6 1. Cubic Equations Finally, the desired solution x of the cubic equation x3 + px + q = 0 is obtained from Cardano's formula l / f y / / / / A y y / / / u Figure 1.2. Depicted here is the geometric basis of the bi- nomial equation (it + v)3 = 3uv(u + v) + (w3 + f 3 ), similar to Cardano's presentation in his Ars Magna. The large cube can be decomposed into two subcubes and three rectangular parallelepipeds, all with side lengths w, v, and u + v. If we apply this result to our problem x3 + x — 6 = 0, we obtain x = W 3 + •», , i, - •«, w 3V 3 V 3V 3 ' whose decimal value is approximately 1.634365. 1.3 In his Ars Magna Cardano also solved cubic equations involv- ing quadratic terms.6 We have already seen, in the introduction, an 6 Ars Magna, Chapter XXIII.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2006 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.