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Rolle, Michel


(b. Ambert, Basse-Auvergne, France, 21 April 1652; d. Paris, France, 8 November 1719)


The son of a shopkeeper, Rolle received only a very elementary education. He worked first as a transcriber for a notary and then for various attorneys in his native region. At the age of twenty-three he moved to Paris, Married early and burdened with a family, he had difficulty earning sufficient money as master scribe and reckoner. But by independent study he learned algebra and Diophantine analysis. In the Journal des sçavans of 31 August 1682 Rolle gave an elegant solution to a difficult problem publicly posed by Ozanam: to find four numbers the difference of any two of which is a perfect square as well as the sum of the first three. Ozanam had stated that the smallest of the four numbers would have at least fifty figures. Rolle provided a solution in which the four numbers were expressed by homogeneous polynomials in two variables and of degree twenty. The smallest numbers found in this fashion each had only seven figures.

This brilliant exploit brought Rolle public recognition. Colbert took an interest in him and obtained for him a reward and, it was said, a pension. Rolle later enjoyed the patronage of the minister Louvois. He gave lessons in elementary mathematics to the latter’s fourth son, Camille Le Tellier, abbé de Louvois (1675–1718). Rolle even received an administrative post in the ministry of war, from which he soon resigned.

Rolle entered the Académie des Sciences in 1685 with the title—rather disconcerting for us—of élève astronome. When the Académie was reorganized in 1699 he became pensionnaire géomètre, a post that assured him a regular salary. In 1708 he suffered an attack of apoplexy. He recovered, but a second attack in 1719 proved fatal.

Although it was his skill in Diophantine analysis that made Rolle’s reputation, his favorite area was the algebra of equations, in which he published Traité d’algèbre (1690), his most famous work. In this book he designated, following Albert Girard (1629). the nth root of a number a, a, not as √a, as was usually done before him. His notation soon became generally accepted. He retained the Cartesian equality sign until 1691. when he adopted the equal sign (=), which originated with Robert Recorde (1557).

In 1691, Rolle adopted, in advance of many of his contemporaries and in opposition to Descartes, the present order relation for the set of the real numbers: “I take –2a for a greater quantity than –5a.”

Rolle’s Algèbre contains interesting considerations on systems of affine equations. Following the techniques established by Bachet de Méziriac (1621), Rolle utilized the Euclidean algorithm for resolving Diophantine linear equations. He employed the same algorithm to find the greatest common divisor of two polynomials, and in 1691 he was able to eliminate one variable between two equations.

The Traité d’algèbre, the language of which is so special, has remained famous, thanks notably to the method of “cascades.” Rolle used this method to separate the roots of an algebraic equation. He justified it by showing (1691) that if P(x) = 0 is the given equation, and if it admits two reals roots a and b. then P′(b)= (ba) Q(b), where Q is a polynomial. P′(x), a polynomial derived from P(x), is what Rolle called the “first cascade” of the polynomial P(x). The second cascade is our second derivative, and so on.

Arranging the real roots in ascending order, Rolle showed that between two consecutive roots of P(x) there exists a root of P′(x). His methods of demonstration are elaborations of the method utilized by Jan Hudde in his search for extrema (1658).

In 1846 Giusto Bellavitis gave Rolle’s name to the present theorem: if the function f(x) is defined and continuous on the segment [ab], if f(a) = f(b), and if f′(x) exists in the interior of the segment, then f′(x) is equal to zero at least once in the segment.

In 1699 the three pensionary geometers of the Academy were the Abbé Jean Gallois, a partisan of Greek mathematics; Rolle, an autodidact but very well versed in Cartesian techniques; and Pierre Varignon, who favored the ideas of Leibniz. L’Hospital was an honorary academician; in 1696 he had published Analyse des infiniment petits. The Academy was very divided over infinitesimal analysis. Rolle—incited, it was said, by influential persons—vigorously attacked infinitesimal analysis and strove to demonstrate that it was not based on solid reasoning and led to errors. Among the examples he chose were the curves

Varignon defended the new methods and pointed out the paralogisms that Rolle displayed in discussing these examples. The latter, too plebeian to control himself, created an uproar. A commission established to resolve the matter was unable to come to a decision. The dispute lasted from 1700 to 1701, and then continued in the Journal des sçavans in the form of exchanges between Rolle and a newcomer, Joseph Saurin. The Academy again intervened, and in the fall of 1706 Rolle acknowledged to Varignon, Fontenelle, and Malebranche that he had given up and fully recognized the value of the new techniques.

Rolle also displayed a certain vigor in the field of Cartesian geometry. In 1693, in the Journal des sçavans, he offered a prize of sixty pistoles for the solution, without the use of his methods, of the following problem: construct the roots of an equation by utilizing a given arc of an algebraic curve. Before leaving Paris, Johann I Bernoulli had given a solution to this problem in Latin to L’Hospital and had requested him to submit a French translation to the Journal. The solution did not meet with Rolle’s approval, and the resulting polemic lasted for five numbers of the Journal; the sixty pistoles remained in the donor’s coffers.

Another of Rolle’s achievements is an observation that, though initially paradoxical, was recognized as correct by Saurin: two arcs of algebraic curves the convexity of which is in the same direction can have a large number of common points.

Rolle was a skillful algebraist who broke with Cartesian techniques; and his opposition to infinitesimal methods, in the final analysis, was beneficial.


I. Original Works. Rolle’s books are Traité d’algèbre, ou principes généraux pour rèsoudre les questions de mathématique (Paris, 1690); Démonstration d’une méthode pour resoudre les egalitez de tous les degrez; suivie de deux autres méthodes dont la première donne les moyens de résoudre ces mémes égalitez par la géométrie et la seconde pour résoudre plusieurs questions de Diophante qui n’ont pas encore esté resoluës (Paris, 1691); and Mèthode pour résoudre les équations indéterminécs de l’algèbre (Paris, 1699). See also two papers in Mémoires de l’Académie royale des Sciences: “Règles pour l’approximation des racines des cubes irrationnels” (read 31 Jan. 1692), 10 ; and “Méthode pour résoudre les égalités de tous degrés qui sont exprimés en termes généraux” (read 15 Mar. 1692), ibid., 26–33.

Rolle’s later papers in the Mémoires of the Académie des Sciences include “Remarques sur les lignes géométriques” (1702), 171–175, (1703), 132–139; “Du nouveau système de l’infini” (1703), 312–336; “De l’inverse des tangents” (1705), 222–225; “Méthode pour trouver les foyers des lignes géométriques de tous les genres” (1706), 284–295; “Recherches sur les courbes géométriques et mécaniques, où l’on propose quelques règles pour trouver les rayons de leurs développées” (1707), 370–381; “éclaircissements sur la construction des égalitez” (1708), 339–374 (1709), 320–350; “De l’évanouissement des quantités inconnues” (1709), 419–451; “Règles et remarques pour la construction des égalités” (1711), 86–100; “Remarque sur un paradoxe des effections géométriques” (1713), 243–261, read 12 July 1713; and “Suite des remarques sur un paradoxe des effections géométrique” (1714), 5–22, read 10 Jan. 1714.

Rolle also published many articles in the Journal des sçavans beginning in 1682.

II. Secondary Literature. See the éloge by Fontenelle, read at the Académie des Sciences on 10 Apr. 1720, Niels Nielsen, Géomètres français du dix-huitième siècle (Copenhagen, 1935), 382–390; Gino Loria, Storia dells matematiche, 2nd ed. (Milan, 1950), 670–673; J. E. Montucla, Histoire des mathénmatiques, II (Paris, 1758), 361–368, 2nd. ed., III (Paris, 1802; repr. 1960), 110–116, (which contains a detailed account of the polemic over the infinitesimal calculus—Montucla had access to Varignon’s MSS); Cramer, ed., Virorum Celebrium G. G. Leibnitii et Johan Bernoullii commercium philosophicum et mathematicum, 2 vols. (Lausanne, 1745)—see index and II, 148 for Cramer’s note containing a list of the articles in the Journal des sçavans concerning the dispute between Rolle and Saurin; O. Spiess, ed., Der Briefwechsel von Johann I Bernoulli (Basel, 1955)—see index and p. 393 for a note on the articles in the Journal des scavans concerning the prize offered by Rolle and claimed by Johann I Bernoulli; Malebranche, Oeuvres completes, XVII, pt. 2 (Mathematica) (Paris, 1968); Pierre Costabel, Pierre Varignon (1654–1722) et la diffusion en France du calcul différentiel et intégral, in the series Conférences du Palais de la Découverte (Paris, 1965); Petre Sergescu, Un episod din batalia pentru triumful calculului diferential: polemica Rolle-Saurin 1702–1705 (Bucharest, 1942), repr. in Essais scientifiques (Timisoara, 1944); D. E. Smith, A Source Book in Mathematics (1929; repr. New York, 1959), 253–260, which contains Rolle’s theorem and partial English translations of the works of 1690 and 1691; L. E. Dickson, History of the Theory of Numbers,II (New York, 1934), 45 (on linear Diophantine equations) and p. 447 (on the problem posed by Ozanam that Rolle solved); Dickson nowhere cites the passage of the 1691 work concerning Diophantine analysis; Jean Prestet, Nouveaux élémens des mathécuatiques, II (Paris, 1689), 238, a solution, by a different procedure, of Ozanam’s problem and a criticism of Rolle’s method—this work is not cited in Dickson; and Jakob Hermann, “Observationes in schediasma quod Dn. Rolle cum hac inscriptione: éclaircissements sur la construction des égalitez,” in Commentariis Academiae Regiae Scientiarum (1708, published in 1727), Miscellanea Berolinensia, III (Berlin, 1927), 131–146.

Jean Itard

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