TABLE OF CONTENTS. Introduction. Containing a brief explanation of the purpose of algebra, and of some of the signs 1. Questions producing simple equations, in which the un- known quantity is multiplied by known quantities II. Questions producing simple equations, in which the un- known quantity is divided by known quantities III. Questions producing simple equations, in which the un- known quantity is added to known quantities IV. Questions producing simple equations, in which quanti- ties consisting of two or more terms are to be multiplied V Questions producing simple equations, in which quantities consisting of two or more terins are to be divided by a VI. Questions producing simple equations, in which quanti- ties consisting of two or more terms, some of them having the sign — before them, are to be subtracted from other Case of fractions to be subtracted, when some of the terms in the numerator have the sign - VII. Examples for exercise in putting questions into equation A precept useful for this purpose VIII. Questions producing equations with two unknown IX. Explanation of some of the higher purposes of algebra, and examples of generalization Addition, multiplication, and subtraction of simple quan- tities 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 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 XXXI. Questions producing pure equations of the third de- XXXII. Extraction of the third root of fractions . 159 XXXIII. Questions producing pure equations of the third XXXIV. Questions producing affected equations of the 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 val- ues of the unknown quantity, also of the positive and negative values of it, in equations of the second degree 177 XXXVI. Of powers and roots in general 193 XXXVIII. Extraction of the roots of compound quantities of any degree XXXIX. Extraction of the roots of numerical quantities of XL. Fractional exponents and irrational quantities - 197 XLII. Summation of series by differences XLIV. Binomial Theorem, continued frorn Art. XLI. 221 XLV. Continuation of the same subject XLVI. Progression by difference, or Arithmetical progression 228 XLVII. Progression by quotient, or Geometrical progression 233 268 - 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 learned 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 6 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 14 - 6 = 8, is read 6 subtracted from 14, or 4 less 6, or 14 minus 6 is equal to 8. Observe that the signs + and — affect the numbers which they stand immediately before, and no others. Thus . 14 — 6+8= 16; and 14 +8—6= 16; and 8 — 6 + 14 = 16; and, in fine, — 6+ 8 + 14 = 16. In all these cases the 6 only is to be subtracted, and it is the same, whether it be first subtracted from one of the numbers, and then the rest be added, or whether all the others be added and that be subtracted at last. (x) (.) An inclined cross, or a point, is used to express multiplication ; thus, 5 X 3= 15, or 5.3= 15. (=) A horizontal line, with a point above and another below it, is used to express division. Thus 15 • 3=5, is read 15 divided by 3 is equal to 5. But division is more frequently expressed in the form of a fraction (Arith. Art. XVI. Part II.), the divisor being made the denominator, and the dividend the numerator. Thus 5 = 5, is read 15 divided by 3 is equal to 5, or one third of 15, is 5,. or 15 contains 3, 5 times.. Example. 6 x 9 + 15— 3=1.8- 1 + 14. This is read, 9 times 6 and 15 less 3 are equal to 8 times 7 less 16 divided by 4, and 14. To find the value of each side ; 9 times 6 are 54 and 15 are 69, less 3 are 66. Then 8 times 7 are 56, less 16 divided by 4, or 4 are 52, and 14 more are 66. In questions proposed for solution, it is always required to find one or more quantities which are unknown ; these, when found, are the answer to the question. It will be found extremely useful to have signs to express these unknown quantities, because it will enable us to keep the object more steadily and distinctly in view. We shall also be able to represent certain operations upon them by the aid of signs, which will greatly assist us in arriving at the result. Algebraic signs are in fact nothing else than an abridgment of common language, by which a long process of reasoning is presented at once in a single view The signs generally used to express the unknown quantities above mentioned are some of the last letters of the alphabet, as 2, y, z, &c. |