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The first meaningful interest in polynomial equations of third degree in Europe came with Leonardo Pisano (otherwise known as Fibonacci, 1178-?), who was at one point asked by Frederick II to solve the cubic equation tex2html_wrap_inline154. Fibonacci showed first that no rational solution could exist, by dividing through by 10 and substituting tex2html_wrap_inline156 supposing the fraction reduced so that hcf(a,b)=1. This gathers
which is impossible as the highest common factor of a and b was supposed to be 1. Hence tex2html_wrap_inline162. Using similar procedures Fibonacci showed that the solution could not even be in any of the forms tex2html_wrap_inline164, tex2html_wrap_inline166, tex2html_wrap_inline166, tex2html_wrap_inline170. He then used an approximate method that allowed him to recover one of the roots to an approximation of about ten digits. It is not known what was the method involved, but it may be Horner's method, which was already known by the Chinese.

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