DKVELOPMENT 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 ^ 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 wiLli examin- 

 ing and strengthening the foimdations 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," 



102) ; the latter was energetically- 

 repudiated by Sylvester in his 

 famous Address to tiie first section 

 of the British Assoc, at Exeter 

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



^ Gauss considered mathematics 

 to be " the (^ueen of the Sciences, 

 and arithmetic the (^ueen 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 (lediichtniss,' by Sar- 

 torius von Waltershausen, Leipzig, 

 1856, p. 79). Cayley's presidential 

 Address to the Britisli Association, 

 1883, has been frequently quoted : 

 "Mathematics connect themselves 

 on one side with common life and 

 the physical sciences ; on the otiier 

 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 liere that 

 the connection (if it exists) of 

 arithmetic and algebra with tlie 

 notion of time is far less obviou.s 

 than that of geometry with the 

 notion of space" ('Mathematical 

 Papers,' vol. xi. j). 130). 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 f/roinefriit 

 sitim, in which he saw large unde- 

 veloped trivcts which could not be 

 concjuered by the existing calculus " 

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



' To this mitrht be added the 

 psychological interest wliich at- 

 taches to matliematical concep- 

 tions. The late Prof. Paul Du 

 Bois - Reymond occupied liimself 



