18 July 2015

Solving the cubic by poetry

At a time when the Egyptians struggled with simple linear equations (they never got much further), the Babylonians had a general method for dealing with quadratic equations (and much more). But from

ax2+bx+c=0
to
ax3+bx2+cx+d=0

i.e., raising the highest exponent by one, it would be a long long way. It was in renaissance Italy, at the university of Bologna, that mathematicians put their minds to it. It's simple enough to reduce the equation to the depressed form
x3+px+q=0

but the subsequent steps took some effort. In a thrilling episode of Math History, featuring del Ferro, Fiore, Tartaglia, Cardano, Ferrari, and involving a challenge, a mathematical contest, oaths of secrecy, betrayal, stolen authorship, mutual insults and violent pamphlets, the formula finally got revealed:


It was eventually named after Cardano, but set to poetry by Tartaglia in 1539. (Here or here the Italian poem by Tartaglia. La cosa = the thing = the unknown, which we denote by x. Translation, explanation + correction here.) Actually, it's a little more intricate than that, because counting the number of possible roots you may end up with as much as nine solutions while only three are possible. Moreover, you run into imaginary numbers, regarded with suspicion by everybody, including the Inquisition. In due time, Cardano (a Franciscan friar) was sentenced as a heretic, not because of black magic with imaginary numbers, but for casting the horoscope of Jesus Christ.

In the half millennium since, we have learned to deal casually with complex numbers, and we can spell the whole thing out in a few lines of high school math.




P.S. For the sake of completeness, we add the solution of the quartic, which was found by Ferrari, one of the actors in the episode described above. Don't expect anything similar for the quintic! We know since Niels Henrik Abel that solution by roots ends with the fourth degree.