The Great Art or The Rules of Algebra
Girolamo Cardano (1501-1576) wrote prolifically on many subjects, including philosophy, mathematics, astronomy, astrology, and gambling – he was himself an inveterate gambler. But his magnum opus is certainly The Great Art, the first full bloom of Renaissance mathematics.
The principal subject of this work is the solution of the 13 arrangements or forms of the cubic equation. Cardano does not consider the single generalized cubic equation, but treats the 13 forms in separate chapters, perhaps as a parallel to the development of the quadratic equation by the algebrist al-Khowarizmi (ca. A.D. 840). Having solved these equations, Cardano regarded the art of algebra as essentially complete, since he regarded as unseverable the geometrical connection between power and dimensionality: “... we conclude our detailed considerations with the cubic, others being merely mentioned, even if generally, in passing. For as position [the first power] refers to a line, quadratum [the square] to a surface, and cubum [the cube] to a solid body, it would be very foolish for us to go beyond this point. Nature does not permit it.” (But, notwithstanding this attitude, Cardano and his pupil Lodovico Ferrari also played an important role in opening up the biquadratic equation to solution. This development is likewise treated in The Great Art.)
Cardano, famous as a teacher, passionately believed in expounding new ideas – his own and other men's. This spirit of openness and intellectual freedom in the discovery and dissemination of ideas was just awakening in the Renaissance, and led to the dispute with the mathematician Tartaglia, resulting in the sobriquet “the so-called Cardano solution.” Tartaglia developed a formula for solving a cubic of the form x3+ax=n, told Cardano about it, and perhaps (or perhaps not) swore him to secrecy. Cardano published the formula in The Great Art, giving Tartaglia full credit, and went on to present the first proof of its correctness. Cardano then extended this method to apply to other forms of the cubic equation. He also considered the possible number of solutions, demonstrated the existence of identical and multiple roots, and emphasized the importance of negative, irrational and complex solutions. He displayed a willingness to accept imaginary numbers in spite of what he called their “sophisticated” nature.
Cardano did not complete the art of algebra, as he himself thought. Rather, his solutions of the 13 forms of the cubic equation can be considered one of the foundation stones upon which subsequent advances in the art and science of mathematics have been built.