236 CONTEMPORARY SCIENCE 



time. Similarly, a different section would have to be used 

 for the observer on Arcturus; but by a suitable selection 

 he would get his own familiar three-dimensional space 

 and his own time. Thus the space defined by Minkowski 

 is completely isotrophic in reference to measured lengths 

 and times, there is absolutely no difference between any 

 two directions in an absolute sense ; for any particular 

 observer, of course, a particular section will cause the 

 space to fall apart so as to suit his habits of measurement ; 

 any section, however, taken at random will do the same 

 thing for some observer somewhere. From another point 

 of view, that of Lorentz and Einstein, it is obvious that, 

 since this four dimensional space is isotropic, the expres- 

 sion of the laws of electromagnetic phenomena take iden- 

 tical mathematical forms when expressed by any observer. 

 The question, of course, must be raised as to what can 

 be said in regard to phenomena which so far as we know 

 do not have an electromagnetic origin. In particular what 

 can be done with respect to gravitational phenomena ? Be- 

 fore, however, showing how this problem was attacked 

 by Einstein ; and the fact that the subject of my address 

 is Einstein's work on gravitation shows that ultimately I 

 shall explain this, I must emphasize another feature of 

 Minkowski's geometry. To describe the space-time char- 

 acteristics of any event a point, defined by its four co- 

 ordinates, is sufficient; so, if one observes the life-history 

 of any entity, e.g., a particle of matter, a light-wave, etc., 

 he observes a sequence of points in the space- time conti- 

 nuum; that is, the life-history of any entity is described 

 fully by a line in this space. Such a line was called by 

 Minkowski a "world-line." Further, from a different 

 point of view, all of our observations of nature are in 

 reality observations of coincidences, e.g., if one reads a 

 thermometer, what he does is to note the coincidence of 

 the end of the column of mercury with a certain scale 

 division on the thermometer tube. In other words, think- 

 ing of the world-line of the end of the mercury column 



