Fermat Peter

FERMAT (PETER equally celebrated as a restorer of ancient mathematics, and an original au thor of modern improvements, was born in 1590.

His public life was occupied by the active duties attached to the situation of a Counsellor of the Par liament of Toulouse, in which he distinguished both for legal knowledge, and for strict integrity of conduct. Besides the aciencep, which were the prin cipal object of his private studies, he was an accom plished scholar, an excellent linguist, and even a re spectable poet.

His

Opera Mathemalica were published at Tou louse, in two volumes folio, 1670, and 1679; they are now become very scarce. The first contains the Arithmetic of Diophantus, illustrated by a commen tary, and enlarged by a multitude of additional pro positions. In the second we find a Method or the Qua drature 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 fluxion, 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 im mediately 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 Reciifica. Lion of Curves, besides a restoration of Apollonius's Plane Loci ; together with the author's correspond.. ence, addressed to Descartes, Pascal, Roberval, Huy. gent, and others.

It was too much Fermat's custom to leave his most important propositions wholly undemonstrated; some times, perhaps, because he may have obtained them rather by induction than by a connected train of rea soning; and, in other cases, for the purpose of pro.. posing them as a trial of strength to his contempora. nes. 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 re quire further illustration. It happened not uncom monly, 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 maxi ma and minima, and could never be persuaded that Fermat's first propositions on the subject were unex ceptionable. Fermat was however enabled to pus. sue his favourite studies with less interruption than Descartes; and the products of his labour were pro portionate, as Lacroix remarks, to the opportunities that he enjoyed, as well as to the talents that he pos sessed.

There is a very ingenious proposition of Fermat, which deserves to be particularly noticed, on ac ' count of the discussion that it has lately excited among mathematical philosophers. He has demon..

strated that the true law of the refraction of light may be deduced from the principle, that it describes that path, by which it can arrive in the shortest pas sible time from any one point of its tract to another; on the supposition, however, that the velocity of light is inversely proportional to the refractive den sity of the medium; and the same phenomena of re fraction have been shown, by Maupertuis, to be de ducible, 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 ; while that of Mauper. tuis is at most only an interesting commentary on the operation of an accelerating force. It was Newton that showed the necessary connection between the action of such a force and the actual law of refrac tion; demonstrating that all the phenomena might be derived from the effect of a constant attraction, per. pendicular to the surface of the medium; and ex cept 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 con stant velocity within the medium, but the refraction would be very different from that which is observed, though the law of Maupertuis would indicate no dif.

Terence: so that the law must be here applied with the tacit condition, that the refractive force is per pendicular to the surface. In M. Laplace's theory of extraordinary refraction, on the contrary, the ta cit condition is, that the force must, not be perpendi. cular to the surface : so that this theory not only re quires the gratuitous assumption of a different velo city 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 modi fied, 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 ve locities, and attracting the light perpendicularly, is unnatural ; and that the law is the more valuable for not being applicable to it : but a mathematical equa. tion is true even with respect to impossible quanti ties; 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 esta blished, by physical evidence sufficiently extensive to render it universal.

Our author died in 1664, or the beginning of 1665, at the advanced age of 74. He left a son, Samuel de Fermat, who was a man of some learning, and published translations of several Greek authors.

((s. r.))