AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 503 



have rejected as misleading. The differential can be nothing else than (A + dA) 

 (B + dB) — AB. But Newton writes instead of this (A + £ dA) (B + \ dB) 

 — (B — i dA) (B — £ dB), thereby making an error in so elementary a process as 

 the multiplication of two binomials ! — But where is Hegel's justification for 

 saying that what Newton is seeking is (A + dA) (B + dB) — AB ? Newton says 

 nothing about differentials at all ; his a is, as we have seen, not the infinitely 

 small increment of A, but an arbitrary multiple of the fluxion of A, which need 



~T> _j_ 7, A 



not be infinitely small. Newton's — - — is, if you please, 



_ TX (A + ^A) (B + c/B)-AB 

 ~ Lt dA ~ > 



but even this, which is very different from what Hegel writes, is simply a 

 different, by no means a more fundamental, view of the problem than Newton's. 

 Dr Stirling tells us that Hegel's expression is what Newton's says his is, 

 " the excess of the increase by a whole dA and dB." But what Newton says is 

 only that when the sides are increased from A — J a and B — |- b, through incre- 

 ments a and b the rectangle increases by aB + bA. That this is true surely 

 cannot be denied. In fact (A + a) (B + b) — AB would have represented not 

 the velocity at value AB, but the average velocity of the rectangle during the 

 interval between values AB and (A + a) (B + b), and therefore the real velocity at 

 a point between these limits which Newton was not wanting. We know, in fact, 



that it would have been the velocity when the sides are = A + ^ and B + „ . 



Instead, therefore, of Newton rejecting a quantity on the ground of relative 

 smallness, we find that Hegel has gratuitously introduced such a quantity. 



Of course, the Hegelian will reply to all this, that our method is " rendered 

 impure by the concrete adjunct of motion." And here, of course, we can say 

 nothing, except that the fluxional calculus is essentially kinematical, and that to 

 construct it apart from motion is as likely a task as to make a geometry without 

 lines. To make bricks without straw is a light task compared with that which 

 Hegel has set himself. 



Happily unconscious of these difficulties, Hegel goes on to moralise with 

 much satisfaction upon Newton's melancholy self-deception, in palming on him- 

 self such a proof. 



After this specimen of Hegel's analytical subtilty, it is perhaps sufficient to 

 confront the assertion which immediately follows (Werke, iii. 313; Stirling, ii. 

 364), that Newton, in finding fluxions by the method of expansions, uses a process 

 analogous to his method of solving approximately numerical equations, con- 

 stantly " neglecting what is relatively unimportant," with the explicit words of 

 the De Quadratura (Introd. § 5)—" Errores quam minimi in rebus mathematicis 

 non sunt contemnendi." The terms omitted are, of course, always terms which we 



vol. xxv. part u. 6 o 



