MEMOIR OF LEGENDRE. 151 



consists ill this : two prime numbers m and n being given, if m be raised to the 

 power n minus 1 divided by 2 and the result be divided by n, then n to the 

 power m minus 1 divided by 2, and the result be divided by m, the remainders 

 of the two divisions, which are alwaj's capable of being expressed by plus 1 or 

 minus 1, will both be of the same sign, or else of the contrary sign, in certain 

 determinate cases ; a result which has found and continues to find numerous 

 applications in researches relating to the properties of numbers. 



M. Legendre, in reproducing, in successive editions of the Theory of Numbers, 

 the demonstration of this theorem as he had given it in 1785, discovered that in a 

 determinate case it presents a lacuna, without tlie theorem itself having been ever 

 found in default. M. Gauss, wdio, by his Disquisitiones Arithmcticce, published 

 in 1801, had placed himself in the first rank of the savants who have dealt 

 with the theory of numbers, gave a demonstration of the theorem of reciprocitv 

 wdiich left nothing further to be desired. M. Legendre reproduced this demon- 

 stration in his Theory of Numbers in 1830, observing that it is the niore remark- 

 able as resting on the most elementary principles, and at the same time gave 

 another yet more simple, proposed by M. Jacobi. Still later, M. Lionville and 

 other eminent geometers have given other demonstrations of the same law. The 

 exactness of the law of Legendre is therefore more than sufficiently demonstrated; 

 but here the inventor has left to those who have followed him the privilege of 

 contpJetinr/ his discovery. 



This circumstance recalls, somewhat remotely, the fate of the remarkable 

 theorems on numbers which Fermat left without demonstration ; all, with the 

 exception of a single one, have been demonstrated within a century and a half 

 after the death of their author, by Euler, Lagrange, and Legendre ; this one, the 

 last theorem of Fermat, without having ever been found in default, still awaits 

 a demonstration, though the Academy has, in late years, several times proposed 

 it as the subject of a prize to tlie emulation of geometers. 



But if M. Legendre took delight, like Euler, in tiie combinations, so arduous 

 in appearance, of the theory of numbers, like Euler, he excelled also in the 

 research of the integrals of differential quantities, a researcli which is itself not 

 directed by any certain ruk>, and in whicli the inquirer is conducted to the result 

 only by a certain intiutive prevision of the combinations and reductions which 

 will be available in the formulas and figures. The finest integrals appear often 

 to have been found by hazard : but these are hazards, as M. Legendre said in 

 speaking of Euler, whic/t never oeciir to cmij but those ivlio laioiv how to create 

 them. This remark, insufficient doubtless to make us comprehend how a difier- 

 ential expression is integrated, Avill enable us perhaps to conceive how the mind 

 may be stimulated to this pursuit, as to that of the properties of numbers, and 

 how these two kinds of research, which seem to call into play analogous facul- 

 ties, were the two dominant passions of Euler and Legendre. 



A differential quantity given by a problem of geometr}', mechanics, or physics, 

 docs not always correspond to an analytic expression existing in the science, and, 

 in order not to leave certain problems without solution, it becomes an object to 

 enrich analysis Avith new functions. After having exhausted expressions purely 

 algebraic, we succeed in integrating a great number of difierentials by means of 

 arcs of the circle and of logarithms which are the most simple of transcendent 

 quantities ; but, in order to extend still further the applications of the integral 

 calculus, it was necessary to have recourse to transcendents of a more composite 

 order. 



Euler thought that instead of ]>eing limited to the circle, other curves of the 

 second degree, especially the ellipsis and hyperbola, might be considered, and that 

 tables analogous to the tables of logarithms and to tliose of circular functions 

 might be drawn up in reference to them. By one of those happy combinations, 

 which seem almost fortuitous, he found under a purely algebraic form the com- 

 plete integral of a differential equation composed of two separate but similar 



