Equations of the third degree, by definition, include at least one unknown cubed, the general form of the equation being:
Solving these types of equations was a topic that haunted mathematicians for a long time.
Throughout the Renaissance, this problem acquired romantic overtones and the protagonists involved created a combative atmosphere where betrayals, deceptions and even the occasional murder were not lacking.
The starting point of this story can be dated to 1515. It was in Bologna where the mathematician Scipione dal Ferro (1465-1526), the son of a paper manufacturer, achieved the desired solution. Far from publishing it and making it known to the rest of the university community, he kept it in absolute secret, he only shared it with Hannibal of Navia, his son-in-law, and with Antonio Maria Fiore, one of his disciples.
When dal Ferro passed away, his disciple decided to take advantage of the conclusions and with the solution under his arm he left Bologna and went to Venice, where another great mathematician was based, Niccolo Tartaglia, who, coincidentally, had been working in this field of mathematics for some time.
There would be no other opportunity equal to Fiore, if he was able to beat him in that field his name would appear forever engraved in golden letters in the History of Mathematics. He would be remembered for generations to come as one of the great Renaissance mathematicians. He had to be publicly humiliated, for that reason I challenge him to a mathematical duel.
Now the existence of this type of joust may surprise us, but in Renaissance Italy it was a very common practice. They were used to settle intellectual seizures or to dispute chairs. The best of society came to the duel and, despite the fact that the contestants played the most precious thing they had, their personal and academic prestige, many times there were bets involved.
The mathematical duel between Fiore and Tartaglia was established with the following conditions: each of the adversaries would pose thirty problems to his adversary and the one who solved the most correctly and in the shortest time would win.
Fiore did not think twice, all his problems would be on the same subject, the resolution of third degree equations. That was precisely his big mistake, because before accepting the challenge Tartaglia – named for his stuttering – he had already discovered the method to solve them.
The duel went to the side of Tartaglia who solved all the problems correctly in less than two hours, while Fiore was humiliated by being unable to solve even one. This duel took place on February 12, 1535 and was long remembered by the entire Venetian cloister.
The most boastful of the Renaissance
One of Tartaglia’s disciples was Gerolamo Cardano who, in turn, had a no less brilliant disciple, Ludovico Ferrari, who was able to reduce fourth degree equations to third degree equations, using what is known as cubic resolvers.
Cardano was a man as extraordinary as he was extravagant and a Renaissance in the fullest sense of the term. He devoted himself to medicine, philosophy, astrology, astronomy and, of course, mathematics. But to all this he added an unbridled ego. It was precisely the latter who led him to publish, without the consent of the authors, the knowledge of his teacher and that of his disciple in a book entitled “Ars Magna.” That was a scandal in capital letters.
Tartaglia challenged Cardano to solve that injury with a duel, but not mathematical but of swords. The second rejected the offer, but, against all odds, the other aggrieved, Ferrari, who challenged Tartaglia to a mathematical duel, picked up the glove.
This new confrontation took place in Milan, there Tartaglia was humiliated to the extreme of being forced to leave the city the same night of the defeat.
Meanwhile, Cardano, an inveterate gambler, continued to do his thing and almost ended up behind bars on more than one occasion as a result of his quarreling habits. Long ago he had predicted, through a tangled astrological study of his own, that his death would take place on September 21, 1575.
When the fateful day arrived, there was great expectation and Cardano himself debated between defrauding his followers or committing suicide. He opted for the latter, so that the man died but the “scientist” triumphed.
Pedro Gargantilla is an internist at the Hospital de El Escorial (Madrid) and the author of several popular books.