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
+ 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
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.
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