Chapter 3

Now we put the question, What are the laws of disposition of the disc-shadows _L'_ on the plane _E_? Evidently they are exactly the same as the laws of disposition of the discs _L_ on the spherical surface. For to each original figure on _K_ there is a corresponding shadow figure on _E_. If two discs on _K_ are touching, their shadows on _E_ also touch. The shadow-geometry on the plane agrees with the the disc-geometry on the sphere. If we call the disc-shadows rigid figures, then spherical geometry holds good on the plane _E_ with respect to these rigid figures. Moreover, the plane is finite with respect to the disc-shadows, since only a finite number of the shadows can find room on the plane.

At this point somebody will say, "That is nonsense. The disc-shadows are _not_ rigid figures. We have only to move a two-foot rule about on the plane _E_ to convince ourselves that the shadows constantly increase in size as they move away from _S_ on the plane towards infinity." But what if the two-foot rule were to behave on the plane _E_ in the same way as the disc-shadows _L'_? It would then be impossible to show that the shadows increase in size as they move away from _S_; such an assertion would then no longer have any meaning whatever. In fact the only objective assertion that can be made about the disc-shadows is just this, that they are related in exactly the same way as are the rigid discs on the spherical surface in the sense of Euclidean geometry.

We must carefully bear in mind that our statement as to the growth of the disc-shadows, as they move away from _S_ towards infinity, has in itself no objective meaning, as long as we are unable to employ Euclidean rigid bodies which can be moved about on the plane _E_ for the purpose of comparing the size of the disc-shadows. In respect of the laws of disposition of the shadows _L'_, the point _S_ has no special privileges on the plane any more than on the spherical surface.

The representation given above of spherical geometry on the plane is important for us, because it readily allows itself to be transferred to the three-dimensional case.

Let us imagine a point _S_ of our space, and a great number of small spheres, _L'_, which can all be brought to coincide with one another. But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved away from _S_ towards infinity, and this increase is to take place in exact accordance with the same law as applies to the increase of the radii of the disc-shadows _L'_ on the plane.

After having gained a vivid mental image of the geometrical behaviour of our _L'_ spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of our _L'_ spheres. Then we shall have a vivid representation of three-dimensional spherical space, or, rather of three-dimensional spherical geometry. Here our spheres must be called "rigid" spheres. Their increase in size as they depart from _S_ is not to be detected by measuring with measuring-rods, any more than in the case of the disc-shadows on _E_, because the standards of measurement behave in the same way as the spheres. Space is homogeneous, that is to say, the same spherical configurations are possible in the environment of all points.* Our space is finite, because, in consequence of the "growth" of the spheres, only a finite number of them can find room in space.

* This is intelligible without calculation--but only for the two-dimensional case--if we revert once more to the case of the disc on the surface of the sphere.

In this way, by using as stepping-stones the practice in thinking and visualisation which Euclidean geometry gives us, we have acquired a mental picture of spherical geometry. We may without difficulty impart more depth and vigour to these ideas by carrying out special imaginary constructions. Nor would it be difficult to represent the case of what is called elliptical geometry in an analogous manner. My only aim to-day has been to show that the human faculty of visualisation is by no means bound to capitulate to non-Euclidean geometry.

End of Project Gutenberg's Sidelights on Relativity, by Albert Einstein