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1535-1545 CEReputable source · 2 sourcesDebated

Tartaglia Solves the Cubic, Then Cardano Breaks His Oath and Publishes It

A secret formula, a sworn promise never to reveal it, and the book that broke the promise anyway

On the timeline · around 1535-1545 CE · The Scientific RevolutionThe Scientific RevolutionTartaglia Solves the Cubic, Then Cardano Breaks His Oath and Publishes It14501500155016001650

Quick facts

Del Ferro's initial solution
c. 1515, form x^3 + mx = n
Tartaglia's contest win
1535, against Antonio Fior
Ars Magna published
1545, by Cardano
Quartic solved by
Lodovico Ferrari, Cardano's student

What happened

Scipione del Ferro, who held the chair of arithmetic and geometry at the University of Bologna, worked out how to solve cubic equations of the form x cubed plus mx equals n sometime around 1515, but kept the method secret until just before his death in 1526, when he revealed it only to his student Antonio Fior. In 1535 Fior challenged Niccolo Fontana, known as Tartaglia, the stammerer, to a public problem-solving contest, but Tartaglia had independently worked out how to solve a wider class of cubics and, with only eight days left before the deadline, found a general method covering all of Fior's problem types, winning the contest outright. The mathematician and physician Gerolamo Cardano later persuaded Tartaglia to share the method, swearing by God's holy Gospels never to publish it and promising to record it only in code so no one could read it after his death. In 1543 Cardano learned that del Ferro had solved the same problem years earlier, and used that prior claim as justification to break his oath: in 1545 he published Ars Magna, the first major Latin treatise on algebra, disclosing the cubic solution along with his student Lodovico Ferrari's method for solving the quartic equation.

Why it matters

The Renaissance solution of the cubic and quartic equations extended algebra well past what ancient and Islamic mathematicians had solved and introduced European mathematicians to complex numbers, since Cardano's formula sometimes requires taking the square root of a negative number even when the final answer is a real one. The dispute over credit between del Ferro, Tartaglia, and Cardano is one of the earliest and most bitter priority disputes in the history of mathematics, foreshadowing the far larger fight over calculus a century and a half later.

How we know

Tartaglia's own account of Cardano's sworn oath survives in his later writings, and Cardano's student Ferrari documented seeing del Ferro's original manuscript directly, in the hand of Cardano's father-in-law, giving historians corroborating testimony from multiple people directly involved in the dispute.

Sources

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