OF

ARITHMETIC

BY

HOMERSHAM COX, M.A.,

FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

CAMBRIDGE:

DEIGHTON, BELL AND CO.

LONDON: GEORGE BELL AND SONS.

1885

PREFACE.

THE object of the present work is to give an account of the principles of Arithmetic, omitting all merely mercantile applications. I have endeavoured, as far as possible, in the explanation of the different methods and results to follow the order of historical development. In doing this I have been guided mainly by Cantor's "Geschichte der Mathematik," the chief work on the history of mathematics. I have also consulted Hankel's "Vorlesungen über die Geschichte der Mathematik," and Nesselmann's "Algebra der Griechen," and have verified many special statements by reference to the original authorities. The conception of the subject as a whole, and many of the details have been taken from the mathematical portions of the works of Auguste Comte and in especial from his last great work the "Synthèse Subjective." HOMERSHAM COX.

CAMBRIDGE,

August 13th, 1885.

PRINCIPLES OF ARITHMETIC.

INTRODUCTION.

ARITHMETIC is as the word implies the science which treats of the relations between numbers. In order to define this science we must then first of all consider what is the kind of questions to which numbers are applied and what are the fundamental relations which exist between them.

Numbers are used for two purposes, to count and to measure. In the first case numbers are employed to enumerate distinct objects and even thoughts or events; in the second case they express some continuous quantity such as a length or an interval of time in terms of some other quantity of the same kind. The latter use of numbers depends on the former, which has indeed been always the more common and familiar. It is in this former application that we naturally first consider numbers.

A number has no meaning apart from the objects to which it is applied. It is like any other abstract term, such as length or mass, not capable of definition. We must therefore define the property of two groups which are said to contain the same number of objects. It is