A for a non-Euclidean space. Thus the surface of a sphere is an example of a two-dimensional continuum which is finite but unbounded. body, or a ray of light, moving on the surface of the sphere in any direction, would nowhere meet with a barrier to its further progress. It could move for ever without ever leaving the surface of the sphere. Nevertheless, the total area of the spherical surface the total amount of “ space "-is finite. Similarly, it is mathematically possible for a three or four-dimensional continuum to possess such metrical properties that it nowhere exhibits a boundary and yet has a total volume which is finite. It is of little advantage for us to try to picture such a continuum. We must be content with the logical consequences of certain premises. If the laws governing the behaviour of our measuring appliances-which may be freely moving bodies and rays of light— are of a certain kind, then we can deduce, mathematically, that the continuum having those metrical properties is finite and unbounded. A crude estimate of the size of the finite universe propounded by Einstein shows that a ray of light would go all round it in about a thousand million years. On its journey, however, the ray would be somewhat deflected by the gravitational fields through which it passed and would probably also suffer a certain amount of absorption. For these reasons it is unlikely that it would return, accurately focussed, to its starting point. Otherwise we might suppose that some of the stars are really "ghosts at the places where stars used to be. Einstein's finite universe is such that its radius is dependent upon the amount of matter in it. Were more matter to be created, the volume of the universe would increase. Were matter to be annihilated, the volume of space would decrease. Without matter space would not exist. Thus the mere existence of space, besides its metrical properties, depends upon the existence of matter. With this conception it becomes possible to regard all motion, including rotation, as purely relative. The centrifugal force developed by a rotating body, which Newton referred to absolute space, may now be referred to the presence of the other masses in the universe. A body's inertial mass becomes wholly conditioned by the presence of other bodies in the universe. An isolated rotating body would develop no centrifugal forces. In the opinion of some writers Einstein's finite universe gives altogether too important a rôle to matter. Matter is taken as fundamental, and everything else as derivative from it. Eddington's view, as we have seen, is very different. He starts with a four-dimensional continuum, possessing a certain degree of structure. He then makes matter to be nothing but an exhibition of certain peculiarities of this structure. The fact that matter has an atomic constitution, however, has not yet been explained by this theory. In fact, the whole theory of relativity has thrown almost no light upon the problem presented by the atomic constitution of matter. A finite universe, having a somewhat different structure from Einstein's, has been developed by De Sitter. This universe illustrates very well how widely the universe of science may differ from the universe of common-sense. Thus, in De Sitter's universe, natural processes occurring at a distance from the observer will seem to be taking place more slowly than similar processes in his neighbourhood. This slowing down of all natural processes takes place in such a way that at a certain distance—the "horizon "-everything would appear to be at a standstill. This phenomenon is illusory in the sense that if the observer were transported to the horizon he would find everything in his new position proceeding normally, and the horizon would now appear to be at the place he had left. Some evidence for this theory is provided by the spiral nebulæ, the most distant objects in the universe. The spectra of a curiously high percentage of these bodies show a displacement towards the red end of the spectrum. This has usually been taken to indicate a high velocity of recession on the part of these bodies. It may be, however, that this displacement is partially due to the general slowing down of all processes, including atomic vibrations, becoming perceptible at these great distances. Unlike Einstein's finite universe, De Sitter's space-time continuum does not appear to be dependent upon the existence of matter. It has an inherent structure of its own. CHAPTER XI NEW PROBLEMS We have seen that the attempt to describe the intimate structure of matter mathematically led to the Rutherford-Bohr model of the atom. This model achieved, as we have said, some striking successes. By the year 1925, however, it became apparent that certain experimental results could not be accounted for by this model. Calculation could not give these experimentally observed data, not because the calculations were too difficult, but because the Bohr atom did not furnish the necessary premises. The Bohr atom, it could be shown, was too simple to explain certain spectral effects. It did not permit, within itself, a sufficient degree of variety. It became necessary, therefore, to investigate the assumptions made in constructing the Bohr atom, in order to determine whether any of them implied restrictions which could conceivably be loosened. As a result of this examination it was suggested that the notion of the " electron," as assumed by Bohr, could be further complicated. The only motion that Bohr had attributed to the electron was a motion of translation. The electron could move in straight or curved paths, but that was the only kind of motion it was capable of. But the electron was not a point; it was a body of finite dimensions. It was conceivable, therefore, that it could execute a spinning motion about an axis passing through it. This notion of the spinning electron introduces an additional complication into the Bohr atom, and gives it a greater degree of variety. The additional variety so introduced was found to be just what is required to explain certain experimental results. Independent of this development, however, an entirely new attack, which is still being vigorously pursued, has been launched upon the whole problem. We have previously said that a methodological rule which is becoming of more and more importance in physics is that none but observable factors shall be invoked in the construction of a scientific theory. Now the Bohr atom only very partially obeys this criterion. The only observable atomic quantities are those characteristic of its radiations. On Bohr's theory the atom only radiates when it jumps from one state to another. It is the passage of electrons from one orbit to another within the atom that is responsible for the observable radiation effects. But Bohr's theory professes to tell us what is happening in the meantime, when the atom is in a steady state. The electrons are then revolving steadily in their orbits without executing jumps. Thus Bohr's description of the atom includes an account of atomic states that can never be observed. Another unsatisfactory feature of Bohr's atom is that it contains within itself no prophecy of its future. If we are given a complete specification of the Bohr |