4oo POPULAR SCIENCE MONTHLY 



worthy? It is because certain noteworthy natural bodies, the solid 

 bodies, undergo motions almost similar. 



And then when we ask: Can one imagine non-Euclidean space? 

 that means : Can we imagine a world where there would be noteworthy 

 natural objects affecting almost the form of non-Euclidean straights, 

 and noteworthy natural bodies frequently undergoing motions almost 

 similar to the non-Euclidean motions ? I have shown in ' Science and 

 Hypothesis ' that to this question we must answer yes. 



It has often been observed that if all the bodies in the universe 

 were dilated simultaneously and .in the same proportion, we should 

 have no means of perceiving it, since all our measuring instruments 

 would grow at the same time as the objects themselves which they serve 

 to measure. The world, after this dilatation, would continue on its 

 course without anything apprising us of so considerable an event. In 

 other words, two worlds similar to one another (understanding the 

 word similitude in the sense of Euclid, Book VI.) would be absolutely 

 indistinguishable. But more; worlds will be indistinguishable not 

 only if they are equal or similar, that is, if we can pass from one to the 

 other by changing the axes of coordinates, or by changing the scale to 

 which lengths are referred; but they will still be indistinguishable if 

 we can pass from one to the other by any ' point-transformation ' what- 

 ever. I will explain my meaning. I suppose that to each point of 

 one corresponds one point of the other and only one, and inversely; 

 and besides that the coordinates of a point are continuous functions, 

 otherwise altogether arbitrary, of the corresponding point. I suppose 

 besides that to each object of the first world corresponds in the second 

 an object of the same nature placed precisely at the corresponding 

 point. I suppose finally that this correspondence fulfilled at the ini- 

 tial instant is maintained indefinitely. We should have no means of 

 distinguishing these two worlds one from the other. The relativity of 

 space is not ordinarily understood in so broad a sense; it is thus, how- 

 ever, that it would be proper to understand it. 



If one of these universes is our Euclidean world, what its inhabit- 

 ants will call straight will be our Euclidean straight; but what the 

 inhabitants of the second world will call straight will be a curve which 

 will have the same properties in relation to the world they inhabit and 

 in relation to the motions that they will call motions without deforma- 

 tion. Their geometry will, therefore, be Euclidean geometry, but their 

 straight will not be our Euclidean straight. It will be its transform 

 by the point-transformation which carries over from our world to theirs. 

 The straights of these men will not be our straights, but they will have 

 among themselves the same relations as our straights to one another. 

 It is in this sense I say their geometry will be ours. If then we wish 



