ARITHMETIC: EMBRACING THE SCIENCE AND APPLICATIONS OF NUMBERS BY CHARLES DAVIES, LL. D., PROFESSOR OF HIGHER MATHEMATICS IN COLUMBIA COLLEGE; AND A. S. BARNES & COMPANY, NEW YORK AND CHICAGO. 1873. NEW YORK DEN LENOX FOUNDATIA NEW SERIES OF ARITHMETICS, 1904 By CHARLES DAVIES, LL.D., AUTHOR OF THE WEST POINT COURSE OF MATHEMATICS. The following named volumes are entirely new works, written within the past ten years, to conform to all modern improvement, and take the place of the author's older series. NO CONFLICT OF EDITIONS is possible, if patrons will be particular to order the book they want by its exact title. Whenever any change is made so radical as to be likely to cause confusion in classes, THE NAME OF THE BOOK IS CHANGED. Teachers using any Arithmetic by DAVIES not hereinafter enumerated, are not availing themselves of the advantages offered by THE NEW SERIES. Primary, Intellectual, and Practical Arithmetics constitute the Series proper. Other volumes are optional. DAVIES' PRIMARY ARITHMETIC. The elementary combinations, by object lessons. DAVIES' INTELLECTUAL ARITHMETIC. DAVIES' ELEMENTS OF WRITTEN ARITHMETIC. DAVIES' PRACTICAL ARITHMETIC. DAVIES' UNIVERSITY ARITHMETIC. ALSO, DAVIES' NEW ELEMENTARY ALGEBRA. AND A FULL COURSE OF HIGHER MATHEMATICS. Entered according to Act of Congress, in the year 1862, by In the Clerk's Office of the District Court of the United States for the Southern District of New York. P'L. A. PREFACE. ARITHMETIO embraces the science of numbers, together with all the rules which are employed in applying the principles of the science to practical purposes. It is the foundation of the exact and mixed sciences, and the first subject, in a well-arranged course of instruction, to which the reasoning powers of the mind are directed. Because of its great uses and applications, it has become the guide and daily companion of the mechanic and man of business. In the present work, a few general principles are laid down, to which all the operations in numbers may be referred: 1st. The unit 1 is regarded as the base of every number, and the consideration of it is the first step in the analysis of every question relating to numbers. 2d. Every number is treated as a collection of units, or as made up of sets of such collections; each collection having its own base, which is either 1, or some number derived from 1. 3d. The series of numbers expressing the relation between two different units of a number, is called the SCALE; and the employment of this term enables us to generalize the laws which regulate the formation of numbers. rin 4th. By employing the term "fractional unit," the same pr ciples are made applicable to fractional numbers; for all fractions are but collections of fractional units, these units having a known relation to 1. 5th. The presentation of the fractional units to the minds of young pupils, by means of a diagram, as exhibited in the Primary and Intellectual Arithmetics, has greatly simplified the operations in fractions; and they may now be placed before Denominate Numbers, where, in a purely scientific arrangement, they properly belong. In the preparation of the work, two objects have been kept constantly in view: 1st. To make it Educational; and, 2d. To make it Practical. To attain these ends, the following plan has been adopted: 1. To introduce every new idea to the mind of the pupil by simple question, and then to express that idea in general terms nder the form of a definition. 2. When a sufficient number of ideas are thus fixed in the mind, they are combined to form the basis of an analysis; so that all the principles are developed by analysis in their proper order. 3. The work has been divided into sections, each containing a number of connected principles; and these sections constitute a series of dependent propositions that make up the entire system of principles and rules which the work develops. Great pains have been taken to make the work PRACTICAL in its general character, by explaining and illustrating the various applications of Arithmetic in the transactions of business, and by connecting, as closely as possible, every principle or rule, with all the applications which belong to it. I have great pleasure in acknowledging my obligations to many teachers who have favoreu me with valuable suggestions in regard to definitions, rules, and methods of illustration. Their generous_reciation of my labors has been both an encouragement and a reward. A Key to this volume has been prepared for the use of Teachers only. |