TIME GEOMETRY 135 



should be noticed. After measuring with a scale from 

 A to B you can turn your scale round and measure from 

 B to A, obtaining the same result. But you cannot turn 

 a clock round, i.e. make it go backwards in time. That 

 is important because it decides which two sides are less 

 than the third side. If you choose the wrong pair the 

 enunciation of the time proposition refers to an im- 

 possible kind of measurement and becomes meaningless. 



You remember the traveller (p. 39) who went off 

 to a distant star and returned absurdly young. He was 

 a clock measuring two sides of a time-triangle. He 

 recorded less time than the stay-at-home observer who 

 was a clock measuring the third side. Need I defend 

 my calling him a clock? We are all of us clocks whose 

 faces tell the passing years. This comparison was simply 

 an example of the geometrical proposition about time- 

 triangles (which in turn is a particular case of Einstein's 

 law of longest track). The result is quite explicable in 

 the ordinary mechanical way. All the particles in the 

 traveller's body increase in mass on account of his high 

 velocity according to the law already discussed and 

 verified by experiment. This renders them more slug- 

 gish, and the traveller lives more slowly according to 

 terrestrial time-reckoning. However, the fact that the 

 result is reasonable and explicable does not render it the 

 less true as a proposition of time geometry. 



Our extension of geometry to include time as well as 

 space will not be a simple addition of an extra dimension 

 to Euclidean geometry, because the time propositions, 

 though analogous, are not identical with those which 

 Euclid has given us for space alone. Actually the dif- 

 ference between time geometry and space geometry is 

 not very profound, and the mathematician easily glides 

 over it by a discrete use of the symbol V-i. We still 



