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* 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. 4) A horizontal line, signifies subtracted from. It is sometimes read less or minus. Minus is Latin, signifying less. Thus XIII. Multiplication of compound quantities XIV. Division of algebraic quantities Multiplication of Algebraic fractions XVI. Division of algebraic fractions To multiply fractions by fractions XVII. Reducing fractions to lower terms Division when part of the factors of the divisor are con- XVIII. Addition and subtraction of fractions To reduce fractions to a common denominator XIX. Division of whole numbers by fractions and fractions by XX. Division of compound quantities XXI. A few abstract examples in equations XXII. Miscellaneous Questions producing simple equations 104 XXIII. Questions producing simple equations involving more 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 made upon the known quantities XXVII. Questions producing equations of the second degree 131 XXVIII. Extraction of the second root XXIX. Extraction of the second root of fractions XXX. Questions producing pure equations of the second de- XXXI. Questions producing pure equations of the third de- XXXII. Extraction of the third root of fractions XXXIII. Questions producing pure equations of the third de- XXXIV. Questions producing affected equations of the second General formula for equations of the second degree 174 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 177 XXXVI. Of powers and roots in general XXXVIII. Extraction of the roots of compound quantities of XXXIX. Extraction of the roots of numerical quantities of XL. Fractional exponents and irrational quantities XLII. Summation of series by differences XLIV. Binomial Theorem, continued from Art. XLI. XLV. Continuation of the same subject XLVI. Progression by difference, or Arithmetical progression 228 XLVII. Progression by quotient, or Geometrical progression 233 PREFACE. The first object of the author of the following treatise has been to make the transition from arithmetic to algebra as gradual as possible. The book, therefore, commences with practical questions in simple cquations, such as the learner might readily solve without the aid of algebra. This requires the explanation of only the signs plus and minus, the mode of expressing multiplication and division, and the sign of equality ; together with the use of a letter to express the unkuown quantity. These may be understood by any one who has a tolerable knowledge of arithmetic. All of them, except the use of the letter, have been explained in arithmetic. To reduce such an equation requires only the application of the ordinary rules of arithmetic; and these are applied so simply, that scarcely any one can mistake them, if left entirely to himself. One or two questions are solved first with little explanation in order to give the learner an idea of what is wanted, and he is then left to solve several by himself. The most simple combinations are given first, then those which are more difficult. The learner is expected to derive most of his knowledge by solving the examples himself; therefore care has been taken to make the explanations as few and as brief as is consistent with giving an idea of what is required. In fact, explanations rather embarrass than aid the learner, because he is apt to trust too much to them, and neglect to employ his own powers ; and because the explanation is frequently not made in the way, that would naturally suggest itself to him, if he were left to examine the subject by himself. The best mode, therefore, seems to be, to give examples so simple as to require little or no explanation, and let the learner reason for himself, taking care to make them more difficult as he proceeds. This method, besides giving the learner confidence, by making him rely on his own powers, is much more interesting to him, because he seems to himself to be constantly making new discoveries. Indeed, an apt scholar will frequently make original ex. planations much more simple than would have been given by the author. |