Cardano's Solution

This is a term paper I wrote for my Math History class in college, which was an optional course offered once every 2 or 3 years. I felt lucky to take it, and it was very interesting going through the building blocks of modern day mathematics from a "math class" point of view, rather than the point of view of a "history class".

The paper answers the question, "Is Cardano's Solution a great theorem?"

Original Post: And now for your reading pleasure: my math history paper that is due in 15 minutes:

Cardano's Solution, a Great Theorem

Is Cardano's 'solution of the cubic' a Great Theorem? Cardano's solution to the cubic took great algebraic insight. In the book, Journey Through Genius, William Dunham includes Cardano's proof, along with eleven other proofs as the greatest mathematical achievements of all time. Each chapter in his books exhibits rigorous proofs of our modern notational methods of proving, but for the original mathematicians to prove these theorems, there was no notation. Original proofs had geometry and just words to illustrate the methods of finding such answers.

So what makes a Great Theorem great? Something that all the great theorems in Journey Through Genius clearly illustrate is mathematical genius not seen in their time. Some of these mathematicians were so immersed in the problems, that it was a matter of time before they found the answer. Others solved these problems because their process of thought saw the problem differently than everyone else. Euclid's proof of the 'infinitude of primes' showed his concept of infinity, before anyone decided to conceptualize the name.

Archimedes' 'determination of circular area' showed his concept of infinity and limits well before their time. Heron's 'area of triangular space' provided a drastically different approach to area than what was considered at the time. After these proofs were available to all mathematics, conventional thought then changed as well. So, where does Cardano's 'solution to the cubic' fit into this?

I can understand why Cardano's 'solution to the cubic' would be considered a weak Great Theorem. By today's concept of mathematics Cardano's proof is extremely long, tedious, and doesn't work for all the solutions to the problem. Cardano was limited by the tools of his time, therefore imaginary, and other possibly real answers could not be obtained by his proof. Great Theorems aren't about how useful they are today, though.

A Great Theorem is something that redefined mathematical thought at the time; that changed history into the direction we are now. Within the solution to the cubic, it took great insight to notice the substitution of x to y - b/3a would get rid of the squared term in the general equation. It also took great resource skills to make expert use out of all the tools available to him. This theorem influenced the thought of solving multiple polynomials, by reduction of the middle terms. Cardano later solved the quadric in this manor. Three hundred years later, this theorem brought about Abel's proof that you cannot exceed quadric equations through this method; a new method of solving must be devised.

I agree that Cardano's 'solution to the cubic' is a Great Theorem, because it influenced mathematics and brought about new ideas in their time. William Dunham, in Journey Through Genius, doesn't give a specific definition of what a Great Theorem is, but he provides great information on how the mathematician came about his discovery and his proof. From Dunham's example I think a Great Theorem is measured by its impact on the mathematical society, and the illustration through different mathematical thought.