When the insect gets very near the North Pole, its shadow, now of enormous dimensions, has moved to a very great distance; and when finally it reaches the Pole, its shadow becomes infinitely great and thus stretches to infinity.
But let the insect wander on along the meridian, past the North Pole, down towards the South. At the moment when it passes the upper Pole its shadow jumps from the right side to the left. Its shadow now emerges from an infinite distance to the left, and, instead of being infinite size, again becomes finite in dimensions as it approaches. It contracts as it approaches, and, in short, the same process as occurred during the first half of the journey now occurs in the reverse order.
[If we fix on the critical moment of the jump from the right to the left, that is, from plus infinity to minus infinity, we may encounter difficulties. For the surface-creature pursues its way without interruption and continuously, and we experience a wish to ascribe to it a shadow-path that is also unbroken and continuous. This is possible only if we assume the two points at infinity to be connected, that is, if we consider them identical. This assumption will seem more natural if we reason as follows. In the profile-picture the table is represented as a straight line, and it is along this line that the shadow travels. We may regard this line as an infinitely great circle, for an infinitely great circle has zero curvature, just as the straight line, from which it is therefore indistinguishable. The infinitely great circle has, however, only one point situated at an infinite distance, that is, it associates together the two apparent points at infinity of the straight line with which we identify it. Accordingly, we preserve the continuity of the shadow-journey, too. Einstein considers it allowable to say that the right and the left portion each represent a half of the infinite projection, which becomes complete only when the two ends are joined.]
Now we must be prepared for an effort of thought which will need considerable help from our imaginations. Firstly, instead of one surface-creature, we shall suppose several crawling about on different meridians, so that a series of shadows will be moving about along straight lines radiating from the South Pole. Next, let us imagine the whole picture to have its dimensions increased by one, that is, we transform the plane-picture into a space model. The phenomena are to remain the same, except that they are to be strengthened by one dimension, surface conditions becoming space conditions, and surfaces becoming solids.
What we now see are actual insects with round bodies (if we retain our original type of creatures), or, since there is no restriction as to their size—the shadows have assumed all possible sizes—we may assume any solid bodies whatsoever, stars or even star-systems. Their motions take place in exactly the same way as those of the shadows previously thrown by the flat bodies.
This means that, if a stellar body moves, its size increases until it reaches the spherical boundary of space, where it becomes infinitely great, and, at the same moment, passes from plus infinity to minus infinity, that is, it enters the universe from the opposite direction; then, if it continues moving in its original direction (as it has been doing all along), it gradually becomes smaller in size until, finally, it reaches its original position and its original size. If we suppose the body to be endowed with the power of sensation, it would not be able to observe its own changes of size, since all its scale-measures would be altered in the same proportion. This whole complex of phenomena would still be taking place in an infinite world of space, but, according to the General Theory of Relativity, the geometry that is valid in this world would no longer be that of Euclid; it is replaced by a system of laws that arise from physics as a geometric necessity. In this new geometry, a circle described with unit radius is a little smaller than it would be in Euclidean geometry, with the result that the greatest conceivable circle in this world cannot assume an infinite size.
Thus we have to imagine that our solid bodies, say stars, arrive at a point in their travels which we may term only "enormously distant." If we call the directions right and left instead of positive and negative, then the process reduces itself to this: the moving body reaches the point, which is enormously distant on the right, and which is identical with the point enormously distant on the left; this means that the body never moves out of the space continuum of this world, but returns to its initial point of departure even when it moves ever onward in what is apparently a straight line. It moves in a "warped" space.
Einstein has succeeded in finding an approximate value for this non-infinite universe, from the fact that there is a determinable gravitational constant. In the constitution of the universe it denotes the same for the mass-relationships of the earth as the gravitational constant of the earth denotes for us, namely, the quantity from which we can calculate the final velocity attained by a freely falling body during a unit of time. He also assumes a probable average for the density of distribution of matter in the universe, by supposing that it is about the same as that of the Milky Way. On this basis Einstein has arrived at the following result by calculation:
The whole universe has a diameter of 100 million light-years, in round numbers. That amounts to about 700 trillion miles.
M.: Does this follow from the discussion you entered on just now?