Chapter 1

Cubic Equations

Find a number that when added to its cube root yields 6.

1.1 Problems like the one given above have "entertained" genera-

tions of schoolchildren. Such problems are at least several hundred

years old. They appear as the first thirty problems that were posed to

Niccolo Fontana (1499 or 1500-1557), better known as Tartaglia (the

stutterer), in a mathematical competition. His challenger was Anto-

nio Fior (1506-?), to whom Tartaglia also posed thirty problems.1

As usual, the path to a solution begins with finding an equation

that represents the problem. In our example, with x representing the

cube root in question, we obtain the equation

x3

+ x - 6 = 0.

But how are we to solve it? Quadratic equations can always be solved

by "completing the square." Then one simply takes the square root

and out pops the solution. That is, in the general case of a quadratic

equation

x2 + px -f q = 0,

1A complete listing of the thirty problems set by Fior can be found in Re-

nato Acampora, "Die Cartelli di matematica disfida." Der Streit zwischen Nicold

Tartaglia und Ludovico Ferrari, Institut fur die Geschichte der Naturwissenschaften

(Reihe Algorismus, 35), Munich, 2000, pp. 41-44. See also Friedrich Katscher, Die

kubischen Gleichungen bei Nicolo Tartaglia: die relevanten Textstellen aus seinen

"Quesiti et inventioni diverse" auf deutsch iibersetzt und kommentiert, Vienna, 2001.

1

http://dx.doi.org/10.1090/stml/035/01