DEFINITIONS I35 



rolls is rigorously plane. Should this not be the case, if there 

 are hollows and bumps, ascents and descents, we cannot 

 speak of equal distances as we do not know the value of the 

 angles of the route in the plane of observation. A yardstick 

 appears to be only 3-6 inches long if looked at from a certain 

 angle. In this case, if an ant takes two minutes to go from 

 one end to another of this yardstick we will be under the 

 impression that it has taken two minutes to cover 3-6 inches, 

 that is at a rate ten times slower than the reality. Let us take 

 another illustration. An observer on the moon seeing a scenic 

 railway on earth as a simple railroad could not explain the 

 variations of speed observed even if the cars were propelled 

 by a motor imparting a uniform velocity to them. He can 

 only see the projection of the undulated track on the ground, 

 the trace of his plane of vision. In order to detect the undula- 

 tions he would have to look sidewise, namely to move in the 

 third dimension. One can conceive that it would be easy to 

 give this observer the impression that the speed is uniform 

 by imparting to the wagon, by means of a motor, such speeds 

 as to make up for the apparent slowing up due to the descents 

 and ascents. The three dimensions are thus necessary to 

 allow the measurement of time just as time is necessary for 

 the perception of dimensions; that is to say, of matter. 



I trust that the notion of the 'four-dimensional continuum' 

 is now clear in the mind of the reader. In all the formulae 

 of Relativity the fourth dimension is not expressed by the 

 symbol of time, r, but by another, proportional to it, obtained 

 by multiplying r, for practical reasons which I cannot explain 



here, by the square root of -i (ctV-i); that is by the symbol 



of the 'imaginary' numbers.^ 



If the idea of the four-dimensional continuum, our universe, 



^ The symbol t introduced by Minkowski does not express time in 

 current units, that is to say, in seconds, but in units represented by 

 the reciprocal of the velocity of light. These mathematical expedients, 

 by which it was possible to materialize Minkowski's conception, have 

 given it all its value. We are reminded by Houllevigue (Revue de 

 Paris, April 15th 1935) that Pfltiger when mentioning this discovery 

 said, 'One must be a mathematician to appreciate fully the aesthetic 

 joy given by this conception.' 



