26 LECTURES AND ESSAYS [1869- 



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 defini 

 tions 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 conceives to be the mathe 

 matical infinite. &quot; 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&quot; (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 further, the relation itself, in 

 so far as it is a quantum, vanishes. ; The limit of a 

 quantitative relation 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 as 

 qualitative relation of quantity.&quot; This sentence must 

 mean that in the equation 



the left-hand side vanishes as quantum in the same sense 

 in which Sx and 8y vanish, or, as Hegel often puts it, ~ 



u&amp;gt;X 



is &quot; infinite,&quot; just as truly as 8y and 8x. Now, we are told 

 again and again that the &quot; infinity &quot; of the Sx and 8y 



