702 SCIENTIFIC THOUGHT. 



In the investigation of those higher functions which 

 the purely analytical methods of Abel and his followers 

 had forced upon the attention of mathematicians, the 

 methods of Riemaim proved to be eminently useful and 

 suggestive. But these novel methods themselves had 

 been imported into the pure science from the side of its 

 application in physics. The value of such ideas has 

 always been questioned by another class of thinkers who 

 aim at building up the edifice of the science by rigorous 

 logic, without making use of practical devices which could 

 only be legitimately employed when once their validity 

 had been thoroughly proved and its limits defined. The 

 merit of having done this in the whole domain of those 

 conceptions which, since the age of Descartes, Newton, 

 and Leibniz, had been introduced as it were from the 

 outside into analysis, belongs to the school of mathe- 

 maticians headed in Germany by Karl Weierstrass. 

 so. Eiemann had grown up in the traditions of the school 



Weierstrass. 



of mathematical thought which was inspired by Gauss 

 and Weber in Gottingen. Geometrical representation 

 and physical application, including the immediate evi- 

 dence of the senses, formed a large and important factor 

 in the body of arguments by which scientific discovery 

 and invention was carried on in that school; though 

 Gauss himself made logical rigour the final test of 

 maturity in all his published writings, abstaining in 

 many cases from communicating his results when they 

 had not satisfactorily passed that test in his own mind. 

 Through this self-imposed restriction he had permitted 

 important discoveries, which led to large increase of 

 mathematical knowledge, to be anticipated by others. 



