SPACE 665 



there are toises divided into six feet. Let us look at the question a 

 little more closely. I assume that the straight line in Euclidean space 

 possesses any two properties, which I shall call A and B ; that in non- 

 Euclidean space it still possesses the property A, but no longer pos- 

 sesses the property B; and, finally, I assume that in both Euclidean 

 and non-Euclidean space the straight line is the only line that pos- 

 sesses the property A. If this were so, experiment would be able to 

 decide between the hypotheses of Euclid and Lobatschewsky. It would 

 be found that some concrete object, upon which we can experiment 

 for example, a pencil of rays of light possesses the property A. 

 We should conclude that it is rectilinear, and we should then 

 endeavor to find out if it does, or does not, possess the property B. 

 But it is not so. There exists no property which can, like this pro- 

 perty A, be an absolute criterion enabling us to recognize the straight 

 line, and to distinguish it from every other line. Shall we say, for 

 instance, " This property will be the following : the straight line is a 

 line such that a figure of which this line is a part can move without 

 the mutual distances of its points varying, and in such a way that all 

 the points in this straight line remain fixed " ? Now, this is a pro- 

 perty which in either Euclidean or non-Euclidean space belongs to 

 the straight line, and belongs to it alone. But how can we ascer- 

 tain by experiment if it belongs to any particular concrete object? 

 Distances must be measured, and how shall we know that any concrete 

 magnitude which I have measured with my material instrument really 

 represents the abstract distance? We have only removed the diffi- 

 culty a little farther off. In reality, the property that I have just 

 enunciated is not a property of the straight line alone; it is a pro- 

 perty of the straight line and of distance. For it to serve as an ab- 

 solute criterion, we must be able to show, not only that it does not 

 also belong to any other line than the straight line and to distance, but 

 also that it does not belong to any other line than the straight line, and 

 to any other magnitude than distance. Now, that is not true, and if 

 we are not convinced by these considerations, I challenge any one to 

 give me a concrete experiment which can be interpreted in the Eucli- 

 dean system, and which cannot be inteipreted in the system of Lobat- 

 schewsky. As I am well aware that this challenge will never be ac- 

 cepted, I may conclude that no experiment will ever be in contradic- 

 tion with Euclid's postulate: but, on the other hand, no experiment 

 will ever be in contradiction with Lobatschewsky's postulate. 



5. But it is not sufficient that the Euclidean (or non-Euclidean) 

 geometry can ever be directly contradicted by experiment. Nor could 

 it happen that it can cnly agree with experiment by a violation of 

 the principle of sufficient reason, and of that of the relativity of space. 

 Let me explain myself. Consider any material system whatever. We 

 have to consider on the one hand the " state " of the various bodies of 



