In Folha, the duels of the cubic equation in the Renaissance.

Niccolò Tartaglia. Imagem: Wikimedia Commons

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

At the beginning of the 16th century, the Italian Scipione del Ferro (1465–1526) discovered a method for finding the solutions of any special cubic equation x³+mx=n. Del Ferro earned his living solving mathematical problems, and this equation was his great achievement. He kept the secret until his death, when he left it to his apprentice Antonio Fior.

Every cubic equation ax³+bx²+cx+d=0 can be reduced to the special form and, therefore, without knowing it, he had solved a much more general problem, dating back to 2000 BC. Without knowing it, because to perform the reduction it is necessary to use both positive and negative coefficients, and at that time negative coefficients had not yet been discovered.

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In 1530, a competitor emerged: Niccolò Tartaglia (1500–1557) announced that he also knew how to solve cubic equations. Concerned, Fior challenged him to a duel: each would propose equations to their rival, and whoever solved the most would win the bet. Fior proposed equations of a special type, which both knew how to solve. The astute Tartaglia, however, opted for equations of the type x³+mx²=n, which Fior was unable to solve.

Enter one of the most fascinating figures of the Italian Renaissance: Gerolamo Cardano (1501–1576). A mathematician, physician, biologist, chemist, astronomer, astrologer, philosopher, and writer, he was also an avid gambler. His interest in games of chance led him to be one of the pioneers of probability theory.

Although it wasn't a breach of the agreement, strictly speaking, Tartaglia felt betrayed and challenged Cardano to a mathematical duel. Cardano refused, but was replaced by his disciple Lodovico Ferrari (1522–1565), who won the contest, ruining Tartaglia's career.

Ferrari was no ordinary disciple. In 1540, he discovered the solution to the quartic equation ax⁴ + bx³ + cx² + dx + e = 0, which Cardano also published in Artis Magnae.

The expectation that higher-degree equations would soon follow was dashed at the beginning of the 19th century, when the Italian Paolo Ruffini (1765–1822) and the Norwegian Niels Henrik Abel (1802–1829) showed that from degree 5 onwards, such solutions do not exist.

To read the full text, visit the newspaper's website.