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PHYSICAL ARITHMETIC.

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Aleiander
A. MACFARLANE, M.A., D.Sc., F.R.S.E.,

EXAMINER IX MATHEMATICS IN THE UNIVERSITY OF EDINBURGH.

“Every quantity is expressed by a phrase consisting of two components,
one of these being the name of a number, and the other the name of a thing
of the same kind as the quantity to be expressed, but of a certain magni-
tude agreed on among men as a standard or unit."

- CLERK-MAXWELL, Heat, p. 75.

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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 advanced 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 of reduction 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 equivalence 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 Rule.

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 confirmed me in the opinion that an elementary

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