Monday, December 14, 2009


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.



Mahesh Mahadevan said...

Thalai, you are shaping into a Douglas Hofstadter or Bill Bryson. Seriously, history of science FTW!
You could even write a Hindu mythology story about it and call it TrimUla dahanam :-D

I had only heard about Cardano and Ferrari (who also developed a methods for solving biquadratics) before, and the full story seems much more brilliant!

Revenge is ice cream: ROTFL! On a related note, I guess my idea of a finishing PhD party to be called a dessertation party, seems to now have undertones...

Aditya said...

How I look forward to every post of yours!

"Every single fart of every single ant". LOL

Reminds me of a story we had in our school English textbook in 7th standard - A Sound of Thunder. If you haven't read this already, I think you'll enjoy it!

doodlesandmusings said...

"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."

"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."

Beautiful... I thoroughly enjoyed this article. Keep going!

shreevatsa said...

Nice story, and nice post, of course.
(Cardano actually published a book just two months after Tartaglia had revealed his secret, which nearly gave the latter a heart attack, but the book turned out not to contain the method. So when it was finally revealed six years later, T. ought to have been prepared.)

Not to spoil the spirit, but minor comments:

Ouch, he doesn't seem much like a villain anymore :(
Why? Because his father was murdered and his jaw sliced, he's not responsible for his actions?

(Why did he seem a villain in the first place? Because he was clever and kept his methods secret, like every mathematician of the time?)

I don't agree that in Indian mythology there is no evil; this seems like a postmodern attempt to blur lines even when they exist. :P
There clearly are evil actions. Of course, there is no "Evil", and it is understood that "good people" or "bad people" can only be spoken of after the fact (rather than being intrinsic attributes) but this doesn't mean everyone is equivalent.

Also, the beauty of mathematics is not that it is useful. (I feel tempted to move the "not" closer to the end of the sentence!) The idea of finding roots by successive approximation — which is what graphing is — was already known ever since Archimedes, surely? The question of solving cubic equations "exactly" was worthwhile for its own sake, because it pushed the boundaries of what we knew to do. (BTW, the closed form is ugly; they were after the method.) If nothing else, it led to the magnificent Galois theory (and group theory!), with another gripping story of genius, intrigue, accidents, love, politics, duels, and tragedy.

[BTW: some coincidental — or barely incidental — parallels with the classical Indian way of doing mathematics: Tartaglia wrote down his solution as a nice poem, and (unrelated) Brahmagupta had challenged others with "he who can solve x²-92y²=1 in a year is a mathematician". :-)]

Mohan K.V said...

@Thalai and AJ, thank you, I very humbly appreciate your continuous support! An idealized finite-length uniform roller BC would shrivel in shame and quietly convert to a pin joint if it comes face-to-face with the uniformity of the support of your patronship :-)

@Thalai: Desertation, LOL :D

@AJ, thanks man! You have a million times more patience than me, to actually read all that I write! :-)

I've read the story, it's very nice. I read it after seeing The Butterfly Effect and reading the wiki article :-)

@Gopal, thank you, the Life-is-a-movie line is one of my favorite quotes, and I'm glad you liked it too. And welcome to my blog!

@shreevatsa, thanks, I'm glad you liked it. That 'of course' is one of the most tickling things I've read in days :-)

(I was actually waiting for your comment :-) ) All the points you raise are extremely important, and I had thought of some of them when writing this. First, in this entire post, I wrote down my first reactions or what I instinctively felt instead of any kind of reasoned judgment. The part which made me feel T was a villain was him bribing Cardano's son to tell on him.

What you say about him not deserving to be absolved of villainy is absolutely true, but that is not how I _feel_ when I read about him having a troubled childhood. As far as my (highly irrational) feelings are concerned, it is almost as if there exist something like 'tragedy brownie points'. If a bad guy has had a hard life, he certainly seems to be more deserving of sympathy than an equally bad guy who's had a good life. Is this even remotely logical or sensible? Absolutely not, but that's not my first reaction! I specifically wanted to highlight that here. Almost the entirety of news media today exploits this kind of short-sighted not-reasoned knee-jerk judgments, and I wonder if I my first reactions to things will change even when I know for sure they are wrong.

Stepping back, how can we draw any judgment on anyone after 500 years anyway? Who is to know what was recorded, and what was not? Who is to tell this wasn't the only moral-slip of poor old T, and that he was a very virtuous man otherwise, making this just a silly non-incident? At best we can speak about an idealized, closed set of 'facts' to attribute blame, but I've never been comfortable with carrying forward a purely intellectual game of judging things into the real world, upon real people.

Another thing that I deliberately did was to try to make T seem like a villain just because he was lucky in that contest. That's another common pitfall in my first-reactions. The only ground on which I justify merrily going about indulging in such pernicious opinionation is my blog's spectacular insignificance :-)

I completely agree with what you say about evil in our mythology; certainly everyone is not equivalent. All I want to point out is that labeling someone as 'bad' strongly depends on what time-frame you are looking it. In the grand cosmic scale, there is no Evil. I'm curious, what lines gave you an impression that I'd gone pomo? :D

LOL @moving the not. In fact I prefer the latter version! It has become my latest hobby to argue that everything is pointless, I wonder how long this phase will last. It is precisely the point you noted - a search for beauty, instead of usefulness - that gives the story a strange kind of allure. People devoting their entire lives and fighting and conspiring and bickering and arguing about which scrawls on paper were more 'beautiful' - wow! Just wow!

And Galois - ah, I need to read more on him. I've read the wiki page on him many times, but I need to spend more time. I've never actually studied anything that he was involved with, so that's been something of a deterrent.

As always, thanks a bunch for the very nice links!