next up previous contents
Next: The theory of permutations Up: Effects of the Ars Previous: Effects of the Ars

François Viète

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 tex2html_wrap_inline230. Horner's method for approximating roots is similar in concept to the Newton-Raphson method. Take for example a quadratic equation tex2html_wrap_inline232. Substitute tex2html_wrap_inline234 where tex2html_wrap_inline236 is known to be a good approximation of x. If tex2html_wrap_inline240 is small, i.e. if tex2html_wrap_inline242 is small, then tex2html_wrap_inline244 will be very small, and may be cancelled. So we get tex2html_wrap_inline246. Proceeding inductively we can approximate any root to any degree of accuracy. Notice that tex2html_wrap_inline248 is the derivative of tex2html_wrap_inline250. The new substitution for solving the cubic involved tex2html_wrap_inline252, which brought to solving tex2html_wrap_inline254, a quadratic in tex2html_wrap_inline256.

Leo Liberti
Thu Feb 26 17:04:11 CET 1998