In this equation there are two unknown quantities; and, in general, any two numbers of which the proportion is that of 13 to 6, will agree to it. But, from the nature of the question, 13 and 6 are the only two that can give the proper answer, viz. 131. 6 s. for its reverse 61. 13 s. is just its half. The ratio of x and y is expressed in the lowest integral terms by 13 and 6 ; any other expression of it, as the next greater 26 and 12, will not satisfy the problem, as 12 l. 26 s. is not a proper notation of money in pounds and shillings. CH A P. A LGEBRA may be employed for the demonstration of Theorems, with regard to all those quantities concerning which it may be used as an analysis, and from the general method of notation and reasoning, it possesses the same advantages in the one as in the other. The three first sections of this chapter contain some of the most simple properties of series which are of frequent use; and the last, miscellaneous examples of the properties of algebraical quantities and nụmbers, 1. 1. Of Arithmetical Series, Def. When a number of quantities increase or decrease by the same common difference, they form an' Arithmetical Series. Thus a, atb, a+2b, at3b, &c, x, x-b, x—26, &c. Also, 1, 2, 3, 4, 5, 6, &c.; and 8, 6, 4, 2, &c. Prop. In an arithmetical series, the sum of the first and laft terms is equal to the sum of any two intermediate terms, equally distant from the extremes. Let the first term be a, the last x, and b the common difference; then a+b will be the second, and x-b the last but one, &c. Thus a, a-t-b, a+2b, a+3b, a+46, &c. x, x-b, x-2b, x-3b, *—46, &c. It is plain, that the terms in the same perpendicular rank are equally distant from the the extremes, and that the sum of any two in it is a+x, the sum of the first and last. Cor. 1. Hence the sum of all the terms of an arithmetical series is equal to the sum of the first and last, taken half as often as there are terms. Therefore, if n be the number of terms, and s the sum of the series ; s=a+**** n nx If a=o, then s= 1 Cor. 2. The same notation being understood, since any terin in the series consists of a, the first term, together with b taken as often as the number of terms preceding it, it follows that x=a+n-Ixb, and hence s=2a+n_1xbx“; or by multi 2an+nbnb plication, s= Therefore, from the first term, the common difference, and number of terms being given, the sum may be found. Ex. Required the sum of 50 terms of the series 2, 4, 6, 8, &c. R . 2 2x2x50+50** 2—50*2=5100=2550. 2 Cor. 3. Of the first term, common difference, sum and number of ternis, any three being given, the fourth may be found by resolving the preceding equation; a, b, s, and n being successively considered as the unknown quantity. In the three first cases the equation is simple, and in the last it is quadratic. II. Of Geometrical Series. Def. When a number of quantities increase by the same multiplier, or decrease by the same divisor, they form a Geometrical Series. This common multiplier or divisor is called the common ratio. Prop. I. The product of the extremes in a geometrical series is equal to the pro duct |