Pagina 175 de 212

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 = 4πr2; if it is spherical,

then F is always less than 4πr2. 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.