EINSTEIN'S LAW OF GRAVITATION 235 



a space-time point using the method of measuring space 

 and time intervals as outlined by Einstein; and, if an 

 observer on Arcturus used a similar set of axes and the 

 method of measurement which he naturally would, the 

 set of axes of the latter could be obtained from those of 

 the observer on the earth by a pure rotation (and naturally 

 a transfer of the origin). This is a beautiful geometrical 

 result. To complete my statement of the method, I must 

 add that instead of using as his fourth axis one along 

 which numerical values of time are laid off, Minkowski 

 defined his fourth coordinate as the product of time and 

 the imaginary constant, the square root of minus one. 

 This introduction of imaginary quantities might be ex- 

 pected, possibly, to introduce difficulties ; but, in reality, it 

 is the very essence of the simplicity of the geometrical 

 description just given of the rotation of the sets of axes. 

 It thus appears that different observers situated at dif- 

 ferent points in the universe would each have their own 

 set of axes, all different, yet all connected by the fact 

 that any one can be rotated so as to coincide with any 

 other. This means that there is no one direction in the 

 four dimensional space that corresponds to time for all 

 observers. Just as with reference to the earth there is no 

 direction which can be called vertical for all observers 

 living on the earth. In the sense of an absolute meaning 

 the words "up and down," "before and after," "sooner or 

 later," are entirely meaningless. 



This concept of Minkowski's may be made clearer, p&r- 

 haps, by the following process of thought. If we take a 

 section through our three dimensional space, we have a 

 plane, i.e., a two-dimensional space. Similarly, if a sec- 

 tion is made through a four-dimensional space, one of 

 three dimensions is obtained. Thus, for an observer on 

 the earth a definite section of Minkowski's four dimen- 

 sional space will give us our ordinary three-dimensional 

 one; so that this section will, as it were, break up Min- 

 kowski's space into our space and give us our ordinary 



