1. Cubic Equations 5 secret. Nonetheless, Cardano published the solution procedure in his Ars Magna, a book describing the current state of algebraic knowl- edge.5 The solution of cubic equations is based on the cubic binomial formula (u + ^) 3 = 3uv(u + v) + (u3 + v3) , which Cardano was able to derive in an analogous manner to the geometric method used by al-Khwarizmi for the quadratic equation, though in this case the argument of course used three-dimensional figures and volumes (see Figure 1.2). However, this identity can also be interpreted as a cubic equation, where the sum u -\- v yields a solution x of the cubic equation x3 + px + q 0 if the conditions Suv = —p, u3 + v3 = -q, are satisfied. The cubic equation x3 + px + q = 0 can then be solved if one can find suitable quantities u and v. But that is a relatively simple task. Since both the sum and product of the quantities u3 and v3 are known, one can solve the quadratic equation w2 +qw - (-J = 0 to obtain u3 and v3 as the two solutions -N(W. so that u and v can be determined from the two equations •=fbW^W- -f!-i/(!)'+(i)"- 5 The result of Cardano's alleged breaking of his word led to a great falling out between Tartaglia and Cardano. It is thanks to the publications around this dispute (see the works referred to in an earlier footnote) that we have knowledge of the history leading up to the Ars Magna.
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