i8 7 3] HEGEL AND THE CALCULUS 41 



may always be made less than unity by taking i&amp;lt; 



Hegel, however, asserts that -/&quot; (x +j)&amp;lt;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 abscissae of whose points of section is less than unity, 

 since for the chord through (x, y) cutting the curve again 

 at (x + i), mi = 0. 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 + i) n , has no right 

 even to form for every curve the expansions 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. &quot; 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.&quot; I present this proposition, which is entirely due 

 to Hegel, and in the development of which my share has 

 been &quot; purely mechanical,&quot; 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 accelera 

 tion, which in the form of harmonic motion is by far the 

 most common in nature, is quite ignored. 



Again, criticising the assertion that j- represents the 



u&amp;gt;t 



velocity at any point of a course, he tells us that it is 



