TABLE OF CONTENTS. ntroduction. Containing a brief explanation of the purpose of algebra, and of some of the signs I. Questions producing simple equations, in which the unknown IV. Questions producing simple equations, in which quantities. V. Questions producing simple equations, in which quantities consisting of two or more terms are to be divided by a num- VI. Questions producing simple equations, in which quantities This mode has also the advantage of exercising the learner in reasoning, instead of making him a listener, while the author reasons before him. The examples in the first fifty pages involve nearly all the operations, that are ever required in simple numerical equations, with one and two unknown quantities. In the ninth article, the learner is taught to generalize particular cases, and to form rules. Here he is first taught to represent known quantities by letters, and at the same time the purpose of it. The transition from particular cases to general principles is made as gradual as possible. At first only a part of the question is generalized, and afterwards the whole of it. When the earner understands the purpose of representing known quantities as well as unknown, by letters or general symbols, he is considered as fairly introduced to the subject of algebra, and ready to commence where the subject is usually commenced in other treatises. Accordingly he is taught the fundamental rules, as applied to literal quantities. Much of this however is only a recapitulation in a general form, of what he has previously learnt, in a particular form. After this, various subjects are taken up and discussed. There is nothing peculiar in the arrangement or in the manner of treating them. The author has used his own language, and explained as seemed to him best, without reference to any other work. A large number of examples introduce and illustrate every principle, and as far as seemed practicable, the subjects are taught by example rather than by explanation. The demonstration of the Binomial Theorem is entirely original, so far as regards the rule for finding the coefficients. The rule itself is the same that has always been used. The manner of treating and demonstrating the principle of summing series by difference, is also original.* Proportions have been discarded in algebra as well as in arithmetic. The author intended to give, in an appendix, some directions for using proportions, to assist those who might have occasion to read other treatises on mathematics. But this volume was already too large to admit it. It is believed, however, that few will find any difficulty in this respect. If they do, one hour's study of some treatise which explains proportions will remove it. * See Boston Journal of Philosophy and the Arts, No. 5, for May, 1825 In order to study this work to advantage, the learner should solve every question in course, and do it algebraically. If he finds a question which he can solve as easily without the aid of algebra as with it, he may be assured, this is what the author expected. If he first solves a question, which involves no difficulty, he will understand perfectly what he is about, and he will thereby be enabled to encounter those which are difficult. When the learner is directed to turn back and do in a new way, something he has done before, let him not fail to do it, for it will be necessary to his future progress and it will be much better to trace the new principle in what he has done before, than to have a new example for it. The author has heard it objected to his arithmetics by some, that they are too easy. Perhaps the same objection will be made to this treatise on algebra. But in both cases, if they are too easy, it is the fault of the subject, and not of the book. For in the First Lessons, there is no explanation; and in the Sequel there is probably less than in any other books, which explain at all. As easy however as they are, the author believes that whoever undertakes to teach them, will find the intellects of his scholars more exercised in studying them, than in studying the most difficult treatise he can put into their hands. When the learner feels, that the subject is above his capacity, he dares not attempt any thing himself, but trusts implicitly to the author; but when he finds it level with his capacity, he readily engages in it. But here there is something more. The learner is required to perform a part himself. He finds a regular part assigned to him, and if the teacher does his duty, the learner must give a great many explanations which he does not find in the book. 1* 9 ALGEBRA. Introduction. THE operations explained in Arithmetic are sufficient for the solution of all questions in numbers, that ever occur; but it is to be observed, that in every question there are two distinct things to be attended to; first, to discover, by a course of reasoning, what operations are necessary; and, secondly, to perform those operations. The first of these, to a certain extent, is more easily learnt than the second; but, after the method of performing the operations is understood, all the difficulty in solving abstruse and complicated questions consists in discovering how the operations are to be applied. It is often difficult, and sometimes absolutely impossible to discover, by the ordinary modes of reasoning, how the fundamental operations are to be applied to the solution of questions. It is our purpose, in this treatise, to show how this difficulty may be obviated. It has been shown in Arithmetic, that ordinary calculations are very much facilitated by a set of arbitrary signs, called figures; it will now be shown that the reasoning, previous to calculation, may receive as great assistance from another set of arbitrary signs. Some of the signs have already been explained in Arithmetic; they will here be briefly recapitulated. (=) Two horizontal lines are used to express the words "are equal to," or any other similar expression. (+) A cross, one line being horizontal and the other perpendicular, signifies "added to." It may be read and, more, plus, or any similar expression; thus, 7+5 12, is read 7 and 5 are 12, or 5 added to 7 is equal to 12, or 7 plus 5 is equal to 12. Plus is a Latin word signifying more. (-) A horizontal line, signifies subtracted from. It is sometimes read less or minus. Minus is Latin, signifying less. Thus Table of Contents. To reduce fractions to a common denominator XXIV. Negative quantities, explanation of them XXV. Explanation of negative exponents XXVI. Examination of general formulas, to see what values the unknown quantities will take for particular suppositions General formula for equations of the second degree XXXV. Demonstration of the principle that every equation of the second degree admits of two values for the unknown Discussion concerning the possible and impossible values of the unknown quantity, also of the positive and nega- tive values of it, in equations of the second degree |