DEVELOPMENT OF MATHEMATICAL THOUGHT. 699 



one side, and Newtonian forces on the other ; still more 

 when Fourier, Lame, and Thomson (Lord Kelvin) pointed 

 to the further analogy which existed between the distri- 

 bution of temperature in the stationary flow of heat 

 and that of statical electricity on a conductor, and ex- 

 tended the analogy to hydrostatics and hydrodynamics, 

 it became evident that nature herself pointed here 

 to a mathematical dependence of the highest interest 

 and value. Many eminent thinkers devoted themselves 

 to the study of this subject, but it was reserved for 

 Bernhard Eiemann to generalise the mode of reasoning 

 peculiar to these researches into a fundamentally novel 

 method for the explanation and definition of mathe- 

 matical function or dependence. 1 



1 Although Riemann's original 

 method of dealing in a general way 

 with algebraical functions is here 

 introduced as a generalisation of 

 certain ideas suggested by mathe- 

 matical physics, it was not in this 

 way that they were introduced to 

 the mathematical world. This was 

 done in his very abstract and difficult 

 memoir, ' Theorie der Abel'schen 

 Functionen ' (published in 1857 

 in vol. liv. of Crelle's 'Journal'). 

 In .this memoir the connection 

 which existed with mathematical 

 physics was not patent, and it 

 took a long time before his 

 methods, which seemed to be a 

 development of Cauchy's earlier 

 researches, were understood and 

 fully appreciated. It was only 

 after he had lectured repeatedly on 

 the subject, and initiated a num- 

 ber of younger mathematicians, 

 who now occupy many of the chairs 

 at the German universities, that 

 the discoveries and inventions of 

 Riemann received their deserved 

 appreciation. Even in his own 

 lectures on mathematical physics 



notably on partial differential 

 equations (including harmonics) 

 and the theory of the potential 

 he did not lead up to the funda- 

 mental ideas which he developed 

 in his lectures on the theory of 

 the Abelian functions. Some light 

 is thrown on the subject of the 

 genesis of Kiemaun's ideas by his 

 dissertation written in the year 

 1851, though even the biographical 

 notice attached to the 1st edition 

 of his works (1876) did not deal 

 with the origins of his theory. 

 It seems, therefore, correct to- 

 date the adequate recognition of 

 Riemann's work inlwider circles from 

 the publication in 1882 of Prof. 

 F. Klein's tract mentioned above. 

 Like several other short treatises 

 of this eminent living mathema- 

 tician, it must have thrown quite 

 a new light upon the subject ; 

 and, like several of his other writ- 

 ings, it revealed connections be- 

 tween regions of thought which to 

 many students must have appeared 

 isolated. " Through the treatment 

 initiated by Klein, the theory of 



