François Budan de Boislaurent

Ferdinand François Désiré Budan de Boislaurent (28 September 1761 – 6 October 1840) was a French amateur mathematician, best known for a tract, Nouvelle méthode pour la résolution des équations numériques, first published in Paris in 1807, but based on work from 1803.

He then proceeded to Paris where he studied medicine, receiving a doctorate for a thesis entitled Essai sur cette question d'économie médicale : Convient-il qu'un malade soit instruit de sa situation?

Budan explains in his book how, given a monic polynomial p(x), the coefficients of p(x+1) can be obtained by developing a Pascal-like triangle with first row the coefficients of p(x), rather than by expanding successive powers of x+1, as in Pascal's triangle proper, and then summing; thus, the method has the flavour of lattice path combinatorics.

Taken together with Descartes' Rule of Signs, this leads to an upper bound on the number of the real roots a polynomial has inside an open interval.

Budan's book was read across the English Channel; for example, Peter Barlow includes mention of it in his entry on Approximation[permanent dead link‍] in his Dictionary (1814), although grouping it with the method of Joseph-Louis Lagrange as being accurate, but of more theoretical interest than practical use.