DEVELOPMENT OF MATHEMATICAL THOUGHT. 657 



received tardy recognition. Such speculations can hi 

 carried on either as fascinating exercises of mere 

 ingenuity, or for practical purposes to improve the 

 refined instruments of mathematical calculation, or in 

 the philosophical interest of arriving at the fundamental 

 processes of human thought .ind intuition.^ Many 

 persons think tliat only the second of these three in- 



' Already Euler had remarked on 

 the different interests that prompted 

 mathematical research. Referring 

 to tiie wTitings of Count Fagnano, 

 he says in the introduction to 

 the tirst of his memoirs on Elliptic 

 Integrals (1761, ([Uoted by Brill & 

 Nuther in ' BerichL der Deutschen 

 Matheniatiker-Vereinigung,' vol. iii. 

 p. 206) : " If one looks at mathe- 

 matical speculations from the point 

 of view of utility, they can be divided 

 into two classes : first, those which 

 are of advantage to ordinary life 

 and other sciences, and tiie value 

 of which is accordingly measured by 

 the amount of that advantage. The 

 other class comprises speculations 

 which, without any direct advant- 

 age, are nevertheless valuable be- 

 cause they tend to enlarge the 

 boundaries of analysis and to exer- 

 cise the powers of the mind. Inas- 

 much as many researches which 

 promise to be of great use have to 

 be given up owing to the inade- 

 (juacy of analysis, those speculations 

 are of no little value which i)romise 

 to extend the province of analysis. 

 Such seems to be the nature of 

 observations which are usually made 

 or found a posteriori, but which 

 have little or no chance of being 

 discovered a priori. Having once 

 been established as correct, methods 

 more easily present themselves 

 which lead up to them, and there 

 is no doubt that thrcjugh the search 

 for such methods the domain of 

 analysis may be considerably ex- 



VOL. II. 



tended." The school of mathema- 

 ticians headed by Abel and Jacobi 

 pursued mathematics from purely 

 scientific interest, and was criti- 

 cised on this ground bj- eminent 

 contemporary mathematicians in 

 France : see a letter of Jacobi to 

 Legendre, dated July 2, 1830, in 

 which he refers to a Report of 

 Poisson on his great work, but 

 adds : " M. Poisson n'aurait pas 

 du reproduire dans son rapport 

 une phrase pcu adroite de feu M. 

 Fourier oii ce dernier nous fait 

 des rej)roches, a Abel et h, moi, de 

 ne pas nous etre occupes de pre- 

 fdrence du mouvement de la chaleur. 

 II est vrai que M. Fourier avait 

 I'opinion ijue le but principal des 

 mathcmatiques ot^iit I'utilite pub- 

 lique et lexplication des i)hcno- 

 menes naturels ; mais un philosophe 

 comme lui aurait du savoir que le 

 but uni<iue de la science, c'est 

 I'honneur de I'esprit Immain et que 

 sous ce titre, une question de 

 nombres vaut autant ([u'une (jues- 

 tion du systemedu monde." In the 

 sequel he adds : " Je crois entrevoir 

 que toutes ces transcendantes " (i.e., 

 the elliptic and Abelian functions) 

 "jouissent des jiroprictcs admir- 

 ables et inattendues auxquclles on 

 l)eut etre conduit par le theorfcme 

 d'Abel. . . . J'ai retlcchi aussi de 

 temps en temps sur une methode 

 nouvelle de traiter les perturbations 

 celestas, methode dans laciuelle 

 doivent entrer les theories nou- 

 velles des fonctions elliptiques." 



2 T 



