22 SECTIONAL ADDRESSES. 



a highly complex assumption which could scarcely be used in an axiomatic 

 treatment of the subject until it has been dissected into a considerable 

 number of elementary axioms, some of them perhaps of a disputable 

 character. 



It seems to me that we should abandon measuring-rods and accurate 

 clocks altogether, and begin with something more primitive. Let us then 

 take any system of reference for events — a network of points to each of 

 which three numbers are assigned — which can serve as spatial co-ordinates, 

 and a number indicating the succession of events at each point to serve 

 as a temporal co-ordinate. Let us now refer to this co-ordinate system, 

 the paths which are traced by infinitesimal particles moving freely in 

 the gravitational field. Then it is one of the fundamental assumptions 

 of the theory that these paths are the geodesies belonging to a certain 

 quadratic differential form 



A 9 m dx v dx a- 



The truth or falsity of this assumption may, in theory at any rate, be 

 tested by observation, since if the paths are geodesies they must satisfy 

 certain purely geometrical conditions, and whether they do or not is a 

 question to be settled by experience. 



Granting for the present that the paths do satisfy these conditions, 

 let us inquire if a knowledge of the paths or geodesies is sufficient to 

 enable us to determine the quadratic form. The answer to this is in the 

 negative, as may easily be seen if we consider for, a moment the non- 

 Euclidean geometry defined by a Cayley-Klein metric in three-dimensional 

 space. In the Cayley-Klein geometry the geodesies are the straight lines 

 of the space ; but a knowledge of this fact is not sufficient to determine 

 the metric, since the Absolute may be any arbitrary quadric surface. 



In order to determine the quadratic form in general relativity we must 

 then be furnished with some information besides the knowledge of the 

 paths of material particles. It is sufficient, as Levi-Civita has remarked, 

 that we should be given the nidi geodesies, i.e. the geodesies along which 

 the quadratic form vanishes. In the Cayley-Klein geometry these are 

 the tangents to the Absolute ; in general relativity they are simply the 

 tracks of rays of light. 



So from our knowledge of the paths of material particles and the 

 tracks of rays of light we can construct the quadratic form 



i g pq dx p dx 



a 



and then we are ready for the next great axiom, namely Einstein's 

 Principle of Covariance, that ' the laws of nature must be represented 

 by equations which are covariantive for the quadratic form 



IhQ 



<9iin dx i> dx a 



with respect to all point-transformations of co-ordinates.' 



The theory is now fairly launched and I need not describe its axiomatic 

 development further. The point I wish specially to make is that in the 



