A.— MATHEMATICAL AND PHYSICAL SCIENCES. 25 



surface of a sphere in ordinary three-dimensional Euclidean space ; and 

 the easiest way of constructing a model of the De Sitter World is to take 

 .a pseudo-Euclidean manifold of five dimensions in which the line-element 

 is specified by the equation 



_ &« = dx 2 + df + dz* - du 2 + dv 2 



and in this manifold to consider the four-dimensional pseudosphere whose 

 equation is 



x 2 + y 2 + z 2 — a 2 + v 2 = R 2 . 



The pseudospherical world thus defined has a constant Riemannian 

 measure of curvature — 7^ 2 - 



The De Sitter World may be regarded from a slightly different stand- 

 point as having a Cayley-Klein metric, governed by an Absolute whose 

 equation in four-dimensional homogeneous co-ordinates is 



x 2 + y 2 + z 2 — u 2 + v 2 = 



where u is time. Hyperplanes which do not intersect the Absolute are 

 spatial, so spatial measurements are elliptic, i.e. the three-dimensional 

 world of space has the same kind of geometry as the surface of a sphere, 

 differing from it only in being three-dimensional instead of two-dimen- 

 sional. In such a geometry there is a natural unit of length, namely the 

 length of the complete straight line, just as on the surface of a sphere 

 there is a natural unit of length, namely the length of a complete great 

 circle. 



We are thus brought to the question of the dimensions of the universe : 

 what is the length of the complete straight line, the circuit of all space ? 

 The answer must be furnished by astrophysical observations, interpreted 

 by a proposition which belongs to the theory of De Sitter's world, namely 

 that the lines of the spectrum of a very distant star should be systematically 

 displaced ; the amount of displacement is proportional to the ratio of 

 the distance of the star from the observer to the constant radius of curva- 

 ture R of the universe. In attempting to obtain the value of R from this 

 formula we meet with many difficulties : the effect is entangled with the 

 ordinary Doppler effect due to the radial velocity of the star ; it could 

 in any case only be of appreciable magnitude with the most distant 

 objects ; and there is the most serious difference of opinion among 

 astronomers as to what the distance of these objects really is. Within 

 the last twelve months the distance of the spiral nebula M 33 Trianguli 

 has been estimated by Dr. Hubble of the Mount Wilson Observatory at 

 857,000 light-years, and by Dr. Perrine, the Director of the Cordoba 

 Observatory, at only 30,000 light-years ; and there is a similar uncertainty 

 of many thousands per cent, in regard to all other very remote objects. 

 Under these circumstances we hesitate to assign a definite length for the 

 radius of curvature of the universe ; but it is millions of light-years, though 

 probably not greater than about a hundred millions. The curvature of 

 space at any particular place due to the general curvature of the universe 

 is therefore quite small compared to the curvature which may be imposed 

 on it locally by the presence of energy. By a strong magnetic field we 

 can produce a curvature with a radius of only 100 light-years, and of 



