OF ALGEBRA, BEING AN ABRIDGMENT OF DAY'S ALGEBRA, ༣ ADAPTED TO THE CAPACITIES OF THE YOUNG, AND THE METHOD OF INSTRUCTION, IN SCHOOLS AND ACADEMIES. BY XJAMES B. THOMSON, A. M. FOURTH EDITION. NEW HAVEN: DURRIE & PECK. PHILADELPHIA: SMITH & PECK. 1844. Entered according to Act of Congress, in the year 1843, by JEREMIAH DAY and JAMES B. THOMSON, in the Clerk's Office of the District Court of Connecticut. N. B. The Key to this work will shortly be published for the use of teachers. An abridgment of LEGENDRE'S Geometry, by the same author, will also be published for the use of Schools and Academies. Stereotyped by RICHARD C. VALENTINE, 45 Gold-street, New York. 12-13-40 mic PREFACE. PUBLIC Opinion has pronounced the study of Algebra to be a desirable and important branch of popular education. This decision is one of the clearest proofs of an onward and substantial progress in the cause of intellectual improvement in our country. A knowledge of algebra may not indeed be regarded as strictly necessary to the discharge of the common duties of life; nevertheless no young person at the present day is considered as having a "finished education" without an acquaintance with its rudiments. The question with parents is, not "how little learning and discipline their children can get through the world with ;" but, "how much does their highest usefulness require ;" and "what are the best means to secure this end?" It has long been a prevalent sentiment among teachers and the friends of education, that an abridgment of Day's Algebra, adapted to the wants of schools and academies, would greatly facilitate this object. Whilst his system has been deemed superior to any other work before the public, and most happily adapted to the circumstances of college students, for whom it was especially prepared; it has also been felt, that a smaller and cheaper work, combining the simplicity of language and the unrivalled clearness with which the principles of the science are there stated, would answer every purpose for beginners, and at the same time bring the subject within the means of the humblest child in the land. 254946 In accordance with this sentiment, such a work has been prepared, and is now presented to the public. The design of the work is to furnish an easy and lucid transition from the study of arithmetic to the higher branches of algebra and mathematics, and thus to subserve the important interests of a practical and thorough education. Its arrangement, with but few exceptions, is the same as that of the large work. For the sake of more convenient reference, the division by compound divisors, and the binomial theorem, both of which were originally placed after mathematical infinity, are brought forward, the former being placed after division of simple quantities, and the latter after involution of simple quantities. The reason for deferring the consideration of compound division in the original, was the fact that some of the terms contain powers which it is impossible for pupils at this stage of their progress to understand. To avoid this difficulty in the present work, whenever a power occurs, instead of using an index before it has been explained, the letter is repeated as a factor in the same manner as in multiplication, and also in dividing by a simple quantity. (Arts. 80, 94.) Afterwards, under division of powers, copious examples of dividing by compound quantities which have indices, are given. As continued arithmetical proportion and arithmetical progression are one and the same thing, they are placed contiguously in the same section. For the same reason continued geometrical proportion and geometrical progression are placed in a similar manner. Mathematical infinity, roots of binomial surds, infinite series, indeterminate co-efficients, composition and resolution of the higher equations, with equations of curves, are subjects which belong to the higher and more difficult parts of algebra, and it has been thought advisable to omit them in the abridgment. Those who have leisure and are desirous of acquiring a knowledge of these subjects, will find them explained with all the author's accustomed clearness and ability in his large work, to which they are respectfully referred. The similarity between the operations in addition, subtraction, multiplication and division of radical quantities, and those of the same rules in powers; also between involution and evolution of radicals, and of powers, has been more fully developed, and the rules of both are expressed in as nearly the same language as the nature of the case would admit. It has also been attempted to illustrate the "Binomial Theorem," on the principles of induction; the second method of completing the square in quadratic equations has been demonstrated; and other methods of completing the square pointed out, which, so far as the author knows, are original. This It was a cardinal point with the distinguished author of the large work, never to use one principle in the explanation of another, until it had itself been explained, a characteristic of rare excellence in school-books and works of science. plan has been rigidly adhered to, in the preparation of the abridgment. After the principles have been separately explained, and illustrated by examples, they have then been carefully summed up in the present work, and placed in the form of a general rule. This, it is thought by competent judges, will be found very convenient and useful both to teachers and scholars. By this means the peculiar advantages of the inductive and synthetic modes of reasoning have been united, and made subservient alike to the pleasure and facility both of imparting and acquiring knowledge. As a guide to the attention of beginners to the more important principles of the science, a few practical questions are placed at the foot of the page. They are intended to be merely suggestive. No thorough teacher will confine himself |