Physical Arithmetic

404 Pages·1885·46.44 MB·English

by  MacFarlane

PHYSICAL
ARITHMETIC.
BY
A. MACFARLANE, M.A., D.Sc., F.R.S.E.,
EXAMINERINMATHEMATICSINTHEUNIVERSITYOFEDINBURGH.
"Everyquantityisexpressed byaphraseconsistingoftwocomponents,
oneofthesebeingthenameofanumber,andtheotherthenameofathing
of thesamekind as thequantitytobeexpressed,butofacertain magni
tudeagreedonamong menasastandardorunit."
--CLERK-MAXWELL,Heat,p.75.
MACMILLAN AND
CO.
1885.
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Mr
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A^AA^MI
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PREFACE.
THIS book may be described as a treatise on applied arithmetic,
the applications being chiefly in physical science. Knowledge of
the elements of pure arithmetic is assumed, but the more ad
vanced methods are explained when their application happens to
occur.
The progress of physical science has caused the idea of the
unit to become more prominent in text-books on arithmetic and
;
the old form of rule of three has been replaced to a large extent
by what is called the unitary method, or the method ofreduction
to the unit. That method, in my opinion, very imperfectly
represents the reasoning process involved.
The method developed in this work may be called the equival
ence method. Each quantity is analysed into unit, numerical
value, and, when necessary, descriptive phrase. The rate, or
law, or convention, according to which one quantity depends on
one or more other quantities, is expressed by an equivalence.
These equivalences are of two kinds, absolute and relative the
former expressing the equivalence of dependence, the latter the
equivalence of substitution or replacement. Finally equivalences
are combined according to a form which is a development of the
Chain Eule.
The present work is a development of notes which I began to
take when a student on the subject of units and the reasoning-
processes involved in elementary calculations. My experience as
examiner has continued me in the opinion that an elementary
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PREFACE.
vi
work on the subject is a desideratum. In the carrying out of
this rather laborious task, I have received much encouragement
from Professor Tait and from Professor Chrystal, who has
pointed out the want of a text-book on physical arithmetic in his
reports as examiner ofHigh Schools.
The writings which I have studied most in this connection are
those of Clerk-Maxwell, Sir W. Thomson, Professor Tait, and
Professor Everett while for numerical data I have consulted
principally the Tables ofLandolt and Bornstein. I have gone on
the principle of comparing the values given by the different
authorities so as to retain the common figures, and cut off the
uncertain remainder. As a general rule every one of the figures
given may be relied on as significant, with the exception of per
haps the last,
The examples have been partly selected from the recent
examination papers of the several Universities, and partly pre
pared by myself either for examinations or to illustrate kinds of
calculation which were otherwise not illustrated.
ALEXANDER MACFAELANE.
EDINBURGH, 31stJuly, 1884.
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CONTENTS.
CHAPTER FIRST.
FINANCIAL.
HKCTION
rA(:I,
I. Value, - 1
II. Price, - 10
III. Profit and Loss, - ... is
IV. Mixture, - 22
V. Simple Interest, - 27
VI. Compound Interestand Annuities, - 34
VII. Sharesand Stocks, - 43
VIII. Exchange, - 50
CHAPTER SECOND.
GEOMETRICAL.
IX. Length, - 60
X. Angle, - 68
XL
Surface, 77
XII. Surface, continued, 84
XIII. Volume, 92
XIV. Volume, continued, 96
CHAPTER THIRD.
KlNEMATICAL,
XV. Time, - 101
XVI. Speed, - 108
XVII. RelativeSpeed, 113
XVIII. Velocity, 118
XIX. Angular Velocity, 121
XX. Rate of Change of Speed, 125
XXL
Acceleration, 132
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CONTENTS.
CHAPTER FOURTH.
DYNAMICAL.
SECTION PAGK
XXII. Mass, - 138
XXIII. Density, 143
XXIV. Specific Mass(Gravity), 148
XXV. Mass-vector, 153
XXVI. Momentum,- 156
XXVII. Force, - 15!)
XXVIII. Compositionof Forces, - 166
XXIX. Deflecting Force, - 170
XXX. Specific Gravity, - 176
XXXI. Pressure, 183
XXXII. Pressure ofaGas, 188
XXXIII. Work, - 192
XXXIV. Kinetic Energy, - 195
XXXV.
Power, 202
XXXVI. MechanicalAdvantage,- 204
XXXVII. Momentof aForce, 208
XXXVIII. Gravitation,- 214
CHAPTER FIFTH.
THERMAL.
XXXIX. Temperature, - 219
XL. Heat, - 222
XLI. Thermal Capacity, - 226
XLII. Latent Heat, 234
XLIII. Expansion of Solidsand Liquids, - - 239
XLIV. Expansion of Gases, - 246
XLV. Conductivity, - 251
CHAPTER SIXTH.
ELECTRICAL.
XLVI. Magnetic, - - 256
XLVII. Electrostatic, 263
XL
VIII. Electromagnetic, - - 270
XLIX. Resistance, - - - 276
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CONTENTS.
x
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CHAPTER SEVENTH.
ACOUSTICAL. ....
SECTION pAO,,
L. Musical Sound, - 287
LI. Velocity of Sound, . 291
CHAPTEE EIGHTH.
OPTICAL.
LIL Physical, - - - 297
LIII. Geometrical, - 395
CHAPTER NINTH.
CHEMICAL.
LIV. Composition, 309
LV. Atom and Molecule, - - 316
LVL
Combination, 321
LVIL Equivalence, 327
ANSWERS TO THE EXERCISES, - 332
INDEX, - - 350
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PHYSICAL ARITHMETIC.
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