234 CONTEMPORARY SCIENCE 



the famous address of Minkowski, in 1908, on the sub- 

 ject of "Space and Time." It would be difficult to over- 

 state the importance of the concepts advanced by Minkow- 

 ski. They marked the beginning of a new period in the 

 philosophy of physics. I shall not attempt to explain his 

 ideas in detail, but shall confine myself to a few general 

 statements. His point of view and his line of develop- 

 ment of the theme are absolutely different from those of 

 Lorentz or of Einstein; but in the end he makes use of 

 , the same transformation formulae. His great contribution 

 consists in giving us a new geometrical picture of their 

 meaning. It is scarcely fair to call Minkowski's develop- 

 ment a picture ; for to us a picture can never have more 

 than three dimensions, our senses limit us ; while his pic- 

 ture calls for perception of four dimensions. It is this 

 fact that renders any even semi-popular discussion of 

 Minkowski's work so impossible. We can all see that 

 for us to describe any event a knowledge of four co- 

 ordinates is necessary, three for the space specification and 

 one for the time. A complete picture could be given then 

 by a point in four dimensions. All four coordinates are 

 necessary : we never observe an event except at a certain 

 time, and we never observe an instant of time except with 

 reference to space. Discussing the laws of electromagnetic 

 phenomena, Minkowski showed how in a space of four di- 

 mensions, by a suitable definition of axes, the mathemati- 

 cal transformation of Lorentz and Einstein could be de- 

 scribed by a rotation of the set of axes. We are all 

 accustomed to a rotation of our ordinary cartesian set of 

 axes describing the position of a point. We ordinarily 

 choose our axes at any location on the earth as follows : 

 one vertical, one east and west, one north and south. So 

 if we move from any one laboratory to another, we change 

 our axes ; they are always orthogonal, but in moving from 

 place to place there is a rotation. Similarly, Minkowski 

 showed that if we choose four orthogonal axes at any 

 point on the earth, according to his method, to represent 



