the two terms a and b of the first fraction by b+m, and those of the second by b. Now, the multiplication of a by b+m consists in taking a as many times as there are units in b plus as many times as there are units in m, which gives a b+ a m. It might be proved in the same manner, that the product of ab+am b by b+m is b2 + b m, which gives for the first fracb2+bm tion. In like manner, by multiplying the two terms of the second The two numerators, a b+am, ab+bm, have a common part a b; and the part b m of the second is greater than the part a m of the first, since b>a. is greater than the first. Q. E. D. Hence the second fraction We see, moreover, from the preceding reasoning, that must be a proper fraction in order that the theorem may be true; for otherwise we would have ab+bm<a b+am, and the new fraction would be less than the first. 7. The preceding questions ought to be sufficient to give the beginner a clear idea of the object of algebra. By reflecting upon the solutions of these questions, he will perceive that the employment of algebraic signs must give rise to certain rules which are common to many questions. It is thus, for example, that in the second and third questions, we have been required to perform the multiplication of a sum a+b, by a number a; of a sum a+b, by a number b; of a number a, by a sum b+m. Hence, by establishing general rules for finding the results of the operations that we may have to perform with algebraic quantities, we will have fixed methods for resolving all questions relating to numbers, by algebraic symbols. This part of algebra is entitled, the manner of performing arithmetical operations upon algebraic or literal quantities; that is, upon numbers represented by algebraic symbols. CHAPTER I. OF ALGEBRAIC OPERATIONS. Preliminary Definitions. 8. EVERY quantity written in algebraic language—that is, with the aid of algebraic signs, is called an algebraic quantity, or the algebraic expression of the proposed quantity. Thus, 3 a is the algebraic expression of three times the numbera; 5 a2 is the algebraic expression for five times the square of a; 7 a3b2 is the algebraic expression for seven times the product of the cube of a, by the square of b. 3 a 5b, is the algebraic expression of the difference between three times a and five times b. 2 a3 —3 ab +4 b2, is the algebraic expression for twice the square of a, diminished by three times the product of a by b, augmented by four times the square of b. When an algebraic quantity is not connected with any other by the sign of addition or subtraction, it is called a monomial, or quantity composed of a single term, or simply a term; an algebraic expression composed of two or more parts, separated by the signs or, is called a polynomial, or quantity involving two or more terms. Thus, 3 a, 5 a2, 7 a3b3 are monomials; 3 a 5 b, 2 a2-3ab + 462, are polynomials. The first of these two expressions is called a binomial, because it is composed of two terms; the second is called a trinomial, because it is composed of three terms. 9. The numerical value of an algebraic expression, is the number which would be obtained, by giving particular values to the letters which enter it, and performing the arithmetical operations indicated. This numerical value evidently depends upon the particular values attributed to the letters, and will generally vary with them. Thus, the numerical value of 2 a3 is 54, when we make a = 3; for the cube of 3 is 27, and twice ADVERTISEMENT. The whole of the fourth, ninth, and tenth chapters of the original, with the exception of the article upon Figurate Numbers, and the Series upon which they depend, have been omitted in the translation; also, that part of the seventh chapter which relates to the Theory of Symmetrical Functions. Some of the articles of the 8th chapter have also been omitted. In consequence of these omissions, the numbers which designate the different articles do not succeed each other in regular order. 4c+2c2 d, are homogeneous polynomials. 8a3-4 ab+c is not homogeneous. 12. Terms composed of the same letters, and affected with the same exponents, are called similar terms. Thus, 7 ab and 3 ab, 4a3 b, and 5 a3 b2, are similar terms. 8 a2 b and 7 a b2 are not similar terms, for although they are composed of the same letters, yet the same letters are not affected with the same exponents. It often happens that a polynomial contains several similar terms; it may then be simplified. Take the polynomial 4 a2 b-3a2c+9 ac2 2 a2 b + 7 a2 c-6b3; it may (9) be written thus: 4 a2 b-2ab+ 7 a2 c - 3 a2 c + 9 ac2 6b3; now, 4 a2 b to 2 a2 b. 7 a c-3 ac reduces to 4 a nomial becomes 2 a2 b + 4 a2 c + 9 ac2 3 4 a b c2, +6 a3 bc2, 3 -2a2 b reduces c; hence the poly 6b3. When we have, in any polynomial, the terms + 2 a3 bc2, Sa be2, +11a3 bc2; the sum of the additive terms may be reduced to +19 a3 bc2; and the sum of the subtractive terms to 12 a3 bc2; hence the five proposed terms reduce to 19 a3 bc2-12 a3 be2, or 7 a3 bc2. bc2 It may happen that the sum of the subtractive terms exceeds the sum of the additive terms. In this case, subtract the positive coefficient from the negative, and prefix the sign to the result. Thus, when +5 a b is the sum of the positive, and 8 a b the sum of the negative terms, as 8a2 b is the same as 5 a2 b 3 a2 b, it follows that + 5 a2 b. -8 a2 b is equivalent to +5 a2 b-5 a2 b— 3 a2 b, or - 3 a2 b. Hence the following rule for the reduction of similar terms. b. Form a single additive term of all the terms preceded by the sign+; which is done by adding together the coefficients of these terms, and annexing to this sum the common literal part. In the same manner form a single subtractive term with all the terms preceded by the sign; then subtract the least sum from the greatest, and give the result the sign of the greater. (It should be observed that the reduction affects only the coefficients, and not the exponents.) From this rule we find that 4a2 b-8 a2 b. 9a2 b+11 a2 b reduces to 7 a b c2 8abc2 + 6abc - 2 a2 b; reduces to |