# Fermat sentence example

The dispute on the latter point between

**Fermat**and Descartes was continued, even after the philosopher's death, as late as 1662.In another question connected with this, the problem of drawing tangents to any curve, Descartes was drawn into a controversy with Pierre (de)

**Fermat**(1601-1663), Gilles Persone de Roberval (1602-1675), and Girard Desargues (1593-1661).**Fermat**and Descartes agreed in regarding the tangent to a curve as a secant of that curve with the two points of intersection coinciding, while Roberval regarded it as the direction of the composite movement by which the curve can be described.Between Roberval and Descartes there existed a feeling of ill - will, owing to the jealousy aroused in the mind of the former by the criticism which Descartes offered to some of the methods employed by him and by Pierre de

**Fermat**; and this led him to criticize and oppose the analytical methods which Descartes introduced into geometry about this time.A reprint of 1670 is only valuable because it contains P. de

**Fermat**'s notes; as far as the Greek text is concerned it is much inferior to the other.AdvertisementIn the theory of numbers he furnished solutions of many of P.

**Fermat**'s theorems, and added some of his own.Pascal and P. de

**Fermat**had initiated he brought very nearly to perfection; but the demonstrations are so involved, and the omissions in the chain of reasoning so frequent, that the Theorie analytique (1812) is to the best mathematicians a work requiring most arduous study.It was proposed by Pierre de

**Fermat**to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians.A solution was also given by

**Fermat**in his Relation.Diophantine problems were revived by Gaspar Bachet, Pierre

**Fermat**and Euler; the modern theory of numbers was founded by**Fermat**and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and**Fermat**, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and others.AdvertisementThis problem was proposed by Pierre de

**Fermat**first to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians.Now the direction and phase of the light are those of the ray which reaches the eye; and by

**Fermat**'s principle, established by Huygens for undulatory motion, the path of a ray is that track along which the disturbance travels in least time, in the restricted sense that any alteration of any short reach of the path will increase the time.As, however, our terrestrial optical apparatus is now all in motion along with the matter, we must dealt .with the rays relative to the moving system, and to these also

**Fermat**'s principle clearly applies; thus V+ (lu'--mv'-Fnw') is here the velocity of radiation in the direction of the ray, but relative to the moving material system.To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de

**Fermat**, and which was called by Gauss the " gem of arithmetic."He was a great mathematician in an age which produced Descartes,

**Fermat**, Huygens, Wallis and Roberval.AdvertisementThe cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes,

**Fermat**, Roberval and Torricelli, who had in turn exhausted their skill upon it.The mathematical theory of probability and the allied theory of the combinatorial analysis were in effect created by the correspondence between Pascal and

**Fermat**, concerning certain questions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Mere.Sir Christopher Wren, the famous architect, determined the length of the arc and its centre of gravity, and Pierre

**Fermat**deduced the surface of the spindle generated by its revolution.At the same time he challenged Roberval and

**Fermat**to construct the tangent; Roberval failed but**Fermat**succeeded.It was further investigated by John Wallis, Christiaan Huygens (who determined the length of any arc in 1657), and Pierre de

**Fermat**(who evaluated the area between the curve and its asymptote in 1661).AdvertisementThe extraordinary advances made by him in this branch of knowledge were owing to his happy method of applying mathematical analysis to physical problems. As a pure mathematician he was, it is true, surpassed in profundity by more than one among his pupils and contemporaries; and in the wider imaginative grasp of abstract geometrical principles he cannot be compared with

**Fermat**, Descartes or Pascal, to say nothing of Newton or Leibnitz.Hesse, "they are, like P.

**Fermat**'s theorems, riddles to the present and future generations."**Fermat**'s Theorem, if p is prime and a is prime to p then a p-1 -1 is divisible by p, was first given in a letter of 1640.**Fermat**'s Problem is that x n +y n =z n is impossible for integral values of x, y and z when n is greater than 2.**Fermat**was for some time councillor for the parliament of Toulouse, and in the discharge of the duties of that office he was distinguished both for legal knowledge and for strict integrity of conduct.AdvertisementHe left a son, Samuel de

**Fermat**(1630-1690) who published translations of several Greek authors and wrote certain books on law in addition to editing his father's works.The Opera mathematica of

**Fermat**were published at Toulouse, in 2 vols.The Ouvres of

**Fermat**have been re-edited by P. Tannery and C. Henry (Paris, 1891-1894).His great work De maxims et minimis brought him into conflict with Rene Descartes, but the dispute was chiefly due to a want of explicitness in the statement of

**Fermat**(see Infinitesimal Calculus).**Fermat**, Roberval and Desargues took exception in their various ways to the methods employed in the geometry, and to the demonstrations of the laws of refraction given in the Dioptrics and Meteors.