WHILE engaged as a teacher of elementary mathematics, the author has often felt the want of a Treatise on Algebra, which might serve as an introduction to the higher, and more difficult parts of that science; a treatise which, commencing with the first principles of Arithmetic, might conduct the learner by gradual and easy steps, from the expression of quantity by numbers, to the investigation of the relations of quantity, by algebraic symbols. In most of our public schools, the pupil commences the study of Algebra, with a very imperfect knowledge of arithmetic. He has been taught arithmetic, it is true, but mercantile arithmetic only. He has been taught to perform operations upon numbers, by rules, the principles of which, he is rarely required to investigate, and which, for the most part, the books that are put into his hands, do not even profess to demonstrate. He thus begins the study of algebra, unacquainted with the first principles of mathematical science, and finds himself perplexed at the same time with a kind of reasoning, to which he is wholly unaccustomed, and new symbols for the expression of quantity, the nature and use of which, he finds it difficult to comprehend. With these difficulties to encounter, it is not strange that all seems dark and confused; or that he often turns, with disappointment and disgust, from a study, upon which he was ill prepared to enter. To remove these difficulties, is the object of the short treatise on arithmetic, with which this volume commences. In it, an attempt has been made to explain, with clearness and brevity, the connexion between numbers, and the quantities they are used to express; and also to give concise and satisfactory demonstrations of the rules for numerical calculations. The second chapter is necessarily more full than any other part of the arithmetic, inasmuch as a thorough knowledge of fractions, both vulgar and decimal, is an indispensable preparation for the study of algebra, and the same demonstrations apply both to numerical and algebraic fractions. Having accustomed the pupil to reasoning upon numbers, in the first three chapters, letters, as the representatives of numbers, are first introduced in the fourth ; and the same reasoning applied to them, which immediately before, had been applied to the numbers themselves, thus preparing the way for the use of algebraic symbols, and making the transition from arithmetic to algebra easy and pleasant. The treatise on algebra, which forms the principal part of the present volume, is not so closely connected with the arithmetic as to render it necessary that both should be studied together. If the pupil be well versed in the principles of arithmetic, he may proceed at once with algebra ; yet he will find it convenient to have always at hand, a treatise on arithmetic to which he can refer, when he would recall some principle which he may have forgotten. It was the author's intention, that the treatise on algebra should be eminently practical, and for this purpose a great number of examples have been introduced. The examples are so selected, as to illustrate, most fully, the rules to which they are applied, and so arranged as to form a series of progressive exercises, rendering the passages from the more "simple, to the most complicated algebraic operations, as easy as possible, and preparing the student, by making him familiar with the practical part of algebra, to enter with advantage upon the higher branches of mathematical science. Throughout the whole, the reasoning is adapted to the progress of the student. In the first part of the work, the rules are given and applied, and then followed by demonstrations; but as the student is supposed to become more familiar with mathematical reasoning, the analytic method is gradually introduced; and in the last two chapters, used to the exclusion of every other. This is an important part of the plan of the work, it being originally designed to introduce the student, by a series of progressive exercises, to the more abstruse analytic reasoning of the higher branches of algebra. It is hoped, however, that while it meets the wants of students who are commencing an extensive course of mathematics, it will be particularly useful to those who have less leisure for study, or who, on account of their particular profession, or occupation in life, are desirous only of obtaining a knowledge of the more practical and useful parts of mathematical science. Preliminary remarks and definitions . o e e Division - - o - e - e e CHAPTER II. Fractions—Definitions - - e - To find the greatest common divisor of the two numbers denominator e e e e e e To reduce a mixed number to an improper fraction e To reduce an improper fraction to a whole or mixed number . o e e e To reduce a compound fraction to a simple one . e To reduce a complex fraction to a simple one e To reduce the lower denominations of a compound number, to a fraction of a higher denomination . e |