DA PRACTICAL TREATISE ON ARITHMETIC, WHEREIN EVERY PRINCIPLE TAUGHT IS EXPLAINED IN A SIMPLE AND CONTAINING NUMEROUS QUESTIONS, AND COMBINING THE USEFUL PROPERTIES OF FORMER WORKS, BEING A COMPLETE SYSTEM. TO WHICH IS ADDED TWO METHODS OF BOOK-KEEPING, WITH EXAMPLES FOR EXERCISE. By GEORGE LEONARD, Jr. FIFTH EDITION, STEREOTYPED. BOSTON; OTIS, BROADERS, AND COMPANY. NEW YORK, ROBINSON, PRATT, & CO, AND COLLINS, KEESE, & CO.; 1841. Entered according to Act of Congress, in the year 1840, by in the Clerk's office of the District Court for the District of Massachusetts. CAMBRIDGE; STEREOTYPED BY FOLSOM, WELLS, AND THURSTON. PREFACE. THE manner of teaching arithmetic was formerly very different from that employed at the present time. Certain arbitrary precepts or rules were stated, according to which the scholar performed the examples, remaining in entire ignorance of the propriety of his operations. Such rules are soon forgotten; no person regards them, but solves the questions that occur in business, by means of principles suggested by common sense. There seems to be an obvious improvement, then, in late works, where the scholar, in learning the science, is taught to investigate and apply those principles on which he must depend in practice. This treatise combines the conciseness of the old system with the advantages of the new. It commences in a very simple manner, so as to be readily understood by a person of moderate capacity, having no previous knowledge of the subject. As it advances, the examples and questions are so arranged, that the scholar is led by unperceptible degrees to discover new principles. The reasons for every rule and operation are made obvious, and when explanations are necessary, great care has been taken to render them very lucid and concise. The subjects are arranged and discussed in a more natural order than that usually adopted; for instance, even in the late improved arithmetics, Fractions are partially described in Division; Federal Money follows immediately after Division, so that many of the principles of Decimal Fractions are employed before they can be well explained or understood; Compound Numbers, being usually put before Fractions, cannot be described in one place, and are resumed again in the latter part of Fractions, under heads called Reduction of Fractions and Reduction of Decimals. When subjects are divided in this way, and dissimilar ones jumbled together, the learner is greatly confused and retarded. On the contrary, in this treatise, Federal Money and Compound Numbers succeed Common and Decimal Fractions; whence the scholar, in Federal Money, learns no new rule, but merely applies the principles that govern Decimal Fractions; and in Compound Numbers, he employs Common and Decimal Fractions in the same manner as in other cases. There are many similar improvements; no subject being introduced until every thing necessary to be known before, has been explained in its proper place. No article is mutilated or superficially described; but every thing important to be known concerning it is fully and fairly stated. As examples to be performed in the mind furnish a very useful and necessary exercise, we have given an adequate number, and have interspersed them throughout the work, in appropriate places, so as to afford a pleasing variety, and illustrate the different parts. At the bottom of each page are copious questions, having a phraseology similar to that of their respective answers, so that the learner sees at once what should be committed to memory. The great number of questions, their arrangement at the bottom of each and their pepage, culiar adaptation to the required answers, save much labor and vexation to the instructer, as well as to the scholar; and in connexion with the simplicity of the work, and the regular gradation by which it proceeds from the obvious to the more abstruse, render it very convenient for the purposes of self-instruction. This Arithmetic is well calculated for the wants of the farmer and mechanic, being short, plain, and practical. The merchant will find no work that can be studied with greater advantage, or which contains more that is really useful for his purpose, while the mode of reasoning and the general plan are well suited to the scientific or literary student. The articles on Square Root, Cube Root, Mensuration, and Simple Machines, are explained, it is hoped, with much greater clearness and precision than in any similar work. Book-Keeping, and the Forms of Notes, Bonds, Orders, Receipts, &c., are treated in a manner quite new and original; for there is not only a clear and accurate account of their use, with the necessary examples, but the scholar is required to write, in a proper form, many of the transactions usual in business. This work is intended to be a complete treatise on arithmetic. It contains every useful rule that can be introduced with propriety, and commences in a manner so simple as to render the study of an introduction unnecessary. However, an introduction for small scholars may be useful, to familiarize their minds with the subject, and preserve a more valuable book from being torn and defaced. The person who desires a competent knowledge of arithmetic should study every part of this work, the commencement, as well as the rules of more direct application; however, if the scholar intends to become a farmer or mechanic, and has but a short time to devote to the subject, he can omit all after Fellowship, except BookKeeping, as of secondary importance. Still, the articles on Mensuration and Simple Machines, as well as many others, are very useful in practice, and furnish an excellent discipline for the mind. Even those on Money, and |