TO THE NATIONAL ARITHMETIC, ON THE INDUCTIVE SYSTEM, COMBINING THE ANALYTIC AND SYNTHETIC METHODS; IN WHICH THE PRINCIPLES OF THE SCIENCE ARE FULLY EXPLAINED AND ILLUSTRATED. DESIGNED FOR COMMON SCHOOLS AND ACADEMIES. BY BENJAMIN GREENLEAF, A. M., NEW ELECTROTYPE EDITION, WITH ADDITIONS AND IMPROVEMENTS. BOSTON: PUBLISHED BY ROBERT S. DAVIS & CO NEW YORK: MASON, BAKER, & PRATT, 142 GRAND STREET. PHILADELPHIA: J. A. BANCROFT & COMPANY. ST. LOUIS: HENDRICKS & CHITTENDEN. CHICAGO: S. C. GRIGGS & CO. 1874. GREENLEAF'S NEW COMPREHENSIVE SERIES. AN ENTIRELY NEW MATHEMATICAL COURSE, FULLY ADAPTED TO THE BEST METHODS OF MODERN INSTRUCTION. GREENLEAF'S NEW PRIMARY ARITHMETIC. GREENLEAF'S NEW ELEMENTARY ALGEBRA. GREENLEAF'S NEW ELEMENTARY GEOMETRY. GREENLEAF'S ELEMENTS OF TRIGONOMETRY. Entered according to Act of Congress, in the year 1842, by Ir the Clerk's Office of the District Court of the District of Massachusetts. Entered according to Act of Congress, in the year 1848, by In the Clerk's Office of the District Court of the District of Massachusetts. Entered according to Act of Congress, in the year 1856, by In the Clerk's Office of the District Court of the District of Massachusetts Entered according to Act of Congress, in the year 1870, for renewal, by LUCRETIA K. GREENLEAF, widow, and BETSEY P. KENDALL, EMILY A. TEWKSBURY, and LYDIA K. SYKES, children, of BENJAMIN GREENLEAF, In the Office of the Librarian of Congress, at Washington. HARVARD COLLEGE LIBRARY GIFT OF WARREN H. MANNING MAR 2 1929 PREFACE: THE present edition of this work has been thoroughly revised and re-written, and also improved by the addition of much valuable new material, rendering it a sufficiently complete practical treatise for the majority of learners. The arrangement is strictly progressive; the aim having been to introduce subjects in an order most in accordance with the laws governing the proper development of mind. The rules have generally been deduced from the analysis of one or more questions, so that the reasons for the methods of solution adopted are rendered intelligible to the pupil; no knowledge of a principle being required, that has not been previously illustrated and explained. In preparation of the rules, definitions, and illustrations, the utmost care has been taken to express them in language simple, precise, and accurate. The examples are of a practical character, and adapted not only to fix in the mind the principles which they involve, but also to interest the pupil, exercise his ingenuity, and inspire a love for mathematical science. The reasons for the operations are explained, and an attempt is made to secure to the learner a knowledge of the philosophy of the subject, and prevent the too prevalent practice of merely performing, mechanically, operations which he does not understand. Analysis has been made a prominent subject, and employed in the solution of questions under most of the rules in which it could be used with any practical advantage. All the most important methods of abridging operations, applicable to business transactions, have been given a place in the work, and so introduced as not to be regarded as mere blind mechanical expedients, but as rational labor-saving processes. Old rules and distinctions, which modern improvements have rendered unnecessary, and which, deservedly, are becoming obsolete, have been avoided. Rules for finding the Greatest Common Divisor of Fractions, |