708 



SCIENTIFIC THOUGHT. 



the labours of the two great analysts is nowhere better 

 shown than in the special manner in which Weierstrass 

 succeeded in strengthening the foundations l on which 

 much of Riemann's work rests. 



The labours of the great analysts Gauss, Cauchy, 

 Eiemann, and Weierstrass all tended to increase our 



publication Weierstrass withdrew 

 from the press an extensive memoir 

 which he had presented in the year 

 1857 to the Berlin Academy, be- 

 cause, as he himself says (Weier- 

 strass, 'Math. Werke,' vol. iv. p. 10) : 

 "Riemann published a memoir on 

 the same problem which rested on 

 entirely different foundations from 

 mine, and did not immediately 

 reveal that in its results it agreed 

 completely with my own. The 

 proof of this required investigations 

 which were not quite easy, and took 

 much time ; after this difficulty 

 had been removed a radical remod- 

 elling of my dissertation seemed 

 necessary," &c., &c. The mutual 

 influence of Riemann's and Weier- 

 strass's work is also referred to by 

 Weierstrass in a letter to Prof. 

 Schwarz, dated 1875, in which 

 he utters what he calls his con- 

 fession of faith: "The more 

 I ponder over the principles of 

 the theory of functions and I 

 do this incessantly the stronger 

 grows my conviction that it must 

 be built up on the foundation of 

 algebraical truths, and that, there- 

 fore, to employ for the proof of 

 simple and fundamental algebraical 

 theorems the ' transcendental,' if I 

 may say so, is not the correct way, 

 however enticing prvma, vista the 

 considerations may be by which 

 Riemann has discovered many of 

 the most important properties of 

 algebraical functions. It is a mat- 

 ter of course that every road must 

 be open to the searcher as long as 

 he seeks ; it is only a question of 



the systematic demonstration " 

 (Weierstrass, 'Werke,' vol. ii. p. 

 235). 



1 This refers mainly to Weier- 

 strass's investigation of the principle 

 called by Riemann "Dirichlet's 

 principle," but which had been 

 stated already with great generality 

 by Thomson (Lord Kelvin) in the 

 year 1847. The validity of this 

 method depended on a certain 

 minimum theorem. Weierstras* 

 has shown that the existence of 

 such a minimum is not evident, and 

 that the argument used is not con- 

 clusive. He laid before the Berlin 

 Academy, in the year 1870, a com- 

 munication giving a test - case to- 

 prove that Dirichlet's method was 

 not generally valid ('Werke,' vol. 

 ii. p. 49). "Through this," Prof. 

 Klein says (loc. cit., p. 67), "a 

 great part of Riemann's develop- 

 ments become invalidated. Never- 

 theless the far - reaching results 

 which Riemann bases upon the 

 prinsiple are all correct, as was 

 shown later on exhaustively and 

 with all rigour by Carl Neumann 

 and H. A. Schwarz. Indeed we 

 must come to the conclusion that 

 Riemann himself arrived at these 

 theorems by a physical intuition r 

 and only afterwards resorted to the 

 principle referred to in order to 

 have a consistent mathematical line 

 of reasoning " (loc. cit., p. 67). See 

 on this also Poincare" (loc. cit., 

 pp. 10 and 15), who gives other 

 instances where the work of Weier- 

 strass supported that of Riemann. 



