i 3 o SCIENCE IN GRECO-ROMAN ANTIQUITY 



since, in order to establish this, it is necessary, in 



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to neglect the term as insignificant. Up to what 



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point is it right to do this ? That is the question. 



Instead of admitting the possibility of an infinite 

 division, let us suppose that this division has a limit 

 and that there exist ultimate elements of space, time 

 and motion (whether in finite or infinite number, it 

 matters little). To this Zeno replies with the paradox 

 of the arrow. The extremities and the body of an arrow 

 in flight must coincide at each instant with the points 

 which compose its traj ectory ; but if there be a coin- 

 cidence for however brief an instant of time, there is 

 immobility. Then the movement of the arrow is re- 

 duced to a sum of instantaneous immobilities, which 

 is absurd. If we attempt to avoid this objection by 

 affirming that each instant corresponds, not to a certain 

 position of the arrow, but to the passage from each 

 position to the next, Zeno appeals to the argument 

 of bodies which moving inversely to one another cross 

 one another's paths, and he shows that the speeds 

 supposed to be different are in reality equal, since by 

 dichotomy the sum of the instants of which these 

 speeds are composed can always be reduced to the 

 same number, that is to infinity. 



The arguments of Zeno in fact amount to the proof 

 by reductio ad absurdum that a geometrical body is 

 not a sum of points, that time is not a sum of instants, 

 that motion is not the sum of passages from one point 

 to another. They had the result of establishing once 

 for all the infinite divisibility of space. Henceforward 

 the discussion relating to divisibility dealt with matter, 



