FER3IAT, PIERRE DE, was born at Toulouse, about 1595, and was brought up to the profession of the law. We have but few inci dents of his private life, except that be became a counsellor of the parliament of his native town, was universally respected for his talents, and lived to the age of seventy years. His works were published in 1670 and 1679, in folio : the last volume contains his correspoudence, besides some original scientific papers.
Fermat restored two books of Apollonius, and published Diophan tus, with a commentary. The whole of the actual works of Fermat fill an exceedingly small space ; nevertheless they contain the germs of analytical principles which have since come to maturity. In fact they may be regarded, generally speaking, as announcements of the results to which he had arrived, without demonstrations, or any indications of the processes employed.
The properties of numbers were the subject of his enthusiastic researches, and no single individual has added more that is both curious and useful to this branch of mathematics than Fermat : the theorem now commonly called Format's is but a particular case of a much more general one given in hie works.
His method for finding Maxima and Minima has only the merit of a moderate ingenuity, before the differential calculus was discovered; the analysts of that day hovered on the brink of that beautiful process of analysis which has been rather ridiculously termed the greatest disco very of the human mind. A method not very remote from Fermat's was practised by other analysts of his day; and in spirit also by the ancient geometers; but it certainly is not the differential calculus, and Laplace has no ground for his attempt to snatch ,from the claims of the English and German nations this grand step of analysis in order to appropriate it to his own.
In Fermat's correspondence with Father Meraenne, we find him, in a bungling manner, contesting with Roberval the first principles of mechanics, and maintaining that the weight of bodies is least at the surface of the earth, increasing both within and without, which is the direct opposite to the truth ; and in one of his letters, when greeted by Mersenne with the retraction of his errors, he very disingenuously attempts to deny them, asserting that no body has a centre of gravity, with many similar trifles, which place in bold relief the immortal discovery of -Sir Isaac Newton of the law of universal attraction, and add lustre to his predecessor Galileo, who escaped from similar para doxes, from which common sense ought to have guarded both Fermat and Descartes.
The correspondence of Fermat is sufficiently replenished with vanity, which was also well fed by some of his compatriots, who lauded hie propositions as the finest things which had ever been dis covered. But it Is justly suspected that the discovery of many of his properties of numbers was effected by a tentative process, he himself roettessing uo demonstration, as no vestige remains in the works published by his son of any peculiar analysis for arriving at them; while there are abundant proofs that he and Frenacle, a young Parisian, employed the methods of tabulation and trial, to suggest properties, and by further trials observe if they could generalise them. In a subject less barren thau the theory of numbers this talent and industry would have produced more useful results; for what are the theorems of Fermat to the laws of Kepler? Fermat conjectured that the path of light, in passing from air to denser medium, ought to be such as to describe the shortest pos sible course. This is a particular case of the principle of least action,
and requires some remark. First, we see that Fermat's method for finding maxima and minima was not the differential calculus, for though importuned from various quarters to try this principle he was deterred, as he says himself, for two or three years, by the dread of the asymetries of the process, though any tyro acquainted with the first principles of the differential calculus, with the proper data given, would now do it in five minutes : when Fermat at last did this, it was in a geometrical manner. Secondly, during the life of Descartes, he seems to have disbelieved this' law of refraction. The foundations of both their reasonings in natural philosophy were of the slenderest description, if indeed we can at all use such a term as reasoning to the methods of Descartes, whose followers had the greatest faith when he employed the least of that useful faculty. But the law is truly attributable to Suellius, and, though this is well known, many French writers still ridiculously talk of the Cartesian law of refraction. Thirdly, Fermat did not attribute the truth of the principle to any mechanical laws, of which he seems to have known nothing, but to tho pseudo-physical principlo that nature should take the shortest course in performing its operations-for which indeed he was subjected to several cases of objection, to which he has given good answers, considering the position in which such an hypothesis placed him.
To give a more exact idea of the man,' we shall give one of his problems, entitled Problem by P. de Fermat. To Wallis, or any other mathematician that England may contain, I propose this problem to be resolved by them.
To find a cube number which, added to its aliquot parts, will give a square number I Example 343.
If Wallis and no English mathematician can solve this, nor any analyst of Belgic or Celtic Gaul, then an analyst of Narboune will solve it.' Wallis gives an account of this in the Commercium Epistolieum,' the correspondence having been conducted through Sir Keneltn Digby. The works of Fermat contain also the tangents to some known curves, and some centres of gravity.
Though thus strongly endowed with the faculty of self-esteem, and of that cunning which seeks to hide the tracks of discovery, we must still place Fermat among such men as Pascal, Barrow, Brouncker, Wallis; but he had none of the masculine mind of Descartes, nor a particle of the penetrating spirit of the glory of his age and nation, Newton.
It would be wrong to omit here the most curious of the theorems of Fermat relative to numbers. To make it more generally intelli gible we may state, that a triangular number means the sum of any number of terms from the first of the natural numbers 1, 2, 3, 4,5, • thus 1, 3, 6,10, &c., are triangular numbers; the square numbers are 1, 4, 9, 16, &c., and are the sums of the progression 1, 3, 5, 7, ese. ; pentagonal numbers in like manner are the sums of the numbers 1,4, 7, 10, &c., namely, 1, 5, 12, 22, &c. The theorem consists in this, that 'every number' is the sum of 1, 2, or 3 triangular numbers; every number is the sum of 1, 2, 3, or 4 square numbers, and so on. In the works of Euler, Legendre, and Barlow, the demonstration of the first two cases may be ' ; and though Legendre and Csuchy have both laboured to prove it more generally, yet our impression is that the general theorem is still without proof.