500 MR W. ROBERTSON SMITH ON HEGEL 



as qualitative relation of quantity." This sentence must mean that in the 

 equation 



LtJM 



dx X 



the left hand side vanishes as quantum in the same sense in which dx and dy 



vanish, or, as Hegel often puts it, -/ is "infinite," just as truly as dy and dx. 



Now, we are told again and again that the " infinity" of the Sx and dy 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 admitting that -. 



is also indeterminate ; for both numerator 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 



q, 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 "a qualitative relation 

 of quantity," 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 



" expresses a certain value which lies in the function of variable magnitude." In 

 asking this question, he still supposes himself to be criticising Newton and the 

 mathematicians, and accordingly proceeds, with much severity of manner, to 



knock down the indeterminate j x 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 



to = P + 1 to + , &c, 



and then letting 8x and dy vanish Lt£=p ;— not as it should have been = «. 



This, of course, is sadly inconsistent ; for instead of our fine qualitative deter- 

 mination, here is a stubborn quantum turning up. Now, says Hegel, the 



