FERMAT, Pierre de, equally celebrated as a restorer of ancient mathematics, and an original author of modern improvements, was born at Toulouse about 1595. His public life was occupied by the active duties attached to the situation of a counsellor of the parliament of Toulouse, in which he was distinguished both for legal knowledge and for strict integrity of conduct. Besides the sciences, which were the principal objects of his private studies, he was an accomplished scholar, an excellent linguist, and even a respectable poet.

His Opera Mathematica were published at Toulouse, in two volumes folio, 1670 and 1679; they are now become very scarce. The first contains the Arithmetic of Diophantus, illustrated by a commentary, and enlarged by a multitude of additional propositions. In the second we find a Method or the Quadrature of Parabolas of all kinds, and a Treatise on Maxima and Minima, on Tangents, and on Centres of Gravity ; containing the same solutions of a variety of problems as were afterwards incorporated into the more extensive method of fluxions, by Newton and Leibnitz; and securing to their author, in common with Cavalleri, Roberval, Descartes, Wallis, Barrow, and Sluse, an ample share of the glory of having immediately prepared the way for the gigantic steps of those illustrious philosophers. The same volume contains also several other treatises on Geometric Loci, or Spherical Tangencies, and on the Rectification of Curves, besides a restoration of Apollonius' 's Plane Loci ; together with the author’s correspondence, addressed to Descartes, Pascal, Roberval, Huygens, and others.

It was too much Fermat’s custom to leave liis most important propositions wholly undemonstrated; sometimes, perhaps, because he may have obtained them rather by induction than by a connected train of reasoning; and, in other cases, for the purpose of proposing them as a trial of strength to his contemporaries. The deficiency, however, has in many instances been supplied by the elaborate investigations of Euler and Lagrange, who have thought it no degradation to their refined talents to go back a century in search of these elegant intricacies, which appeared to require further illustration. It happened not uncommonly, that the want of a more explicit statement of the grounds of his discoveries deprived Fermat, in the opinion of his rivals, of the credit justly due to him for accuracy and originality. It was thus that Descartes attempted to correct his method of maxima and minima, and could never be persuaded that Fermat’s first propositions on the subject were unexceptionable. Fermat , was however enabled to pursue his favourite studies with less interruption than Descartes; and the products of his labour were proportionate, as Lacroix remarks, to the opportunities he enjoyed, as well as to the talents he possessed.

There is a very ingenious proposition of Fermat, wnι deserves to be particularly noticed, on account of the discussion that it not long ago excited among mathematic [7:9:519]philosophers. He has demonstrated that the true law of the refraction of light may be deduced from the principle, that it describes the path by which it can arrive in the shortest possible time from any one point of its tract or course to another; on the supposition, however, that the velocity of light is inversely proportional to the refractive density of the medium: and the same phenomena of refraction have been shown by Maupertuis to be deducible, upon the opposite supposition with respect to the velocities, from the law of the minimum of action, considering the action as the product of the space described into the velocity. But the law of Fermat is actually a step in the process of nature, according to the conditions of the system to which it belongs in its original form; whilst that of Maupertuis is at most only an interesting commentary on the operation of an accelerating force. It was Newton who showed the necessary connection between the action of such a force and the actual law of refraction; demonstrating that all the phenomena might be derived from the effect of a constant attraction, perpendicular to the surface of the medium: and, except in conjunction with such a force, the law of Maupertuis would even lead to a false result. For if we supposed a medium acting on a ray of light with two variable forces, one perpendicular to the surface, and the other parallel to it, we might easily combine them in such a manner as to obtain a constant velocity within the medium, but the refraction would be very different from that which is observed, though the law of Maupertnis would indicate no difference; so that the law must be here applied with the tacit condition that the refractive force is perpendicular to the surface. In Laplace’s theory of extraordinary refraction, on the contrary, the tacit condition is, that the force must not be perpendicular to the surface; so that this theory not only requires the gratuitous assumption of a different velocity for every different obliquity, which is made an express postulate, but also the implicit admission of the existence of a force, determinate in direction and in magnitude, by which that velocity is modified, and without which the law of Maupertuis would cease to be applicable. It may indeed be said, that the supposition of a medium exhibiting unequal velocities, and attracting the light perpendicularly, is unnatural, and that the law is the more valuable for not being applicable to it; but a mathematical equation is true even with respect to impossible quantities, and a physical law, however useful it may be, requires physical proof; and it will not be asserted that the law of Maupertuis has been or can be established by physical evidence sufficiently extensive to render it universal.

Fermat died at Toulouse in the end of 1664, or beginning of 1665, at the age of seventy. He left a son, Samuel de F ermat, who was a man of some learning, and published translations of several Greek authors. (l. l.)