1873] STIRLING AND HEGEL 85 



to name. If Hegel had mastered this distinction, he could 

 not have made that unlucky suggestion about differentiat 

 ing Kepler s law of periodic times. Nevertheless, Dr. 

 Stirling holds that the great merit of Hegel in this depart 

 ment is just the stress he lays on the qualitative difference 

 between continuous and discrete quantity ; and he can 

 not conceal his amazement that I should have asserted 

 the opposite (p. 124). Obviously there is some mis 

 understanding here as regards the meaning of words. In 

 fact, the idea which Dr. Stirling attaches to continuity is 

 not that of quantity running through a continuous series 

 of values, but that of quantity hanging together ; of a 

 connected space, for example, as contrasted with a series 

 of unconnected lines. That is, continuity is not to his 

 mind associated with the idea of variability, but with the 

 idea of squares and cubes, in which we pass from relatively 

 discrete lines to continuous area, and from relatively 

 discrete planes to continuous solids. It is with transitions 

 of this sort that Hegel and Dr. Stirling conceive the 

 calculus to be interested. We have not yet reached a 

 point where we can ask what value pertains to this way of 

 looking at the matter. But as Dr. Stirling and his master 

 are clearly quite unconscious that they have here diverged 

 in language as well as in thought from the mathematicians, 

 it is quite plain that they have not apprehended the 

 meaning which the latter attach to continuous variation. 



As to the limit, I can be more brief. Mathematical 

 tyros almost invariably stumble at this notion, and Hegel 

 is no exception to the rule when he speaks of the calculus, 

 in contrast to the method of indivisibles, as beginning with 

 &quot; limits between which the independently determinate 

 thing (das Fursichbestimmte) lies, and at which it aims as 

 its goal&quot; (iii. 353). No more absolute misapprehension 

 is possible. A mathematical limit is not the boundary 

 within which something lies, but the actual value of a 

 quantity under certain extreme circumstances. For 

 example, the line joining the points in which two circles 



