From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.
Adrian Marie Legendre was born at Toulouse on September 18, 1752, and died at Paris on January 10, 1833. The leading events of his life are very simple and may be summed up briefly. He was educated at the Mazarin College in Paris, appointed professor at the military school in Paris in 1777, was a member of the Anglo-French commission of 1787 to connect Greenwich and Paris geodetically; served on several of the public commissions from 1792 to 1810; was made a professor at the Normal school in 1795; and subsequently held a few minor government appointments. The influence of Laplace was steadily exerted against his obtaining office or public recognition, and Legendre, who was a timid student, accepted the obscurity to which the hostility of his colleague condemned him.
Legendre's analysis is of a high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original. His chief works are his Géométrie, his Théorie des nombres, his Exercices de calcul intégral, and his Fonctions elliptiques. These include the results of his various papers on these subjects. Besides these he wrote a treatise which gave the rule for the method of least squares, and two groups of memoirs, one on the theory of attractions, and the other on geodetical operations.
The memoirs on attractions are analyzed and discussed in Todhunter's History of the Theories of Attraction. The earliest of these memoirs, presented in 1783, was on the attraction of spheroids. This contains the introduction of Legendre's coefficients, which are sometimes called circular (or zonal) harmonics, and which are particular cases of Laplace's coefficients; it also includes the solution of a problem in which the potential is used. The second memoir was communicated in 1784, and is on the form of equilibrium of a mass of rotating liquid which is approximately spherical. The third, written in 1786, is on the attraction of confocal ellipsoids. The fourth is on the figure which a fluid planet would assume, and its law of density.
His papers on geodesy are three in number, and were presented to the Academy in 1787 and 1788. The most important result is that by which a spherical triangle may be treated as plane, provided certain corrections are applied to the angles. In connection with this subject he paid considerable attention to geodesics.
The method of least squares was enunciated in his Nouvelles méthodes published in 1806, to which supplements were added in 1810 and 1820. Gauss independently had arrived at the same result, had used it in 1795, and published it and the law of facility in 1809. Laplace was the earliest writer to give a proof of it; this was in 1812.
Of the other books produced by Legendre, the one most widely known is his Éléments de géométrie which was published in 1794, and was at one time widely adopted on the continent as a substitute for Euclid. The later editions contain the elements of trigonometry, and proofs of the irrationality of and . An appendix on the difficult question of the theory of parallel lines was issued in 1803, and is bound up with most of the subsequent editions.
His Théorie des nombres was published in 1798, and appendices were added in 1816 and 1825; the third edition, issued in two volumes in 1830, includes the results of his various later papers, and still remains a standard work on the subject. It may be said that he here carried the subject as far as was possible by the application of ordinary algebra; but he did not realize that it might be regarded as a higher arithmetic, and so form a distinct subject in mathematics.
The law of quadratic reciprocity, which connects any two odd primes, was first proved in this book, but the result had been enunciated in a memoir of 1785. Gauss called the proposition ``the gem of arithmetic,'' and no less than six separate proofs are to be found in his works. The theorem is as follows. If p be a prime and n be prime to p, then we know that the remainder when is divided by p is either +1 or -1. Legendre denoted this remainder by (n/p). When the remainder is +1 it is possible to find a square number which when divided by p leaves a remainder n, that is, n is a quadratic residue of p; when the remainder is -1 there exists no such square number, and n is a non-residue of p. The law of quadratic reciprocity is expressed by the theorem that, if a and b be any odd primes, then
thus if b be a residue of a, then a is a residue of b, unless both of the primes a and b are of the form 4m + 3. In other words, if a and b be odd primes, we know that
and, by Legendre's law, the two ambiguities will be either both positive or both negative, unless a and b are both of the form 4m + 3. Thus, if one odd prime be a non-residue of another, then the latter will be a non-residue of the former. Gauss and Kummer have subsequently proved similar laws of cubic and biquadratic reciprocity; and an important branch of the theory of numbers has been based on these researches.
This work also contains the useful theorem by which, when it is possible, an indeterminate equation of the second degree can be reduced to the form ax² + by² + cz² + 0. Legendre here discussed the forms of numbers which can be expressed as the sum of three squares; and he proved [art. 404] that the number of primes less than n is approximately .
The Exercices de calcul intégral was published in three volumes, 1811, 1817, 1826. Of these, the third and most of the first are devoted to elliptic functions; the bulk of this being ultimately included in the Fonctions elliptiques. The contents of the remainder of the treatise are of a miscellaneous character; they include integration by series, definite integrals, and in particular and elaborate discussion of the Beta and the Gamma functions.
The Traité des fonctions elliptiques was issued in two volumes in 1825 and 1826, and is the most important of Legendre's works. A third volume was added a few weeks before his death, and contains three memoirs on the researches of Abel and Jacobi. Legendre's investigations had commenced with a paper written in 1786 on elliptic arcs, but here and in his other papers he treated the subject merely as a problem in the integral calculus, and did not see that it might be considered as a higher trigonometry, and so constitute a distinct branch of analysis. Tables of the elliptic integrals were constructed by him. The modern treatment of the subject is founded on that of Abel and Jacobi. The superiority of their methods was at once recognized by Legendre, and almost the last act of his life was to recommend those discoveries which he knew would consign his own labours to comparative oblivion.
This may serve to remind us of a fact which I wish to specially emphasize, namely, that Gauss, Abel, Jacobi, and some others of the mathematicians alluded to in the next chapter, were contemporaries of the members of the French school.
This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).