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both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a
constant value p, which is independent of the diameter of the circle. On their spherical surface our flat beings
would find for this ratio the value
eq. 27: file eq27.gif
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 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
PART III 41
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 [ Pobierz całość w formacie PDF ]