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 queftion, 13 and 6 are the only two that can give the proper answer, viz. 13 1. 6 s. for its reverse 61. 13 s. is juft its half. The ratio of x and y is expreffed in the lowest integral terms by 13 and 6; any other expreffion of it, as the next greater 26 and 12, will not fatisfy the problem, as 12 1. 26 s. is not a proper notation of money in pounds and fhillings. СНАР. CHA P. VII. Demonftration of Theorems by Algebra. A LGEBRA may be employed for the demonftration of Theorems, with regard to all thofe quantities concerning which it may be used as an analysis, and from the general method of notation and reasoning, it poffeffes the fame advantages in the one as in the other. The three firft fections of this chapter contain some of the moft fimple properties of feries which are of frequent ufe; and the laft, mifcellaneous examples of the properties of algebraical quantities and numbers. I. Of Arithmetical Series. Def. When a number of quantities increase or decrease by the fame common difference, they form an' Arithmetical Series. Thus a, a+b, a+2b, a+3b, &c, x, x—b, x-2b, &c. Also, 1, 2, 3, 4, 5, 6, &c.; and 8, 6, 4, 2, &c. Prop. In an arithmetical feries, the fum of the first and laft terms is equal to the fum of any two intermediate terms, equally diftant from the extremes. Let the firft term be a, the laft x, and b the common difference; then a+b will be the second, and x-b the last but one, &c. Thus a, a+b, a+2b, a+3b, a+46, &c. x, x-b, x-2b, x—3b, x—4b, &c. It is plain, that the terms in the fame perpendicular rank are equally diftant from the the extremes, and that the fum of any two in it is ax, the fum of the first and last. Cor. 1. Hence the fum of all the terms of an arithmetical feries is equal to the fum of the first and laft, taken half as often as there are terms. Therefore, if n be the number of terms, n and s the sum of the series; s=a+××2. Cor. 2. The fame notation being underftood, fince any term in the feries confifts of a, the first term, together with b taken as often as the number of terms preceding it, it follows that x=a+n-1xb, and η 2a+n—1×b×7; or by multi hence s the firft term, the common difference, and number of terms being given, the fum may be found. Ex. Required the fum of 50 terms of the feries 2, 4, 6, 8, &c. R Cor. 3. Of the firft term, common difference, fum and number of ternis, any three being given, the fourth may be found by refolving the preceding equation; a, b, s, and n being fucceffively confidered as the unknown quantity. In the three first cases the equation is fimple, 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 fame divifor, they form a Geometrical Series. This common multiplier or divifor is called the common ratio. Prop. I. The product of the extremes in a geometrical feries is equal to the pro duct |