SPACE IN MODERN THEORY OF RELATIVITY 235 



appears to itself to degrade itself into inert-material — that is, into 

 nothing, for the qualities of inertia are negative ones. Here it is 

 obvious that our discussion is becoming very frankly metaphysical , 

 and must stop. 



APPENDIX II 

 SPACE IN THE MODERN THEORY OF RELATIVITY 



Some of the leading ideas in the theory of relativity are not at 

 all difficult to grasp, and they are of extreme importance. 



First, then, we note that the Euclidean system of geometry is 

 insufficient for speculation upon the nature of the universe. It 

 is based largely on the parallel postulate — that no more than one 

 straight line can be drawn through a point that lies outside a 

 given straight line and still be parallel to that given straight line. 

 Euclid assumed this, but could not prove it, and no one since 

 him has been more successful. We can assume the contrary — 

 that more than one straight line can be drawn parallel to a given 

 straight line. Then we can prove a number of propositions that 

 seem to be absurd, but are really quite consistent and free from 

 contradiction. The non-Euclidean geometries describe the 

 ordinary things that we see quite satisfactorily, but they are not 

 so easily worked as the classic system, so we use the latter in our 

 everyday affairs. 



Second. — ^The Euclidean geometry of three dimensions does 

 not describe events. A thing is not simply there, so to speak — 

 it always happens. A molecule of water is not really a thing — it 

 is things in motion (electrons). Therefore, we need the idea of 

 time in describing nature. We say that a thing is somewhere — 

 it is so much distant from the plane of X, so much from the plane 

 of Y, and so much from the plane of Z. Its space description 

 involves three variable numbers — X, Y, and Z — but since its 

 description also requires a statement of the time at which it 

 happens, we want a fourth variable, T. (We want the when as 

 well as the where.) 



Thus our universe must, at least, be four-dimensional. Things 

 are really and actually "specified" by four variables: X, Y, 

 Z, T. This is the space-time continuum of Minckowski, adopted 

 by Einstein. 



Third. — The four-dimensional (but still Euclidean) geometry 

 is insufficient. The latter is to be thought about as a three- 

 dimensional one of three co-ordinate planes — X, Y, Z — which are 



