1. Cubic Equations

5

secret. Nonetheless, Cardano published the solution procedure in his

Ars Magna, a book describing the current state of algebraic knowl-

edge.5

The solution of cubic equations is based on the cubic binomial

formula

(u + ^) 3 = 3uv(u + v) + (u3 + v3) ,

which Cardano was able to derive in an analogous manner to the

geometric method used by al-Khwarizmi for the quadratic equation,

though in this case the argument of course used three-dimensional

figures and volumes (see Figure 1.2). However, this identity can also

be interpreted as a cubic equation, where the sum u -\- v yields a

solution x of the cubic equation

x3 + px + q — 0

if the conditions

Suv = —p,

u3 + v3 = -q,

are satisfied. The cubic equation

x3

+ px + q = 0 can then be solved

if one can find suitable quantities u and v. But that is a relatively

simple task. Since both the sum and product of the quantities

u3

and

v3 are known, one can solve the quadratic equation

w2 +qw -

(-J = 0

to obtain

u3

and

v3

as the two solutions

-N(W.

so that u and v can be determined from the two equations

•=fbW^W-

-f!-i/(!)'+(i)"-

5The

result of Cardano's alleged breaking of his word led to a great falling out

between Tartaglia and Cardano. It is thanks to the publications around this dispute

(see the works referred to in an earlier footnote) that we have knowledge of the history

leading up to the Ars Magna.