distances may perhaps be of the order a million light years. If natural phenomena are slowed down there, the vibrations of an atom are slower, and its characteristic spectral lines will appear displaced to the red. We should generally interpret this as a Doppler effect, implying that the nebula is receding. The motions in the line-of-sight of a number of nebulae have been determined, chiefly by Prof. Slipher. The data are not so ample as we should like; but there is no doubt that large receding motions greatly preponderate. This may be a genuine phenomenon in the evolution of the material universe; but it is also possible that the interpretation of spectral displacement as a receding velocity is erroneous; and the effect is really the slowing of atomic vibrations predicted by de Sitter's theory.
Prof. Einstein himself prefers a different theory of curved space-time. His world is cylindrical—curved in the three space dimensions and straight in the time dimension. Since time is no longer curved, the slowing of phenomena at great distances from the observer disappears, and with it the slight experimental support given to the theory by the observations of spiral nebulae. There is no longer a barrier of eternal rest, and a ray of light is able to go round the world.
In various ways crude estimates of the size of the world both on de Sitter's and Einstein's hypotheses have been made; and in both cases the radius is thought to be of the order 1013 times the distance of the earth from the sun. A ray of light from the sun would thus take about 1000 million years to go round the world; and after the journey the rays would converge again at the starting point, and then diverge for the next circuit. The convergent would have all the characteristics of a real sun so far as light and heat are concerned, only there would be no substantial body present. Thus corresponding to the sun we might see a series of ghosts occupying the positions where the sun was 1000, 2000, 3000, etc., million years ago, if (as seems probable) the sun has been luminous for so long.
It is rather a pleasing speculation that records of the previous states of the sidereal universe may be automatically reforming themselves on the original sites. Perhaps one or more of the many spiral nebulae are really phantoms of our own stellar system. Or it may be that only a proportion of the stars are
