Note 156 — CIII. The Invention of a General Method for determining the Sum of every 2d, 3d, 4th, or 5th, &c. Term of a Series, taken in order; the Sum of the whole Series being known. By Thomas Simpson, F.R.S.

Because -ẋ ⁄ √(1 - xx) and -Ẋ ⁄ √(1 - XX) are known to express the fluxions of the circular arcs whose co-sines are x and X, it is evident, if those arcs be supposed in any constant ratio of 1 to n, that nẋ ⁄ √(1 - xx) = Ẋ ⁄ √(1 - XX), and consequently that nẋ ⁄ √(xx - 1) (= nẋ ⁄ √-1 × √(1 - xx) = Ẋ ⁄ √-1 × √(1 - XX)) = Ẋ ⁄ √(XX - 1). From whence, by taking the fluents, n × Log. (x + √xx - 1) (or Log. (x + √xx - 1)ⁿ) = Log. X + √XX - 1; and consequently (x + √xx - 1)ⁿ = X + √XX - 1: whence also, seeing x - √xx - 1 is the reciprocal of x + √xx - 1, and X - √XX - 1 of X + √XX - 1, it is likewise evident, that (x - √xx - 1)ⁿ = X - √XX - 1. Hence, not only the truth of the above assumption, but what has been advanced in relation to the roots of the equation zⁿ - 1 = 0, will appear manifest. For if x ± √xx - 1 be put = z, then will zⁿ (= (x ± √xx - 1)ⁿ) = X ± √XX - 1: where, assuming X = 1 = co-s. 0 = co-s. 360° = co-s. 2 × 360° = co-s. 3 × 360°, &c. the equation will become zⁿ = 1, or zⁿ - 1 = 0; and the different values of x, in the expression (x ± √xx - 1) for the root z, will consequently be the co-sines of the arcs, 0 ⁄ n, 360° ⁄ n, 2 × 360° ⁄ n, &c. these arcs being the corresponding submultiples of those above, answering to the co-sine X (= 1).——In the same manner, if X be taken = -1 = co-s. 180° = co-s. 3 × 180° = co-s. 5 × 180°, &c. then will zⁿ = -1, or zⁿ + 1 = 0; and the values of x will, in this case, be the co-sines of 180° ⁄ n, 3 × 180° ⁄ n, 5 × 180° ⁄ n, &c.