1873] HEGEL AND THE CALCULUS 23 



of velocity which the demonstration in the Principia 

 implies, the law of equal areas can be deduced without 

 even that apparent use of the infinitely little which, as 

 Newton himself warns his readers, is always merely 

 apparent (Thomson and Tait s Natural Philosophy, 234) . 

 In one word, Newton s proofs are always physical through 

 out, and really belong to the essence of the thing to be 

 proved ; while Hegel first shuts his eyes to the real import 

 of the fluxional method, insisting that it must be made 

 purely analytical, and then rails at Newton for using the 

 method to do work for which, if it had been purely alge 

 braic, it would not have been fit. A Hegelian calculus, 

 as we shall see, would certainly have been of little service 

 to physics ; but the doctrine of fluxions is itself a part of 

 physics, and absolutely indispensable in some form or 

 other to the right understanding of physical problems. 



We have still, however, to see how it is that Newton s 

 system comes to have anything at all to do with the 

 infinitely little which, as he himself says (Introd. ad Quad. 

 Curv. n), it is the peculiar merit of that system to 

 render unessential. The reason is simply, as we are told 

 in the scholium at the end of the first section of the 

 Principia, that he was anxious to provide for ease of 

 conception, and also to introduce all legitimate abbrevia 

 tions in his arguments. When Newton is called upon to 

 justify his method, he always refers to the simple fact that 

 a velocity definite, yet never for the shortest space of time 

 uniform, is a notion really furnished by nature, and that 

 the true measure of that velocity is to be got by allowing 

 the motion at any point to become uniform for a unit of 

 time. But if one wishes, as Hegel would say, to substitute 

 for this notion a convenient &quot; Vorstellung &quot; to assist the 

 imagination, Newton is ready, by means of the doctrine 

 of prime and ultimate ratios, to point out a way in which 

 we may avail ourselves of the method of indivisibles, 

 always remembering that this method shall have merely 

 a symbolic value, and so must be used with caution, 



