substantial bodies; the remainder are optical ghosts revisiting their old haunts. It is, however, unlikely that the light rays after their long journey would converge with the accuracy which this theory would require. The minute deflections by the various gravitational fields encountered on the way would turn them aside, and the focus would be blurred. Moreover there is a likelihood that the light would gradually be absorbed or scattered by matter diffused in space, which is encountered on the long journey.
It is sometimes suggested that the return of the light-wave to its starting point can most easily be regarded as due to the force of gravitation, there being sufficient mass distributed through the universe to control its path in a closed orbit. We should have no objection in principle to this way of looking at it; but we doubt whether it is correct in fact. It is quite possible for light to return to its starting point in a world without gravitation. We can roll flat space-time into a cylinder and join the edges; its geometry will still be Euclidean and there will be no gravitation; but a ray of light can go right round the cylinder and return to the starting point in space. Similarly in Einstein's more complex type of cylinder (three dimensions curved and one dimension linear), it seems likely that the return of the light is due as much to the connectivity of his space, as to the non-Euclidean properties which express the gravitational field.
For Einstein's cylindrical world it is necessary to postulate the existence of vast quantities of matter (not needed on de Sitter's theory) far in excess of what has been revealed by our telescopes. This additional material may either be in the form of distant stars and galaxies beyond our limits of vision, or it may be uniformly spread through space and escape notice by its low density. There is a definite relation between the average density of matter and the radius of the world; the greater the radius the smaller must be the average density.
Two objections to this theory may be urged. In the first place, absolute space and time are restored for phenomena on a cosmical scale. The ghost of a star appears at the spot where the star was a certain number of million years ago; and from the ghost to the present position of the star is a definite distance
