How Math Explains the World.pdf

goes into establishing a result such as the solution of the depressed cubic,
a nd most of t h at is er ror. We k now t h at g reat comp osers suc h a s B e e t hoven
made sketchbooks of their ideas, and we can read them to discover some
of the passages Beethoven considered using before arriving at the final
version. Many mathematicians do the same—they keep records of their
failed attempts, because sometimes what doesn’t work for one problem
might well solve another. However, these records generally don’t make it
into the archives, and as a result we have no idea how long it took del
Ferro to discover his approach. Using modern notation, del Ferro’s even-
tual successful solution isn’t that difficult to follow.
We can divide our depressed cubic by the coefficient of x^3 to arrive at an
equation that has the form

x^3 CxD 0

Instead of presenting the solution in the form it is usually given in a text-
book, let’s try to reconstruct what del Ferro did. Mathematicians often try
different things in the hope of getting lucky, and so del Ferro tried as-
suming the solution had the form xs
t. There’s a valid reason for try-
ing something like this, as using two variables (s and t) rather than one
introduces an extra degree of freedom into the problem. This is a stan-
dard weapon in the mathematician’s arsenal of problem-solving tech-
niques, as the price one must pay for having to solve for additional
variables may be more than offset by the ease of the solution. After mak-
ing this substitution, the depressed cubic becomes

(s
t)^3 C(s
t)D

(s^3 
 3 s^2 t 3 st^2 
t^3 )C(s
t)D

(s^3 
t^3 D)
 3 st(s
t)C(s
t)

(s^3 
t^3 D)(C
 3 st)(s
t)

At this point, del Ferro undoubtedly realized he might have hit the jack-

pot. If he could find s and t such that s^3 
t^3 D 0 and C
 3 st0, the last

equation would become

0  0 (s
t) 0

and xs
t would be a root of the depressed cubic. So del Ferro was led to
the system of two equations

3 stC

t^3 
s^3 D

We are now left with the problem of finding s and t that satisfy the two
equations; but here’s how the bucket of water gets moved from under the

The Hope Diamond of Mathematics 85