The name of Michel Rolle is primarily associated with Rolle's Theorem, which concerns the position of roots in an equation. He also developed the modern expression for the term nth root of x, and presented what he called his "cascade" method for separating the roots in an algebraic equation. Rolle, whose most famous work was Traité d'algèbre (1690), is also remembered for his opposition to techniques pioneered by René Descartes (1596-1650).
The son of a shopkeeper, Rolle was born in the French town of Ambert on April 21, 1652. His origins prevented him from obtaining a formal education, and instead he went to work in his teens as a scribe. By the age of 24 he was in Paris, earning his living as a secretary and an accountant. He married and had children, and was sufficiently successful in his profession that from the late 1670s (when he was in his late twenties) onward, he was able to devote considerable time to his avocation of mathematics.
A prominent self-taught mathematician of the time—and no doubt something of a model for Rolle—was Jacques Ozanam (1640-1717). Recreational mathematics, such as puzzles and tricks, were popular among educated Frenchmen of that era, and Ozanam was a master, putting forth problems such as the following: find four numbers such that the difference between any two is a perfect square, and is also the sum of the first three. It was Rolle's solution to this problem in 1682 that first brought him to the attention of Ozanam and to the larger mathematical public. The publication of his solution in Journal des scavans, the leading scientific journal, led to his earning an honorary pension, as well as his appointment to tutor the son of an influential government official.
In 1690, Rolle published Traité d'algèbre. The work contained what was then the novel use of notation for the nth root of a number, as well as Rolle's method of cascades. The latter used principles first put forth by Dutch mathematician Johann van Waveren Hudde (1628-1704) for finding the highest common factor of a polynomial, a method Rolle used to separate the roots of an algebraic equation. When other mathematicians complained that the book contained inadequate proofs, Rolle published Demonstration d'une methode pour resoudre les egalitez de tous les degrez (1691). This book included Rolle's Theorem, a special case of the mean-value theorem in calculus.
Beginning in 1691, Rolle began speaking out against errors in Cartesian methodology—first with regard to Descartes's ordering of negative integers on the same path as positives, such that -2 was smaller than -5. In 1699, the same year he was awarded a geometry pension by the Académie Royale des Sciences, he published a significant paper on indeterminate equations. He also weighed in against the validity of infinitesimal analysis, and though he was eventually forced to accept the discipline, his critique helped its supporters work out difficulties in their methodology.
Rolle suffered a stroke in 1708, and was never as strong thereafter. A second attack in 1719 proved too much, and he succumbed on November 8 of that year.