IS EUCLID'S GEOMETRY MERELY A THEORY? 563 



Going to and fro along the track, the little man would judge that 

 the rails are not equidistant at all. Applying his shrinkable foot-rule, 

 he would decide that the space between the rails varied in width at dif- 

 ferent places. He would come to the same conclusion that we have set 

 down, namely, that parallels may at first approach but, following them 

 further, they diverge more and more. All this he would discover and 

 never suspect that his own variable dimensions were the cause of a 

 deception, for would not his surroundings shrink always in proportion 

 to himself ? Beltrami's illustration thus attains its purpose by making 

 solid objects expand and contract in place of allowing space itself to 

 grow any more roomy than our Euclidean notions permit. 



If Euclid were to return to earth to-day, he would find many geom- 

 etries, but the three here described would probably interest him above all 

 others. A fitting task for Euclid would be to coordinate this trilogy of 

 systems. With the three volumes spread open before him, he could 

 write a dictionary by the aid of which a student could translate any 

 proposition stated in one volume into the corresponding proposition 

 given in either of the other two. It is at bottom a matter of words, or 

 at least the facts lend themselves to that interpretation. "With intense 

 satisfaction Euclid could still contemplate his own geometry. Where 

 long and involved phrases are necessary to convey the idea presented in 

 the other systems, his own ideas are always lucid and tersely expressible. 

 His system is consequently by far the most convenient, so much more 

 convenient, that if we should ever discover any discrepancy between it 

 and the facts of the physical universe, we would probably prefer to 

 change our laws of physics or mechanics rather than to adjust ourselves 

 to a less convenient system of geometry. 



