84 THE FOURTH DIME-NSION. 



curately expressed, in a shortest line, would sometime, though per- 

 haps after having traversed a distance which to us is inconceivable, 

 ultimately have to arrive from the opposite direction at the place 

 from which it set out, just as a point which moves forever onward 

 in the same direction on the surface of a sphere must ultimately ar- 

 rive at its starting point, the distance it traverses being longer the 

 greater the radius of the sphere or the smaller its curvature. 



It will seem, at first blush, almost incredible, that the space of 

 experience possibly could have this property. But an example, 

 which is the historical analogue of this modern transformation of our 

 conceptions, will render the idea less marvellous. Let us transport 

 ourselves to the age of Homer. At that time people believed that 

 the earth was a great disc surrounded on all sides by oceans which 

 were conceived to be in all directions infinitely great. Indeed, for 

 the primitive man, who has never journeyed far from the place of 

 his birth, this is the most natural conception. But imagine now that 

 some scholar had come, and had informed the Homeric hero Ulysses 

 that if he would travel forever on the earth in the same direction he 

 would ultimately come back to the point from which he started ; 

 surely Ulysses would have gazed with as much astonishment upon 

 this scholar as we now look upon the mathematician who tells us 

 that it is possible that a point which moves forever onward in space 

 in the same direction may ultimately arrive at the place from which 

 it started. But despite the fact that Ulysses would have regarded 

 the assertion of the scholar as false because contradictory to his 

 familiar conceptions, that scholar, nevertheless, would have been 

 right ; for the earth is not a plane but a spherical surface. So also 

 the mathematician may be rigjit who bases this more recent strange 

 view on the possible fact that the space of experience may have a 

 measure of curvature which is not exactly zero but slightly greater 

 than zero. If this were really the case, the volume of the space of 

 experience, though very large, would, nevertheless, be finite ; just 

 as the real spherical surface of the earth as contrasted with the 

 Homeric plane surface is finite, having so and so many square miles. 

 When the objection is here made that a finiteness of space is totally 

 at variance with our modes of thought and conceptions, two ideas, 



