34 LTMITS AND FLUXIONS 



of quantities infinitely little. " 1 consider mathe- 

 matical quantities in this place not as consistiiig of 

 very small parts, " says Newton.^ Also " the very 

 smallest errors in mathematica! matters are not to 

 be neglected, "- and "in the method of fluxions 

 there is no necessity of introducing figures infinitely 

 small into geometry. "^ In view of these statements 

 the symbol o used in the Quadrature of Curves, a 

 ''quantitas ad modum parva,"* must be interpreted 

 as a small finite quantity. In this connection De 

 Morgan's remarks are of interest : ^ "In 1704, 

 Newton in the Quadratura Cuì'vai'um renounced 

 and abjured the infinitely small quantity ; but he 

 did it in a manner which would lead any one to 

 suppose that he had never held it. . . . And yet, 

 there is something like a recognition of some one 

 having used infinitely small quantities in Fluxions, 

 contained in the following words : volui ostendere 

 quod in Methodo Fluxionum non opus sit figuras 

 infinite parvas in Geometriam introducere : nothing 

 is wanted except an avowal that the some one vvas 

 Newton himself The want of this avowal was 

 afterwards a rock of offence. Berkeley, in the 

 Analyst, could not or would not see that Newton 

 of 1687 ^^""d Newton of 1704 were of two different 

 modes of thought. " 



We do not interpret Newton's expressions of 

 1704 as declarations that a logicai exposition of 



1 See our §§ 27, 34. 2 See our §§ 30, 39. 



» See our §§ 33, 41. " See our §§ 44, 45. 



•■* De Morgan, Philosophical Magazine, 4 S., voi. iv. 1852, 

 p. 328. 



