EINSTEIN'S LAW OF GRAVITATION 237 



and the world-line of the scale division, what we have 

 observed was the intersection or crossing of these lines. 

 In a similar manner any observation may be analyzed; 

 and remembering that light rays, a point on the retina 

 of the eye, etc., all have their world lines, it will be recog- 

 nized that it is a perfectly accurate statement to say that 

 every observation is the perception of the intersection of 

 world-lines. Further, since all we know of a world-line 

 is the result of observations, it is evident that we do not 

 know a world-line as a continuous series of points, but 

 simply as a series of discontinuous points, each point 

 being where the particular world-line in question is 

 crossed by another world-line. 



It is clear, moreover, that for the description of a 

 world-line we are not limited to the particular set of four 

 orthogonal axes adopted by Minkowski. We can choose 

 any set of four-dimensional axes we wish. It is further 

 evident that the mathematical expression for the coinci- 

 dence of two points is absolutely independent of our se- 

 lection of reference axes. If we change our axes, we will 

 change the coordinates of both points simultaneously, so 

 that the question of axes ceases to be of interest. But 

 our so-called laws of nature are nothing but descriptions 

 in mathematical language of our observations; we observe 

 only coincidences ; a sequence of coincidences when put 

 in mathematical terms takes a form which is independent 

 of the selection of reference axes ; therefore the mathe- 

 matical expression of our laws of nature, of every char- 

 acter, must be such that their form does not change if we 

 make a transformation of axes. This is a simple but far- 

 reaching deduction. 



There is a geometrical method of picturing the effect 

 of a change of axes of reference, i.e., of a mathematical 

 transformation. To a man in a railway coach the path of 

 a drop of water does not appear vertical, i.e., it is not 

 parallel to the edge of the window; still less so does it 

 appear vertical to a man performing maneuvers in an 



