DEVELOPMENT OF MATHEMATICAL THOUGHT. 631 



ceases as soon as discussions arise which cannot benefit 

 those who use the instrument for the purposes of 

 application in mechanics, astronomy, physics, statistics, 

 and other sciences. At the other extreme we have 

 those who are animated exclusively by the love of pure 

 science. To them pure mathematics, with the theory of 

 numbers 1 at the head, is the one real and genuine 

 science, and the applications have only an interest in 

 so far as they contain or suggest problems in pure 

 mathematics. They are mainly occupied with examin- 

 ing and strengthening the foundations of mathematical 

 reasoning and purifying its methods, inventing rigorous 

 proofs, and testing the validity and range of applicability 

 of current conceptions. We may say that the former 

 are led by practical, the latter by philosophical, interests, 

 and these latter may be either logical or ontological, 2 



102) ; the latter was energetically 

 repudiated by Sylvester in his 

 famous Address to the first section 

 of the British Assoc. at Exeter 

 (1869, 'Report,' &c., p. 1, &c.) 



1 Gauss considered mathematics 

 to be " the Queen of the Sciences, 

 and arithmetic the Queen of Mathe- 

 matics. She frequently conde- 

 scends to do service for astronomy 

 and other natural sciences, but to 

 her belongs, under all circum- 

 stances, the foremost place " (see 

 ' Gauss zum Gedachtniss,' by Sar- 

 torius von Waltershausen, Leipzig, 

 1856, p. 79). Cayley's presidential 

 Address to the British Association, 

 1883, has been frequently quoted : 

 "Mathematics connect themselves 

 on one side with common life and 

 the physical sciences ; on the other 

 side with philosophy in regard to 

 our notions of space and time and 

 the questions which have arisen as 

 to the universality and necessity of 



the truths of mathematics, and the 

 foundation of our knowledge of 

 them. I would remark here that 

 the connection (if it exists) of 

 arithmetic and algebra with the 

 notion of time is far less obvious 

 than that of geometry with the 

 notion of space " (' Mathematical 

 Papers,' vol. xi. p. ISO). In addi- 

 tion to founding higher arith- 

 metic, Gauss occupied himself with 

 the foundations of geometry, and, 

 as he expected much from the 

 development of the theory of num- 

 bers, so he placed "great hopes on 

 the cultivation of the geometria 

 situs, in which he saw large unde- 

 veloped tracts which could not be 

 conquered by the existing calculus" 

 (Sartorius, loc. cit., p. 88). 



2 To this might be added the 

 psychological interest which at- 

 taches to mathematical concep- 

 tions. The late Prof. Paul Du 

 Bois - Reymond occupied himself 



