AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 509 



superfluous" (p. 347). One word in explanation of these. Lagrange takes an 

 abscissa (x + i), and gets 



i 2 

 fix + i) =fx + if x + g-/' (x +f) , 



and 



i 2 

 J?(x + i) = ¥x + iF'x + -^Y'{x+:f), 



or for the straight line given above, 



= fx + ifx . 



Thus the difference of the ordinates of the curve and straight line with abscissa 



i 2 

 x + i is q- fix +j). For any other straight line the difference may be written mi. 



Now, the ratio of these increments is i m , which may always be made less 



than unity by taking i < ,»,,, .x . Hegel, however, asserts that «-/"(# +j) < mi, 



whenever i is a proper fraction, which is an obvious analytical absurdity, and, 

 in fact, is equivalent to saying that it is impossible to draw a chord to a 

 curve, the difference of the abscissas of whose points of section is less than unity, 

 since for the chord through (x, y) cutting the curve again at (x + i), mi = . In 

 the face of this absurdity, it is scarcely necessary to add, that Hegel having 

 resolved to simplify matters, as we saw, by getting his derived functions from 

 the expansion of (x + if, has no right even to form for every curve the expan- 

 sions on which Lagrange's proof depends. 



I shall, in passing from the subject of geometry, merely enunciate a simple 

 deduction from Hegel's result in an intelligible form. " At any point of a curve 

 there are an infinite number of tangents, which may be got by uniting that point 

 with any other point on the curve whose abscissa is not different by a quantity 

 greater than unity." I present this proposition, which is entirely due to Hegel, 

 and in the development of which my share has been " purely mechanical," for 

 the admiration of all Hegelians whatsoever. 



Hegel's account of the application of the calculus to mechanic is much briefer, 

 and presents less interest after what Whewell has written on a connected point. 

 I cull only one or two illustrative points. For the purposes of the calculus, 

 Hegel classes motion as uniform, uniformly accelerated, and motion returning 

 into itself, alternately uniformly accelerated and retarded. Variable acceleration, 

 which in the form of harmonic motion is by far the most common in nature, is 

 quite ignored. 



US 



Again, criticising the assertion that -ji represents the velocity at any point of 



a course, he tells us that it is " schiefe Metaphysik" to speak of the velocity at 

 the end of a part of time. u This end must still be a part of time ; if it were not, 



