Scipione dal Ferro
Scipione dal Ferro
Scipione dal Ferro left behind no published writings, and were it not for papers found after his death, his role in unlocking one of the key mathematical challenges of his day might never be known. At the time, mathematicians were struggling with the solution to third-power equations, and the frustration associated with the quest had prompted Luca Pacioli (1445-1517) to suggest that such a solution was impossible. Unbeknownst to his colleagues, however, dal Ferro had found a way to solve such problems, but he kept this knowledge to himself.
The son of Floriano, a papermaker, and Filippa dal Ferro—the family name is sometimes rendered as Ferreo, Ferro, and del Ferro—dal Ferro was born in Bologna on February 6, 1465. He probably attended the University of Bologna, Europe's oldest institution of higher education; but other than this supposition, virtually nothing is known about his early life. It was at Bologna in 1496 that dal Ferro obtained what turned out to be lifetime employment as a lecturer in arithmetic and geometry. At some point he must have married, though there is no record of such, and fathered a daughter, Filippa, who grew up to marry one of his students, Hannibal Nave or Annibale dalla Nave.
Dal Ferro made his most important discovery some time between 1505 and 1515, but the fact that he was so secretive about it makes it difficult to pinpoint the date. Since the time of the Babylonians, mathematicians had known how to solve quadratic equations, ones in which (despite the somewhat deceptive name) the highest power is 2; but the solution to cubic equations, or those in which the highest order is 3—e.g., ax3 + bx2 + cx + d = 0—had eluded them.
After Pacioli, with whom dal Ferro was acquainted, made his statement in 1494, there followed many years of competition between the leading mathematicians of the day, each of them eager to find a solution to the cubic equation. Meanwhile, dal Ferro worked out the solution to what was called the depressed cubic, or one in which the second-power term was missing. Though this did not completely solve the larger problem, it would make such a solution possible—yet dal Ferro kept his discovery under his hat.
His reason for this may have been the then-common practice of public challenges, in which two prominent mathematicians engaged in a sort of intellectual boxing match. Often a mathematician's patronage, or his continued economic support by a wealthy benefactor, was at stake; and dal Ferro, who greatly feared the threat of a challenge, may have held out his secret as a sort of trump card to ensure victory.
But dal Ferro, who died on November 5, 1526, did not take his secret to the grave. He had written it in a notebook, which his son-inlaw Hannibal kept; and he had shared it with his lackluster assistant, Antonio Fior. Fior was even more paranoid about a challenge than his former mentor, and he recklessly engaged the superior mind of Nicolò Fontana, a.k.a. Tartaglia (1499-1557). As a result of the competition, which he won, Tartaglia discovered the depressed cubic method and appropriated it as his own.
Only in 1543, when Girolamo Cardano (1501-1576) and his assistant Ludovico Ferrari (1522-1565) paid a visit to Hannibal and saw the notebook, did the truth come out. Tartaglia had deceived Cardano into believing that he was the original discoverer of the depressed cubic solution, but now Cardano published the new information. Unlike Tartaglia, Cardano was eager to give credit where credit was due, referring to dal Ferro's work as "a really beautiful and admirable accomplishment." Ironically, however, neither dal Ferro nor Tartaglia ultimately received credit; rather, the depressed cubic is known today as Cardano's formula.