496 MR W. ROBERTSON SMITH ON HEGEL 



reference to a generation by flux, is that which has for its geometrical type 

 systems of straight lines; and thus geometers were tempted to introduce the 

 fiction of indivisibles, in order to reduce higher problems to this type. 



But these higher problems are not simply complicated cases of the rectilineal 

 type ; on the contrary, that type is produced by one of the two essentially distinct 

 elements (generated and generating quantity), which usually appear side by side, 

 ceasing to be explicitly manifest. 



Take, for example, Newton's own instance at the beginning of the " De 

 Quadratura." Suppose the abscissa 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 func- 

 tion of the abscissa. Then Newton shows that the 

 reason why the determination of the tangent at C 

 is a difficult problem, is that the ratio of the ordinate 

 BC to the sub-tangent VB is the graphical represen- 

 tation 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 all attempts to give it another explanation without reference 

 to motion simply ignore the real gist of the problem, and of course end in diffi- 

 culties 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 foundation of the doctrine of fluxions as here developed. 

 "The thought," says he {Werke, iii. 302; Stirling, ii. 354), "cannot be more 

 correctly determined than Newton has given it ; that is, the conceptions of move- 

 ment and velocity (whence fluxion) being withdrawn as burdening the thought 

 with inessential forms and interfering with due abstraction" — 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 " early still naive period of the calculus" in which 

 " mathematicians sought to express, in words and propositions, results of the 

 newly invented calculus, and to present them in geometrical delineations," 

 assigning to the " definitions and propositions so presented a real sense per se" 

 in which sense they were "applied in proof of the main positions concerned." If 

 there is any meaning at all in these statements, which are the gist of a somewhat 

 lengthy discussion (Werke, iii. 324; Stirling, ii. 375), that meaning must be 

 that Newton and others first differentiated a function, then sought a geometrical 

 construction to suit, and finally invented a physical proposition to correspond. 



