MERITS AND DEFECTS 279 



236. After the development of the theory of 

 limits by the English mathematicians and by such 

 Continental writers as D'Alembert and Lacroix, it 

 would hardly seem necessary even for the sake of 

 brevity to reintroduce the old-time infinitesimal 

 which could be *'dropped" whenever it was very 

 small, yet stood in the way. But at ali times, and 

 particularly in the eighteenth and beginning of the 

 nineteenth centuries, there have been mathematicians 

 who cared little for the logicai foundations of their 

 science. Fascinated by the ease with which the 

 calculus enabled them to dispose of difficult prob- 

 lems in the theory of curves, ordinary mechanics, 

 and celestial motions, they felt more like poets, and 

 held the sentiments toward logie that a distinguished 

 hard entertained toward pure intellectualism when 

 he contemplated the beauties of the rainbow : 



" Triumphal arch that fiU'st the sky, 

 When storms prepare to part, 

 I ask not proud philosophy 

 To teach me what thou art." 



Defects 



237. AH the eighteenth-century expositions of 

 the foundations of the calculus — even the British — 

 are defective. Without attempting an historical 

 treatment or a logicai exposition of later develop- 

 ments, we desire to point out briefly what some of 

 these defects were. 



In the first place, the doctrine of fluxions was so 

 closely associated with geometry, to the neglect of 



