86 MECHANICAL RECREATIONS [CH. V 



events of the nth. day in the rath year, hence in time the events 

 of any assigned day would be written, and therefore no part of 

 his biography would remain unwritten. This argument might 

 be put in the form of a demonstration that the part of a 

 magnitude may be equal to the whole of it. 



Questions, such as those given above, which are concerned 

 with the continuity of space and time involve difficulties of a 

 high order. Many of the resulting perplexities are due to the 

 assumption that the number of things in a collection of them is 

 greater than the number in a part of that collection. This 

 is axiomatic for a finite number of things, but must not be 

 assumed as being necessarily true of infinite collections. 



Angular Motion. A non-mathematician finds additional 

 difficulties in the idea of angular motion. For instance, there 

 is a well-known proposition on motion in an equiangular spiral 

 which shows that a body, moving with uniform velocity and 

 as slowly as we please, may in a finite time whirl round a fixed 

 point an infinite number of times. To a non-mathematician 

 the result seems paradoxical if not impossible. 



The demonstration is as follows. The equiangular spiral 

 is the trace of a point P, which moves along a line OP, the 

 line OP turning round a fixed point with uniform angular 

 velocity while the distance of P from decreases with the 

 time in geometrical progression. If the radius vector rotates 

 through four right angles we have one convolution of the 

 curve. All convolutions are similar, and the length of each 

 convolution is a constant fraction, say 1/rcth, that of the con- 

 volution immediately outside it. Inside any given convolu- 

 tion there are an infinite number of convolutions which get 

 smaller and smaller as we get nearer the pole. Now suppose 

 a point Q to move uniformly along the spiral from any point 

 towards the pole. If it covers the first convolution in a seconds, 

 it will cover the next in a/n seconds, the next in a/n 2 seconds, and 

 so on, and will finally reach the pole in 



(a + a/n + a/n 3 + a/n s + ) 



seconds, that is, in an/{n — 1) seconds. The velocity is uniform, 



