Solving the cubic sparked secret formulas, betrayal and a maths duel
Renaissance Italy settled mathematical rivalries with public contests, money on the table. The prize problem was the cubic equation. A method guarded as a secret, a broken promise and a published book in 1545 turned the equation into a scandal, and along the way forced mathematicians to confront square roots of negative numbers.
Cubic equations were studied long before Italy. Babylonian clay tablets from the 20th to 16th centuries BC hold tables of cubes and cube roots, and a Chinese classic, The Nine Chapters on the Mathematical Art, includes solution methods. Doubling the cube, the oldest such problem, defeated the Egyptians and the Greek Hippocrates. In the 11th century Omar Khayyam classified cubics and solved them geometrically with intersecting conic sections, admitting he had failed to find an algebraic formula and hoping later mathematicians would.
Early in the 16th century Scipione del Ferro cracked one family of cubics and hid the method, whispering it to his student Antonio Fior only near death. In 1535 Niccolò Tartaglia claimed he could solve cubics too, and Fior challenged him. Each posed problems to the other with a stake at risk and 30 days to answer. Tartaglia had a general method for his questions; Fior's set proved beyond him, and Tartaglia won.
Gerolamo Cardano coaxed the secret from Tartaglia in 1539, swearing not to reveal it. After learning of del Ferro's earlier work, Cardano printed del Ferro's method in Ars Magna, reasoning that this did not break his oath, and credited Tartaglia with an independent discovery. Tartaglia was furious and issued a challenge. Cardano's student Lodovico Ferrari took it up and beat him, costing Tartaglia his reputation and his income.
The formula carried a surprise. It sometimes demanded the square root of a negative number, and Cardano included such a calculation without really understanding it. Rafael Bombelli investigated properly, and for that he is often regarded as the discoverer of complex numbers. Later François Viète found a trigonometric solution for cubics with three real roots, which Descartes extended.
Source: Cubic equation
