The Tale of the Cubic

Once upon a time there was no uniform way to solve all cubic functions--how sad.  It was a problem that puzzled mathematicians up until 1535.  What happened in 1535 you ask?  Well there was a math competition in Italy that a guy named Niccolo Fontana attended.  You might know him by the nickname Tartaglia which means "the stutterer."  Or you might not.  Anyway this guy was an engineer and amateur mathematician--because who doesn't want to do math in their free time?  So he came to this competition and SURPRISE he won it by solving a cubic function using his general formula!  Everyone was shocked because they had thought it to be impossible.  

Since it was such a coveted formula, Tartaglia wanted to keep it to himself, even hiding it by encoding it in a poem.  Here's a picture of the formula from this Vanderbilt website. Can you imagine fitting that whole thing secretly into a poem? 

So Tartaglia wanted all the glory for himself and kept it a secret until the smooth talking Gerolamo Cardano came along and got the formula from Tartaglia which he subsequently published in his book "Ars Magna" in 1545.  Needless to say Tartaglia was not pleased that Cardano broke his promise of keeping it a secret, but got his revenge by helping Cardano get arrested for heresy after Cardano made a horoscope for Jesus.  

When we talked about the cubic formula in class I attempted to use the formula to solve the following cubic.  The (-3,0), (2,0), and (5,0) are the solutions I found from graphing.  My incomplete attempt gave me imaginary numbers after taking the first square root.  I may have messed up a negative somewhere...  But take a look and then take a moment to appreciate graphing calculators, the factor theorem, and long division of polynomials that can find the solutions a whole lot faster!

It makes me wonder what kind of applications they had that needed the solution of a cubic function.  As I thought back on my schooling to try to remember an application we learned that used cubics I couldn't think of a single one.  I think we were always just given a graph and/or the equation and had to find the solutions from there without giving an adequate reason why we should even solve it.  So I went to google and found this page where several people came up with applications of cubics.  Most of them seem to be far more advanced than what we would learn about when first seeing cubic formulas.  I thought the most interesting one was the claim that the typeset letters are formed from cubic functions.  So thank you cubics for helping me type this blog!  Do you know of any other applications of cubic functions?  Any that people would have needed in 1535?

Hey look!  If you want more information about the characters in this tale click here for a more in depth story courtesy of Luke Mastin!