84 MECHANICAL RECREATIONS [CH. V 



Motion. The difficulties connected with the idea of motion 

 have been for a long time a favourite subject for paradoxes, 

 some of which bring us into the realm of the philosophy of 

 mathematics. 



Zeno's Paradoxes on Motion. One of the earliest of these 

 is the remark of Zeno to the effect that since an arrow cannot 

 move where it is not, and since also it cannot move where it is 

 (that is, in the space it exactly fills), it follows that it cannot 

 move at all. . This is sometimes presented in the form that at 

 each instant a flying arrow occupies a fixed position ; but 

 occupying a fixed position at a given instant means that it is 

 then at rest. Hence the arrow is at rest at every instant of its 

 flight, and therefore is not in motion. The usual answer is that 

 the very idea of the motion of the arrow implies the passage 

 from where it is to where it is not. 



Zeno also asserted that the idea of motion was itself incon- 

 ceivable, for what moves must reach the middle of its course 

 before it reaches the end. Hence the assumption of motion 

 presupposes another motion, and that in turn another, and so 

 ad infinitum. His objection was in fact analogous to the 

 biological difficulty expressed by Swift: — 



" So naturalists observe, a f ea hath smaller fleas that on him prey. 

 And these have smaller fleas to bite 'em. And so proceed ad infinitum." 



Or as De Morgan preferred to put it 



"Great fleas have little fleas upon their backs to bite 'em, 

 And little fleas have lesser fleas, and so ad infinitum. 

 And the great fleas themselves, in turn, have greater fleas to go on; 

 While these have greater still, and greater still, and so on." 



Achilles and the Tortoise. Zeno's paradox about Achilles 

 and the tortoise is known even more widely. The assertion 

 was that if Achilles ran ten times as fast as a tortoise, yet 

 if the tortoise had (say) 1000 yards start it could never be 

 overtaken. To establish this, Zeno argued that when Achilles 

 had gone the 1000 yards, the tortoise would still be 100 yards 

 in front of him ; by the time he had covered these 100 yards, 

 it would still be 10 yards in front of him ; and so on for ever. 

 Thus Achilles would get nearer and nearer to the tortoise 

 but would never overtake it. Zeno regarded this as confirming 

 his view that the popular idea of motion is self-contradictory. 



