Viète (1540-1603) worked on what is known as Horner's method for the
approximation of polynomial roots, and also he discovered another substitution
to solve cubics of the form . Horner's method for approximating
roots is similar in concept to the Newton-Raphson method. Take for example a
quadratic equation . Substitute where is known
to be a good approximation of *x*. If is small, i.e. if is
small, then will be very small, and may be cancelled. So we get
. Proceeding inductively we can approximate
any root to any degree of accuracy. Notice that is the derivative of
. The new substitution for solving the cubic involved
, which brought to solving , a quadratic in
.

Thu Feb 26 17:04:11 CET 1998