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 ax^{3}+bx^{2}+cx+d = 0 but before the 17^{th} century this was not the case and equations such as ax^{3}+cx = d and ax^{3}+bx^{2} = d were seen as fundamentally different. Ferro found a general solution for equations of the type x^{3}+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.

Well, poopie on you! I was more or less familiar with the broad strokes of the story, though not some of the subtleties. Twitter is a tough medium to convey details in! The thing that amused me the most is that Cardano seems to have gotten recognition for a solution that he didn’t find himself.

I’m glad to have served as an early muse for your 2nd year! 😉

Cardano gets the credit because he published first.

I couldn’t wish for a better muse!

Now this is the first time I’ve ever heard about this whole controversy but I fail to see how the story proves your point:

1. Ferro discovered solution to one subtype of cubic equations but didn’t publish.

2. Then Tartaglia discovered the general solution to the cubic but didn’t publish.

3. Then Tartaglia gave his solution to Cardano on condition that he doesn’t publish it before Tartaglia.

4. Then Cardano publishes general solution and claims that it’s actually a Ferro’s one and his own generalization!

Clearly Tartaglia is right in the case of general equation – Cardano broke the contract. Once you know the answer you cannot “reinvent” it and claim it’s yours, Cardano claim to a general solution that he made it by generalizing Ferro work is laughable cause he already new what the general solution was and he knew it because Tartaglia worked it out and gave it to him.

Tartaglia did not discover the general solution but only a wider solution than that available to Fiore. I left out the maths but Ferro’s solution actually implicitly contains the general solution including Tartaglia’s extension. The work that Cardano published goes far beyond the work of Tartaglia. Also Cardano never claimed to discovered the solution but credits this to Ferro which is perfectly correct. Anybody with Cardano’s capabilities, and he was a brilliant mathematician, who like Cardano saw Ferro’s solution could like him develop the general solution thereby preempting any future publication of Tartaglia and this is basically what Cardano did.

Isn’t a wider solution a more general solution?

From what I can see on wikipedia Ferro worked out only x3 + mx = n and was unable to generalize this. Tartaglia worked out both x3 + mx = n and x3 + mx2 = n so his method was clearly more general.

The fact that neither Ferro nor Fiore were able to generalize Ferro solution proves that it was certainly not equivalent to Tartaglia work at the time.

Once Cardano learned the more general method from Tartaglia he couldn’t claim to later rederive it from Ferro work, this is not a valid claim as rederiving a result you know is banal compared to obtaining a completely novel result.

If Cardano was so brilliant why didn’t he come up with a solution himself instead of having to pressure Tartaglia to reveal his?

How would you feel if after you have given me your new and unique result on condition I do not publish it I published the same result claiming I have derived it myself from the earlier work and claimed that your result was implicit in the body of mathematics anyway. Every result can be said to be implicit in the body of mathematics but unless it’s derivation is completely trivial and obvious to someone with standard math training (at the time in question) it’s not a defense for not giving credit..

1) Cardano gives full credit for their achievements to both del Ferro and Tartaglia and does not claim to have discovered anything already discovered by them.

2) History is full of brilliant mathematicians who failed to find a solution to a given specific problem. This doesn’t make them any less brilliant.

3) In the field of algebraic equations Cardano, in the Ars magna, is the first mathematician to give a full general analysis of the quadratic, the cubic and the bi-quadratic over the reals. He no longer treats different formulations as different entities but recognises them as variants on a standard form as we do today. He even considers complex roots but unlike Bombelli he is mentally unable to take the final step and accept their existence. His treatment of algebraic equations is way ahead of anything done previously, including by Tartaglia.

4) I hope I have finally made it clear to you that he did not publish the same result claiming anything. He published a complete analysis of second, third and fourth order algebraic equations giving credit for those partial results discovered by others to those who had discovered them, del Ferro, Tartaglia, Ferrari et. al. As Tartaglia result is actually only a fairly trivial extension of del Ferro’s which was in the public arena Cardano can hardly be accused of stealing anything.

“I hope I have finally made it clear to you that he did not publish the same result claiming anything. He published a complete analysis of second, third and fourth order algebraic equations giving credit for those partial results discovered by others to those who had discovered them, del Ferro, Tartaglia, Ferrari et. al. As Tartaglia result is actually only a fairly trivial extension of del Ferro’s which was in the public arena Cardano can hardly be accused of stealing anything.”

OK, if Cardano did not claim credit and gave credit to Tartaglia for his part then I agree that he didn’t steal anything.

The only point of contention is whether he broke his promise to Tartaglia. Here I believe the answer is yes, he did publish Tartaglia work (even credited it as you say), and how insignificant it was in the whole book is completely irrelevant.

That Tartaglia result is “actually only a fairly trivial extension of del Ferro’s” is your interpretation which IMO is not supported by the facts. First neither Ferro nor Fiore two trained mathematicians working in this area were able to derive it even though they certainly tried – this fact alone means it was NOT trivial (unless you have some weird definition of trivial), it may seem trivial now when you already know the answer and further development, besides the fact that the procedure sometimes requires squares of negative numbers (which you imply even Cardano was uneasy with and which had caused a whole controversy of their own) is a further proof that it was in fact a highly nontrivial matter at the time. So however minor it may seem now the work of Tartaglia was a genuine advancement in the field and Cardano broke his word when he published it after promising he won’t do it before Tartaglia.

Now I agree that it was probably a right thing to do and that it benefited mathematics and the society but that still doesn’t change the fact that the promise was broken.

Ok, I’ll leave this discussion at that, I think my stance is also clear now, I enjoyed your post and this discussion.

I think that the promise demanded by Tartaglia implied that Trataglia was about to publish and not that he was doing nothing for years. There’s something like a higher cause of the advancement of knowledge involved here. Apparently, Tartaglia was not the finisher type that would have advanced knowledge any time soon. What was Cardano supposed to do. Die with his insight remaining a secret and in the knowledge that Tartaglia will never come around?

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Speaking of proper credit, Abel was the first to prove that the general quintic was unsolvable by radicals. Ruffini had earlier given a near-proof, i.e. an argument with a significant gap in it. The gap can be patched, but Ruffini did not notice the gap. Galois, finally, proved the unsolvability by radicals for all general equations of degree greater than 4; he also gave a criterion for particular equations to be solvable by radicals. (For a rather trivial example, the equation x^5 = 10 is solvable.)

Michael Rosen has written a very nice article about Abel’s proof, available from the MAA website, “Niels Hendrick Abel and Equations of the Fifth Degree”.

Your essay reminds me of something I’d been wondering. I know of the public challenges to do problems that was common in Cardano and Tartaglia’s time … but only from the one instance of Cardano and Tartaglia’s challenge. I don’t think I’ve run across a mention of it as an incident in any other mathematician’s biography. Have I just been oblivious, or more interested in mathematicians from times and places that didn’t have such challenges? Or was this something so much a part of the finance world that mathematicians don’t even notice it happened outside of this?

Would you be able to recommend a decent review of the way these reckoning challenges developed, and how they faded from common practice?

The point you raise is an interesting one. Like you, I know that such challenges were common in the Renaissance amongst reckoning masters to prove their capabilities and to recommend themselves to potential employers and I am absolutely certain that I have read literature on exactly this. When Michael Brook was writing his book on Cardano,

The Quantum Astrologer’s Handbook, he got in touch with me to ask exactly that question: could I recommend literature on this topic? I thought no problem and began to search. I like Michael and another historian of maths that I know who specialises in Renaissance mathematics all came up blank! If you can find anything I would be eternally grateful!Huh! I’m surprised it’s been hard to find other examples of these challenges. If I do find anything I shall certainly pass it on. Also thanks for the word about a book on Cardano.

You can find my review of Michael’s book here

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