700 



SCIENTIFIC THOUGHT. 



49. 

 Riemann. 



The peculiarity of such dependence, as exemplified in 

 the phenomena of the steady flow of heat or of electric 

 distribution, consisted in this, that if at certain points 

 or in certain regions of space the thermal or electrical 

 conditions were defined and known by actual observation, 

 then the whole distribution in other points and regions 

 was completely determined. Those boundary conditions 

 could therefore be regarded as the necessary and sufficient 

 definition of the whole existing distribution. Translated 

 into mathematical language, this means that functions 

 exist which are completely defined by boundary values 

 and singularities i.e., values at single points. Nature 

 herself had shown the way to define and calculate 

 measured relations when through their intricacy they 

 evaded the grasp of the ordinary operations of algebra. 1 

 Pliicker had already in geometry (following in the lines 

 of Xewton), when attacking the problem of the infinite 

 variety of higher curves, suggested the method of classi- 

 fying them according to their characteristic properties 

 or singularities. What had been done by geometers 

 and physicists in isolated cases with the expenditure 

 of much ingenuity and skill, Eiemann and his school 

 elevated to the rank of a general method and doctrine. 



functions acquires a great degree 

 of clearness and connectedness, 

 which is mainly gained by concep- 

 tions derived from the (physical) 

 theory of the potential, and thus 

 exhibits the intimate relationship 

 of these theories" (Bacharach, 

 ' Geschichte der Potentialtheorie,' 

 Gottingen, 1883, p. 71). 



1 On this subject see Burkhardt's 

 ' Memorial Lecture on Riemann ' 

 (Gottingen, 1892), p. 5, &c.; Bach- 

 arach (loc. tit.), p. 30, &c. The 

 latter especially with reference to 



the theorem called by Clerk-Maxwell 

 " Thomson's theorem " ('Cambridge 

 and Dublin Mathematical Journal,' 

 1848, or ' Reprint of Papers on 

 Electro - statics,' &c., p. 139) ; and 

 abroad 'Dirichlet's Principle,' after 

 Riemann (1857). Further, Brill 

 and Mother's "Bericht" ('Math. 

 Ver.,' vol. iii. p. 247) ; and lastly, 

 a very suggestive address by Prof. 

 Klein (" On Riemann's Influence on 

 Modern Mathematics") to the meet- 

 ing of the German Association in 

 Vienna in 1894 ('Report,' p. 61). 



