1918] on Gravitation and the Principle of Relativity 227 



The purpose of Einstein's new theory has often been misunder- 

 stood, and it has been criticized as an attempt to explain gravitation. 

 The theory does not offer any explanation of gravitation ; that lies 

 quite outside its scope, and it does not even hint at a possiVjle mech- 

 anism. It is true that we have introduced a definite hypothesis as to 

 the relation between gravitation and a distortion of space; but if 

 that explains anything, it explains not gravitation but space, i.e. the 

 scaffolding constructed from our measures. Perhaps the position 

 reached may be made clearer by another analogy. Let us picture the 

 particle which describes a world-line as hurdle-racer in a field thickly 

 strewn with hurdles. The particle in passing from point to point 

 always takes the path of least effort, crossing the fewest possible 

 hurdles ; if the hurdles are uniformly distributed, corresponding to 

 undistorted Minkowskian space, this will, of course, be a straight 

 line. If the field is now distorted by a mathematical transformation 

 such as an earthquake so that the hurdles become packed in some 

 parts and spread out in others, the path of least effort will no longer 

 be a straight line ; but it is not difficult to see that it passes over 

 precisely the same hurdles as bei^ore, only in their new positions. The 

 gravitational field due to a particle corresponds to a more funda- 

 mental rearrangement of the hurdles, as though someone had taken 

 them up and replanted them according to a -law^ which expresses the 

 law^ of gravitation. Any other particle passing through this part of 

 the field follow^s the guiding rule of least effort, and curves its path 

 if necessary so as to jump the few^est hurdles. Now, we have usually 

 been under the impression that when we measured distances by 

 physical experiments we were surveying the field, and the results 

 could be plotted on a map ; but it is now realized that we cannot do 

 that. The field itself has nothing to do with our measurements ; 

 all we do is to count hurdles. If the only cause of irregularity 

 of the hurdles were earthquakes (mathematical transformations) 

 that would not make much difference, because we could still 

 plot our counts of hurdles consistently as distances on a map ; 

 and the map would represent the original condition of the field 

 with the hurdles uniformly spaced. But the more far-reaching 

 rearrangement of hurdles by the gravitational field forces us to 

 recognize that we are dealing with counts of hurdles and not with 

 distances ; because if we plot our measures on a map they will not 

 close up. The number of hurdles in the circumference of a circle * 

 will not be tt times the number in the diameter ; and when we try to 

 draw on a map a circle whose circumference is more than tt times 

 its diameter, we get into difficulties— at least in Euclidean space. This 



* A circle would naturally be defined as a curve such that the number of 

 hurdles (counted along the path of least effort) between any point on it and 

 a fixed point called the centre is constant. To make the vague analogy more 

 definite, we may suppose that the hurdles are pivoted, and swing round auto- 

 matically to face the jumper ; he is not allowed to dodge them, i.e. to introduce 

 into his path sinuosities comparable with the lengths of the hurdles. 



