40 LECTURES AND ESSAYS fo 



ments on Lagrange. It is absurd to write q = a -f bp l as 

 the equation of the line to be compared with the tangent, 

 q = pb being quite general. That the line q = bp would not 

 necessarily pass through the given point of the curve at 

 all is, of course, a trifling consideration ! 



A still greater improvement regards the process by 

 which Lagrange shows that we can always find a point 

 (with abscissa x + i), at which q =fx - xfx + pf x shall be 

 nearer the curve than any other assigned straight line. 

 At this point Hegel begins to dread (not unjustly) that 

 the conception of limit, or rather &quot; das beriichtigte 

 Increment/ is to be employed. However &quot; this appar 

 ently only relative smallness contains absolutely nothing 

 empirical, i.e. dependent on the quantum as such ; it is 

 qualitatively determined through the nature of the 

 formula, when the difference of the moment on which the 

 magnitude to be compared depends, is a difference of 

 powers. Since this difference depends on i and i 2 , and 

 i, as a proper fraction, is necessarily greater than i 2 , it is 

 really not in place to say anything about taking i of any 

 size we please, and any such statement is quite super 

 fluous &quot; (p. 347). One word in explanation of these. 

 Lagrange takes an abscissa (x + i) , and gets 



and 



or for the straight line given above, 



=fx + ifx. 

 Thus the difference of the ordinates of the curve and 



straight line with abscissa x + i is -f&quot;(x+j)&amp;gt; For any 

 other straight line the difference may be written mi. 



Now, the ratio of these increments is i* ^ x+ ^ which 



2m 



1 Hegel uses p = aq + b, but I keep Lagrange s own letters throughout. 



