**François Budan**was born in Limonade, a village 18 km south east of Cap-Français in Saint-Domingue. Today the town of Cap-Français is named Cap-Haitien, and Saint-Domingue is Haiti. Cap-Français was founded by the French in 1670 as the capital of Saint-Domingue (although the country was only given that name in 1697). Budan's family had long had connections with the Saint-Domingue with his great-grandfather Jean Budan settling there early in its time as a French colony after his marriage in Nantes. François Budan's father Michel Budan and his mother Marie Minière owned property in Saint-Domingue.

When François was eight years old he was sent to France to be educated. On 4 August 1769 he entered the College of Juilly as a boarder. The College, close to Paris in the diocese of Meaux, was run by priests of the Oratory and much patronized by the prominent families. The Congregation of the Oratory of Jesus and Mary Immaculate, often called the Oratorians, was founded by Pierre de Bérulle in 1611 and approved in 1613. It had a fine reputation for providing a good education with particular emphasis on the classical languages. 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. He studied for eight years there and showed great liking and aptitude for the sciences, but as these were not part of the standard teaching, he received special mathematical tuition two days per week given on a voluntary basis by the mathematician J-C Farcot. 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. On 3 October 1778 he entered the Maison de l'Institution situated in the Rue Saint-Honoré in Paris where he underwent religious training for a year before being sent to Nantes where members of his family lived.

In Nantes the Oratorians ran the College of Saint-Clément and Budan was attached to the College from October 1779 until 1787. He undertook various teaching duties at the College, including logic and physics in his last two years there. During these years he was also a member of the Faculty of Arts at the University of Nantes, which again was an Oratorian institution. During the following years he seems to have been sometimes in Montmorency but mostly in Nantes. 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?* At around this time he was submitting mathematical works to the Academy of Sciences which we will discuss in more detail below.

Budan was appointed Inspector General for Public Instruction in 1803. He married Thérèse-Désirée de Piolenc at Nantes on 13 February 1809. He was 47 and his wife was 30 years of age. François and Thérèse-Désirée Budan had three daughters and one son: Louise-Victorine (born 25 June 1810), Antoinette (born 28 August 1811), Charles-Albert (born 13 September 1812), and Marie-Thérèse (born 22 December 1813). All four children were born in Paris. On the death of the wife of Antoine-Athanase Royer-Collard in 1815, Budan acted as tutor to their five children. 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 *n* real roots between two given numbers. Akritas writes in [3]:-

Budan's rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807 inIn the early19(^{th}century F D Budan and J B J Fourier presented two differentbut equivalent)theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval.

*Nouvelle méthode pour la résolution des équations numerique d'un degré quelconque*. In it Budan wrote:-

Grabiner writes [1]:-If an equation in x has n roots between zero and some positive number p, the transformed equation in(x - p)must have at least n fewer variations in sign than the original.

A paper giving a proof was presented to the Academy of Sciences in Paris in 1811 and he published it in 1822. 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. Fourier had independently discovered a rule which he taught in a course from 1797 but Fourier's version was not published until 1831, after his death. Lagrange, however, did not appear to know of Fourier's result since he described Budan's result as a new one. Grabiner writes [1]:-He quoted Lagrange to show that it would be useful to give the rules for solving numerical equations entirely by means of arithmetic, referring to algebra only if absolutely necessary. Budan's goal was to solve Lagrange's problem - between which real numbers do real roots lie? - purely by methods of elementary arithmetic. Accordingly, the chief concern of Burdan's Nouvelle méthode was to give the reader a mechanical process for calculating the coefficients of the transformed equation in(x - p). He did not appeal to the theory of finite differences or to the calculus of these coefficients, preferring to give them "by means of simple additions and subtractions".

Let us note that Charles-François Sturm in his famous paperBudan's success in discovering a correct rule and giving a reasonably satisfactory proof of it shows that, at the beginning of the nineteenth century, it was still possible for one without systematic training in mathematics to contribute to its progress ... The professionals were about to take over. Fourier's simultaneous and independent discovery, using derivatives, exemplifies the powerful methods available to one thoroughly schooled in mathematics.

*Mémoire sur la résolution des équations numériques*published in 1829 completely solved the problem of determining the number of real roots of an equation on a given interval. Hermite wrote:-

Hermite was wrong. 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. Fashions in mathematics change and solving a problem which Lagrange had deemed important does not guarantee your solution will achieve fame.Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance.

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. He also contributed to law, medicine and poetry, writing a Latin ode on the birth of the son of the Duke of Burgundy. He was honoured for his contributions being named Chevalier of the Légion d'Honneur on 20 November 1814.

**Article by:** *J J O'Connor* and *E F Robertson*