AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 493 



difficulty lies "in the concrete side," in the deduction from these generalities of 

 the practical rules of the method. To this subject Hegel devotes his second 

 note, professing to point out a purely analytical method whereby, without any 

 application of the doctrine of limits, everything necessary for practice can be 

 deduced. If we can demonstrate that the analytical method is radically unsound, 

 producing results mathematically false, it will surely be vain to appeal in defence 

 to any " deficiency in the judgment of a pure mathematician." 



The plan that suggests itself is therefore the following : — First, to consider 

 the real character of Newton's method, and to show what may, I think, be made 

 quite clear to an unprejudiced mind, that that great man really did know what 

 he was doing; and, in the second place, to show that Hegel having refused to be 

 instructed by Newton's real knowledge, but having acutely enough caught sight 

 of something like the ghost of an idea, which he could not for want of solid 

 knowledge really make his own, was first ensnared by the plausible but fallacious 

 method of Lagrange, and then, in attempting to improve that method, lost any 

 glimpse of the truth that he had before, and was swamped in hopeless absurdity. 



The ingenuity of a great deal that Hegel has said on this subject I do not wish 

 to dispute. No doubt he, 



" with as delicate a hand, 



Could twist as tough a rope of sand" 



as any man that ever lived. But the question is, after all, one of plain truth and 

 error ; and however much we may admire the chivalry with which Hegel rushes 

 into an unequal encounter with so gigantic an antagonist as Newton, it will 

 never do to 



" Coin a formal lie on't 

 To make the knight o'ercome the giant." 



We must begin, then, by examining the principles on which Newton based his 

 doctrine of Fluxions. In doing this, it is not necessary to inquire how far 

 Newton's own views varied during his life. Hegel knows Newton's method 

 from the Principia only, and a quotation from the second Lemma of the Second 

 Book ( Werke, iii. 305) shows that it was the current text of the Principia (that 

 of the second edition) which he had before him. In fact, Hegel's acquaintance 

 with Newton's writings was clearly of the most superficial character, embracing 

 apparently little if anything beyond the section on Prime and Ultimate Ratios, 

 and the Lemma just referred to. These facts make all merely bibliographical 

 inquiries superfluous in dealing with Hegel's objections. I may refer, however, 

 to a paper by Professor de Morgan, in the " Philosophical Magazine" for 1352, 

 on the " Early History of Infinitesimals in England," in which it is shown " that 

 Newton never varied in his meaning of do j" or, in other words, that Newton 

 " held to the conception of the velocity or fluxion," although he at first " used 

 the infinitely small increment" (only of the first order, however), " as a means of 



