390 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Einstein. The theorems of the principle of relativity which correlate 

 space and time appeared, however, far less bizarre and unnatural 

 when Minkowski showed that they were merely theorems in a four 

 dimensional geometry. 



Suppose that a student of ordinary space, habituated to the inter- 

 pretation of geometry w ith the aid of a definite horizontal plane and 

 vertical axis, should suddenly discover that all the essential geometri- 

 cal properties of interest to him could be expressed by reference to a 

 liew plane, inclined to the horizontal, and a new axis inclined to the 

 Vertical. Whereas formerly he had attributed special significance 

 to heights on the one hand and to horizontal extension on the other, 

 he would now recognize that these were purely conventional and that 

 the fundamental properties were those such as distance and angle, 

 which remain invariant in the change to a new system of reference. 



Let us now consider a four dimensional manifold formed by ad- 

 joining to the familiar x, y, z axes of space a t axis of time. Any 

 point in this manifold will represent a definite place at a definite time. 

 Space then appears as a sort of cross section through this manifold, 

 comprising all points of a given time. For convenience we may 

 temporarily ignore one of the dimensions of space, say z, and discuss 

 the three dimensional manifold of x, y, t. This means that we will 

 consider only positions and motions in a plane. The locus in time of 

 a particle which does not change its position in space, that is, of a 

 particle at rest, will be a straight line parallel to the t axis. Uniform 

 rectilinear motion of a particle will then be represented by a straight 

 line inclined to the t axis. 



3. If we adopt the view that uniform motion is only relative, we 

 may with equal right consider the second particle at rest and the first 

 particle in motion. In this case the locus of the second particle must 

 be taken as a new time axis. What corresponding change this will 

 necessitate in our spacial system of reference will depend entirely 

 upon the kind of geometry that we are led to adopt in order to make 

 the geometrical iuA'ariants of the transformation correspond to the 

 fundamental physical invariants whose occurrence in mechanics and 

 electromagnetics has led to the principle of relativity. 



It is immediately evident that if vmiform motion is to be repre- 

 sented by straight lines, the statement that all motion is relative shows 

 that the transformation must be of such a character as to carry 

 straight lines into straight lines. In other words, the transformation 

 must be linear. Further we must assume that the origin of our space 

 and time axes is entirely arbitrary. 



