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## The disputes for the solution of the cubic

Luca Pacioli (1445-1514) wrote in his Summa de arithmetica, geometria, proportioni et proportionalità that it was not possible to solve the cubic. Ironically, the solution of the cubic is now attributed to Scipione del Ferro (1465-1526) who was living at the same time as Luca Pacioli. Scipione del Ferro never published his findings, but revealed them to a student of his, Antonio Maria Fiore. Before long rumours started that the solution to the cubic was known, and this pushed Niccolò Tartaglia (1500-1557) to search for the solution, which he found independently of Scipione del Ferro. A contest was organized between Fiore and Tartaglia, where each of them had to solve 30 problems posed by the other contestant. It is told that a few days before the contest Tartaglia managed to find the general method of solving another kind of cubic, so the problems he posed to his opponent were all of this second type that Fiore did not know how to solve. Apparently Tartaglia managed to solve all the 30 problems given by Fiore in a couple of hours, while Fiore did not manage to solve any of Tartaglia's problems. Tartaglia was the winner, and when news of this contest reached the ears of Girolamo Cardano (1501-1576) the latter invited Tartaglia to Milan with the vague promise of introducing him to a nobleman who could have given him a pension. Instead Cardano asked Tartaglia to reveal to him his method for solving cubics so that he could include it in his book, the Ars magna. Tartaglia at first told him that he himself wanted to write a treatise of algebra, so he would have published the method in his book, but he successively disclosed his method to Cardano under somlemn oath of keeping it secret until he had published his treatise. Following this meeting Cardano went to Bologna to study Scipione del Ferro's papers, and he found the solution that Cardano had revealed to him. Cardano felt therefore that the oath was not binding as the solution was not original, and published it in the Ars magna.

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Leo Liberti
Thu Feb 26 17:04:11 CET 1998