Duels of the Cubic Equation in the Renaissance – 03/30/2021 – Marcelo Viana

In the early 16th century, the Italian Scipione del Ferro (1465–1526) discovered a way to find solutions to any particular cubic equation x³ + mx = n. Del Ferro made a living solving math problems, and that equation was his great accomplishment. 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 a special form, and therefore, without knowing it, he had solved a much more general problem that lasted up to 2000 BC. BC reaches back. Without knowing it, for the reduction it is necessary to use both positive and negative male coefficients, and at the time the negatives had not yet been discovered.

In 1530 a competitor appeared: Niccolò Tartaglia (1500–1557) announced that he also knew how to solve cubic equations. Worried, Fior challenged him to a duel: everyone suggested equations to their rival, and whoever made the most decisions won the bet. Fior proposed equations of the special type that both could solve. The wise Tartaglia opted for equations of the type x³ + mx² = n that Fior could not solve.

One of the most fascinating figures of the Italian Renaissance enters the scene: Gerolamo Cardano (1501–1576). The mathematician, doctor, biologist, chemist, astronomer, astrologer, philosopher, and writer was also a die-hard gambler. His interest in gambling made him one of the pioneers of probability theory.

In 1539 Cardano convinced Tartaglia to reveal his method to him and promised not to reveal it to anyone. In his 1545 book Artis Magnae (Great Art in Latin), however, he published the method of Scipione del Ferro, which he had also learned.

While this wasn’t strictly a violation of the agreement, Tartaglia felt betrayed and challenged Cardano to a math duel. He refused, but was replaced by his student Lodovico Ferrari (1522-1565), who won the competition and ruined Tartaglia’s career.

Ferrari wasn’t just any student. In 1540 he discovered the solution of the quarterly equation ax⁴ + bx³ + cx² + dx + e = 0, which Cardano also published in Artis Magnae.

The expectation that the higher equations would soon follow was dashed in the early 19th century when the Italian Paolo Ruffini (1765–1822) and the Norwegian Niels Henrik Abel (1802–1829) showed that it was from the 5th grade onwards there are no such resolutions.

The work of the French Évariste Galois (1811-1832) published posthumously in 1846 culminated in this adventure and created a general theory for solving polynomial equations.

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