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.
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