PRACTICAL TREATISE ON ALGEBRA DESIGNED FOR THE USE OF STUDENTS IN HIGH SCHOOLS AND ACADEMIES. BY BENJAMIN GREENLEAF, A.M., AUTHOR OF THE “ NATIONAL ARITHMETIC,” ETO. Xmproved Stereotype Edition. BOSTON: PUBLISHED BY ROBERT S. DAVIS & CO. 1853. V Entered according to Act of Congress, in the year 1852, by BENJAMIN GREENLEAF, 1 1 Entered according to Act of Congress, in the year 1853, by BENJAMIN GREENLEAF, GREENLEAF'S SERIES OF MATHEMATICS. 1. MENTAL ARITHMETIC, upon the Inductive Plan; designed for Primary and Intermediate Schools. Revised and enlarged edition. 144 pp. 2. INTRODUCTION TO THE NATIONAL ARITHMETIC ; OR, COMMON SCHOOL ARITHMETIC. Improved stereotype edition. 324 pp. 3. THE NATIONAL ARITHMETIC, designed for advanced scholars in Common Schools and Academies. Improved stereotype edition. 360 pp. COMPLETE KEYS TO THE INTRODUCTION AND NATIONAL ARITHMETIC, containing Solutions and Explanations, for Teachers only. 2 vols. 4. PRACTICAL TREATISE ON ALGEBRA, for High Schools and Academies, and for advanced Students in Common Schools. New edition, revised and stereotyped. KEY TO THE PRACTICAL ALGEBRA, containing the Answers, and full Solutions and Explanations, for Teachers only. PREFACE. The following Treatise is designed to present a system of theoretical and practical Algebra. It is intended to be both elementary and comprehensive, and adapted to the wants of beginners, as well as those who are advanced in the study. In the course of his labors the author has consulted the most approved European treatises on the subject, and availed himself of whatever he thought might add to the interest and usefulness of his work. It has been the aim of the author, throughout his investigations, to give to it a practical character, that those who study it may know how to apply their knowledge to useful purposes. The demonstrations connected with the several Roots, will greatly aid those who wish for a complete and thorough knowledge of Evolution in Arithmetic. The method of solving Cubic Equations by completing the square, the author believes, will be very useful. This method will not apply to all problems; but, wherever it will apply, it not only very much abridges the labor, but the result is perfect accuracy, which is not always the fact by the common method of approximation. The Table of Logarithms at the end of the work, will be often found convenient and useful. The examples, of which a large number have been placed under each Rule, are intended to be neither too numerous nor too difficult; and all who may use the work, either by themselves or in connection with a class, are advised to solve all the problems, in the order in which they are given. No labor on the part of the pupil will be productive of more intellectual and practical benefit. The answers to several questions have been designedly omitted, that the pupil may try his skill as upon an original problem. One who has a thorough knowledge of Arithmetic, will find the study of Algebra a most pleasing, and, generally, not a difficult task. As a mental exercise, it is admirable for its effect upon the logical powers of the mind, assisting one to think and reason closely and conclusively. As Mr. Locke has remarked, in his Essay on the Human Understanding, “ Nothing teaches a man to reason so well as Mathematics, which should be taught to all those who have time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures." BENJAMIN GREENLEAF. BRADFORD, January 23, 1852. ADVERTISEMENT TO THE STEREOTYPE EDITION. In revising this work for a second edition, the author has made such changes and additions as he believed would better adapt it to its purpose. Every part of it has been carefully and critically examined, and many portions have been entirely re-written. In a few cases, where improvement in that respect seemed desirable, the arrangement of articles has been somewhat altered. The new articles which have been inserted, will, it is hoped, add materially to the interest, as well as to the value, of the treatise. The theory of Equations has been more fully developed, and illustrated by a variety of carefully prepared examples. A brief space has been given to Indeterminate Analysis, a subject which, though usually omitted in elementary works on Algebra, the author believes to be one of no small practical importance. It gives the student the command of a class of problems which cannot possibly be solved by the rules of Arithmetic, nor by the more familiar principles of Algebra. In the revision of the work, the author has availed himself of the suggestions of several teachers who have used it as a text-book since its first publication ; and he would take this opportunity to express his gratitude for their kindness. He feels himself especially indebted to Mr. Hagar, of Roxbury, and to Mr. Rolfe, of Dorchester, who have afforded him much valuable aid in his efforts to make the work more worthy of the favor with which it has been received. April 26, 1853. Definitions of different Fractions To find the greatest Common Measure of the terms of a Fraction 48 To reduce Fractions to their lowest terms To reduce a Mixed Quantity to the form of a Fraction To represent a Fraction in the form of a Whole or Mixed Quantity 53 To reduce a Complex Fraction to a Simple one |