THE SCIENTIFIC SPIRIT IN FRANCE. 



103 



termed the synthetical method, and has in modern times 

 survived principally in England, where inductive reason- 

 ing, based upon observation of detail, has since the age 

 of Lord Bacon been most successfully cultivated. 1 These 

 different ways of approaching the same subject will fre- 

 quently engage my attention in the course of this survey : 

 the greatest mathematicians of modern times have recog- 

 nised the importance of both aspects, and the enormous 

 progress of the science itself has depended, no doubt, on an 

 alternating employment of them. Leibniz clearly foresaw 

 this when, in his correspondence with Huygens and others, 

 he urged the necessity of not abandoning the purely geo- 

 metrical view, or entirely sacrificing the older for the 

 modern methods. 2 There can, however, be no doubt that 



is. 



Older syn- 



1 See on this point the opinion of 

 an authority, Hermann Hankel, in 

 his highly interesting and sugges- 

 tive lecture, ' Die Entwickelung der 

 Mathematik in den letzten Jahr- 

 hunderten' (Tubingen, 1869, re- 

 published by P. du Bois-Reymond, 

 1884). Speaking of the age of 

 Leibniz he says : "Though on the 

 Continent mathematicians were not 

 eo conservative as in England, where 

 a purely geometrical exposition was 

 considered to be the only one worthy 

 of mathematics, yet the whole spirit 

 of that age was directed to the sol- 

 ution of problems in geometrical 

 clothing, and the result of the cal- 

 culus had mostly to be retranslated 

 into geometrical forms. It is the in- 

 estimable merit of the great mathe- 

 matician of Basel, Leonhard Euler, 

 to have freed the analytical calculus 

 from all geometrical fetters, and 

 thus to have established analysis 

 as an independent science. Analy- 

 sis places at its entrance the con- 

 ception of a function, in order to 

 express the mutual dependence of 



two variable quantities. . . . The 

 abstract theory of functions is the 

 higher analysis. . . . The concep- 

 tion of a function has been slowly 

 and hesitatingly evolved out of spe- 

 cial and subordinate conceptions. 

 It was Euler who first established 

 it, making it the foundation of the 

 entire analysis, and hereby he in- 

 augurated a new period in mathe- 

 matics" (p. 12, &c.) 



2 To Huygens, 16th September 

 1679: "Je ne suis pas encor con- 

 tent de PAlgebre, en ce qu'elle ne 

 donue ny les plus courtes voyes, ny 

 les plus belles constructions de Ge"o- 

 metrie. . . . Je croy qu'il nous faut 

 encor une autre analyse proprement 

 ge'ome'trique ou lineaire, qui nous 

 exprime directement situm, comme 

 1'Algebre exprime magnitudinem. 

 Et je croy d'en avoir le moyen, 

 et qu'on pourroit repre'senter des 

 figures et mesures des machines et 

 mouvements en caracteres, comme 

 PAlgebre reprdsente les nombres 

 ou grandeurs" (Leibniz, Mathem. 

 Werke, ed. Qerhardt, vol. ii. p. 19). 



