Miscellanea Curiosa, Vol. 1 / Containing a collection of some of the principal phaenomena in nature, accounted for by the greatest philosophers of this age

by Unknown

But if in the aforegoing Equation the Signs are alternately Affirmative and Negative; or which is the same thing if the Series be after this manner,

ny + 1 - nn / 2 × 3 ny³ + 1 - nn / 2 × 3 × 9 - nn / 4 × 5 ny⁵ + 1 - nn / 2 × 3 × 9 - nn / 4 × 5 × 25 - nn / 6 × 7 ny⁷, &c. = a.

The Root of it will be equal to

(1) y = ½ n√ a + √ aa - 1 + ½ / n√ a + √ aa - 1 or,

(2) y = ½ n√ a + √ aa - 1 + ½ n√ a - √ aa - 1 or,

(3) y = ½ / n√ a - √ aa - 1 + ½ n√ a - √ aa - 1 or,

(4) y = ½ / n√ a - √ aa - 1 + ½ / n√ a + √ aa - 1

Here it is to be noted, that if n - 1 / 2 be an odd Number, the Sign of the Root found must be contrary to it.

Let an Equation be propos'd 5y - 20y³ + 16y⁵ = 6, whence n = 5, and a = 6, and the Root will be

= ½ 5√ 6 + √ 35 + ½ / 5√ 6 + √ 35