In sixteenth century Italy, a mathematician could lose his job in an afternoon. University positions were defended through public challenges: two opponents each submitted a list of problems, both worked in private for a fixed period, and whoever solved more problems in front of witnesses kept the chair and the salary. In that environment, a method nobody else possessed was not knowledge to be shared. It was capital.
This is the setting for the most famous priority dispute in the history of algebra, and for the solution of a problem that had resisted mathematicians for more than three thousand years.
The Problem That Would Not Yield
Quadratic equations had been solved by the Babylonians, by the Greeks in geometric form, and systematically by Al-Khwarizmi in ninth century Baghdad. Every serious mathematician knew how to handle them.
The cubic, an equation containing an unknown raised to the third power, was different. Omar Khayyam had given geometric constructions for particular cases in the eleventh century using intersecting conic sections, but a general algebraic formula, a recipe that turns the coefficients into the solution, did not exist. Luca Pacioli, who wrote the standard mathematical textbook of the Italian Renaissance, stated in 1494 that the problem was impossible.
A further complication made the work harder than it needs to be today. Negative numbers were not accepted, so the cubic came in several separate cases depending on which terms fell on which side of the equation, and each case had to be solved individually.
Scipione del Ferro’s Secret
Around 1515, Scipione del Ferro, professor at the University of Bologna, solved one of those cases: the equation now written as x³ + px = q, the so called depressed cubic with no square term.
He told nobody. He recorded the method in a private notebook and kept it for the rest of his life as insurance against a challenge. On his deathbed in 1526 he passed it to a student, Antonio Maria Fior, who was not a strong mathematician but now owned something valuable.
In 1535 Fior used it. He challenged Niccolò Fontana, known as Tartaglia, and set him thirty problems, all of them depressed cubics.
Tartaglia
Tartaglia was born poor in Brescia around 1500. When the French sacked the city in 1512 he was a boy, and a soldier’s sabre split his jaw and palate. He survived with a permanent speech impediment, which is where the nickname comes from: tartagliare means to stammer. He taught himself mathematics and made a living teaching it in Venice.
Fior’s challenge was a serious threat, because Tartaglia had claimed he could solve cubics of a different form, and Fior’s thirty problems were all of the form Fior could solve and Tartaglia could not.
On the night of 12 February 1535, eight days before the deadline, Tartaglia found the method. He solved all thirty problems in about two hours. Fior, who could only handle the single case he had inherited, solved none of Tartaglia’s. Tartaglia declined the prize dinner and took the reputation instead.
The Oath
News of the victory reached Gerolamo Cardano in Milan. Cardano was a physician by profession, one of the most successful in Europe, and also a mathematician, an astrologer, and a compulsive gambler who wrote the first serious treatise on the mathematics of games of chance, a century before Pascal and Fermat.
He was preparing a book on algebra and wanted the cubic. He wrote to Tartaglia repeatedly and was refused. In 1539 he tried a different approach and invited Tartaglia to Milan, dangling an introduction to the Spanish governor, a potential patron for Tartaglia’s work on ballistics.
Tartaglia came, and after some pressure he handed over the method. He did it in the form of a poem, twenty five lines of verse encoding the procedure, which was a common way of making a formula memorable and slightly obscure at the same time. He extracted an oath in return: Cardano swore not to publish it.
What Cardano and Ferrari Did Next
Cardano worked out why the formula was correct, which Tartaglia had never explained, and extended it to the other cases of the cubic. His assistant Lodovico Ferrari, who had joined the household as a servant at fourteen and turned out to be a mathematical prodigy, then solved the quartic, the fourth degree equation.
Ferrari’s method reduced any quartic to an associated cubic. This created an awkward situation: the two men now had a major result that they could not publish, because using it required the cubic formula, and the cubic formula was under oath.
The deadlock broke in 1543. Cardano and Ferrari travelled to Bologna and were shown the papers of the late Scipione del Ferro by his son in law, Annibale della Nave. There, in a notebook written years before Tartaglia’s sleepless night, was the solution.
Cardano concluded that his oath covered Tartaglia’s discovery, not del Ferro’s, and that he was free to publish the earlier work.
Ars Magna
The Ars Magna appeared in 1545 and is one of the founding books of modern algebra. It gives complete solutions to the cubic and the quartic in all their cases, with worked examples and geometric justifications.
It is worth noting that Cardano did not conceal his sources. Chapter XI states plainly that Scipione del Ferro solved the case first, that Tartaglia rediscovered it and communicated it to him, and that Ferrari solved the quartic. By the standards of the period this was scrupulous attribution.
Tartaglia did not see it that way. He published a furious attack in 1546, accusing Cardano of perjury. Ferrari responded on his master’s behalf, and the exchange escalated through twelve printed pamphlets over two years. It ended in a public debate in Milan in August 1548, in a church, before a large crowd. Accounts of what happened are partisan, but Tartaglia left the city before the second day and the position he had been hoping for went to Ferrari. He returned to Venice and died poor.
The Accident That Produced Complex Numbers
The mathematically important consequence of the Ars Magna had nothing to do with the quarrel.
Cardano noticed that in certain cubics with three perfectly ordinary real solutions, the formula demands that you take the square root of a negative number partway through the calculation. The negative quantities cancel by the end and the correct real answer appears, but the intermediate steps are impossible under the rules as they stood. This case is called the casus irreducibilis, and it cannot be avoided.
Cardano recorded the difficulty, called such quantities as subtle as they are useless, and moved on. Rafael Bombelli took it seriously in 1572, worked out consistent arithmetic rules for these objects, and showed that treating them as legitimate numbers made the whole procedure coherent.
That is the origin of complex numbers, and it is a genuinely unusual episode. They were not introduced to solve unsolvable equations. They forced their way into mathematics as a necessary detour on the road to answers everyone already agreed were real.
The End of the Story
After the quartic, the obvious next target was the fifth degree equation. Mathematicians pursued it for nearly three centuries without success.
The answer, when it came, was not a formula. Paolo Ruffini and then Niels Henrik Abel proved that no general formula in radicals can exist for the quintic. Évariste Galois then explained precisely why, by connecting the solvability of an equation to the structure of the group of symmetries among its roots, and in doing so created a branch of mathematics far more important than the problem that prompted it.
Between Cardano and Galois, the notation also changed beyond recognition. Viète introduced letters for coefficients, Descartes fixed the modern conventions, and Gauss proved that every polynomial equation has as many roots as its degree, a result he returned to four times across his career and which appears in his working notebooks.
Why It Mattered
The Ars Magna marks the first point at which European mathematics went beyond what it had inherited from the Greeks and the Islamic world. Everything up to that moment had been recovery, translation, and commentary. The solution of the cubic and the quartic was new.
Cardano says as much in his own preface, and the confidence it produced is visible in the century that follows. A mathematical culture that has just solved a problem Pacioli declared impossible behaves differently from one that assumes the ancients knew best.
The formula itself is now a minor tool, rarely used in practice because numerical methods are faster. What survived is everything that came with it: complex numbers, the beginnings of group theory, and the demonstration that a problem can be closed not by finding an answer but by proving that no answer of the expected kind exists.
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