HEGEL AND THE CALCULUS 21 



AB of a curve to flow uniformly, in which case it may be 

 taken as the graphic representation of the independent 

 variable, i.e. of time, while the ordinate BC is of course 

 a function of the abscissa. 

 Then Newton shows that the 

 reason why the determination 

 of the tangent at C is a diffi 

 cult problem is that the ratio 

 of the ordinate BC to the 



sub-tangent VB is the graphical representation of the 

 fluxion of the ordinate. In fact, the meaning of the 

 tangent is, that it is the direction in which the curve is 

 flowing at the point C ; and ah 1 attempts to give it another 

 explanation without reference to motion simply ignore 

 the real gist of the problem, and of course end in difficulties 

 that can be escaped only by violent assumptions. It is 

 only in the straight line where the fluxion of the ordinate 

 is constant, or the tangent sinks into the curve, that the 

 conception of rate can be dispensed with. 



Before we go farther, it is proper to remark that in 

 criticising Newton, Hegel coolly ignores the whole founda 

 tion of the doctrine of fluxions as here developed. &quot; The 

 thought,&quot; says he (Werke, iii. 302 ; Stirling, ii. 354), 

 &quot; cannot be more correctly determined than Newton has 

 given it ; that is, the conceptions of movement and 

 velocity (whence fluxion) being withdrawn as burdening 

 the thought with inessential forms and interfering with 

 due abstraction &quot; i.e. because Hegel thought that the 

 calculus should be based, after the manner of Lagrange, 

 on purely analytical considerations, it never enters his 

 head that if Newton thought otherwise there might be 

 some deeper ground for this course than a want of insight 

 into his own method. On the contrary, Hegel comments 

 in the most edifying manner on the &quot; early still naive 

 period of the calculus &quot; in which &quot; mathematicians sought 

 to express, in words and propositions, results of the newly 

 invented calculus, and to present them in geometrical 



