**Michel Rolle**'s father was a shopkeeper. Michel had little formal education being largely self-educated after receiving some elementary schooling. He worked as a transcriber for a notary and then as an assistant to several attorneys in the district around his home town of Ambert. In 1675, probably seeking a better life, he went to Paris where he worked as a scribe and arithmetical expert. However, quite soon after he arrived in Paris he married and children quickly followed. His income was not sufficient to support his growing family but he had been studying higher mathematics on his own and it was the skill that he had developed in this discipline which provided the breakthrough.

In 1682 he achieved a certain fame by solving a problem which had been publicly posed by Jacques Ozanam. Jean-Baptiste Colbert, the controller general of finance and secretary of state for the navy under King Louis XIV of France, rewarded Rolle for this achievement. Colbert arranged a pension for Rolle which started him on the road to financial security, but there were other equally important consequences [6]:-

Of equal importance with the financial benefits to which Rolle's prize led, was the fact that it brought him to the notice of Louvois who happened to be looking for someone to teach mathematics to one of his sons. Rolle was hired by Louvois, who came to be greatly impressed by his pedagogic and mathematical skills. From1783onwards Louvois was in a position to confer further scientific patronage on his protégé and did so when he made Rolle a member of the Académie in1685.

The problem which Rolle solved was posed in the *Journal des sçavans* on 31 August 1682 by Jacques Ozanam; it was the following:

Find four numbers the difference of any two being a perfect square, in addition the sum of the first three numbers being a perfect square.

Ozanam stated that the smallest of the four numbers with these properties would have at least 50 digits, but Rolle found four numbers satisfying the conditions with each number having seven digits. He made his solution known through publishing it in the* Journal des sçavans*. Louvois, who is referred to in the above quote, was François Michel le Tellier, Marquis de Louvois, the French Secretary of State for War. He employed Rolle to tutor his fourth son, Camille le Tellier (1675-1718). The Marquis de Louvois arranged for Rolle to have an administrative post in the Ministry of War, but Rolle disliked the work and soon resigned. Rolle was elected to the Académie Royale des Sciences in 1685 and the impressive mathematical work he produced following his election fully justified the Marquis de Louvois' faith in him.

Before going on to discuss the interesting mathematical contributions Rolle made, let us give a couple of further facts about his life. He became a Pensionnaire Géometre of the Académie des Sciences in 1699. In 1708 Rolle suffered a stroke. He recovered his health fairly well but his mental capacity was diminished and he made no further mathematical contributions after this stroke. He survived for eleven years after this first stroke but in 1719 he suffered a second stroke which proved fatal.

Let us now look at Rolle's important mathematical contributions. He worked on Diophantine analysis, algebra (using methods of Claude Gaspar Bachet de Méziriac involving the use of the Euclidean algorithm) and, to a lesser extent, on geometry. He published his most important work *Traité d'algèbre* in 1690 on the theory of equations. In this treatise, he invented the notation ^{n}√*x* for the *n*th root of *x* and, as a consequence, it became the standard notation; it is used today. In *Traité d'algèbre* Rolle used the Euclidean algorithm to find the greatest common divisor of two polynomials. He also used it to solve Diophantine linear equations. Perhaps the most significant part of the work, however, is where he introduces the notion of 'cascades'. Let us see how this idea worked: If *P*(*x*) = 0 is a given polynomial equation with real roots *a* and *b* then he constructs a polynomial *P*'(*x*), which he called the 'first cascade,' so that *P*'(*b*) = (*b* - *a*)*Q*(*b*) where *Q*(*x*) is a polynomial of lower degree. Of course in our terminology *P*'(*x*) is the first derivative of *P*(*x*). Rolle then constructs the 'second cascade' which is the second derivative, and continues in this fashion. Between any two consecutive roots of *P*(*x*) there is a root of *P*'(*x*), between any two consecutive roots of *P*'(*x*) there is a root of *P*''(*x*), etc. His method is to start with a given polynomial, make a linear transformation to obtain a polynomial all of whose roots are positive (he never proves that his transformation always works but it does), then to continue to construct the cascade of polynomials until a linear polynomial is reached. One can then move back up the cascade, finding approximately the roots of each polynomial. Julius Shain writes:-

The method of cascades has an important historical significance. Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method. It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives. Rolle's theorem, an important proposition of the calculus, also owes its origin to the method.

In fact Rolle is best remembered for 'Rolle's Theorem' which was published in *Démonstration d'une Méthode pour resoudre les Egalitez de tous les degrez* in 1691. This work was written to provide proofs of certain methods (in particular the method of cascades) which he gave without justification in *Traité d'algèbre*. His proofs were based on methods introduced by Johann van Waveren Hudde. The familiar Rolle's Theorem states:

If

f(a) =f(b) = 0 thenf'(x) = 0 for somexwitha≤x≤b.

The name 'Rolle's Theorem' was given to this basic result by Giusto Bellavitis in 1846. In his 1691 work Rolle adopted the notion that if *a* > *b* then -*b* > -*a*. It seems strange today to realise that this was not the current practice at the time but was in opposition to the ordering of the real numbers used by Descartes and others. Other notation Rolle used in his 1691 work was the equals sign '=.' This notation was not invented by Rolle, rather it was invented by Robert Recorde, but Descartes had used a different notation and the sign = was not in common use. Rolle published another important work on solutions of indeterminate equations in 1699, *Méthode pour résoudre les équations indéterminées de l'algèbre*.

It might be assumed from what we have just written about Rolle's work that he was developing the infinitesimal calculus. This would be a serious error, for Rolle described the infinitesimal calculus as a collection of ingenious fallacies and he believed that the methods could lead to errors. He begins his memoir *Du nouveau systême de l'infini* (1703) as follows (see for example [4]):-

Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. Among its principles one could only find true axioms and all the theorems and problems posed were either soundly demonstrated or capable of sound demonstration. And if any false or uncertain propositions were slipped into it they would immediately be banned from this science. But it seems that this feature of exactness doe not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes.

Now we should note that this memoir, although only published by the Academy of Sciences in 1703, actually contained material which had been used in the early stages of a vigorous debate which took place in the Academy of Sciences in 1700-01. This vigorous disagreement was between Rolle and Pierre Varignon and it ended in uproar (some say that Rolle's lack of breeding showed in his bad behaviour). The Academy set up a commission to decide which of the two mathematicians was correct but it failed to come to a definite conclusion. Michel Blay [2] writes:-

This opposition, extremely active from1700on, was led by Michel Rolle. The burden of his critique rested on two arguments, one stressing the inadequacy and the lack of logical rigour of the fundamental concepts and principles of the new calculus, the other pretending to show(with the aid of cleverly selected examples)that the new calculus led to error, insofar as it did not yield the same results obtained in the classical, algebraically inspired methods of Fermat and, more especially, Hudde.

In particular Rolle suggested that there were difficulties associated with the following axiom given by de l'Hôpital in *Analyse des infiniment petits pour l'intelligence des lignes courbes* (1696):

Grant that two quantities whose difference is an infinitely small quantity may be taken(or used)indifferently for each other; or(which is the same thing)that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.

For example, Rolle addressed the Academy on 12 March and 16 March 1701. He asked his audience to consider the curve

y-b= (x^{2}- 2ax+a^{2}-b^{2})^{(2/3)}/a^{(1/3)}.

He then applied the new method of the infinitesimally small and found that

dy= 4(x-a)dx/3(ax^{2}- 2a^{2}x+a^{3}-ab^{2})^{(1/3)}

so that this method showed that the curve has a turning point at *x* = *a*. Rolle then applied Hudde's method to show that the curve has turning points at three points, *x* = *a*, *x* = *a* - *b* and *x* = *a* + *b*. This is a cleverly constructed example, but Varignon was able to see the subtle error in Rolle's analysis. He later replied [2]:-

One will obtain first x = a by setting dy =0; and second, x = a - b or x = a + b by making dy infinite in relation to dx or dx =0. When one sees that if the desired curve has some maximum or minimum that meets a tangent parallel to the[x-axis], this can only be at the extremity of x = a; and that if it has one that meets ...[tangents]perpendicular to this axis, this can only be at the extremity of x = a - b and of x = a + b.

After the Academy decided that no further discussion of this topic was take place, Rolle continued the argument in the pages of the *Journal des sçavans* opposed by Joseph Saurin. To his eternal credit, Rolle eventually conceded that he was wrong. He acknowledged this to Varignon, Fontenelle and Malebranche. Jean Itard ends his article [1] with the following assessment:-

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

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

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