82 MYSTICISM AND LOGIC 



age of Greece) which has a more convincing proof to offer 

 of the transcendent genius of its great men. Of the three 

 problems, that of the infinitesimal was solved by Weier- 

 strass ; the solution of the other two was begun by 

 Dedekind, and definitively accomplished by Cantor. 



The infinitesimal played formerly a great part in 

 mathematics. It was introduced by the Greeks, who 

 regarded a circle as differing infinitesimally from a polygon 

 with a very large number of very small equal sides. It 

 gradually grew in importance, until, when Leibniz in- 

 vented the Infinitesimal Calculus, it seemed to become 

 the fundamental notion of all higher mathematics. 

 Carlyle tells, in his Frederick the Great, how Leibniz used 

 to discourse to Queen Sophia Charlotte of Prussia con- 

 cerning the infinitely little, and how she would reply that 

 on that subject she needed no instruction the behaviour 

 of courtiers had made her thoroughly familiar with it. 

 But philosophers and mathematicians who for the most 

 part had less acquaintance with courts continued to 

 discuss this topic, though without making any advance. 

 The Calculus required continuity, and continuity was 

 supposed to require the infinitely little ; but nobody 

 could discover what the infinitely little might be. It was 

 plainly not quite zero, because a sufficiently large number 

 of infinitesimals, added together, were seen to make up a 

 finite whole. But nobody could point out any fraction 

 which was not zero, and yet not finite. Thus there was a 

 deadlock. But at last Weierstrass discovered that the 

 infinitesimal was not needed at all, and that everything 

 could be accomplished without it. Thus there was no 

 longer any need to suppose that there was such a thing. 

 Nowadays, therefore, mathematicians are more dignified 

 than Leibniz : instead of talking about the infinitely 

 small, they talk about the infinitely great a subject 



