698 SCIENTIFIC THOUGHT. 



tions of all the more advanced and of some even of the 

 elementary operations which analysts had become accus- 

 tomed to use without a previous knowledge of the range 

 of their validity. AH applications of mathematics con- 

 sist in extending the empirical knowledge which we 

 possess of a limited number or region of accessible 

 phenomena into the region of the unknown and inac- 

 cessible ; and much of the progress of pure analysis con- 

 sists in inventing definite conceptions, marked by symbols, 

 of complicated operations ; in ascertaining their proper- 

 ties as independent objects of research ; and in extending 

 their meaning beyond the limits they were originally 

 invented for, — thus opening out new and larger regions 

 of thought. 

 48. A brilliant and most suggestive example of this kind of 



The 



potential, reasoning was afforded by a novel mode of treating a large 

 class of physical problems by means of the introduction of 

 a special mathematical function, termed by George Green, 

 and later by Gauss, the " Potential " or " Potential func- 

 tion." ^ All the problems of Newtonian attraction were 

 concentrated in the study of this formula : and when the 

 experiments of Coulomb and Ampere showed the analogy 

 that existed between electric and magnetic forces on the 



"o' 



^ See vol. i. p. 231 of this work. algebraischen Functionen' (Leipzig, 



The history of the subject has been 

 written by Todhunter (' History of 

 the Theories of Attraction and the 

 Figure of the Earth,' 2 vols., 1873) 



1882, trans, by F. Hardcastle, 



Cambridge, 1893) ; Prof. Carl Neu 



mann's ' Untersuchungen tiber das 



Logarithmische uud Newtouische 



;l 



for the earlier period down to 1832. Potential' (Leipzig, 1877); Dr 



For the later period see Bacharach's ' Burkhardt's ' Memorial Lecture on 



'Abriss der Geschichte der Poten- Riemann ' (Gottingen, 1892); and 



tialtheorie,' Gottingen, 1883 ; for i jointly with Dr F'ranz Meyer, the 



the connection of the theory with ! same author's chapter on " Poten- 



Riemann's mathematical methods, tialtheorie " in the 2nd volume 



especially Prof. F. Klein's tract, i (p. 464) of the ' Encyclopildie der 



' Ueber Riemann's Theorie der Math. Wiss.,' 1900. 



