THE LAW OF CURVATURE 139 



All explanations of gravitation on Newtonian lines 

 have endeavoured to show why something (which I have 

 disrespectfully called a demon) is present in the world. 

 An explanation on the lines of Einstein's theory must 

 show why something (which we call principal curvature) 

 is excluded from the world. 



In the last chapter the law of gravitation was stated 

 in the form — the ten principal coefficients of curvature 

 vanish in empty space. I shall now restate it in a slightly 

 altered form — 



The radius of spherical* curvature of every three-di- 

 mensional section of the world, cut in any direction at any 

 point of empty space, is always the same constant length. 



Besides the alteration of form there is actually a little 

 difference of substance between the two enunciations; 

 the second corresponds to a later and, it is believed, more 

 accurate formula given by Einstein a year or two after 

 his first theory. The modification is. made necessary by 

 our realisation that space is finite but unbounded (p. 

 80). The second enunciation would be exactly equiva- 

 lent to the first if for "same constant length" we read 

 "infinite length". Apart from very speculative esti- 

 mates we do not know the constant length referred to, 

 but it must certainly be greater than the distance of the 

 furthest nebula, say io 20 miles. A distinction between 

 so great a length and infinite length is unnecessary in 

 most of our arguments and investigations, but it is 

 necessary in the present chapter. 



♦Cylindrical curvature of the world has nothing to do with gravita- 

 tion, nor so far as we know with any other phenomenon. Anything 

 drawn on the surface of a cylinder can be unrolled into a flat map without 

 distortion, but the curvature introduced in the last chapter was intended 

 to account for the distortion which appears in our customary flat map; it 

 is therefore curvature of the type exemplified by a sphere, not a cylinder. 



