brown lines on good, smooth, hard paper. The
*' Nigrosine" and other so-called black-line prints, furnished by dealers, usually
give perishable purple lines on a gray and somewhat glossy ground, and on brittle,
unserviceable paper.

(c) Francis I^eCl^re, 21 North 13th Street, Philadelphia, furnishes excel-
lent black-line prints to order at 10 cents per square foot. The lines are
perfectly black and permanent, and the prints are made on good drawing
paper, the color and durability of which are not affected by the process. He
also furnishes fine blue-line prints, on similar paper, at 5ceuts per square foot.

*Either carbonate ('* wasbing-soda") or bicarbonate ("baking-soda") wUl
answer.

^

PRICE LIST AND BUSINESS DIRECTORY. 983

FBIOE LIST AND BUSINESS DISEGTOBT.

For a work of this kind, any attempt to present a list of exact or even of
closely approximate prices would be useless. We aim merely to give indica-
tions of the average costs or of the ranges of cost. For actual quotations,
apply to those named in the business directory, following the list of prices.
See the numbers given in the line or lines immediately following each title
in the price list, and referring to said names.

In selecting names for the business directory the aim has been merely to
furnish a useful (thou^ by^ no means exhaustive) list of representative
names. No other consideration has been entertained.

AbbreTlated Outline of Classification.

For principle of classification, see Bibliography, p. 1008.

1.0 Materials and Elementary Shapes.

1.1 Chemicals^ etc. 1.13, Preservatives; Paints, Impregnating, etc. 1.14,

Explosives.

1.2 Wood, Lumber, Poles, Posts, and Piles.

1.3 Stone, Concrete, Asphalt, etc. 1.34, Cement. 1.35, Brick, Tile,

Glass, etc.

1.4 Iron and Steel. 1.45, Nails, Rivets, Screws, Bolts, etc., Chains. 1.46,

Tubes. 1.47, Wire, etc.

1.5 Other Metals and Alloys.

1.6 Paper. 1.7, Ropes, etc. 1.8, Packings, Gaskets, Belting, Lag^ng,

etc.

2.0 Constructions. *

2.1 Earthwork. 2.12, Dredging. 2.13, Foundations.

2.2 Masonry. 2.21, Brick. 2.22, Stone. 2.23. Concrete.

2.3 Metal Structures. 2.31, Bridges. 2.32, Turntables. 2.33, Tanks,

Stacks, etc. 2.34, Boilers. 2.35, Fireproofing, Concrete Metal Con-
struction.

2.4 Paving. 2.5, Sewers. 2.6, Chimneys. 2.7, Wharves, Docks, Har-

bor Improvement,

3.0 Machinery.

3.1 Electrical Machinery.

3.2 Tools. 3.22, Machine Tools.

3.3 Engines, Locomotives, Cars. 3.35, Water Enj^nes and Motors, Tur-

bines. 3.36, Cars. 3.37, Wagons.

8.4 Blowing and Pumping Machinery. 3.44, Wind Mills. 3.45, Hy-

dramic'Rams. 3.46, Pumps.

3.5 Hoisting and Conveying Machinerv. 3.51, Power Transmission.

8.6 Excavators, Dredges, Machinery for Road aftd General Construction.

3.65, Diving Apparatus. 3.66, Pile Drivers. 3.67, Wells and Weil
Driving Machinery. 3.68, Road Making Machinery.

3.7 Heating. Ventilating, and Refrigerating.

4.0 Engineering, Surveying, and Scientific Instruments and Supplies. 4.1,

Testing Machines. 4.2, Surveying Instruments. 4.3, Computing
Instruments. 4.4, Drawing Insts and Materials. 4.5, Heliography.
4.8, Testing Laboratories.

0.0 Miscellaneous Supplies (Arranged according to class of work).

9.1 Railroad Supplies.

9.2 Hydraulic Supplies. 9.22, Filters. 9.24, Water Meters. 9.25, Pipe

and Hose. 9-26, Hydrants and Valves.

984 PRICE LIST.

, PRICE LIST.

1.0 . Materials and Elementary Shapes.

1.1 Chemicals, etc.

1.13 Preservatives.

1.13^ Coatings, Paints.

36, 76, 199, 274, 280. 327. 361, 433, 502. 564. 586. 635.
Paints, in .oil', $1 to $1.50 per gal.

In cts per lb :

Lead: White, foreign, 8 to 10; American, 7. Red, foreign, 8; Ameri-
can, 6.

Zinc: American, 5; Paris, 9 to 10; Antwerp, 7 to 8.

Lampblack, 12 to 14.

Blue, Chinese, 40; Prussian. 35; ultramarine, 15; brown, Vandyke,
10 to 13. Green,, chrome, 10 to 12. Sienna, burnt and raw, 10
to 13. Umber, burnt and raw, 10 to 12.

Metal coatings, $1.50 to $2.50 per gal.

Preservatives, fillers, oils, etc., 25 to 50 cts per eal.

Graphite pipe-joint compound, 13 to 20 cts perlb.

Linseed oil^ 60 to 70 cts per gaJ. Turpentine, 40 cts per gal.

Plain varnish, 30 cts per gal.

Carbolineum avenarius. 80 cts per gal. Woodiline or spirittine, 25
cts per gal. Creosote oil, 1 to li cts per lb.

1.133 Creosotlng, Impregnating, ete.

48, 63. 133, 149.5. 229, 319. 373.5. 442, 453, 578, 612.5. 634. 663.

Creo-resinate and creosote process, 13 to 19 Cts per cu ft.

Creosoting, 20 to 60 cts per cii ft of material .treated, depending chiefly on

d^^ee of saturation, and exclusive of cost of timber; «* $16 to $50 per

1000 ft B M.
Kyanizing (mercury bichloride process), 8 to 9^ cts per cu ft.
Barschall or Hasselmann process, 8 cts per cu ft.
Wellnouse (zinc- tannin process).. 12 to 19 cts pef tie.
Bumettizing (zinc chloride process), 8 to 18 cts per tie.
The treatment of ties is usually cheaper per cu ft than that of larger liunber.

1.14 Explosives.

214.311.346,452,492,517.

Gunpowder, 16 cts per lb.

Smokeless powder, 60 cts per lb.

Rackarock, 18 to 25 cts per lb.

Dynamite, 13 to 21 cts per lb for different grades, varying between 20%

and 75% nitroglycerine.
Percussion caps, 30 to 60 cts per M.
Blasting machinery, see 3.1231.
Drills, see 3.23.

1.2 Wood, Lumber, Timber.

16.5, 268, 271, 330, 599, 636.
' Lumber, in dollars per 1000 ft board measure (B M) :
Yellow pine, short leaf. 12 to 13; flooring, 20 to 35; long leaf, 19 to 20;

flooring. 22 to 25.
Walnut. 1 10 to 130. Poplar. 25 to 40. Ash. 60 to 65.
Oak, culls, 20; common, 28; plain sawed, 40; boards, 60 to 70; lO-inoh

and wider, 100 to 125; plank, 40.
Hemlock joists and boards, 15 to 20.
Spruce, 30 to 40.

Shingles, cypress, per 1000, 8 to 11.
Studding, joists, rafters, etc., hemlock, 15 to 18.
Clearing and grubbing, see 2.11.
Wood pipe, see 9.254. Ties, 9.14. Piles, 1.23.

1«23 Poles, Posts, Piles.

408, 494, 599.

Piles, 15 to 25 cts per linear ft of pile.

Piling, round or sheet, 30 to 50 cts per linear ft of pile.

i

MATERIALS. 985

Stone, Concrete, Asphalt, etc.

Earthwork, dredging, foundations, see 2.1.

1.32 Stone.

92, 368, 388, 303, 620.

Sand and gravel (within 100 miles of seashore), $1 to $2 per cu yd.

Broken stone, 75 cts to (2 per cu yd.

Rip-rap, $1 to $3 per cu yd.

Trap rock, 70 cts per ton of 2000 lbs.

Ordinary building stones, $1 to $5 per cu yd.

Granite, S15 to $45 per cu yd

Slate roofing, 12 to 25 cts per sq ft.

1.33 Asphalt.

17, 51, 61, 27?, 437, 444, 449, 559, 649.
Paving, see 2.4.

1.34 Cement.

22, 23, 30. 64, 86, 97, 101, 106, 149, 170, 177, 179, 227, 252, 258, 310, 326,
327, 336. 352. 358, 360, 366, 418, 445, 456, 569. 588. 616. 626, 642, 657.

Portland (artincial) cements, per bbl'of about 400 lbs gross: German,
$2.25 to $3.00; American. $1.10 to $1.60.

Rosendale (natural) cements, per bbl of about 300 lbs net: From Rosen-
dale Township and vicinity, Ulster Co., N. Y., 95 cts to $1.10; other
Rosendales, 75 to 85 cts.

About $1 to $2 worth of cement mortar required per cu yd of masonry m
buildings, $1.50 to $3 per 1000 bricks.

Lime, 60 to 90 cts per bbl of about 250 lbs.

About 60 cts worth required per cu yd of masonry in buildings. $1 to $1 .50
per 1000 bricks.

Plaster, $1.50 to $2 per bbl of varying weight.

Concrete construction, see 2.35.

1.35 Brick, Tile, Glass, etc*

Sewer pipe, see 9.255.

1.351 Brick.

217, 232, 291. 299, 383, 448, 468, 505, 615.

Paving, see 2.4.

BuUding bricks, per 1000: Salmon, $5 to $7; hard, $7 to $9; stretcheret

$9 to $14; pressed, $17 to $20; colored. $20 to $30; hx>n spots, $30;

Pompeiian, $35.
Fbe-biick. $20 to $24 per M.
Vitrified paving brick. $15 to $25 per M.
Sewer pipe, see 9.255. Paving, 2.4.

1.359 Tiltns.

Floors and walls, 35 and 40 cts per sq ft and upward.

Tile, 380, 0.13 to 0.8 ct per cu in of materiaL

Roofing tUe, $7 to $30 or more per square. 1 square — 100 sq ft.

1.353 Glass.

557.'

American window. $ per box of about 50 sq ft. IMscount. 80% to 85%.
••United inches" 25 60 80 100

Single AA 32 38 49

Double AA 43 66 68 88

Smgle A 27 32 45

Double A , 38 60 62 80

Single B 26 30 39

Double B 36 46 66 75

Or, say, single thick, A to i ct per united in: double thick, i to i ct per
united in: where the number of united ins equals the sum of the two
dimensions. Ihus, a sheet of glass 24 X 36 contains 60 united ins.

986 PRIOE LIST.

1.4 Iron and Steel.

^.lPs.}^JSh}^AJP* 1®^' 1^^' 233. 242. 283. 301, 310. 329, 526. 63Sk
OOO. 612, 640. 662, 670.

Scrap iron and steel. 112 to $19 per ton of 2240 Ibe.

1.41 Cast Iron and SteeL

71, 145, 161, 186, 189, 192, 218, 249. 280. 281. 329, 372, 377, 463.2, 612,

624, 652.
Cast-iron pipe, see 9.251.
Piff iron, per ton of 2240 lbs: Foundry, $13 to $15; Bessemer, $16; gny

zorge, $14; Lake Superior charcoal. $17.

1.42 Forged Iron and SteeL

16, 29, 69, 79, 128, 132, 137, 147, 163, 176, 219, 255, 269, 273, 312, 320;
341, 343, 463.2. 463.6, 550, 553, 580, 597. 622, 662, 665, 668. 677.

1.43 Boiled and- Structural Iron and SteeL

16. 28, 29, 34, 41, 45, 73, 84, 137, 176, 248. 263. 324, 329. 340, 377, 43i

435, 466, 547, i^60, 571, 606.
Iron and steel, ots per lb :

Refined iron bars. and steel bars, ordinary siies.

Angles, ordinary sizes. T 8hai)e8.

Beams and channels, structiiral shapes.

Tank plates, structural plates.

Bessemer machinery steel.
Steel rails, $28 per ton of 2240 lbs. Old, $15.

Iw431 Sheet and Plate Iron and SteeL

38, 41, 44, 73, 84, 189, 263. 324, 329, 382, 395, 435, 463.2, 471.5, 673, 675^
681.,

Galvanized iron sheets:
Discount, 60% to 80%.

Gage, 14 to 17 22 25 28 29 30

Cts per lb, 12 to 13 14 16 17 19 21

Extra, for additional widths, 36 to 48 in, 1 to 4 cts per lb.
Black iron, gage 16, 3 cts per lb ; gage 28, about 4 cts.

1.44 Bar SteeL

45, 84, 98, 189, 283, 301, 324, 329, 331, 342, 429, 435, 466. 518, 560. 612,
662.

1.45 Fastenings.
28,309.

1.451 Nails and Spikes.

40, 44, 45, 170.6, 202, 283, 329, 356, 434, 518, 621, 640.
Nails, etc.; cts per lb: Cut, 2 to 2i; wire, 2^.
Spikes, railway. If to 2 cts per lb.

1.452 Blvets.

28. 40, 81, 123, 154, 170.6, 262. 305, 310. 329, 356. 491. 529, 621.

1.453 Screws.

40, 81. 518, 529.

1.454 Bolts and Nuts.

28, 34, 40, 152, 237, 305, 309, 329, 356, 431, 518, 519, 529.

Bolts and nuts for machines, price per 100, square or button heads. Lengih
under head, 2 ins. Discount, 70% to 75%. See list, p. 884.

Diameter, ins i ^ f 1

Price per hundred $1 .78 $3.86 $7.70 $16.00

Extra per in over 2 ins.. 0.16 0.52 1.00 1.80

1.455 Tumbuckles.

157, 409, 518.

Open, price each. Discount, 67%.

Rod. ins ^12

With ends $0.80 $1.60 $5.35

Without ends 0.60 1.10 8.10

With upset ends, 30% extra.

ELEMENTABY SHAPES. MATERIALS. 987

1.406 Washers.

163. 306, 356, 629.

1.457 Chains.

28, 96. 100, 160, 262, 329, 398, 462, 685.

American ooil chain :

Inch A i i itoU

Ctsperlb 8 6 4 3.6

1.46 Tubes.

10, 321, 407. 424. 434, 556, 662. See also 9.26, etc.

1.47 Wlre» Wire Bope, and Fencing.

44, 332, 342, 366, 369, 384, 407. 630.6, 621, 623. 646.
Hoisting and conveying machinery, aee 3.6, etc.

1.471 Wire.

44, 342, 407, 621, 623.

Cts per lb :

Iron. Tinned*. Cast Stbeu

Nos. 0 to 9 : 2.6 3.7 8.0

No. 18 4.0 4.6 13.0

1.472 Wire Fencing.

332, 356.

For the simpler patterns, 30 cts to $1 per rod (16i ft), aoeording to style
and finish.

1.473 Wire Bope.

36, 44, 369, 384, 646. See pages 976, 977. Discount, 30% and 7i%;
galvanised, 26% and 7i%.

1*5 Other Metals and Alloys.

15. 46, 77, 407, 459, 471.6, 483, 609.

Boll and sheet brass, random lengths. Discount. 20% to 30%.

Width in ins 2 to 18 18 to 24 24 to 32 32 to 40

Ctsperlb.. 22 to 30 29 to 39 36 to 49 60 to 75

Extra quality. 4 to 7 cts per lb extra. Bronze metal, 7 cts per lb extra.

Soft sheet copper, 20 to 23 cts per lb net for 16 oks to the sq ft in thickness,
base sizes.

Lead, pig, 4.3 to 4.4 cts per lb ; sheet, 6 cts. Sheet .sine, 6 to 7 cts per lb.
Spelter, 3.76 to 4 cts per lb. Antimony, 9 to 11 cts per lb. Nickel, 55
to 60 cts per lb. Mercurv, about $1.60 per lb in large lots.

Tin plates, per box of 112 lbs:

American charcoal plates, $6 to $8.

American coke plates, Bessemer. $6 to $7.

American teme plates, per box of 224 lbs, $10 to $12.

1.6 Paper.

Tar, 2 cts per lb, 60 to 70 cts per roll of 108 sq ft.
Tarred felt. 2i cts per lb. Straw paper. If cts per lb.
Rosin sised sheathmg, 30 to 60 cts per roll of 600 sq ft.

1.7 Bopes, etc.

36, 246.

Rope, cts per lb: Manila, plain or tarred, 10 to 12; Sisal, 7 to 9; hay, 7 to

8; cotton, 9 to 14: jute, 6 to 7.
Twine, 7 to 10. Oakum, best, 6 to 7.

1.8 Packing, Gasket, Belting, Lagging, etc.

26, 90. 266. 326, 334, 459, 632.

Belting, rubber, cts per in of width, per ft of length: 2-ply, 7.5; 5-ply,
13.5; 8-ply, 21.
Leather:

Width, ins ... 1 6 12 24 72

Per ft $0.12 $0.92 $1.88 $4.20 $15.00

Pipe jointing supplies, see 3.26.

988 PRICE LIST.

2.0 Constructions.

Water-works supplies, see 0.2. Railroad supplies, 9.1.

2.1 Earthwork, Dredslng, Foundations.

2*11 Excavation and Embankment.

See also pp. 800, etc.

Clearing and grubbing, $50 to $150 per acre.

Earth excavation, 20 cts to $1 per cu yd. In trenches, $1 to $5, according

to depth.
Pipe trenching, see also pp. 658. 650.

Embankment, 50 cts to $1.50 per cu yd. Rolled, $1 to $2.
Cut and fill, 10 to 25 cts per cu yd.
Sodding, 20 to 40 cts per sq yd.

Rock excavation, $1 to $6 per cu yd, depending on hardness of rock, etc
Rock filling, $1 to $4 i)er cu yd.

2.12 Dredging.

10 cts to $1 per cu yd» according to length of haul, etc.
See also pp. 631, 632.

2.13 Foundations.

402, 419. 539, 600.
Piles, see 1.23.

2.2 Masonry.

2.21 Brick Masonry.

$10 to $20 per cu yd. $5 to $10 per 1000 + cost of bricks. '

Bricks, see 1.351.

2.22 Stone Masonry.

Dollars per cu yd: Rubble, dry, 2 to 6; in cement, 3 to 6.
Ashlar, 7 to 30. Granite coping, 30 to 45.
Plain cellar work, 3 to 4.
Stone, see 1.32.

2.23 Concrete and Cement Masonry.

In place, $4 to $10 per cu, yd.

Cement, see 1.34. Floors and sidewalks, 2.4.

2.24 Plastering.

Three-coat work, 25 to 50 cts per sq yd.

2.3 Metal Structures.

146. 215, 600, 603.

2.31 Bridges.

28, 70, 75, 91, 143, 146, 160, 161, 167, 190, 215, 216, 218, 328, 339, 344, 378.

391, 394, 402, 419, 426, 439, 465, 466, 471, 472, 474, 482, 626, 639,
544, 598, 600, 603, 638, 639, 642.5.
Riveters, see 3.27. Foundations, 2.13.

2.32 Turntables.

28, 91, 167, 190, 344, 394. 419, 466, 474, 482. 617, 639, 672.

2.33 Tanks, Stacks, etc.

64, 126. 146, 228, 250, 266, 302, 335, 463.4, 625, 542, 664, 616, 617, 633.

648.
Pipes, see 9.26« Wind mills, 3.44.
Stand-pipe, 22 X 60 ft, including foundations, $4000 to $7000.

2.34 Boilers.

21. 56, 180, 257, 266, 288, 294, 302, 336, 464, 467. 525, 646, 679, 617, 638.
Boilers. Upright tubular. Price with base and fixtures :

HP 4 12 30 50

Dollars 150 226 876 660

Riveters, see 3.27. Engines, 3.3.

CONSTBUCTION8. MACHINERY. 989

2.35 Fire-proofliig» Cpnerete-metal Construction.

60, 142, 223, 406, 410, 411, 630, 683.

Concrete-metal construction : Concrete and wire, flooring, 16 to 26 cts pei
sq ft, exclusive of floor beams. For covering oolimms and girders, 10
cts per sq ft; if concreted, 16 cts. Walls, 70 cts to $1.40 per sq yd.
Wall furring, 40 to 60 cts per sq yd. Wire lathing, 12 to 20 cts per sq yd.

2.36 Wharves, Docks, Harbor Improvement.

212, 330, 402, 633.

Excavators, etc., see 3.6. Cement, 1.34.

Dredging, 2.12.

2.4 Paving.

411,683.

Per sq yd. Macadam, 66 cts to $1 ; brick, $1 to $2 ; Belgian block, $2 to

$3 ; asphalt, $2 to $3.
Cellar floors, $1.26 to $2; sidewalks, $3 to $6.60. Cement, see 1.34.

2*5 Sewers.

From 6 to 18 ins diameter, dollars per ft run :
Depth,

ft ... 6 10 13 16 17 20

0.30 to 0.90 0.66 to 1.00 0.90 to 1.20 1.30 to 1.40 1.60 2.00

Laying sewer pipe, cts per ft, exclusive of excavations: 16 in, 30 to 60;

4 in, 20 to 30.
Brickwork in sewers, $8 to $12 per cu yd.

2.6 Chimneys.

181. .

2.7 Booflng.

Slate, 7 cts and upward per sq ft. Slag, 4 cts. Tin, 6 to 8 cts. Shingle,

10 cts.
Skylights, complete and erected, 60 cts per sq ft and upward.

3.0 Machinery*

Testing machines, see 4.1. Surveying instruments, 4.0. Miscella-
neous, 9.0.

3.1 Electrical Machinery.

253,267,464,691,660.

Power transmission, mechanical, see 3.61. Blasting apparatus. See
3.231.

3.2 Tools, Machine Tools.

Road-making machinery, see 3.68. Pile drivers, 3.66

3.21 Small and Wood Tools.

63,240,498,5^9.
Shovels, see 3.62.

3.22 Machine Tools.

103.6, 376, 451, 498. 662.6, 674.

3.23 DrUls.

32, 121, 148, 169. 194. 306, 316, 346, 400, 478, 481, 610, 631, 596, 606.

Air compressors, see 3.43. Explosives, 1.14.

Rock drills, percussion :

Diam cylinder, ins li 3\ 4^

Length stroke, ins 3f 6f 7i

Depth hole, ft 1^ 10 to 16 20 to 30

Bottom diam hole, ins 1 If 2i

Boiler H P required 3 10 16

Price, complete $160 $300 $460

Rock drills, diamond (rotary). Discount. 10%.

Depth hole, ft 4000 1600 1000 600 400

Diam hole, iris 2^% 2^9 2^ lA 1ft

Diam core, ins 2 1$ li 1 1

Boiler H P required 26 16 12 to 16 10 hand

Card price, dollars 4000 2600 1900 1400 426

Pumo. extra, dollars 3400 2800 1900

990 PRICE UBT.

8.831 ' Blastliic Machinery*

346, 492.

Bhwiting nuMshizi6 :

Tofire20to aOholes : $25

To fire 76 to 100 holes 76

Connecting wire, 40 ots per lb.
Leading wire, li cts per ft.

8.24 Presses.

172, 242.5, 645.

3.95 Punches, Shears.

74, 375.

8.36 Pipe-cuttliiff» Tapping, and Jointing Machinery.

8.361 Cutters.

404.541.

8.963 Tappers.

224. 466.5, 563.

For dry pipe, S20 to $30 each. For pipes under pressure, $100 to $200
each.

8.363 Jointers.

643
2-in. $2 each; 12-in. $9; 36-in, $16; 72-in, $44.

8.37 Rivet ers.
12. 16, 292, 478, 674.

8.8 Engines, Locomotives, Cars.

Pumps, see 3.4. Road rollers, 3.681. Boilers, 2.34.

Portable engines:

H P Ctlindbr. Boiler Dxam. Price.

10 7 X 8 ins 29 ins $600

25 9X10 " 36 " 860

50 13X15 " 44 " 1300

8.81 Stationary and Hoisting Engines.

33, 125, 136, 171.4, 200. 204. 231. 234, 241, 251, 257, 259, 333, 364, 370,

397, 427, 512. 536. 545, 572. 602.
Hoisting en^nes:

Sin^e cylinder, friction drum, but no foot brake, 4 to 6 H P, $250 to $400.

Double cylinder:

12 HP 20HP 60HP

Friction drum and foot brake, or re-
vereible $600 $1100 $1300

Same, with boiler 1000 2000 3000

2-winoh engine 600 ' 100 1300

6-winch engine 900 1400

Single cylinder stationary engines: i

HP 12 30 50 100 200 300

From $300 $450 $650 $1100 $2000 $3100

To 450 600 800 1400 2000 3400

With base^ about 10% extra.
Portable engines on wheels, complete, 6 to 15 H P, $700 to $f500.

3.33 Liocomotives.

102, 103. 124, 171.6. 199. 371. 387, 486, 493. 520. 524. 543.

3.33 Gas and Gasoline Engines.

47, 57, 172.2, 184, 247, 415, 440, 463.4. 581. 584. 593. 595, 601, 661. 674.

3.34 Oil Engines.
500.

3.35 Water Engines and Motors, Turbines.

18, 57. 365,469. 514, 522, 665, 590, 610, 633, 674.
Water-works supplies, see 9.2. Pumps. 3.4.

8.36 Cars.

29, 53.5, 102. 108, 278, 351. 405, 486, 499, 509, 571, 672.

MACHINERY. ENGINES. 991

8.37 Waffons.

160,644.

8*4 Blowing and PnmpliiK Maehinery.

3.41 Blowers, Forges.

572, 604.

3.49 Mechanical Draft.

604.

3.43 Air Compressors.

47, 156, 170.2, 315. 346, 353. 400. 443. 457, 510. 552, 567, 504, 605.

3.44 Wind Mills.
250, 633.

3«45 Hydraulic Bams*

207, 265, 497.

Ihive pipe:

Diam, ins 1 2 4 6

Dollars, net 5 to 7 10 to 12 35 to 40 60 to 70

Certain manufaeturers make much heavier and more elaborate machines
at several times these prices.

8«46 Pumps.

20, 58, 59, 62, 82. 83, 129, 182, 183, 186, 187, 188, 200, 208, 221. 222, 231«
261, 279. 284, 296, 297.5, 303, 319.5, 320.5. 345, 348, 353, 359. 396. 423,
503. 504, 507, 508, 522, 532. 538, 551, 555, 567, 568, 572, 594, 674, 676.

Prices in dollars each :

Capacity in fi^ls

per min 5 20 100 500 2500 10,000

Sinfl^e cylinder:

Boiler feed .. 50 200 400

Tank and low

lift 150 350 600

Duplex:

High pressure 250 750

Low pressure 150 500

Centrifugal . 50 to 75 80 to 175 200 to 450 800 to 1300

8.5 Hoisting and Conveying Machinery.

234, 257, 344, 370, 373, 622.

Electrical, see 3.1. Excavating machinery, 3.6. Hoisting engines,
3.31. Cars, 3.36. Wire rope, 1.473.

8«51 Power Transmission.

173, 205. 373.

3.52 Cranes, Derricks.

33, 67, 104, 136. 140. 148, 171.4, 204, 293. 314, 333, 344. 674.

8.53 Pulley Blocks and Trucks.
89, 333.

8JS4 Elevators, Hoists.

65. 88, 148. 171.4, 322, 370. 373, 390, 441.

Hoistmg crabs on winches, i to 2^ tons capacity, $35 to $100.

Differential hoists, i to 3 tons capacity, $10 to $15 to $40 each.

3.55 Jacks.

148,211.455,645.

8JS6 Tramways, Conveyors.

65, 88. 104. 127, 171, 251, 307. 822. 333, 344, 351, 354, 369. 370, 373. 405,

441, 527.
Traek, see 9.1. Cars, 3.36. Wire rope, 1.473.

8.6 Excavators, Dredges, Boad and General Construction

Machinery, etc.
Hoisting and conveying, see 3.5. Wagons. 3.37. HoistinK engines,
3.31. Drills. 3.23. Explosives, 1.14. Excavation and embankment,
2.11.

992 PRICE LIST.

3*61 Excavators.

67, 107, 333, 389, 405, 613, 618. 678.

3.62 Trenching Maehinery.

67, 139. 333, 389, 421, 496.

3.63 Scrapers, Plows, etc.

127, 160, 333, 658, 607, 658.

Wheeled scrapers. Discount, 20%. $40 to $60 each.

Drag scrapers. Discount, 50% to 60%.

Ordinary, $10 to $15 each. Fresno or back, $35 to $40 each

Plows. Discount 20%.

Horses 2 4 6 8 10 12

Each, dollars, 15 to 30 30 35 40 45 55

Hardpan plow, $70.

3.64 Dredging Machinery.

52, 67, 94, 107, 222, 241, 293, 389, 447, 463, 539, 566.
Pumps, see 3.4.

3.65 Diving and Diving Apparatus.

236, 425, 546.

Air pumps. Discount 10%. Each, $125 to $400, according to depth

Heliuets, $100. Rubber suits, 40. Weights, underwear, line, tubinc »

pair materials, etc., per outfit. $110 to $146.
Complete outfit: Deep sea, $700 to $750; moderate depths. $560 to $600:

shallow water, $350 to $400. ^^ ^^

8.66 Pile-driving Machinery.

136, 314, 333, 641.

3.67 Wells and Well-driving Machinery.

47, 171.8, 282, 480, 484, 666.
Drills, see 3.23.

3.68 Boad-making Machinery, etc.

13,39,160,545.

Rock drills, etc., see 3.23.

3.681 Boad Boilers.

110, 230. 290, 295, 320, 333, 337, 545.

3.682 Concrete Mixers.

164, 171.2, 209, 230, 245, 260, 320, 333, 511, 545.
Each, $225, $500 and up\»rard to about $12£0.

3.683 Bock and Ore Crushers.

55, 105, 155, 160, 194, 238, 264, 275, 333, 346, 390, 513, 658, 664, 680.

Receiving Capacity, H P Price

Capacity, Tons per Required. Dollars
Inches. Hour.

UX3 Hand 30

7 X 10 4 to 6 8 to 10 500

10 X 20 12 to 16 18 to 20 1000

13 X 60 40 to 60 40 to 60 4000

24 X 72 160 to 200 150 to 180 13000

3.7 Heating, Ventilating, and Befrigerating.

26, 318, 604.

4.0 Engineering, Surveying, and Scientific Instruments and

Supplies.

14, 68, 72. 99, 120, 196, 201, 239, 243, 285, 297, 804, 340, 347, 360, 364. 367.
386, 473, 501, 506, 523, 528, 569.5, 582, 625, 667, 679. ''•"*• *'"'»
Prices, see below.

4.1 Testing Machines.

235, 460, 475, 521.

4.2 Surveying Instruments.

Dealers, see 4.0. Prices, see below.

INSTRUMENTS. SUPPLIES. 993

4.!31 Transits, Plane Tables, Compasses, etc.

Transits, plain, $150 to $200. Engineers', complete. $200 to $300. Min-
ing, $200 to $300 and $700. Mountain, $150 to $300.

Theodolites and portable alt-azimuth astronomical instruments, $500 to
$1000.

Solar attachments, $50 to $70. Sextants, $60 to $150. Pocket sextants,
$50.

Plane tables, complete, $150 to $300. Compasses, pocket, $10 to $25.

4k.22 Lefvels.

Engineers' levels, $100 to $200. Hand levels, $5 to $18 ; usually about $0.

4.23 Bods, Tapes, etc.

381, 450.

Leve^ng rods, $13 to $16 each. Range poles, $2 to $5.
Tapes, from 5 to 10 cts per ft, depending upon graduation and length.
Chains, 8 to 12 cts i>er ft.

4.29 Miscellaneous.

Current meters, $65 to $100. Direction meters, $200 to $250. Velocity

register and timepiece, $50 to $60.
Hook gages, $15 to $60.

4.3 Computing Instruments.

317.

Planimeters, $25 to $125.

idlide rules, plain Mannheim, $1 to $5 ; other forms, $1 to $50.

Computing machines, $100 to $300.

4.4 Drawing: Instruments and Materials.

24, 201, 243, 340, 386, 506, 569.5, 667.

Drawing instruments, $10 to $30 per set, very elaborate sets as high as $60

and even $100. Drawing pens, $1 to $3 each. Compasses (drawing),

$3 to $9. Dividers, $2 to $4.50.
Triangular boxwood scales, ordinary, 12 ins long, $1 to $3.
Metal straight edges, 36 ins long, $4 to $5; shorter, as low as $1.50.
Tnsquares, 36 ins, with celluloid edges, about $2.25; wood, $1 to $2.40;

steel, $7 to $10, usually with adjustable angle.
Triangles, celluloid, 6 ins, 50 cts; 12 ins, $1.10. French curves, celluloid,

50 cts to $1.50.
Protractors, German silver, 4 ins, 50 cts to $1.60; 6 ins, about $3; with

arms, 6 ins, $9 ; 8 ins, $20.
Drawing inks, 25 cts per bottle.

4.5 Heliography.
363, 438, 569.5, 573.

Black-line prints, Le Clare's, 10 cts per sq ft. Nigrosine, 8 cts.

Blue prints, 3 cts per sq ft.

Blue*print paper, 30 ins wide, 10 to 20 cts per yd.

Blue-print frames, hand, from $2 for 10 X 12 ins to $40 for 36 X 60 ins.

4.6 Tents, etc.
428, 637.

Wall tents. Discount, 55%.

Size, Height,

Ft. Ft.

9X9 7i

12 X 12 8

18 X 24 11

24X28 13 105 206)^,^

30X70 15 266 616 J "^P®**'

U S folding canvas cot, $1.50 to $2.

4.7 Drawing: Tables.

19, 226, 357.

4.8 Testing Laboratories.

308, 399, 488.

0.0 Miscellaneous Supplies.

0.1 Railroad Supplies.

108, 168, 202, 300, 361, 405, 479. 508.5, 655, 664.5, 672.
Bridges, see 2.31. Earthwork, 2.1. Locomotives, 3.32. Cars, 3.36.
Tramways, 3.56.

63

8-oz.

16-oz.

Duck.

Duck.

$11

$26

16

36

.44

96

106

206)

265

616;

994 PRICE LIST.

9.11 Bails.

45, 329, 434, 472, 495.

Steel rails, (28 per ton. Old, $15.

0.12 Joints.

87, 128» 170.8, 329, 461, 653.
1.5 to 2.5 cts per lb.

9>13 Si¥itches» Frogs, and Slgrnals.

60, 158, 225, 286, 472, 509, 587, 611, 628, 655, 664.5. 672.

9.14 Ties.

66, 494, 599, 636.

White oak, 50 to 70 cts each; chestnut, 35 to 45; jrellow pine, 45 to 65.

Tie plates, 5 to 15 cts each.

9.15 Spikes.

202, 283, 329, 356, 518, 640. li to 2 cts per lb.

9.3 Hydraulic Supplies.

Gaskets, see 1.8. Stand pipes, tanks, 2.33. Pipe-cutting, tapping, sod
jointing machines, 3.26.

9.21 Water-softening Plants.
313, 654.

9.22 Filters.

193. 288, 376. 385, 446, 534, 542.

9.23 Water-wheel Governors.

374.

Water-wheels, see 3.35.

9.24 Water Meters.

109, 122, 244, 298, 432, 436, 487, 590, 614, 629, 676.

All types (except piston ; see below) :

Nom'l diam, pipe, ins | or i 1 3 6 12

Dollars each, from 9 18 95 300 1000

"to 12 26 125 500 1500

Reciprocating piston meters, about 60% more,

Venturi meters, see pp. 532, etc. Current meters, 4.29.

9.25 Pipe.

37, 95, 169, 206, 362, 490, 614.
Pipe-cutting machinery, etc., see 3.26.
Pipe laying, see pp. 668, 669.
Lead- and tin-lined iron pipe :

Lead-unkd, 6 Ctb per Lb. Tin-lined, 10 Ots pbr L».

Discount, 26% 10%

Size Price Price

IN Ins per Ft. per Ft.

1 $0.30 $0.65

2 0.65 1.35
4 1.72 3.00
6 3.28 4.26

Block tin pipe, 35 cts per lb.

Lead pipe, 6 cts per lb. ^ ^^^ _, . __„

Boiler tubes. Discount, 40% to 60%. Prices, see p. 882.

Seamless brass tubes, base price, 20 to 25 ctsper lb. Extras, see p. 919.

Spiral riveted pipe, in dollars per linear ft. Discount, 40%.

DiAM. Ins. Black. Asphalted. Galtanxzbd.

Thickness, 0.022 in,
3 0.20 0.28 0.30

6 0.33 0.39 0.50

12 0.68 0.80 1.06

Thickness, 0.049 in.
3 0.34 0.37 0.46

6 0.67 0.63 0.85

12 1.16 1.27 1.65

Thickness, 0.109 in.
6 1.25 1.31 1.90

12 2.60 2.62 3.26

24 4.70 4.94 6.25

HYDRAULIC SUPPLIES. 995

9.251 Cast-iron Pipe.

29, 162, 203, 264, 270, 392, 416, 630. 640, 647, 674.

Cast-iron water pipe, $24 per ton. Pipe and laying, $30 to $35 per ton.
See also pp. 658, 659, 875, 876.

9.252 Steel Pipe.

1 1, 42, 138, 270. 403, 469, 542.
Steel pipe. 4 cts per lb at works.

9.253 Wrought-iron Pipe.

424,434.

Wrought-iron pipe, per ft. Discount, li-in and smaller, 50% ; 2-in and
larger, 60%. Prices, see p. 882.

9.254 Wood Pipe.

Prices in cts per ft :

Inner diam, ins.. 1 3 6 10 16 30 84

Plain, square ... 5 15 40

Plain, round .... 5 20 45
Strengthened for

40 lbs persq in . 12 25 50

160 " " " " .18 30 70
Woodstave pipe,

401bspersqin. ' 50 90 200 350 1200

160 •* " " " . 70 140 270

9.255 Sewer Pipe, etc.

31, 80. 130, 176, 197, 232, 291. 416, 420, 448. 608, 619, 631.
Sewer pipe. Discount, 70 to 80%. Prices, see p. 575.

9.256 Hose.

90.

Water hose, price per in of internal diameter, per ft of length :

2-PLT 4-PLT 6-PLT

$0,333 $0.50 $0.75

Other plies at nearly proportional rates.
Air, hot water, and steam hose, price per in of internal diam, per ft of
length:

4-PLY 6-PI.T 8-PLT

$0.83 $1.24 $1.66

Other plies at nearly proportional rates.

9.26 Hydrants and Valyes.

93, 141, 144, 162, 165, 166, 172.8, 210, 220, 254, 323. 338, 379, 401, 412,
458, 489, 490, 514, 516, 563, 615, 627, 669.

Gate valves, iron body, bronse mounted, bell or spigot ends. Discount,
40%.

Diam. ins 4 6 12 18 24 36 48

Dollars each 20 30 90 200 350 900 2000

Single gate, iron body, low pressure steam and water. Discount, 40%.

Diam, ins 4 6 12 18 24 48

Screw ends $15 $25 $85

Flange ends 20 30 80. $160 $280 $1600

Double gate, iron body, brass mountings, steam or water. Discount, 55%.

Diam, ins 2 4 6 12

Screw or flange $10 $20 $30 $100

Double gate, iron body, composition or bronze mounted, for heavy pres-
sures &bout 500 lbs per sq in) :

Diam, ins 2 4 6 12

From $10 $20 $35 $90 Discount, 75%

To 15 30 60 150 Discount, 60%

Double gate valves, all bronae. Discount, 60%.

Screw ends $1.40 $2.36 $6.25 $34.00 $76.00

Flange ends $3.40 4.16 11.00 43.00 88.00

Fire hydrants. Discount, 30%. 6i-in stand pipe, $38; 6i-in, $47.

Compression fire hydrants. Discount, 20%. Length from pavement to
bottom of connections, 5 ft. 1 to 2i-in noazles, $28; 2 to 2i-in, $33. Add
or deduct $1 for each 6 ins dififerenoe in height from 5 ft. 2i-m nossles,
$2. Frost case, $5.

9.27 Anti-biirstins Devices.
48.j|.

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BUSINESS DIRECTORY. 997

70 Bellefontaine Bridge and Iron Co., Bellefontaine, Ohio.

71 Belle City Malleable Iron Co., Racine, Wis.

72 Berger, C. L. — & Sons, Boston.

73 Berger Mfg. Co., Sheet Flooring, Canton, Ohio.

74 Berlin-Erfurt Machine Works, 66 and 68 Broad St., New York.

75 Berlin Iron Bridfi^e Co., East Berlin, Conn.

76 Berry Bros., Limited, Varnishes, Detroit, Mich.

77 Bethlehem Steel Co.. South Bethlehem, Pa.

79 Billings & Spencer Co., Hartford, Conn.

80 Blackmer & Post Pipe Co., St. Louis, Mo.

81 Blake & Johnson, Waterbury, Conn.

82 Blake^eorge F.*— Co., 9 1 Liberty St ., New York.

83 Bliss, E. W.— Co., Brooklyn, N. Y.

84 Boker, Hermann — A Co., 103 Duane St., New York.

86 Bonneville Portland Cement Co., 1307 Real Estate Trust Bldg., Philm;

87 Boniano Rail Joint Co., 22 S. Fifteenth St., Philadelphia.

88 Borden & Selleck Co., Chicago.

89 Boston and Lockport Block Co., 146 Cqmmercial St., Boston.

90 Boston Belting Co., 256 Devonshire St., Boston.

91 Boston Bridge Works, 70 Kilby St.. Boston.

92 Bound Brook Stone Crushixig Co., Trap Rock. Bound Brook, N. J.

93 Bourbon Copper and Brass Works. 618 and 620 £. Front St., Cin.. O.

94 Bowers, A. B.— , Hydraulic Dredging, First Nat. Bank Bldg., San

Francisco, Cal.

95 Bowes A Co., 23 Lake St., Chicago.

96 Bradlee ft Co., Philadelphia.

97 Bradley Pulveriser Co., Boston.

98 Braeburn Steel Co., Braebum. Pa. *

99 Brandis, F. £.— Sons ft Co., 812 Gates Ave., Brooklyn, N. Y.

100 Bridgeport Chain Co., Bridgeport-, Conn.

101 Brier Hill Iron and Coal Co., Youngstown, Ohio.

102 Brm, J. G.— Co., Philadelphia.

103 Brooks Locomotive Works, Dimkirk, N. Y. See 34.5.
103.5 Brown ft Sharpe Co., Providence, R. I.

104 Brown Hoisting Machinery Co., Incorporated. Cleveland, Ohio.

105 Buchanan, C. G. — , 141 Liberty St., New York.

106 Buckeye Portland Cement Co. Bellefontaine, Ohio.

107 Bucsrrus Co., South Milwaukee, Wis.

108 Buda Foundry and Machine Co., Hand Cars, Crossing Gates, Harvey;

1 09 Buffalo Meter Co., 363 Washington St . , New York.

110 Buffalo Pitts Co., Buffalo, N. Y.

120 Buff ft Buff, Transits and Levels. Boston.

121 Bullock, M. C— Mfg. Co.. Diamond Drills, Chicago.

122 Builders' Iron Foundry, Venturi Meter, Providence, R. I.

123 Burden Iron Co., Troy, N. Y.

124 Bumham, Williams ft Co., Baldwin Locomotive Works, Philadelphia.

125 Byers, John F. — Machine Co., Ravenna, Ohio.

126 Caldwell. W. E.— Co., Louisville. Ky.

127 California Wire Works, 9 Fremont St., San Francisco, Cal.

128 Cambria Steel Co., PhiUdelphia.

129 Cameron Steam Pump Works, New York.

130 Camp, H. B. — Co., Akron, Ohio.

131 Canton Steel Co., Tool Steel, etc.. Canton, Ohio.

132 Cape Ann Tool Co., Pigeon Cove, Mass.

133 Carbolineum Wood P*reserving Co., 13 Park Row, New York.

134 Carbon Steel Co., Pittsburg, Pa.

135 Carborundum Co., Niagara Falls, N. Y.

136 Carlin's. Thomas— Sons Co.. Allegheny, Pa.

137 Carnegie Steel Co., Carnegie Bids., Pittsburg, Pa.

138 Carroll-Porter Boiler and Tank Co., Pittsburg, Pa.

139 Carson Trench Machine Co.. Boston.

140 CMC Mfg. Co., Columbus, Ohio.

141 Casnita Wheel and Foundry Co., Sayre, Pa.

142 Central Fire-proofing Co., Fire-proofing System, 874 Broadway, New

York.

143 Champion Bridge Co., Wilmington, Ohio.

144 Chapman Valve Mfg. Co., Indian Orchard, Mass.

998 BUSINESS DtBEXrrOST.

146 Chester Steel Cutings Co.. 407 Library St., PhiladelphU.
14ft Chiengo Bridge and Iron nn. Dhirumi
-- -■ [oDropForgi

Jc *■"*
ChicBgoPoTtland Cement. Co., 61:) Stock Eichante B!dg., Chici
5 Chicago Tie Freaerviug Co., Cbicago.
ChUlcott-Evwia Chain Co., AUeBhenv, Pa.
Chrome Steel Worke. Crucible Steel C^tincs, Brooklyn, N. Y.
Church, Isaac— , Eipanoion Bolta, Toledo, Ohio,
fe ^ .„.. "o.. Auburn, N. Y.

MCortUndtSt., NewYor;
fe.,CIeve- ■ "■

— keand Kirkiana i

uimai Road Uuhine Co_^ Marathon, N. Y.

Clinion Bridge and Iron Works, CTinton, Iowa.

Clow, JameaB. — & Sons, Hpe and Specials, Lake and Franklin Sts.,

Chicago.
Cobb iDrew.Washeifl, etc., Plymouth, Mass., and Book F»ll9. 111.

169 CleveUi

!., Clevelnnd, Ohio.

N.J.
kiffin Val-

., 240 EleveDth St., Jersey cit;,

(Co., 1

Il-Wilcon Co., Newbui^h, N. Y.

J ia Bridge Co., Pittsburg, Pa.

Columbiao PowderCo. Hamilton Bldg^ Pittsburg, Pa.

Colwell Lead Co.. 63 CeuU

Commercial Wood and Ccd

2 Compressed Air Co.. GZl B

^*yor^.

* Fifth Ave., New York.

lew York,

)., Bourse Bldg.. Philadelphia,

Conaolidsted Telpherage Co., 20-22 Broad St.. New York.
2 Contraolors' Plant Co., 172 Federal St.. Boston.
4 Contractors' Plant Mfg. Co., Buffalo. N. Y.
6 CookeLocomotiveamlMachinoCo., Paterson. N.J. Bee 34.6.
8 CookWe!lCo,.Bys(emofWells,at.L ' •'

Cornell, J. B. A J. M.—, Hydraulic Pi

etc.. Twenty-siith Bt. uid

172.2 Cornell Mac

S Comine Brake Shoe (
8 Crane Co.. Chicago.

Cresson. GeorRe V.— Co., Einhteenth St. and Allegheny Ave., PhiU

Crosby Steam Gaae and Valve Co., Boston.

Crown Fire Clay Co.. Akron, Ohio.

Crucible Steel Co. of Amerira -^

Cumberiand Hydraulic Cemt

Cummer, F. D.— & Son Co.. Driers, i^ievpiana, uniu.

Cumminm Cement Co., Eilieott Square Bldi., Buffalo. N. Y,

Cunnincham Iron Co., Summer, B. and Fargo Sta , Boston.

CustodS, Alphona — Chimney Construction Co.. Bennett Bldg., New
York.

194 Diamond Drill and Machin.

BUSINESS DIRECTORY. 9^9

105 Diamond State Steel Co., Wilmington, Del.

196 Dibble, F. J. — , Weir Gages, Peabody, Mass.

197 Dickey, W. S.— Clay Mfg. Co., Kansas City, Mo.

198 Dixon, Joseph — Crucible Co., Graphite ripe Joint Compound, etc.,

Jersey City, N. J.

199 Dickson Locomotive Works, Narrow Gage Locomotives, Soranton,

Pa* See 34.5.

200 Dickson M^. Co., Scranton, Pa. See 19.5.

201 Dietagen, Eugene — Co., 149, 151 Fifth Ave., New York.

202 Dilworth, Porter & Co., Limited, Spikes, Rail Braces, etc., Pittsbui^g.

203 Dimmick Pipe Co., Birmingham, Ala.

204 Dobbie Foundry and Machme Co., 138 Dey St., New York.

205 Dodge Mfg. Co., Mishawaka, Ind.

206 Donaldson Iron Co., Emaus, Lehigh Co.,^a.

207 Douglas, W. & B. — , Middletown, Conn.

208 Downie Pump Co., Deep-well and other Pumps, Downieville, Pa.

209 Drake Standard Machine Works, 298 W. Jackson Boulevard, Chicago.

210 Drummond, M. J. — <fe Co., 192 Broadway, New York.

211 Duff Mfg. Co., Allegheny, Pa.

312 Dutton Pneumatic Lock and Engineering Co., Yonkers, N. Y.

214 Dupont, E. I.— & Co., Wilmington, Del.

215 Eastern Bridge and Structural Co., Worcester, Mass.

216 Eastern Construction Co., Brooklyil, N. Y.

217 Eastern Paving Brick Co., Catskills, N. Y.

218 Easton Foundrv and Machine Co., Easton, Pa.

219 Eccles, Richard--, Auburn, N. Y.

220 Eddy Valve Co., Waterford.N.Y.

221 Edson Mfg. Co., 132 Commercial St., Boston.

222 Edwards, Joseph — 6c Co., Centrifugal Pumpsand Hydraulic Dredging

Machinery, 414 Water St., New York.

223 Electric Fire-proofing Co., Fire-proofing System, 119 W. Twenty-third

St., New York.

224 Eley, Philip N.— , Bayonne, N. J.

226 Elliot Frog and Switch Co., East St. Louis, III.

226 Emerson. F. W.— Mfg. Co^ Rochester, N. Y.

227 Empire Portland Cement Co., Warners, N. Y.

228 Enterprise Boiler Co., Youngstown, Ohio.

229 Eppinger db Russell Co., Creosote and Creosoting, 66 Broad St., N. Y.

230 Erie Machine Shops, Asphalt Rollers and Mixers, etc., Erie, Pa.

231 Erie Pump and Enjrine Co., Erie, Pa.

232 Evens & Howard Fire Brick COm St. Louid, Mo.

233 Ewald Iron Co., Staybolt Iron, St. Louis, Mo. •

234 Exeter Machine Works, Pittston, Pa.

235 Fairbanks, The — Co.; Philadelphia.

236 Falcon, Jos. G. — , Submarine Work, Flexible Joint, Evanston, III.

237 Falls Hollow Staybolt Co., Cuyahoga Falls, Ohio.

238 Farrel Foimdry and Machine Co., Havemeyer Bldg., New York.
289 Fauth & Co., 108 Second St., S. W., Washington, D. C.

240 Fay, J. J.— & Egan Co., Wood Tools, 202 W. Front St., Cin., Ohio.

241 Featherstone Foimdry and Machine Co., 1215 Monadnock Block,

Chicago.

242 Federal steel Co., Empire Bldg., New York. See Illinois and Lorain

Steel Cos.
242.5 Ferracute Machine Co., Bridgeton, N. J.

243 Ferrari, Giudo — , S. E. Cor. Seventh and Chestnut Streets, Phila.

244 Ferris, Walter—, Ferris-Pitot Water Meter, Drexel Building, Phila.
246 Fisher A Saxton, 123 G St., N. E^ Washington, D. C.

246 Fitler, Edwin H.— A Co., Rope, Philadelphia.

247 Foos Gas Engine Co., Station F, Springfield, Ohio.

248 • Fort Pitt Bridge Works, Pittsburg, Pa.

249 Flagg, Stanley G.— & Co., Small Steel Castings, Philadelphift.

250 Flint & Walling Mfg. Co., Kendallville, Ind.
261 Flory, S. — Mfg. Co.. Cableways, Bangor, Pa.
252 Fort Scott Cement Association, Kansas City, Mo.

263 Fort Wayne Electric Works, Fort Wayne, Ind.

264 Fox, John — A Co., Special Castings, Water, Gas, and Flange Pipes,

New York.

1000 BUaiNGSS DIKECTOBT.

2ES Fnoktord BtMt and ForinDS Co^ EthalUns, EUwood City, Pk.

see Frukliu Hfg. Co.. FrankUn, P&.

267 Fnuer A Cb&lmera, Chicuo. See lS.fi.

858 French. Swn'l H.— A Co.. Cementa, Plaster, etc, Ytak. Ave. ud

CaUowhill St.. PhiluJelphLa.

2E9 FultoD Iron and EDSina Works, Detroit, Mich.
SM St., Chicago.

use Bids-, FittBbuis, Pa.

a Asphalt, W^nwrisht Bide, BL

'ottBtown, Pa.

t8qiuni.Pa.
rued. New York

282 Ortgory, Elieha— . Artesian Wellii, 60 Liberty St., New Yo*.
383 Qreen Ridae Iron and Spike Works, Spikea, Smubm. P)L.
284 Guild A OarriBOD, Brooklyn, N. Y.

y-Btth and Gias^a Fam

mbbuigiPa.

d» Ave., PhDadeibiUa/
. JaffeiKia St., CUeaao.

ilwayi, 1 Broadway, Kv*
infit..KewYoik.

308 sry, Chieaco.

809 ..NewYMt

810 Illinois Steel Co. Cement Department, Steet Portland Cta

Bookery, ChioBBo,

ail Independent Powder Co., Terre Haute. Ind.

812 Indianapolis Drop Forving Co.. Indiaimpolie, lod.

B13 Industr&l Water Co., 15 Water St., New York.

314 Industrial Works, Bay City. Micli.

SIS Ingeraoll-Senteant Dr.ll Co., Park Place, New York.

317 Inlemational Aritboiacbine Co.. 141 LaSalleSt.Cbieaao.

M8 tnteraationalCoolinsCo.. 32PiiieSt., NewYork.

BUBIHE88 DIBBOTORT.

81B.5 Intern

890 iRiqud

320 6 jMksoD. Byron— Maohins Co., Cenliif . Pumpa. Sas Fnndaco, CH

821 Juiney, Stemmsti & Co., Di«iel Bldg., FhiUdalpbia.

322 JoareyMfB.Co.,Coliunbus.Ohio.

823 JhiIudi Bros.. New York, Philadelphia, ChioMio, Boatim.

SZ4 Joaop, Wm.~ & Bona, ei John Bt., New York.

828 JohDB, H. W.— Mfg. Co., Packings, 100 William St., New York.

82a JohnnHi A Wilsoa.lMNauauSt.. New York.

327 Johnson Cement Coating Co., Coating for Wall*. 160 Fifth Avs., New

York.
32S Joliet Bridse and Trnn fi,.. JnVu-t. TIL
320 Jones A Laugliliiii

331 Janes. B.' M.—' & Co., S~l Hilk 8t~, Boston^

332 Jonffi National Fence Co., Columbus, Ohio.

333 Kalteabaph A Qri«»i KtlS WUliamBon^Bldg.. Cleveland. OUo.

Portland Cement.

8M Lan*. M.— 4 Bona. Pittaburg, Pa.

367 Lau^ilm-Housh Co., 30 Broi3 St., New York.

3E8 Lawrenoe Cement Co.. 1 Broadway, New York.

300 Lawnnoe Machine Co., Lawrence. Musa.

MO LawtenoeviUeCemeDt Co., 26 Cortlandt Bt, New York.

361 Lawrenoe, W. W.— A Co., Pittaburg, Pa-

sea Lead-lined Iron Pipe Co., Lead and Tm-Uned Pipe. Wakefield. Hast.

303 Le a*™, F.— , Black-line Prints from Traoings, 21 N. Thirteenth SI

Philadelphia.

mt Leddor, O. O,— . 302 Waehington St.. Boston.

sea LeSeL Junes— A Co.. Automatio Engines, Springfield, Ohio.

8M Lehigh Portland Cement Co., Allenlown, Pa.

367 Leili,A.— Co.,422BMiamentoSt.,88nFranraBoo.Cftl.

868 Leopold, J. — A. Co., Qranite and Trap Rock, 18 Broadway, Ne

York.

869 Lesahen, A.— & Sons Co.. 020-022 N. Main St.. Bt- Louis. Mo.
370 LidgerwDod Mfg. Co.. 06 Liberty Bt,, New York.

871 Lima Looomottve and Machine Co.. Lima, Ohio.

372 Lima Bteel Casting Co., Lima, Ohio.

373 Link Belt Engineering Co., Nicetown. Philadelphia ,

1002 BUSINESS DIRECTORY.

375 Long & Allstatter Co., Hamilton, Ohio.

376 Loomi»-Manning Filter Co., 420 Chestnut St., Philadelphia.

377 Lorain Steel Co., Steel Castings, Lorain, Ohio.

378 Louisville Bridge and Iron Co., Incorporated, Louisville, Ky.

379 Ludlow Valve Mfg. Co., Troy, N. Y.

380 Ludowici Roofing Tile Co., 508 Chamber of Commerce Bldg., Chicaga

381 Lufkin Rule Co.. Measuring Tapes, Saginaw, Mich.
882 Lukens Iron and Steel Co., Coatesville, Pa.

383 Mack Mfg. Co., New Cumberland, W. Va.

384 Macomber & Whyte Rope Co., 21 South Canal St., Chicago.

385 Maignen Filtration Co., 1310 Arch St., Philadelphia.

386 Manasse, L. — Co., 88 Madison St., Chicago.

387 Manchester Locomotive Works, Manchester, N. H. See 34.5.

388 Manhattan Trap Rock Co., 11 Broadway, New York.

389 Marion Steam Shovel Co., Marion, Ohio.

390 Martin, Wm. R. — , Iron Works, Screens, Lancaster, Pa.

391 Massillon Bridge Co., Massillon, Ohio.

392 Massillon Iron and Steel Co., MassUlon, Ohio.

393 McClenahan & Bros., Granite, Port Deposit, Md.

394 McClintic-Marshall Construction Co., Pottstown and Pittsburg, Pa.

395 McCuUough Iron Co., Sheet Iron and Steel, Wilmington, Del.

396 McGowen, John H. — Co., Cincinnati, OhiOk

397 Mcintosh, Seymour & Co., Auburn, N. Y.

398 McKay, Jas.— & Co., Pittsburg, Pa.

399 McKenna, Chas. F.— , 221 PearlSt., New York.

400 McKieman Drill Co., 120 Liberty St., New York.

401 McLean, John — , 298 Monroe St., New York.

402 McMuUen, Arthur— & Co., 13 Park Row. New York.

403 McNeil^ James — & Bro., Twenty-ninth and Railroad Sts., Pittsbuigi

404 McWilliams & McConnell, Shamokin, Pa.

405 Mead, John A. — Mfg. Co., 7 Broadway, New York.

406 Melan Arch Construction Co., 13 Park Row, New York.

407 Merchant & Co., Metals, Seamless Tubes, etc., Philadelphia.

408 Meredith, J. P. — Cedar Co., Memphis, Tenn.

409 Merrill Bros., 467 Kent Ave., Brooklyn, N. Y.

410 Merritt ^ & Co., Expanded Metal Construction, Philadelphia.

411 Metalloid Sidewalk Co., Concrete and Expanded Metal Cement Walk^

606 Century Bldg., St. Louis, Mo.

412 Michigan Brass and Iron Works, Detroit, Mich.

413 Michigan Lubricator Co., Detroit, Mich.

414 Michigan Pipe Co., Stand Pipe, Bay City, Mich.

415 Mietz, A.—, 128-138 Mott St., New York.

41 6 Millar, Charles— & Son Co., Utica Pipe Foundry Co., Utica, N. Y.

418 Milwaukee Cement Co., Milwaukee, Wis.

419 Missouri Vallej' Bridge and Iron Works, Roofs. Leavenworth, Kan.

420 Monmouth Mining and Mfg. Co., Monmouth, 111.

421 Moore Mfg. Co., Syracuse^ N. Y.

422 Moran Flexible Steam Joint Co., Incorporated, Louiff\nlle, Ky.

423 Morris Machine Works, Centrifugal Piunps, Baldwinsville, N. Y.

424 Morris, Tasker & Co., Incorporated, Philadelphia.

425 Morse. A. J. — & Son, Diving Outfits, 140 Congress St., Boston.

426 Mount Vernon Bridge Works, Mt. Vernon, Ohio.

427 Mundy, J. S.— , Newark, N. J.

428 Murray & Co., Tents, 329 S. Canal St., Chicago.

429 Nash, Geo.— & Co., 15 Piatt St., New York.

430 Nathan Mfg. Co., 92 Liberty St., New York.

431 National Elastjc Nut Co., Milwaukee, Wb.

432 National Meter Co., New York.

433 National Paint Works, Williamsport, Pa.

434 National Tube Co., Pittsburg, Pa., New York, Philadelphia, Boston,

Chicago.

435 NationalSteel Co., Battery Park Bldg., New York.
435.5 National Wood Pipe Co., Los Angeles, Cal.

436 Neptune Meter Co., Jackson Ave. and Crane St., New York.

437 Neuchatel Asphalt Co., Limited, Val de Travers Rock Asphalt. 265

Broadway, New York.

438 New Blue on White Process Co., Cincinnati, Ohio.

439 New England Structural Co., Boston.

"^

BiraiNEaS DIBECrORT. 1003

New Era Iron Works, Gu Enjinw. DayMm, Ohio.

New Jersey Foundry and MBcbiue Co., Overheiul Tnokue Systems.

28 Cortlandt St., New York.
N«w OrleaDS Wood Preflenring Works, CiWHOte and CfWiflOtinff, Ne9

New York 'Air 'compressor Co., 9S Liberty 8t„ New York.

New York and Bennude. Co., Bowling Greeo BIdg., New York.

New York and RoseDdale Cement Co., 200 Broadway, New York.

New York Continental Jewell Filtration Co^ Ifi Broad St., New York
_ _ . New York Dredging Co., Park Row Bldg., New York.
448 [jg„ York "'—^^--=- ■"'----"- "'--'- '■■-—-"
Mfl New York

450 Nichols Engineering and Contracting Co.. Steei Uesaurius Tape, Mo,

IS38 Monadnock Block. Chicago.

4fil Niles Tool Works Co., Hamillonljhio.

4B2 Nitro-powder Co., Kingston, N, Y.

4A3 Norfolk Creosoting Co., Norfolk. Va.

464 Northern Electrical Mfg. Co., Madison, Wis.

465 Norton, A. O.— , 167 ORvbt 8t.. Boston.

466 Norton, F. O.— Cement Co., »2 Broadway, New York,
4G7 Norwalk Iron Works Co.. South Norwalk, Conn.

468 Norwood Engineering Co.. Florence, Mass.

468 Ogden, J. Edward— Co., Yani, Pig Lead. 147 Cedar St., New York.
4e0 Oben, Tinius^ & Co.. 600 N. Tw^th St., Philadelphia.

461 100 Percent. Splice Co., 803 Land litle Bldg,, Philadelphia.

462 Oneida Community. Limited, Nia«a™ FalWM, Y.

463 Osmod Dr«dge Co., Albany, N. Y.
403.2 Otu Steel Co., Cleveland, Ohio.

463.4 OttoGas Engine Works, Philadelpbiaand Chicago.

Parker. Thatcher A.—, Terre Haute,

N. Darien St.. Philadelphia.
D Girard Bids., Philadelphia.
Oo., Philadelphia.

488 , „ -^ ^-^.j.^ ..jaS Water St., Pittsburg. Pa.

489 Pittsburg Valve, Foundry, and Construction Co., Empu* Bid*., Pitts-
burg,!^,

490 Pleuger A Heager Mfg. Co., St. Louis. Ho.
4B1 Plymouth Mills, Washeri, Plymouth, Mass.

492 Pollard. J. G.— , Firing Tools. 141 Raymond St.. Brooklyn, N. Y.

493 Porter, H. K.— Co.. Light and Compressed Air Looomotivea. 6:

Wood St.. Pittsburg. Fa.
404 Pnrtpr Mnrse Co . Hnn^nw Mir\,

» Bldg.. Pittsburg, Fa.

DIItECTOKT.

«7 Power Bpeoialty Co., 128 Liberty St.. New York.
4S8 Prstt AWbitneyCo., H&rtfonl, ConD,
499 Presnd Steel Car Co,. Pitt)'

..- Priesltniui *Co., 1
BOr Prince, L.M.- ""

luri, Pa.

ited. Bourse Buildlni, PbUadel|ihia.

irtb8t.,Cincipiiati,OMo.

I , see aOS.S, Railros
ki.. 1010 Chestnut 8t
'm. E.— , Screw Pun
igineering Co., Ride

S08.S Rallroul Supply Co.. Bedford Bidg., Chicago.

SOB lUmapo Iron Works, Hillburn, Rockland Co., N. Y,

BIO RaodDriUCo, 128 Broadway, New York.

sn Raasome ± Smith Co., Brooklyn, N, Y.

■12 Rawsoa ic Morrison Mfg. Co., Cambridgeport (Boston), Mass.

113 Raymond Broa. Impact Pulveriier Co., 1402 Monadiiocli Blook, Chi-

B«e 34.S.
iUd«I|dui.

Doluznbus,

Ohio,

529 Rockford Bolt Works. Rocktord, III.
630 Rocbling Construction Co.. 121 Liberty St., New York.
B30.6 Roebling's Sons Co., John A. — . Trenton, N. J.
681 Roger^ohn M.— . GlouceiWr. N. J.

632 Roots. P. H. A F. M.— Co., Connersvillejind.

633 Rosa. Sanford P.—, Incorporated. 277 Wasblnglon St.. Jen
N.J.

ar Pumps. Seneca

689 San Francisco Bridge Co.. 220 Market St.. San Fianclsoi, Cat.

640 Sargent Co., Chicago.

. 641 Baundera, D.— Sons, Pipe Thnsding Machinery, Yonkers. N. Y.

642 Scaife.Wm.B.— ASons, Pitlsburg, Pa.

643 Schenectady Locomotive Works, Schenectady. N. Y, See 34.S.

644 Scherier Rolling Lift Bridge Co., Monadnock Block, Cbicsgo.
648 Scholl, Julian— * Co., 128 Liberty St., New York,

646 Schradcr's Son, A..~. Diving Apparatus, 32 Rose St^New York.

MS Scoit.ChariEs— Spring Co.l Philadelphia.
MO Scranton FDr(cinKCo..^cr8nton, Pa.
— " SteamTumi " "

-Fishci ~

-.», ./m.—

aeward, M.— 4

9haw, ThomfcH— (Quimby fenginecring Co., Successor). Compaiwd
Propeller Pump, SlS Ridge Ave., Philadelphia.

BUSINESS DIRBCTOBT.

Fedsnl St., Boston.

BYork.
'York.

whill Sts., PhiU.
r York.

a-hcarth Strr],
smile. Tex.

1006 BU8INEB8 DIRECTORY.

623 Trenton Iron Co., Trenton, N. J.

624 Troy Malleable Iron Co., Troy, N. Y.

625 Ulmer tic HofF, 224 Champlain St.. Cleveland. Ohio.

626 Union Akron Cement Co.. 141 Erie St.. Buffalo. N. Y.

627 Union Hydraulic Works (J. Thompson & Co.), Philadelphia.

628 Union Switch and Sijznal Co., Swissvale, Pa.

629 Union Water Meter Co.. Worcester, Mass.

630 United Stages Cast-iron Pipe and Foundry Co., 217 La Salle St.

Chicago.

631 United States Clay Mfg. Co., Empire Bldg., Pittsburg, Pa.

632 United States Mineral Wool Co., 143 Liberty St., New York.

633 United States Wind Engine and Pump Co., Batavia, lU.

634 United States Wood Preserving Co., Creo-resinate and Creoaote Pro*

cess, 29 Broadway, New York.

636 Valentine & Co., Varnishes, New York.

636 Vanderbilt A Hopkins, Lumber, Timber Preservation, 120 Liberty

St.. New York.

637 Vanderherchen. M. F. — & Co.. Vine and Water Sts.. Philadelphia.

638 Variety Iron Works Co., Cleveland, Ohio.

639 Virginia Bridge and Iron Co., Roanoke, Va.

640 Virginia Iron, Coal, and Coke Co., Spikes, Pig Iron, Bristol, Tenn.

641 Vulcan Iron Works. Chicago.

642 Vulcanite Portland Cement Co., Vulcanite Bldg., 1710 Market St.,

Philadelphia.

642.5 Wabash Bridge and Iron Works, Wabash, Ind.

643 Watkins, Thomas — , Johnstown, Pa.

644 Watson Wagon Co., Canastota, N. Y.

645 Watson-Stillman Co., 453 Rookery, Chicago.

646 Waterbury Rope Co., 69 South St., New York.

647 Warren Foundry and Machine Co., Phillipsburg, N. J., 160 Broadway,

New York.

648 Warren City Boiler Works, Warren, Ohio.

649 Wamer-Omnlan Asphalt Co., Trinidad Asphalt, 4 Warner Bldg., Syn-

cuse, N. Y.

652 Weaver-Hirsh Co., Gray Iron Castings, Allentown, Pa.

653 Weber Railway Joint Mfg. Co., Empire Bldg., 71 Broadway. New

York.

654 Wefugo Co., Cincinnati, Ohio.

655 Weir Frog Co., Cincinnati, Ohio,

656 Wells Light Mfg. Co., Welk Light, 44 Washington St., New York.

657 Western Cement Co.. Louisvilfe, Ky.

658 Western Wheeled Scraper Co., Aurora, 111.

659 Westinghouse Air Brake Co., Pittsburg, Pa. ■•

660 Westinghouse Electric and Manufacturing Co., Pittsburg, Pa.

661 Westinghouse Machine Co., Pittsburg, Pa.

662 Westmoreland Steel and Mfg. Co., Tool Steels, Pittsburg, Pa.

663 West Pascagoula Creosote Works, West Pascagoula, Miss.

664 West Pulverising Machine Co., 220 Broadway, New York.
664.5 Wharton Railroad Switch Co., Philadelphia.

665 Wilcox, D. — Mfg. Co., Mechanicsburg, Pa.

666 Williams Bros., Aiachinists, Ithaca, N. Y.

667 William. Brown & Earle, 918 Chestnut St., Philadelphia.

668 Williams, J. H.— & Co., Brooklyn, N. Y.

669 Williarasport Valve and Hydrant Co., Williamsport, Pa.

670 Wilmot & Hobbs Mfg. Co , Steel, Strip-steel, Crucible-steel, Bridge*

port. Conn.

672 Wonham & Magor, Switches, 29 Broadway, New York.

673 Wood, Alan— Co., Sheet Iron and Steel, 519 Arch St., Philadelphia.

674 Wood, R. D.— & Co., 400 Chestnut St.. Philadelphia.

675 Worth Bros. Co., Coatesville, Pa.

676 Worthington, Henry R.—, 120 Liberty St., New York.

677 Wyman & Gordon, Worcester, Mass.

678 Wyoming Shovel Works, Wyoming, Pa.

679 Young & Sons, 43 North Seventh St., Philadelphia.

680 Young-Brennan Crusher Co., Ill Hancock St., Brooklsoi, N. Y.

681 Youngstown Iron and Steel Roofing Co., Trough Floor, Youngatown;

Ohio.

^

1008

BIBLICX^RAPHY.

BIBUOGBAPHT.

^ The, following list of books makes no pretensions to oonatpleieneBS. It
aims simply to oe usefully suggestive to tne general oivU engineer. A few
works, believed to be specially useful, are designated bv sing^le and double
stars. The list is arranged according to the I>eoimal Classincation of Mel-
ville Dewev. In this classification (see outline below) all subjects are u*
ranged under ten general heads, and to each of these heads is assigned a num-
ber in the hundreds place, as Natural Science, 600; Useful Arts, 600; etc
Then each general head is divided into ten sub-heads, to each of wiuoh a num-
ber in the tens place is assigned. Thus, Natural Science (500) is subdivided
into Mathematics, 510; Astronomy, 520: Phsrsics, 530; etc. AsaiH, eadi
of these is subdivided, and deciikiairy numbered, and this successive siu>divi-
sion and decimal enumeration may be continued indefinitely. To find a sub-

t'ect, it is best to inquire, first, under which of the ten general heads it
>elongs, then under which sub-head, and so on. Thus, Plane Geometry is
seen to belong (1) under Natural Science, 500, (2) und^ Mathematics, 510,
and (3) under Geometry, 513. However, matter on a special subject is
often contained in books on a more general subject which embraces the spe*
oial one. Thus, matter pertaining to Geometry (513) is found in many
works which would be classified under Mathematics (510). Ck>nver8ely, in
looking for books on Mathematics, the sub-heads, Algebra, Geometry^ etc.
should be consulted as well. In general, it is advisable to look unoer all
heads that may contain the information wanted. Thus, information oa
Locomotives may be found under Mechanical Engineering, 621, or under
Railroads, 625. To avoid duplication in such cases, however, one such head
has usually been selected, anci reference to it made under the other.

Outline of CJlaMiillcatloii.

500

510

510.8

512

513

514

515

516

517

519

520

526

526.9

530

531

531.2

531.3

531.4

531.6

531.7

532

533

535-6

537-8

540

550

551.5

600

620

620.C

620.1

620.1

621

621.1

621.13

621.18

621.2

621.3

621.4

621.6

Natural Science
Mathematics

Tables Math. Insts.
Algebra
Greometry
Trigonometry
Descriptive Geometry
Analytical Geometry
Calculus
Least Squares
Astronomy
Geodesy
Surveying
Physics
Mechanics
Statics ^

Dynamics. Kinetics
Work. Friction.
Energy
Power
Hydraulics
Pneumatics
Light and Heat
Elec V and Magnetism
Chemistry
Geology
Meteorology
Useful Arts
Engineering;

Civil Engineering
Indexes

Stren^h of Materials
Mechamcal Engin'p;
Steam Engineering
Locomotives
Steam Generation
Water Eng'4k Motors
Electrical Engin'g
MiscellaneousMotors
Pumps & Blowers

621.8 TransmissionM'oh'm

622 Mining

624 BridgM and Roofs
624.0s Specifications
624.a Trestles. Viaducts

624.2 Girders. Beams

624.3 Trusses

624.6 Arches

624.7 Compound Bridges

624.8 Draw Bridges

625 Roads and Railroads

625.1 Route. Track
625.1a . R.R. Survesdng
625. lao R.R. Chirves
625. lae R.R. E'rthwoi^

625.2 Trains. C!ars
625.8 Roads & Pavements
626-7 Hydraulic Engineerini
627.8 Dams

628 Sanitary Engineerins

628.1 Water Works

628.12-13 Reservoirs

628.15 Pipes

628.16 Purification

628.17 Meters.

660 Chem. Teoh'y.Explosivet

670 Msmufactures. Iron db Sti

690 BuUding

691 Materials
691.7 Iron and Steel

693 Masonry

694 Carpentry

697 Heating and Vent'c
700 Fine Arts

720 Architecture

721 Archi Construotloa
721.1 Foundations

725 PubUoBuUdinci

740 Drawing

770 Photography

1

BIBLIOGRAPHT. 1009

Abbreviations.

A D. Appleton & Co., 1, 3 <fe 5 Bond St., New York, N. Y.

B Henry Carey Baird & Co., 810 Walnut St., PhiladelphU. Pa.

C G Charles Griffin <& Co., Limited, Exeter Street, Strano, London.

C H Chapman & Hall, 11 Henrietta St., Covent Garden. London. W. C.

C L Crosby Lockwood & Son, 7 Stationers' Hall Court, London, E. C.

E N Engineering News Publishins Co., 220 Broadway, New York, N. Y»

L J. B. Lippincott Co., Philadelphia, Pa.

L G Longmans, Green & Co., 91 & 93 Fifth Ave., New York, N. Y.

M The Macmillan Company, 66 Fifth Ave., New York, N. Y.

R G Railroad Gazette, 83 Pulton St., New York, N.Y,

S C Spon and Chamberlain, 12 Cortlandt St.. New York, N. Y.

S L Sampson, Low, Marston & Co., London.

V N D. Van Noetrand Co., 23 Murray St., New York, N. Y.

W John Wiley & Sons, 43 & 46 E. 19th St., New York, N. Y.

600 Natural Science.

510 Mathematics.

♦♦Bledsoe, A. T. — . The Philosoijhy of Mathematics. L.
Hutton, Charles — . Mathematics. 1818.

Merriman. Mansfield — and Robert C. Woodward. Higher Mathematics.
1vol. 8vo. Cloth. $5.00. W.

510.8 Tables and Mathematical Instruments.

Babbage, Chas. — . Logarithms of Noa. from 1 to 108.000. $3.00. S C.
Barlow^ Tables of Squares, Cubes, Square Roots, Cube Roots, ReciprO'

cals, to 10,000. $2.60. S C.
Buchanan, E. E.—. Tables of Squares. $1.00. S C.
♦Chambers's Mathematical Tables. Logarithms of Numbers from 1 to

108,000, Trig., Nautical and other Tables. 8vo. Cloth. $1.76. V N.
Compton, Alfred G. — . Manual of Logarithmic Computations. 3d ed.

12mo. Cloth. $1.60. W.
Hall, John L.—, Tables of Squares. $2.00. EN.
♦Hering, Carl — . Tables of Equivalents of Units of Measurement. W. J.

Johnston, New York.
Holman. Silas W. — . Computation Rules and Logarithms. 8vo. Cloth.

$1.00. M.
♦Hutton, Charles — . Mathematical Tables. London. 1822.
Johnson, J. B. — . Three-Place Logarithmic Tables, Numbers and Trigo-
nometric Fimctions. 15c. each, $5.00 per 100. W.
Jones, G. W. — . Logarithmic Tables. 8vo. $1.00. M.
Ludlow, Lt., H. H. — and Edgar W. Bass. Logarithmic, Trigonometric,

and other Mathematical Tables. 8vo. Cloth. $2.00. W.
Osborn, Frank C. — . Tables of Moments of Inertia and Squares of Radii

of Gyration. 176 pp. $3.00. V N and E N.
Pickworth. Charles N.— . The Slide Rule. 12mo. Flexible cloth.

$0.80. V N.
Rankine, W, J. M. — . Rules and Tables. C G.

Skidmore and Vidal. Table of Tangents. 4to. Cloth. $2.00. V N.
Unwin, W. C. — . Short Logarithhnic and other Tables. Small 4to.

Qoth. $1.40. S C.
Tables. See also under subiect in question.
Surveying Instruments. See 526.91.

5118 Algebra.

Chrystal, G.— . Algebra. Part I. 8vo. $3.75. Part II. 8vo. $4.00. M.
Ltkbsen, H. B. — . Mathematics Self Taught. Arithmetic and Algebra.
Translated and published by H. H. Suplee, 120 Liberty St., New York.
Smith, Charles—. Algebra. $1.90. M.

Todhunter, I. — . A Treatise on the Theory of Equations. $1.80. M.
Todhunter, I. — . Algebra. 16mo. $0.75. M.
Wentworth, G. A. — . Algebra. G.

613 Geometry.

♦Chauvenet. William — . Geometry. L. .
Halstead. Geo. Bruce — . Elements of Geometry. 8vo. Cloth. $1.76. W.
Wentworth. G. A. — . Geometry. G.
Analytical Geometry. See 516.
Descriptive Geometry. See 616.

♦ ♦* Believed to be specially useful.
64

1010 BIBLIOGRAPHY.

514 Trigonometry.

Todhunter, I. — . Trigonometry. Plane — . M.
Todhunter. I. — . Trigonometry. Spherical — . M.

aia Descriptive Geometry.

See 744. Drawing.

516 Analytical Geometry.

Johnson, W. W. — . Curve Tracing in Cartesian Co-ordinates. 12ma

Cloth. $1.00. W.
Todhunter. I. — . Treatise on Plane Co-ordinate Geometry. SI. 80. M.
Todhunter. I. — . Examples of Anal. Geom. of Three Dimensions. SI. M.

517 Calculus.

♦Barker, Arthur H. — . Graphical Calculus. 197 pp. 8vo. S1.50. L G.
Greenhill, A. G. — . Differential and Integral Calculus. S2.60. M.
Johnson, W. W. — . An Elementary Treatise on the Integral Calculus.

Small 8vo. 243 pp. Cloth. $1.50. W.
Rice. J. M. — and Johnson. W. W. — . An Elementary Treatise on the

Differential Calculus. Small 8vo. 485 pp. Cloth. $3.00. Same.

abridged. 208 pp. $1.50. W.
Todhunter, I.—. Calculus, Differential— Integral. $2.60 each. M.
Wansbrough, Wm. D.— . The A. B. C. of the Differential Calculus.

12mo. Cloth. $1.60. V N.

510 Probabilities. Least Squares.

Johnson. W. W. — . The Theory of Errors and the Method of Least

Squares. 12mo. 182 pp. Cloth. $1.50. W.
Merriman, Mansfield — . The Method of Least Squares. 7th.ed. 8to

Cloth. $2.00. W.
520 Astronomy.

Doolittle, C. L. — . A Treatise on Practical Astronomy. 8vo. 652 pp.

Cloth. $4.00. W.
Loomis, Elias — . Practical Astronomy. Harper A Bros., New York.
Young, C. A. — . Astronomy. G.

520 Geodesy.

Comstock, George C. — . A Text-Book of Field Astronomy for EngineersL

8vo. 213 pp. Qoth. $2.50. W.
Gore, J. H.—. Elements of Geodesy. 8vo. Cloth. $2.60. W.
Hasrford, John F. — . Geodetic Astronomy. 8vo. Cloth. $3.00. W.
Merriman , Mansfield — . The Elements of Precise Surveying and Geodesy-

8vo. Cloth. $2.50. W.
Least Squares. See 519.
62S.9 Surveyins.

Burt, W. A. — . Key to the Solar Compass, and Surveyor's Companion.

5th ed. Pocket-book form. $2.50. V N.
Clevenger. S. R. — . A Treatise on the Method of Government Surveying

as prescribed by the U. S. Congress and Commissioner of the Qenoiu

Land Office. 16mo. Morocco. $2.50. V. N.
Gillespie, W. M. — LL.D. Treatise on Land-Surveying. A.
Gribble, Theodore Graham — . Preliminary Survey and Estimates. 8vo.

Qoth. 415 pp. LG. '

*Johnson, J. B. — . The Theory and Practice of Survesring. Revised and

enlarged. 900 pp. 15th ed. Small 8vo. Cloth. $4.00. W.
Merriman, Mansfield — and John P. Brooks. Hand-Book for Surveyors.

2d ed. 16mo. Moroccp. $2.00. W.
Nugent, Paul C. — . Plane Surveying. 8vo. 693 pp. 320 figuies.

Cloth. $3.50. W.
Plympton, Geo. W. — . The Aneroid Barometer: Its Construction and

Use. 18mo. $0.50. V N.
♦Williamson, R. S. — . On the Use of the Barometer on Surveys and Ree-

onnoissances. 4to. Cloth. $15.00. V N.
Winslow, Arthur—. Stadia Surveying. 18mo. $0.60. V N.
Railroad Surveying. See also 626.1a.

526.91 Instruments.

Baker, Ira O. — . Engineers' Surveying Instruments. 2d ed. 400 pp.
12mo. Cloth. $3.00. W.
♦United States Coa.st and Geodetic Survey. The Plane Table: Its Ua
in Topographical Surveying. 8vo. Cloth. $2.00. V N.

* ♦♦ Believed to be specially useful

BIBLIOGRAPHY. 1011

Webb, W. L. — . Problems in the Use and Adjustment of Engineering
Instruments. Revised and enlarged. 16mo. Morocco. $1.25. W.

5186.94 Leveling.

Baker, Ira'O. — . Leveling: Barometric, Trigonometric, and Spirit.
10.60. V N.

5^6.08 Topography.

*Haupt, Lewis M. — . The Topographer: His Instruments and Methods.
J. M. Stoddard, PhUa., 1883.

Reed, Lt., Henry A. — . Topographical Drawing and Sketching. In-
cluding Photography Applied to Surveying. lUus. 4th ed. 4to.
Cloth. $5.00. W.

Specht. George J. — , A. S. Hardy, John B. McMaster, H. F. Walling. Topo-
f;rapnical Surveymg. 18mo. $0.50. V N.

Wilson, Herbert M. — . Topowaphic Surveyinpj, including Geographic,
Exploratory and Military Mapping, with Hmt^ on Camping, Emer-
gency Surgery and Photography. 8vo. Cloth. $3.50. W.

530 Physics.

^Barker, George F. — . Physios, Advanced Course.

*Deschanel, A. Privat — . Elementary Treatise on Natural Philosophy.

Translated by J. D. Everott. A.
*Ganot. Physics. Translated by E. Atkinson, Ph.D. Wm. Wood & Co.,

New York.
Tait, P. G.— . Properties of Matter. 3d ed. $2.25. M.
Meteorology. See 551.5.

531 Mechanies.

Ball, Robert Stawell — . Experimental Mechanics. M.

Church, I. P. — . Mechanics of Engineering. 8vo. Cloth. $6.00. W.

Church, I. P. — . Notes and Examples in Mechanics. 135 pp. 8vo.
Cloth. $2.00. W.

DuBois, A. Jay — . Mechanics. Vol. I, Kinematics, $3.50; Vol. II, Sta-
tics, $4.00: Vol. Ill, Kinetics, $3.50. 8vo. Cloth. W.

DuBois, A. Jay — . The Mechanics of Engineering. Small 4to. Cloth.
Vol. I, 669 pp. $7.50. Vol. II. 632 pp. $10.00. W.

Goodeve, T. M. — . Principles of Mechanics. L G.

Greene, Chas. E. — . Structural Mechanics. 271 pp. $3.00. E N.
"^Lansa, Prof. G. — . Applied Mechanics and Resistance of Materials. 7tb
ed. 8vo. Cloth. $7.60. W.

Mach, Dr. Ernst — . The Science of Mechanics. Translated by Thos. J..
McCormack. 12mo. 534 pp. $2.50. Open Court Publishing Co.
*BfaxweIl. J. Clerk— . Matter and Motion. 18mo. $0.50. V N.
*Merriman, Mansfield — . Text-Book on Mechanics of Materials. 8th ed^
8vo. Cloth. $4.00. W.

Michie. Peter S. — . Elements of Analytical Mechanics. 4th ed. 8vo.
aoth. $4.00. W.

Morin, A. — . Fundamental Ideas of Mechanics and Experimental Data.
Revised and translated by Joseph Bennett. 1860. A.

Moeeley, Henry — . The Mechanical Principles of Engineering and Archi-
tecture. With additions by D. H. Mahan. W.

Nystrom, John W. — . A New Treatise on Elements of Mechanics. 352
pp. 8yo. Cloth. $3.00. B.

Perry, John — . Practical Mechanics. Edited by W. R. Ayrton. Si
shillings. V N.

Rankine, W. J. M. — . Mechanical Text-Book. 9 shillings. C G.

Rankine, W. J. M. — . Applied Mechanics. 8vo. Cloth. $5.00. C G.

Weisbach, Julius — . Mechanics of Engineering and Machinery. Trans-
lated by J. F. Klein. 2d ed. 8vo. Cloth. $5.00. W.

Weisbach, Julius — . Mechanics of Engineering. Tittnslated by Eckley
B. Coxe. 1vol. Large 8vo. 1112 pp. 902 illus. $10.00. V N.

Weisbach, Julius — . A Manual of Theoretical Mechanics. Translated by
Eckley B. Coxe. 1100 pp. 8vo. Cloth. $10.00. V N.

Wood, De Volson — . The Elements of Analytical Mechanics. 6th ed.
500 pp. 8vo. Cloth. $3.00. W.

Ziwet, Alexander — . An Elementary Treatise on Theoretical Mechanics
— Kinetics — Statics — Dsmamics. $2.25 each. In one vol., $5.00. M

♦ ** Believed to be specially useful.

1012 BIBLIOGRAPHY.

Hydromechanics. See 532.

Machinery and Applied Mechanics. See also 621.8.

Strength of Materials. See 620.1.

531.2 Statics.

Johnson, L. J. — . Statics by Algebraic and Graphic Methods. 8va

141 pp. Cloth. $2.00. W.
*Lock. J. B. — . Elementary Statics. M.

Todhunter, I. — . A Treatise on Analytical Statics. $2.60. M.
Arches. See 624.6.
Testing and Strength of Materials. See 620.1.

531.3 Dynamics. Kinetics.

Gamett, William — . A Treatise on Elementary Dynamics. 6 shillings.

GB.
*Lock, J. B. — . Dynamics for Beginners. M.
Tait. P. G.— . Dsmamios. $2.50. M.

531.4 Work. Friction.

^Thurston, Robt. H. — . Treatise on Friction and Lost Work in Machinery
and Mill Work. 5th ed. Svo. Cloth. $3.00. W.

531.6 Energy.

Stewart, Balfour — . The Conservation of Energy. 12mo. Cloth.
$1.50. A.

531.7 Power.

Flather, J. J. — . Dynamometers, and the Measurement of Power. 394
pp. 12mo. Cloth. $3.00. W.

532 Hydraulics.

Bovey, Henry T. — . A Treatise on Hydraulics. Svo. Cloth. S5.00. W.

♦♦Coffin, Freeman C. — . The Graphical Solution of Hydraulic Problems.

80 pp. 16mo. Morocco. $2.60. W.

Fidler, T. Claxton — . Calculations in Hydraulic Engineering. 167 pa

8vo. $2.50. LG.

♦Merriman, Mansfield-^. A Treatise on Hydraulics. Svo. Clotk

$5.00. W.
♦Unwin, W. C. — . Hydromechanics. Encyclopedia Britannica.
Hydraulic Engineering. See 626 and 627.
Hydraulic Motors. See 621.2.
Waterworks. See 628.1.
Bee also under 531.

.532.5 Fluids in Motion.

Bazin, H. — . Experiments upon the Contraction of the Liquid Vein
Issuing from an Orifice. Translated by John C. Trautwine, Jr. 8vo.
Cloth. $2.00. W.
♦Flynn, P. J. — . Flow of Water in Open Channels, Pipes, Conduits,

Sewers, etc. With Tables. 18mo. $0.50. V N.
♦♦Francis, Jas. B. — . Lowell Hydraulic Experiments. 4to. Cloth.
$15.00. V N.
Ganguillet, E. — and W. R. Kutter. A General Formula for the Uniform
Flow of Water in Rivers and Other Channels. Translated by Rudolph
Hering and John C. Trautwine, Jr. 2d ed. Svo. Cloth. $4.00. W.
♦Herschel, Clemens — . One Hundred and Fifteen Experiments on Uie
Carrying^ Capacity of Large, Riveted, Metal Conduits. Svo. Cloth.
$2.00. W.
Moore, C. S. — . New Tables for the Complete Solution of Ganguillet and

Kutter's Formula. 6" X 9", 231 pp. 15 shillings. B.
♦Weston, Edmund B. — . Tables Showing Loss of Head Due to Friction of
Water in Pipes. 170 pp. $1.50. VN and Elf.
See also under 628.1, Waterworks.

533 Pneumatics.

Blowers. Pumps. See 621.6.
Meteorology. See 551.5.
See also under 531 and 532.

* *♦ Believed to be specially useful.

^

BIBLIOGRAPHY. 1013

535 Light.

Tait, P. G.— . Light. $2.00. M.

536 Heat.

Barr, Wm. M. — . A Practical Treatise on the Combustion of C3oal. 307

pp. 8vo. Cloth. $2.60. B.
Garnett, William — . An Elementary Treatise on Heat. 3 s. 6 d. G B.
Maxwell, J. Clerk — . Theory of Heat. 357 pp. 12mo. $1.50. L G.
Tait, P. G.— . Hea*. $2.50. M.
*Tyndall, John — . Heat Considered as a Mode of Motion.
Steam. See 621.1.
Heating and Ventilating. See 697.

537-8 Electricity and Magnetism.

Jenkin, Fleeming — . Electricity and Magnetism. 415 pp. 12mo.

$1.26. L G.
Electrical Machines and Instruments. See 621.3.

540 Chemistry.

Phillips, Joshua — . Engineering Chemistry. 2d ed. 8vo. Cloth.

$4.00. V N.
Remsen, Ira — . Chemistry. Henry Holt & Co., New York.
Sexton, A. H. — . Chemistry of the Materials of Engineering. 12mo.

Cloth. $2.50. V N.
Potable Waters. See 628.1.

550 Geology.

Merrill, G. P. — . Rocks, Rock Weathering, and Soils. 8vo. $4.00. M.

Stockbridge, Horace Edward — . Rocks and Soils. Their Origin^ Compo-
sition, and Characteristics; Chemical, Geological, and Agricultural.
With 15 full-page plates. 2d ed. 8vo. Cloth. $2.50. W.

Tarr, R. S. — . Economic Geology of the United States. 8vo. Cloth.
$3.50. M.

Tillman, S. E. — . A Text-Book of Important Minerals and Rocks. 8vo.

, Cloth. $2.00. W.

Building Stones. See 691.2.

Mining. See 622.

551.5 Meteorology.

Ferrel, William — . A Popular Treatise on the Winds. 2d ed. 8vo.

aoth. $4.00. W.
^Russell, Thomas — . Meteorolonr. 8vo. Cloth. $4.00. M.
Williamson, R. S. — . Practical Tables in Meteorology and Hypsometry in

Connection with the Use of the Barometer. 4to. Cloth. $2.50. V N.

600 Useful Arts.

620 Engineering.

Brooks, Robt. C. — . A Bibliography of Municipal Problems and City

Conditions. 346 pp. Cloth. $1.50. Reform Club.
Carpenter, R. C. — . Experimental ^Engineering. 5th ed. Revised 1897.

8vo. Cloth. $6.00. W.
Crehore, Wm. W. — . Tables and Diagrams for Engineers and Architects.

$0.25 to $0.50 each. Set, $7.50. E N.
Cross, C. S.->. Engineers' Field Book. 4th ed. 166 pp. $1.00. E N.
Dawson, Philip — . The "Engineering" and Electric Traction Pocket-

Book. 1056 pages. ;1300 ilTus. 16mo. Morocco flap. $5.00. W.
Goodhue, W. F. — . Municipal Improvements. 3d ed. 12mo. Cloth.

$1.75. W.
Haupt, L. M. — . Engineering Si>ecifications and Contracts. J. M.

Stoddard A Co., Phila.
Hurst. Tables and Memoranda for Engineers. Vest-pocket edition.

64mo. Rpan»' $0.50. SC.
♦♦Johnson, J. B. — . Engineering Contracts and Specifications. $3.00. E N.
■■■ Kempe, ,^^:K.■rr^' The Engineer's Year-Book. 670 pp. Crown 8vo.

Leather. $3.00. C L.
» . » ' — I

♦ ♦♦ Believed to be specially useful.

1014 BIBLIOGRAPHY.

•Molesworth and Hurst. The Pocket Book of Pocket Books. 32ma

Russia. $5.00. S C.
Philbrick, P. H. — . Field Manual for Engineers. 16mo. Morocco.

$3.00. W.
Rankine, W. J. M. — . Useful Rules and Tables for Engineers and Others.

8vo. Cloth. $4.00. V N.
*Shunk, W. F.— . The Field Engineer. 11th ed. 12mo. Morocco,
. tucks. $2.50. V N.
Smart, Richard A. — . A Hand-Book of Engineering Laboratory Practice.

290 pp. 12mo. aoth. $2.60. W.
Wait, John Cassan — . Engineering and Architectural Jurisprudence.

085 pp. 8vo. Cloth, $6.00; sheep, $6.50. W.
Wait, John Cassan — . The Law of Operations Preliminary to Constmo-

tion in Engineering >nd Architecture. 720 pp. 8vo. Cloth. $5.00;

sheep, $5.50. W.
Weale, John — . A Dictionary of Terms Used in Architecture, Buildinc,

Engineering, Mining, Metallurgy, Arohseology, the Fine Arts, etc. 5tE

ed. 12mo. Cloth. $2.50. V N.
Retaining Walls, etc. See 721.1.
River Embankments. See 627.
Railroad Earthwork. See 625. lac.

690.0 Civil Ensineerinir.

Butts, Edward — . The Civil Engineers' Field Book. 16mo. Morooeo.

$2.50. W.
Mahan, D. H. — . A Treatise on Civil Engineering. Revised by De Vol-

son Wood. New chapter on River Improvements by F. A. Mahan.

8vo. Cloth. $5.00. W.
♦Pat ton, W. M. — . A Treatise on Civil Engineering. 8vo. Half leather.

$7.50. W.
Rankine, W. J. M. — . A Manual of Civil Engineering. 20th eSi. 8vo.

aoth. $6.50. CO.
Trautwine, John C. — . The Civil Engineer's Pocket-Book. Revised and

enlarged by John C. Trautwine, Jr., and John C. Trautwine 3d. 18th

ed., 70th thousand. 16mo. Morocco,, gilt edges. $5.00. W, C H.-
Wheeler, J. B. — . An Elementary Course of Civil Engineering. 8vo.

Cloth. $4.00. W.

620.1 Indexes.

♦Assn. Eng. Soc., John C. Trautwine, Jr., Sec'y. Descriptive Index of Cur-
rent Engineering Literature. Vol. I, 1884-1891. $5.00. 267 S. 4th
St., Phila.

Enirineering Magasine. Descriptive Index of Current Engineering Litera-
ture. ♦Vol. II, 1892-1895, $6.00. ♦♦Vol. Ill, 1896-1900. 120 Liberty
St New York.

Galloupe, F. E.— . Index to Engineering Periodicals. 1888-1892.

$3.00. R G.

6!30.1 Strength of Materials.

Baker, B.— . On the Strengths of Beams, Columns, and Arches. V N.
Bovey Henry T. — . Theory of Structure and Strength of Materials. 3d

ed. 830 pp. 8vo. Cloth. $7.50. W. ^ . . . „ .

Burr W. II. — . Elasticity and Resistance of Materials of Engineering.

6th ed., Re-written. 772 pp. 8vo. Cloth. $7.50. W.
Johnson, J. B.— . The Materials of Construction. Large 8vo. 810 pp.

Martens", Adoiph—. Hand-Book on Testing Materials. Part I : Methods.
Machines and Auxiliary Apparatus. Translated by Gus. C. Henning.
ME. 2 vols. 8vp. Cloth. $7.50. W
♦Merriman, Mansfield—. The Strength of Materials. 2d ed. 12mo.

Cloth. $1.00. W. , , . ^ J ,. J ^ ,

Moore and Kidwell. Tables of Safe Loads for Wooden Beams and Col-
umns. 57 pp. $0.50. EN. . . « .
Spangenburar, Ludwig — . The Fatieue of Metals under Repeated Strains.
Translated. 18mo. $0.50. V N.

♦ ♦♦ Believed to be specially useful.

BIBLIOOBAFHY. 1015

Thurston, Robert H.-*-. A Text-Book of the Materials of Construction.

8vo. 785 op. Cloth. S5.00. W. . m

Unwin. W. C. — . The Testing of Materials of Construction. 500 pp.

8vo. $6.00. LG.
Wood, De Volson — . A Treatise on the Resistance of Materials, and an

Appendix on the Preservation of Timber. 7th ed. 8vo. Cloth.

$2.00. W.
See also 721, Architectural Construction; 624, Bridges; 691. etc.; 532,

Mechanics; 532.2, Statics.

621 Mechanical' Englneerinff.

Clark, D. Kinnear — . The Mechanical Engineer's Pocket-Book of
. Tables, Formule, Rules, and Data. 3d ed. 700 pp. 8vo. Leather.

6 shillings. C L.

Clark, D. Kinnear — . A Manual of Rules, Tables, and Data for Mechani-
cal Engineers. 1012 pp. 6th ed. 8vo. Cloth. $5.00. V N.

Kennedy. Alex. B. W. — . The Mechanics of Machinery. M.
**Kent, William — . The Mechanical Engineer's Pocket-Book. 1100 pp.
6th ed. 16mo. Morocco. $5.00. W .

Lockwood's Dictionary of Terms Used in the Practice of Mechanical Engi-
neering. Edited by Joseph G. Homer. 6000 definitions. 8vo. Cloth.

7 8. 6 d. C L.

Rankine, W. J. M. — . A Manual of Machinery and Millwork. 580 pp.

♦♦Reuleaux, F. — . The Constructor. Translated by H. H. Suplee. $7.50.
312 pp. 1200 illustrations. Eng. Mag., 120 Liberty St.. New York.
♦Unwin, W. C. — . Elements of Machine Design. Part I, General Princi-
ples. 476 pp. $2.00. Part II, Engine Details, 306 pp. $1.50. L G.
Weisbach, Dr. Julius — and Prof. Gustav Herrmann. The Mechanics of
Hoisting Machinery, including Acctunulators, Excavators and Pile
Drivers. Translated by Karl F. Dahlstrom. 8vo. Cloth. 332 pp.
177 illus. $3.75. M.

621.1 Steam Engineerlnfl:.

*Holmee, Geo. C. V.—. The Steam Engine. 12mo. $2.00. L G.
*Pray, Thomas — Jr. Steam Tables and Engine Constants. Compiled

from Regnault, Rankine, and Dixon. 8vo. Cloth. $2.00. V N.
Rankine, W. J. M. — . A Manual of the Steam Engine and other Prime

Movers. C G.
Spangler, H. W. — ; Greene, W. M.— ; Marshall, S. M. — . The Elements

of Steam Engineering. 8vo. Cloth. 273 pp. $3.00. W.
Thurston, Robert H. — . Handy Tables. From the ' 'Steam-Engine Man-
ual." 8vo. Cloth. $1.50. W.
Thurston, Robert H. — . A Manual of the Steam-Engine. Part I, Struc-
ture and Theory; Part II, Design, Construotion, Operation. Elach
Eart, 8vo. 1000 pp., $6.00. Two parts, $10.00. W.
itham, JayM. — . Steam-Engine Design. Svo. doth. $5.00. W.

621.13 liOComotlves.

♦♦Forney, Mathias N. — . Catechism of the Locomotive. $3.50. V N &

Meyer, J. G. A. — . Modern Locomotive Construction. 4to. Cloth.

$10.00. W.
Reagan, H. C. — . Locomotives: Simple, Compound and Electric. 12mo.

617 pp. Cloth. $2.60. W.
"Railroad Gazette." Modem Locomotives. $7.00. R G.

621.18 Steam Generation.

Barr, Wm. M. — . A Practical Treatise on High Pressure Steam Boilers.
456 pp. 8vo. Cloth. $3.00. B.
♦Heine Safety Boiler Co. "Helios." (Tables.) Published by author.
Peabody, Cecil H. — and Miller, Edward F. — . Steam Boilers. 8vo.

384 pp. Cloth. $4.00. W.
Thurston, R. H. — . A Manual of Steam-Boilers : Their Designs, Construo-
tion, and Operation. 5th ed. 8vo. Cloth. $5.00. W.
♦Wilson, Robert — . A Treatise on Steam-Boilers. Enlarged and Illus-
trated by J. J. Flather. 3ded. 12mo. Cloth. $2.50. W.
Steam Heating. See 697.

♦ ♦♦ Believed to be specially useful.

1016 BIBLIOGRAPHY.

631 .2 Water Engines and Motors.

♦♦Francis, Jas." B. — . Lowell Hydraulic Experiments. 4to. Cloth.
$15.00. V N.
♦FriBell. J. P.—. Water Power. 684 pp. 8vo. Cloth. $5.00. W.
Weisbach, Julius — . Hydraulics and Hydraulic Motors. Translated by

A. J. Du Bois. 2d ed. lUus. Svo. aoth. $5.00. W.
Wood, De Volson — . Turbines. Old ed^Svo, cloth, $1.00; 2d ed., re-
vised and enlarged, Svo. cloth, $2.50. W.
See also 532. Pumps, 621.6.

621.3 Electrical Enginefering.

♦Foster, H. A. — . Electrical Engineer's Pocket-Book. V N.

Haupt, Herman — . Street Railway Motors. 213 pp. 12mo. doth.
$1.75. B.

Rosenberg, E. — , Haldane Gee, W. W. — , Kinzbrunner, Carl — . electri-
cal Engineering. Svo. 275 pp. Cloth. $1.50 net. W.

See also 537.

621.4 Air, Gas, and Other Motors.

Goldingham, A. H. — . The Design and Construction of Oil Engmes. 196

pp. . $2.00. S C.
♦Kennedy, A. B. W. — and W. C. Unwin. Transmission by Air-Power.

ISmo. $0.50. V N.
Richards, Frank — . Compressed Air. 12mo. .Cloth. $1.50. W.
Saunders, W. L. — . Compressed Air Production. 58 pp. $1.00. E N.
Wolff, A. R.— . The Windmill as a Prime Mover. 2d ed. Svo. Cloth.

$3.00. W.

621.6 Blowing and Pnmplng Engines.

Barr, Wm. H. — . Pumps. L.

Weisbach, Julius — and Gustave Hermann. The Mechanics of Pumping
Machinery. Translated by K. P. Da^Istrom. Svo. $3.75. M.

621.8 Transmission Mechanism.

Cooper, John H. — . A Treatise on the Use of Belting for the Transmiaaion

of Power. Svo. Cloth. $3.50. B.
Flather, J. J.— . Rope Driving. 12mo. Cloth. $2.00. W.
Kerr, E. W. — . Power and Power Transmission. Svo. 368 pp.

aoth. $2.00. W.
Stahl, Albert W. — . Transmission of Power by Wire Ropes. ISmo.

$0.60. V N.
Principles of Mechanism. See also 531, Mechanics. See also 621,

Mechanical Engineering.

eiZ2 Mining.

Bowie, Aug. J. — Jr. A Practical Treatise on Hydraulic Mining in Cali-
fornia. 5th ed. Small 4to. Cloth. $5.00. V N.
♦Drinker, Henry S. — . Tunneling, Explosive Compounds, and Rook
Drills. lOOOIllus. 3d ed. 4to. Half-bound. $25.00. W.

Hermann, E. A. — . Steam Shovels and Steam Shovel Work. 60 pp.
$1.00. EN.

Ihlseng. M. C— . A Manual of Mining.' Svo. 585 pp. Cloth. $4.00. W.

PreUni, Charles— . Tunneling. 311pp. 6" X 9i''. $3.00. V N.

Simms, W. F. — . Practical Tunneling. 4th ed. Svo. Cloth. $12.00.
VN.

Wilson, E. B. — . Hydraulic and Placer Mining. 12mo. Cloth. $2.00.
W. •

Retaining Walls, etc. See 721.1.

Explosives. See also 660.

624 Bridges and Boots.

Baker, B. — . Long-Span Railway Bridges. 97 pp. 12mo. Cloth.

$1.00. B.
Bender, Charles E. — . Proportions of Pins Used in Bridges. ISmo.

$0.50. V N.
Boiler, A. P. — . Practical Treatise on the Construction of Iron Highway

Bridges. (Written in popular language.) 4th ed. Svo. Cloth.

$2.00. W. "e -» /

♦ ♦♦ Believed to be specially useful.

BIBLIOGRAPHY. 1017

Boiler, A. P. — . The Thames River Bridge. Limited edition. lUus. 4to.
Paper. $5.00. W.

Burr, W. H. — . Stresses in Bridges and Roof Trusses, Arched Ribs, and
Suspension Bridges. 0th ed. Revised. Plates. 8vo. Cloth. $3.50.
W.

Chanute, O. — and George S. Morison. The Kansas City Bridge. 4to.
Cloth. $6.00. V N.
♦Fidler, T. Claxton — . A Practical Treatise on Bridge-Construction. 2d

ed. 8vo. 30 shillings. C G.
♦Green, Chas. E. — . Graphics for Engineers, Architects, and Builders.
Part I: Roof Trusses. Diagrams. New revised ed. 8vo.' Cloth.
$1.25. Part II: Bridge Trusses. New revised ed. Svo. Cloth. $2.50.
Part III: Arches in Wood, Iron, and Stone. 3d ed. Svo. Cloth. $2.50.
W.

Johnson, J. B. — , Bryan, C. W. — , Turneaure, F. E. — . The Theory and
Practice of Modem Framed Structures. Small 4to. 538 pp. Cloth.
$10.00. W.

McMaster, John B. — . Bridge and Tunnel Centers. 18mo. $0.50. VN.
•♦Merriman, Mansfield — and Henry S. Jacoby. A Text-Book on Roofs and
Bridges. Part I: Stresses in Simple Trusses. 5th ed.^ revised and en-
larged. 8vo. Cloth. $2.50. Part II: Graphic Statics. 3d ed., en-
larged. With. 6 folding plates. Svo. Cfloth. $2.50. Part Hit
Bridge Design. 3d ed. Svo. Cloth. $2.50. Part IV: Cantilever,
Contmuous, Draw, Suspension and Arch Bridges. 2d ed. Svo.
Cloth. $2.50. W.

Morison, George S. — . The Memphis Bridge. $10.00. W.

Ritter, August — . Iron Bridges and Roofs. Translated by H. R. San-
key. S C.
♦Waddell, J. A. L. — . De Pontibus. A Pocket-Book for Bridge Engi-
neers. 416 pp. 16mo. Morocco. $3.00. W.

Whipple, S. — . An Elementary and Practical Treatise on Bridge Build-
ing. Svo. Cloth. $3.00. VN.

Wood, De Volson — . A Treatise on the Theory of the Construction of
Bridges and Roofs. lUus. 6th ed. Revised and corrected. Svo.
Cloth. $2.00. W.

Retaining Walls, Foundations, etc. See 721.1. See also 531.2, Statics.
Architectural Construction. Strength of Materials, see 620.1.

6!34.0s Specifications for Bridges.

♦Bouscaren, G. — . Specifications for Railway Bridges and Viaducts of

Iron and Steel. 9 pp. $0.25. EN.
♦Cooper, Theodore — . Specifications for Steel Highway Bridges. 25 pp.

$026. EN.
♦Cooper, Theodore — . Sp>ecifications for Steel Railroad Bridges. 24 pp.

$0.25. E N.
♦Osborn Co. Specifications for Metal Highway Bridge Superstructure

12 pp. $0.25. E N.
♦Osborn Co. General Specifications for Railway Bridges. 10 pp.

$0.25. E N.
♦Thacher, Edwin — . General Specifications for Highway Bridges. 8 pp.

$0.25. E N.
♦Thomson, G. H. — . Standard .Specifications for Structural Steel for

Modem Raihoad Bridges. $0.10. E N.
♦Waddell, J. A. L. — . Specifications for Steel Bridges (from "De Ponti-
bus"). 12mo. Cloth. $1.25. W.

624.a Trestles. Viaducts.

♦Foster, Wolcott C. — . A Treatise on Wooden Trestle Bridges. 4to.
Cloth. 271 pp. $5.00. W.
Katte, W. — . Specifications for Standard Pile and Timber Trestle
Bridges. $0.05. E N.

624.3 Girders.

Birkmire, Wm. H. — . Compound Riveted Girders as Applied in Build-
ings. Svo. Cloth. $2.00. W.

♦ ♦♦ Believed to be specially useful.

1018 BIBLIOGRAPHY.

Philbriek, P. H. — . Beams and Girders. Practical Formulas for Tlieii
Resistance. ' 18mo. $0.50. V N.
• ♦Stoney, Bindon B. — . The Theory of Stresses in Girders and Similar
Structures. 777 pp. 8vo. $12.50. V N.

634.3 Trusses.

Ricker, N. C. — . Elementary Graphic Statics and the Construction (A
Trussed Roofs. 4th ed. 8vo. Cloth. $2.00. £ N.

624.6 Arches.

Buck, G. W.— . Oblique Bridges (Arches). Revised by J. H. W. Bock.

With plates. C L.
Cain, William — . Voussoir Arches Applied to Stone Bridges, Tunnels,

Culverts, and Domes. ISmo. M.50. V N.
Howe, Malverd A. — . A Treatise on Arches. 371 pp. 8vo. Cloth.

$4.00. W.
Woodbury, D. P.—. Stability of Arches. V N.

624.7 Compound Bridges.

Bender, Charles — . Practical Treatise on the Properties of ContinuooB
Bridges. ISmo. $0.50. V N«

624.8 Draw Bridges.

Wright, Chas. H. — . The Designing of Draw-Sipans. Part I: Plate Gir-
der Draws. Part II: Rlveted-Truss and Pin-Connected Long-Sptf
Draws. lUus. 8vo. Cloth. $3.60. W.

625 Boads and Bailroads.

Berg, Walter G. — . Building and Structures of American Railroads.

Large 4to. Cloth. $6.00. V N.
Cleemann, Thos. M. — . The Railroad Engineer's Practice. 4th ed.

12mo. Cloth. $1.50. V N.
Dredge, James — . History of the Pennsylvania Railroad. Ekigravinss;

Map, Plates, etc. Folio. Half-morocco, $10.00. Paper, $5.00. W.
Godwm, H. C. — . Railroad Engineer's Field Book. (An fhcploro's

Guide.) 2ded. 16mo. Morocco. $2.50. W.
*Henck, John B. — . Field Book for Railroad Engineers. 1896. S2.50. A.
Nagle, J. C. — . A Field Manual for Railroad Engineers. 2d ed. Idmo.

Morocco. $3.00. W.
Paine, Charles — . The Elements of Railroading. $1.00. RG.
♦Paine, George H. — . The New Roadmaster*s Assistant. $1.50. R G.
Vose, G. L. — . Manual for Railroad Engineers. Two vols., text ud

plates. Lee A Shepard, New York, 1873.
Bridges. See 624.
Tunnels. See 622.
Electric Railways. See 621.3.
Locomotives. See 621.13.

625.1 Boute, Track. Fixed Equipment.

Katte, W. — . General Specifications for Cross Ties. $0.05. E N.

Katte, W. — . Specifications for Track-laying. $0.10. E N.
""Parsons, W. B. — Jr. Track. A Complete Manual of Maintenance of
Way. 8vo. Cloth. $2.00. EN.

Pratt, Mason D. — and C. A. AI<ien. Street-Railway Roadbed. 8to.
Ck)th. $2.00. W.

Raih-oad Gazette. Block Signaling. $2.00. R G.
♦Tratman, E. E. R.— . Railway Track and Track Work. 502 pp. 200

illus. $3.00. 1901. E N.
*Tratman, £. E. Russell — . Metal Railroad Ties. Report on the Use of.
Preservative Processes and Metal Tie-Plates for Wooden Ties. Pub-
lished by U. S. Department of Agriculture.

Webb, Walter Loring — . Railroad Construction. 16mo. 601 pp.
Morocco. $6.00. W.

Railroad Stations. See 725.

Bridges. See 624.

* ♦♦ Believed to be specially useful.

BIBLIOGRAPHY. 1019

695.1a B. B. SnrYeylns.

Allen, 0. Frank — . Railroad Curves and Earthwork. 194 pp. Pocket-
book Form. $2.00. S C.
Brooks, John P. — . Hand-Book of Street' Railroad Location. 16mo
Morocco. $1.50. W.'
♦Gribble, T. G. — . Preliminary Survey and Estimates. 480 pp. 12mo.
$2.50. L G.
♦♦Searles, Wm. H. — . Field Engineering. Railway Surveying, Location,
and Construction. 16th ed. 16mo. Morocco. $3.00. W.
*Shunk, Willtam F. — . A Practical Treatise on Railway Curves and Loca-
tion, for Young Engineers. 12mo. Cloth, tucks. $2.00. B.
♦Wellington, A. M. — . Economic Theory of Railway Location. 980 pp.
$5.00. W &EN.
See also 526.9.

695.1 ac B. B. Ciirres.

Clark, Jacob M. — . A New System of Laying Out Railway Turnouts
Instantly, by Inspection from Tables. 12mo. Leatherette. $1.00.
VN.
♦Crandall, Chas. L. — . The Transition Curve. Revised and enlarged.
16mo. Morocco. $1.50. W.

Fox, Walter G.—. Transition Curves. 18mo. $0.50. VN.

Gieseler, E. A. — . Scales for Turnouts. Stiff cardboard. $0.25. R G.

Howard, Conway R. — . The Transition Curve Field Book. 16mo.
Morocco. $1.50. W.
**Searles, Wm. H.— . The Railroad SF)iral. The Theory of the Compound
Transition Curve Reduced to Practical Formulas and Rules for Applica-
tion in Field Work. 6th ed. 16mo. Morocco. $1.50. W.

Torrey, A. — . Switch Layouts and Curve Easements. $1.00. R G.

Trautwine, John C. — . The Field Practice of Laying out Circular Curves
for Railroads. Revised by John C. l^utwine, Jr. 13th ed. 12mo.

• Morocco. $2.50. W, C H.

695.1ae B. B. Earthwork.

Allen, C. F. — . Tables for Earthwork Computation. 8vo. Cloth.
$1.50. VN.
♦Crandall, Chas. L. — . Railway and Other Earthwork Tables. 8vo.

aoth. $1.50. W.
♦Hudson, J. R. — . Tables for Calculating the Cubic Contents of Excava-
tions and Embankments by an Improved Method of Diagonals and Side
Triangles. New edition with adoitional tables. 8vo. Cloth. $1.00.
W.

Johnson, J. B. — . Stadia and Earthwork Tables. 8vo. Cloth. $1.25.
W.

Katte, W. — . Specifications for Grading and Masonry. 16 pp. $0.25.
EN.

Taylor, Thomas V. — . Prismoidal Formula and Earthwork. 8vo.
Cloth. $1.50. W.

Trautwine, John C. — . A Method of Calculating the Cubic Contents of
Excavations and Embankments by the Aid of Diagrams. Revised and
enlar^d by John C. Trautwine, Jr. 9th ed. 8vo. Cloth. $2.00. W.

Trautwine. John C. — Jr. Cross-Section Sheet. To be Used with Traut-
wine's Excavations. Sheet form. $0.25. W.

TrautwincLJohn C. — Jr., and Woodson, D. Meade — . Cross-Section Sheet.
$0.50. Williams. Brown & Earle, Phila.

Earth Handling. See 622.

Foundations. See 721.1.

695.2 Trains. Boiling Equipment.

♦Railroad Gasette. Car-Builder's Dictionary. $5.00. R G.
Locomotives. See 621.13.

025.8 Boads and PaTements.

Aitken, Thomas — . Road Making and Maintenance. Cloth, 7" x 9".
440 pp. 139illus. $6.00. L.

♦ ♦♦ Believed to be specially useful.

1020 BIBLIOGRAPHY.

Baker. Ira O. — . A Treatise on Roads and Pavements. 8vo. 663 pp

Cloth. S5.00. W.
Byrne, Austin T. — . Highway Construction. 8vo. Cloth. S5.00. W.
Gillmore, Q. A. — . Practical Ireatise on the Construction of Roads,

Streets, and Pavements. 12mo. Cloth. $2.00. V N.
Herschel, Clemens — and £. P. North. Road Making and Maintenance.

156 pp. $0.50. EN.
Spalding, Fred. P. — . A Text-Book on Roads and Pavements. 12mo.

Cloth. $2.00. W.
Stone, Gen. Roy — . New Roads and Road Laws in the United States.

200 pp. 12mo. Cloth. $1.00. V N.
Tillson, George W. — . Street Pavements and Paving Materials. 600 pp.

8vo. Cloth. W.

626-7 Hydraulic Engrineerins.

Flynn. P. J. — . Irrigation Canals, andT Other Irrigation Works, etc.

Author, San Francisco. Cal.
Hewson, Wm. — . Principles and Practice of Embanking Tiands from

River Floods, as Applied to the Levees of the Mississippi. 8vo. Cloth.

$2.00. V N.
Hill.C.S.— . Chicago Main Drainage Channel. 129 pp. $1.50. EN.
Newell, Frederick Haynes — . Irrigation in the United States. S2.00.

Thos. Y. Crowell & Co., New York.
^Starling, Wm. — . Floods of the Mississippi River. 57 pp. $0.50. E N.
*U. S. Geological Survey. Water Supply and Irrigation Papers. About

50 pamphlets have been issued, and more are to follow. 6'' x 9", U. S.

Geol. Surv., Wash., D. C.
Wilson. Herbert M. — . Manual of Irrigation Engineering. 3d ed.

Small 8vo. Cloth. $4.00. W.
Retaining Walls, etc. See 721.1.

627.8 Dams.

Gould, E. Sherman — . Specifications for Dams and Reservoirs. 11 pp.

$0.25. EN. .

Gould, E. Sherman — . High Masonry Dams. 18mo. $0.50. V N.
Leffel, James — & Co. The Construction of Mill Dams. 312 pp. 8vo>

Cloth. $2.50. B.
*Wegmann, Edward — . The X>esign and Construction of Dams. 4th ed.

revised and enlarged. 4to. Cloth. $5.00. W.
Reservoirs. See 628.13.

628 Sanitary Enslneerlns.

*Adams, J. W. — . Sewers and Drains for Populous Districts. 5th ed.

8vo. Cloth. $2.50. VN.
*Baker, M. N. — . Sewerage and Sewage Purification. 18mo. $0.50.

VN.

♦Baumeister, R. — . The Cleaning and Sewerage of Cities. Adapted from

the German by J. M. Goodell. 2d ed. 291 pp. 8vo. Cloth. $2.00.

VN.

*FolwelI, A. Prescott — . Sewerage. The Designing, Construction, and

Maintenance of Sewerage Systems. 445 pp. 8vo. Cloth. $3.00. W.

♦Kiersted, Wynkoop— . Sewage Disposal. 12mo. Qoth.. $1.25. W.

*Merriman, Mansfield — . Elements of Sanitary Engineering. 2d ed.

8vo. Cloth. $2.00. W.
*Ogden, H. N.— . Sewer Design. 234 pp. 12mo. Qoth. $2.00. W.
*Rafter, G. W.~ and M. N. Baker. Sewage Disposal in the United
States. 598 pp. $6.00. VN dc E N.
Rideal. Samuel—. Sewage and the Bacterial Purification of Sewage.

8vo. Cloth. $3.50. W.
Sedgwick, William T.— . Sanitary Science and the Public Health.

Cloth. 8vo. $3.00. M.
Swaab, S. M. — . Tables and Diagrams for Making Estimates for Sewerage

Work. 20 pp. $0.50. EN.
Waring, Geo. E. — Jr. Modem Methods of Sewage Disposal for Towns,
Pubno Institutions, and Isolated Houses. 2d ed. 260 pp. Cloth.
$2.00. V N.
Waring, Geo. E. — Jr. Sewerage and Land Drainage. 3d ed. Quarto.

Cloth. $6.00. V N.
Ventilation and Heating. See 697.

* ** Believed to be specially uaeful.

r

BIBLIOGRAPHY. 1021

028.1 Water Works.

♦♦Baker, M, N. — . Manual of American Water- Works. (Descriptive list of

works, with names of officers.) 700 pp. $3.00. E N.
♦Baker, M. N. — . Potable Water and Methods of Detecting Impuritie!:

18mo. $0.50. V N.
Croes, J. J. R. — . Statistical Tables of American Water Works. $2.00.

EN.
Fanning, J. T. — . A Practical Treatise on Hydraulic and Water-Supply

Engineering. 14th ed. 8vo. Cloth. $5.00. V N.
♦Folwell. A. Prescott — . Water-Supply Engineering. Water-Supply Sys-
tems. 562 pp. 8vo. Cloth. $4.00. W.
♦Fuertes, James H. — . Water and Public Health. 70 figures. 12mo.

Cloth. $1.60. W.
♦Fuertes, James H. — . Water Filtration Works. 300 pp. Cloth.

$2.50. W.
Goodell, John — . Water Works for Small Cities and Towns. 300 pp.

Engineering Record.
Gould, E. Sherman — . The Elements of Water Supply Engineering. 168

pp. $2.00. EN.
Mason. William P. — . Water Supply. With special reference to health-
fulness of . 8vo. Cloth. $5.00. W.
McPherson, J. A. — . Water Works Distribution. 6" x 8". 154 pp.

lUus. Cloth. $2.50. VN.
Nichols, Wm. Ripley — . Water Supply. Considered mainly from a

chemical and sanitary standpoint. With plat«s. 4th ed. 8vo.

Cloth. $2.50. W.
♦Tumeaure, F. E. — and Russell, H. L. — . Public Water Supplies. 8vo.

760 pp. $5.00. W.
Turner, J. H. Tudsbery — and A. W. Briehtmore. The Principles of

Waterworks Engineering. Large 8vo. Cloth. $10.00. S & C.
Wegmann, Edward — Jr. The Water Supply of the City of New York

from 1658-1895. 4to. Cloth. $10.00. W.
Pumps. See 621.6.
Dams. See 627.8.
Flow in Pipes, Channels, etc. See 532.

628.1S-13 Stand Pipes. Tanks. Reservoirs.

Hazlehurst. J. N. — . Towers and Tanks for Water-Works. 8vo. Cloth.
$2.60. W.

Jacob, Arthur — . On the Designing and Construction of Storage Reser-
voirs. 18mo. $0.50. V N.
♦Pence, W. G. — . Standpipe Accidents and Failures. 195 pp. $1.00.
EN.
♦♦Schuyler, James Dix — . Reservoirs for Irrigation, Water-Power and
Domestic Water Supply. Revised, 1901. 432 pp. Large octavo.
Cloth. $5.00. W.

Retaining Walls, etc. 721.1.

See also 627.8, Dams.

628.15 Pipes.

Barstow, C. D. — . Cost of Laying Water Pipe. 16 pp. $0.10. EN.
Weston, Edmund B. — . Tables for Estimating the Cost of Laying Cast-
Iron Water Pipe. 12 pp. $0.25. E N.
Flow in Pipes. See 532.

628.16 Purification.

♦Fuertes, James H.— . Water Filtration Works. Cloth. 5" x 8''. Illus.

$2.50. W.
♦♦Hazen, Allen— . The Filtration of Public Water Supplies. 333 pp. 8vo.

Cloth. $3.00. W.
♦Hill, John W.— . The Purification of Public Water Supplies. 304 pp.

8vo. Cloth. $3.00. V N.
♦Kirkwood, Jas. P. — . Report on the Filtration of River Waters for the

Supply of Cities, as Practised in Eurojje, made to the Board of Water

Commi^ioners of the city of St. Louis. 4to. Cloth. $7.60. V N.
Rideal, Samuel — . Water and Water Purification. Crown 8vo. 7s. 6d.

CL.

♦ ♦♦ Believed to be specially useful.

1022 BIBLIOGRAPHY.

628.17 Use and Waste. Meters.

Browne, Rom E.' — . Water Meters: Comparative Testa of Accuracy, De-
livery! etc. . 18mo. $0.60. V N.
Kent, W. Q.— . The Water Meter. 8vo. Cloth. S C.

630 Agriculture, Forestry.

Green, Samuel B. — . Principles of American Forestry. 12ino. Cloth.

$1.50. W.
Pinohot, Giflford — . A Primer of Forestry. Bulletin 24, Div. Forestry,

U. S. Dept. Agr.

660 Chemical Technology. Explosives.

^Eissler, Manuel — . The Modern High Explosives — Nitro-glyoerin and
Dynamite. 3ded. Plates. 8vo. Cloth. $4.00. W.
Sanford, P. Gerald—. Nitro-Explosives. 270 pp. 8vo. Cloth. $3.00.

VN.
Wisser, John P. — . Explosive Materials. 18mo. $0.50. V N.
Metallurgy. See 670, Manufactures.

See 601, Buildinc( Materials.
Explosives. See also 622, Mining.

670 Manufactures. Iron and Steel.

*Bauerman, H. — . A Treatise on the Metallurgy of Iron. 515 pp. 12mo.

Cloth. $2.00. B.
Bolland, Simpson — . The Encyclopedia of Founding and Dictionary of

Foundry Terms Used in the Practice of Moulding. 12mo. Cloth.

$3.00. W.
Bolland, Simpson — . "The Iron Founder." Supplement. 400 pp. 12mo.

Cloth. $2.50. W.
Campbell, H. H. — . Manufacture and Properties of Structural SteeL

The Scientific Pub. Co., N. Y. and London.
Overman, Frederick — . The Manufacture of Steel. 285 pp. 12mo.

Cloth. $1.50. B.
West, Thomas D. — . American Foundry Practice. 10th ed. 12mo.

Cloth. $2.50. W.

West, Thos. D. — . Moulder's Text-Book, Being Part II of American

Foundry Practice. 7th ed. 12mo. Cloth. $2.60. W.
Iron and Steel. See also 691.7, Building Materials.

690 Building.

See 721, Architectural Construction.

691 Materials and Preservatives.

Bjrme, Austin T. — . Inspection of the Materials and Workmanship Em-
ployed in Construction. 556 pp. 16mo. Cloth. $3.00. W.
♦Jonnson, J. B. — . The Materials of Construction. 795 pp. 3ded. 8vo.
Cloth. $6.00. W.

Terry, George — . Pigments, Paint, and Painting. 12mo. Cloth. S3.00.
SC.
♦Thurston, Robt. H. — . Materials of Construction. 6th ed. 8vo. Cloth.
$5.00. W.

Strength of Materials. See 620. 1 .

See ako Tratman, under 625.1.

691.1 Wood.

Boulton, S. B. — . The Preservation of Timber by the Use of Antiseptics.
18mo. $0.50. V N.

Snow, Chas. H. — . The Principal Species of Wood. Their Character-
istic Properties. Large 8vo. 214 pp. Cloth. $3.50. W.

691.2 Natural Stone.

♦Merrill, George P. — . Stones for Building and Decoration. Illus. 2d cd.
8vo. Cloth. $5.00. W.
See also 693, Masonry.

* *♦ Believed to be specially useful.

BIBLIOORAPHY. 1023

091.3-5 Artificial Stone» Concrete, Cement.

Butler, D. B.-r-. Portland Cement. 360 pp. S6.00. E N.
♦Gillmore, Gen. Q. A. — . Treatise on Limes, Hydraulic Cements, and
Mortarst. 8vo. Cloth. $4.00. V N.

* Jameson, Charles D. — . Portland Cement. 8vo. Cloth. $1.50. V N.
*Newman, John — . Notes on Concrete and Works in Concrete. 12mo.

aoth. $2.50. S C.
Spalding, Frederick P. — . Hydraulic Cement. 12mo. Cloth. $2.00.

• W.

091.7 Iron and Steel.

Birkmire, Wm. H. — . Architectural Iron and Steel. 3d ed. 8vo.

aoth. $3.50. W.
Davies, James — . Galvanized Iron: Its Manufacture and Use. Svo.

aoth. $2.00. S C.
Greenwood, W. H. — . Steel and Iron. 536 pp. 12mo. Cloth. $1.75. B.
Keep, William J.—. Cast Iron. Svo. 238 pp. Cloth. $2.50. W.
♦Metcalf, William—. Steel. 12mo. Cloth. $2.00. W.
Thurston, Robt. H.—. Iron and Steel. 6th ed. 8vo. Cloth. $3.50. W.
See also 670, Manufactures.

683 Masonry.

*Baker, Ira O. — . A Tx«atise on Masonry Construction. 9th ed. Svo.
Cloth. $5.00. W.
Macinnis, Owen B. — . Bricklaying. Svo. aoth. $2.00. V N.
Sieoert, Jno. S. — and F. C. Biggin. Modern Stone Cutting and Masonry.

Svo. aoth. $1.50. W.
Dams. See 627.8.
See also 691.2, Stone.

897 Heating and Ventilation.

^Carpenter, Rolla C. — . The Heating and Ventilating of Buildings. 400
pp. Svo. Cloth. 4th ed. $4.00. W.

700 Fine Arts.

7130 Architecture.

721 Architectural Construction.

Birkmire, Wm. H. — . Skeleton Construction in Buildings. 2d ed. Svo.

aoth. $3.00. W.
Birkmire, Wm. H. — . The Planning and Construction of High Office

BuUdings. Svo. aoth. $3.50. W.
Black, W. M.— . The United States Public Works. Summary of

Methods of Construction, Materials, and Plants under War and Treasury

Departments. Ob. 4to. Cloth. $5.00. W.
Bovey, Henry T. — . Theory of Structures and Strength of Materials.

Svo. Cloth. 830 pp. $7.50. W.
Christie, W. Wallace — . Chimney Design and Theory. 12mo. Cloth.

$3.00. VN.
Fowler, Chas. £. — . Greneral Specifications for Steel Roofs and Buildinga

12 pp. $0.25. EN.
Freitag, Joseph K. — . Architectural Engineering. High Building Con-
struction. Svo. Cloth. 2d ed. Re-written. $3.50. W.
Freitag, Joseph K. — . The Fireproofing of Steel Buildings. Svo. aoth.

$2.50. W.
♦♦Johnson, J. B. — , W. H. Bryan and F. E. Turneaure. The Theory and

Practice of Modern Framed Structures. 7th ed. Revised and en-
larged. Small 4to. Cloth. $10.00. W.
♦Kidder, F. E.— . The Architect's and Builder's Pocket-Book. 1030 pp.

500 engravings. 13th ed. Revised and greatly ehlarged. 16mo.

Morocco. $4.00. W.
Weyrauch, J. J. — . Strength and Calculations of Dimensions of Iron and

Steel Construction, with reference to the Latest Experiments. 12mo.

Cloth. $1.00. V N.
Winslow, Benj. E. — . Diagrams for Calculating the Strength of Wood,

Steel, and Cast-iron Beams and Columns. 19 plates llf x 9^".

$2.00. E N.
Bridges, 624.
Strength of Materials, 620.1.

* ♦♦ Believed to be specially useful.

J

1024 BIBLIOORAPHY,

7131.1 Foundation!.

Baker, Benjamin — . The Actual Lateral Pressure of Earth-Work. 18ma

$0.60. V N.
Fowler, Charles Evan — . The Coffer-Dam Process for Piers. 173 pp.

100 illus. Svo. Cloth. $2.50. W.
Howe. Malverd A. — . Retaining Walls for Earth. 3d ed., rewritten aad

enlarged. 12mo. Cloth. $1.25. W.
Jacob, Arthur — . Practical Designing of Retaining-Walls. 2d ed

18mo. $0.60. V N.
Newman, John — . Earthwork Slips and Subsidences Upon Public Works.

234 pp. 12mo. Cloth. $3.00. S C.
Osbom Co. General Specifications for Bridge Substructure. 10 pp.

$0.25. E N.
*Patton, W. M. — . Practical Treatise on Foundations. 429 pp. 8n>.

Cloth. $5.00. W.
Wellington, A. M.—. Piles and Pile-Driving. $1.00. EN.
Embankments. See 627.

726 Public Buildings.

Dillenbeck, Clark — . Standard Specifications for Railroad Structurei,
Brick Passenger Stations, Brick Freight Houses, Frame Passenger Ste>
tions. Frame Freight Houses. $0.40 each. £ N.

740 Drawing.

♦Jacoby, H. &.—. Text-Book on Plain Lettering. 82 pp. S3.00. E N.
Mahan, D. H. — . Industrial Drawing. Revised and enlarged by D. F.

Thompson. 30 plates. Svo. Cloth. $3.50. W.
*Reinhardt, C. W. — . Lettering for Draughtsmen, Engineers, and Stu*

dents. 32 pp. $1.00. E N.
^Reinhardt, C. W. — . Tecbnic of Mechanical Drafting. 36 pp. $1.00.

E N.
Smith, R. S. — . Manual of Topographical Drawing. Revised and ^•

larged by Chas. McMillan. 12 folding plates, -dd ed. 8vo. Cloth.

$2.50. W.
Warren, S. Edward — . Drafting Instruments and Operations. Th(i^

oughly revised, with additions. 12mo. Cloth. $1.25. W.
Warren, S. Edward — . Elements of Plane and Solid Free Hand Geomet-
rical Drawing. Plates and wood-cuts. 12mo. Cloth. $1.00. W.
Descriptive Geometry, 515.
Blue Printing, etc. See 770.
Topographical Drawing. See also 526.08.

770 Photography.

♦Leitjse, Ernest — . Modem Heliographio Processes. 2d ed. 8vo. Cloth.

VN.
Pettit, James S. — . Modem Reproductive Graphic Processes. ISmo.

$0.50. V N.
Photographic Surveying. See 526.9.

* ** Believed to be specially useful.

OliOeSAKT OF TEBHS. 1026

QLOSSABY OF TEAMS.

Akmeua ; ths flat square member on top of a oolamn.

AhaeiM or obBeiua ; any portion of the axis of a curre, flrom the vertez to anr point firom whloh
A line leaves the axis at right angtos, and extends to naeet the enrve ttseif; said line being oallad an
prdimatt. An abeolss and ordinate together are oalled co-or«notss.

AoMtUyi an upward slope, or asoent of ground, Ac

AdU : a horiiontal passage into a mine, so.

AdMt; a well-known onrved euttlng. instrument, for dressing or ehippins horlsontal surfaces.

AU^rnattng moMmi up and down, or baokward and forward. Instead of nroiTing, Ao.

Anglt'htttd, or plaster bead : a bead nailed to prctJeoting angles in rooms, to protest the plaster oa
their edges f^om u^ury.

AngU'block; a triangular blook againnt which the ends of the braces and counters abut'in a Howe
bridge*

Atmtal; to toughen some of the metals, glass, *e, by first beating them, and then oauaing them t«
tool Terr slowly. This prooess howaver lessens the tensile strength.

ArUteUnal oarfs ; in geology ; a line from which the strata of rooks slope away downward in.opp*>
■ite diraetlons, like the slates on the roof of a house ; the ridge of the roof representing the asda.

4|M9; m point In either chord of a truss, where two web members meat.

Ap>ron; a oorering of timber, stone, or metal, to protect a snrfaoe against Um notion of water flow*
ing oTer it. Has many other meanings.

AriMtr. See Journal.

ArcAftrovs ; that part of an entablature which Is next above the eolumns. AppUee also when there
are no columns. Also, the mouldings around the sides and tops of doors and windows, attaohed to
either the inner or outer fhoe of the wall.

Arria ; a sharp edge formed by any two surfaces which meet at an angle. The edges of a brlok are
arrises.

AMhUr ; a ftustng of ent stone, applied to a backing of rubble or rough masonry, or Mekwork.

Attragal ; a small moulding, about semi-cironlar or semi-elllptio, and either plain or ornamented by
OMrring.

AxiM ; an imaginary line passing through a body, which may be supposed to revolve around it: as
the diam of a sphere. Any pleoe that passes through and supports a body which revolves ; in which
oase it is called an axle, or shaft.

Aarie-ftox. See Journal-box.

Aadttrf ; an axle whioh remains fixed while the wheel rerolves around It, as in wagons, ko.

Aaimuth. The aiimuth of a body is that aro of the horizon that is included between the meridian
eirole at the given place, and another great eirole passing through the body.

Backing; the rough masonry of a wall faced with finer work. Earth deposited behind a retaining-

BaLaino»'Uam»: the long top beams of look-gates, by whloh they are pushed open or shut.

BaSk; a large beam of timber.

BaUaat; broken stone, sand or gravel, ko, on whieh railroad eross-ties aro laid.

BaU-eoek; a cistern valve at one end of a lever, at the other end of which is a floating ball. The
ball rises and falls with the water in the cistern ; and thne opens or shuta the valve.

BaO'Vakte. Bee Yalve.

BargaboairdM ; boards nailed against the enter Ihee of a wall, along the slopes of a gable end of a
honse, to bide the rafters, Ac ; and to make a neat finish.

BtueuU bridgt: a hinged lift-bridge furuiahed with a oounterpolse.

Mmtimr, (sometimes aflbotedlv icrtlr,) or talwe ; the sloping baokward of a fhee of masonry.

Bmg; on bridgee, Ao, sometimes a panel ; sometimes a span.

B*ad ; an ornament either oompoeed of a straight eyllndrieal rod ; or carved or oast in that slu^w
eaany surfhee-

Btaring ; the course by a eompass. The span or length in the clear between the points of support
ef a beam, Ac. The points of support themselvee of a beam, shaft, axle, pivot, ko.

Btd-motMUng* ; ornamental mouldings on the lower faoe of a pr^eeting cornice, ko.

Btd-pUtU : a large plate of iron laid as a foundation for something to rest on.

BtetU; a heavy wooden rammer, such as paven use.

BM-tramk. See Orank.

Btnek-marh; a level mark out at the foot of a tree for fnturo roferanoe, as being moro permanent
than a stake.

Hams, or h«fm» : a horinntal snrfaoe, as if for a pathway, and forming a kind of step along the fhoe
•r sloping ground, in canals, the level vop of the embankment opposite and corresponding to thi
towpath is called the berm.

Bessemer steel is formed by forolng air into a mass of melted oast iron ; by whioh means the excess
of carbon in the iron is separated from it, until only enough remains to constitute oast steel. The
oarbon is ehmiieaUp united with the steel, but mechanieaUg with the iron.

Btton; concrote of hydraulic oement, with broken stone and bricks, gravel, ko.

Bwti; the slope formed by trimming away a sharp edge, as of a board, ko, Bdges of common
drawing rulers and aoales are usually bevelled.

BtvA g*ar; cog-wheels with teeth so formed that the wheels can work Into each other at an angle.

BUge ; the nearlv flat part of the bottom of a ship on eaeh side of the keel. Also, the swelled part
•r a barrol, ke. To bilge is to spring a leak in the Ulge, mr to be broken thero.

BUt* : the small boring points used with a brace.

Bla»t-pip«a ; in a locomotive : those through whioh the waste steam passes from the oylinder into
1Bb» smoke-pipe, and thus creates an artificial draft in the chimney, or smoke-pipe.

Jfoa««W0 ; dressing stone with a broad chisel oalled a boaster, and mallet. The boaster gives a
naoother surface after the use of the point, or the narrow chisel called a tool.

Modg; the thiokness of a lubricant or otber liquid. AUo, the mea§ura of itiat thickness, exprvsaed
In the number of seconds in which a given quantity of the oil, at a givwi tempwaturo. floa-e throogb
a given aperture.

65

1026

GLOSSARY OF TBftHtt.

BoUUr; a ttmoer, or a thick Iron plat« iaoed betwtott tbe end of a bridge aad Its aeac on (ki
abaUneni.

Bond : the dUpoting of tbo blooka of atc^o or brtekubrk ao as to for A tko whole into a firm ttn^
taro, bj a Jadicioni overlapping of eaeh other, ao as to break Joint. Applies also to timber, iee, k
TarioBS ways.

Boruut; a oap over the and of a pipe, Ac. A oast*lron plate bolted down as a ooreriag over m
apertore.

Mtr» ; Inner diameter of a bellow ^Under.

MorrimfU f a pit dag in order to obtain malarial Ibr an embankmnnt.

JMs; an Inerease of the diameter at any part of a shaft for amy pnrpeae. A proifeetioii in sii^pi
•r a segment of a sphere, or somewhat so, whether for nse or for ornament; oilea earved, or <

Boat-drain ; a square or reetaugular drain of masoorj or tlisber, ander a railroad, *e.

Bruce ; a kind of eonvd handle used for boring holes with bitts. The head of the braoe
stationary, being pressed against by the body of the person osing it, while the other part with the
bitt is turned ronnd by his hand. Also, an tDCllned beam, bar, or strat, tar sastaintng romproisiM

Brtidbet; a prqJeotiDg pieoe of board, Ac, frequently triangular, the vertical leg attached to tte
fhoe of a wall, and the horisontal one snpporUog a shelf, Ac. Often made in ornamental ahapee Ik
supporting busts, clooks, Ac. Also, the supports for shafting ; as pendent, wall, and pedestal braetott-

Brake ; an arrangement for preventing or diminishing motion by means of friction. The Metim
is usually applied at the eironmferenoe of a revolving wbeti, by means of levers. On raUroada. At
ear-brakes should be worked by steam, as those of Loughridge, Westtnghonse, and Creamer. Iba
eaeh a handle as that of a oommon pump.

^oss is oompeeed of oopper and sine.

Braeeee; fittings of brass in many plummer-bleeks, and In other positions, for dimlnlsbing Iht
frietion of revolving Journals which rest upon them.

Brame; to unite pieoes of iron, oopper. or brass, by means of a hard solder, called spelter seMer.
and composed, like brass, of oopper aud sine, but in other proportions.

Break jekUf to so overlap ptoees that the jdats shall not ooonr at the same place, and thus pi»
duoe a bu bond.

Bre€ut'»ummer t a beam of weed, iron, or stone, supporting a wall over a door or other <q>en]ag;
a Und of lintel.

Breaet-waU; one built to prevent the fhlling of a verthsal face out into the natural soil; In db-
tinotion to a retaining* wall or revetment, which is built to sustain earth depatUed behind It.

Sreeeh; Um hind part of a cannon. Ae.

Bridge, or bridge-jtieee, or bridge-bar; a narrow strip placed across an opening, for anpportiBK
•omething wlthont closing too much of the opening.

BronMe is composed of copper and tin.

Bvtkkead; on ships, Ac, the timber partitions across them. Also, a long fhoe of wharf paraiia
Id the stream. .. ^ ^

Buof ! a floating body, Ihstoned by a efaatn or rope to some sunk body, as a guide for finding tk
laHer. Sometimes also used to Indicate channels, shoals, rooks, fto.

J?»mM»; to polish by rubbing; chiefly applies to metelB. ^ ^. _ ^ .

Btuh ; to line a cironlar hole by a ring of metal, to prevent the hole flrom wearing larger, lln,
when a piece la cot out, and another piece neatir inserted into thtf cavity, the last pieoe is aometiBe
■aidtobebnihedlnj sometimesitiseaUedaplng. »,,,,. u^ . , ,

BuUrioint: one in which the ends of the two pieoes abut together without overlapping, and m
Joined by one or more separate pieoes called covers or welts, which reaeh aorosa the Joint aad sn
fastened to both pieces. ..... ^

Buttreee; n vertical projecting piece of brickwork or masonry, built In front of n waO »

oSSSn; a large wooden box tHth sides that may be detached and floated away.

Caliber ; the inner diameter, or bore. j i u *,

OaOpere ; compasses or dividers with curved legs, for measuring outside and inside diameCera.

CaJk. or can/*: to fill seami or Joints with something to prevent leaking.

Calking iron , a lool Tur forcing calking into a Joint.

Clim&, or cam, or wiper: a piece Axed upnn a revolving shaft in suoh a manner as to prodnee sa
alternating or reciprocating motion in Rometbing in contact with the cam. An eccentrio.

Oamber ; a slight upward curve given to a beam or truss, to allow for Mttiing.

Camel: a kind of barges or hollow floating vessels, which, when filled with water, are fastened «
the sides of a ship ; and the water being then pnmped out, they rise by their buoyantly ; and lift tki
ship BO that she can float in shallower water.

dmtHeffera ; projecting pieces for Bupporting an upper baloeny<fto. ^ , ^ . , _,

Cante, rime, or akrottditiaa ; the pieces forming the ends of the buckets of water* wheela« to preveat
the water from spilling endwise. ... . _w.um- m - ....w

Captan: a long hnllow mpe-drnm surrounding a strong vertical pivot, npbn the bead of whioah
lesu; and around irhieh it turns. Its top is a thick prqjecting oirottlar piece, having helea arotMd in
outer edge or droamfbrenee, for the insertion of the ends of levers ; or capstan-bars. It is a kiB4 «

vertical windlass. .,,.,.*....,.„.

OMs-hordsn ; to convert the outer surfkoe of wrought iron into steel, by heaUng It while In ooatsd

wHh charcoal. ...... * ^

Caeemate ; in fortification ; the small apartment in which a oannon stands.
Oaetore ; rollers usually combined with swivels; as those used under heavy nBmltnre, Ae.
Caiueemttif ; a raised footway or roadway.

Cavetto; a moulding consisting of a receding quadrant of a droM.

OemenUMon ; the process of converting wrought iron into steel, by heaUng it in eontaot wtthehir
iroduoea blisters on the steel bars ; hence hiieter steel . These are removed, aa4

eoal. This process produoea

the steel compacted, by reheating It, and then subjeoting It to a tilt>hammer. It is then tiiud atsti
or ehear steel. Or if the blister steel is broken up, remelted In a omelble, and then mn into Ingsli
or blocks, It is called cruci&le, cost, or ingot ateel ; which is harder and closer-grained than tilted stesl>
It may be softened, and thus become less brittle, by annealing. The IngoU nu^y he converted InM
bars by either rolling or hammeriDg. the ssme as shear and blister steeL
Center; the supports of an arch while being built.

OLOBSARY OP TERMS. 1027

Oenttr of percuuUm, In a movlnf body, Is that polot whiob would strike an opposing body wUfc
•reiter fonse than any other point would. If the opposing body is immorable. it will receive ofl th«
•toi-c) of a rtgld mofing body which strikes with its center of peroussfon. See Pendulum, page MR.

C-t^pocl; a shallow well for reoeiring waste water, filth, fte.

Chamftr; means mooh the same as berel ; but applies more espeoially when two edges are out away
*o %n to form either a chamfer-groove, (see 14. p 735, of Tmsses.) or a projecting sharp edge.

Ckeek$: two flat parallel pieces oonflning something between tnem.

ChOUiu, eMU-k0rd9tUng, or cAlU-ecuKno; giring great hardness to the outside of oast-Iron, bf
pouring it into a mould made of Iron instead of wood. The iron mould causes the outside or skin of
the casting to cool very rapidly ; and this for some unknown reason increases its hardnesa. This pnr
«e4S Is frequently confounded with case-hardening.

<^ck ; any piece used for filling up a chance hole, or vacancy.

Chtuck; the arrangement attached to the revolving shaft, arbor, or mandril of a lathe, for holding
tbe thing to be turned.

Chwrn-driU; a long iron bar. with a cutting end of steel ; much used in quarrying, and worked by
raising It and letting it fall. When worked by blows of a hammer or sledge it is called a jumper.

Cima, or eyma ; a moulding nearly in shape of an S. Wh^ tbe upper part is oonoave, it Is ealM
ft otDsa reota ; when oonvax, a oima revena.

Clack valrt. See Talve s.

Clan^ ; a piece fkstened by tongue and groove, transversely along the end of others, to keep than
ftom warping. A kind of open collar, which, being closed by a elamp-sorew, holds tight what It sur.
rounds. See Cramp.

(7I<:9 6oari{S; short thin boards, shingle.shaped, and need instead of shingles.

Claw , a split provided at the end of an iron bar, or of a hammer, tut, to take hold of the heads of
nails or spikes for drawing them out ; as in a common claw-hammer.

Cleat; a piece merely bolted to another to serve as a support for something else ; as at 7, 8, 10,
lie, p. 7S&. or Trusses. Often used on shipboard for fastening roixs to, as at 11. Also a piece of
board nailed across two or more other boards, fbr holding them together, as Is often done in tempo-
rary doors, Ae.

CUviM. See Shackle.

fJUek. See Batehet.

OUf : a fkstening like that on the topa of tbe Ts of a spirit level ; being a kind of half eoUar openllg
by a hinge.

Clvtch ; applied to various arrangements at the ends of separate shafts, and whieh by olotehlng tt
efttohing Into each other cause both shafts to revolve together. A kind of coupling.

OM; a kind of valve for the disoharge of liaulds, air, steam, Ac.

Ootmdani; or a Conttant of frietion. saflety, or strength, fte, may usually be taken to be a nom-
ber which siiows the proportion (or ratner tbe ratio) which friction, safetv. tensile strength, Ac, bear
lo a certain something eUe whioh la not generally expressed at the time, but la well understood. Thna,

when we say that the ooeff of (riotlon of one body upon another is ■^, ka, it is understood that the

Motion is In tbe proportion of -«^th of the jtreatwn whieh produces it. A ooeff of safety of S, meani

that the safety has a proportion or ratio of 3 to I to tbe theoretl''xH breaking load. A ooeff of &00 lbs,
er of 20 tons, to, of tensile strength of any material, denotes that said strength Is in tbe proportion
of 500 lbs. or of 90 tons, Sus, to each aqrtare inch of traneverae aeetion. to. Same as Mbdulue.

Oojfer-dam ; an enclosure built in tbe water, and then pumped dry, so as to permit masonry or
other work to be carried on inside of It.

Cog ; the tooth of a cog-wheel.

Collar; a flat ring surrouodiag anything closely.

OoUar-beam ; a horiiontal timber stretehing from one to another of two rafters which meet at top;
but above the main tie-beam.

Concrete ; artificial stone formed by mixing broken stone, gravel, to, with common lime. When
kydranlio oement is used instead of Hme, the mixture is called beton. The terms " lime conorete"
and " oement concrete " would be convenient.

Connecting-rod ; a piece which connects a orank with something which moves It, or to whieh It

res motion.

ConeoU ; a kind of ornamenul bracket, somewhat in shape of an S ; much used In cornices, fte,
fbr supporting ornamental mouldings above it.

Coping ; flat plates of stone, iron, to, placed on the tops of walls exposed to the weather.

Cbrbsc ; a horiiontal projecting piece whioh assists in supporting one resting upon It whloh proieots
sllll farther.

Cfere; anything serving as a moold for anything else to be formed around. A term much used in
Dsmndries.

dprnice; the ornamental projection at the eaves of a building, or at the top of a pier, or of any other
•trncture.

Cotter-boU, or luy-boU; a bolt which, instead of a screw and nut at one end, has a slot cut through
it near that end, for the insertion of a wedge-shaped key or cotter, for keeping It in its place. Some,
limes the ends of these keys are split, so as to spread open after being inserted, so as not to be Jolt^
out of piaoe.

Covitaerfort ; vertical projections of masonry or brickwork built at intervals along tbe back of a wall
to iAtrengtheu it ; and generally of very little use.

Oounter-ehafi ; a secondary shaft or axle which receives motion from the principal one.

Counteraunk. See Beam.

Countor-toeigkt ; or counter-balance ; any weight used to balance another.

CkvpUnge; a term of very general appUoation to arrangemeuu for connecting two shafts ao that
Ikej ahall revolve together.

Cbeer; see " bott-Joint.**

Coter ; in re-rolling iron and steel from plies of small pieees, a largo bar or slab, called a oover, of
the same width and length as the pile, is employed to form the bottom of the pile, and a similar slak
ler tbe top. The covers serve to hold the pile together ; and, after rolling, th^ form unbroken tef
and bottom iinrfaeee of the ftnlshed plate, bar, rail, I beam, tc

1028

QLOG&A.BY OF TEBSCS.

Orub : a ilmrt shaft or axle, whioh mitm ai a rope-dram in raising weighta j and is revolredcitMi
Iv ooc-wli»i)a, a winch, or hy ]ev«n or liandspikes, inserted in holes aroond iu oircuiufiBreaoe Ukea
v^Bdlasa, or eapsUn. of whlob it is a rariefr. It maj be either vertical or horUontaL It is ofim
i«k in a frame, to be carried from place to place. Also tbe wkoU machw is oaUed a crab.

OrmdU ; applied to yarioos kinds of timber aapporta, which partly Miolose the mass snatainea.

Oromv: • short bar of metal, having lu two ends bent downward at right angles for iaseruonuli
two adUoinlng pieces of stone, wood, &o, to hold them together. Much need at the ends of ooping-stoMk
▲lao a similar bent piece, with a set-screw pasting through one of the bent enda, for bolding uuap
tight between it and the other end. This last Is also called a clamp. , ,,^ ^

Orane; a hoiatiog machine consUting of a revolving vertical post or slott ; a prqjoetuurMi ; saA
a •(<» for sustaining the outer end of the Jib. The stay may be either a strut or a tie. There sn
also cog-wheels, a rope drum or barrel, with a winch, ropes, pulleys. Ac. In a crane the post,p,
and stay do not change their relative positionB, ai they do in a derrick. . ^ ^ , ^ . i

OratJt ; a double bend at right angles, somewhat Uke a Z, at the end of a shaft or axle, &nd fonsiBp
a kind of handle by which the axle may be made to revolve. Sometimes, as in common ^ndatoMi.
this crank Is formed of a separate piece removable at pleasure. That part of this pieoe which has Os
Muare opening in it for fitting it to the square end of the axle, is called the erank-arm ; and the otMr
part the a-ank-handU. ▲ hM-erank oonsisu of 4 bends at right angles at the center of aa axle, fon-
iDg in it a kind of U. A douhU citm^ consists of two beU cranks arranged thns, jj^. The bend is
-. _ - . . ^ ^.^» .«»/«« <r h« trnrm belUcrank is applied also to those used In flxing common dwsB-

trcAn^h^?^iVrUS\h*t «j:m>r;^^ -Me ft lUble to npeet easi^;.

"*s;ur;hTtirp!S?oV^Tro^Jr\1;i,^^^^^^^ , ,

^■^'^iB^^T^r^ir^.'^i^^^^ - right angle, to.

J^r^A a ?£?of ro?VrJis"*Oflen seen on piston rods, which they serve to keep in pl»»^

wsUng on the slides. <>' pWj- , j^^ ^^^us purposes ; often pointed at one end.
aS:i%i ^«;a^"•iT?^-whS^L%hlS^the^terSL^^ not Son its outer oircumffer«i« «

■eual, but »P«» **;« fi*"* ?f *Sni'Sf%ood iron or stone, placed under the bottoms of oirtmlar walh.
..^a ie'l'S^hlftViTp^Vei^u^^^^ into the wsUs at intenrala. for the nm

*"SS;o,^. \n^m^li!^rl^^^i off the steam f^om a cvUnder'hefore the piaton has m^m

Out-offj »° f"*°«^™®": ent thronlh a narrow neck of land, to straighten the course of a rfv«.
'^Sura1i..o^r'riin\"?£r^^^^^^ a. to sOlowwa*

^AS."*rrooro?v:lv1 ^i^SlTtirtlmisBion of air U>a furnace, stove, *c.

"'/>eanaVio«, of the sun, or of a star. U its angle north or «>uth of the earth's equator at the UN
ef observation. j *.«

DecKvMv ; a downward slope or descent of gronna, •»•. ^,. . ,. „_. i_t^--i- uxMit thev- ■■■laiW'

Which forms the stay may be let out or hauled in at P*®""®' •""" ^^r* ,,i_ed .Mt This caaiMt

'SSirafla^'^rt"^^^^^^

en an'ediJo?e«S oVwhSS is hoS^Dwed o«t a Wclwulju: half of » short fe^^^^^

Dlates are put in contact they form a complete female screw, like that »5 a nnt , ana wmg

Ee'd JoSthM by an iron bo^ng called the die-stocks, wWc»i have long h«dlM for revolrtog ^

wostuSte a mo\dd or cutter for forming thread, on a male •«*r-„-^-^«« 'J%7tf i hSfaJnufToTSt

di^ioiVn^?pU!'Sf;a5i%«^

eompass-needle rests on its pivot after being magnetized.

^^; In^artKreSSTsure. either partial or total, in which ?"?• and ogj Tjeel. are plss.1
fbr being loaded or unloaded, or repaired. The first is a irst <|o«* 'J^i"' ' fT,^**.. rf-vj ,,*i

SlK5. it to^^ill . 4o,.iron wl.« Mill «B. end U b«it down wd volnHd for drlTlnf. ••

GLOSSARY OF TERMS. 1029

•Hmt end being formed Into an eye or a handle by whleh the pieoe into which the other end la drlyen
m^ be hauled or towed away.

3onke]f-»ngine; a email iteam engine attaahed to a large one, and fed from the same boiler. It i4
■aed for pumping water into the boiler.

Double erank. See Crank.

DovaiaU; a Joint like aO. page 736 1 it Ib a poor one for timber when there ii mach strata,

being then apt to draw out more or less.

Dowti; a straight pin of wood or metal, inserted part way into each of two faces whioh It nnltes.

Draft; the depth to which a floating Tessel sinks in the water; in other words the water it draws^

Draught; a drawing. A narrow level stripe which a stonecutter first outs around the edges of %
rough stone, to guide him in dressing off the face thus enclosed by the draught.

Draw-plate; a plate of very hard steel, pieroed with small circular holes of different diameters,
through which in succession rods of Iron are drawn, and thus lengthened out into wire. Sometimes
ihe holes are drilled throogh diamond or ruby, Ac, Instead of steel.

Drift; a horizontal or inclined passage-way, or small tunnel, in mines, Ac. To float away with a
•nrrent. Trees, Ice, oarried along by fi-eshets.

Drip ; a small ehaanel eat under the lower prq}ecttng edge of coping, ke, so that rain when it
reaches that point will drip <nr fall off, instead of finding its way horisontally beneath to the wall,
which it would make damp.

i>rop ; short pieces of nearly complete cylinders, placed at small distances apart, in a row llk«
teeth, as an ornament to oornioas, Ac.

Drum; a reTolving cylinder around which ropes or belts either travel or are wound. When nar*
row and used with belts they are called pull^s.

Dry-rot; decay In such portions of the timber of honses, bridges, Ac. as are exposed to dampness^
especially in oonfined warm situations. The timber in cellars and basement stories is more liable to
it than in other parts, owing to the- greater dampness absorbed by the brickwork ftrom the ground.
CoD«^^«ot with lime or mortar hastens dry rot. The ends of girders, joists, Ac, resting on damp walls,
nay be partially protected by placing pieces of slate or sheet iron under them. The painting or tar>
"^g of untHuoned timber expedites internal dry rot. ▲ thoroogh soaking of timber in a eolntioii of
38 grains of quicklime to 1 gallon of water is said to be a prerenuTe of dry-rot ; bat the best proeees
for that purpose is sataration with ereoaote or carbolic acid.

Dyke ; mounds of earth, Ae, bollt to prevent overflow tnan rivers or the sea. A kind of geologleal
irregularity or disturbance, eonsistlng of a stratum of rock iAJeeted as it were by volcanic action, be^
tween or across strata of rooka of another kind. ▲ levee.

Beeentrie ; a circular plate or pulley, surrounded by a loose ring, and attached to a revolving
shaft, and moving around with it, but not having the same center ; for producing an alternate motion.
Often used instead of a crank, as they do not weaken the axle by requiring it to be bent. There art
many modifications.

Mtearpment ; a nearly vertical natural face of rock or solL

J!ic«<eA«on; the little outside movable plate that protects the keyhole of a lock ftom dust.

Ar«; a circular hole in a flat bar, Ac, for receiving a pin, or for other purposes.

Ifye and wtrap; a hinge common for outside shutters, Ac, one part consisting of an iron strap oim
end of which is forged into a pin at right angles to it; and the other part, of a spike with an eye,
through whioh the pin passes. When the eye is on the strap, and the pin on the spike, it is called ft
book and strap. Such hinges are sometimes called " baokflaps."

Mjfe'boU: a bolt which has an eye at one end.

ne*-waR; one built to sustain a face cot Into natural earth, in distinction to a retaining- wall,
which supports earth deposited behind it.

Fatt; the rope used with pulleys In hoisting.
' JUss-ieorJks ; the scaflbld. center, or other temporary supports for a strncture while it Is being
Irallt. In very swift streams it is sometimes necessary to sink cribs filled with stone, as a base for
telM-works to foot upon.

P^eine»; bundles of twigs and small branches, for forming foundations on soft ground.

•^sMfirue; of materials; the increase of weakness produced by frequent bending} or by sustaining
heavy loads for a long time.

Faucet; a short tube for emptying liquids fh>m a oask, Ac; the flow Is stopped by a spigot. The
wider end of a common cast-iron water or gas pipe.

Feather; a slightly projecting narrow rib lengthwise of a shaft, and which, catching into a corre-
sponding groove in anything that surrounds and slides along the shaft, will hold it fast at any required
part of the length of the feather. Has other applications.

Feather-edge; when one edge of a board, Ac, is thinner than the other.

Felloe^ or feUg; the circular rim of a wheel, into whioh the outer ends of the spokes fit; and which
la often surrounded by a Ure.

Felt : a kind of coarse fabric or cloth made of fibres of hair, wool, coarse paper, Ac, by pressure,
and not by weaving.

Fender; a piece for protecting one thing ftom being broken or injured by blows from another:
frequently vertical timbers along the outer faces of wharves, to prevent injury ftrom the rubbing of
vessels.

Fender-pHee; piles driven to ward off accidental floating bodies.

Ferrule ; a broad metallic ring or thimble put around anything to keep It ftrom splitting or breaking,
A small sleeve.

FtUet; a plain narrow flat moulding in a oomioe, Ac. See Platband.

Fleh; to Join two beams, Ac, by fastening other long pieces to their sides.

Flagt; broad flat stones for paring.

Flange ; a projecting ledge or rim.

FUuhinge; broad strips of sheet lead, copper, tin, Ac, with one edge inserted Into the Joints ef
brickwork or masonry an inch or two above a roof, Ac ; and projecting out several inches, so as to be
flattened down close to the roof, to prevent rain fk-om leaking through the Joint between the roof and
the brick chimney, Ac, which projects above it.

Ftaeka; upper and lower; the two parts of the box which contains the mould into which melted
Iron is poured for castings.

Flatting ; causing painting to have a dead or dull, instead of a glossy finish, by nslng tnrpentlae
Instead of oil in the last coat.

FUere; a straight flight of steps in a sUirway.

Fleodgau ; a gate to let off excess of water in floods, or at ether timee.

1030

OLOBSABY OF TERICS.

Flumsf k dltoh, trough, or other ohannd of moderate ■lie for coodaotlog water. The dltehea oc
eulverts throngh which •arplna water passes from an upper to a lower reach of a eanal.

FluMh ; forming an even continuous line or surface. To olean out a line of pipes, aewera, gattora
tu, by letting on a sudden rush of water. The splitting of the edges of stones nnder preaiiare.

Fhimu! Tarlons snbetanoes nsed to prevent the Instantaneous formation of rast when welding two
cieoes of hot metal together. Such rutft would cause a weak weld. Borax is nsed for wrought iron;
A mixture of borax and sal ammooiao for steel ; chloride of siuo for sine i sal ammoniae for copper
or brass ; tallow or resin for lead.
Fljf 'Wheel; a heavy revolving wheel for equalizing the motion of maehinery.
Ibumingf an undue amount of boiling, caused bj grease or dirt In a boiler.
FoUower; any cog-wheel that is driven by another; that other is the leader.
Foreepe; any tools for holding things, as by pincers, or pliers.

Forehay, or penetock; the reservoir from which the water passes immediately to a water- whoeL
Forge; to work wrought iron into shape by first softening it by heat, and then hanDmering it Inte
the required form.
Forge-hammer ; a heavy hammer for forging large pieces ; and worked by machinery.
Foiiaili a thin wedge inserted into a slit at the lower end of a pin, so that as the pin Is driven
down, the wedge enters it and causes it to swell, and bold more firmly.

Frame ; to put together pieces of timber or metal so as to form a truss, door, or other ■troetoie.
The thing so ftvraed.

FrietUm-roUere ; hard cylinders placed under a body, that it may be moved more readilj ttian I7
sliding.

Frletie^viheele ; wheels so placed that the Journals of a shaft may rest upon their rims, and thas
be enabled to revolve with diminished nriction.

Friexe; in architecture, the portion between the arohitraTe and oomioa. The term is oflen applied
when there is no architrave.
Fulcrum; the point about which a lever turns.

Furringe ; pieces placed upon others which are too low, merely to bring their upper iiirfaoeB op ts
ft required level ; as is often done with joists, when one or more are too low ; a kind of chock.

Fuxe, orfuee; to melt. A slow match, which, by burning for some time before the fire reaches the
fowder, gives the men engaged in blasting, time to get out of the way of flying fiiigmenta of atoce.
Oaeket; rope-yam or hemp, nsed for stufllng at the Joints of water-pipes, Ac.
Oearing; a train of cog-wheels. Now much rapplanted by belt*.

Oib; the piece of metal somewhat of this shape, L— J, often nsed in the same hole with a wedf»-
Shaped key for confining pieces together, tn common use for fastening the strap to the stob-end of
the connecting-rod of an engine.

Oin; a revolving vertical axis, nsnally fhrnlshed with a rope-drum, and having one or uMre long
arms or levers, by means of which it Is worked by horses walking in a circle around It. Used te
lolsting. Cotton-gin, a machine for separating cotton from its seeds.
Oirder: a beam larger than a common Joist, and nsed for a similar purpose.
dad*; in fortiflcadon, an easy slope of earth.
Okmd. See Stnfflng-box. Also, a kind of conpUng for shafts.

Glue; a cement for wood, prepared chiefly from the gelatine famished bj boiling the parings cf
bides. Oood glue will hold two pieces of wood together with a force of nrom 400 to 750 lbs per sq ia.
Oovemor; two balls so attached to an upright revolving axis as to fly outward by their oentrirB|iI
force, and thus regulate a valve.
Qrapnel; a kind of compound hook with several curved points, for finding things in deep water.
OriUage; a kind of network of timbers laid crossing each other at right angles; frequeptlj pIsMd
OD the heads of piles, for supporting piers of bridges, and other masonry.

Oroin; an arch formed by two segmental arches or vaults intersecting each other at right angles.
Also, a kind of pier built from the shore outward, to intercept shingle or gravel.

Groove; a small channel. A triangular one la ealMi

Aamfisred groove.

€hrout%d-ewell ; waves which eontinne after a storm has oeaaed ; or eansed by storms at a 4istaaea
Grout; thin mortar, to be poured into the interstices between stones or bricks.
Gudgeone; the metal Journals of a horisontal shaft, such as that of a water-wheel. Per moderaM
ipeeds _«_________»______

D lam, ins ( _ V' Weight in Iba un one gudgeon
if of cast-iron ) "^ IQ

Vor wiwnght-iroB, add one-iwenlMh.

Own metak or bronze ; a compound of copper and tin, sometimes nsed for cannon. Also, a qnalltj
ef cast iron fit for the same purpose.

Gueeete; plain triangular pieces of plate iron, riveted by their vertical and horizontal legs to tkt
sides, tops, and bottoms of box-girders, tubular bridges, to. inside, for strengthening their angles.
Guge ; ropes or chains used to prevent anything flrom swinging or moving about.
Gyrate ; to revolve around a central axis, or point.

Halving ; to notch together two timbers which cross each other, so deeply that the Joint thieknen
shall equal only that of one whole timber.

Hammer dreee ; to dress the face of a stone by slight blown of a hammer with a cutting edge. TM
paterU hammer for snch purposes has several such Mges placed parallel to each other, eaeh of whlek
may be removed and replaoed at pleasure.
Hand-lever; in an engine, a lever to be worked by hand instead of bv steam.
Handepihe ; a wooden lever for working a capstan or windlass ; or otner purposes.
Handwheel ; a wheel used instead of a spanner, wrench, winch, or lever of any kind, for serswlaf
nnts, or for raising weights, or for steering with a rudder, te.

Hangere, or pendent brackett ; fixtures prqfeeting below a eeillng, to support the Journals ef lost
lines of shafting ; and for other purpose. Should be " self-adlJastlng."
Haep ; a piece of metal with an apening fbr folding It over a staple.
Hatchway; a horizontal opening or doorway in a floor, or In the deok ef a veasel.
Haunehee; the parts of an arch from the keystone to the skewbaek.
Head-UocM ; a block on which a pillow-block rests.

Header; a stone or brick laid lengthwiRe at right angles to the faoe of the masonry.
Heading ; in tonnelling, a small driftway or passage excavated in advanoe ef the main body ef thi
tunnel, but formlns part of it; for fScllitatiog the work.
Headway ; the clear height overhead. Profrress.
Heel-poet ; that on which a look gate tnms on its pivet.
Helve t the hunrll** nf an axe.

GX'OSSABY OF TE^M^. 1031

Min§9 f thoie oommonly uted on th« doors of dwellings are called buttn, or butt binges. (Kye and
Bt*»9, .) JtUing hingei are such •■ oatue the dodr to rise a Utile as it Is opened, and thus cause

fne door to shut itself.

M^ roof, or hipped roof; one ttiat slopes four ways ; thus forming angles called hips.

Hoarding; a temporarj olose fenoe of board*, placed aroutid a work In progress, to ezolade
•tracglers.

Holding-platu, or anchor* ; strong broad plates of iron sunk into the ground, and generally snr-
r*anded by mtutoury ; for resisting the pull of the cables of suspension bridges ; and for other simi-
larpurposes.

Hook and atrap. See Eye and strap.

J9»r*ea; the stopiug tliiiberd wbidb carry the steps In a staircase.

HouHngt; iu roUiug mills, &o, the vertical supports for tbe boxes In which the journals rerolTe.

Hub, or natfe; tlie central part of a wheel, through which the axletree passes, and f^om which
«fae spokes radiate.

Impo»t} the upper part of a pier from which an arch springs.

Ingot ; a lump of caat metal, generally somewhat wedge-shaped. A pig of east iron is an ingot.

Invert; an inverted aroh frequently built nnder openings, in order to distribute the pressure more
evenly over the foandation.

Jadi; a raising instrument, oonsisting of an Iron rack, in connection with a short stout timber
which supports it, and worked by cog-wheels and a winch. A »crgte-jaek is a large screw working
in a strong frame, the base of which serves for it to stand on ; and which is caus^ to revolve and
rise, carrying the load on top of it, by turning a nut, or otherwise.

Jack-ra/t«r0, or oonunon rafters ; small rafters laid on the purlins of a roof, for supporting tha
shingling laths, Ac.

Jag-apike; a spike whose sides are Jagged or notched, with the mistaken idea that it* holding power
is thereby mvoh increased. If a spike or bolt Is first put into its place loosely, and then has mdtcd
Isad run around it, the jagging does assist; .but not when it is driven into wood.

Jamlt*; the sides of an opening through a wall, to ; as door, window, and fireplace Jambs.

Jdti^-lining* ; the facing of woodwork with which jambs are covered and hidden.

^w; an opening, often T-shaped, the Inner edges of which are for holding something in place.

Jtttie, OTJettg; a pier, mound, or mole projecting into the water; as a wharf-pier, Ao.

J0; the upper projecting member or arm of a crane, snpported by the stay.

Hg-aaw ; a very narrow thin saw worked Terdcally by machinery, and need for sawing curved
•maments in boards.

JoggU ; a Joint like that at 8 or 4, ftc, p T85, of TnuMs, for receiving the pressure of a strat at
right angles or nearly so. Also applied to slquared blocks of stone sometimes inserted between
•durses or masonry to prevent sliding, be.

'JoUt ; binding joists are girders for sustaining common Joists. The oommon ones are then called
bridging iointM. Ceiling joinu are small ones nnder roof trusses, or nnder girders, and for sustain*
lag merely the plastered ceiling.

Joumat-b&x; a fixture upon wbloh a Journal rests and reTolves, instead of a plnmmer-block.

JotamaU ; the cylindrical supporting ends of a horisontal revolving shaft. Their length is usually
about 1 to IH times their diam. In lines sf shafting 4 diams. To find the diam, see Gudgeon.
^Jumper; a drill used tor bdrtsg holes in stoiks by aid of blows of a sledge-hammer.

J^edae ; ammHtaohor. . ..... ,. -.

Xetmere ; the pieces of metal or wood which keep a sUding bolt in its plase, and guide it in sUoing.

fyfff tibe opening or narrow slit mide in sawing.

Key-bolt. See Cotter*bolt.

Keyatone ; the oentsr stone of an arch.

Kibble; the bucket used for raising earth, stone, «o, ftrom shafts or mines.

Xing-poat, king-rod ; the center post, vertical piece, or rod, in a truss ; all those on each side of it
■re queen-posU, or queen-rods. Frequently called simply kings and queens.

Knee ; a piece of metal or wood bent at an angle ; to serve as a bracket, or as a means of uniting
two surfaces which form with each other a similar angle.

Lagging, or aheeting ; a covering of loose plank ; as that placed upon centers, and supporting the
•nAstolies. Also, an buter wooden easing to locomotive boilers and others.

Landing; the resting-plaee at the end of a flight of stairs.

Lantern tokssl. See Trundle. . . . , ^t %^ ^a »k-» -r *i,. «*i..*

Lap : to place one piece upon another, with the edge of one reaching beyond that of 'he other.
Lap-welding: welding together pieces that have first been lapped ; in distinction to hu«'reldlng
Lead, (nrono.m'-ert /e«r» Mln steiTm-engines, a certain aaiouBt df opening of the port-valve befoie
gf^ itroke ef the piston begins. The distance to which earth is hauled or wheeled.
Jt^eader : a cog-wheel that ri«^f " motion to the next one or follower.
r«<MHno-6««m; i««Mi<n«7.piie; one placed as a guide for placing others.
jSJSSS-wAsSi; iua locomotive, those frequently placed in ffont of the driving-wheels.

i^^'*a plrt?rojeSSlrover 41ke a shelf; a rock so prqjectimr. A narrow strip of board nailed

•^7 other boards, to hold them together, as in temporary ledge-doors.

^SUi!; an arrsniement composed of 2 or 8 pieces of meUl let into a wedge-«haped hole in a block
•r st6ne, by wUirti to raise the blook. .k„»-

Liahtir ; a scow, raft, or other vessel, used for unloading vessels out trom the shore.

lAnehpin: a pin near the end of an axle, to hold the wheel on.

TAnk • onrn of the divisions of a chain : or a pieoe shaped like one.

^m^; a devtee for regulating the movement of the main or port valve in a steam-eD^lne.

iSfeT-Thorisontal beam wross an opening in a wall, as seen in windows, doors, 4c. When of
-m1 «!ln Md .nnnortlMheavv brickworVor masonry, it is called a breast-summer, or bressummer.
"^SdF^ thJL "mmon door^^^^ ^-oealed within the thickness of tbedoor^ ar.

wSd licSfw SSs ; thosVwhioh are screwed against the faoe of a door, rim locks. It must be remem.

^^'^V^SnTot'y':^^''A^oC, fluently at the top. of roofs of depots. *c. provided with hor

iMutal slats, which permit ventilation, and exclude rain.

TMMmnae: the shape of a rhomb: often called dlamond-snapea. .v «- «—

iS^^oiting. imaU projections f^om the general surface, and for :^"»«" P"T«ws. such as fol

UI«S the body ; or for a flange for joining it to another ; or for a support tor something else.
JHWM ; the wooden hammer used by stonecutters.

1032 QL08SART OP TERMS.

Mandra ; aa Iran rad OMd m • oora around whiflh • flat piMe may bt beat Into a oTllndri^ i**^
ilM> ihe shaft that oarriat the ohtiok of a lathe. ^ ojnnwnMX aupa

jran*ol« ; an opealof bj which a mao can ent«r a boUMr, oalvwt, *e. to olaaa or repair It.
Mattotk: a klod of plok with broad odgei for digging. *^

Maul ; » heary woodon hammor.
Jftan, oWtJtfiMMeal; half th« sain of two nnmtwra.

" , gtomttrvnU; the aq rt of iho produot of two nrnttben.
Mtan-proportional ; the aame aa the Momatrloal meaa.
Meridian ; a north and south lina. noon.

Mitn-ioint: a joint formod along tho diagonal lino wbora the enda of two pieeea nra nnitadatat
ancle with each other.
MRtf-»a; the ■111 agalnat whieh the look gatea of a oaaal shnt.
Modtdu* : a datum aerrlng aa a moaaa of oomporiaon. Same aa etmKtant or co^gkcUiU.

Moment I tendenej of foroe aetiag with leverage.

Moment o/nip(iir«, or of bending i the tendency whieh anj load or forae exerta to break or bend a
body by the aid of leverage. Ita amonnt is found In foot-pounds by moUiplying tho foroo iu lbs, In
the length of leverage in feet between It and that part of the body upon whioh the tendency Inemrtal

JfonJkey ; the hammer or ram of a pUe-driver.

Monkeji-wreneh, or screw-wrendk; a spanner, the gripping end of whieh eaa be adjoated by means
of a serew to fit ejects of different sixes.

Mooring*; fixtures to which ships, ko, can make fast.

Mortiio; a hole out in one piece, for reoeiving the tenon whieh prctjeota fh>m another pieoe.

Muck : soft snrfaoe soil oonuinlng much vegetable matter.

Muntint, or mutUont : the vertical pieoes whioh separate ihe panes in a wlndow-saeh.

Jfailing-bloeka ; blocks of wood inserted in walls of stone or brick, for nailing washboarde, fte, te.

Naot; the main body of a building, having connecting wings or aisles on eaoh side of it. The hab
«r a wheel.

Newel ; the open space surrounded by a stairway.

Newelrpoet ; a vertical post sometimea used for sustaining the outer enda of stepe. Alao the laqi
baluster often placed at the foot of a stairway.

JHpper* ; pincers. An arrangement of two curved arms for oatehing hold of anything.

Xbrmal; perpendicular to. Aooordlng to rule, or to oorreet prlndptoa.

Noeing ; tiie slight projection ofttn given to the fk«nt edge of the tread ot a atep ; naoally ronndai.

mu, or biwrr; the short pieoe w^th a oentral female aeraw, need on the end of a aerew-bolt. Jte, ftv
keeping it in place.

Ogee ; a moulding in shape oT an S, the same aa a olma.

OrdifMte / a line drawn at right anglea f^om the asda of a onrre, and extending to the eurve.

Oteittate; to swing baokwnrd and forward like a pendulum.

Out ofvindt pronounced iiynd ; perfootlv straight or flat.

Oeolo ; a projfoting oon'«x moulding of quarter of a eircle ; when it iAeonoave it la a eavetle, tf
hollow.

J'ueHng ; tho material placed in a atnffing-box, *e, to prevent leaka.

Paddngpieeee ; short pieces inserted between two others whieh are to be riveted or bolted togathg
to prevent their coming in contact with each other.

PmU, or pawl. Sv9 Batehet.

Parapet; a wall or any kind of Ibnce or railing to prevent persons from fklllng off.

IVireel ; to wra^ canvas or rags round a rope.

Parge ; to make the inside of a flue smooth by plastering it.

Patent kammer ; a hammer with several parallel aharp edaee for dreasiog stone.

Pay. To cover a surface with tar, pitch, ko. A ship word.

Pag out. To slacken, or let out rope.

Pediment ; the triangular space in the face of a wall that is inelnded between the two alopinc aidM
of the roof «nd a line Joining the eaves.

Penetoch. See Forebay.

Pier : the support of two a^aeent arobee. The wall apaoe between windows, ko. A struotare built
ant into the water.

Pierre-perdue; lost stone; roruiom eUme, or rough stones thrown into the water, and let flnd Asir
own plope.

Pitaeter; a thin flat projection from the face of a wall, as a kind of ornamental substitute far a
oo\nmn.

Pile-plaiike ; planks driven like pllea.

PiUow-Uoek, or plummer-hlock ; a kind of metal chair or support, upon whioh the Joumals of bar-
if ontal shafts are generally made to rent, and on which they revolve.

Pinien; a small cog-wheel which gives motion to a larger one.

PUMe ; a vertical pnifectlng pin like that often placed at the tops of orane-poato, and orar
the holding rings at the tops of the wooden guys flt. Also, such aa is aaed for the kiiifaa ofi—
sr of wlndow-ehutters to turn around.

GLOeeART OF TEBHS.

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ffap-bott- Sfa Jif-aplba.

1034

GLOSSARY OF TERMS.

Bt-tntering augU ; an angle or corner prvjectlDg inward. See Salient, below.

M4VMtm€nt I tteep facing of itone (o the videe of a ditob or parapet in fortifloatioD. A retainiag-walL

Rib i tbe curved pieoea which form the arches of iron or wooden bridges, Ac. AUo, cboae to wbiflfe
the outer planking of a ■ailing vee§el, Ac, are faatened.

Eidn» of a roof ; its peak, or the sharp edge along its very top. Has various similar appUeations.

Sidg»-pol€, ridge-jriece, or ridge-plate ; ibe highest horisontiU timber in a rooT, ezteoding from top
to top of the several pairs of rafters of the irusses : for supporting tbe beads of tbe jaclt-rarten.

Sight and left ; a lock which in ibi proper position suits one flap of a pair of folding daors, win
not suit if fastened to the other flap ; nor even to the same (lap if nHiulred to open tA th« right in-
stead of to the left, or vice versa, according to whether it is a right or a left-band look. And so with
many otuer things, as, for instance, certain arrangementai for working railway swiicbaa, Ao. Bight
ind left boots and shoes are a familiar illostration ; also, right and left sorews. TiJerefore, in ordsr-
Ing several of anything, it is necessa^ to consider whether they may ail be of tbe same pattwn, «r
whether some must be r1ght*hand, and others left-band ones.

Right shore of a river ; that which is on tbe right hand when desoending tbe river.

Right-eolid body ; one which has iu axis at right angles to its base ; when nui mo, it ia ohtigmt.

Ring-bolt ; a bolt with an eye and a ring at cue end.

Rip-rap. See Random stone.

Roadttead ; anchorage at some distance from shore.

Roeh-ehaft ; a shaft which only rooks or makes part of a revolntion aaob way, instead of revolviig
entirely around.

Rockwork; squared masonry in which the face is left rough to give a nutio appearance.

Rubble ; masonry of rough, undressed stones. Seabbled rubble has only tbe roa^eat IrregvlaxidBi
knocked oif by a hammer. Ranged rabble baa the itonea tn eaeb oonrw rudely dreased to nea>^ s
■niform height.

Rttndle, or round ; the step of a ladder.

Rustic ; much tbe same as rockwork.

Saddle; the rollers and fixtures on top of tbe piers of a. suspension bridge, to aooommodate fx-
pausiori aod contraction of tbe cables. Tbe top pleee of a stone oornlee of a pediment. Has muf
ether applications.

Sag ; to bend downward.

Salient ; projecting outward. See Re-entering, above.

Sandbag ; a bag filled with sand for stopping leaks.

Seabble . to dress off the rougher projections of stones for rubble maaonry, with a stosie-axe, «
wabbling hammer.

Scantling : the depth and breadth of pieces of timber; tbus we say, a aoantlmg of 8 by 10 ins, Aa

Scarf; tbe uniting of two pieces by a long Joint, aided by Imlta, ko.

Scarp t % steep slope, fn fortification tbe inner slope of a ditch.

Scotia ; a receding moulding consisting of a semi-oircle or semi-ellipae, or similar flgare.

Sereedt : long narrow strips of plaster put oo horizontally along a wall, and earefuUj fisoed oat if
wind, to serve as guides for afterwnrd plsiiteriug tbe wide intervals between them.

Screw-boft, a bolt with a screw cut on one end of it.

Screw-jack. See Jack.

Serew-wreneh. See wrench.

Merits ; to trim ofT tbe edge of a board, *e, so as *• make It lit elossly at all potau. to aa Imgiltf
Burfaee. Tbe lower edge* of an open caisson are scribed to tit the in«galarities of a rooky rirer bottoa*

Scroll; an ornamental form consisting of volntes or spiials arranasd somewhat in tbe abape of &

Scupper naile ; nalU with broad heads for nailing down canvas, «o.

Seuppert ; on shipboard, holes for allowing water to flow off from the deck into the sea.

Scuttle ; a small hatchway. To make holes in a vessel to cause sinking.

Sea-waU; a wall built to prevent encroachment of tbe aea.

Secret nailing ; so nailing down a floor by nails along the edges of tlie boards, that tbe natl-h«sdl
do not sbow.

Smnte ; to wrap twine or yam, Ao, closely ronnd a rope to keep It t^m rnbbiog.

Set-eerew, or Hghtening-eereui ; a sorew fbr merely pressing one thing tightly against another it
will ; such as that which confines tbe movable leg of a pair of dividers in its socket.

Shackle, or elevie; a link in a obaia shaped like a U, and so arranged that by drawing oat a Ml
or pin, which fits into two holes at the ends of the U, tbe chain can be separated at that point.

Shaft ; a vertical pit like a well. The body of a column. A. Urge axle.

Shank ; the body of a bolt enloslTe of itt head. Tbe long straight part of many things, as of M
anchor, a key, Ac.

Sheara, or sJteers; two tall timbers or poles, with their feet some distanoe apart, and their tspi
fastened together; and supporting hoisting tackle.

Sheave ; a wheel or roand block with a groore around its dronrafbrenee for guiding a rope.

Sheeting, or theathing; covering a surface with boards, sheet iron, felt, Ac.

Shingle f the pebbles on a seashore.

Shoe* ; certain fittings at the ends of pieces ; as tbe pointed iron shoes for piles.* The wall tk**
Into which the lower ends of iron rafters generally fit, Ae.

Shore ; a prop.

Shot ; the edge of a board is said to be shot when it is planed perfeotly straight.

Shrink. When an Iron hoop or band is first heated, and then at onoe placed apon tbe body wbish
It is intended to surround, it shrinks or contracts as it cools, and therefore clasps tbe body more flraif.
This is called ehrinking on the hoop.

Shuttle ; a small gate for admitting water to a water-wheel, or ont of a canal look, Ac.

Siding; a short piece of railroad track, parallel to the main one, to serve as a passing-place.

SKlt ; soft fine mud deposited bv rivers, Ac.

^hon culvert; a cnlrert built In sha|ie of a U, fbr earrylng a stream under an obsucle. and allev>
ing it afterward to rise again to its natural level. The term la improper, inssmooh as tbe prindpis
•f tbe siphon is not involved. ,

Skewback ; the inclined stone f^om which an arch springs.

Skida ; vertical fendera. on a ship's sides. Two parallel timbers fbr rolling things

Skirting} narrow boards nailed along a wall, as the washboards in dwellings.

Sledge; a heavy hammer.

Sleeper; any lower or foundation piece in onotact with tbe ground.

Jlseve ; a hollow oyllnder slid over two pieces to hold them together.

QLOSaARY OF T£BM8. )035

8Ud»-har», or »Ude$t bars for aojtbios to slide aioog; u those for tb« cross-beads of piston-rods,
te. Often called giddes.

SUngt; pieces of rope or obidn to be put aronnd stonea. Ac, for raising tbem by.

S'Up;'%h» sliding dowu of the sides of eartb-cnts or banks. A long narrow water space or dock
between two wharf- piers.

Slop»-wiM; a wall, generally thin and of rubble stone, used to preserve, slopes from the action of
water in the banks of canals, rivers, reservoirs, Ac ; or f^om the action of rain.

Slot; a long narrow bole cut through anything.

Sluice: a water-channel of wood, masonry, Jcc; or a mere trench. The flow is usually regulated
by a sluice-gate.

9mok»-hox; in locomotives, that space in front of the boiler, through which the smoke passes to
the chimney.

Snag ; a lug with a bole tbrough it, for a bolt.

Socket; a cavity made in oue piece for receiving a prqjectlon from, or the end of, another ptaoe ; as
that into which the movable leg of a pair of dividers fits.

SiiffiX ; the lower or underneath surface of an arch, cornice, window, or door-opening, ko.

Solder ; a compound of different metals, which when melted is used for uniting pieces of metal also
heated. S^ft solder is a compound of lead and Un, and is used for uniting lead or tin. Tbere are
▼arions hard solders, aiicb as spelter solder, composed of copper and sine, for uniting iron, copper, or
brass.

SoU ; that linlDg around a water*wheel which forms the bottoms of the bnckets.

Spandrel ; tbe space, or the masonry, Ac, between the back or extrados of an arch and the roadway.

J^anner ; a kind of wrench, oonaisOng of a handle or lever wiib a square eye at one end of it ; much
•sed for tightening up tbe nuts upon screw-bolts, fto. Tbe ere fits over or surrounds the nut.

Sdot; a beam; but generally applied to round pieces like mat^ts, &c.

Spelter; zinc.

Spigot ; tbe pin or stopper of a faucet. The smaller end of a common oast-iron water or gas pipe.

apindle ; a thin delicate shaft or axle.

tplag ; to widen or flare, like tbe Jambs of a common fireplaoe, or those of many windows ; or like
the wing- walls of most onlverts.

Spltce; to unite two pieces firmly together.

Aringer: tbe lowest stone of an arch.

Sproelcttwheel, or rag-ieheet ; one with teeth or pins which catch In the links of a chain.

£bur-wheel;^% common cog-wheel. In which tbe teeth radiate flrom aoommon cen, like those of a spur.

Square; in roofing ; lOO square feet.

Square-head; a square termination like that npon which a watch-key fits tbr winding ; or thai
«pon which the eye M tbe handle of a common grindstone fits for turning it. &c.

Staging ; tbe temporary flooring of a soaflbld, platform, Ac

^lUincAton ; a vertical prop or strut.

Standing-hoU, or etud-holt: a bolt with a screw cut upon each end : one end to be screwed perma>
nently into something, and tbe other end to hold by means of a nut something else that may be >•>
onired to be removed at times.

Staple ; a kind of double pin In shape of a XT ; its two sharp points are driven into Umber, aad
enrved part is left prqjectlng. to receive a hoop, pin. or hasp, ftc.

Starltnge ; tbe projecting up and down-stream ends or cutwaters of a bridge pier.

Stay; variously applied to props, struts, and ties, for staying anything or keeping it in place.

Stay-bolte ; long bolts pisced across tbe inside of a boiler, Ac, to give It greater strength.

Steam-chest; the iron box in locomotive engines and others, through which tbe steam is admitted
to the cylinders.

Steatn-pipe ; the one which leads steam ftrom a boiler to the steam-chest.

Step ; a cavity in a piece for receiving tbe pivot of an upright Rbaft ; or the end of any upright piece.

i8m«s ; the flat vertical plecer between and at the sides of the panels in doors, ftc.

Stock ; the eye with handles for turning it, in which the dies for the cutting of screws are held.

Stove-up, or etoved. or «;>««( ; when a rod of iron is heated at one end, and then hammered end-
wise so that that part becomes of greater diameter or stouter than tbe remainder. The heads of bolts
are frequently made in one piece with tbe shank In this way ; and tbe screw ends of long screw-rods
are often upset, so that the cutting of tbe threads of the screw may not reduce tbe strength of the bar.

Strap ; a long thin narrow piece of metal bolted to two bodies to hold tbem tog<>ther. A strap-
hinge is a strap fastened to a shutter, Ac, and having an eye or a pin at one end for fitting it to the
•Cber part of the hinge which is attached to tbe wall.

J^atum ; a layer, or bed ; as tbe natural ones in rooks, Ae.

Stretcher ; a brick, or a block of masonry laid tengthwiee of a wall. A f^ame for stretching any
thing npon.

AretcAer-eoiirse; a coarse of masonry all of stretAiers, without any headers.

Strike ; an imaginary horisontal line drawn npon the inclined face of a stratum of rocks. Thus,
If the slates or shingles on a roof represent inclined strata of rocks, then either the ridge or the eaves
•f the roof, or any horizontal line between them, will represent their strike. Tbe inclination is
tailed tbe dip of tbe strata; and the strike is always at right angles to it by compass.

luring ; variously applied to longitudinal pieces.

String-board; tbe boarding (often ornamented) at the outer ends of steps in staircases. It hides
the horuee, as tbe inclined timbers which carry the steps are called.

^ring-oowne; a long horisontal eonrse of brick or masonry projecting a little beyond the others;
■Bd often introduced for ornament.

Skringeri any longitodinal timber or beam, Am,

1036

OL06SART OF TERMS.

Strut ; % prop. A pleoe that nutalni oompiVMloo, whether vertloBl or InoUnaA.

StnU'H; or ti«-»trut ; » piece ■4Upied to •attain both teniion and oompresaion*

JKuA-end; a bluDt end.

Stud; a ahort itout projecting pin. ▲ prop. The Tcrtleal pieoei in a stud partltloiL.

Stud-bolt. See Standing-bolt.

Stujfbig-box ; a tmall boxing on the end of a eteam cylinder, and sarroanding the piston-rod liki
a oollar ; or in other poaitions wbere a rod li required to moTC baokward and forward, or to reToIre,
in an opening through any kind of partition, without allowing the eaoape of eteam, air, or water, ka,
as the case may be. The box la filled with greaaed hemp or other packing, which is kept pressed eloac
■aronnd the moving rod by means of a top-pieoe or kind of cover called the glamd, which may be
aorewed down more or leas tightly upon it at pleasure. The rod pasaea throngh the gland also.

Jbayt, or $uHip; a draining well into whleh rain or other water may be led by Uttia ditohaa tnm
dilhrent parta of a work to which it would do iujnry.

JFurftoae ; the inside horizontal mouldlnga Juat under a window-silL Also those around tfaa top of i
pedestal, or of wainacocing, 4c.

Stnge, or awtdgt; a kind of hammer, on the faoe of which is a semi-cylindrical, or other shaped
groove or indenution ; and which, being held upon a piece of hot iron and struck by a hearj h^wimTt
leaves the ahape of the indentation upon the iron.

SwUch ; the movable tongue or rail by which a train is directed f^m one track to another.

Swiv^ ; devices for permitting one pieoe to turn readily in various directions upon another, witfe
out danger of entanglement or separation.

SyfiMnal azU ; in geology, a valley axla, or one toward which the strata of rooks slope downward
ftom opposite directions. The line of the gutter in a ralley roof may represent such an axis.

r« ; pieces of metal in that shape, whether to serve as straps, or for other purposes. So also wiA
L'a. 8*a, Wa, +'a. 4c

TmekU; a combination of ropea and pulley i.

Talua ; the aame aa batter.

Tamp ; to fill up with aand or earth, 4o, the remainder of the hole in which the powder haa hbea
poured for blaating rook. To oompaot earth generally, aa under oross-ties, ftc.

Tap ; a kind of screw made of hard steel, and having a square head which may be grasped by a
wrench for turning it around, and thna forcing it through a hole around the inaide of whioli itcnta aa
interior acrew. To atrike with moderate force. To make an opening in the side of any Teasel.

TbfwsC; a pin or short arm prctfeoting from a revolving abaft ; or from an alternating bar, and in*
tended to come into contact with, or Up, something at each revolution or stroke.

Teeth ; or ooga of wheels.

Temper ; to change the hardneaa of metala by first heating, and then plunging them into water, ol^
4o. To mix mortar, or to prepare eiay for bricks, 4o.

Ten^let: the outline of a moulding or other article, cut out of Sheet metal or thin wood, tosem
aa a pattern for atoneoutters, earpentera. 4o.

Tenon ; a proJeoUng tongue fitting into a oorreaponding cavity called a mortise.

Terra eotta ; baked clay. Brick is a coarae kind.

TMmble : an iron ring with ita outer faoe curved Into a continnona groove. A rope being doubled
around this and tied, the thimble acta aa an eye for It, and preventa that part of the rope trom ««ar>
ing. Alao, a abort piece of tube alid over another piece, or over a rod, 4o, to strengtiien a Joint, 4e.

Thretid ; the continuous spiral projection or worm of a acrew.

Threugh-etone ; a atone that extenda entirely through a wall.

Throw; the radiua, or diatanoe to which a crank " throwa out" Its arm. Applies in the same way
to lathea. Some use it to express the diameter instead of the radiua. To avoid mistakes, the tenia
** single " and " double " throw might be used.

n* ; any piece that sustains tension or pull.

ns-atntt ; a pieoe adapted to auatain either tcnalon or oompresaloa.

Tlghtnioff-eerew. See Set- acrew.

3Vr«; the iron ring placed around the outer circumference of the felloe of a wheel.

Tongue; a long alightly projecting atrip to be inserted into a corresponding groove, aa in toocued
and grooved floors.

Tooling ; dressinff stone by means of a tool and mallet; the tool being a chisel with a eutttng edge
•f I to 2 Inches wide. Tooling is generally done in parallel stripes serosa the stone.

Torua; a projecting semi-oirenlar, or aemi-elliptic moulding; often naed in the bases of eolamafc
It la the reverse of a sootla.

DraiUng-wheel* ; in a locomotive, those aometimea placed behind the driving-wheels.

Train; a number of cog-wheels working into each other.

\

GLOSSARY OP TERMS. 1037

Traruom ; a beam across the opening for a door. fte. Also, a borlEontal piece dividiug a high
window into two stories, to, Ae. Also, an opening above a door, for veDtilatlon or light.

Tread ; the horizontal part of a step.

Treadle} a kind of foot- lever, tor taming a lathe, grindstone, ko, \jj the foot.

Treenail; along wooden pin.

Trimmer; a short oross-timber framed into two Joists so as to sastain the ends of intermediate
lolsts, to prevent the latter from entering a chlmnej-Sue, or Interfering with a window, &o.

Trip-hammer, or tiU-hammer; a large hammer worked by oamb machinery, and nsed for heavy
Iroa work, especially for hammering irregolar masses into the shape of bars, Ac.

lYuek; a kind of small wagon consisting of a platform on two or more low wheels. Also, those
frames and wheels usually placed ander railroad oars and engines, and which, bv means of a pintle
•onneoting the two, allow them to vibrate or move laterally to some extent independently of each other.

Tr%iindle, lantern-wheel, or weMotoer ; nsed instead of a oog-wheel, and consisting of two parallel
otroalar pieoes some distanoe apart, and united by a central axis, and by cylindrical rods placed
aronnd and parallel to the axis, to serve instead of oogs or teetlu

Trunk; a long wooden boxing forming a water channel.

Trunniona ; cylindrical projections, as at the sides of a cannon, forming as It were an interrnpted
•xle or shaft for supporting the cannon on its carriage ; and allowing it to revolve vertically through
some distanoe.

Tumbler; a kind of spring catch, which at the proper moment falls or tumbles into a notch or
tMie ivepared for it In a piece ; thus holding tlie pieoe in poeiUon until the tumbler ia lifted oat ef tJM
notch.

TumhXinghay ; see *' waste- weir.**

TwaMing-ehaft i in looomotives, a shaft used in (he " link motion.**

TurmtaUe ; the well-known arrangement for taming looom«tlTM at reat.

XTndermine ; to excavate beneath anything.

Underpin; to add to the height of a wall already oonstrnoted, by excavating and building beneath
It. Also, to introduce additional support of any kind beneath anything already completed.

^s«t. See Stove-up.

Yalvee; various devices for permitting or stopping at pleasure the flow of water, steam, gas, Ac
A SAFBTT VALvs Is ouc SO balanced as to open of itself when the pressure becomes too great for
safety. A sudk valv* is one that slides backward and forward over the opening through which the
Abw takes place. A ball va.lv>, or spherical valve, is a sphere, which in any position fits the open,
ing. When the pressure below it raises It off from Its seat, it Is prevented ftrom rolling away by
^jwms of a kind of open caging which surrounds it. A oowical or rurrvt valvs is a horisontal slice
flff 'a cone, which fits into a corresponding conical seat made in the opening. In rising and falling It
la kept in position by a vertical valve-stem or spindle, whieh passes through its oenter, and whloh
plays through guide- holes in bridge-pieces placed above and below the valve. A trap, clack, vlap.
or DOCK valvb, is a plate with hinges like a door. When two such valves are nsed, with their hinged
•dges adjacent to each other, so that In opening and shutting they flap like the wings of a butterfly,
tfi^ eonstltute a butterfly valve. A thbottlb valvs Is one whloh when closed forms a partition
aeroas a pipe : and opens by partially revolving upon an axis placed along Its diameter. A botaht
TALTs works like a common stopoodk. A sntVTiiro valvb is one which lets out steam under water ; and
la so called from the snifting noise thereby produced. The post valvb Is the sliding one which ad-
■Its steam fh>m the steam-chest into the cylinders. A doublb sbat, or doitblb-bbat valvb is a pe-
enliar one with two seats, one above the other ; and so arranged that the pressure of steam or water
against it when shut, does not oppose its being opisned. A citp valvb is in shape of an inverted
•ylindrioal cup, with a length somewhat greater than its diameter. Its lower or open edge is ground
to fit the seat over whieh it rests. As this cup rises and falls, it is kept in place by a oyllndrioal
•aging closed at top, and having for Its sides four or more vertical pieoes, against the Inner sides of
whloh the sides of the eup play. A chbck valvb is any kind so placed as to check or prevent the
return of the fluid after its passage through the valve into the pipe or vessel beyond it.

Voitli; an arch long in comparison with Its span. The spaoe covered by such an arch.

Feneer; a very thin sheet of ornamental woml glued over a more common variety.

Wminecot : a wooden fkelng to walls In rooms, Instead of plaster, or over a fkoing of plaster ; usually
not more than 3 or 4 feet high above the floor.

WaUe ; long longitudinal timbers in the sides of a shlp,eoflbr-dam, oaisson, ke.

WaXUne; a water-wheel, Ao, la said to wallow when it does not revolve evenly on its Journals.

WaU9uer. See Trandle,

Watl-l^ate, or raieing-plate; a timber laid along the tops of walls for the roof trusses or rafters to
raat on, so as to distribute their weight more equuly upon the wall.

Warped; twisted, as a board, or the face of a stone, fto, which is not perftoUy flat. To warp; to
haul a vessel ahead by means of an anchor dropped some ^stanoe ahead. To flood an- extent of
ground with water for a short time to increase its fertility.

Waekhoarde ; boards nailed around the walls of rooms at the flooc so as to prevent injury to tlM
plaster when washing the floors.

ITosAert ; broad pieoes of metal snrronndlng a bolt and placed between the faces of the timber
through which the bolt passes, and the head and nut of the bolt, so as to distribute the pressure over
a larger surface, and prevent the timber from being crushed when the bolt Is tightly screwed up.

Waete-ueir; an overfall provided along a oanal, ke, at which the water may discharge itself in
ise of beooming too high by rain, ko. Sometimes oalled a tumbling-bay.

WiiOek-taekU ; ropea rannlng in dilbrent direetlons from a boat, and oaad la briagini it into a
'poaiUon.

1038

GLOBSABY OF TBR103.

WMmr-»ked: the •loping croond from wtaloh mln-water deMend* loto a stream.

W!aur-ttM9; a ■licht prq|«ollon of (bt lowor maionrj or brickwork on the outside of a wall, uA
feaohlng to a few feet above the groand rarfaoe, aa a partial proteetioo agalnit rain, or aa omameat

Waijf ; the Inellned timbers along whlob a resMl glides when being launched.

irs«<A«r>ftoar(ifl ; boards used Instead of brleks or masonrj for the oatsldes of a bolldinc, or bridga
ko. Thej are nailed to vertloal and Inclined Indoor timbers ; and maj be either TertTeal or her.
Whflo hor, ther are so plaeed that the lower edge of one oTorlaps the opper edge of the one beloT.
When Tert, thdr edges should be tongued and grooTod ; and narrow slips be nailed orer the Tert joints,
to keep oat rain, *e.

WMr, or tHer; a dam. or an OTerfall.

WUd; to join two pieoes of metal toaether by first softening them by beat, and then hammeriaf
them in oontaet with eaoh other. In this operation fluxes are need.

ITett; see " bntfe-Joint."

Whmrt; a lorel spaoe apon whieh veesels lying along its sides can diseharge their cargoee ; or tarn
hioh tMy 01

whioh taw oaB reeeiTe i

ir^st-oigus ; the distance ftrom oentar to center f^m the extreme ftront wheels, to the extreme lilal
ones In a looomotive, oar, Ac.

Wicket ; a small door or gate made in a larger one ; as the shottle or Talve in a lock-gate, for lettfag
•at the water.

Winch ; a handle bent at right angles, and need for turning an axis ; that of a common grlndstoM.

mud. See Out of wind.

Finders; thoee stsps (often triangular) in a staircase by which we wind, or turn angles.

WhtdUau ; the wheel and axle, or winch and drum, as often used In common wells. Alao. a hoii
MBtal shaft on shipboard, by whioh the anchor la raised ; the windlass being reyolred by m«>»w of
wooden levers called handtpikct.

Wing-dam I a prcijectlan carried out part way tcross a shallow stream, so as to force all the watv
to tlow deeper through the channel thus contracted.

Winaa ; applied in many ways to prqjeotions. The flanges which radiate out f^m a gpdceoa : awi
by which It Is fastened to the shaft. Small bnlldings pr«||ectiag fh>m a main one. The wfags «r
flaring wing- walls of a culvert or bridge.

Wing-waiU; the retaining- walls which flare out ftom the ends of bridges, eulveru, Ac.

Wiper. See Camb.

Worktng-b0am, or waUdng-heam; a beam vibrating vertically on a rock-shaft at its center, as seeb
in some steam-engines ; one end of it having a connection with the piston-rod ; and the other end with
a crank, or with a pump-rod, Ac.

Form ; the so-ealled endless screw, whioh by revolving without advanoing gires motion to a eog>
wheel (worm-wheel), the teeth of which catch in the thread of the screw.

FrsneA ; a long handle having at one end an eye or jaw which may catch hold of anything to be
twisted or turned around, as a screw-nut, Ac. when it has a jaw which by means of a screw is
adaptable to nuts, Ac, of different slses, it is a monkey-wrench, or screw-wrench.

INDEX.

The nambers refer to the iMiflfes. In the aJphabetical arrangemeBt,
minor words, as "and," "betweea," "in," "on," "through," etc., are
nearlected. See also Glossary, pp. 1025, etc., and Table of Contents, pp.
XXV, fete.

Abaeos—AmortiBatloii.

A.

Abacus, defined, 1025.
Abrasion

of cements, 937.

by streams, 577, 582.
Abscissa, defined, 1025.
Absorbent bodies,

specific gravity of — , 211.
Absorbents for nitro-glycerine, 949,

951.
Absorption

by bricks, 927.

by earth, etc., 329.
Abutment, Abutments,

of arch, 617.

batter. 619.

courses in — , inclination of — ,620.

of dams, 645.

foundations for — , 582.

masonry, 623.

piers, 619.

to proportion — , 617, 618.
Accelerated tests for cements, 941.
Acceleration, 334.

of gravity, 335, 336, 348, 350, 539.
ejjuivalents of — , 250.

on inclined planes, 349.

units of — , conversion of — , 250.
Acid fumes, effect of — on roofs, 970.
Acre, Acres,

area of—, 222.

equivalents of — , 233.

-foot, equivalents of — , 235.

required for railroads, 254.
Action,

line of—, 359.

and reaction, 333.
Addition of fractions, 36.
Adhesion

of cement, 934.

of glue, 922. 1030.

of locomotives, 413, 860.

of mortar, 926.

of nails and spikes, 818.
Adit, defined. 1025.
Adjustable counters, 721.

Adjustment. See the several in'

struments.
Adjutages, flow through — , 540.
Admiralty knot, 220.
Age, effect of — on cements, 932.
Agonic line, 301.
Air, 320.

buoyancy, 513.
chambers, 663.
compressed — . 320, 597, 681.
breathing — , 320, 597, etc.
in diving bells, 321.
in foundations, 596.
in rock drills, 681.
compressors, 681.

manufacturers, 991.
density, 320.
locks, 597.*
pressure, 320, 502.

barometer, levelling by — , 312.
of compressed — , 320, 597, etc.
on water surface, 502.
in siphons, 521.
slacking, 925.
in tunnels, 812.
valves, 662.
ventilation, quantity required

for—, 320.
vessel, 663.
volumes of unit weights of — ,

conversion of — , 242.
weight, 320.
weights of unit volumes of — ,

conversion of — , 242.
wind, 321.
Alcohol, weight of — , 212.
Alieth. 285.
Allardyce process, 955.
Alligation, 40.
Alphabet, Greek — , 34.
Alternating stresses, 465. 761.
Alternation of ratios, 38.
Altitude. See Height.

of the pole, 284.
Alumina, in cement, 930.
Aluminum

oxide, in cement, 930.
weight, 212.
Amortization, 43.

1039

1040

INDEX.

Amount, in interest, etc., 41.
Anchorage, Ancnorages.
of suspension bridges, 770.
wind — in bridges, 759.
Angle, Angles, 92.

arc, angle subtended by — , 181.

780, etc.
blocks. 736.
in built sections, 723.
chords subtending — , 143, 780, etc.
circular measure of — , 34.
complement and supplement, 94.
co-secants of — , 97.
cosines of — , 97.
co-versed sines of — , 97.
deflection—, 780, 784-789, 840.
defl^rees in — , decimals of — , 95.
of direction, 765.
of friction, 409.

in arches, 432.

in dams, 433.
frog—, 835, 839.
hour—, 285.
iron, 497, 896-899. 912.
of maximum pressure, 607.
to measure

with the hand, etc., 96.

with the sextant, 152.

with the tape line, 152.

with the two-foot rule, etc., 96.
minutes and seconds in decimals

of a degree, 95.
plates for rail joints, 820-823.
in polygons, 148.
rule, 2 ft. — , to measure — by, 96.
secant of — , 97.
seconds in decimals of a degree,

95.
mnes of — , 97.

table, 98.
of slope, 255-257.
on sloping groimd, 151.
steel, 896. 898.

test. 753.
subtended by arc, 181, 780, etc.
supplement and complement, 94.
switch—, 827.
symbol for — , 33.
tangent of — , 97.
tangential — , tables, 784-786.
in triangles, 148.
versed sines of — , 97.
An^lar velocity. 351.
Animal power, 685.
Anneal, defined, 1025.
Annual

earnings of railroads, 867. etc.
expenses of railroads, 867, etc.
magnetic variations, 301.
Annuity, Annuities, 43.
equations, 44.

required to redeem $1000, 46.
Antecedent, 38.
Anthracite,

heat from — ^ 317.

space occupied by — , 215.

weight, 212, 215.

Anti-bursting device. 665.
Anticlinal axis, 1025.
Anti-component, 362.
Anti-friction rollers, 417, 725, 751,

846.
Antilogarithms, 71.
Antimony,
strength, 920.
weight, 212.
Anti-resultant, 362.
Apertures,

contiguous — , flow through — , 542.
shape of — , effect on flow, 541.
in tnin partition, 541.
Apex, apex distance. 780.
Apothecaries'

measure, 223, 224.
weight, 220.
Apparent solar time, 265.
Application of force, 332, 347.

point of—. 333. 359.
Applied and imparted forces, 372.
Approach, velocity of — . 556,
Aqueduct. Aqueducts,
flow in—. 560.
Kutter's formula, 563.
Arc. Arcs.

circular — , 179.

angles subtended by — , 181,

780, etc.
center of gravity, 391.
chords, 143, 179, 780, etc.
co-secant, 97.
cosine, 97.
cosine, table, 98.
co-versed sine, 97.
graduated—. 292.
lante — . to draw — , 181.
orcTmates, 180, 784, 817, 840.
radii, 180, 781. etc.
rise. 180.
secant. 97.
sine, 97.

table, 98.
tables, 183, 185.
tangent, 97.
table, 98.
time equivalents of — , 265, 200.
versed sine, 97*
elliptic—, 189.
parabolic — , 192.
semi-elliptic — , 189.
Arch, Arches, 424, 430, etc., 613. 740.
abutments, 617.
angle of friction in — , 432.
brick—. 629, 632.
bridges. 613.
centers for — , 681.
concrete — , 615, 616.
design of—. 431. 432, 613, ©te.
elliptic — , 616.

joint in — . to draw — , 190,
kesnstone, 613, 615.
line of pressure, 430.
line of resistance. 430.
line of thrust, 430.
mechanics of — t 430. 432.

1

INDEX.
Areli->Beam.

1041

Arch, Arches — continued.

moments in — , 424.

practical considerations, 432.

pressure in — , 430, 614.

pressure, ^ne of — , 430,.

radius, to find — , 614,

resistance line, 430,

roofs, 740, 742.

rubble—, 616.

settlement of, — 432.

statics Of—, 430, 432,

stones, 613.

chamfering, 634.

pressure in — , 430, etc., 614.

pressure of — , on centers, 633.

theory of—, 430, 432.

thrust line, 430, 432.
Archimedes screw, 687.
Architrave, defined, 1025.
Area, Areas,

of a circle, to find — , 161.

of a circle, table, 163-178.

contraction of — in iron and steel,
752, 754, 873.

crippling — , of rivet, 775.

of pipes, 526.

reduction of — , 752, 754, 873.

of sections of beams, 468, 802,
etc.

of surfaces. See the surface in
guestion.

unit — , equivalents of — , 233.
Arithmetic, 35.

Arithmetical complements, 71,
Arithmetical progression, 39.
Arm, lever — , 360.
Arris, defined, 1025.
Arroba, 227.
Artesian wells, 671.
Artificial

horizon, 298.

stone, concrete, 943.
Ascent. See Grade, Height, Slope.

efiFect of — on power of horses,
683.

e£Fect of — on power of locomo-
tives, 860.
Ash wood, strength, .476, 957,

958.

weight, 212.
Ashlar masonry, cost, 601, 602.
Asphaltum, weight, 212.
Atlas powder, 951.
Atmosphere, See Air, 320.

bubyancy, 513.

(unit of pressure), 240.

weight, 212, 320.
Augers for earth and sand, 670.
Aune, 226.

Avoirdupois weight, 220.
Axis, See the given surface or
solid.

of buoyancy, 514.

of equilibrium, 514.

of flotation, 514.

neutral — , 466.

of symmetry, 514,

66

Axle, friction, 416, 417.
Azimuth, Azimuths, 284-290,

B.

Back, Backs,

of^rch, defined, 613.

of retaining walls, 603.
Backing of walls, 603.
Backstays for suspension bridges,

766.
Bag scoop or spoon, 581.
Bag^ge cars, 865.
BaUmg by bucket, day's work, 686.
Balk, defined, 1025.
Ballast,

for railroads, 815.
cost of — , 855.
Balloon, principle of — , 613.
Balls, weight, 874,877, 879. 918.
Baltimore truss, 694.
Bar, Bars,

in built-up sections, 723,

dredging, 580.

iron — , weight, 877, 878.
Barbed fence. 854.
Bargeboards, defined, 1025,
Barometer, 312, 320.
Barrel, contents, 223, 224,
Barrow. See Wheelbarrow,
Barschall process, 955.
Bascule or lift bridges, 697,
Base, Bases,

of logarithmic systems, 72.

wheel — of locomotives, etc., 856.
Batten plates, 724.
Batter,

of abutments, 619.

defined, 1025.

of retaining walls, 603, etc.
Bazin's formula, 552.
Beam, Beams,

breaking loads, coefficients, 476.

channel—, 497, 894. 912.

coefficients for breaking loads,
476.

concrete — . strength of — , 945.

continuous — , 489.

curved — , 446.

deflections, 481, 959.
loads for given — , 480.
under sudden loading, 460, 959.

elastic limit, 482.

elasticity, modulus of — , 457.

end reactions, 439.

equilibrium of — , 437.

floor—, 720, 749.

forces acting upon — , 437.

franite — , 924.
— and channel — , 892, etc., 912.
in fire-proof floors, 894.
as pillars, 497, 902, 912.
separators for — , 900,
tables, 892, etc.
inclined — , 445.

1042

INDEX.

Beam— Boxed limber.

Beam, Beams — continued.

iron — , safe loads and deflections,

960, 962.
limit of elasticity. 482.
loads, 437, etc., 466. etc., 476, 760.
762, 892, etc.
for cpiven deflection, 480.
withm limit of elasticity, 482.
suddenly applied — , 460, 959.
mechanics of — , 437, etc., 466,

etc.
modulus of elasticity, 457.
moments, 440, 443, 445.
reactions, 439.
rolled — . See Beams, I — ;

Beams, Channel — ; etc.
shear, 446.
steel and iron — , loads, 892, etc..

960. 962.
stone—, 476, 924.
strength. See Beams, Loads,
suddenly loaded, 460. 959.
timber — , See Beams, Wooden — .
and trusses, comparison, 689.
of uniform strength, 486.
wooden—, 760, 762, 764, 969, etc
deflections, table, 959.
loads, tables, 959-962.
Bea^inff, Bearings,
of piles, 590.
power of soils, 583, 593.
and reverse bearing, 277.
stresses, permissible — , 762.
in trusses, 721, 725, 750, 751.
Beats, of clocks and watches, 266.
Beaum^ hydrometer, 211.
Bed plates, 721, 750.
Beech-wood, strength, 476, 957,

958.
Beetle, defined, 1025.
Bell,

diving — , pressure in — , 321, 597.
joint for pipes, 660. .
Belting, cost and mfrs., 987.
Belts, leather — , strength, 922.
Bench mark, 1025.
Bending

of beams. See Beams, Deflec-
tions,
and compression, combined, 724.
moments, 466, etc.
stresses in bridge members, per-
missible— t 762.
tests, 871.

bridge steel, 752.
iron and steel, 873.
steel castings, 754.
Bends in water pipes, 537.
Bents in trestles, 814.
Berm, defined, 1025.
Beton. defined, 1025.

See Concrete.
Beveled joints for rails, 819.
Bibliography, 1008.
Bilge, defined, 1025.
Birch, strength, 476, 957, 958.

Birmingham gauges, 887. 890.

Bismuth,

strength, 920.

weight, 212.
Bitumen, weight, 212.
Bituminous coal,

space occupied by — , 215.

weight of—, 212, 222.
Blaok^ine prints, 982.
Blasting. 600, 953.

apparatus, 952.
Bled timber, strength, 057.
Blocks, angle — , 736.
Bloom ton, 216.
Blue-line prints, 982.
Blue prints, 979.
Board measure, table. 269.
Boasting, defined, 1025.
Boat, canal — , 684.
Body, Bodies. See the body il
question.

defined, 330.

expansion of — by heat. 317.

falling—, 348, 539.

floating — , 513.

rigid—, force in—, 330. 358.
Boiler. Boilers.

cost and mfrs., 988.

incrustation, 327.

iron—, 872,

thickness, 511.

tubes, 882.
Boiling point, 326.

levepjig by — , 314.
Boiling tests for cement, 941.
Bollman truss, 695.
Bolster, defined. 1026.
Bolts, 883, etc.

cost and mfrs., 986.

expansion — t 884.

iron—, table; 886.

strength and weight, tabhv
886.

stresses, permissible — , 762.
Bonnet, denned, 1026.
Bonsano rail-joint, 823.
Books, 1008.
Boring,

artesianr wells, 671.

augers for earth — , 670, 671.

test—. 582, 670.

wells. 671.
Borrow pit,

defined, 1026.

to measure — , 195. 196.
Boss, defined. 1026.
Bottom.

heading, 812.

of stream, scour on — , 577.

velocity. 560.
Bowstring,

centers. 637, 638.

truss, 695. 699.
Box

cars, 865.

drains, 627.

sextant, 297.
Boxed timber, strength. 957.

INDEX.

1043

Braetn§f— Camel.

Bracing.

in bridges, 091, 710. 748, 749.

counter — , See Counter-bracing.

cross — , 691.

for dams, 502.
Brad spikes, 818.
Brake friction, 412.
Branches in pipes, 661.
Brass,

oalls, weight, 9l8.

ductility, etc., 459.

effect of mortar, etc., on — , 936.

effect of water on — , 327.

expansion, by heat, 317.

friction, 411, 415.

pipes and tubes, seamless — , 919.

strength. 476, 920, 921.

tubes, seamless — , 919.

weight, 212, 876-878, 887.

wire, 887.
Brasses, defined, 1026.
Braze, defined, 1026.
Breast-wall, defined, 1026.
Breathing,

air consumed in — , 320.

in diving-bells, 320.
Brick. Bricks, 926, 927.

arches, 629, 632.

cost and mfrs., 985.

cylinders, sinking of — , 599.

dust, 925.

friction, 411.

incrustations, 929, 936.

laying, 927.

strength, 476, 922, 923.

weight, 212, 213.

work, 927.

mortar required for — , 931.
weight of—. 213.
Bridge, Bridges. See also Arch,
Beam, Girder, Trestle, Truss,
etc.

arch — , 613.

brick—, 629, 632.

Brooklyn — , foundations, 598.

camber, 726, 746.

centers, 631.

clearance, 746.

combination — , specifications, 763.

crossnaection, 746.

design, 720, 745.

electric railway — , specifications,
745.

erection of — , 743, 763.

friction rollers, 725, 751.

gage on — , 746..
eadway on — , 746.
highway — , specifications, 745.
joints (connections), 774.
manufacturers, 988.
painting, 763, 764.
protection, 763, 764.
railroad — , specifications, 745.
roadways, drainage, 628.
specifications, digest of — , 745.
stone — , 613. See also Arch,
stone — , centers. 631.

Bridge, Bridges — continued.

suspension — , 765.

cables of—, 765, 891, 976, 977.

test of completed — , 753.

trusses, 689.

weight of—, 731, 738.

Wissahickon — , Phila., 640.

wooden — , specifications for — «
763.
Briggs logarithms, 70, 78, 80, etc.
Bri^uett^ cement — , 938. etc.
British

Imperial measure, 223, 224.

rod of brickwork, 222, 928.
Broken

bubble tube, to replace — , 296.

cross-hairs, to replace — , 296.

joints, 819.

stone. See also Rubble, 583.
for concrete, 943, 945.
foundations, 583.
voids in—, 688, 943.
Bronze,

phosphor — wire, strength, 920,

weight, 212.
Brooklsm bridge, foundations, 598.
Bubble tube, to replace — , 296.
Buckle plates, 750, 885.
Builder's level, to adjust — , 311.
Building, Buildings,

specifications for — , 764.
Built beams, 479, 734.
Built sections, truss members, 722.
Bulk, increase of — , broken stone.

etc., 688, 943.
Buoyancy,

of air, 513.

of liquids, 210, 513, 515.
Burkli-Ziegler formula, 575.
Burned clay, 925.
Burnettizing, 955.
Burr truss, 695.
Bursting,

anti— device, 665.

of pipes, 513, 663, 665, 668.
Bush, defined, 1026.
Bushel, 223, 224, 234.

volume of — , 223.
Business directory, 983, 996.
Butt-joint, 773.
Buttresses, 612.

c.

Cable, Cables,

number of wires in — , 891.

stays, 766.
Caisson, 585.

Brooklyn bridge, 598.

work in — . See Diving-bell, 321.
Calcium carbonate, in cement, 930.
Calcium oxide, in cement, 930.
Calking, 660.
Camber, 696, 726. 746.
Camel, defined. 1026.

1044

INDEX.

Canal— Cliaek.

Canal, Canals,

boats, 684.

flow in—, 660.

Kutter's formula, 523. 563.

leakage from—, 329, 561.

traction on — , 683.
Cantara, 227.

Cantilever. Cantilevers. See also
Beams.

bridges, 606.

end reactions, 439.

equilibrium of — , 437.

forces acting upon — , 437.

moments, 440, 442, 445.

reactions, 439.

of uniform strength, 486.
Cap, for blasting, 952.
Capitalization, 41, 44.
Car, Cars, 865.

derrick-, 808.

earthwork, 807.

friction, 417.

mfrs., 990.

pressed steel — , 865.

resistance, 417.

wrecking, 808.
Carat, 219, 220.
Carbon,

dioxide, in cement, 930.

in steel, 872.

in steel castings, 754.
Carbonic acid, in cement, 930.
Carnegie beams, channels, etc., 892,

etc.
Cart, Carts,

earthwork, 800, 802, 805.

excavating (wheeled scrapers),
805.

road, repairs of — , 801.

rock, removing, 810.

traction, 683.
Cartridge,

dynamite — , 950.

rack-a-rock, 951.
Cassiopeia, 285.
Castellano, 227.
Castelli's quadrant, 561.
Casting, Castings,

safety—, 824, 833.

steel — , requirements, 754, 872.

weight of — , by size of pattern,
875.
Cast-iron. See Iron, Cast — .
Cattle cars, 865.
Cedar, strength, 476, 957, 958.
Cement, 930-942.

brick-dust — , 925.

concrete, 943.

cost and mfrs., 985.

and iron pipes, 657.

for leaks, 971, 973.

moisture, effect on — , 929.

gjf^nr 940

Strength, 923, 932, etc.
weight, 212, 934, etc.
Center, Centers,
for arches, 631.

Center, Centers — continued.

of buoyancy. 514.

of circle, to nnd — , 161.

for fire-proof floors, 895.

of force, 399.

of gravity, 386.

of gsrratioc, 496.

of moments, 361.

of oscillation, 351.

of percussion, 351.

of pressure, 399, 501, 506^ 514.
Centigrade thermometer, 318.
Centigram, 226, 236.
Centifiter. 224. 225. 235.
Centimeter, 225, 231, 233.

cubic — , weights, 212.
Centistere, 224, 225.
Central forces, 354.
Centre. See Center.
Centrifugal force, 354, 711. 758.
Centripetal force, 354.
Chain, Chains,

cost and mfrs., 987.

equivalents of — , 232.

Gunter's— , 220.

iron — , 915.

loaded—, 428.

pump, 687.

rivetme, 774.

strength of — , 915-

surveying, 274, 282.

of suspension bridges, 765.

weight of — , 915.
Chainmg, 282.
Chalk,

strength, 923.

weight, 212.
Chamfer, defined, 1027.
Chamfering arch stones, 634.
Channel, Channels,

flow in—, 523, 560.

Kutter's formula, 523, 563.

steel — as pillars, 497, 912.
table, 894.
Channeling in rock, 681.
Characteristics, 70-79.
Charcoal, weight, 212.
Chart, Charts,

isogonic — % U. S.. 300.

logarithmic — , 73.
Checking of cement, 938.
Cherry wood, weight, 212.
Chestnut. See Wood.
Chilling, defined, 1027.
Chock, defined, 1027.
Chord, Chords,

of arcs, to find — , 179.

in circles, 162, 179.

in curves, 780, etc.

increments, 701.

long — , table, 787.

members. See Trusses.

to radius 1, table, 143.

stresses, 701, 709.

of trusses, 689.
Chronometers, behavior, 266.
Chuck, defined, 1027.

INDEX.

1045

Cftmni Dpl11i«gr--€oiiiiM»Hent.

Ohurn drilling, 600.
Cippoltitti trapezoidal notcfau 559.
Circle. Circles, 161, etc. See also
Circular.

angles in — , 94.

chords, 162, 179.

great—, 184, 284.

radius, 161, 180.

tables, 163-178.

vertical — , 284.
Circular

arcs. See Arc, Circular — .
tables, 183. 185.

curves for railroads, 780.

inch, 222, 889.

lune, 186.

measiire, 222.
of angles, 34.
of wires, etc., 889,

motion, 351.

ordinates, 180.

tables, 784, 786, 817, 840.

plates, strfngth, 493.

rings, 186.

sector, 186.

center of gravity, 393.

segment, 186.

center of gravity, 394.
table, 187.

spindle, 209.

zone, 166.
Circulating decimals, 38.
Circumference

of circle, to find—, 161, 171, 178.

of ellipse, 189.
Cisterns, 512, 851, 854.
Civil day, month, time, year, 266.
Clamp, Clamps,

pouring — . for pipe joints, 660.

rod, switch, 825.
Clay,

effect on mortar, 925, 926, 935.936.

in foundations, 583.

loosening of — , 800.

swelling of — by absorption, 583.
Clearance,

in hi^hwav bridges, 726.

in railroad bridges, 746.
Clearing, cost. 855.
Cleat, defined, 1027.
Climate, effect of — on rainfall, 322.
Clinometer, 256, 311.
Clips, for cement briquettes, 939.
Clock,

to reflate — by star, 266.
Close piles, 590.
Cloth, tracing — , 978.
Coal,

cars, 865.

consumption of — by locomotives,
860, etc.

corrosive fumes from — , 880, 916.

for locomotives, 856.

oil, weight (petroleum), 214.

space occupied by ton of — , 215.

ton of — , volume. 215, 222.

weight, 212, 222.

Cocks, corporation — , 664.
Coefficient, Coefficients,
of contraction, 542.
defined, 1027.
of deflection, 483.
of friction, 408.
of kinetic friction, 409.
for loads within elastic limit,

482.
of roughness, 523, 564, 565.

for pipes, 523.
of' safety. See Safety, Factor of — .
of stability, 423.
Coffer-dam, 585, 586.

defined, 1027.
Cohesion, 957.
Coin, Coins, 218, 219.
Coke, weight. 212.
Cold,

effects of — ,

on cement, 932.
on explosives, 948, 950.
on iron, 819, 874.
on mortar, 928.
-rolled iron, 920.
Colinear forces, 361, 363.
Collision, impact, 347.

Sosts, 695.
Dgarithm, 71.
Colors,

draughtsmen's, 978.
Column, Columns (pillars). *

See Pillars.

Gray—, 906.

Phoenix— 497, 904, 912, 913.

water — , 852.

Z-Bar— . 901-903.
Combination

bridges. 738.

specifications, 763.

and permutation, 40.
Combined stresses, 493, 724.
0)mmercial

measures, size of — by weight of
• water, 224.

weight, 220.
Common

denominator, 35.
to reduce to — , 36.

divisor, 35.

factor, 35.

fraction, 36, 37.

logarithms, 70-91.

measure, 35.

multiple, 35.
Compass,

to adjust, 298.

declination, 301.

variation, 301.
Compensating reservoir, 653.
Compensation, water, 653.
Complement and supplement, 94.
Component, Components,

equations for — , 365.

of force, 362.

normal — , 370.

rectangular — , 369. _

1046

INDEX.

Coiiiponent<-<;o-Uiii|c«iit.

Component, Components — cont'd.

stress — , 371.

summation of — , 466.

tangential — , 369.
Composition

of couples, 405.

of forces. 362, etc.

of ratios, 38.

of steel, 753.
Compound

interest, 42.
equations, 44.
table, 43.

levers, 420.

locomotives, 856.

stresses, 762.
Compressed

air, 320, 597, 681.

gun-cotton, 951.
Compressibility

of air, 320.

of liquids, 326.

of sand, 935.
Compression

a-nd bending, combined, 493, 724.

members, 721, 722, 732. 747.

and tension, 359.
Compressive strength

of cements, 934.

of concrete, 943.

of metals, 921.

of stone, etc., 923.

of timber, 958.
Compressors, air — , 681, 991,
Concentrated

excess loads, 705.

loads, deflections, 484.

loads, moments due to — , 444.

loads, shears in beams, 446.
Concrete, 943.

beams, strength of — , 945.

defined, 1027.

-metal construction, oost, 989.

mixers, 943.

cost and mfrs., 992.

strength, compressive — , 923,
943, etc.

strength, transverse, 945.
Concretions in pipes, 655.
Concurrent forces, 361, 364, 380
Cone, Cones, 200.

center of gravity of— =-, 395, 397.

frustum, 201, 395, 397.
Conical rollers, 846.
Connections, pin and riveted, 721.
Conoid,

frustum of — , 209.

parabolic — , 209.
Consequent, 38.

Consolidation locomotives, 857.
Constants. See Coefficients.
Construction,

bridge — , 720.

railroad — , cost, 855.
Consumption

of fuel by locomotives, 860.

of water, 649.

Contiguous openings, flow thro— ^,

542.
Continued proportion, 38.
Continuous beams, 489.
Contoiii^ lines, 302.
Contracted vein, 541.
Contraction,

of area, iron and steel, 752,
754, 873.

coefficient of — , 542.

by cold, 317, 819.

incomplete — , 540, 644.

of outflow, 541.

of rails, 819.

of waterway, 623.
Contractor's profit, 801.
Conversion tables of units of

weights and measures, etc., 228.
Conveying machinery, cost and

mfrs., 991.
Cooper's standard loading, 755.
Co-ordinates in resolution, of forces,

372. ^

Coping,

defined, 1027.
Coplanar forces, 361.
Copper,

balls, weight, 918.

compressibility, etc., 459.

cost, 919, 987.

effect of cement, mortar, etCi
on—, 926, 936.

effect of water on — , 327.

expansion of — by heat, 317.

pipes, seamless — , 919.

roofs, 918.

sheets, 918.

strength, 499, 500, 920, 021.

sulphate, for wood, 955.

tubes, seamless — , 919.

weight, 212, 875. 878, 886. 887.
918
Corbel, defined, 1027.
Cord,

loaded—, 428.

mechanics of the — , 425.

polygon, 377, 428.

of wood, 222, 234.
Cork,

weight, 212.
Corporation cocks or stops, 664.
Corrections for tapes, 283.
Corrosion,

by acid fumes, 970.

by coal fumes, 880, 916.

by water, 327, 594.
Corrugated

flooring, 914.

sheet-iron, 880.
Co-secants, 97.
Cosines, 97, 98.

logarithmic — , 72.
Cost, Costs. See articles in quas"
tion.

price list, 983.

of operation of railroads, 867.
Co-tangent, 97, etc.

INDEX.

1047

Ck^tter^bolt— BAm.

Ck>tter-bolt, defined. 1027.
Counter, Counters,

adjustable — , 721.

-bracing, 690, 706, 712, 721, 738,
746.
of centers, 634.

-scarp revetment, 612.

sloping revetment, 612.
Counterforts, 612, 1027.
Couples, 404.
Couplings for pipes and tubes, 657,

882.
Courses

of masonry, 603, 620.
Cover

in a butt-joint, 773.

plate, 723.
Co-versed sines, 97.
Crab, defined, 1028.
Cradle, defined, 1028.
Cramp, defined, 1028.
Cranej defined, 1028.
Creepmg

of rails, 819, 820.
Creo-resinate process, 955.
Creosote, 815. 954, 984.
Crescent truss, 695.
Crib, Cribs,

coffer-dam, 645.

dams, cost, 645.

foundations, 584, 585.
Criterion,

for maximum chord stresses, 709.

for maximum web stresses, 706.
Critical velocity, 415.
Cross

bracing. 691, 710, 748.

S'rts, turntable, 846.
airs, in level, 306.
-hairs, to replace — , 296.
section of bridge, 746.
section paper, 978.
logarithmic, 73.
-shaped beam, 492.
ties, 815, 855.
Crowd, weight of—, 726.
Crown,

of arch, defined, 613.
(coin), value, 218.
Crushers, stone—, 943, 992.
Crushing

loads, 923, 934, 943, 958.
in timber construction, 732.
Cube, Cubes, 55, 194, 195.

roots, 54, etc. See also Powers,
of decimals, to find — , 67.
of large numbers, to find — ,

66.
tables, 54, etc.
tables, 55.
Cubic

centimeter, foot, inch, etc. See

Conversion Tables,
measure, 222.
metric, 225.
meter, etc. See Meter, etc.
Culmination of Polaris, 287, 288.

Culmination

of a star, 284.
Culvert, Culverts,

arches for — , 613.
' box—, 627.
foundations of — , 627.
lengths of — , 622.
quantity of masonry in — , 622.
Curbs

in highway bridges, 750.

Current meters, 562, cost, etc., 993.

Curvature of the earth, table, 153.

Curve, Curves. See Arc, Circle,

Ellipse, Parabola, etc.

effect of — on distribution of live

load on bridge, 712, 756.
elastic — , 483.
railroad — , 780.
gauge on — , 789.
tables of—, 784-789.
in tunnels, 812.
in turnouts, 840.
in water pipes, 537.
Curved

beams, 446.
chords in trusses, 695.
Curvilinear motion, 351.
Cuttings,

level—, 790.
Cutwater, defined, 1028.
Cycloid, 194.

center of gravity, 894.
Cylinder, Cylinders, 196.

contents, table, .197, 223, 525.
in foundations, 593, 594, 596,

599. See also Foundations,
of locomotives, 856, 861.
pneumatic process, 596.
pressure in — , 511.

locomotive — , 861.
stren^h of — , 611.
Cylindrical

beams, deflections, 485.
pillars, 497, 912, 913.
unfi^ula, 199.

center of gravity of — , 397.
Cjmia, to draw — , 191.

D.

Dam, Dams, 400, etc., 430, etc., 433,
etc., 502, 576, 642.
center of pressure against — , 400.
coffer—, 685, 586.
construction, 585, 642.
danger in — , 436.
deflection, 436.
discharge over — ^ 547.
height of water, 554.
leakage through—, 329, 651.
masonry — , 400, etc., 430, etc.,

433, etc.
practical considerations, 436.
stability, 433, 508.

I8DEX.
in--IMBehBr(e.

Day. Dsyi, 236, 265, 266.
Dejid load. Dead loads,

tor brldaee, 690, 755.

ID eanrses.

1. 226. 231. 236.

224, 225, 235,

K

this, 70.

ter. 226, 233.

re, 225.

ux^. 602.

tioQ, 284.

mugnetio — . 301.

alRT. formula, 200.

DMpeoed beams. 479, 734.

DeflSct

on. Deflection^

aogl

, 780, 784-780, 840

oh. 438,

of b<

of

[tile vera of u^orm

ooB&iient, 483.

otcr

OSS-shaped beaniB, 4

of dam. 436.

Doluutrani,
DekSwr. ;
Dekameter

228, 231. 236.

Jepartures aud latitudes, 374.
Jepreciatioo 43, 864.
!>ebth. Depths,

converaloD table, 23S.

Derri

defined, *1028.
Detrusion, 4»9.
Dew-point, 321.
Diagonal. EKi^oDals.

of paraUelogram. 96. 1S7.
of trspetoij; etc., 158.
in trusses, 689.
Diagram. Diurams,

fnr dead load atressea, 703.

for pi
for Kuti

for

■a formula.

). 706.

Hhear, 440.

actual and nominal-^, 52«. 881.

of rivets for safely, 775.

BO. roots of—, 526.

.of wire. 887-881.
DiamoDd drill, 675.
Dike, defined, 102B.

IKD£X.

1049

IMa^iagyc ISOrth.

Discharge, Discharges — continued.

through notches, 559.

through orifices, 539, 546.

through pipes, 516, etc.

through sewers, 574.

through short tubes, 540.

tables of—. 261-265.

through thin partition, 541.

units of rateSs of — , eonveraion
of— ^243.

over wlirs, 547, etc.
Disks, centrifugal force in^, 355*.
Distance, Distances,

deflection — , tangential — , 781.

frog — , 839.

polar — , 284.

by sound, 316.
Distributing reservoirs, 653.'
Distribution of pressure, 400.
Diurnal magnetic variations, 301.
Diving

apparatus, cost and mfrs., 902.

-beU, 321.

dress, 992.
Division

of decimals, 37.

of fractions, 36.

by logarithms, 71.

by logarithmic chart, or slide
rule, 75.

of a modified logarithm, 72.

of ratios, 38.
Divisor, common, 35.
Dodecagon, 148.
Dodecahe<h*on, 194.
Dog-iron, defined, 1028.
Dollar,

U. S. — , weight, etc., 219.

value of—, 218.
Dolomitic limestones, 931.
Dome, pneumatic — , 665.
Donkey engine, defined, 1029.
Double

float, 561.

intersection trasses, 694.

rivetinje. 772.

rule Of three, 39.

shear, 499, 774.
Dovetail, defined, 1029.
Dowels defined, 1029.
Draft

of horses, 683, 685.

of locomotives, 860.

of vessels, 515.
Drag

scrapers, earthwork by — , 805.

of train on bridge, 711, 758.
Drain, Drains,

area drained by — , 675.

box — , 627.

foundations of — , 627.

pipe, 575.
Drainage

of roadways of bridges, 628.

sewers, ■674.

of tunnels, 812.
Draw-bridges, 696.

Drawing

instruments, cost and mfrs., 993.

materials, 978.
Drawn pipes and tubes, 919.
Dredge, Dredges, 580.

land—, 808.

mfrs., 992.
Dredging, 580.

by screw-pan, 596.
Dress, diving — , 992.
Dressmg of stone, 601.
Driftj defined, 1029.
Driftmg test, 752.
Drill, Drills,

cost and mfrs., 989.

rock—, 600, 676.
Drilling,

artesian well — , 671.

rock—, 800, 670, 675.

tunnel—, 812.
Driving

wheels, 856.
weights on — , 706, etc., 755,
etc., 856, etc.
Drop

tests, 871.

timbers, 644.
Drowned or submerged weirs, 554.
Dry

drains, 627.

measure, 223.

rot. defined, 954, 1029.
Dualin, 952.
Dubuat's formula, 555.
Ducat, value of — , 218.
DuctiUty, 455. 459.
Dump-cars, 865.

DuoaecimaJs, duodenal or duo-
denary notation, 47.
Duplicate ratio, 38.
Dyke, defined. 1029.
Dynamics, 330.
Dynamite, 949, 984.

E.

E and W line, to run — ^ 277.
Earnings of railroads, 867.
Earth,

augers, 670.

bearing power, 583.

blasting, 950.

boring, 670.

cars (dump-cars), 865.

curvature, table, 153.

friction, 612, 683.

hauling, 801.

heat of—, 320.

leakage through — , 329, 651.

leveling of — , 801.

loosening of — , 800.

natural slope, 419, 607, 610.

pressure, 607.

resistance of — , 683.

shoveling of — , 800.

shrinkage, 799.

1060

IVDBX.

Earth— Ex|Muisioii«

Earth — continued.

dope of—, natural — , 607, 610.
supporting power, £83.
weight, 212.
-work, 790-811.
cost. 800, 855, 988.
in tunneU, 812.
volume of — , 790, etc.
East and west line, to run — , 277.
Eastern elongation, 284.
Easting. 274.
Eccentric,
defined. 1029.
loads, 712.

deflections, 484.
Efflorescence, 929, 936.
Effort, total — of force, 371.
Elastic
curve, 483.

deflection, trusses, 718.
limit. 458, 459, 482.
brid^ steel. 752.
cast iron, 874.
iron and steel, 459, 754, 873.
modulus. 456, etc., 459.
ratio. 458. 461.
Elasticity.

Umit of—. 468, 469, 482.

in beams. 482.
modulus of — . 456, 469.
cast iron, 874.
Electric

blasting machine, 952.
railroad bridges, loads for — , 757.
Electricity in compass box, 302.
Elevation of outer rail on curves,

787
Ellipse. 189. 190.

false — , to draw — , 191.
ordinate, 189.

tangent to — , to draw — , 190.
Ellipsoid, 208.
Elliptic
arc, 189.

ordinate^ 189.
table, 190i
arch, 616.
joints in — , to draw — , 190.
Elm wood.

strength. 476, 957, 958.

weight, 212.
Elongation,

bridge steel, 752.

by heat, 317.

polar distances and azimuths of

Polaris at — . table of — , 290.
of Polaris, location of meridian

by—. 286.
of Polaris, times of — . 288.
required, iron and steel, 873.
of a star, 284.
in steel castings, 754.
under tension, 455.
of truss members. 718.
Embankment, 790-811.
cost. 800.
shrinkage, 799.

End

poet, design, 723.

reactions, 360, 439, 699, 702, 714
Energy. 343.

kinetic — , 343.

potential — , 346.
Engine, Engines,

cost and mfrs., 990.

locomotive — , 856.
dimensions, 856.
performance, 860. *
weight, 856.

pumpmg — , 852.

wheel loads, 706.
Entry head, 616.
Equal

altitudes, location of meridian by
any star at — , 287.

shadows from the sun, looatlon oi
meridian by — , 288.
Equality of ratios, 38.
Equation

of payments, 42.

of time, 265.
Equilibriimi, 358.

of beams and trusses, 487, 466,
698.

of couples, 406.

of floating bodies,
axis of — , 514.

indifferent—, 387, 614.

in levers, 419.

of moments, 360.

polygon, trusses, 707.

stable—, 387, 514.

unstable—, 387, 614.

vertical of — , 514.
Equipment, railroad — , cost, 865i

867.
Equivalence of work,

in trusses, 718.
Equivalents. See Conversion Tables,

230. 231, etc.
Erection of bridges, 743, 763.
Erie R. R. locomotive standard,

858.
Establishment of a port, 328.
Evaporation, 329, 661.

by locomotives, 864.
Even joints, 819.
Evolution by logarithms, 71.
Excavating carts (wheeled aozAponX

805.
Excavation, 790-811.

cost of—, 800, 866, 988.

in tunnels, 812.

volume, 790.
Excavators,

mfrs., 992.

steam — (land dredge), 808.
Excess loads, concentrated — , 70&
Expansion

bearings. 721, 726, 761.

bolts, 884.

of cement, 937.

by heat, 317. See Heat,
of rails, 819.

INDEX.

1051

Expense— Flume.

Expense, Expenses,

locomotive running — , 864.

railroad-^, 867.
Exploder, Exploders, 952.
Explosive, Explosives, 948.

cost and mfrs., 984.
Express cars, 865.
Extrados, 613.

Extreme fiber stress, permissible — ,
' 759.

Extremes, ratio and proportion. 38.
Eye-bars, 721, 747.

design. 722.

full size — , test of — , 753;

F.

Face of arch, 613.
Face wall, 603.
Facing switch, 824.
Factor, Factors,

common — , 35.

friction — , 531.

and multiples, 35.

safety — ,

for piles, 593.
for pillars. 909, 912.
See also Siafety, factor of — .
Fahrenheit thermometer, 318.
Fall, Falls,

defined, 1029.

required for a given discharge,
527, 566. 573.

in sewers, 674.
Falling

bodies, 348. 539.

water, horse power, 678.
Fabe

ellipse, to draw — , 191.

-works, 743.
defined. 1029.
Fanega, 227.
Fascines, 599.

defined. 1029.
Fathom, 220. 232.
Fatigue of materials, 465.

defined, 1029.
Faucet in pipe joint, 660.
Feather, defined, 1029.
Feet. See Foot.
Felloe or Felly, defined, 1029.
Fence, 854.
Fencing, 987.
Ferris-Pitot meter, 636.
Ferrule,

defined. 1029.

for water pipe, 664.
Fiber

reactions. 466.

stress, 466. 467, etc.

• and deflection. 481.
permissible — , 759.
Field tests for cements, 942.
Fifth powers and roots, 67-69.
Figure. Figures, 148.

areas of — , 160.

Figure, Figures— continued.

defined. 92.

to draw — . 159.

to enlarge — . 160.

irregular — , to find area of — , 160.
Filler m pin joints, 725.
Filling, spandrel — , 613.
Filters, mfrs., 994.
Fineness,

of cement, 938, 940.

of sand and cement, 937.
Finish, hard—, 968.
Fink truss, 695.
Fir, strength. 957, etc.
Fire, Fires,

heat of — -. 317.

hydrant (fire-plug), 669.

-proof floors. 894.

-proofing, cost. 989.

protection, wat6r for — . 650.
Firing, simultaneous — of blasts, 952.
Fish-plates, 820.
Fittings for pipes, 656, 882.
Flagging,

strength of — , 476.
Flashings, defined, 1029.
Flasks, casting — . defined, 1029.
Flats, in built-up sections, 723.
Flexible joints for pipes. 661.
Floating

bodies, 513.

mills, 578.
Floats, 560, 561.
Floor, Floors.

beams, 720, 749.
connections, 730.

bridge—, 720, 749.

corrugated — , 914.

fire-proof — , 894.

glass — , 974.

sections, rolled — , 914.

systems of bridges, 720, 749,

trough-, 750, 914.

wooden — in bridges, 750.

Z-bar—, 914.
Florin, value, 218.
Flotation, 513.
Flow,

through adjutages, 540.

in channels, 560.

through contiguous openings, 542L

full—, 540.

Kutter's formula, 523, 563, 664.

obstructions to — . 537, 676, 578.

through orifices, 639, 646.

in pipes, 616.

in sewers, 674,

through short tubes, 640.

in streams, 560.

in syphon, 620.

through thin partition, 541.

in trough, 544.

over weirs, formula, 649.
Fluid, Fluids. See also Liquid.

friction of—, 415, 623, 624.
factor of — , 530, 531.
Flume, defined, 1030.

■

J

1052

INDEX.

Flnsli—FrletlAM.

Flush, d«fined, 1030.
Fluxes, defined, 1030.
Fly-wheels, centrifu^l force, 366.
Follower, in pile driving, 694.
Foot, Feet,

cubic — ,

equivalents of — , 222, 234.

of substances, weight of — , 212.

equivalents of — , 232.

inches reduced to decimals of — ,
221.

of mercury (pressure), equiva-
lents of — , '241.

per mile, equivalents of — , 237.

per second, equivalents of — =■, 242.

-pound, 237, 341.
Force, Forces, 330, 332, 368.

acting upon beams and trusses,
437.

application of — , point of — , 333.

applied and imparted — , 372.

center of — , 399, 606, 614.

centrifugal — , 354.
on bridges, 768.

centripetal — , 354.

olassincation, 361.

colinear — , 363.

component, 362.

composition of — , 362, 364.

defined, 332.

difiFusion of — through liquids, 606.

on inclined planes, 349.

internal — in beams, 466.

living—, 343.

measure of — , 338.

parallel—, 382.
couples, 404.
resultant of — , 399.

parallelogram, 364.

parallelopiped, 380.

point of application of — , 333.

polygon, 374, 377.

resolution of — , 362, 364.

resultant of — , 362.

in rigid bodies. 330, 368.

total effort, 371.

transmission, 358.

triangle, 367.

units of — , 368.

conversion of — , 235.
Forcite, 952.
Forebay, defined, 1030.
Foreign

coins, 218.

explosives, 952. .
Forgings, steel — . requirements, 872.
Formula. See aUo the given prol^
lem.

Gordon's — , 495.

Kutter's— , 623, 563, 664.

prismoidai — , 203.
Foundations, 582.

of arches, 613.

artificial islands, 600.

brick cylinders, 599.

caissons, 585.

for centers, 631.

Foundations — continued.

in clay, 583.

close piles, 590.

coffer-dams, 585, 586.

crib — , 585.

of culverts, 627.

cylinders, 504, 506, 597, 590,600.

of drains, 627.

fascines, 599.

on gravel, 583.

grillage, 690.

iron piles, 594.

islands, artificial — , 600.

loads for — , 683.

masonry — , cylinders, 599.

Nasmyth pile-drivers, 591.

Pierre perdue, 583.

pile — . See Pile, Piles.

plenum process, 597.

pneumatic process, 596.

random stone, 683.

resistance of — , 583, 592.

of retaining walls, 612.

rip-rap, 583.

on sand, 582.

sand piles, 599. 670.

sand pump, 599.

screw piles, 594.

sheet piles, 590.

sustaining power, 583, 592.

for trestles, 814.

for turntables, 846.

vacuum process. 596.
Four-way stojj-valve, 667,
Fourth proportional, 38.
Fractions, 35.

logarithms of-^— , 72.
Frames, blue-print — . 980.
Framing, timber — , 734.
Framework, steel — ^ 8i>ecifications,

764.
Franc, value of — , 218.
Francis's formula, 550.
Franklin Institute standard dimen-

sions of bolts, etc., 883.
Free end reaction, roof trusses, 715.
Freezing, 326, etc.

of dynamite, 950.

effect of — on cement, 932.

of explosives, 948.

of mercury, 318.

of mortar, 928.

of nitro-glycerine, 948.

in pipes, 656, 665.

behind retaining walls, 604.

in stand pipes, 663.

in track tank, prevention, 853.

of water, 326.
Freight,

cars, 865.

earnings. 867.

locomotives, 866.

ton-mile, 867.
Friction, 407.

an^le of — , 409.
in arch, 432.
in dams, 433.

IKDEX.

1053

Frietion— ^rade.

Friction — continued,
axle — , 416.
of cars, 417.
coefficient of — , 408.
of earth, 612.

fluid—, 415, 623, 524, 527, etc.
factor, 530, 531.
head, 516, 527.
on inclined planes, 350.
of iron cylinders, 593.

i'ournal — , 416.
:inetic — , coefficient, 409.

launching — , 415.

longitudinal — of revolving shafts,
419.

of masonry, 411, 612.

Morin's laws, 410.

of piles, 593.

rollers. 417, 725, 751, 846.
defined, 1030.

of walls, 608.

of water, 415. •

Frictional stability, 409.
Frog, Frogs, 834-840.

angle of—, 835, 839.

distance, 839.

graphic method, 842.

length, 835.

number, 835, 840.

point, 835.
Frost

jacket in fire hydrant, 669.

-proof tank, 852.
Frustum,

of cone, 201.

of parabola, 192.

of paraboloid, 209.

of prism, 195.

of pyramid, 201.
Fteley and Stearns's formula, 552.
Fuel consumption, locomotives, 861.
Fulcrum, 419.
FuU

flow, 540.

size eye^bars, tests of — , 763.
Fimies, acid — , effect on roofs, 970.

coal — , effect on iron, 880
Funds, sinking-y, 43.
Funicular machine, 427.
Furlong, 220, 232.
Furrings, defined, 1030.
Fuse, defined, 1030.

G.

G. C. D., 35.
Gage, Gages,

Birmingham—, 887, 890.

hook — , 548.

narrow — cars, 865.

narrow — locomotives, 857.

railroad — , 827,
on bridges, 746.
on curves, 787, 789.

rain—, 324.

stubs — , 890.

stuff, 968.

Gage, Gages — continued.

wire—, 887-891.
Gaging of streams, 560.
GaUon, 223, 224, 234.
Galton's experiments, 412.
Galvanic action in water pipes, 656.
Galvanized

iron, 880.

pipes, 664.
Gas

en^nes, mfrs., 990.

weight, 211.
Gasket, 660.

defined, 1030.

to prevent washing — into pipe,
661.
Gate valves, 666, cost, etc., 995.
Gates for water pipes, 666.
Gauge, Gauges. See Gage, Gages.
Gauging of streams, 560.
Gauthey's pressure plate, 561.
Gearing, ratio of power and wt., 420.
Gelatine, explosive — , 952.
Geographical mile, 220.
Geometrical progression, 39.
Geometrical similarity, 92.
Geometry, 92.
Giant powder, 951.
Gib, defined, 1030.
Gin, 686.

defined, 1030.
Girders,

details, 728.

erection, 743.

plate— 731, 747.
bracing, 749.

and trusses, comparison, 689.
Glass, 973.

cost and mfrs., 974, 985.

dimensions, etc., 973.

expansion by heat, 317.

friction, 411.

strength, 476, 922, 923, 974.

weight, 212.
Glazing, 973.
Globe. 204. 205.
Glossary of terms, 1025.
Glue,

adhesion of — , 922.

defined, 1030.
Glycerine, nitro — , 948.
Gneiss, weight, 213.
Gold,

strength, 920.

value—, 219.

weight, 213, 219.
Gondola cars, 865.
Gordon's formula. 495.
Grade. Grades, 255-257.

contour lines, 300.

etc., conversion of — , 237.

defined, 255, 256.

effect on horses, 683.

effect on locomotives, 860.

hydraulic — , 519, 521.

percentage, 255.

resistance, 683, 860.

1054

n^DEX.

Grade— Heetometor.

Grade, Grades — continued.

of roads. 255, 683.

of sewers, 574.

tables, 265-257.

traction on — , 683.

in tunnels, 812.

on turnpikes, 255.

of water-pipes, 653.
Gradient, hydraulic — , 519, 521.
Grading, cost. 800, 855.
Grain (a weight), 220, 226, 235.
Gram, or Gramme, 217, 226.

equivalents of — , 236.
Granite,

beams, 924.

cost of — blocks, 601.

expansion by heat, 317.

rubble, cost, 602.

strength, 476, 923, 924.

weisjht, 212.
Graphic

method, truss stresses, 703, 706.

representation of couples, 405.

statics, 428-431, 435.
Gravel,

boring in — , 670.

in concrete, 943.

dredging in — , 580.

for foundations, 582.

natural slope of — , 610.

weight, 213.
Gravity,

acceleration of — , 335, 336, 348,
349, 539.

center of — , 386.

on inclined planes, 349.

line of—, 389.

plane of — , 389.

specific — , 210.
Gray column, 905.
Great

bear, constellation, 285.

circle, 284.
Greatest common divisor, 35. .
Grillage, 590, 1030.
Groin, defined, 1030.
Gros, 226.
Gross ton, 216.
Ground lever, 826.
Grout, 926.

defined, 1030.
Grubbing, cost, 855.
Guard, Guards,

rails, 750, 828, 833, 835.

wheel — , 750.
Gudgeon. 416.

defined, 1030.
Guide-rails, 828, 833, 835.
Guldinus theorem, 194.
Gun

-cotton, compressed, 951.

metal, strength, 920.

-powder, 953.
pile-drivers, 591.
weight (under Powder), 214.
Gunter's chain. 220. 232, 282.
Gusset, defined, 1030.

Gutta-percha

pipe, 657.

weifirht, 213.
Gsrpsum, weight. 213.
Gjrration,

center of — , 496.

radius of—, 362, 496, 892, etc.

H.

H. C. F., 35.

H. P. See Horse-power.

Hair,

cross — , to replace — , 296.

stadia — , 293.
Half-section, equivalents of — , 233.
Hand

level, 310.

spike, defined, 1030.
Hard finish. 968.
Hardening of cement, 930.

rate of—, 932.
Hasselmann process, 955.
Haul, mean — , 801.
Hauling, 683, 685, 801, 805.
Haunches, defined, 1030.
Head, Heads,

block, 826.

of bolts, 883.

due to a given velocity, 539.

entry — , 616.

friction—, 616, 627.

for a given velocity, to find — ,
627.

for piles, 693.

plate, 826.

pressure — , 258, etc., 518.

theoretical — , 639.

tripod—, 292.

velocity—, 616, 639.

of water, 616.

for water supply, 664.
Header, defined, 1030.
Heading. 812.

defined, 1030.
Headway in bridges. 746.
Heat.

of the air. 320.

conduction of — , by air, 320.

expansion of air by — , 320.

expansion of rails by — , 819.

expansion of solids by — , 317.

expansion of surv. chains by — ,
274, 283.

of fires, 317.

subterranean — , 320.

thermometer, 318.

and work, units of — , conversion
of—, 237.
Hectare,

equivalents of — , 225, 234.
Hectogram, 226, 236.
Hectoliter. 226, 236.
Hectometer,

equivalents of — , 225, 233.

INDEX.

1065

Heel— Inclined.

Heel of frog, 835.

of switch, 825, 828, 839.
Height,

effect on temperature, 320.

effect on weight, 336, 348.

to find — by barometer, 312.

to find — by boiling point, 314.

to find — by trigonometry, 151.

of locomotive smoke-stack, 856.
Heliography, 979.
Helve, defined, 1030.
Hemlock,

strength, 476, 499, 958, 965.

weight, 213.
Heptagon, 148.
Hexagon, 148, 159.
Hickory,

strength, 476, 957. 958.

weight, 213.
High explosives, 948.
Highest common factor, 35.
Highway bridges, 745, etc.
Hip

roof, defined, 1031.

suspender, 709, 746.
Hogshead, 223.
Hoisting

engines, cost and mfrs., 990.

machinery, cost and mfrs . of — , 99 1 .
Holes,

for blasting, 600.

boring — in earth, 670.

boring— in rock, 600, 670, 675.
Homogeneity, t^ts for — , 871.
Hook-head spikes, 818.
Hopkins's pneumatic dome, 665.
Horizon, artificial — , 298.
Horizontal,

defined, 153.

forces, summation of — , 466.

loads in trusses, 710.

shear in beams, 478.
Horse, Horses,

power of—, 683, 852.

-power, 342. 685.

equivalents of — , 244.

■ of falling water, 578.
-hour, equivalents of — , 237.
metric — . See under Metric,
of running streams, 578.

pumping, day's work, 852.

weight, 685.
Hose, cost, 995.
Hour, hours,

angle, 285.

defined, 265.

equivalents of — , 236.
House, engine — , cost, 850.
Howe tnoss, 692, 736, 738.
H. P. See Horse-power.
Hundredweight, 216, 220.
Hydrant, Hydrants,

cost and mfrs., 995.

fire (fire-plug), 669.
Hydraulic, Hydraulics, 616. See
also Water, Flow, Velocity, Dis-
charge, etc.

Hydraulic, Hydraulics — continued-
cement. See Cement.

grade line, 519, 521.

mdex, 930.

lime, 930.

mean depth, 523, 564.

radius, 523, 564.

ram, 578.

cost and mfrs., 991.
Hydraulicity of cement, 930.
Hydrogen, specific gravity — , 213.
Hydrometers. 211.
Hydrometric pendulum, 561.
Hydrostatic, Hydrostatics, 501.

paradox, 501.

press, 506.
Hyperbolic logarithms, 72.

I.

I-beams. See also Beams, I — .

in fire-proof floor, 894.

as pillars, 497, 912.

separators for — , 900.

table, 892.
Ice, 326; etc.

adhesion to piles, 594.

blastine of — , 950.

in stand pipes, 663.

strength, compressive — , 923.

weight, 213, 326.
Icos^hedron, 194.
Illumination

of cross-hairs, 286.

of stake, surveying, 286.
Impact, 347.

of trains on bridges, 711, 758.
Imperial

gallon. See Gallon.

measure, British, 224.
Impost, defined, 1031.
Impulse, 337.
Inch, Inches, 216, 220, etc.

equivalents of—, 221, 232.

circular—. 222.

cubic — , equivalents, 222, 234.

in decimals of a foot, 221.

per foot, equivalents of — , 237.

of mercury (pressure), equiva-
lents of—, 241.

miner's — , 546.

spherical — , equivalents of — , 222.

square — , equivalents of — , 233.
Inclination. See Grade.

of courses in masonry, 603, 620.

tables of—, 255-257.

in tunnels, 812.
Inclined

beams, 445, 485.

plane, 349, 369.
descent on — , 349.
ropes for—, 976-977.
stability on — , 424.
tables, 255-257.
velocity on — , 349.

1066

INDEX.

Ineomplete eontraetion, 544.
Increments, chord — , 701.
Incrustation,
of boilers. 327.
of walls, 929. 936.
Indeterminate stresses, 720.
Index, loflparithms, 70.
India rubber, weight, 213.
Indifferent equilibrium, 387, 614.
Inertia, 338.

moment of — , 351, 468.
Infinity, symbol for — , 33.
Influence diagrams, 403, 449, 702.
Ingot, defined. 1031.
Instability, 514.
Interest. 40.

Internal forces in beams, 466.
Internat'l metric screw thread, 883.
Intersections.

in railroad curves, 780.
in trusses, 694.
Intrados, defined, 613.
Inverse proportion, 39.
Inversion of ratios, 38.
Invert, defined, 1031.
Involution. 54-69.

by logarithms, 71.
Iron,

baUs, weight, 874, 876, 877. 879,

918
bars, weight, 877, 878.
beams. See Beams, iron — .
bending tests, 873.
blasting of — , 950.
bolts. 883, 886.
in bridges, requirements, 754.
cast — ,

balls, weight of — , 918.

cohesive strength, 920.

compressive strength, 874, 921.

elastic limit, 459. 874.

expansion by heat, 317.

friction, 411.

malleable — , stren^h, 874.

modulus of elasticity. 459, 874.

pillars, 495.

pipes, flow in — , 522.

pipes, weight, 656, 876.

requirements, 874.

salt water on — , 327, 594.

shearing strength, 499.

strength. 459, 476, 499, 500,

874, 920, 921.
tensile strength, 874, 920.
torsional strength, 500.
transverse strength, 476. 874.
weight, 213, 875. 918.
casting, weight, 875.
and cement, pipes of — , 657.
chains, 915.

channels, as pillars, 497, 912.
cohesive strength of — , 920.
cold, effect on—, 274, 819, 874.
cold-rolled — , 920.
columns. See also Pillars, iron — .

495.
compressive strength, 921.

Iron — continued.

contraction of — by odd, 274, 819.

corrosion of — by coal fumes, 880.

corruirated sheet — , 880.

cost, 986.

crushing strength, 921.

cylinders, bursting pressure in — ,

611. 612.
eylinders, foimdations, etc. See

also Foundations, 593-698.
ductilitv of — , 469.
effect of cement on — , 936.
effect of cold on — , 274, 819, 874.
effect of heat on — , 274, 317, 819.
effect of mortar on — , 926, 936.
effect of water on — , 327, 694.
elastic limit. 469. 872, 874.
expansion of — by heat, 274, 317,

819.
friction of — , 411.
galvanised — , 880.
heat, effect on—, 274, 317, 819.
limit of elasticity, 459, 872, 874.
malleable cast — , strength, 874.
manufacture, 870.
manufacturers, 986.
modulus of elasticity, 459, 874.
net, 774.

paints for preserving — , 763, 972.
piles, 594. See also Foundations,
pillars. 495, 497, 901-013. See

also Pillars,
pipes,

cast — , weight, 666.

fittings for—, 882.

flow m— , 622.

galvanised — , 664.

joints for — , 656, 660.

thickness, 612, 666.

wrought — , 666.

diams, actual and nominal — ,
526. 882.
plates, buckled — , 886.
porosity of — , 512.
prices — , 986.
re-rolled — , 920.

rolled — . See Iron, wrought — .
rolled — for bridges, requirements,

754.
roofs. See Roofs,
salt water, effect on — , 327, 694.
shearing strength, 499.
sheet—, 880.
specific gravity, 213.
specifications, 870.
spikes, 818.
strength, 459, 476, 499, 600, 870,

872, 874, 907, 920. 921.
stretch of — , 469.
T— . 497, 898. 912.
tensile strength, 920.
tests, bending — , 873.
torsional strength, 600.
transverse strength, 476, 874.
tubes, 882.
water, effect on — , 327, 694.

salt—, effect on—, 327, 594.

INDEX.

1057

iTOn— I4A|».

Iron— continued.

weight, 213, 875-882. See also
Iron, cast — ; Iron, wrought — .
wire, 891.

rope, 976, 977.
-wood (Canadian), strength, 476.
wrought — ,

bars, weight, 877. 878.
cohesive strength, 873, 920.
compressive strength, 921.
elastic limit, 459, 872.
expansion by heat, 274, 317,

819.
friction of — , 411.
pillars. See also Pillars, 495.
pipes, 656, 657.

diams, actual and nominal^ — ,

526. 882.
fittings for—, 882.
joints for—, 656, 660.
weight, 656, 882.
prices, 986.

shearing strength, 499.
strength. 476, 499. 500, 872,

920, 921.
tensile strength. 872, 920.
torsional strength, 500.
transverse stren^h, 476.
tubes, weight, £§2.
water pipes, 656.
weight. 213. 877-882.
Island, artificial — ,> for foundations.

600.
Isogonic chart and lines, 300, 301.

J,

Jack, defined. 1031.

rafters, defined. 1031.
Jag-spike, 818, defined, 1031.
Jaw-plate, 724.
Jet

pile driving, 595.
Jetty, defined, 1031.
Jig-saw, defined, 1031.
Jomt, Joints,

in arches, 629.

bell — for pipes, 660.

in bridges, 724.

butt-;-, 773.

in chimneys, etc., cement for — ,
971, 973.

distribution of pressure in — , 400.

end — , roof trusses, 733.

flexible — for pipe, 661.

lap—, 773, 778.

masonry — .
distribution of pressure in — ,

400.
inclination of — , 603, 620.
lead in—, 634, 921.

net—, 774.

pin — , 747.

pin and riveted—, 721.

for pipes, 656, 660, 882.

67

Joint, Joints — continued.

rail—, 819.

cost and mfrs.. 994.

riveted—, 721, 749, 772.

in roofs, 733. 916, 971.

timber — , 734.

toggle—, 427.
Joule,

equivalents of — , 237.

per second, equivalents of — , 245.
Journal friction, 416.
Jumper,

defined, 1031.

driU, 600.

. K.

Key frog, 836.
Keystone, 613. 615.

pressure on — , 614.
Kieselguhr, 949.
Kilogram,

centigrade, equivalents of — , 237.

equivalents of — , 226, 236.
Kilogrammeter, equivalents of — ,
237.

per second, equivalents of — , 246.
Kiloliter, 224, 225, 236.
Kilometer,

equivalents of — , 225, 233.

per hour, etc., equivalents of — :
243.
Kilowatt, equivalents of — , 245.
Kinetic

energy, 843.

friction. 407.

coefficient of — , 409.
King,

post, defined, 1031.

truss, 691.
Knife-edge, strength, 921.
Knot (nautical), 220.
Kutter's formula, 523, 563, 564.
Kyanizing, 955.

L.

L. C. D., 35.
L. C. M., 35.

Lacing, 722.
Lagging.

for centers, 631, 639.

defined. 1031.
Laitance, 947.
Land.

dredge, 808.

measure. 222, 233.
metric — . 225.

required for railroads, 254.

section of — , area of — , 222, 233.

surveying. 274.

ties, 612.
Lap

joint, 773, 778.

welded boiler tubes, 882.

welded pipe, 656, 882.

1058

INDEX.

iMr^—IArre,

Lard,

as a lubricant, 415.

weight, 213.
Lateral

bracing, 691, 720.
timber trusses, 737.
Laths. 968, "969.
Latitude, Latitudes,

astronomical — , 284.

degree of — , length, 220.

and departures, 274.

effect of — on barometer, 312, 314.

effect of — on gravity, 336, 348.
Lattice

bars, 747.

truss, 694.
Latticing, 722.
Launching, friction of — , 415.
Laying

bricks, 927.

out of turnouts, 839.

pipe, cost, 658.

track, cost, 855.
Lead,

balls, weight, 918.

defined, 1031.

effect of cement, mortar, etc., on
— , 926. 936.

elasticity, etc., 459.

expansion by heat, 317.

in masonry joints, 634, 921.

paint, 971.

pencils, 978.

pipe, 513, 918.

for pipe-joints, 658-661.

roofs, 918.

sheets, 918.

strength, 920, 921.

weight, 213. 875-878, 887, 918.

white — cement for leaks, 971.

white — paint, 971.
Leaded tin, 916.
League, 220, 226.
Leak in roof, to stop — , 971, 973,
Leakage, 329, 561, 642, 650, 651.
Leap year,

defined, 266.

equivalents of — , 236.
Least

common denominator, 35.

common multiple, 35.
Leather,

friction, 415.

strength, 922. i

Legua, 227.
Length,

per time, units of — , conversion
of—, 242.

units of — , conversion of — , 232.
Level, Levels, 306.

builder's — , to adjust — , 311.

cost and mfrs., 993.

cuttings, 790.

engineer's — , 306.

hand — . Locke — , 310.

lines, defined, 153.

note-book, form of — , 309.

Level, Levels — continued.

Y— , 306.
Levelling,

by barometer, 312.

by boiling point, 314.

of earth on embankment, 801.

screws, 292.
Lever, Levers, 419.

switch — , 826, 830.
Leverage, 360, 419.
Libra, 227.
Life,

average — ,
of cars, 865.
of shingles, 971.
of ties, 815.
Lift bridges, 696.
Ligne, 226.
Lignum vitae,

strength, 476, 957.

weight, 213.
Lime, 925.

in cement, 930.

hydraulic—, 930.

paste, 926.

quick — , 930.

weight, 213.
Limestone, 213, 923, 930.
Limit,

elastic—, 458. 459, 482.
cast iron, 874.
iron and steel, 752. 873.

of elasticity. See Limit, elas-
tic— .
Limnoria, 954.
Linch pin, defined, 1031.
Line, Lines, 92.

of action, 359.

agonic — , 301.

center of gravity of — , 391.

contour — , 302.

of gravity, 389.

hydraulic grade — , 519.

isogonic — , 301,

of no variation, 301.

parallel — , to draw — , 94.

of pressure, 399, 430.

resistance—, 430, 432. 434-4.36.

of resultants, 430, 432, etc.

thrust—, 430, 432, 434-436.
Lining of tun'nels, 812.
Link, equivalents of — , 232.
Liquid, Liquids. See Water.

buoyancj^ of — , 513, 514.

compressibility, 326.

flow, 516, 523, 524.

friction, 415, 623, 524.

measure, 223.

pressure, 500, 518.

transmission of — . 506.

specific gravity, 211.
Liter, 224, 225. 235.
Lithofracteur, 952.
Little bear, constellation, 285, 286.
Live lo£ul. See Loads, live — .
Living lorce, 343.
Livre, 226.

1

INDEX.

1059

Load, Loads,

on bridges, 726, 755.

cart — , of earth, 800.

chord stresses, 709.

on columns, 495, etc., 901, etc.,

963, etc.
dead—, 690.
on driving-wheels, 705, etc., 755,

etc., 856-859. 861.
on earth, safe — , 583.
for given deflection, 480, 481, 483.
line, 707.
Uve—, 690, 705, 709, 726, 755,

856-859. 861.
locomotive — . See Loads oi^

driving wheels,
moving. See Loads on driving

wheels,
permissible — on beams, 473.
for permissible deflections, 485.
on piles, 592.

on pillars. 495. 901. etc., 963, etc.
on roofs, 321. 713.
on roof-trusses, 713, 764.
on sand, 582.
stresses, 705.

graphic method, 706.
iddei "

suddenlv applied — , 460, 486, 959.

on wooden bridges, 764.
Loaded

chain. 428.

cord, 428.
Loading,

standard — , 705, 755.

of trusses, 690.
Local time, 287.
Location of the meridian, 284.
Lock,
. air — , 597.

gates, spacing of cross-bars, 506.

nut, 821, 885.
Locke level, 310.
Locomotive, Locomotives. 856.

adhesion of — . 413.

house, cost, 850.

mfrs., 990.

statistics. 867.

tonnage rating of — . 862.

turntables for — . 845.

water for—. 327. 851.

wheel-load. 705. etc., 755. etc.,
856-861.
Locust, strength, 476, 957, 958.
Logarithmic

chart. 73.

plotting. 74.

sines, tangents, etc., 72.
Logarithms. 70-91.
Long

chords, table, 787.

measure, 220, 225, 232.

ton, 216.
Longitude, degree of — , length, 220,

221.
Longitudinal and transverse stresses

combined, 493, 724.

Lower

chord, design, timber trusses,
733.

chord splice, 736.

culmination. 284.
Lowering of centers, 631, etc.
Lowest terms, 36.
Lubricants, 415.
Lubrication of tiu*ntables. 846.
Lumber. See also Wood, Timber.

cost and mfrs., 984.
Lunation, 266.
Lune, circular — , 186.

Machine,

drill, 675.

funicular — . 427.

riveting, 775.

for tapping pipes, 657, 664.
Magnesia in cements, 931.
Magnetic

declination, 301.

variation. 301.
Magneto-electric blasting. 952.
Mahogany,

strength. 476, 957, 958.

weight, 213.
Mail

cars, 865.

earnings, 867.
Man power, 686.
Mandrel, defined. 1032.
Manganese, in steel, 872.
Mantissa, logarithms, 70.
Manufacture of iron and steel, 870.
Manufacturers, list of, 996.
Map, to reduce or enlarge — . 160.
Maple-wood,

strength, 476, 957.

weight, 213.
Marble,

cost, 602.

expansion by heat, 317.

strength. 476. 923.

weight, 213.
Marc, 226.
Mark,

German — . 218. 246, etc.

Spanish — , 227.
Masonry,

in abutments, quantity. 623.

adhesion of mortar to — , 926.

inarches, quantity, 622-628.

and concrete, 945.

cost, 601, 988.

courses.

inclination of — . 603, 620.
lead between, 634, 921.

dam. 433.

foundations. loads on — , 750.

friction of—. 411. 603. 620.

incrustation of — . 929, 936.

joints, distribution of pressure
on — , 400.

mortar required for — , 931.

1060

IirD£Z«

Masonry— continued.

in piers, quantity, 628.

quantity

in arches, 622-62S.
in piers, 62S.
in retaining walls, 610.
in walls of wells, 198.
in wing-walls, 624.

railroad — , cost, 855.

in retaining walls, 603.

strength, compressive-*, 923.

weight. 213.
Mass, 334. 330.
Material, Materials,

fatiipie, 465.

particle, 358.

point, 358.

strength, 454.

weight, 210, etc.
Mathematical symbols, 33.
Mathematics, 33.
Matter, defined, 330.
Mattock, defined, 1032.
Biaximum

intensity of rainfall at points in
U. S., table, 323.

and min. stresses in truss mem-
bers, 712.

pressure,

angle, etc., of — , 607.

velocity, 560.
Mean,

depth, hydraulic—, 523, 564.

haul, 801.

proportional, 38.

radius. 523. 564.

of ratio and proportion, 38.

solar time, defined, 265.

sun, 265.

velocity, 522, 560.
Means, defined, 1032.
Measure. Measures, 216.

apothecaries', 223. 224.

circular — of angles, 34.

circular — of wires, 889,

commercial — , size of — , by weight
of water, 224.

common^, 35.

conversion tables, 228, etc.

cubic — , 222. 234.

fluid—, 223. 224.

long—, 220, 232.

metric—, 217, 225, etc., 228,
etc.

Russian — , 227.

Spanish—, 227.

square — , 222. 233.

weights, etc., conversion tablei» of
units of — , 228.

wine—, 223. 224.
Measuring weirs. 547, 646.
Mechanics, 330.

of arch, 430-432.

of beam, 437, etc., 466, etc.

of masonry dam,. 430. 433-436.

of trusses. 698. etc.
Melting points, 317.

Mercury,

barometer, 312, 320.

foot of — , etc. (pressure), eqiiiv»
lents of—. 241.

fieeaing-point, 318.

thermometer, 318.

weight, 213.
Meridian,

location of—, 284.

of longitude, 220. 221.

variation of compaae^ 296, 301.
Metacenter, 514.

Metal, Metab. See also Iron, and
Steel.

blasting of—, 950.

cohesive strength, 920.

comj^ressibility, 459.

compressive strength, 921.

ductility, 459.

effect of cement on — , 926, 996.

effect of heat on — , 317, 819.

effect of lime on—, 926, 936.

effect of mortar on — , 926. 936w

effect of water on — , 327, ^94.

elastic limit, 459.

expansion by heat, 317, 819.

limit of elasticity, 459.

modulus of elasticity, 459.

preservation of — by cement, 936b

roof trusses, 740. •

shearing strength, 499.

sheet-, 880, 881. 887. 916-919.

strength. 454, 459, 476, 499, 500,
920, 921.

stretch of — , 459.

tensile strength, 920.

torsional strength, 5O0L

transverse strength. 476.

weight, 210, etc.. etc.
fi^eter,

equivalents of — , 225, 233.

Ferris-Pi tot— . 536.

length, 217, 225, 233. etc.

Pitot— , 536, 561, 562.

radii, etc., of curves in — ^ 78&

Venturi — , 532. etc.

water — , 649, cost and mfrs., 994.

wheel, 562.
Metric.

atmosphere. 240t 320.

horse-power, equivalents of — ^i
245.

horse-power hour, equivaleoti
of— 237.

measures, 217, 225, etc., 228, etc.

railroad curves, tables, 786.

screw thread* international—.
883.

ton, equivalents of — , 236.

weights. }s«« Metrie measures.
Mica, weight, 213.
Middle

ordinates. ISa 784, 786, 788. 817.
840

third. '402.
Mikron, equivalents of — ^ 232.

IN1>EX.

1061

MH'-JlHrrow^mm^e,

MU.

equivalents of — , 282.
Mile, Miles,

equivalents of — , 220, 232.

freight ton-mile, 867.

geographical — , 220.

p€rnour,etc., equivalentsdf— ,242.

land and sea—, 220,^233.

nautical — , 220.

passenger — , 867.

sea — , 220.

square — (section), 222, 233.

ton — , 867.
MiUier, 226.
Milligram, 226, 236.
MmiUter, 225, 235.
Millimeter, 225, 233.
Mills, floating—, 578.
Miner's friend powder, 951.
Miner's itich, 546.
Minim, 223, 224.
Minimum

and maximum stresses, 712.

sections, 722.
Minute, Minutes,

in decimals of a degree, 95.

equivalents of — , 236.

of time, 265.
Mitred joints for rails, 819.
Mitre-joint, defined, 1032.
Mixing cement for briquettes, 998.
Mizar, 285, 287.
Models,

of beams,strength and weight, 478.

in force composition, 380.
Modern explosives, 948.
Modified logarithms, 72.
Modulus. See Coefficient, Strength.

defined, 1032.

of elasticity, 456, 459.

of flow, 540.

of resilience, 460.

of rupture, 468.

section—, 467-8, 473, 892 to 898.
Mo|^ locomotives, 856.
Moisture,

effect of — ,

on cement, 934.
on sound, 316.
on zinc, 917.
Molded concrete, 945.
Molds for cement briquettes, 939.
Molecular action, 358.
Moment, Moments, 360.

in arches, 424.

in beams, 440, 443.

in cantilevers, 440, 442.

in continuous beams, 489.

of couple, 405.

defined. 360. 1032.

diagrams, 479.
trusses, 707.

of inertia, 351.
in beams, 468.

influence diagrams, 449.

in levers, 419.

Moment, Moments — continued.

live load — , 709.

maximum bending — , 474.

of non-coplanar forces, 381.

resisting — , 467.

and shear, relation of — , 452.

of stability, 422, 508, 514, 608.

summation 'of — , 466.

in trusses, 440, 701.
Momentum, 338, 345.
Money, 218.
Monkey-switch, 826.
Mont Uenis tunnel, 812.
Month, civil — , sidereal — , synodro*-;

266.
Morin's laws of friction, 410.
Mortar,

adhesion of — , 926.

in arches, 616, 629, 683.

bricks, etc., 925.

cement — , 931.

clay, e£Fect on — , 926, 935, 936.

effect on iron, 770, 926, 936.

effect on wood, 926.

^out, 926.

m retaining walls, 604.

rubble, cost, 602.

weight. See Masonry, 213.

salt, effect on — , 926, 936.

sand for—, 925, 926. 935, 986.

strength, tensile — , 933.

weight, 213, 926.
Mortise, defined, 1082.
Motion, 331.

circular — , 351.

quantity of — , 338.

relative — , 331, 358.
Mould. See Mold.
Movable bridges, 696.
Moving load. See Load, liv«".
Muck, defined. 1032.
Mud,

penetrability, 593.

in reservoirs, 651.

weight, 213, 581.
Multiple, common — , 35.
Multiples and factors, 35.
Multiplication

of decimals, 37.

of fractions, 36.

by logarithms, 71.
chart or slide rule, 75.
Muskrats, 651.
Myria^ram, 226, 236.
Myriahter, 225.
Myriameter, 225.

N.

Nails,

cost and mfrs., 986.

shingling, 971.

slating, 970.
Napierian logarithms, 72.
Narrow-gauge

locomotives, 857.

railroad cars, 865.

1062

INDEX.

NMwnjrtli— Panib^Ue.

Nasmyth pile-driver, 591.
Natural

cements, 030, 037, 040.

logarithmB, 72.

mnee, 07, 08.

slope, 410. 604, 600, 610.
Nautical mile, 220.
Neat cement tests, 04'!.
Needle,

compass — , 203, 200.
variation of — , 301.
Negative

characteristics, logarithms, 72.

exponents, logarithmic chart, 76.

numbers, logarithms, 72.
Net

earnings of railroadsj 867.

iron, net plate, net joint, 774.

section of tension members, 750.

ton, 216.
Neutral

axis, 466.

surface, 466.
Newel, defined, 1032.
Niagara cantilever, 606.
Nicholson hydrometer, 211.
Nickel steel, 872.
Nicking test, 752, 871.
Nitro-glycerine, 048.
Nonagon, 148.
Non-concurrent forces, 375.
Non-coplanar forces, 380.
Normal component,. 370.
North

point, etc., 284.

star. 285.
Northing, 274.
Number, Numbers,

and equivalents in common use,
conversion tables, 231.

of frog, 835, 840.

prime — t 35.

by wire gage, 887-801.
Numerus logarithmi, 71.
Nut, Nuts, 883.

locks, 821, 885.

o-

Oak,

strength, 476, 409, 057, 058.

weight, 214.
Oblique, ObUques,

lines, 02.

pillars, 408.

pressure, 372, 504. 607.
Obstacles in surveying, to pass — ,

281.
Obetruction, Obstructions,

to flow, 575.

by {)iers, 575.

in pipes, to prevent — , 655.
Octagon,

area, 148.

to draw — , 159.
Octahedron, 104.
Ooree. defined. 1032.

Oil, Oils,

coal — (petroleum), weight, 214.

weight, 214.

wells, nitro-glycerine, 948.
Once, usuel, 226.

Open channels, flow in — , 523, 560.
Open hearth steel, requirements.

872.
Openings,

flow through — , 540-542.
Ordinate, Ordinates,

defined, 1032.

elliptic—, 180.

to find—, 180.

middle—, 180, 784-789, 817, 840.

parabolic — , 102.

tables, 784-780, 817, 840.
Orifices, flow through — , SSiO, 546.
Oscillation, center of — , 351.
Ounce. 220, 235.

equivalents of — , 235.

fluid—, 223, 224, 235.
Outer rail, elevation of — , 787.
Outflow, velocity of — , theoretical

— , 530.
Outlet valves, 653.
Oval, to draw — , 101.
Overfall

dams, 642.

discharge over — , 547.

for reservoir, 652.
Overturning,

effect of wind, 710.

work of—, 422.

P.

Packing,

defined. 1032.

erf eye-bars, 722.

piece, 775.
Paint, Paints, 071.

cost, 084.

for iroiii, 763, 072.

on zinc, 880.
Painting, 071.

of bridges. 763, 764.
Panel, Panels,

diagonal of — , to find length of—,
160.

points, 602.

reactions, 702.

in trusses, 602.
Paper, 078.
Parabola, 102, 103. See ParaboUe.

center of gravity, 804.

to draw — t 103.

ordinates, 102.

semi — , . center of gravity of— i
304.

tangent to — , to draw — , 193,
Parabolic

arc, 102, 103.

conoid, 200.

frustum of — , 200.

curve. 102. 103.

frustum, 102.

INDEX.

1063

Parabolic— Pillar.

Parabolic — continued.

ordinates, 192.

zone, 192.
Paraboloid, 209.

center of- gravity of — , 398.
Paradox, hydrostatic — , 501.
Parallel

forces, 382, 514.
couples, 404.
defined, 361.
resultant of—, 382, 399.

lines, to draw — , 94.

plates, 292.
Parallelogram, Parallelograms, 95,
157.

force — , 364.
Parallelopiped, 195.

force — , 380.
Parlor cars, 865.
Partial contraction, 540, 544.
Particle, material — , 358.
Partition, thin — , flow through — ,

541.
Passenger
■ cars, 865.

earnings, 865, 867.

locomotives, 856, etc.

mile, 867.
Paste, lime — , 926.
Patterns, #

weight of casting, 875.
Paving,

Belgian—, 602.

brick—, 927.

cost, 989.
Payments, equation of — , 42.
P. C, P. I., P. T., 780.
Peck, 223, 224.

Pedestals, bridge—, 721, 750.
Pencils, lead. 978.
Pendulum, Pendulums, 350.

hydrometric, 561.

seconds — , 216, 351.
Pennsylvania R. R.

locomotives, 857, 859.

track-tank, 853.
Pennyweight, 220.
Penstock. See Forebay.
Pentagon, 148.
Per, Percentage, etc., 40.

of grade, 255.

interest, annuities, 40.
Perch, 222.

linear, 220.

cubic—, 222, 235.
Percussion, center of — , 351.

drills, 676.
Perimeter. See also Circumference,

wet—, 523, 563.
Permanent way, 815.
Permutation, 40.
Perpendicular, to draw — , 93.
Perpetual snow, limit of — , 324.
Persian wheel, 687.
Petroleum, weight, 214.
Phrpnix .segment-columns, 497,

904, 912, 913.

Phosphor bronze,

permissible load, 762.

requirements. 754.

wire, strength, 920.
Phosphorus, 753, 764, 872.
Pi, symbol and value, 34.
Picks, wear, 801.
Pied, 226.
Pier, Piers,

abutment — , 619.

foundations, 582.

masonry, quantity in — , 628.

obstructions by — , 575, etc.

of suspension bridges, 768.
Pierre perdue, 683.
Piezometer, 518.
Pig iron ton, 216.
Pile, Piles, 589, etc.

adhesion of ice, 594.

bearing — , 590.

blasting of — , 950.

cost, 984.

in cylinders, 600.

drivers, 590, 591, 687.
gunpowder — , 690.
mfrs., 992.
steam — , 591.

driving, 590, etc.
by jets, 595.

factor of safety, 693.

foundations, 589, etc.

friction, 593.

grillage, 590.

head for— , 594.

hollow — , 596.

ice, adhesion to — , 694.

iron — , 594.

jet driving, 595.

loads for — , 692.

resistance of — . 592.

sand—, 599, 670.

screw — , 694.

sheet — , 590.

shoes for — , 593.

sustaining power, 692.

water jet for driving — , 595.

withdrawal, 594.
Pillar, Pillars, 495, etc., 760, 761,
901, etc., 907, etc., 963, etc.

of angle-iron, 497, 912.

capitals of — , shapes of — , 498.

Carnegie Z-bar—, 901-903.

of channel-iron, 497, 912.

ends of — , shapes, 495, 498.

factor of safety, 909, 912.

Gordon's formula, 495.

hinged ends, 495.

of I-beams, 497, 912.

iron—, 495, etc., 760, 901-913.
factor of safety, 909, 912. .

masonry — , strength, 923.

oblique — , 498.

Phoenix segment. 497, 904, 912.

pin-ended — , 495.

radius of gyration, 496.

with rounded ends, 495.

safety factor of—, 909, 912.

1064

INDEX

Pillar— Plaster.

Pillar, Pillars — continued,
segment — . See Phoenix — .
steel — . See Pillar, iron — .
"straight-line" formulas, 761, 902.
strength—, 495, 760, 901-913,

963, etc.
T-iron— . 497, 912.
wooden—. 761, 764, 963, etc.
Z-bar-r, 901-903.
Pillow-block, defined, 1032.
Pin, Pins,

connections, 721, 724, 725, 747,

762.
-end columns, 495.
surveying — , 282.
Pine,

pillars, 963, etc.

strength, 476, 499, 957, 958, 963.
weight. 214.
Pinions and wheels, 420.
Pint, 223, 224.
Pintle, defined, 1032.
Pipe, Pipes,

air- valves for — , 662.
areas and contents — , 197, 526.
bends in — , 537.
branches. 661.
brass and seamless — , 919.
bursting of—, 513, 663, 665, 668.
thickness required to pre-
vent— , 511. 513.
bursting pressure in — , 518.
cast-iron—, 653, 658, 662, 876.
cost of — and laying, 658.
weight, 656, 658, 876.
cement and iron — , 657.
concretions in — , to prevent — ,

655.
contents and areas, 197, 526.
copper seamless — , 919.
cost of — . 658.
cost of laying — , 658.
cost and mfrs. — , 994.
coupliujgs for-, 656, 660, 882.
cracks in — , 661.
curves in — . 537.
diam. of—, 524. 653, 654, 656.
actual, nominal — , 526, 882.
for water-supply, 653.
square roots of — , 526.
discnarge from — , 616, 522.
drain — , 575.
drawn brass — , 919.
ferrules for — , 664.
flexible joints for — , 661.
flow in — , 516. etc.

Kutter's formula, 523, 563, 564.
galvanic action in—, 656.
galvanized — , 664.
gates for — , 666.
gutta-percha — , 657.
iron — ,

cast—, 653-656, 658-662, 876.

weight, 656, 658, 876.
and cement, 657.
fittings for — , 661, 882.

Pipe, Pipes — continued,
iron — , continued.

joints for—, 660, 661, 882.

flexible—, 661.
laying — , 658, etc., 660.
lead—. 664. 918.

thicknesses of — , 513.
mat^ial of — , effect on velocity,

523.
to mend — , 661.
obstructions in — , to prevent — ,

655.
pressure of water in — , 511, 518.
seamless — , 919.
service-^, 657, 664, 918.
sleeves for — , 661.
stand — , 663.
steam — , 882.
stop-valves for — , 666.
street — , 653. ^
tapping of — , 657, 664.
terra-cotta — , 575.
thickness required, 511, 513, 656.
valves for — , 666.
of varying diameter, discharge

through — , 531.
velocity in—, 522-524.
water—, 653, 657, 876.

cost of — and laying, 658.
freezing, anti-bursting device,
665.
weight, 656, 658, 876, 882.

oT water in — , 525.
wooden — , 657.
wrought iron—, 656, 657. 882.
diam. actual and nominal',

526. 882.
weight, 656, 882.
Pitch,

defined, 1033.

effect of — , on wind pressure, 714.
of rivets, 776.
of roofs, 970.
of screw, 436.
weight — , 214.
Pitman, defined, 1033.
Pitot's tube, 536, 561.
Plane, Planes, 148.
of couple, 404.
of flotation, 514.
of gravity, 389.
inclined — .

See Inclined Plane,
of moment, 360.
surfaces. 148.
trigonometry, 150.
Plank.

board measure table, 269.
in foundations, 582.
sheet piling, 590.

thickness for a given pressura,
586, 648.
Plaster of Paris. 968.
effect on metals, 936.
price — , 985.
strength, 922, 923.
weight, 214.

INDEX.

1065

Plastering^— Pressure.

Plastering, 968.
Plate, Plates,

bed—, 750.

buckle—, 750, 885.

fish—, 820.

frog—, 837.

girders, 747.

bracing in — , 749.
details, 728.

^ass, 974.

iron — , prices, 986.

net — , 774.

parallel — , transit, 292.

resistance of — , 492.

steel — , tinned, 916.

strength, 492.

terne — , 916.

tie—, 816.

tin—, 916.
Platform,

cars, 865.

revolving — , 850.
Platinum, 214. 920.
Plenum process, 697.
Plows, cost and mfrs., 992.
Plug, fire— (fire-hydrant), 669.
Plumb level, to adjust — , 311.
Plumbago as a lubricant, 415.
Plummet, defined, 1033.
Plunger, defined, 1033.
Plus, 33, 782.
Pneumatic

dome, Hopkins' — , 665.

foundations, 596.
Pocket sextant, 297.
Point, Points,

ancien, 226.

of application of force, 333, 359

boiling^, 314, 326.
levelling by — , 314.

of curve, 780.

freezing — , 326.

frojs— , 835-843.

of intersection, 780.

material — , 358.

melting — , 317.

position of — , to find — , 156.

switch—, 828, 830.

of tangent, 780.
Pointers — , astronomy, 285.
Pointing with cement, 936.
Polar distance, 284.

and azimuth of Polaris, 290.
Polaris, 285-290.
Pole, linear measure, 220.
Pole, ndrth— , 284.
Polygon, Polygons, 148.

cord—, 377, 428.

force — , 374, 377.

irregular — , to find area of — , 160.

to reduce to a triangle, 159.

regular — , to draw — , 159.
Poljoiedron, Polyhedrons, 194.
Pond, discharge of — , time required

for — , 545.
Pony trusses, 692.
Pood, 227.

Poplar, strength, 476, 957, 968.
Porous bodies, specific gravity, 211.
Port, establishment of — , 328.
Portal bracing, 691.
Portland cement, 930, etc.
Posts, 359. See also Pillars.

design of — , 722, 733.

fence — , 854.

pivot — in turntables, 846.

and ties, 689.
Potential energy, 346.
Pouce, 226.
Pound *

equivalents of — , 235.

Fahrenheit, equivalents of — , 237.

sterling, value, 218.

weight, 216, 220.
Pouring-clamps for pipe joints, 660.
Powder, 214, 963.
Power, Powers^ See Steam, Water,
Wind, Animal, etc.

animal — , 685.

defined, 342.

fifth—, 67-69.
sq. rts. of — , 69.

finding — by log. diart, 76.

finding — by logarithms, 71.

finding — by slide rule, 76.

of horse, 683, 685, 852. See also
Horse-power.

in levers, 419.

of locomotives, 860, 861.

man — , 686.

second and third — , tables. 65, etc.

tractive — , 683.

units of — , conversion of — , 244.
Pratt truss, 692.
Precipitation (rainfall), 322.

in the U. S., table of details, 325.
Present worth 41, 42, 44.
Preservation

of metals by cement, 936.

of timber, 954.
Press, Presses, hydrostatic — , 506.
Pressed

brick, 927.

steel cars, 865.
Pressure. See Load.

of air, 320, 597.

in arches, 614, .616.

of atmosphere, 320.

barometer — , 320.
' levelling by — , 312.

center of — , 399, 501, 506, 614.

on centers of arches, 633.

in dams, 648.

distribution of — , 400.

of earth, 603, 607.

on foundations, 683.

head, 258, 618.

hydrostatic — , 501, etc.

on inclined planes, 349.

line of — , 430, etc.

maximum — ,

angle, prism, slope of — , 607.

in pipes, 511, 518.

plate, Gauthey's, 561.

1066

INDEX.
Preasn re— Ram mlny.

Pressure — continued,
in reservoirs, 651.
on retaining walls, 603, 607.
of running streams, 578.
of running water in pipes, 518.
steam cylinder — , 861.
transmission of — through liquids.

506.

unit — , conversion of — , 240.

of water, 501, etc., 516, etc.
in cylinders, 511.
in pipes, 511, 518.
plank to resist — , $86, 648.
running, 518, 578.
walls to resist — , 508.

of wind, 321.

on roof trusses, 714.
Price list, 983.
Prime,

defined, 1033.

number, 35.
Principal, in interest, 41.
Prints,

black-line—, 982.

blue—, 979.
Prism, Prisms, 195.

center of gravity of — , 395.

frustums of — , 195.

center of gravity of — , 395.

of max. pressure, 607.
Prismoid, 202.
Prismoidal formula, 202.
Profile, Profiles, 304.

paper, 978.

transformation of — , 611.
Progression, 39.
Projection, defined, 1033.
Proportion,

by logarithms, 71.

and ratio, 38.
Proportionals, 38.
Protection of bridges, 763, 764.
Protracting by chords, 143.
Puddle,

defined, 1033.

walls, 651.
Pug-mill, defined, 1033.
Pun, 359.

on tapes, surveying, 282.
Pulley, 428.
Pump, Pumps, 852.

chain — , 687.

cost and mfrs., 991.

day's work at — , 686, 852.

sand—, 599, 670.
Purlins, 713, 1033.
Push and pull, 359.
Puzzolan cement, 940.
Pyramid, Pyramids. 200.

frustum of — , 201.

^uart. Quarts, 222, 223, 224.
juintal, 226.

R.

Rabbet, defined, 1033.
Race, defined, 1033.
Rack-a-rock, 951.
Radii, Radius,

to find—, 161. 179.
of gyration, 352, 353, 495, 496,
892, etc.
square of — , 496.
mean, 523, 564.
of railroad curves, 784-786.
of turnouts, 840.
Rail, Rails, 817.

bending — , ordinates for — , 817.
cost and mfrs., 994.
creeping of — , 819, 820.
elevation of outer — , 787.
expansion by heat, 317, 819.
fence — , 854.
frog, 835.
guard or guide—, 750, 828, 833,

835.
joints, 819.

ordinates for bending — , 817.
outer — ,

elevation of — , 787.
requirements, 870, 872.
roads, 780-869.

acres required for — , 254.

ballast, 815.

bridges. See Bridge, Trusa,

Arch, etc.
construction, 855.
cost, 855.
cross-ties, 815.
resistance on — , 417.
roadway. 815.
shops, cost, 850.
slopes, 256-257.
spikes, 818.
switch, 824.
ties, 815.

time, standard — , 267.
track tank, 853.
traction on — , 860.
turnout, 824.
water stations, 851.
safety—, 833.
stock—, 828.
switch-length, 830.
way. See Railroad.
Rain, 322.
fall, 322

depths, equiv. volumes, 250.
equivalent of snow, 324.
gages, 324.

reaching sewer, rate, 675.
and snow, 322.
water, 327.
Rainy days, av. number of — , 325.
Ram, Rams,

hydraulic — , 678.

cost and mfrs., 091.
water — , 513, 663, 668.
Ramming concrete, 946.

INDEX.

1067

Random — ^Roller.

Random stone, 683.

defined, 1033.
Ratio,

elastic — , 458, 461.

and proportion, 38.
Reaction, 333.

end—, 860, 439.

in trusses, 699, 702, 714.

of fibers, 466.

of soils, elastic — , 593.
Reaumur thermometer, 318.
Recii>rocal, Reciprocab, 48-53.

or inverse proportion, 39.

on logarithmic chart, 76.

by lojKarithms, 71.

by slide rule, 77.
Rectangle, Rectan^^es, 157.
Rectangular

components, 369.

plates, strength of — , 492.
Recurring decimals, 38.
Reduction

of area, 752, 754, 873.

of figures, 160.
Redundant members, 720.
Iteflection, to measure heights by — »

155.
Refraction and curvature tables,

153.
Regular figures, 148.
Regular solids, 194.
Regulation of time-pieces by the

stars, 266.
Reinforcing plates, 724, 747.
Relative densitv, 210.
Renewal of bridges, 743.
Repair, Repairs,

of bubble-tube, 296.

of cars, 865.

of cross-hairs, 296.

of pipe, 661.

in reservoirs, 652.

of road, 801.

of rolling stock, 865.
Repeating decimals, 38.
Reservoir, Reservoirs, 650.

evaporation from — , 329.

for railroads, 852.
Resilience, 460.
Resistance

of cars, 417.

to flow, 523, 537, 563.

of foundations, 583, 592.

on grades, 860.

line, 430, 432, 434-436.

of piles, 592.

of plates, 492.

on railroads, 417.
Rnsolutes, 369.
Resolution of forces, 362, etc.
Rest, relative — , 358.
Resultant,

of forces, 362.

line of—. 430, 432.

of moments, 360.

of parallel forces, 382, 899.

sense of — , 366.

Retaining walls, 603.

masonry in — , quantity of — , 610L
612.

surcharged, 605.

theory of, 606.

transformation of profile, 611.
Reverse bearing, 277.
Revetment, 612.

defined, 1034.
Revolving bodies, 851.
Rhombj 157; 195.
Rhombic pnsm, 195.
Rhombohedron, 195.
Rhomboid, 157.
Rhombus, 157.
Rhumb-line, 277.
Ridge-pole, defined, 1034.
Right angle, to draw, 93.
Rigid bodies,

force in — , 330, 858.
Rigidity in bridges, 721.
Ring, Rings, circular, 186, 209.
Rip-rap. 583.

defined, 1034.
Rise of arch, 613.
River, Rivers. See Water, Rain, etc.

dams, 642.

fiow in — , 560.

scour of — , 577.
Rivet, Rivets, 772.

stresses in — , permissible — , 762.
Riveted

connections, 721.

joints. 749, 772.
Road, Roads,

cart — , repairs, 801.

grade. 255, 683.
tables, 255-257.

maintenance, 801.

rail — . See Railroad.

rollers, mfrs. of — , 992.

traction on — , 683.

-way, acres required for — , 254.
drainage of — , in arches, 628.
Rock, Rocks,

blasting, 948, etc.

broken, voids in—, 688, 810, 943.

channeling, 681.

crushers, cost and mfrs. of — , 992.

drill, 600.
hand — , 681.
machine — , 675.

removal 810. 811.

weight. 212, etc.

work in tunnels, 812.
Rocker bearings, 725, 730.
Rod. Rods,

of brickwork, 928.

cost and mfrs., 993.

equivalents of—, 220, 232.

upset — , 886.
Rolled iron. See Iron, wrought—,
and Steel.

cost and mfrs., 986.
Roller, Rollers,

anti-friction—, 417, 761, 846.

bearings, 725-728.

1068

INDEX.

Rollings— (Scrapers.

Rolling

friction, 414.

lift bridges, 697.

load. See Load, live — .

stock, 856. 865, 867.
resistance of — , 417.
Roof, Roofs,

acid fumes on — , 880, 970.

copper — , 918.

iron for — , 880.

lead—, 918.

leak in — , to stop — , 971, 973.

paintine, 764, 880, 972.

pitch of—, 916, 970.

sheet-iron — , 880.

shingle — , 971.

slate — , weight, 970.

tin—, 916.

trusses, 698, 713. See Truss,
loads on — , 764.
metal — , 740.
specifications for — , 764.

wind on— , 321, 713.

zinc — , 916.
Roofing,

cost of — , 989.
Root, Roots,

cube and square — , tables, 54.

of decimals, to find — , 67.

fifth—, 67, 68.

finding — by logarithmic chart, 76.

finding — by Icfgarithms, 71.

finding — by slide rule, 76.

of large numbers, to calculate — ,
66.

square — , tables, 64.
of diameters, 526.
of fifth powers, 69.
Rope, Ropes, 975.

cost and mfrs., 987.

strength, 922, 975.

wire—. 976, 977.
Rosendale cement, 931.
Rosin, weight, 214.
Rot, dry—, 954.
Rotary

drills, 676.

motion, 351.
Rotating bodies, 351.
Rotundity of the earth, 163.
Rough casting, 973.
Roughness,

coefficients of — , 523, 564, 565.
Rubble,

adhesion to mortar, 926.

arches, 616.

cost, 602,

defined. 1034.

proportion of mortar in — , 213.

retaining walls, 610.

strength, 923.

voids in — , 688, 799.

weight, 213.
Rule, Rules,

of three, 39.

t\vo-foot — , to measure angles
by—, 96.

Run-oflf, equivalents of — , 251, 252
Rupture, modulus of — , 468.
Russian weights and measures^

227.
RUtger's process, 955.

s.

Sachine, 227.
Safety,

castings, 824, 833.

factor of — ,
for beams, 959.
for piles, 593.

for pillars, 495, 909, 912, 913.
for retaining walls, 605.
for suspension bridges, 767.

rail, 833.
Sag,

of tape, correction for — , 282.

in trusses, 718.
Salt,

effect of — on mortar, 926, 936.

water,

effect of — on iron, 327, 594.
weight, 326.

weight, 214.
Sand,

augers, 670.

blasting of — , 950.

cement, 937.

in cement, effect, 931, etc.

for cement, quantity, 935.

for centers, striking, 633.

in concrete, 943, etc.

cost, 985.

dredging, 580.

effect of- — in cement, 932, etc.

excavating in — , 800.

for foundations, 582.

natural slope, 419, 610.

penetrability, 593.

pUes, 599, 670.

in plaster, 968.

pressure, 603.

price of — , 985.

pump, 599, 670.

required in mortars, 931,

retaining walls for — , 603.

slope of — , natural — , 419, 610.

specific gravity, 211, 214.

stone, expansion by heat, 317.
strength, 476, 922, 923.
weight, 214.

sustaining power, 583, 593.

voids in—, 214, 935.

weight, 211, 214.
Sandage, 216.
Sap 954

Scabble, defined, 1034.
Scale, Scales, track — , 854.
Scantling, defined, 1034.
Scour of streams, 577.
Scrapers,

cost and mfrs., 992.

earthwork by — , 806.

INDEX.

1Q69

Stereedins— StatUUiir-

Soreedins, 968.
Screw, Screws, 436.

Archimedes, 687.

cylinders, 594.

levelling—, 292, 307.

piles, 59'^

standard dimensions, 883.

for striking centers, 633.

thread, metric — , 883.
Scribe, defined, 1034.
Sea,

mile, 220.*

tides, 328.

water, 326, 328, 594.

worms, 954.
Seamless,

pipes and tubes. 919.
Seasoning, 954.
Secant, 97.
Second, Seconds,

in decimal of a degree, 95.

equivalents of — , 236.

to estimate — , 266.

pendulum, 216.

of time, defined* 265.
Section,

equivalents of — , 222, 233.

of land, area, 222, 233.

of members, minimum — , 722.
in timber trusses, 733.

method by — , stresses in truases,
700.

modulus, 467, 468, 473, 892, 894,
896, 898.

net — in tension members, 759.
Sector,

center of gravity, 393.

circular, 186.

spherical — , center of gravity,
396.
Secular magnetic variation, 301.
Sediment in reservoirs, 651.
Segment, Segments,

circular—, 186, 187, 394.

colunms. Phoenix — , 497, 904,
912, 913.

spherical — , 208.

center of gravity, 395.
Sellers' standard dimensions of

bolts, etc.. 883.
Semi-
circle, center of gravity, 391.

parabola, center of gravity,
394.
Sense,

of force, 359.

of moment, 360.

of resultant, 366.
Separators for I-beams, 900.
Series, arithmetical and geometri-
cal— , 39.
Service pipe, 657, 664, 918.
Set-screw, defined, 1034.
Setting

of cement, 930, etc.
Settling

of arch. 432.

Settling — continued,
of backing, 604.
of centers, 633, 640.
of embankment, 799.
Sewer, Sewers,
cost, 989.
flow in — , 574.

Kutter's formula, 523, 563. 564.
rain water, rate of reaching — ^

575.
velocities in — , 574. 575.
Sextant,

angles measured by — , 152.
bpx or pocket — , 297.
center of gravity of — , 393.
Shackle, defined, 1034.
Shadows, equal — from the sun,

location of meridian by — , 288.
Shaft,

revolving — , longitudinal friction,

419.
of tunnel, 812.
Shafting,

friction, 416.
strength, 500.
Shale, weight — , 214.
Shapes, structural — , tables, 892-
'898.
T— . 898.
Sharpening tools, cost, 801.
Shear, Shears,
in beamus, 446.
in continuous beams, 489.
defined, 1034.
diagrams, 479.

trusses, 702, 706.
double and single — , 499, 774.
horizontal — in beams, 478.
influence diagrams for — , in

beams, 450.
and moments, relation between — ,

452.
in trusses, 702.
web stresses, 706.
Shearing,

of rivets, 774.
strength, 499.

of cements, 934.
stresses, permissible — , 762.
in timber construction, 732.
Sheet, Sheets,
copper — , 918.
iron—, 880.

corrugated — , 880.
galvanized — , 880.
roof,. 880.

and steel, cost and mfrs., 986
lead, 918.

metals, thickness, 887-890.
piles, 590.
zinc, 916.
Sheeting of centers, 631, etc., 639.
Shell,
-lime, 926.
spherical — , 208.
weight, 875. 877.
Shilling, value, 218.

1070

INDEX.

81ilnffle«— filpberieal.

Shingles, 971.
Shoes,

bridge—, 721.

for piles, 593.
Shops, railroad — , cost, 850.
Shore, defined, 1034.
Short ton, 216.
Shoveling earth, 800.
Shovels, wear of — , 801.
Shrinkage of embankment, 799.
Sidereal, day, month, time, year,

266.
Sieves, for cement tests, 938.
Signal target, 826, 829. 833.
Silica cement, 937.
Silicate of alumina, 930.
Silver,

coins, etc., 218.

strength, 920.

weight, 214, 219.
Similarity, geometrical — , 92.
Simple interest, 41.
Sine, Sines, 97.

logarithmic — , 72.

natural — , defined, 97.
table. 98.

by slide rule, 77.
Single

riveting, 772.

rule of three, 39.

shear, 499, 774.
Sinking fund, 43.
Siphon, 520.
Skew ■

back, 613.
defined, 1034.

bridge, 697.
Skidding of wheels, 413.
Slacking of lime, 925, 926, 930.
Slag cement, 940.
Slaking, 925, 926, 930.
Slate, 969.

expansion by heat, 317.

roofs, weight, 970.

strength,

compressive — , 923.
tensile—, 922.
transverse — , 476.

weight, 214.
Sleeping cars, 865.
Sleeves for pipes, 661.
Slide rule, 73.
Slope, Slopes,

angle of — , 255, 256.

description of — , 255, 256.

earthwork, description of — , 256.

hydraulic—, 523, 564.

instrument, 256, 311.

Kutter's formula, 523, 563, 564.

of maximum pressure, 607.

natural—, 419, 604, 606, 610.

railroad — , 255, 256.

structures built upon — > 424.

tables, 255-257.

of tapes and chains, corrections
for—. 283.

Slope, Slopes — continued.

in tunnels, 812.
Sloping weirs, 558.
Sluice, defined, 1035.
Sluices in dams, 645.
Snow, 214, 323.

load, 713.

rainfall equivalent, 324.
Soakage, loss by—, 329, 561, 651.
Soap,

as a lubricant, 415.

stone, weight, 214.

wash for walls. 928.
Soffit, defined, 613, 1035.
Soil, Soils,

boring in — , 670.

dredging in — , 580.

excavation of — , 800.

leakage through — , 329, 661, 65L

penetrability, 593.

pressure of — , 603.

reaction of — , elastic — , 593.

scour, 577.

sustaining power of — , 583, 593.

weight, 212. See under Earth.
Solar time, mean and apparent-—.

265.
Solid, Solids, 194.

center of gravity, 396.

defined, 92.

expansion by heat, 317.

floors. 721, 750, 914.

measure. 222, 234.
metric- 225. 234.

mensuration of — , 194.

specific gravity. 210.

surface of — ,

center of gravity, 395.
Sound, 316.

Soundness of cement, 940.
Southing, 274.
Sovereign, 218.
Span, 613, 759.
Spandrel, 613-618.

defined, 1035.
Spanish weights and measures,

227.
Spanner, defined, 1035.
Specific gravity, 210, etc.
Specifications,

for bridges and buildings, 745.

for combination bridges, 763.

for iron and steel, 870.

for roof trusses, steel framework
and buildings, 764.

for wooden bridges, 763.
Speed, Speeds,

o! teams, 801, 806.

of trains, to estimate — , 866.
Spfelter. See Zinc.
Sphere, Spheres. See Spherical,
204, 205. 222, 396, 875, 877, 91&

volume of — , 222.
Spherical

sector, center of gravity, 896.

segment, 208.
center of gravity, 395.

INDEX.

1071

Spberical— Steel.

Spherical — continued .
shell, 208, 875, 877.
zone, 208.

center of gravity, 396.
Sphericity of the earth, 153.
Spheroid, 208.

center of gravity, 395.
Spigot, defined, 1035.

in pipe joint, 660.
Spikes, 818.
Spindle,

circular — , 209.
torsional stress in — , 500.
Splice bars, requirements, 872.
Splice, timber — , 736.
Split switch, 828-830.
Spreading of earth, 801.
Spring, Springs,
of arch, 613.
in foiuidations, 583.
frog, 838.
Spruce,

strength, 476, 499, 957.

958.
weight, 214.
Spudding, 672.
Spur-wheel, defined, 1035.
Square, Squares. See also Powers,
area, 157.

equivalents of — in circles, 161.
center of gravity, 391.
measure, 222, 233.
conversion table, 233.
metric — , 225.
mensuration of — , 157.
of radius of gyration, 496.
of roofing, 970, 1035.
roots, 54.

of decimals, to find — , 67.

of diameters, 526.

of fifth powers, 69.

of large niuubers, to calculate

— , 66.
tables, 54.
sides of—, 157, 161.
tables of — , 55.
StabUity, 422, 514.

of arches, 430, 432, 620.
of dams, 433, 510.
frictional — , 409.
on inclined planes, 424.
of retaining walls, 603.
Stable equilibrium, 387, 514.
Stadia hairs, 293.
Stand, switch — , 826.
Stand pipes, 663.
for railroad water-station, 852.
ior water-works, 663.
Standard
railway time, 267.
wheel loads, 705, etc., 755, etc.
Starlings, defined, 1035.
Stars, to regulate a watch, etc.,

by—, 266.
Static friction, 407.
Statics, 330, 358.
of arch, 430, 432.

Statics — continued.

of beam, 437, etc., 466, etc.

graphic-, 428-431, 435.

of masonry dam, 430, 433-436.

of trusses, 698, etc.
Station, Stations,

in surveys, 309.

water — , 851.

way — , cost, 854.
Stationary engines, cost and mfr&

of—, 990.
Stays, cable, 766.
Steam,

dredges, 580.

engines,

locomotives, 856.
pumps — , 852.

excavator, 808.

pile drivers, 590, 691.

pipes, 882.

rock-drill, 675.
Steel,

angles, 896, 898.

beams, 476, 892.
. bending tests, 873.

in bridges, requirements, 751.

cars, 865.

castings in bridges. 754.
requirements, 872.

channels, 894.

cohesive strength of — , 920.

columns. See Pillars, iron — .

composition of — , 763, 872.

compressibility, 459.

compressive strength, 921.

cost and mfrs., 986.

ductility, 459.

elastic limit, 459.

expansion by heat, 317.

forgings, requirements, 872.

framework, specifications, 764.

friction, 411.

I-beams, 892.

manipulation, 751.

manufacture, 751, 870.

modulus of elasticity, 459.

open hearth — , requirements, 872,

pillars. See Pillars, iron — .

plates, tinned, 916.

price, 986.

rails, frogB of — , 835.

requirements, 751, 872.

roof trusses, 740.

rope, 976, 977.

shearing strength, 499.

shop work on — , 751.

specifications, 870.

strength, 476, 499, 500, 870, 920,
•921.

stresses in — , permissible — , 760.

stretch, 459.

structural — , requirements, 872.

tensile strength, 920.

tests, bending — , 873.

torsional strength, 500.

transverse strength, 476.

weight, 214, 877, 878.

1072

INDEX.

St«el~S(rae(<unil*

Ste«l — cont inued.

wire, 891.

rope, 976, 977.

yard, 383.
Stere, 224, 22.5, 235.
BtifFeners, 748.
Stiffness in bridges, 721.
Stirrups, timber framing, 734.
Stock rails, 828.
Stone, Stones.

arch—, 613.

in arches, quantity, 622.

artificial — , 943.

ballast, 815. 855.

beams, 476. 924.

bridges, 613.

centers for — , 631.

broken — , voids in — , 688, 943.

cohesive strength, 922.

compressive strength. 923.

cost, 985.

crushers, 943.

cutter's day's work, 601.

dams, 400, etc., 430, etc., 433,
etc., 508, 510.

dressing. 601.

drilling. 600.

expansion by heat, 317.

friction. 411.

key—. 613.

quantity of — , in arches, etc.,
622.

quarrying, 600. 601.

random — , 583.

strength, 476, 922, 923.

tensile strength, 922.

transverse strength, 476.

weight, 212, etc.

work, 600, 809.

mortar required for — , 931.
strength, 923.
weight, 213.
Stop, Stops,

corporation — for pipes, 657, 664.

valves for water pipes, 666.
Storage reservoirs, 652.
Stove-up, defined, 1035.
Strain, Strains, 454. 455.
Straps, timber framing, 735.
Stratum, defined, 1035.
Stream, Streams,

abrasion by — . 577. 578.

flow in—. 560.

-flow and precipitation, relation
between — , 323.

to gauge — , 560.

horse-power of — , 578.

pressure of running — , 578.

scour of — . 577. 578.

virtual head, 578.
Strength. Strengths. See also
article in question.

of arches, 368. 430, etc., 613.

of beams. 466. 473, 476, 478, 892.

of cast iron. 874.

of cement. 932. etc.

tests for — , 939, 940, etc.

Strength, Streiurths — continued.

of chains, 915^

of channels, 894.

cohesive—, 454, 920, 922, 957.

compressive—, 454, 921, 923, 968.

of concrete, 944.

of cylinders, 511.

of iron, 476, 499, 500, 870, 907,
etc., 920.

of materials, 454.

of piles, 592.

of pillars, 495, etc., 901, etc.,
907, etc., 963, etc.

of plates, 492.

of retaining walls, 603.

of riveted joints, 772, etc.

of shafting, 500.

shearing — , 499.

of steel, 476, 499, 500, 870, 920.
etc.

tensile—, 454, 920, 922. 957.

of timber, 476, 499. 500, 764
957, 968.

torsional — , 499.

transverse—, 466, 473, 476, 478,
892.

uniform — , beams and canti-
levers of — , 486.

of wood, 476, 499, 500, 764, 957,
958.
Stress, Stresses, 359, 454, etc.

alternating — , 761.

bearing — , permissible — , 762.

bending — in bridge members, per-
missible— , 762.

in bridges, permissible — , 759.

combined longitudinal and trans-
verse—, 493, 724, 762.

components, 371.

compound — , 493, 724, 762.

fiber—, 466, 467, etc.
and deflection, 481.
permissible — , 762.

in bridge trusses, 759, 764.
in roof trusses, 764.

range of — , 465.

repeated — , 465.

shearing — , permissible — ^ 762.

in truss members, 698.

in trusses, graphic method, 703.

unit — , 456. See also Stress,
fiber — .
in beams. 467.

wind— in bridges, 710. 768.
Stretch,

of materials, 459.

of tape, correction for — , 282.

of truss members, 718.

unit — , 456.
Stretcher, defined, 1036.
Strike, defined, 1036.
Striking of centers, 631, 633, 640.
Stringers,

in trusses, 713, 720, 749.
Structural

shapes, tables, 892, etc., 986.

steel, 872.

INDEX.

1073

S(ra*— Tensile.

Strut, Struts, 689.

design, trusses, 722, 733.
and ties, criterion for — , 359, 699.
Stub switch, 824, 825.
Stubs gauge, 890.
Stucco, 968.

Stuffing box, defined, 1036.
Stumps,

blasting of—, 960.
Sub-delivery,

cost, 855.
Submerged weirs. 554.
Subterranean temperature, 320.
Subtraction of fractions, 36.
Sub-verticals, 694.
Suddenly applied loads, 460. 486.
Sulphur,

in steel, 753, 872.
weight, 214.
Summation

of deflections in trusses, 720.
of forces, 466.
Sump, defined, 1036.
Sun,

dial, to make — , 268.

equal shadows from — , location

of meridiaa by — , 288.
mean — » 265.
Superelevation, 787.
Supplement of angle, 94.
Supported joints, 819.
Surcharged walls. 605. 609.
Surface, Surfaces,
neutral — , 466.
per length,

conversion of — , 238.
pressure of water against — , 501,

etc.
units of — ,

conversion of—, 233, 238.
velocity, 560.
Surveying, 274.
Suspended joints, 819.
Suspender,

hip — , stress in — , 709.
Suspenders of suspension bridges,

770.
Susp>ension bridges, 765.
Swage, defined, 1036.
Sway bracing. 691. 710, 749.
Swing bridges, 696.
Switch, Switches, 824.
Swivels, defined, 1036.
Sycamore,

strength, 476, 957, 958.

weight, 214.
Symbols,

mathematical — , 33.
Symmetry,

axis of — , 514.
Synclinal axis, defined, 1036.
Ssmodic month, 266.
Syphon, 520.
System, metric — , 225.
Systfeme,
ancien, 226.
usuel, 226.

6S

T.

T, Ts,
'defined. 1036.
iron, 896, 898. ^12.
rails, 817.

shapes, 896, 898, 912.
Table, Tables. See the article in
question,
conversion — of units of measurea,

weights, etc.. 228.
tnrn~-~ 844
Tackle, defined. 1036.
Tallow, 214, 415.
Talus. 612.
Tamp, defined, 1036.
Tamping, nitro-glycerine. 948.
Tangent, Tangents, 97. 98.
to circles, to draw — , 162.
to an ellipse, to draw — , 190.
logarithmic — , 72.
natiuul— , 97, 98.
to a parabola, to draw — , 193.
screw, 293, 307.
by slide rule. 77.
Tangential

angles, table, 784-786.
component, 369.
distance, table, 784-786.
Tank,

of tender, capacity. 856, 860.
thickness, 506, 854.
track — , 853.
water — , 851, 856, 860.
Tapes,

surveying — , 282.
cost and mfrs., 993.
Tapping.

of pipes, 657, 664.
of trees, effect on timber, 957.
Tar, weight, 214.
Target, signal—, 826. 829, 833.
Tarpaulin, 946.
Teams, speed of—, 801, 806.
Temperature. See Heat. 317.
of air, 320.

altitude, effect on — , 320.
corrections for tapes, 283.
effect of — on
cement, 932.
evaporation, 329.
metals, etc., 317.
rails, 317, 819.
strength of iron, 874.
surveying chains, 274, 283.
velocity of sound, 316.
weight of water, 326.
subterranean — , 320. -
thermometers, 318.
Templet, defined, 1036.
Tender, Tenders, 853, 856.

scoop, 853.
Tensile

strength, 454, 920, 922, 932, 957.
of cement, 932, etc., 939.
of chains, 915.
of riveted joints, 772, etc.

1074

INDEX.

TeBslon— TtacIk.

Tension,

and compression, 359.

members, 722, 732,746.
flexible and rigid — , 721.
net section, 750.

in tapes, 282.
Tents, cost and mfrs. of — , 993.
Teredo. 964.
Terne plates, 916.
Terra-cotta pipes, 575.
Test, Tests. See Requirements.

bending — , iron and steel, 873.

borings, 582, 670, etc.

of cement.

Am. Soo. C. E., 937.
U. S. Eng'rs. 940.

of completed bridges, 753.

of full-sise eye-bars, 753.

-pieces, iron and steel, 870.

of surveying instruments, 293,
etc.
Testing machine for cements, 941.
Tetrahedron, 194.
Thawing, effect of — on cement,

932.
Theodolite, 296.
Thermometers, 318. *
Thimble, defined, 1036.
Thin partition, flow through — , 541.
Third,

middle—, 402.

proportional, 38.
Three,

rule of — , 39.

-throw switch, 830.
Throat of frog, 835.
Through trusses, 692.
Throw,

defined, 1036.

of switch, 827.
Thrust,

in arch, 430, 432.

line. 430, 432, 434-436.
Tides, 328.
Tie, Ties,

cost of — , 994.

cross — , 815, 855.

land—, 612.

plates, 816.

and struts, criterion for — •, 369,
699.

in trusses, 689.
Timber. See also Wood, Wooden,
etc.

beams, 760, 762, 764, 959, 962.

bled — , strength, 957.

board measure, table, 269.

bridges, 732-740.

cohesive strength, 957.

columns. 96.3, etc.

compressibility, 459.

compressive strength, 958.

cost, 984.

crushing strength, 958.

dams. 642.

decav of — . QUA.

ductility, 459.

Timber — continued.

elastic limit, 469.

friction, 411.

joints, 733, etc.

limit, elastic — , 459.

modulus of elasticity, 459.

pillars, 761, 764, 963, etc.

preservation, 954.

requirements, 754, 760, 764.

roof trussed, 716, 732. 742.

shearing strength, 499.

strength, 476. 499, 500, 760, 764.
967, 968.

stresses in — , 760, 764.

stretch, 459.

tensile strength, 957.

for ties, 815.

torsional strength, 500.

transverse strength, 476.

trestles, 813.

turntables, 848.

weight, 212.
Time, 266.

effect of — , on strength of ce-
ments, 932.

local—, 287.

-piece, to re^^ulate — by stars, 266.

standard railway — , 267.

units of — , conversion of — , 236.
Tin, 916.

elastic limit, etc., 459.

expansion by heat, 317.

leaded—. 916.

roofing — , 916.

strength, 920, 921.

weight, 215, 877.
Toe of switch, 826, 828, 830.
Toggle joint, 427.
Toise, 226.
Ton, 216, 220.

of coal, volume of — , 215, 222.

(2240 lbs.), equivalents of — , 236.

-mile, 867.

net — , 216.
Tonelada, 227.
Tongue,

of frog, 834.

switch, 828.
Tonite. 951.

Tonnage rating of locomotives, 862.
Tonne, or metric ton, 226, 236.
Tonneau, 226.
Tools, wear of — , 801.
Top heading. 812.
Torpedoes, nitro-glycerine — , 948.
Torsion. 499.
Towers,

of suspension bridges, 768. 770.

valve—, 652.
Towne lattice truss, 694.
Tracing cloth and paper, 978.
Track. See Rail.

gauge. 827.

laying, cost, 866.

scales. 854.

tank. 853.

trough. 853.

^

INDEX.

1075

Traction— Unit.

Traction, 683.

of cars, 860.

on grades. 860.

of horses, 683. 685.

of locomotives, 860.
Trailing switch, 824.
Train,

centrifugal force of — , 758.

drag of — on bridge, 758.

earthwork by — , 807.

-shed roof, Broad St., Phila., 740.

speed of — , to estimate, 866.
Transit, Transits,

the engineer's, 291.
cost and mfrs., 993.
Transmission,

of force, 358.

of pressure in liquids, 506. ■
Transportation of bridges, 743.
Transverse

and longitudinal stresses com-
bined, 493, 724, 762.

strength, 466.
of concrete, 945.
Trap rock, weight, 214.
Trapezium, 158.

center of gravity, 392.
Trapezoid, 158.

center of gravity, 392.
Trapezoidal notch, 559.
Tread

-wheel, 590, 686.

of wheel, 821.
Trees, blasting of — , 950.
Trembling of dams, 648.
Tremie, 946.

Trenching machine, mfrs. of — , 992.
Trenton wire gauge, 891.
Trestles, 813.
Triangle, Triangles, 148.

in or about a circle, 161. .

element of truss, 690.

force — , 367.

mensuration of — , 148.

right-angled — , 150.
Triangular truss, 692.
Trigonometric functions, 97, 98, etc.

logarithmic, 72.
Trigonometry, plane — , 150.
Trimmer, defined, 1037.
Trip-hammer, defined, 1037.
Tripod, 292.
Trough

floors, 721, 750, 914.

flow through — , 544.

track—, 853.
Troy weight, 220.
True or apparent solar time, 265.
Trundle, defined, 1037.
Trunnion, friction of — , 416.
Truss. Trusses, 689.

ana beams, comparison, 689.

bracing in—, 691, 710, 748.

for centers, 636, etc.

counterbracing, 690, 705, 712, 721,
738, 746.

diagonals, to find lengths of — ,160.

Truss, Trusses — continued.

end reactions, 439, 699, 702, 714

equilibrium of — , 437.

forces acting upon — , 437.

loads on — , moving — ,
See Loads, live — .

members, stresses in — , 698.

moments in — , 440, 443.

moving loads on — ,
See Loads, live — .

rafters of — , 691, 713, etc.

reactions, 439.

roof — ,

loads on — , 764.
specifications for — , 764.

specifications for — , 745.

in suspension bridges, 765.

weights of—, 731, 738.
Tube, Tubes. See also Pipes, Flow,
etc.

boiler—, 882.

brass seamless drawn — , 919.

bubble — , to replace — , 296.

copper seamless drawn — , 919.

flow in — , 516.

iron—, 882. ,

Pitot's— , 536, 561.

pressure of water in — , 511, 518.

seamless — , 919.

short — , flow through — , 540.

welded—, 882.
Tumbling lever, 826, 830, 833.
Tun, 216, 223.
Tunnel, 812.
Turbines, mfrs., 990.
Turf, weight, 215.
Turn"buckle, Turn buckles, 986.
Turnouts, 824, 839.
Turnpike, grades on — , 255.
Turntables, 844.
Turpentine, 957, 972.
Twaddell hydrometer, 211.
Tympan, 687.
Typical wheel loads, 705, etc., 755r

etc.

u.

U. S. See United States.
Undecagon, 148.
Underpin, defined, 1037.
Ungula, cylindric — , 199, 397.
Uniform,
live load, 705.
loads,

deflections, 485.
influence diagram, 703.
moments due to — , 444,
shears in beams due to, 447.
strength, beams and cantilevers,

486.
velocity, 331.
Unit, Units,

of force, 338, 358.
of measures, weights, etc., con-
version tables of — , 228.

1076

ISTBBX.

VuUr-Ymumoir,

Unit. Units— eontiinied.

of moment of inertia, 468.

pressures, conversion of — , 240.

of rate of work. 342.

stress. See Stress, unit — .

stretch. See Stretch, unit — .

of work, 341.
United States,

average precipitation in — , 822.

coins, 219.

gallon. See Gallon.

measures, 223.

railroad statistics, 867.

standard dimensions of bolts, etc.,
883.
Unstable equilibrium, 387, 514.
Unsymmetrical loading, 690.
Upper chord, 723. 733.
Upper culmination, 284.
Upset rods, 88€>.
Ursa minor and major, 285.

V.

for sinking

Vacuum process

cylinders, 596.
Value

per length, surface, time, voltrme,
weight, work, etc., conversion
of units of — . See Conversion
tables, 246. etc.

present. 42, 44.
Valve, Valves,

air — , 662.

cost and rafrs., 995.

defined, 1037.

four-way — , 667.

outlet — . 653.

stop — , 666.

tower — , 662.

for water-pipes, 666.
Vara, 227.
Variation,

of compass. 301.

line of no — , 300.

magnetic, 301.

vernier, 296.
Vegetation

in reservoirs, 652.
Vehicles, friction, 414.
Vein, contracted^ — , 541.
Velocity, Velocities,

of abrasion, 577.

accelerated — . 331.

through adjutages, 540.

affected by material of pipes,
523.

angular — , 351.

of approach, 556.

in channels, 560.

Kutter's formula, 563, 564.

critical — , 415.

defined, 331.

due to 8 given head, 539.

effect of — on friction, 412.

Velocity, Velocities — continued,
.equivalent to discharge per sor-

face, 253.
of falling bodies. 348, 539.
bead, 516.

for a given velocity, to find — .
527.
on inclined planes. 349.
Kutter's form«Ua, 523. 563. 564.
material of pipe, effect on — ,

523.
mean — , 522, 560.
through orifices, 530, 546.
of outflow, 539.
in pipes, 5'16; etc.
retarded — , 331.
in rivers, 560.
in sewNs, 674.
in short tubes^ 540.
of sound, 316.

theoretical — , of outftow, 539.
uniform — , 331.

units of — , conversion ot — ^ 2i2.
of wind, 321.
Vena contracta» 541.
Ventilation,

air, quantity required, 32a.
of tunnels, 812.
Venturi meter, 532, ete.
Vernier, 293.

variation — . 296.
Versed sines, 97.
Verst, 227.
Vertical, Verticals,
of buoyancy, 514.
circle, astronomy, 284.
defined, 92.
of equilibrium, 514.
of flotation, 514.
Vessel, Vessels,
air—, 663.

contents of~~, 19ft, 223.
floating — , 514, 515.
metallic — , effect of water on — ,
327, 917.
Viaduct, Viaducts. See Trestles,
and Trasses,
erection, 743.
Vibrating bodies, 350.
Vibration, 350.
Vinculum, symbol, 33.
Virtual head. 539, 578.
Vis viva, 343.
Voids,
in broken stone. 688, 943.
in concrete, 943.
in rubble, 799. 925. •
in sand, 214, 935.
Volume, Volumes,
of air, weights of — , convendoB

of—, 242.
equivalent depths, 251.
increase of*- of stone, 943.
occupied by eoal, 215.
unit — f conversion of—. 234.
of water, weights of — , 241.
Voussoir, 613.

INDEX.

1077

w.

Wagons, friction, 414.
Wales, defined, 1037.
Wall, Walls,

backing of—, 603.

battered, 605.

bricks, number in a sq. ft. of — ,

92^^-927.
cost, 601, 602.
dams, 508.
face—, 603.
foundations for — , 582.
incrustation of — , 929, 936.
to resist water pressure, 508.
retaining — , 603.
soap-wash for — , 928.
spandrel — •, 613.
stabUity of—, 508, 606.
surcharged — , 605, etc.
water,

to render impwvious to, 928.

to resist pressure of — , 508:
wharf — , 615, 611.
wing — , 624.
Walnut,

strength, 476. 957, 1)68.
weight, 215.
Warp, defined, 1037.
Warren truss, 692.
Washers, 883.
defined, 1037.
lock-nut or nut lock, 885.
Washes for walls, 928, 972.
Waste of water, 649.
Waste-weir,
defined, 1037.
for reservoirs, ^2.
Watch, to regulate — , by the stars,

266.
Water, 326. See also Pipes, Flow,

etc.
for boilers, 327.
boiling — , to measure heights

by—, 314.
brick work, to render impervious

to — , 928.
buoyancy, 513.
in cement, 932. 936, 938, 941.
cisterns, 512, 851-854.
column, 852.
compensation, 653.
composition of — , 326.
compressibility, 326.
in concrete, 944.
concrete under — , 946.
consumption of — , 649.
corrosion by — , 327, 594.
dams for — , 608, 642.
discharge. See Discharge,
effect on

cement, 932, 936, 938, 941.

dynamite, 950.

iron, 327, 594.

lime. 925, 926:
effect of sine on — , 328, 917.
evaporation, 329.

Water — continued,
for fire protection, 650.
flow of — . See Flow,
foot of — , etc. (pressure), equiva-
lents of—, 240.
foundations in — , 583.
freexing of—, 326, 328.

gates, 666.
ead of—, 258-260, 516.
horse-power. 578.
jet for pile-driving, 595.
leakage, 329, 561, 642. 649, 651.
for locomotives. 327, 852.
masonry, to render impervious

to—, 928.
meters, 532, 536, 662, 649.

cost and mfrs., 994.
motors, mfrs., 990.
pipe, prevention of bursting of — ,

by freezing. 665. \

in pipes. See Pipes, Velocity,

Flow, Discharge, Pressure, etc.
pipes, 653, etc.. etc.

cost of — , 658.
power, 578.
pressure. 501, etc.. 518.

in cylinders, 611.

in pipes, 511. 618.

plank, to resist — , 586, 648.

running — , 578.

still — , 601, etc.

on surfaces, 501, etc.

wall to resist — , 508.
rain-. 322, 327.
ram, 513, 663, 668.
resistance to moving bodies, 678.
running — , pressure. 678.
salt — , effect on metals, 327, 594.
scouring action, 577.
shed, defined, 1038.
sixe of commercial measures by

weight of—, 224.
stations, 851.
stonuEO of — , 650, etc.
supply—, 822, 649.
tank, thicknesses, 864.
traction on — , 683.
in tubes, flow of — , 616.
velocity. See Velocity,
volumes of unit weight of — , con-
version of — ,' 242.
walls, to render impervious to — ,

928.
walls to resist pressure of — , 608,

515.
waste of — , 649.

way, contraction of — , 575, 623.
weight, 241. 326.

in pipes, 625.

size of commercial measures
by—. 224.
wheel, 578.
works, 649.

depreciation, 46.
Watt,

equivalents of — , 245.

-hour, equivalents of — , 287.

1078

ISD£X.

Wax— Wood.

Wax. weight, 215.
Way,

permanent — , 815.

station, ooiit, 854.
Wear,

of 'cars, 865.

of locomotives, 864.

of tie8, 815.

of tools. 801.
Web. Webs.

members, 689.

in plate girders, 748.

plates, permissible shear, 762.

stresses, 702.
live load—, 706.
Wedge, Wedges,

mensuration of — , 203.

striking — , for centers, 631, 632,
640.
Week, 236, 265.

Weight, Weights. See also article
in question, 212.

of air, 320.

conversion of unit volumes,
242.

of beams,

compared, 478.
as load, 477.

of bridges, 731, 738.

of cement, 939.

of centers for arches, 639.

on driving wheels, 705, etc., 755,
etc., 856.

French—, old—, 226.

in levers, 419.

and measures, 216.

conversion tables of units of — ,
228.

metric—, 217, 226, 228, etc.

of roof covering, 713.

of roof trusses, 713.

Russian — , 227.

of snow, 323.

Spanish—, 227.

of steel railroad bridges, 731.

of substances, table, 212. See
also the article in question.

unit — , conversion of — , 235.

of water in pipes, 525.

of wooden bridges, 738.
Weir, 547.

discharge, formula for — , 549.

measunng — , 547, 646.

submerged — , 554.
Well, Wells,

artesian — , 671.

boring, 670.

contents, 197.

.machinery, mfrs., 992.

masonry, quantity in walls, 198.
Wellhouse process, 955.
Western elongation, 2^4.
Westing, 274.
Wet perimeter. 623, 563.
Wheel, Wheels,

barrows, earthwork by — , 803,
810.

Wheel. Wheels — continued.

base, 787, 789, 856, 1038.

centrifugal force, 355.

diac^am, 706.

driving — . 856.

loads on — , 705, etc., 755, etc.,
856-859.

guards, 750.

loads, 705, etc., 755, etc., 856-
859.

of locomotives, 856.

loads on — , 705, etc., 766, etc.,
856-859.

meters, 562.

Persian—, 687.

and pinions, 420.

skidding of — , 413.

tread—, 590, 686.

tread of—, 821.

water — , 578.
Wheeled scrapers, 805.
Whipple truss, 694.
White

effervescence on walls, 929, 036.

lead paint, 971. '

wash, 973.
Whitworth screw thread, etc., 883.
Winch, 686.
Wind, 321.

effect on suspension bridges, 766.

loads, 710.

mills, 852.

pressure on roofs, 321.

on roof trusses, 713.

stresses, 710, 758.
Wine measure, 223, 224.
Wing,

dam, defined, 1038.

defined, 1038.

of frog, 834.

walls, 624.
Wire, 887-891.

circular measurement of — , 888.

cost and mfrs., 987.

fence, 854.

f^uges, 887-891.

iron — , 891.

rope, 976. 977.

steel—, 891.

strength, 920.
WShler's law, 465.
Wood.

See Timber, Wooden, etc.

board measure, table, 269.

cohesive strength, 957.

compressibility, 459.

compressive strength, 958.

cost, 984.

crushing strength, 958.

ductility. 459.

effect of lime and mortar on — , 926.

elastic limit of — , 459.

friction, 411.

limit of elasticity, 459.

modulus of elasticity, 459.

pipe, cost, 996.

preservation, 954.

INDEX.

1079

Wood— Zone.

Wood — continued .

shearing strength, 409.

shingles, 971.

specific gravity, 212.

strength, 459, 476, 499, 600, 760
764. 967, 968, 969-967.

stretch of — , 469.

tensile strength, 967.

torsional strength, 600.

transverse strength, 476, 959.

weight, 211.
Wooden. See also Wood.

beams. See Beams, wooden — , 476.

bridges, 732, etc.

specifications for — , 763.

dams, 642, etc.

floors in bridges, 750.

joints, 734.

pillars, 761, 764, 963.

pipes, 657.

roof trusses, 716, 742.

trestles, 813.

turntables, 848.
Work, 341.

equivalence of — , trusses, 718.

of friction, 418.

and heat, conversion of units
of—, 237.

of overturning, 422.

per time [power], conversion of
units of — , 244.

units of — , 341.

useful — , 342.
Working beam, defined, 1038.
Worm,

defined, 1038.

fence, 854.

sea — , 954.
Worth, present — , 42, 44.
Wrecking car, 808.
Wrought iron. See Iron, wrought — .

pillus, breaking loads, 010.

Y-

Yard, Yards, 216, 220.

cubic — ,

of earthwork, 790.
equivalents of — , 222.

equivalents of — , 232.
Year,

civil—, 266.

equivalents of-^, 236.

sidereal — , 266.
Yield point, 459, 871, 873.

z.

Z-bars, 901-903.

flooring, 914.
Zenith, 284.
Zigsag riveting, 774.
Zinc, 916.

cost, 987.

effect of cement, mortar, etc.
on — , 926, 936.

effect of — on water, 917.

effect of water on — , 328.

expansion of — by heat, 317.

paint, 971.

paint on — , 880.

price, 987.

roofing, 916.

sheets, 916.

strength, 920.

compressive — , 921.

weight, 216, 875, 877, 878, 887.
Zone, Zones,

circular — , 186.

of circular spindle, 209.

parabolic — , 192.

spherical — , 208.

center of gravity of — , 396.

rH£ END.

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BAZIN'S WEIR EIPERMENTS.

Recent Emeriments on tlie Flow of Water oyer Weirs, etc.

By M. BAZIN,

Inspector-General des Fonts et Chausse'es.

TRANSUTED FROM ANNALES DES FONTS ET GHAUSSEES»
By ARTHUR MARICHAL and JOHN C. TRAUTWINE, Jr.

PUBLISHED IN THB

Proceedings of tlie Engineers' Club oL Philadelphia,

EXPERIMENTS

UPON

CONTRACTION OF THE LIQUID VEIN
ISSUING FROM AN ORIFICE,

AND UPON THE

Distribution of the Velocities Within 1

By H. BAZIN,

Inspecteur G6a6ral des Fonts et Chauas^es.

Tfaoslated by JOHN C. TRAUTWINE. Jr.. C E.

JOHN WILEY & SONS, Scientific Publishers.

43 AND 45 E. NINETEENTH ST., NEW YORK.

26

Trautwine's Civil Engineer's Pocket-Book

18th Edition, 1907

NOTES AND <X>fiRECTIONS

'Page S4, in table j

For ^^cironnifeFenoe -i- radiiis,**
Teaid *^ oironnif ereuoe + diameter.''

Page 74y^ lines 7 etc. below table;

The bantion doe^ not apply to powers where the expoiieiit
is a whole namber, such as squares, cubes, etc.

Page 198t last linei

read **dpne away with entirely. In a given volume, the
No. oif perches of 25 on ft, nlult by 0.926, -= the No. at
cu yds ; and No. of ou yds, div by 0.926, = No. of perchi^
of 25 ou ft."

Page 459» line ft;

Read ** stietota or compresaion, m feet^ in a length ci 10
feet."

Page 466, Paragr. 4;

The words *'ntar the middle** ^pply only to heamk. In a
cantilever y th# bending moment is greatest ii< the mppori.

Page 482, Foot nolef;

For * * horizontal, ' ' read ' ^ fixed in position. ' '

Page 484, line 10;

read "8 = 2»."

Page 524, lines 1 to 12;

Use, instead, rules "3d" and "4tli.** pp. 565-6.
(Diameter = 4 R = 4 X mean raclius.)

Page 580t table in foot note;

Under ** Diameter 12 inch."

lor "Slope, 0.04," read "Slope, 0.4."

pBj 16, first line t>elow first equation;

ciM '-^pes," reud "seven slopes."

Pai '

iim itm above Fig* 88;

r "page 7:W," reiid "page 788;"

lib Une below Fig. 88;

*T **page 7^," read "page 784."

June 11, 1907.

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