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 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 of High 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 of Landolt and Börnstein. 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 perhaps the last The examples have been partly selected from the recent examination papers of the several Universities, and partly prepared by myself either for examinations or to illustrate kinds of calculation which were otherwise not illustrated. EDINBURGH, 31st July, 1884. ALEXANDER MACFARLANE. |