32 LECTURES AND ESSAYS [1869- 



(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, 



2 2 



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 &quot; rendered impure by the concrete adjunct 

 of motion.&quot; 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 himself such a 

 proof. 



After this specimen of Hegel s analytical subtilty, it 

 is perhaps sufficient to confront the assertion which im 

 mediately 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 approxi 

 mately numerical equations, constantly &quot; neglecting what 

 is relatively unimportant,&quot; with the explicit words of the 

 De Quadratura (Introd. 5) &quot; Errores quam minimi in 

 rebus mathematicis non sunt contemnendi.&quot; The terms 

 omitted are, of course, always terms which we know to 

 become not relatively but absolutely zero in proceeding 

 to the limit. The motive for using such expressions as 

 &quot; minuatur quantitas o in infinitum,&quot; instead of simply 

 saying, let o = zero, is merely to show that o becomes zero 

 not by a discontinuous process, as subtraction, but by a 

 continuous flow. Nay, cries Hegel, for in the 3rd Problem 

 of Book ii. of the Principia, Newton fell into an error, by 



