DEVELOPMENT OF MATHEMATICAL THOUGHT. 739 



also such apparently algebraical notions as those of 

 irrational and complex quantities. This attempt is an 

 outcome of the school of Weierstrass, which has done 

 so much to banish vagueness and introduce precision 

 into modern text-books. 



Opposed to this so-called arithmetising l tendency is 

 the equally emphatic view, strongly urged by the late 

 Prof. Paul Du Bois-Keymond in his general theory of 

 Functions, that the separation of the operations of 

 counting and measuring is impossible, and, if it were 

 possible (as, since the publication of his work, the fuller 

 expositions of Kronecker and his followers have tried to 

 show that it is), would degrade mathematics to a mere 

 play with symbols. 2 He tries to show that such is philo- 

 sophically impossible, and finds a support for his view in 

 the historical genesis of the idea of irrational numbers in 

 the incommensurable magnitudes of Euclid and ancient 

 geometry. Prof. Klein in his address favours the 

 arithmetical tendency as destined to introduce logical 



be attached to the signs as if they 

 were the figures on the chessboard 

 or on playing-cards. However amus- 

 ing such a play might be, nay, 

 however useful for analytical pur- 

 poses the solution would be of the 

 problem, to follow up the rules of 

 the signs which emanated from the 

 conception qf magnitude into their 

 last formal consequences, such a 

 literal mathematics would soon 

 exhaust itself in fruitless efforts ; 

 whereas the science which Gauss 

 called with so much truth the 

 science of magnitude possesses an 

 inexhaustible source of new ma- 

 terial in the ever-increasing field 

 of actual perceptions," &c., &c. 

 (' Allgemeine Functionen-Theorie,' 

 1882, p. 54.) 



1 The term seems to have been 

 coined by Kronecker. See Prof. 

 Pringsheim in the ' Encyklop. 

 Math. Wiss.,' vol. i. p. 58, note 40. 

 Kronecker's position is set forth 

 in Journal fur Math., vol. ci. pp. 

 337-355, 1887. 



2 " The separation of the con- 

 ception of number and of the 

 analytical symbols from the con- 

 ception of magnitude would reduce 

 analysis to a mere formal and 

 literal skeleton. It would degrade 

 this science, which in truth is a 

 natural science, although it only 

 admits the most general properties 

 of what we perceive into the domain 

 of its researches ultimately to the 

 rank of a mere play with symbols, 

 wherein arbitrary meanings would 



