In short, Mach’s views do not conflict with causality; for we may always say that our efforts have been exerted on our body and not on the stars. But we must recognise that our efforts have set our body in rotation with respect to the star-space,and not with respect to a space which transcends the existence of the material universe.
Now, although a number of thinkers were in sympathy with Mach’s attitude, owing to their belief that all motion in a perfectly empty universe must be meaningless, yet the physical difficultiesconfronting his idea were so great that classical science had never taken it seriously. Euler had discarded the possibility of any such stellar action, and even quite recently Planck had criticised Mach for his stand. It was not that in theory a possible stellar action should be excluded on general principles; for physics had made us acquainted with foreign actions of a similar nature, the best-known of which would be illustrated by the gravitational action of the sun on the planets. Then again, when we wish to cut through an apparently empty space with a knife, we may experience considerable difficulty if the space happens to be permeated by a powerful magnetic field. We cannot, therefore, exclude the possibility that the stars might in some way generate a field endowing space with a structure which we had mistakenly taken to represent the absolute intrinsic structure of space. Science, however, cannot content herself with vague qualitative analogies; and before a mechanics such as Mach’s could be entertained, it would be necessary for it to succeed in accounting for the existence of centrifugal and Coriolis forces on a body in rotation with respect to the stars. This accounting would have to be made,not merely in some vague qualitative way, but quantitatively also, to a high order of precision. Such, indeed, are the requirements that all hypotheses in physics must satisfy. Now, all that Mach had done had been to postulate a possible stellar influence; he had not even attempted to justify his ideas rigorously. Indeed, even had he investigated the problem more deeply, he would have failed; for in his day space-time was unknown.
Furthermore, Mach’s ideas presented insuperable difficulties. Newton’s law of gravitation stated that the attractions exerted between bodies depended solely on their masses and mutual distances; the conditions of relative motion or rest of the bodies being considered irrelevant. But if Mach’s ideas are accepted, we must assume that in addition to the ordinary Newtonian attraction, a supplementary field of attraction, hitherto unsuspected, should arise if the bodies were in a state of relative acceleration, as in the case of the stars rotating round the earth, or the planets moving round the sun.[115]Mach’s mechanics would entail, therefore, the downfall of the Newtonian law of gravitation. Now, when Mach suggested his mechanics, Newton’s law appeared to be justified indirectly by the accuracy with which it had permitted mathematicians to foresee numerous astronomical occurrences (with the possible exception of Mercury’s motion and certain lunar disturbances). In addition, Newton’s inverse-square law of the attraction of matter on matter had been verified by direct measurements in the laboratory, in what is known as Cavendish’s experiment. So far as experiment could detect, there was no trace of an increase in the mass of a bodywhen other bodies were placed in its proximity; hence, on this point it appeared impossible to detect the variations in mass demanded by Mach’s mechanics. As for attempting Cavendish’s experiment with masses mutually accelerated, technical difficulties rendered its success impossible. For all these reasons the majority of scientists viewed Mach’s relativity of rotation with the utmost suspicion.
But when we come to consider the theory of relativity, the situation changes. In the first place, Newton’s law of gravitation is seen to be only approximate, and the relative motion existing between bodies is found to modify their apparent mutual attractions. This in itself is, of course, insufficient to substantiate Mach’s mechanics, but when we examine the implications necessitated by Einstein’s cylindrical universe, we find that Mach’s views may turn out to be correct after all.[116]However, before discussing this aspect of Einstein’s theory, we must mention that even in classical science the relativity of accelerated and rotationary motion could be accepted in a restricted sense for mechanical phenomena.
Consider, for instance, the earth rotating among the stars. The space defined by a frame attached to the earth would be permeated with a field of centrifugal and Coriolis forces. Now we were always perfectly at liberty to regard the earth as non-rotating, and the stars as rotating round it, provided we retained the existence of this special field of force. If we consider this field to extend beyond the stars, calculation shows that under the action of these forces the stars would undoubtedly circle round the earth as they appear to do. To this extent, therefore, the relativity of rotation, when restricted to mechanical phenomena, could be defended in classical science. The general principle of relativity goes one step farther by permitting us to extend this relativity of rotation to all manner of phenomena—optical and electromagnetic as well as mechanical. But, even so, the relativity of rotation is not radical, for we have rendered it possible only by retaining this curious field of force surrounding the earth; and the query naturally arises: Whence does this field originate? Newton replied: “From rotation in absolute space.” Einstein would have replied: “From rotation in absolute space-time.” In other words, the hypothesis of an absolute extension possessing dynamical propertiesper sewas adhered to both by classical science and by the theory of relativity prior to Einstein’s speculations on theform of the universe. Such being the case, to regard the earth as non-rotating was a mere mathematical fiction.
Now, when Einstein formulated his postulate of equivalence, the classical conception of centrifugal force as arising from a rotation in absolute space gave rise to serious difficulties. The postulate of equivalence asserted that forces of inertia and of gravitation were of the same nature; but then it followed that they should be traceable to one same origin. But gravitation, as we know, is due to matter; hence in a world totally devoid of matter there could be no such thing as gravitation. But then, according to the inferences deduced from the postulate of equivalence, neither should there be any such things as forces of inertia, centrifugal forces and inertial mass. It would follow that if some tiny test-body were made to rotate in an otherwise empty universe, no trace of centrifugal force would be observed at its equator. Thus we see how the postulate of equivalence, which is deduced from the well-established identity of the two types of masses and which constitutes the starting point of Einstein’s theory of gravitation, must inevitably tend to lead us towards views similar to those of Mach.
The equivalence postulate also appears to point to the cylindrical universe; for if we accept the hypothesis of an infinite quasi-Euclidean universe with its nucleus of stars, we shall have to assume that the rim of a rotating disk of sufficiently large proportions could be conceived of as extending well beyond the farthest stars of the nucleus. Yet, the larger the disk and the more remote the rim from the stars, the greater would become the centrifugal force acting on the rim. It would appear as though the source of the centrifugal force would have to be traced to the empty regions at infinity, so that this force could never be attributable to the star-matter concentrated in the nucleus. In other words, gravitation would subsist only in and around the nucleus, whereas forces of inertia would subsist everywhere and have their source at infinity. Not only would such a conception be exceedingly unsatisfying by removing to infinity the causes of forces which we observe at finite distances, but it would also appear to be in utter conflict with the inferences deduced from the postulate of equivalence, establishing, as it would, a duality between the nature of inertia and gravitation.
There seems to be only one way to remove these difficulties; namely, to accept the hypothesis of Einstein’s cylindrical universe conditioned entirely by matter. In this self-contained universe the disk, if extended indefinitely, would finally curl round on itself, enveloping the universe; its rim would never extend to spatial infinity. Calculation would show that as the disk was gradually extended, the centrifugal force on the rim would at first grow, then decrease, and finally vanish again. The displeasing necessity of having the cause of centrifugal force removed to infinity would thus be obviated; centrifugal force, just like gravitation, would be traceable to the spatial structure of the universe and hence to the presence of matter at finite distances.
It is to be noted that these views lead us to Einstein’s cylindrical universe, and not to de Sitter’s. For de Sitter’s universe, contrary to Einstein’s, can subsist devoid of all matter; centrifugal force and inertia would be present in it, even in the complete absence of matter and gravitation. It would thus fail to satisfy the requirements deduced from the postulate of equivalence; since once again, as in the infinite universe, centrifugal force and inertia generally would appear as unconnected with matter, hence with gravitation.
All the arguments discussed in the preceding paragraphs tend to prove that Einstein’s theory was leading us insensibly towards Mach’s mechanics and the cylindrical form of the universe; these two, though apparently disconnected, proving themselves to be intimately related. But it would be a grave mistake to assume that Einstein had set his minda priorion justifying Mach’s views. Any one who has followed the arguments outlined will perceive that Einstein is not forcing nature into some preconceived mould. But, even at this stage, we have not discussed the weightier reasons which finally compelled him to feel sympathetic towards Mach’s mechanics. We shall now remedy this omission.
Einstein’s gravitational equations without theterm,i.e., the equations applying to the infinite universe, namely,had received brilliant support in having enabled him to account for the motion of the planet Mercury, and to anticipate such unsuspected phenomena as the double bending of a ray of light and the Einstein shift-effect. It was only natural, therefore, to endeavour to extract from these equations all the remaining treasures they contained. But here integrations of an extremely difficult nature awaited mathematicians.
When the equations were treated in a very approximate way, they yielded Newton’s law of gravitation and the Einstein shift. When the approximation was increased, they yielded the double bending of a ray of light. When the approximation was increased still further, we obtained the precessional advance of Mercury’s perihelion. When treated in a more general way, they enabled Einstein to predict that gravitation would be propagated with the speed of light. Thirring then attacked the equations, carrying his calculations to a still higher degree of approximation, and at last, the effects necessitated by Mach’s mechanics appeared. Thirring found that the rotation of a central body would modify the nature of its attraction on the planets or satellites circumscribing it. (These anticipations would appear to be susceptible of astronomical verification in the case of Jupiter and one of its satellites.) But, most important of all, he discovered that Mach’s belief in the gravitational influence of the stars rotating round a fixed body was finally justified. In particular, the equations proved that even in the hypothesis of an infinite universe, a rotation of matter would produce around a test-body a field of force disposed in exactly the same peculiar way as are the centrifugal and Coriolis forces in the space attached to a body in rotation. Furthermore, the mass of a test-body would be affected by the proximity of surroundingmatter. This last consequence, however, can be obtained without resorting to difficult calculations.[117]
Now the gravitational equations which have yielded these interesting results have most certainly not been doctored up for the sole purpose of justifying Mach’s ideas. As explained in previous chapters, they were discovered by Einstein as a result of a totally different train of thought, long before the relativity of rotation was even considered. Hence, it cannot be denied that by vindicating, as they did, the correctness of Mach’s ideas, a fact of the utmost importance had been established. Mach’s mechanics received at last that rational justification which until then had been lacking; and the first inkling of a possible physical connection between the origin of inertia and gravitation was obtained. However, it is important to note that although from a qualitative standpoint the agreement was perfect, yet when we view the problem quantitatively, calculation shows that the increase of mass and the centrifugal forces generated by the rotation of the stars would produce but an infinitesimal fraction of the magnitudes observed in practice. In order to obtain magnitudes corresponding with observation, it would be necessary to assume that the stars were many trillion times more numerous than we believed them to be, and this supposition would lead us into other difficulties.
The reason for this discrepancy is not hard to discover. Centrifugal force, as we know, is produced when a body is torn away from the space-time geodesic which the metrical field would compel it to follow. To say, therefore, that a rotation of the firmament round a test-body would affect the field of force surrounding the body is equivalent to stating that a rotation of the stars would be capable of affecting the distribution of the metrical field round the body and of dragging the geodesics along. There is no reason to be surprised at some such effect taking place, seeing that the phenomenon of gravitation connotes that matter, whether at rest or in motion, must affect the curvature of space-time, hence the lay of the metrical field. But, on the other hand, until Einstein had discovered the cylindrical universe, the action of matter could only modify the lay of a pre-existing metrical field, and could not create such a field out of nothing. Space-time of itself was assumed to be flat, and matter could only endow it with a certain degree of curvature. The result was that the major part of centrifugal force was due to the intrinsic texture of flat space-time, and only an infinitesimal portion of this force could be attributedto the additional action of the stars. Centrifugal force thus still appeared, in the main at least, to betray absolute rotation in empty, absolute space-time.
None the less, an important point had been established. Mach’s ideas, though still refuted quantitatively, yet seemed to be acceptable qualitatively, whereas in classical science they had appeared utterly impossible from every standpoint. The thin end of the wedge had been driven in. From then on, it became permissible to assume that with an increase in our understanding of the universe the complete relativity of all motion would be established, not only qualitatively, but also quantitatively. In other words, not merely an infinitesimal portion of centrifugal force and of mass, but their totality, might be accounted for by appealing to the interaction of matter and matter. It is easy to understand how this achievement would be realised.
All we should have to do would be to assume that the totality of the metrical field, and not merely a portion of it, was created by the distribution of matter throughout the universe. As we have seen, this condition is precisely the one realised in the cylindrical universe; so that with Einstein’s universe, Mach’s ideas may be vindicated and the bugaboo of absolute rotation dispelled.[118]
Both de Sitter’s universe and the quasi-Euclidean one would be incompatible with Mach’s mechanics, since in either universe the metrical field and the geodesics of space-time would exist in the complete absence of matter. This means that in the empty world of de Sitter, and far beyond the nucleus of stars in the quasi-Euclidean universe, bodies would be submitted to forces of inertia when torn away from their natural geodesical world lines. In either case the space-time void would appear to be endowed with dynamical properties even in the absence of all matter.
The following passage quoted from Einstein’s writings expresses his views on the subject:
“The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained to a first approximation in the equations of the theory of relativity. It follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions and in part upon an independent property of space, Mach’s idea gains in probability. But this idea of Mach’s corresponds only to a finite universe bounded in space, and not to a quasi-Euclidean infinite universe. From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe.”
Here, a number of objections may be urged. It might be claimed that if we assumed the earth to be non-rotating, we should have to attribute velocities greater than light to the stars. The answer to this objection would appear to be similar to that given when the centrifugal force acting on the rotating stars was discussed. We saw that the stars, as a whole, may be regarded as rotating round the earth only in a purely formal sense, for it is the agglomeration of stars which creates the metrical field, hence which creates space; the earth is but an incident in this star-created universe. Loosely speaking, we may say that the stars are always at rest in the universal space which they themselves have created; and this holds regardless of whether or not we consider this space as rotating in some fictitious outer space.
But there is another objection which, to the writer at any rate, appears to have some weight. Suppose all the stars were to be annihilated at a given instant of our earth frame. The gravitational disturbance produced by this annihilation would require a number of years to reach us, since gravitational disturbances travel with the speed of light. Hence, the metrical field, or space-time structure, round the earth would persist for a considerable time. For a number of years, therefore, centrifugal force would still continue to manifest itself. In view of this fact, we should take into consideration the present condition of the metrical field rather than the present positions of the stars. We are thus led to a more dualistic outlook, that of both field and matter. Such appears to be Weyl’s attitude when he writes:
“The falling over of glasses in a dining car that is passing round a sharp curve and the bursting of a flywheel in rapid rotation, are not, according to the view just expressed, effects of ‘an absolute rotation’ as Newton would state, but whose existence we deny; they are effects of the metrical field. In so far as the state of the guiding field does not persist, and the present one has emerged from the past ones under the influence of the masses existing in the world, namely, the fixed stars, the phenomena cited above are partly an effect of the fixed stars,relative to whichthe rotation takes place.... We say ‘partly’ because the distribution of matter in the world does not define the ‘guiding field’ uniquely, for both areat one momentindependent of one another and accidental (analogously to charge and electric field).”
At the present time all these questions relating to the shape of the universe still remain highly speculative; and we need not be surprised to find the most component authorities, such as Einstein, Weyl, Eddington, Silberstein and Thirring, manifesting different tendencies according to their respective temperaments. Eddington manifests a preference for de Sitter’s universe. If this universe be adopted, Mach’s ideas are refuted; for centrifugal forces would be generated in empty space-time devoid of all matter, and the stars would have very little to do with them. De Sitter, of course, defends his own model of the universe and reproaches Einstein with having resuscitated a species of Newtonian absolute space by materialising the inertial frame with respect to which rotation and acceleration are measured. There is notthe slightest doubt that the cylindrical universe accomplishes this result by linking the existence of the inertial frame to a permanent star distribution. But, in the eyes of many, this is one of its greatest advantages, since it leads to the physical relativity of all motion. After all, we cannot deny the existence of centrifugal force and forces of inertia; and surely it is more satisfactory to account for these forces in terms of the acceleration of matter with respect to matter, than in terms of acceleration with respect to that vacuum which we call empty space-time. But these divergencies of opinion arise from mere matters of sentiment, and it is important to note that they concern chiefly the loftiest superstructures of the theory; they do not invalidate in any way its essence and main achievements.
Of course, it would be of the utmost interest to decide the question of the form of the universe by appealing to astronomical observation, and it is highly probable that in a relatively short time the problem will be settled.
Recapitulating, we have three possible hypotheses to consider:
1. An infinite universe of space with a nucleus of star-matter, in which Mach’s mechanics would be impossible. We should then have to consider the void as possessing dynamical properties and a structure, independently of the presence of matter.
2. De Sitter’s universe, which would also be contrary to Mach’s ideas and would present the same unsatisfactory features as the first hypothesis. Furthermore, it would be unstable.
3. Finally, Einstein’s cylindrical universe, which would be stable and whence all the displeasing features mentioned would be removed.
Could we but annihilate all the stars, we should have a means of solving the problem; for in Einstein’s universe the result would be the total disappearance of mass and centrifugal force. In fact, complete chaos would result; for the metrical field of space-time would have vanished, so that light rays and free bodies would no longer know how to move. But as an annihilation of the stars is beyond the range of possibility, we must have recourse to other methods. So we are led to examine the question of stability. Is it true that the stars are in a state of statistical equilibrium? Is it true that their velocities are always small compared to that of light? Eddington casts some doubt on this stability by remarking that the spiral nebulæ appear to be animated with enormous velocities. This fact, coupled with the apparent reddening of their light, leads him to manifest a preference for de Sitter’s universe. But there exists still another method of settling the question.
We remember that Einstein’s original law of gravitation in the case of the infinite universe differed in certain respects from Newton’s, and that these differences were important enough to be detected within the limits of the solar system through their effects upon Mercury’s motion and the double bending of a ray of starlight. But if now we assume the universe to be finite, and hence include theterm in the gravitational equations, calculation shows that a further modification will ensue. It will be as though all space round a starwere filled with negative or repulsive matter of very low density, of the same order as the curvature of the universe. The influence of this factitious negative mass distribution would be to decrease the normal value of the star’s attraction; the decrease, owing to the cumulative effect produced by the distribution, becoming more and more apparent as we considered points farther removed from the star. Needless to say, this additional modification in the law of gravitation would be incredibly small, owing to the minuteness of the world-curvature and hence of the density of the negative mass distribution. It would be idle to hope to detect it unless we studied gravitational phenomena occurring over tremendous astronomical extensions. But the effect might be observed in the following way:
Consider, for instance, gravitating masses such as the sun and its planets. We know that the reason the solar system does not collapse and all the planets fall on to the sun is because the planets are in motion (as referred to an inertial frame attached to the sun). There exists a definite mathematical relationship between the velocity of a planet along a definite orbit and the force of gravitation to which it must be subjected by the sun. The greater the velocity, the greater must be the sun’s attraction in order to maintain the planet in its orbit. Now the globular clusters of stars and the Milky Way constitute precisely such permanent gravitating systems of enormous extent. The individual stars attract one another, and their mean velocities allow us to deduce the resultant gravitational forces to which they must be subjected. If, therefore, on applying Einstein’s original law of gravitation (that corresponding to the infinite universe),[119]we found that the mean velocities of the stars in the Milky Way were smaller than would be expected, we could consider it proved that the law of attraction must fall off more rapidly than Einstein’s original law indicated. This discrepancy would then justify our belief in the finiteness of the universe. The value of,the cause of this discrepancy, could be ascertained, and with it, of course, the value of the universal curvature. Knowing the curvature, we could determine therefrom the size of the universe, the average density of matter, and finally the total mass of the universe. Einstein is of the opinion that astronomical observations of this type will eventually settle this question.
We have exhausted the principal points which prompted Einstein to believe in the finiteness of the universe, but there are yet others connected with the possibility of reducing matter to electricity and gravitation. We shall touch on this additional aspect of the question in a later chapter.[120]For the present, we may pass to a totally different type of problem, also connected with the finiteness of theuniverse. We refer to the standard of gauge.
As Eddington points out, when our equations prove to us that the universe of space must be spherical and have a constant curvature, what else can it mean but that if we measured the radius of curvature with a material rod, we should obtain the same magnitude in every direction? But then it follows that the radius of the universe in any direction constitutes the gauge of length which nature imposes upon us; and that all bodies in equilibriumadjustthemselves automatically so as to maintain some definite fraction of the length of this radius, in whatever direction they be placed. This view suggests that for material bodies to be on a definite scale of size, there must be a curvature of the world. Eddington also suggests that inasmuch as the finite universe is open in a time-like direction, whereas it is limited in space, a possible explanation may be given of the apparent permanency of existence of certain entities such as electrons and atoms, which are limited in spatial extension but are everlasting in time. Weyl’s theory throws further light on these rather hypothetical problems.
The finite universe introduces us to the difficult conception of a spherical space. We have already discussed this type of space in a previous chapter; but in view of its additional importance, now that the real universe is suspected of being one, it may be worth while to examine the problem afresh.
Let us first consider the two-dimensional analogy. If, on the two-dimensional surface of a sphere, we perform measurements with tiny rigid Euclidean rods, we obtain the numerical results, not of Euclid’s but of Riemann’s two-dimensional geometry. Accordingly, we say that the geometry of a spherical surface is Riemannian, and that a Riemann space is curved and finite. But the word “curved” immediately evokes the idea of a third dimension along which the surface or space is curved. In the present two-dimensional case we have no difficulty in visualising this third dimension, because the space of our actual experience happens to be three-dimensional. And so we note that the spherical surface possesses an outside and an inside, and that if we consider solely the points on the surface, we are excluding from our investigation an indefinite number of points of space both inside and outside thesphere. This procedure is unfortunate; for when we generalise our previous two-dimensional illustration to a three-dimensional spherical space, we are inclined to reason by analogy and to conceive, or endeavour to conceive, of a four-dimensional Euclidean space in which our three-dimensional Euclidean universe must be embedded. The result is that we think of points of space outside our universe, as though our universe did not embody the whole of space.
In the present case this attitude would be entirely erroneous, but the fault lies with the peculiar type of two-dimensional analogy we chose to present. In order to eliminate this error, let us proceed as explained in a previous chapter, and let us suppose that the sphere rests at its south pole on an infinite Euclidean plane, while a point of light is situated at its north pole. If the sphere is translucent, our tiny rods displaced on the spherical surface will cast their shadows on the plane; and, of course, if we effect measurements on the plane by means of these shadows of our rods, we shall obtain exactly the same metrical relationships on the plane as we obtained formerly on the spherical surface. In particular, if a hundred tiny rods lying along a meridian span the distance from the south to the north pole on our sphere, the hundred shadows of these tiny rods will extend across the plane to infinity. Likewise, just as a finite number of postage stamps would cover the sphere’s surface, so now a finite number of shadows, the shadows of these stamps, would cover the entire infinite extent of the plane. Thus we see that when measured with our shadows the plane will be finite and its geometry Riemannian, whereas it would have been infinite and Euclidean had we effected our measurements by displacing ordinary foot-rules over its surface.
The advantage of presenting the problem in this alternative manner is that in the present case the introduction of a third dimension is no longer necessary in order to explain curvature, or non-Euclideanism. We realise that regardless of whether the space of the plane is to be considered as finite and Riemannian or as infinite and Euclidean, exactly the same points of space are included in either case. This two-dimensional illustration can, of course, be extended to three dimensions. By following the same line of reasoning, we will say that the universe of three-dimensional space is infinite, if by placing side by side, in rows, columns and superimposed layers, an indefinite number of congruent cubes, we find it possible to go on piling them up for ever and ever. On the other hand, we will say that the universe of space is finite if this process cannot be continued indefinitely, that is, if after a finite number of cubes have been superimposed, further progress is impossible. This last case is, of course, similar to the one discussed previously when we found it impossible to place an indefinite number of shadows of our postage stamps over the plane.
From these examples it is apparent that the finiteness or infiniteness of our space is contingent on the geometrical behaviour of the cubeswhich we regard as congruent or equal. From a purely mathematical standpoint, therefore, the problem of the finiteness of the universe can possess no absolute significance, since congruence, or equality of size, is incapable of being defined in any absolute way. Yet from the standpoint of the physicist the problem assumes a very definite significance, for whereas one uniquerationaldefinition of congruence is impossible, anempiricaldefinition is easily obtained. By congruent cubes the physicist means those which visually and tactually will appear the same in whatever region of space they may be situated, provided the observer accompanies the cubes in their displacements. He means cubes carved out of the substance of those bodies which men have agreed, from time immemorial, to regard as invariable solids, and which are illustrated by the bodies around us, when maintained under conditions of constant temperature and pressure.
It is as referred to such bodies, therefore, or, again, to the courses of light rays, that the universe, according to Einstein, would manifest itself as finite. This is but another way of saying that our bodies would behave like Riemannian solids (over vast areas at least) instead of like Euclidean ones, as was formerly thought to be the case.
Summarising, we see that this same so-called finite universe would manifest itself as infinite were our rods to behave Euclideanly, whence we may conclude that the universe, whether finite or infinite, may contain exactly the same regions of space, just as our plane contained the same points whether measured with yardsticks or shadows. To assume, therefore, that certain regions of space must necessarily be excluded owing to the finiteness of the universe, would be to take a stand in no wise demanded by the actual conditions of the problem.