justification in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio of the circumference of a circle to its radius depends on , but for a given value of it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”
To this two-dimensional sphere-universe there is a three-dimensional analogy, namely, the three-dimensional spherical space which was discovered by Riemann. Its points are likewise all equivalent. It possesses a finite volume, which is determined by its “radius” (). Is it possible to imagine a spherical space? To imagine a space means nothing else than that we imagine an epitome of our ‘‘space” experience, i.e. of experience that we can have in the movement of “rigid” bodies. In this sense we can imagine a spherical space.
Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of these the distance with a measuring-rod. All the free end-points of these lengths lie on a spherical surface. We can specially measure up the area () of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then ; if it is spherical, then is always less than . With increasing values
