Duels—of an intellectual nature—were one method by which up- and-
comers of the time would obtain prestigious academic positions. A chal-
lenger would pose a list of questions, or mathematical problems, to an
established academic, who would counter with a list of his own. After a
predetermined amount of time, the results would be announced—as
might be expected, to the victor belonged the spoils. The depressed cubic
solution was del Ferro’s ace in the hole—if challenged, he would present
a list of depressed cubics to his challenger. As far as we know, del Ferro
never had to play his trump card.^5
A Duel of Wits with Equations As Weapons
Upon his death, del Ferro bequeathed the solution to his student Antonio
Fior, a mathematician of less talent but greater ambition than his mentor.
Del Ferro had kept the solution as a defense, but Fior decided to use it
to make a name for himself, and issued a challenge to the well-known
scholar Niccolo Fontana.
Fontana had been severely wounded as a child when a soldier slashed
his face with a sword. This affected his speech, and led to his being given
the nickname Tartaglia—the Stammerer—the name by which he is
known today. When Fior presented his thirty-problem challenge to Tar-
taglia, Tartaglia countered with a list of thirty problems on a variety of
mathematical topics—only to discover that Fior’s list consisted of thirty
depressed cubics.
It was a classic all-or-nothing situation—Tartaglia was either going to
solve none of the problems, or all thirty, depending upon whether or not
he could generate the solution to the depressed cubic. Tartaglia obtained
the formula
(^3) n/2 + m (^3) /27 + n (^2) /4 − 3 −n/2 + m (^3) /27 + n (^2) /4
for the root of the depressed cubic x^3 mxn. As you can see, you are not
likely to stumble upon this formula using hit-or-miss techniques.
I’ll check out this formula with the depressed cubic x^3 6 x 20. The
result is expressed as x^3 10 + 108 − 3 − 10 + 108. Simplifying this
expression is a good problem for an advanced high-school algebra stu-
dent, but those who prefer modern technology can check with a pocket
calculator that x2, which is indeed a solution of the equation.
There is a subtle deceptiveness to all mathematical textbooks that teach-
ers realize but students generally don’t; an awful lot of trial-and-error
84 How Math Explains the World