Note 155 — 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.
Reading edition
If α, β, γ, δ, &c. be supposed to represent the co-sines of the angles 360° ⁄ n, 2 × 360° ⁄ n, 3 × 360° ⁄ n, &c. (the radius being unity); then the roots of the equation zⁿ - 1 = 0 (expressing the several values of p, q, r, s, &c.) will be truly defined by 1, α + √αα - 1, α - √αα - 1, β + √ββ - 1, β - √ββ - 1, &c. The demonstration of this will be given farther on.