THE MATHEMATICAL SCIENCES 129 



indefinitely or else that it has a limit. Let us suppose 

 in the first place that the division be indefinite. In 

 this case a moving body cannot traverse the length 

 AB because before reaching the point B, it must 



traverse the length — and, before that, — , -^ , etc. 



2 4-8 



The dichotomous division of AB being infinite, one 

 cannot see how the displacement of the moving body 

 can be produced. There is the same difficulty if we 

 consider the relation between two objects in motion. 

 Achilles runs ten times faster than a tortoise, but 

 if he gives it a start of ten yards he will not be able 

 to overtake it. The space he would have to traverse 

 in order to do this is represented by the sum of the 

 following stages, the length of which certainly dim- 

 inishes but never becomes zero : 



10 + 1 H 1 + .... + — ■ + ... 



10 io 2 10 



Each time that Achilles traverses one of these space s 

 the tortoise traverses the following one. It may be 

 objected, it is true, that the meeting point between 

 Achilles and the tortoise can be calculated by the well- 

 known formula giving the limit of the sum of an infinite 

 number of terms of geometrical progression, of which 

 the first term is a and the common ratio r is less than 1, 



S = that is S = = 11 »- yards. 



1 — r 1 



1 



10 



But, as Zeuthen x has pointed out, the very reasons 

 appealed to by Zeno show that even in his time it 

 was known how to effect this summation. What they 

 disputed was precisely the legitimacy of the formula 



1 — r 



1 29 Zeuthen, Histoire des mathematiqucs, p. 54. 



