VI.12 Pierre Fermat
b. Beaumont-de-Lomagne, France, 160?; d. Castres, France, 1665
Number theory; probability theory; variational principles; quadrature; geometry
Fermat, who spent his life as a magistrate in the south of France, contributed decisively to most of the mathematical subjects of his time: from quadrature to optics, from geometry to number theory. Very little is known about his early life—even the date of his birth is uncertain—but by 1629 he had close contacts with VIÈT’S [VI.9] scientific heirs in Bordeaux. His work displays a thorough knowledge of ancient as well as contemporary mathematics and he exchanged problems and mathematical information by correspondence with, among others, RENÉ DESCARTES [VI.11], Gilles Personne de Roberval, Marin Mersenne, Bernard Frenicle, John Wallis, and Christiaan Huygens.
A crucial early-modern topic was the use of algebra to solve geometric problems. Viète and other algebraists before him had used equations in a single unknown to rewrite and solve “determinate” problems (problems admitting a finite number of solutions). In his manuscript Ad Locos Planos et Solidos Isagoge, which circulated in Paris in 1637 (the same year as Descartes’s La Géométrie), Fermat presented a general way of handling and solving indeterminate problems associated with constructions of loci: that is, of sets of points (usually curves) defined by some constraints. He identified the points of such loci by two coordinates linked by an equation (although he chose a different way of taking coordinates from the usual modern x and y coordinates). Moreover, Fermat gave the standard forms of the corresponding equation when the locus to be found was a line, a parabola, an ellipse, etc.
Fermat also used algebraic analysis to solve problems of extrema, including finding the tangent or the normal to a curve at a given point, and determining centers of gravity. His method relies on the principle that a certain algebraic expression takes on the same values twice near the extremum. Although the procedure is purely algebraic, his successors tended to interpret it from a differential perspective, thereby making his work an apparent precursor of the calculus. Fermat applied the method to a variety of problems, including (within the framework of a controversy with Descartes’s followers around 1660) a proof of the law of refraction in optics. Basing his analysis on the principle that “nature acts in the shortest time,” Fermat was able to express the problem as one of extrema and to solve it with his method. The problem of refraction was one of the first complex physical problems to be treated in a thoroughly mathematical way, and Fermat’s approach later led to VARIATIONAL METHODS [III.94].
However, Fermat also showed a perfect mastery of more classical, for instance Archimedean, techniques, which he used when dealing with other types of geometrical questions such as quadrature.
Such versatility also appears in Fermat’s work on numbers. On the one hand, he was happy to apply his algebraic approach to Diophantine analysis in order to obtain solutions for cases previously thought to be insoluble, or to derive new solutions from ones already known. On the other hand, he advocated a theoretical study of the integers, for which the currently available algebraic theory of equations was not sufficient. For example, he gave general properties of the divisors of numbers of the form an ± 1 (among them his now celebrated LITTLE THEOREM [III.58]) and of x2 + N y2 for various N. He invented the method of infinite descent specifically to deal with problems concerning integers. He used this method, which relies on the impossibility of constructing an infinite strictly decreasing sequence of integers, to prove that a4 − b4 = c2 has no nontrivial integer solutions. This is a particular case of his famous LAST THEOREM [V.10], which Fermat only stated in the margins of one of his books: an + bn = cn has no non-trivial integer solutions for n > 2. The first proof of the general case was given by Andrew Wiles in 1995.
In 1654 Fermat exchanged letters with PASCAL [VI.13] on the idea of a “fair game” and on the redistribution of the stakes if a game is interrupted before its end. These letters introduced important concepts in probability, including expected value and conditional probability.
Further Reading
Cifoletti, G. 1990. La Méthode de Fermat, Son Statut et Sa Diffusion. Société d’Histoire des Sciences et des Techniques. Paris: Belin.
Goldstein, C. 1995. Un Théorème de Fermat et Ses Lecteurs. Saint-Denis: Presses Universitaires de Vincennes.
Mahoney, M. 1994. The Mathematical Career of Pierre de Fermat (1601–1665), second revised edn. Princeton, NJ: Princeton University Press.
Catherine Goldstein