MEMOIR OF LEGEI^DRE. 155 



provable; but the fecundity of the nietliods created ))y M. Legondre was sucli 

 that results wliicli he had hardly ventured to anticipate were very soon realized, 

 and I abridg'e, in transcribing', the terms in which he speaks of this event in the 

 advertisement of the third volume : 



A yonusf geometer, M. Jacobi, of Kcenis^sberg:, who could have hatl no knowledf^o of 

 the treatise of elliptical fuuctious, had succeeded, by his own etibrts, in discovering uot only the 

 secoud scale of which we have beeu speakiug, which is relevant to the number ;5, but a third 

 which is relevant to the number 5, and he had already acquired the certainty that there must 

 exist a similar one for every odd number proposed. * * * This theorem being established 

 for every odd number, it was easy thence to conclude that for every integer or simply rational 

 number may be formed a particular scale of modules, which will give rise to an inlinitude 

 of transformations of any one function of the first species, which transformations will be all 

 determinable algebraically. * * * The hopes inspired by the first successes of M. Jacobi 

 have beeu since justified by new publications. * * * It remains for me (says M. 

 Legendre in coutinuatiou) to speak of the admirable researches on the same subject which 

 M. Abel, a rival worthy of M. Jacobi, has published nearly at the same time. The first 

 memoir of M. Abel forms in itself an almost complete theory of elliptical functions considered 

 under the most general point of view. * * * A second memoir of his presents very 

 remarkable results : First, on the division of the particular function of which the modtilns is 

 sin. 45^, and which represents arcs of the lemuiscate; secondly, on the general transformation 

 of functions of the first species, by which, says the author, we are enabled to demonstrate, 

 in a very simple and direct manner, the two general theorems previously published or 

 announced by M. Jacobi. 



We shall not enter into other details (says M. Legendre in conclusion) respecting the 

 labors of these two young geometers, whose talents have dawned upon the learned world 

 with so much brilliancy. It will readily be conceived that the author of the present treatise 

 would be prompted to hail with cordial applause discoveries so greatly promoting that 

 branch of analysis of which he may claim to be in some sort the founder. Hence has originated 

 the design of enriching his own work with a part of these new discoveries, while presenting 

 them under a point of view at once the most simple and most conformed to his own ideas. 

 Such is the object of the two supplements which follow, and of those which, in the sequel, 

 he may unite with them in order to form the third volume of his treatise. 



Rarely has such sincere and emphatic recognition been extended to disciples 

 worthy from the outset of being- counted as rivals; but M. Legeudi-e still fnrther 

 enhanced this recognition by the unaffected and spontaneous warmth with which 

 the paternal tenderness naturally felt for a theory created by himself, and developed 

 during more than 40 years by his single efforts, was reflected on his young c^mipetitors. 

 Persons who, at that epoch, attended the sessions of the Academy will not have 

 forgotten the artless efiusion of feeling with which M. Legendre hastened to 

 communicate to his colleagues the first letters received on a subject so interesting 

 for science and for himself. It might be said that the elliptic functions did no 

 less honor to the nobility of his sentiments than the profundity of his genius. 



These first impressions were not modified by subsequent reflection, and M. 

 Legendre conchules with the following paragraph the third supplement to the 

 Thcorie dcs fondions clliptlqiies, by which that vast labor is closed: 



We shall here terminate the additions which we have proposed to make to our workby 

 taking advantage of the recent discoveries of MM. Abel and Jacobi in the theory of elliptical 

 functions. It will be remarked that the most important of these additions consists in the 

 new branch of analysis which we have deduced from the theorem of iSI. Abel, and which 

 had remained until now wholly unknown to geometers. This branch of analysis to which 

 we have given the name of theory of ultra-dl ijitic functions is infinitely more extended than 

 that of elliptical functions, with which it has very intimate relations; it is composed of an 

 indefinite number of classes, each of which is divided into three species like the elliptic.*- 

 functions, and which have besides a great number of properties. We have been able to 

 enter but partially into this subject ; but that it will be progressively enriched by the Labors 

 of geometers can hardly be doubted, and as little that it will eventually prove one of the 

 most efficient parts of the analysis of transcendents. 



These lines, dated March 4, 1832, maybe regarded as in some sort the scien- 

 'tific testament of M. Legendre, who died within a year thereafter. M. Abel, in 

 whom he reposed such high hopes, had descended to the tomb several years 

 before him; M, Jacobi followed in 1849; but the anticipations of M. Legendre 

 have not the less been realized, as well by the labors of M. Jacobi himself as 

 by those of our learned colleagues, MM. Liouville tmd llermilc, and other distin- 

 guished geometers. 



