TRISECTION

, the dividing a thing into three equal parts. The term is chiefly used in Geometry, for the division of an angle into three equal parts. The Trisection of an angle geometrically, is one of those great problems whose solution has been so much sought for by mathematicians, for 2000 years past; being, in this respect, on a footing with the famous quadrature of the circle, and the duplicature of the cube.

The Ancients Trisected an angle by means of the conic sections, and the book of Inclinations; and Pappus enumerates several ways of doing it, in the 4th book of his Mathematical Collections, prop. 31, 32, 33, 34, 35, &c. He farther observes, that the problem of Trisecting an angle, is a solid problem, or a problem of the 3d degree, being expressed by the resolution of a cubic equation, in which way it has been resolved by Vieta, and others of the Moderns. See his Angular Sections, with those of other authors, and the Trisection in particular by cubic equations, as in Guisne's Application of Algebra to Geometry, in l'Hospital's Conic Sections, and in Emerson's Trigonometry, book 1, sec. 4. The cubic equation by which the problem of Trisection is resolved, is as follows: Let c denote the chord of a given arc, or angle, and x the cord of the 3d part of the same, to the radius 1; then is , by the resolution of which cubic equation is found the value of x, or the chord of the 3d part of the given arc or angle, whose chord is c; and the resolution of this equation, by Cardan's rule, gives the chord