We have yet to show the connection between non-Euclidean geometry of two dimensions defined as the geometry obtained with Euclidean rods on a surface of constant curvature, and non-Euclidean geometry obtained with squirming Euclidean rods, i.e., rigid non-Euclidean ones. For this purpose consider a sphere resting on a plane. We may call the south pole that point of the sphere which stands in contact with the plane; and we shall assume that a point-source of light is located at the north pole. If the sphere is transparent to rays of light all figures traced on the surface of the sphere will cast shadows on the plane. But the same will be true of our little Euclidean rods which we place alongside these figures on the sphere for purposes of measurement. It is easy to see that a circle, for instance, traced on the spherical surface will, generally speaking, have for its shadow an ellipse on the plane. As this circle is displaced on the sphere towards the north pole, its shadow will grow larger and larger, tending finally, when any point of the circle coincides with the north pole, to become a parabola extending to infinity. But inasmuch as the shadows of the little rods with which we measure lengths on the sphere vary in exactly the same way as do the shadows of the figures measured, we shall obtain on the plane exactly the same Riemannian results that endure on the sphere. In short, if we assume that on the plane our standard measuring rods are given by the shadows of the standard measuring rods used on the sphere, we shall obtain Riemannian geometry on the plane. It is scarcely necessary to state that with our Euclidean ideas of congruence it would be impossible for us to accept the statement that these successive squirming shadows of a Euclidean length displaced on the [Pg 68] surface of the sphere were all congruent to one another; but this is not the point at issue. If we were perfectly flat, two-dimensional beings, living on the plane, and if not only all the flat bodies sliding over the plane behaved as did the shadows in our previous example, but also our measuring rods and our own living bodies behaved likewise, we should have no other alternative than to assume that these shadows represented the displacements of bodies that moved while remaining rigid or without change. We should accordingly regard the geometry of our two-dimensional space as Riemannian.