154 MEMOIR OF LEGENDRE. 



pnrel}'^ algebraic, like the celebrated integral of Euler, before refon'ecl to. 

 I^astly, transcendents of tlie third species may also be united to form a snni of 

 v.hicli the valne, without being null or even algebraic, is notwithstanding of a 

 more sim])le uatni'e than each of the former in itself; for it raa^' be expressed 

 by arcs of the circle and logarithms, which are the most simple of transcendents. 



These differences, and several others which exist between the three species of 

 elliptical transcendents, suffice to vindicate the division established by M. Legen- 

 dre ; but, at the same time, tliey do not prevent our perceiving a profound anal- 

 ogy between all these transcendents which justifies their union under the same 

 denomination. The first and second species may be expressed by arcs of the 

 ellipsis; the third is the most compounded, but it has so much analogy with the 

 two others that all three may be regarded as forming but one and the same 

 order of transcendents, the first after arcs of the circle and logarithms. As JM. 

 Legendre elsewhere says, '' the denomination of elliptic function is improper in 

 some respects ; but we nevertheless adopt it on account of the great analogy 

 which exists between the properties of this function and those of arcs of the 

 ellipsis." 



M. Legendre resumed these questions with several others in a great work in 

 three quarto volumes, which he published in 1811, 1816, and 1817, under the 

 title of Exercises cle caleul integral sur divers orders de trunscendantes et sur 

 les quadratures. In this work, part of which was devoted to two classes of 

 definite integrals, to which the author has given the name of integrales eulc- 

 rienneSj he occupied himself also with a great number of questions about the inte- 

 gral calculus, into the details of which it would be difficult here to enter; but the 

 most extensive and in his eyes the most important part was that which treats of 

 elUptic functions, of their application to different problems of geometry and 

 mechanics, and the tables necessary for the use of those functions. Finally, in 

 1825 and 18:26, he combined anew all his results, with the developments and 

 improvements which incessant labor had enalded him to supply, in a work 

 entitled Theorie des functions eUiptiques. This first appeared in two volumes, 

 followed at a later period by three supplements, which constitute the third and 

 last volume. 



Among the improvements which IM. Legendre bestowed on his previous labors 

 when he published them anew in 1825, one of the principal was the discovery 

 of a second scale of modules, different irom that which alone was known at the 

 time of the publication of the exercises on the integral calculus. '' This second 

 scale," as he remarks in the 31st chapter of the first volume, " completed in 

 many respects the labors of the author upon this theory ; it afforded an easy 

 method of arriving at many striking results of analysis which till then it had 

 been imi>racticable to demonstrate except by very laborious integrations. By 

 the combination of the two scales the transformations of functions of the first 

 species could be prodigiously multiplied ; this the author has made evident by 

 constructing a sort of tessellated talde {damicr) infinite in its two dimensions, 

 all the divisions of which might be filled l)y the different transformations of 

 which one and the same function is susceptible." 



The dev(dopment of the properties and uses of elliptical functions, consid- 

 ered with this generality, composed the whole first volume of the publication of 

 1825. The second was devoted, in part, to tables intended to facilitate the 

 conversion of the integrals obtained into numerals. Calcidated by the author 

 himself with the greatest precision, these tables constituted in themselves an 

 immense labor. '' By means of them," said M. Legendre, "the theory of elliptical 

 functions, enlarged and nearly completed by many successive labors, might be 

 ap})lied with almost as much facility as those of circular and logarithmic func- 

 tions, answerably to the wishes and hopes of Euler." 



After the developments which the theory of elliptical functions had received 

 by the discovery of the second scale of modules, further progress seemed scarcely 



