April 21, 2010
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Michel Rolle (April 21, 1652 – November 8, 1719) was a French mathematician. He is best known for Rolle's theorem (1691).

Michel Rolle was a largely self-educated mathematician who taught himself both algebra and Diophantine analysis, a method for solving equations with no unique solution. He won early repute when he solved a problem set by the mathematician Jacques Ozanam, but his real passion lay in the field of the algebra of equations. His most famous work was the Traite d'algebre of 1690, in which he not only invented the modern notation for the nth root of x, but also expounded on his "cascade" method to separate the roots of an algebraic equation. He is best remembered for the theorem which bears his name, Rolle's Theorem, which determines the position of roots in an equation.

Little is known about Rolle's youth. He was born in Ambert, Bass-Auvergne in France on April 21, 1652. His father was a shopkeeper and the young Rolle had only a basic elementary education before he began to earn a living as a scribe for a notary, and then for various attorneys near Ambert. The allure of the big city brought Rolle to Paris by age 24, and an early marriage and family kept him in secretarial and accountancy work. But his curious mind led him to a study of algebra and of Diophantine analysis. The work of the third-century Alexandrian mathematician, Diophantus, had been translated into Latin by Bachet de Meziriac, and Rolle made himself familiar with the solutions to both determinate equations, for which there exists a unique solution, and indeterminate equations. Diophantus had provided a method of analysis for indeterminate equations, and Rolle became an expert in this field.

Rolle was also familiar with the work of another Frenchman, a contemporary of his and also a self-taught mathematician, Jacques Ozanam whose interests were in recreational mathematics--puzzles and tricks--and analysis. In 1682, Rolle published "an elegant solution" to a problem set by Ozanam. The problem was as complex as the solution. Ozanam proposed finding four numbers which corresponded in the following ways: the difference of any two of these numbers not only make a perfect square, but also equal the sum of the first three. Rolle's solution as published in Journal des scavans, brought him notoriety and some patronage. The controller general of finance, Jean-Baptiste Colbert, took notice of him and arranged an honorary pension for Rolle so that he could continue with his mathematical work. Rolle also became the tutor to the son of a powerful minister, Louvois, and for a short time held an administrative post at the ministry of war.

Rolle became a member of the Academie Royal des Sciences in 1685. He published in 1690 his most famous work, Traite d'algebre, a book that has remained well known to this day. Algebra was a departure for Rolle, away from Diophantine analysis in which he had won early recognition, and into the algebra of equations. Rolle broke new ground in many aspects with this work. His new notation for the nth root of a number was first published in Traite d'algebre, and thereafter became the standard notation. Rolle also made important advances in systems of affine equations, which assign finite values to finite quantities, building on the work of Bachet de Meziriac, another contemporary whose work on mathematical tricks and puzzles led to the development of the field of recreational mathematics. But primarily the Traite d'algebre has retained its fame for the exposition of Rolle's method of "cascades." Utilizing a technique developed by Dutch mathematician Johann van Waveren Hudde to find multiple roots of an equation, or the highest common factor of a polynomial, Rolle was able to separate the roots of an algebraic equation. His "cascades" correspond to the derivatives of the highest common factor of a polynomial. Though never clearly defined, it is implied that such "cascades" are the result of the following: if in an equation f(x) = 0, f(x) is multiplied by a progression, then the simplified result equated to zero would be such a "cascade." However, mathematicians of the day complained that Rolle's theory of cascades was given with too little proof.

To rectify that, Rolle published Demonstration d'une Methode pour resoudre les Egalitez de tous les degrez in 1691, essentially a little known work which further demonstrated the method of "cascades." However, in the process, Rolle developed the calculus theorem which bears his name. According to this theorem, if the function y = f (x) is differentiable on the open interval from a to b and continuous on the closed interval from a to b, and if f (a) = f (b) = 0, then f ' (x) = 0 has at least one real root between a and b. It was not for another century and a half that Rolle's name was given to the theorem he developed, a theorem that is a special case of the mean-value theorem, one of the most important theorems in calculus.

Rolle's work encompassed also a certain flair for Cartesian geometry, though he did break with Cartesian techniques in other respects. In 1691, he was one of a vanguard to go against the Cartesian grain in the order relation for the set of real numbers, noting, for example, that -2a was a larger quantity than -5a. He also described the calculus as a collection of ingenious fallacies. Rolle published another important work on solutions of indeterminate equations in 1699, Methode pour resoudre les equations indeterminees de l'algebre, the year that he became a pensionnaire geometre at the Academie. This was a post that carried with it a regular salary which further enabled him to devote more time to mathematics. At this time, members of the Academie were drawing up sides on the value of infinitesimal analysis, in which a variable has zero as a limit. Rolle became one of its most outspoken critics. Though by 1706 he finally and formally recognized the value of the new techniques, his very opposition and criticism had served to help develop the new discipline of infinitesimal analysis.

Two years later, in 1708, Rolle suffered a stroke, which diminished his mental powers. Though he recovered, a second stroke in 1719 killed him.