AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 497 



Purely analytical considerations without any physical basis were held, Hegel 

 thinks, to furnish in this way physical laws. In support of this view, Hegel 

 triumphantly refers to "the Newtonian proof of his fundamental proposition in 

 the theory of gravitation compared with Schubert's ' Astronomy,' where it is 

 admitted that ... in the point, which is the nerve of the proof, the truth is not 

 as Newton assumes it" [!] And so upheld by the dictum of this forgotten 

 astronomer, Hegel goes on to inveigh against the mere jugglery by which 

 Newton, already knowing Kepler's results, avails himself of the " mist of the 

 infinitely little" to bring out apparent mathematical proofs of these results. One 

 does not know whether the singular perversity of this accusation against Newton's 

 moral character, or the incredible ignorance of the argument by which it is 

 supported, is most to be wondered at ; for, not only do the reasonings of the 

 " Principia" rest throughout on the experimental laws of motion on which 

 Newton's first proposition is expressly based, but the proof itself depends not 

 on the interpretation of an analytical process, but on the essentially physical or, 

 more definitely, kinematical considerations above developed. Nay, so little is it 

 the case, that the "mist of the infinitely little" is needed to give a show of 

 plausibility to Newton's process, that the whole gist of the proof lies in the one 

 conception of quantity generated at a definite though variable rate, and that thus, 

 without any change in the spirit of the proof, by simply introducing explicitly a 

 theorem about moments 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 throughout, 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 mus tbe made purely analytical, and then 

 rails at Newton for using the method to do work for which, if it had been purely 

 algebraic, 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. § 11), 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 abbreviations 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 



