THE PROBLEM OF INFINITY 173 



takes a quarter of a minute, and so on, the whole course 

 will take a minute. The apparent force of the argument, 

 on this interpretation, lies solely in the mistaken supposi- 

 tion that there cannot be anything beyond the whole of 

 an infinite series, which can be seen to be false by observ- 

 ing that 1 is beyond the whole of the infinite series J, J, 



J- 15 



8> 1TT> * * * 



The second of Zeno's arguments is the one concerning 

 Achilles and the tortoise, which has achieved more 

 notoriety than the others. It is paraphrased by Burnet 

 as follows : l 



" Achilles will never overtake the tortoise. He must 

 first reach the place from which the tortoise started. By 

 that time the tortoise will have got some way ahead. 

 Achilles must then make up that, and again the tortoise 

 will be ahead. He is always coming nearer, but he never 

 makes up to it." 2 



This argument is essentially the same as the previous 

 one. It shows that, if Achilles ever overtakes the tortoise, 

 it must be after an infinite number of instants have elapsed 

 since he started. This is in fact true ; but the view that 

 an infinite number of instants make up an infinitely long 

 time is not true, and therefore the conclusion that Achilles 

 will never overtake the tortoise does not follow. 



The third argument, 3 that of the arrow, is very interest- 

 ing. The text has been questioned. Burnet accepts the 

 alterations of Zeller, and paraphrases thus : 



" The arrow in flight is at rest. For, if everything is 



1 Op. cit. 



2 Aristotle's words are: "The second is the so-called Achilles. It 

 consists in this, that the slower will never be overtaken in its course by 

 the quickest, for the pursuer must always come first to the point from 

 which the pursued has just departed, so that the slower must necessarily 

 be always still more or less in advance." P/iys., vi. 9. 239B (R.P. 137). 



3 P/iys., vi. 9. 239B (R.P. 138). 



