DEVELOPMENT OF MATHEMATICAL THOUGHT. 707 



corresponds with conditions which we meet with in 

 reality, say in geometry and physics, otherwise our 

 science becomes useless : further, our definitions must be 

 consistent, and follow logically from the fundamental 

 principles of arithmetic, otherwise we run the risk of 

 sooner or later committing mistakes and encountering 

 paradoxes. We have two interests to serve : the ex- 

 tension of our knowledge of functions and the rigorous 

 proof of our theorems. The methods of Eiemann and 

 of Weierstrass are complementary. " By the instrument an <i 



Weierstrass 



of Biemann we see at a glance the general aspect of com P ared - 

 things like a traveller who is examining from the peak 

 of a mountain the topography of the plain which he is 

 going to visit, and is finding his bearings. By the in- 

 struments of Weierstrass analysis will, in due course, 

 throw light into every corner, and make absolute clear- 

 ness shine forth." l The complementary character of 



51. 

 Riemann 



1 Poincare", loc. cit., p. 7. Simi- 

 larly Prof. Klein (loc. cit., 'Vienna 

 Report,' p. 60): "The founder 

 of the theory [viz., of functions] 

 is the great French mathema- 

 tician Cauchy, but only in Ger- 

 many has it received that mod- 

 ern stamp through which it has, 

 o to speak, been pushed into the 

 centre of our mathematical con- 

 victions. This is the result of the 

 simultaneous exertions of two 

 workers Riemann on the one side 

 and Weierstrass on the other. 

 Although directed to the same end, 

 the methods of these two mathe- 

 maticians are in detail as different 

 as possible : they almost seem to 

 contradict each other, which contra- 

 diction, viewed from a higher aspect, 

 naturally leads to this that they 

 mutually supplement each other. 

 Weierstrass defines the functions 



of a complex variable analytically 

 by a common formula viz., the 

 ' Infinite Power Series ' ; in the 

 sequel he avoids geometrical means 

 as much as possible, and sees his 

 specific aim in the rigour of 

 proof. Riemann, on the other 

 side, begins with certain differential 

 equations. The subject then im- 

 mediately acquires a physical as- 

 pect. . . . His starting-point lies 

 in the region of mathematical 

 physics." We now know from the 

 biographical notice of Riemann, 

 attached to his collected works 

 (1st ed., p. 520), that he was 

 pressed (in 1856) by his mathe- 

 matical friends to publish a r6sum6 

 of his Researches on Abelian func- 

 tions " be it ever so crude. " The 

 reason was that Weierstrass was 

 already at work on the same sub- 

 ject. In consequence of Riemann's 



