DEVELOPMENT OF MATHEMATICAL THOUGHT. 657 



received tardy recognition. Such speculations can be 

 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 and intuition. 1 Many 

 persons think that only the second of these three in- 



1 Already Euler had remarked on 

 the differentinterests that prompted 

 mathematical research. .Referring 

 to the writings of Count Fagnano, 

 he says in the introduction to 

 the first of his memoirs on Elliptic 

 Integrals (1761, quoted by Brill & 

 Nother in ' Bericht der Deutschen 

 Mathematiker-Vereinigung,' vol. iii. 

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

 matical speculations from the point 

 of view of utility, they can be divided 

 into t\vo classes : first, those which 

 are of advantage to ordinary life 

 and other sciences, and the 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- 

 quacy of analysis, those speculations 

 are of no little value which promise 

 to extend the province of analysis. 

 Such seems to be the nature of 

 observations which are usually made 

 or found a potei*iori, 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 through 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 by 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 u'aurait pas 

 du reproduire dans son rapport 

 une phrase peu adroite de feu M. 

 Fourier ou ce dernier nous fait 

 des reproches, h, Abel et a moi, de 

 ne pas nous etre occupes de pre- 

 ference du mouvement de la chaleur. 

 II est vrai que M. Fourier avait 

 1'opinion que le but principal des 

 mathematiques ctait 1'utilittJ pub- 

 lique et 1'explication des pheno- 

 menes naturels ; mais un philosophe 

 comme lui aurait du savoir que le 

 but unique de la science, c'est 

 1'honneur de 1'esprit humain et que 

 sous ce titre, une question de 

 nombres vaut autant qu'une ques- 

 tion du systeme du monde." In the 

 sequel he adds : " Je crois entrevoir 

 que toutes ces transcendantes " (i.e., 

 the elliptic and Abelian functions) 

 "jouissent des propriet^s admir- 

 ables et inattendues auxquelles on 

 peut etre conduit par le theore'me 

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

 temps en temps sur une me'thode 

 nouvelle de trailer les perturbations 

 celestes, me"thode dans laquelle 

 doivent entrer les theories nou- 

 velles des fonctions elliptiques. " 



2 T 



