up to a maximum value which is determined by the " world-radius,” but for still further increasing values of r, the area gradually diminishes to zero.  At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a “counter-point” to the starting point.  Under such conditions they have traversed the whole spherical space.  It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface.  It is finite (i.e. of finite volume), and has no bounds.

It may be mentioned that there is yet another kind of curved space:  " elliptical space.”  It can be regarded as a curved space in which the two " counter-points " are identical (indistinguishable from each other).  An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

It follows from what has been said, that closed spaces without limits are conceivable.  From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent.  As a result of this discussion, a most interesting question arises for astronomers and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe.  Our experience is far from being sufficient to enable us to answer this question.  But the general theory of relativity permits of our answering it with a moduate degree of certainty, and in this connection the difficulty mentioned in Section 30 finds its solution.

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter.  Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known.  We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light.  We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.

We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter.  This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe.  But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent