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.
No number of the form 4n+3, or 4n - I, can be the sum of two squares), and goes on to a d, practically, the condition stated by Fermat, "and the double of it [n] increased by one, when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4n - 1," except for the omission of the words "when divided.
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.
This problem was proposed by Pierre de Fermat first to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians.
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."
Pappus gives somewhat full particulars of the propositions, and restorations were attempted by P. Fermat (Ouvres, i., 1891, pp. 3-51), F.
He was a great mathematician in an age which produced Descartes, Fermat, Huygens, Wallis and Roberval.
The 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.
Daguerre for the invention of photography, the grant for the publication of the works of P. Fermat and Laplace, the acquisition of the museum of Cluny, the development of railways and electric telegraphs, the improvement of the navigation of the Seine, and the boring of the artesian wells at Grenelle.
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).
The 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.
PIERRE DE FERMAT (1601-1665), French mathematician, was born on the 17th of August 1601, at Beaumont-de-Lomagne near Montauban.
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.
He 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).
See Paul Tannery, "Sur la date des principales decouvertes de Fermat," in the Bulletin Darboux (1883); and "Les Manuscrits de Fermat," in the Annales de la faculte des lettres de Bordeaux.
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.