**François Budan de Boislaurent** was a Haitian born amateur mathematician best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have *n* real roots between two given numbers.

- When François was eight years old he was sent to France to be educated.
- François received a thorough training in the classics at the College of Juilly and later in his life he often quoted from authors such as Virgil and Horace in his writings.
- During 1775-77 Budan studied in the Royal Academy of Juilly, with rhetoric being the topic of the first of these two years and philosophy as the topic of the second.
- By this time both Budan's parents had died.
- In Nantes the Oratorians ran the College of Saint-Clément and Budan was attached to the College from October 1779 until 1787.
- Budan took up the study of medicine in Paris and, in 1803, received the title of doctor of medicine for a thesis entitled Essai sur cette question d'économie médicale : Convient-il qu'un malade soit instruit de sa situation?
- Budan was appointed Inspector General for Public Instruction in 1803.
- In 1835 Budan retired and he died five years (to the day) later.
- Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have nnn real roots between two given numbers.
- Budan's rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807 in "Nouvelle méthode pour la résolution des équations numerique d'un degré quelconque" Ⓣ(New method for solving numerical equations of any degree).
- Budan's goal was to solve Lagrange's problem - between which real numbers do real roots lie?
- Lagrange was asked to report on Budan's paper of 1811 and he found that it was essentially true but to make it completely rigorous certain gaps had to be filled.
- Lagrange, however, did not appear to know of Fourier's result since he described Budan's result as a new one.
- Charles-François Sturm in his famous paper "Mémoire sur la résolution des équations numériques" Ⓣ(Memoir on solving numerical equations) published in 1829 completely solved the problem of determining the number of real roots of an equation on a given interval.
- Neither Budan's rule, nor Fourier's rule, nor Sturm's rule found a place in textbooks written a few years after they were discovered.
- In total Budan published ten mathematical works and as one further example of his contributions we note that he submitted a paper on the summation of series to the Academy of Sciences in 1802 which was refereed by Biot and Lacroix.

Born 28 September 1761, Limonade, Cap-Français, Saint-Domingue (now Haiti). Died 6 October 1840, Paris, France.

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Origin Haiti

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive