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. All 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." l 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 



1 See voL L p. 231 of this work. algebraischen Functionen' (Leipzig, 



The history of the subject has been 1882, trans, by F. Hardcastle, 



written by Todhunter (' History of Cambridge, 1893) ; Prof. Carl Xeu- 



the Theories of Attraction and the mann's ' Untersuchungen \iber das 



Figure of the Earth,' 2 vols., 1873) Logarithmische und Newtonische 



for the earlier period down to 1832. Potential ' (Leipzig, 1S77) ; 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 jointly with Dr Franz Meyer, the 



the connection of the theory with same authors chapter on " Poten- 



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



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



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



