THE SCIENTIFIC SPIRIT IN GERMANY. 221 



of thought has shown itself in other fields of research, 

 and led to similar innovations. I will here only mention 

 one other line of inquiry, where neither exact nor meta- 

 physical reasoning alone suffices, but where a combination 

 of both is essential. I mean the gradual change which, 41. 



Criticism of 



mainly through the writings of German mathematicians, principles 



' ofmathe- 



has come over our fundamental conceptions in the region inafcics - 

 of geometry, algebra, and the theory of numbers. This 

 subject belongs so essentially to the domain of pure 

 thought that a history of thought seems specially called 

 upon to take notice of it. Accordingly I intend to devote 

 a special chapter to it. At present it interests us mainly 

 because it is an outcome of that peculiar modification 

 which the exact or scientific spirit of thought underwent 

 when, introduced by French and English models, it came 

 in contact with the philosophical and classical ideal of 

 learning in Germany. I will repeat more clearly and 

 concisely what I mean. The exact methods of thought, 

 mainly elaborated in France, and there largely applied, 

 give to science its accuracy and definiteness. In spite 

 of this accuracy and definiteness, it is not immediately 

 clear whether they will lead to completeness of know- 

 ledge, or whether they may not be misapplied. To 

 guarantee completeness, to make sure that in the whole 

 great field no portion has remained untouched and un- 

 explored, that love of detail, that searching and explor- 

 ing spirit, is required which is nursed pre-eminently 

 by historical and classical studies. And to avoid the 

 abuse of existing methods, there is further required that 

 critical spirit which inquires into the value of principles 



