536 A L G E B K A [PEOPOBTION AND Ex. 3. Find the value of-A (6-c)a x = - == . when At the first reduction a divides out, and the fraction is reduced to I . _ Lx. 4. Find the value of . . a v(2a a;) when ic = ^/(i 2 + 4a6) - 6 . By the process explained in this article V o whence the fraction reduces to 45. In arithmetic the square root of a number is another number, which, when multiplied by itself, shall produce the first number. In algebra, where quantity takes the place of number, the definition leads to a less limited result than in arithmetic. In the latter science there can not be two square roots of the same thing ; in the former, there will necessarily be two. For both -I- 2 x + 2 gives 4, and - 2 x - 2 gives 4 ; hence the square root of 4 is - 2 sis well as + 2. And, further, as in algebra, - 2 is a quantity subject to all the operations and definitions of the science, it is clearly competent to express, in some form or other, the result of extracting its square root. That form must of necessity be something very different in character from *J2, whether ^/2 be + or - . For the definition requires that the square root of - 2 shall be such a quantity as when multiplied by itself shall produce - 2. It is then clearly no arith metical quantity either + or - , but some quantity con nected with numerical quantities by its properties, but not by its nature. It is termed an impossible or imaginary quantity, and may be written J - 2 or ^/2 *J - 1, and the same notation applies to the square roots of all negative quantities. The properties of imaginary quantities are almost iden tical with those of surds, and we need not stop to consider them. One example of their application will suffice. It affords strong confirmation of the safety of assuming the commutative law to exist in every branch of pure algebra. Ex. The product of the sum of two squares by the sum of two squares can always be represented under the form of the sum of two squares. For (c 2 + d V- = (ac- -l) (ac-bd- ad + bc J - 1) Cor. (a 2 + 6 2 ) (c 2 + d 2 ) = (ac + bd) z + (ad - Icf, or the pro duct may be represented in two different ways, under the form of the sum of two squares. SECT. V. PROPORTION AND PROGRESSION. roportion 4G. In comparing together any two quantities of the id Pro- same kind in respect of magnitude, we may consider how ession. much the one is greater than the other, or else how many times the one contains either the whole or some part of the other; or, which i.3 the same thing, we may consider either what is the difference between the quantities, or what is the quotient arising from the division of the one quantity by the other : the former of these is called their arithmetical ratio, and the latter their geometrical ratio. These deno minations, however, have been assumed arbitrarily, and have little or no connection with the relations they are intended to express. I. Arithmetical Proportion and Progression. 47. When of four quantities the difference between tho first and second is equal to the difference betAveen the third and fourth, the quantities are called arithmetical propor tionals. Such, for example, are the numbers 2, 5, 9, 12; and, in general, the quantities a, a + d, I, b + d. 48. The principal property of four arithmetical propor tionals is this : If four quantities be arithmetically pro portional, the sum of the extreme terms is equal to the sum of the means. Let the quantities be a, a + d, b, b + d; where d is the difference between the first and second, and also between the third and fourth, the sum of the extremes is a + b + d, and that of the means a + d + b; so that the truth of the proposition is evident. 49. If a series of quantities be such, that the difference between any two adjacent terms is always the same, these terms form an arithmetical progression. Thus, the num bers 2, 4, 6, 8, 10, &c., form a series in arithmetical pro gression, and, in general, such a series may be represented thus: a, a + d, a + 2d, a + 3d, a + 4cZ, a + 5d, a + Gd, &c., where a denotes the first term, and d the common difference. By a little attention to this series, wo readily discover that it has the following properties : 1. The last term of the series is equal to the first term, together with the common difference taken as often as there are terms after the first. Thus, when the number of terms is 7, the last term is a + Gd; and so on. Hence if denote the last term, n the number of terms, and a and d express the first term and common difference, we have z = a + (n l)c?. 2. The sum of the first and last term is equal to the sum of any two terms at the same distance from them. Thus, suppose the number of terms to be 7, then the last term is a + Gd, and the sum of the first and last 2a + Gd; but the same is also the sum of the second and last but one, of the third and last but two, and so on till we come to the middle term, which, because it is equally distant from the extremes, must be added to itself. 3. To find the sum of the series, it is only necessary to observe that, if the progression is written down twice, 1 from the beginning, 2 from the end, the terms of the former increase by the same amount as that by which the terms of the latter diminish; so that the sum of any two terms which stand under each other is always the same, viz., the same as the sum of the first and last terms; hence the double series converts addition into multiplication; so that if s denote the sum of the series, we have 2s = n(a--zJ, and n, N s = -(a-!-z). Ex. The sum of the odd numbers 1, 3, 5, 7, 9, &c., con tinued to n terms, is equal to the square of the number of terms. For in this case a = 1 , d = 2, z = 1 + (n - 1 ) d = 2n m 1, therefore * = - x 2n = K? . 2* II. Geometrical Proportion and Progression. 50. When, of four quantities, the quotient arising from the division of the first by the second is equal to that arising from the division of the third by the fourth, these
quantities are said to bo in geometrical proportion, or are
