DEVELOPMENT OF MATHEMATICAL THOUGHT. 701 



It is a process of generalisation and simplification. 

 Moreover, Riemann's manner of proceeding brought 

 with it the gain that he could at once make the 

 various theorems of the doctrine of the potential useful 

 for purely mathematical purposes: the equation which 

 defined the potential in physics became the definition 

 of a function in mathematics.^ 



1 " One may define Riemann's 

 developments briefly thus : that, 

 beginning with certain differential 

 equations which the functions of 

 the complex variable satisfy, he is 

 enabled to apply the principles of 

 the potential theory. His start- 

 ing-point, accordingly, lies in the 

 province of mathematical physics " 

 (Klein, 'Vienna Report,' he. cit., 

 p. 60). By starting with physical 

 analogies Prof. Klein evades certain 

 difficulties which the purely mathe- 

 matical treatment had to encounter. 

 In the preface to his tract of the 

 year 1882, quoted above, in intro- 

 ducing his method of explaining 

 Riemann's theory, he says: "I 

 have not hesitated to make exactly 

 these physical conceptions the start- 

 ing-point of my exposition. In- 

 stead of them, Riemannn, as is well 

 known, makes use in his writ- 

 ings of Dirichlet's principle. But 

 I cdnnot doubt that he started 

 from those physical problems, and 

 only afterwards substituted Dirich- 

 let's principle in order to support 

 the physical evidence by mathe- 

 matical reasoning. Whoever under- 

 stands clearly the surroundings 

 among which Riemann worked at 

 Gottingeu, whoever follows up Rie- 

 maun's speculations as they have 

 been handed down to us, partly 

 in fragments, will, I think, share 

 my opinion." And elsewhere he 

 says : ' ' We regard as a specific 

 performance of Riemann in this 

 connection the tendency to give 

 to the theory of the potential a 

 fundamental importance for the 



whole of mathematics, and further 

 a series of geometrical construc- 

 tions or, as I would rather say, of 

 geometrical inventions" ('Vienna 

 Report,' p. 61). Klein then refers 

 to the representation on the so- 

 called " Riemann surface," which 

 is historically connected, as Rie- 

 mann himself points out, with the 

 problem which Gauss first attacked 

 in a general way viz., the repre- 

 sentation of one surface on another 

 in such a manner that the smallest 

 portions of the one surface are 

 similar to those of the other : a 

 problem which is of importance in 

 the drawing of maps, and of which 

 we possess two well-known examples 

 in the stereographic projection of 

 Ptolemy and the projection of 

 Mercator. This method of repre- 

 sentation was called by Gauss the 

 " Conformal Image or Representa- 

 tion. " His investigations on this 

 matter were suggested by the 

 Geodetic Survey of the kingdom of 

 Hanover, with which he was occu- 

 pied during the years 1818 to 1830. 

 (See Gauss, ' Werke,' vol. iv., also 

 his correspondence with Schum- 

 acher and Bessel.) A very complete 

 treatise on this aspect of Riemann's 

 inventions is that by Dr J. Holtz- 

 miiller, ' Theorie der Isogonalen 

 Verwandschaften ' (Leipzig, 1882). 

 On the historical antecedents of 

 Riemann's conception, which for 

 a long time appeared somewhat 

 strange, not to say artificial, see 

 Brill and Nother's frequently 

 quoted "Report" ('Bericht der 

 Math. Verein.,' vol. iii.), p. 256 tqq. 



