CONCLUSION 227 



reflected on the earth and then stopped by the celestial 

 vault. 



Another no less interesting analogy to be noted is 

 the following : The so-called theorem of Pythagoras 

 is at the base of the earliest speculations of Greek 

 geometry ; it was this which gave rise to the problem 

 of incommensurables and indirectly to the dialectic 

 of Zeno. Now this dialectic is chiefly concerned with 

 the following problem ; space, according to the Greeks, 

 is an objective reality which is postulated as motionless. 

 How then is it possible to conceive the relation between 

 a moving object such as an arrow and motionless 

 space ? 



The difficulty which gave birth to the physics of 

 relativity, and which the Michelson-Morley experiment 

 has brought fully into light, is quite analogous. A 

 source of light, according as it is motionless or moving, 

 ought to behave differently in relation to the ether 

 supposed to be motionless. But as a matter of fact 

 this is not so. How is this to be explained ? Here 

 comes in the conception of a spatial-temporal interval 

 and the quadratic expression 



ds 2 == dx x 2 + dx 2 2 + dx z 2 + dxf, 



which is only a generalized form of the theorem of 

 Pythagoras. 



Without investigating the metaphysical range and 

 practical use of this fusion of space and time, the 

 important fact remains that the physics of relativity, 

 considered theoretically, is a remarkable attempt to 

 constitute a theory of axioms comparable to that of 

 Euclid. Only this attempt does not aim at establish- 

 ing the domain of a mathematics which is separated 

 from reality ; it tends to unite in one whole the geo- 

 metrical, mechanical and physical properties of the 

 universe. Evidently, as Winter points out, such a 

 science of axioms cannot pretend to create logically 



