Gunfight at the Cubic Corral

My history of science colleague and Internet friend Dr SykSkull at Skulls in the Stars tweets a series of “weird science facts” and I was somewhat surprised the other day to see repeat the most widespread myths in the history of mathematics:

Cardano (1501-1576) stole and published Tartaglia’s solution for cubic equation; now known as “Cardano’s solution”.

Cardano did not steal Tartaglia’s solution and in my naivety I had assumed that everybody with an interest in the history of mathematics already knew the true story, obviously this is not the case so I have decided to retell it here, for once dealing with a couple of real life Renaissance Mathematicae.

That bane of all school children learning mathematics the general solution of the quadratic equation, minus ‘b’ plus or minus the square root of ‘b’ squared minus four ‘ac’ divided by two a, was in principle known to the Babylonians in about 1700 BCE. This knowledge naturally led to speculation about possible general solutions for higher order algebraic equations such as the cubic, the bi-quadratic and the quintic. Over the centuries various solutions of specific higher order equations were found and in the 11^th century the Persian poet and mathematician Omar Khayyám found geometrical solutions to the cubic based on the intersection of conic sections, i.e. parabolas, ellipses, hyperboles etc.

At the end of the 15^th century Luca Pacioli, Leonardo’s maths teacher, claimed in his Summa, the most influential maths textbook in the 16^th century, that a general algebraic solution of the cubic equation was impossible. His timing was wonderful, his book was published in 1494 and a few years later the Bologna professor of mathematics Scipione del Ferro discovered the general solution of one class of cubics. We think of all cubics as being various forms of the general equation ax³+bx²+cx+d = 0 but before the 17^th century this was not the case and equations such as ax³+cx = d and ax³+bx² = d were seen as fundamentally different. Ferro found a general solution for equations of the type x³+px = q. Del Ferro did not reveal his discovery but at his death passed it on to his student Antonio Fiore.

In the 16^th century most professional mathematicians were reckoning masters who earned their living doing the interest calculation etc. for the bankers and businessmen in Northern Italy’s highly commercial society. One such was Niccolò Fontana better know through his nickname Tartaglia, the stutterer. Now, in the 16^th century reckoning master were not in anyway certified and so they lived from their reputations a large pert of which were acquired in the form public challenges that were conducted in a form of slow-motion Renaissance ‘gun fight’. One reckoning master would challenge another and if the challenge was accepted then qualified referees were appointed and the challenge could take place. Each participant presented his opponent with an agreed number of mathematical problems that had to be solved in a set period of time. The winner was the one who managed to solve the largest number of problems. Fiore, del Ferro’s student, was a nobody whereas Tartaglia was a recognised master of his profession so armed with his solution of the cubic Fiore challenged Tartaglia. Now Tartaglia had heard rumours of the solution and as the subject of the challenge was cubics he knew he was on to a hiding to nothing, however he was a brilliant mathematician and so in the period before the contest he set to work and discovered a much wider set of solutions for the cubic than that known to Fiore. On the day of the contest Fiore presented Tartaglia with a set of identical problems all of which required his method of solution and received in return a list that could only be solved with other methods. Tartaglia sat down and almost instantly gave the correct answers to Fiore’s entire list, who was completely unable to solve a single one of Tartaglia’s questions. This whitewash made Tartaglia to a star amongst the reckoning masters, and because it was his trump card he also didn’t reveal his solution to the cubic.

Enter Gerolamo Cardano, doctor, mathematician, astronomer, astrologer, magician and philosopher one of the most colourful characters of the 16^th century. Cardano heard of Tartaglia’s triumph and was naturally interested in discovering his secret. Being a doctor Cardano’s social status was much higher than Tartaglia, a mere mathematician, and it was this that Cardano used to seduce Tartaglia into revealing his secret. In return for introductions to several important figures Tartaglia revealed the solution of the cubic to Cardano in 1539, but made him promise not to publish until he himself had done so. In 1543 Cardano travelled to Bologna with his assistant and pupil, Ludovico Ferrari, and here del Ferro’s nephew, Hannival Nave, showed them his uncle’s solution to the cubic. In 1545 Cardano published his Ars magna in Nürnberg by Johannes Petreius, the publisher of Copernicus’ De revolutionibus, which was an extensive work on the solution of higher order algebraic equations. His book included his own generalisation of Ferro’s solution of the cubic and Ferrari’s general solution of the bi-quadratic equation. In the introduction he gave credit to Ferro, Tartaglia and Ferrari for their contributions to the general solutions. Although Cardano made it very clear that Ferro had priority in solving the cubic and that it was Ferro’s work plus his own generalisation that he was publishing, Tartaglia immediately accused him of breach of trust and intellectual theft, charges of which Cardano was clearly innocent. It should be pointed out that Tartaglia had had six years in which to publish his own solution but had not done so. When he did finally publish his solution it was so inferior to the generalised work on higher order equations that Cardano and Ferrari had created that it went largely unnoticed.

Tartaglia kept trying to draw Cardano into a debate over his grievances but Cardano was not to be baited with his pupil Ferrari taking up the cudgels on his behalf. Ferrari challenged Tartaglia to a mathematical contest but Tartaglia refused until 1548 when he was offered a lectureship in mathematics on the condition that he publicly contest Ferrari. Tartaglia was convinced that he would easily defeat Ferrari but by the end of the first day it was obvious that Ferrari was his superior and so that night he slipped out of town thereby losing the contest and ruining his reputation. With the Ars magna Cardano become acknowledged as the greatest mathematician in Europe and the publication of his book is taken by many historians of mathematics to be the start of modern mathematics, a view that I don’t share.

Cardano and Ferrari’s triumph led, of course, to further speculation about a general solution to the quintic i.e. fifth order algebraic equations and Évariste Galois’ proof, in the 19^th century, that none exists led to the development of group theory but that as they say is another story.