AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 499 



possessing a definite ratio, Newton does not pretend to recognise ^ as a mathe- 

 matical 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 statement 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 ap- 

 proving. 



" Newton," we are told, " only explained what he means by his terms, with- 

 out showing that such a notion has internal truth."* 



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 "notion" to the 

 man who has failed to familiarise himself by actual intuition with the nature of 

 velocity, and acceleration, and the genesis of quantities by flux. But these 

 notions are just as truly capable of being constructed by pure intuition as those 

 of ordinary geometry, and so Newton's definitions enjoy fully the advantage 

 which Kant ascribes to mathematical definitions in general. They cannot err, 

 because they simply unfold a construction by means of which the notion is 

 actually produced. 



If Hegel, however, shut his eyes to Newton's notion, he has got one of his 

 own, which he is sure is just what Newton wanted. I do not intend to attempt 

 to take up anything but the concrete applications of this notion ; but perhaps it 

 may be well to give here part of Hegel's abstract statement of what he con- 

 ceives to be the mathematical infinite. " Das unendliche Quantum . . . ist nicht 

 mehr irgend ein endliches Quantum, nicht eine Grossebestimmtheit, die ein 

 Daseyn als Quantum hatte sondern es ist einfach, und daher nur als Moment ; es 

 ist eine Grossebestimmtheit in qualitativer Form ; seine Unendlichkeit ist als eine 

 qualitative Bestimmtheit zu seyn" (iii. 289; Stirling, ii. 341). Now, says Hegel, 

 this is clearly what Newton needs. His vanishing magnitudes have ceased to 

 exist as quanta, and exist only as sides of a relation ; but farther, the relation 

 itself, in so far as it is a quantum, vanishes. " The limit of a quantitative rela- 

 tion is that in which it both is and is not, or, more accurately, that in 

 which the quantum has disappeared, and there remains the relation only 



* Dr Stirling (ii. 355) seems to have read " Nacli dem damaligen Stande der wissenschaft- 

 lichen Methode wurde nun erklart." In the collected edition of the " Werke," iii. 303, I read 

 " wurde nur erklart," which seems to give a more intelligible sense. 



VOL. XXV. PART II. 6 N 



