sponded with the sixteenth-century equivalent of “Sorry, my agent is
working on a book deal,” but Cardano persisted, and finally persuaded
Tartaglia to leave his home in Brescia and visit Cardano in Milan. During
this visit, Cardano managed to talk Tartaglia into revealing his secret—
but in return, Tartaglia made Cardano take the following oath: “I swear to
you by the Sacred Gospel, and on my faith as a gentleman, not only never
to publish your discoveries, if you tell them to me, but I also promise and
pledge my faith as a true Christian to put them down in cipher so that
after my death no one shall be able to understand them.”^6
Like many of his contemporaries, Cardano placed great stock in dreams
and omens, and was also a practicing astrologer. One night he dreamed of
a beautiful woman in white, and he assiduously (and successfully) courted
the first such woman who crossed his path, despite despairing of his
chances; at the time he was poor as a church mouse. Soon after his meet-
ing with Tartaglia, he heard a squawking magpie and believed that it pres-
aged good fortune. When a young boy appeared at his doorstep looking for
work, Cardano somehow saw this as the good fortune promised by the
magpie and took him in. Maybe there was something to the squawking
magpie theory, as the boy proved to have substantial mathematical ability.
At first the boy, whose name was Ludovico Ferrari, was merely a servant in
Cardano’s household, but gradually Cardano taught him mathematics,
and before Ferrari had reached his twentieth birthday, Cardano had passed
on the secret of solving depressed cubics to him. The two mathematicians
decided to tackle the problem of solving the general cubic.
Cardano and Ferrari achieved two major breakthroughs. The first was
to find a transformation that reduced the general cubic equation to a de-
pressed cubic, which Tartaglia’s technique enabled them to solve. This
transformation moves a different bucket of water from under the table to
the top of the table.
Once again, by dividing by the coefficient of x^3 , we can assume our gen-
eral cubic equation has the form
x^3 Bx^2 CxD 0If we let xyB/3, this equation becomes
(yB/3)^3 B(yB/3)^2 C(yB/3)D 0Expanding the first two terms gives
(y^3 By^2 (B^2 /3)y(B^3 /27))B(y^2 (2B/3)y(B^2 /9))(yB/3)D 0It isn’t necessary to completely simplify the left-hand side to note that
there are only two terms involving y^2 ; the term By^2 that occurs in the
The Hope Diamond of Mathematics 87