CH. V] MECHANICAL RECREATIONS 85 



The fallacy lies in the use of the word " never." The 

 argument shows that during the time occupied by the motion 

 described Achilles will not reach the tortoise. It does not deal 

 with what happens after that time, and in fact Achilles would 

 then overtake and pass the tortoise. Probably Zeno would 

 have stated that the argument and explanation alike rest on 

 the assumption, which he would not have admitted, that space 

 and time are infinitely divisible. 



Zeno's Paradox on Time. Zeno seems further to have 

 contended that while, to an accurate thinker, the notion of 

 the infinite divisibility of time was impossible, it was equally 

 impossible to think of a minimum measure of time. For 

 suppose, he argued, that t is the smallest conceivable interval, 

 and suppose that three horizontal lines composed of three 

 consecutive spans abc, a'b'c', a"b"c" are placed so that a, a', a" 

 are vertically over one another, as also b, b', b" and c, c', c". 

 Imagine the second line moved as a whole one span to the right 

 in the time t, and simultaneously the third line moved as a 

 whole one span to the left. Then b, a', c" will be vertically over 

 one another. And in this duration t (which by hypothesis is 

 indivisible) a' must have passed vertically over the space a"b" 

 and the space b"c". Hence the duration is divisible, contrary 

 to the hypothesis. 



The Paradox of Tristram Shandy. Mr Russell has enun- 

 ciated* a paradox somewhat similar to that of Achilles and the 

 Tortoise, save that the intervals of time considered get longer 

 and longer during the course of events. Tristram Shandy, as 

 we know, took two years writing the history of the first two days 

 of his life, and lamented that, at this rate, material would 

 accumulate faster than he could deal with it, so that he could 

 never finish the work, however long he lived. But had he 

 lived long enough, and not wearied of his task, then, even if his 

 life had continued as eventfully as it began, no part of his 

 biography would have remained unwritten. For if he wrote the 

 events of the first day in the first year, he would write the 

 * B. A. W. Russell, Principles of Mathematics, Cambridge, 1903, vol. i, p. 358. 



