AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 505 



" Aufklarung," which has, from its intrinsic weakness, fallen as fast as it rose. In 

 details, it is true, Hegel is keen enough in detecting the unsatisfactory character 

 of Lagrange's stand-point [see, for example, a note at this very point] ; but that 

 the whole method was artificial he could not see, not for want of mental power, but 

 because, having never studied the subject, he knew nothing whatever about it — 

 had not even mastered its technicalities. Then, again, if it is true that successive 

 differential coefficients have a qualitative difference, how can that be brought out 

 except in virtue of the relations established in mathematics between quantity 

 and quality, relations which are not reached by pure analysis, but only in 

 Newton's way, i.e., by intuition ? And would not these relations be violated, 

 and all mathematics rendered absurd, if the term that is qualitatively important 

 could be quantitatively negligible ? And, last of all, let me challenge Hegel to 

 bring forward any proof on his own principles, that the third term relates to the 

 resistance of forces ; or for that matter, to show that this statement has any real 

 meaning whatever. 



But most men, I imagine, have now had enough of Hegel's criticisms — 

 criticisms which simply show that the " half hour" which he had devoted to the 

 calculus had not sufficed to give him any just idea of that great method. It is 

 certainly much to be regretted that so able a man did not study mathematics 

 thoroughly, for such a course might have proved useful to the theory of mathe- 

 matics, and could not have failed to be profitable to himself. As it is, he has 

 only given us criticisms such as we have seen, and an attempt to which we now 

 proceed to establish the calculus on a new and very inadequate basis. 



The point which we have always found Hegel urging is, that mathematical 

 functions, when they become quantitatively indefinite or " infinite," may still have 

 a real qualitative value. Passing over the fact that this is not the technical 

 sense of infinite in mathematics, we may grant that there is a kind of meaning, 

 however vague, that may be attached to the view. Thus an incommensurable is 

 infinite in the Hegelian sense, not because it can be expressed arithmetically only 

 by an infinite series, but because it is essentially not a sum of units, but, as 

 Hegel vaguely says, a "relation." For relation we should say function, and 

 then we should be able to read in Hegel's words some meaning like this. 

 Algebraic and geometrical functions are qualitatively different from mere arith- 

 metical functions. They imply an entirely different way of looking at quantity, 

 expressing, in fact, steps in time or space [or in kinematics, both in time and 



space]. So, again, the differential coefficient which takes the form ~ ceases to be 



intelligible on the mere arithmetical view, but gives us a real result of a different 

 quality, when we understand it as equivalent to a proposition about the rates of 



the vanishing quantities. But then Hegel does not seem to have seen that ~ has 



a real quantitative value, expressing accurately a definite quantity of a different 



