i8 73 ] HEGEL AND THE CALCULUS 25 



in geometry. Everything is as plainly and undeniably 

 reduced to ordinary geometrical intuition as anything in 

 Euclid, when we once bring with us the fundamental 

 kinematical ideas of velocity and acceleration. It is 



obvious, moreover, that to Newton the fraction -, as 



above explained, means simply the ratio of the rates at 

 which two quantities are flowing at the moment at which 

 they pass together through the point from which we have 

 agreed to reckon their magnitude backwards and forwards. 

 Except where such rates can be assigned possessing a 



definite ratio, Newton does not pretend to recognise - 



as a mathematical reality. 



This outline of Newton s principles is, of course, very 

 meagre. It will probably, however, suffice to enable us 

 to estimate the real value of Hegel s criticisms. 



Hegel highly approves of Newton s statements of what 

 he means by prime and ultimate ratios, viz. that he 

 always deals not with indivisible, but with vanishing 

 divisibles. This is very satisfactory so far, but the next 

 paragraph makes one doubt whether Hegel knew what he 

 was approving. 



&quot; Newton,&quot; we are told, &quot; only explained what he 

 means by his terms, without showing that such a notion 

 has internal truth.&quot; l 



This is an accusation constantly recurring in various 

 forms. Its source is, of course, that determination which 

 we have already noticed in Hegel to pay no regard to 

 considerations of velocity and motion. Now it is quite 

 true that Newton does not condescend to offer any 

 explanation of his &quot; notion &quot; to the man who has failed 

 to familiarise himself by actual intuition with the nature 

 of velocity, and acceleration, and the genesis of quantities 



1 Dr. Stirling (ii. 355) seems to have read &quot; Nach dem damaligen 

 Stande der wissenschaftlichen Methode wurde nun erklart.&quot; In the 

 collected edition of the Werke, iii. 303, I read &quot; wurde nur erklart,&quot; 

 which seems to give a more intelligible sense. 



