The math battles of Italy: Part II

Ahh, math. Such a pity that it has been dulled down into brainless, tedious problems that students toil over for hours at school. In my last blog, I talked about how cubic equations were discovered through mathematical battles — contests where mathematicians challenged each other with problems to see who could solve the most.

We also saw how Niccolò Fontana discovered the secret to solving cubic equations and hid the method inside a poem.

Well, given the immense popularity of mathematics back then, it was only a matter of time before people learned about Fontana’s discovery. One man in particular, Gerolamo Cardano, took great interest. Cardano was a mathematician, physician, and gambler who immediately recognised the importance of Fontana’s work and set out to uncover his method.

Eventually, Cardano confronted Fontana. At first, Fontana refused to share his secret, but Cardano offered to introduce him — a poor and struggling scholar — to a wealthy patron. Fontana finally agreed and revealed his method, but only after Cardano promised never to publish it.

However, Cardano wanted more than just a solution to cubic equations. Alongside his brilliant student, Lodovico Ferrari (not the car), he began expanding on Fontana’s ideas to solve even more complicated problems. Ferrari soon realised that a similar technique could be used to solve quartic equations — equations involving an $x^4$ term, such as:

$3x^4+6x^3-32x^2+5x+765=0$

Cardano was ecstatic.

Then came the turning point. Cardano discovered that another mathematician, Scipione del Ferro(whom we mentioned in the previous blog), had actually discovered a method for solving cubic equations years earlier. Cardano believed this freed him from his promise to Fontana, since the discovery was no longer entirely Fontana’s secret.

So, in 1545, Cardano published Ars Magna (“The Great Art”), one of the most influential mathematics books ever written. In it, he included both Fontana’s cubic method and Ferrari’s quartic solution, while clearly acknowledging their contributions.

Needless to say, Fontana was furious.

A bitter feud erupted between Fontana and Cardano. Eventually, in true Renaissance fashion, the conflict was settled through a mathematical duel between Fontana and Ferrari. Unfortunately for Fontana, Ferrari utterly outclassed him. Realising Ferrari’s superior skill, Fontana withdrew before the contest had even ended.

Afterwards, Fontana’s reputation collapsed, and he died poor and deeply resentful.

And so, friends, this is how quartic equations — one of the great breakthroughs in algebra — were discovered.

But this is not the end. I’ll be back soon with more fascinating stories from the history of mathematics — stories that keep your mind running.

Fun Fact: Niccolò Fontana is better known as Tartaglia (“the stammerer”). On many websites, he is referred to simply as Tartaglia rather than by his real name.