508 MR W. ROBERTSON SMITH ON HEGEL 



it involve the second derived function ? Let us, however, follow our philo- 

 sopher further. " Suppose we have 2 ax — x 2 = y 2 , and take the derived function, 

 we get a ratio a — x : y, — a linear ratio representing the proportion of two 

 lines. The real point is to show that these two lines are the ordinate and sub- 

 tangent." This is very plausible, no doubt ; but let us try a cubic equation, say 

 2 ax — x 2 = y 3 . Now the resulting ratio, to put it in Hegel's way, is 2(a — x) : 3y 2 . 

 Is this a linear ratio ? Yet it still represents the ratio of the ordinate and sub- 

 tangent. Clearly Hegel does not know that when x and y become definite 



co-ordinates of a point on the curve the ratio — ~ ceases to be a linear function 



of variables in any proper sense, and is simply a determinate fraction. This 

 mistake augurs ill for the validity of Hegel's proof, that the two lines, whose 

 ratio is the ratio of the derived functions, are really ordinate and subtangent. 

 But he has Lagrange luckily to help him, who, he sa3^s, has entered on the truly 

 scientific way. We get, therefore, a wordy and loose description, which would be 

 utterly unintelligible to any one who did not know the thing before, of the way 

 in which Lagrange proves that the line q = fx — xfx + pfx lies nearer to the 

 curve y — fx in the neighbourhood of the point (x, y) than any other straight line 

 through that point. Hegel's confusion is not diminished by the fact, that 

 Lagrange deduces this proposition from a general theorem about the contact of 

 curves, and originally writes the straight line as q = F/>. This piece of tactics so 

 puzzles the philosopher that, after all his invective against the differentiation of 

 linear functions, he allows Lagrange, without rebuke, to write fx = Yx. 



In other respects, however, we have great improvements on Lagrange. It 

 is absurd to write q = a 4- bp* as the equation of the line to be compared with 

 the tangent, q = pb being quite general. That the line q = bp would not neces- 

 sarily 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® 

 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 " das 

 beriichtigte Increment," is to be employed. However " this apparently 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 



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



