i.e. a smaller value than p, the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the " world-sphere.”  By means of this relation the spherical beings can determine the radius of their universe (” world “), even when only a relatively small part of their worldsphere is available for their measurements.  But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical " world " and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

Thus if the spherical surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the " piece of universe " to which they have access is in both cases practically plane, or Euclidean.  It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the " circumference of the universe " is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius.  During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole " world-sphere.”

Perhaps the reader will wonder why we have placed our " beings " on a sphere rather than on another closed surface.  But this choice has its 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 c of a circle to its radius r depends on r, but for a given value of r it is the same for all points of the " worldsphere “; 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” (2p2R3).  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 r with a measuring-rod.  All the free end-points of these lengths lie on a spherical surface.  We can specially measure up the area (F) of this surface by means of a square made up of measuring-rods.  If the universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is always less than 4pR2.  With increasing values of r, F increases from zero