1873] HEGEL AND THE CALCULUS 39 



ently the ratio of two straight lines, or a ratio involving 

 only first powers of x and y. Or, again, since the value 

 of the radius of curvature is also on Hegel s principles 

 linear, why does it involve the second derived function ? 

 Let us, however, follow our philosopher further. &quot;Suppose 

 we have zax - x* = 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 subtangent.&quot; This 

 is very plausible, no doubt ; but let us try a cubic equa 

 tion, say zax - x 2 =y*. Now the resulting ratio, to put it 

 in Hegel s way, is 2(0, -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 



a-x 



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 says, 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- xf x + 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 = p. 

 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 

 / #=F *. 



In other respects, however, we have great improve- 



