i8 73 ] HEGEL AND THE CALCULUS 27 



does not lie in their being infinitely small, but in their 

 having ceased to be any determinate magnitude, and only 

 representing the qualitative principle of a magnitude. 

 To this statement Newton would probably not have 

 objected, as his whole use of infinitely small quantities is, 

 as we have seen, merely to help the imagination, and 

 scientific strictness is given to his method from another 

 side. But certainly he would never have dreamed of ad 

 mitting that ? is also indeterminate ; for both numerator 



sv 



and denominator of this fraction are in their nature 

 definite quantities. That the fraction can be expressed 



as - is to Newton by no means the essential point. On 



the contrary, he argues distinctly that - must have a 



definite value, just because this is the form in which 

 certain processes present to us a quantity which, from 

 kinematical grounds, we know to be definite. To Hegel, 



however, the fascinating element is just this -, which 



for his ends would be quite spoiled by being evaluated. 

 That would reduce it to a mere quantum ; but, in the 

 meantime, it is &quot;a qualitative relation of quantity,&quot; 

 which is a far finer thing. Not unnaturally, however, 

 Hegel has now to ask himself, what is to be the practical 



use of this Lt^, which certainly &quot;expresses a certain 



oX 



value which lies in the function of variable magnitude.&quot; 

 In asking this question, he still supposes himself to be 

 criticising Newton and the mathematicians, and accord 

 ingly proceeds, with much severity of manner, to knock 



down the indeterminate -2 which he has just set up 



(p. 318). To apply the conception of limit in the concrete 

 we must determine the limit. This is done by Taylor s 

 theorem, from which if y=f(x) we get 



