9 o LECTURES AND ESSAYS [1869- 



generally.&quot; Hegel, namely, teaches us to regard &quot; the 

 mathematical infinite, whether as in series or as in the 

 fiction of infinitesimals,&quot; as a mere disguise by which it is 

 attempted to conceal the necessary incommensurability of 

 quantities differing in quality (pp. 118, 119). Right lines, 

 for example, are not curves, and so when we attempt to 

 rectify a curve i.e. to treat it as if it were homogeneous 

 with the straight line incommensurability arises (p. 115). 

 Hence the mathematical obscurity of the calculus, the 

 necessity for introducing infinite series to fill up the 

 practical rift between the pieces. It would probably be 

 vain to object that the mathematical doctrine of the 

 incommensurable is by no means connected with the 

 doctrine of infinitesimals in the way here supposed that 

 no mathematician ever used an infinitesimal to express or 

 elude incommensurability. But in view of the distinct 

 assertion that the rectification of a curve always gives rise 

 to infinite series or their equivalents, I will state that the 

 perimeter of the cardioid is four times the length of the 

 straight line which divides the curve symmetrically. It 

 appears, therefore, that the &quot; gist &quot; of Hegel s mathe 

 matical remarks involves a proposition mathematically 

 untrue ; and the inconsistency with fact here brought out, 

 of course vitiates the subsequent argument, that the 

 algebraic exponent of difference of quality is the occurrence 

 of squares or higher powers in the variables, and that 

 therefore the presence of powers higher than the first is 

 the essential character of such variability as invites 

 fluxional treatment. All this only shows that Hegel and 

 his expositor, instead of learning the science on which they 

 desire to theorise, have made a few superficial and in 

 accurate abstractions from half-digested facts, and have 

 covered the nakedness of their scientific induction with 

 the ample and impenetrable mantle of the Notion. 



Instead of dwelling on this point, I shall bring the whole 

 matter to a final issue by taking up the application of this 

 Hegelian notion. We have seen, in dealing with the 



