Of the Rule of Bombelli, for reducing the Resolution of Equations of the Fourth Degree to that of Equations of the Third Degree

765. We have already shewn how equations of the third degree are resolved by the rule of Cardan; so that the principal object, with regard to equations of the fourth degree, is to reduce them to equations of the third degree. For it is impossible to resolve, generally, equations of the fourth degree, without the aid of those of the third; since, when we have determined one of the roots, the others always depend on an equation of the third degree. And hence we may conclude, that the resolution of equations of higher dimensions presupposes the resolution of all equations of lower degrees. 766. It is now some centuries since Bombelli, an Italian, gave a rule for this purpose, which we shall explain in this chapter. Let there be given the general equation of the fourth degree, x 4 + ax 3 + bx 2 + cx + d = 0, in which the letters a, b, c, d, represent any possible numbers; and let us suppose that this equation is the same as ( x 2 + ½ ax + p ) 2 − ( qx + r ) 2 = 0, in which it is required to determine the letters p, q, and r , in order that we may obtain the equation proposed. By squaring, and ordering this new equation, we shall have Now, the first two terms are already the same here as in the given equation; the third term requires us to make, which gives; the fourth term shews that we must make ap − 2 qr = c , or 2 qr = ap − c ; and, lastly, we have for the last term p 2 − r 2 = d , or r 2 = p 2 − d .