Story

You want a story? Here's a story. A formula for the roots of a quadratic equation is simple enough to be taught to sixth-graders. But how about one for a cubic equation? Just one more degree and you have a story full of intrigue, challenge, vagaries of fate, lots of money, misunderstandings, medieval curiosities and finally sweet revenge by a disciple.

(All quotes below are sourced originally from wiki article on Cubic polynomials; I just followed the links to the later parts.)

In the early 16th century, the Italian mathematician Scipione del Ferro found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death in 1526, when he told his student Antonio Fiore about it.

Whoa! Negative numbers not known, formulae kept secret for life, a final change of mind on the deathbed, a lucky student,...

In 1530, another mathematician Niccolò Tartaglia announced that he could solve cubic equations. He was soon challenged by Fiore, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form x3 + mx = n, for which he had worked out a general method. Fiore received questions in the form x3 + mx2 = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Aww.. poor overconfident young Fiore, who lost so badly that his name is now almost ungooglable, and the fiendishly lucky Tartaglia. Grr, do we want revenge, or what!

Later, Tartaglia was persuaded by another mathematican, Gerolamo Cardano to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did reveal a book about cubics, that he would give Tartaglia time to publish. Some years later, Cardano learned about Ferro's prior work and published Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia 6 years to publish his results (with credit given to Tartaglia for an independent solution).

Cardano's promise with Tartaglia stated that he not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano by Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari. Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.

Take that, vile Tartaglia, you beater of poor students in public contests and usurper of their meagre RA stipends, you! Revenge is a dish best served cold, particularly by a disciple! (And apropos of Revenge: "Today, I was thinking about the expression 'revenge is a dish best served cold'. Then I considered that 'revenge is sweet'. I've come to the conclusion that revenge is ice cream. MLIA")

The story ends there. But as all good stories, there are layers upon layers of history and depth to everything, and they lead to more stories of their own. For example, why was the origin of the story, Del Ferro, so secretive?

Instead of publishing his ideas, he would only show them to a small, select group of friends and students. It is suspected that this is due to the practice of mathematicians at the time of publicly challenging one another. When a mathematician accepted another's challenge, each mathematician needed to solve the other's problems. The loser in a challenge often lost funding or his university position. Del Ferro was fearful of being challenged and likely kept his greatest work secret so that he could use it to defend himself in the event of a challenge.

Ha! And to think today's profs whine about losing tenure and not getting an NSF Career award :P

But as every good story-within-a-story, there's a lovely big red button begging to be pushed to take you deeper:

Despite this secrecy, he had a notebook where he recorded all his important discoveries..

Ahhhhhh, now that's a few more hours of wikiing :-)

OK, enough about this old guy. Let's look at another guy with character. What about this Cardano chap? He had to do quite a bit of self-justification and 'miserable pettifogging in the court of his own conscience' when he went ahead and published Tartaglia's work in spite of his promise. What of him?

He was born in Pavia, Lombardy, the illegitimate child of Fazio Cardano, a mathematically gifted lawyer. In his autobiography, Cardano claimed that his mother had attempted to abort him.

Assuming Cardano's psychological problems weren't the cause of this, imagine how it is to live knowing that.

He went on to do a whole lot of work, like being the first to describe Typhoid fever, publishing many results in algebra and probability, building several mechanical devices like the combination lock and universal joint, being instrumental (heh) in the development of high-speed printing presses through his work on hypocycloids (Mukund note), etc. Top notch all-rounder, fits the image of a Enlightened European Engineer+Scientist perfectly.

Significantly, in the history of Deaf education, he said that deaf people were capable of using their minds, argued for the importance of teaching them, and was one of the first to state that deaf people could learn to read and write without learning how to speak first.

And yet, in the same Enlightened Europe, a thought like that was so unconventional that it had to be noted down. But tragedy looms:

Cardano's eldest and favorite son was executed in 1560 after he confessed to having poisoned his cuckolding wife. His other son was a gambler, who stole money from him. He allegedly cropped the ears of one of his sons. Cardano himself was accused of heresy in 1570 because he had computed and published the horoscope of Jesus in 1554. Apparently, his own son contributed to the prosecution, bribed by Tartaglia. He was arrested, had to spend several months in prison and was forced to abjure his professorship.

:-( Grr, that vile bastard Tartaglia. Richly deserved his fate of penury, didn't he? What of him, anyway?

Niccolò Fontana was the son of Michele Fontana, a rider and deliverer. In 1505, Michele was murdered and Niccolò, his two siblings, and his mother were impoverished. Niccolò experienced further tragedy in 1512 when the French invaded Brescia during the War of the League of Cambrai. The militia of Brescia defended their city for seven days. When the French finally broke through, they took their revenge by massacring the inhabitants of Brescia. By the end of battle, over 45,000 residents were killed. During the massacre, a French soldier sliced Niccolò's jaw and palate. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" (stammerer).

Ouch, he doesn't seem much like a villain anymore :( Reminds me of an article a long time ago about how in most of Indian mythology, there is no villain or bad guy or Evil. Good people just assumed the role of the villain temporarily, just to let God have a little bit of leela. In the end, it's one big happy family.

Thinking a bit, what I find most amazing about this whole story and array of characters is that like life itself, it's all spectacularly pointless. What was all the fight for? A closed form solution for cubics? Why? Today, if you give me a cubic equation, the first thing I'll think of doing is to graph it with Wolfram Alpha. Um no, actually the first thing I'll do is to try to convince you that we're both better off by not solving that equation and chilling out instead,



but I'm assuming you didn't fall for that. Graph it, read off the roots and be happy in life. One might object that I am able to do so because of the efforts of all these people, but that is beside the point, that the exact thing these people were fighting for is no longer useful; further, one can always counter that every single fart of every single ant since the beginning of creation has been necessary for me to able to be sitting here and being able to graph cubics. Causality is a very tricky devil.

And that gives me an opportunity to quote one of my most favorite lines:

Life is like arriving late for a movie, having to figure out what was going on without bothering everybody with a lot of questions, and then being unexpectedly called away before you find out how it ends.

:-)
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