FERMAT, far'ine. PIERRE OR (11101-65). A French mathematician, born at 13eautnont-de Lomagne. near -Montauban. He was one of the most versatile mathematicians of his time, and was unsurpassed as a contributor to the theory of numbers. Fermat was educated privately, was of a retiring disposition, and published little dur ing his lifetime. At one time he turned his attention to law, and in 1031 became counselor for the Parliament of Toulouse. The first edition of his works, gathered from his papers. annota tions, and personal letters, was published in two volumes miter the title, Opera If thema t ica ( 167° 79). Copies of this edition have become quite rare. The first volume contains the Arithmetic of Diophantus annotated, and the second, mono graphs on maxima and minima tangents, and centres of gravity, and copies of his correspon dence with Huygens. Pascal, Descartes, and others. His chief contributions to the theory of numbers are found in his commentaries on Dio phantus. Among then; are such well-known propositions as follow: If a is prime to p, p being a prime number, then is divisible by p, or. expressed in the notation of congruences ( q.v. ) . a 0 ( mod. p). A prime greater than 2 can be uniquely expressed as the difference of two squares, where p is prime to q, is not divisible by a prime of the form 4a — 1. If p, q. r, are integers such that then pg cannot he a square. The equation 2=? has a unique solution, and the equation 4 =4/' has two solutions. The equation
x° -;-y° = has no integral root if a is integral and greater than 2. fn the case of particular curves. Fermat obtained the maximum and mini MUM values of their functions; also the sub tanfrents of the ellipse, cycloid. conchoid, and quadratrix. The methods employed so resembled those afterwards developed through the differen tial calculus that some mathematicians. espe cially Laplace and Lagrange, have suggested Fermat as the inventor of the calculus. The rise of the theory of probability (see PRonAnniTv) may he dated practically from the correspondence of Fermat and Pascal (1654). Fermat's an swers to the problems suggested by Pascal re veal his firm grasp on the fundamental prinei pies of probabilities. For further information concerning the life and work of Fermat, consult: Libri. Journal des savants (1839, pp. 539-5G1); Brassinne. Pi-Cris des wurres mathematiques de Fermat (Paris. 1853) ; Hoofer, in the Nonecilc Biographic Universe11c, xvii., 438-51 ; Henry. "Recherches sur les manuscrits de Pierre de Fermat." in the Bulletin, Boncompagni, vols. xii. and xiii.; Paul Tannery, "Sur la date des prin cipales de Fermat." in the Bulletin Darboux, 2d series. vol. vii. (1883) ; and "Les manuscrits de Fermat," in the a males de In acult(' des do Bordeaux. The CEurrcs of Fermat were republished by Tannery and Henry under the auspices of the Alinister of Public Instruction (Paris, 1891-94).