PRACTICAL A RITHMETIC: EMBRACING THE SCIENCE AND APPLICATIONS OF NUMBERS. BY CHARLES DAVIES, LL. D., P_OFESSOR OF HIGHER MATHEMATICS IN COLUMBIA COLLEGE ; AND ALGEBRAS AND ARITHMETICS. ADVERTISEMENT. 67481 The attention of Teachers is respectfully invited to the REVISED EDITIONS of Davies' Arithmetical Series FOR SCHOOLS AND ACADEMIES. 1. DAVIES' PRIMARY ARITHMETIC. The above Works, by CHARLES DAVIES, LL.D., Author of a Complete Course of Mathematics, are designed as a full Course of Arithmetical Instruction necessary for the practical duties of business life; and also to prepare the Student for the more advanced Series of Mathematics by the same Author. The following New Editions of Algebra, by Professor DAVIES, are commended to the attention of Teachers : 1. DAVIES' NEW ELEMENTARY ALGEBRA AND KEY. Entered according to Act of Congress, in the year one thousand eight hundred and sixty-two, BY CHARLES DAVIES, District of New York. 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 number 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. 4th. By employing the term “ fractional unit,” the same principles 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 a simple question, and then to express that idea in general terms under 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 favored me with valuable suggestions in regard to definitions, rules, and methods of illustration. Their generous appreciation of my labors has been both an encouragement and a reward. COLUMBIA COLLEGE, NEW YORK, } CONTENTS. PAGE : FIRST FIVE RULES. Subtraction of Simple Numbers, Multiplication of Simple Numbers, |