expansion of (yB/3)^3 , and the term By^2 that occurs in the expansion of
B(yB/3)^2 ; these terms cancel so the result is a depressed cubic in y,
which del Ferro’s technique enables us to solve for y. Then if xyB/3, x
is a root of the original cubic.
That was the first breakthrough, but the second was even more exciting:
Ferrari discovered a technique for transforming the general quartic equa-
tion (finding the roots of a polynomial of degree four) to a cubic, which
they now knew how to solve. These were the most significant develop-
ments in algebra in millennia—but both advances ultimately rested on
Tartaglia’s solution to the depressed cubic, and Cardano’s oath prevented
them from publishing their results.
Several years later, Cardano and Ferrari traveled to Bologna, where they
read the papers of Scipione del Ferro. These papers contained del Ferro’s
solution of the depressed cubic—which coincided with the solution that
Tartaglia had found. Cardano and Ferrari managed to persuade them-
selves that since del Ferro had previously obtained the solution, using it
would not break Cardano’s pledge to Tartaglia.
Cardano published his classic work, Ars Magna (“the great art”), in
- Algebra was indeed Cardano’s “great art”—though he was an ac-
complished physician (for his time) who had treated the pope, and though
he wrote the first mathematical treatment of probability (Cardano was an
inveterate gambler), his contributions to algebra are the ones for which he
is best remembered. The description given earlier for the procedure used
to solve the depressed cubic is taken from Ars Magna.
In Ars Magna, Cardano gave full credit to the giants on whose shoulders
he stood. The preface to the chapter on the solution of the cubic began
with the following paragraph: “Scipio Ferro of Bologna well-nigh thirty
years ago discovered this rule and handed it on to Antonio Maria Fior of
Venice, whose contest with Niccolo Tartaglia of Brescia gave Niccolo occa-
sion to discover it. He gave it to me in response to my entreaties, though
withholding the demonstration. Armed with this assistance, I sought out
its demonstration in [various] forms. This was very difficult.”^7
Tartaglia did not take this revelation well, accusing Cardano of violating
his sacred oath. Cardano did not reply to these accusations, but Ferrari,
who was known as something of a hothead, did. This culminated in
a challenge match between Tartaglia and Ferrari—but Ferrari had the
home-field advantage and emerged victorious. Tartaglia blamed his defeat
on the vigor with which the onlookers supported the home favorite (there’s
something quaintly charming about a citizenry that will riot in response
to a contest of the intellect rather than, as is the case nowadays, in re-
sponse to the results of a soccer match, but possibly there wasn’t a whole
88 How Math Explains the World