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## Solving the Cubic

Italy was a center of mathematical activity after the publication of Fibonacci’s Liber Abaci (1202) and the Treviso Arthmetic of 1478.  These books formed the foundation for European mathematics.

At some point in the early 1500’s, an Italian mathematician named Scipione del Ferro determined a general solution for what is known as the depressed cubic equation.  This is cubic equation without any x2 terms.  The general form is : x3+px=q.  As it turns out, any cubic equation of the form x3+bx2+cx+d=0 can be written as a depressed cubic, but that came later.

At the time, mathematicians didn’t publish their results, but, instead, kept them secret so that they could win the problem contests that were common at the time in Italy.  As a result, del Ferro didn’t tell anyone about his discovery until shortly before his death in 1526.  He then revealed the secret to a student of his named Antonio Maria Fior.  In 1535, Fior used this knowledge to challenge a better mathematicain named Niccolo Fontana to a problem contest.

Fontana was known as “Tartaglia,” (the stutterer) because of a speech impediment caused by an old sword wound to his jaw.  Tartaglia was a superior mathematician to Fior, but didn’t know how to solve the cubic equation yet.  So, in the time before the contest, he worked feverishly to find a solution.  Finally, he found the same solution that del Ferro had found thirty years before and was able to win the contest.

Word of Tartaglia’s victory spread among mathematicians and Giralamo Cardano decided to see if he could get Tartaglia to reveal his secret to the solution of the cubic.  At first Tartaglia refused, but then told Cardano the formula, but not how to derive it.  He also asked Cardano to promise not to reveal the result to anyone else.  Eventually, Cardano learned that del Ferro had found the solution first.  Cardano also determined the derivation of the formula that Tartaglia had shown him for himself.  So, in 1545, Cardano published the solution of the general cubic in his book Ars Magna.

It is interesting that Cardano encountered the square roots of negative numbers in working with his “cubic formula.”  These numbers, which today are called imaginary, or complex numbers were almost completely unknown at the time.  Cardano was initially flummoxed by these numbers that seemingly had no physical meaning.  However, in true trailblazing spirit, Cardano wrote in his book that although these numbers were unfamiliar, “nevertheless, we will operate,” with them using rules similar to those used for the standard number system.

Before the publication of Ars Magna one of Cardano’s students named Ludovico Ferrari found a solution for the quartic equation.  That is, an equation of the form x4+bx3+cx2+dx+e=0.  This solution was also published in Ars Magna which became known throughout Europe as the foundational text of classical European algebra.

The next step in the story took place over the ensuing 250-300 years.  Finally, in the early 1800’s it was shown that the quintic or fifth degree equation was not solvable by formula.

The stories of these mathematicians is told in An Imaginary Tale, by Paul Nahin and The Equation that Couldn’t be Solved by Mario Livio.  Next I’ll write about Galois and the evolution of Abstract Algebra from the ideas of the Classical Algebra.