OH. V] MECHANICAL RECREATIONS 87 



and yet in a finite time, Q will have traversed an infinite 

 number of convolutions and therefore have circled round the 

 pole an infinite number of times*. 



Simple Relative Motion. Even if the philosophical diffi- 

 culties suggested by Zeno are settled or evaded, the mere idea 

 of relative motion has been often found to present difficulties, 

 and Zeno himself failed to explain a simple phenomenon 

 involving the principle. As one of the easiest examples of 

 this kind, I may quote the common question of how many 

 trains going from B to A a passenger from A to B would 

 meet and pass on his way, assuming that the journey either 

 way takes 4£ hours and that the trains start from each end 

 every hour. The answer is 9. Or again, take two pennies, 

 face upwards on a table and edges in contact. Suppose that 

 one is fixed and that the other rolls on it without slipping, 

 making one complete revolution round it and returning to its 

 initial position. How many revolutions round its own centre 

 has the rolling coin made ? The answer is 2. 



Laws of Motion. I proceed next to make a few remarks on 

 points connected with the laws of motion. 



The first law of motion is often said to define force, but 

 it is in only a qualified sense that this is true. Probably 

 the meaning of the law is best expressed in Clifford's phrase, 

 that force is " the description of a certain kind of motion " — 

 in other words it is not an entity but merely a convenient 

 way of stating, without circumlocution, that a certain kind of 

 motion is observed. 



It is not difficult to show that any other interpretation 

 lands us in difficulties. Thus some authors use the law to 

 justify a definition that force is that which moves a body or 

 changes its motion; yet the same writers speak of a steam- 

 engine moving a train. It would seem then that, according 

 to them, a steam-engine is a force. That such statements 

 are current may be fairly reckoned among mechanical 

 paradoxes. 



* The proposition is put in thia form in J. Richard's Philosophie des 

 MatMmatiques, Paris, 1903, pp. 119—120. 



