Note 20 — Chapter V · AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES

It is to be noted that had the source of light been placed at the centre of the sphere instead of at the north pole, all the great circles on the sphere would have cast straight lines for shadows on the tangent plane. Still, the shadows on the plane would, as before, have yielded Riemann’s geometry (of the elliptical variety). We see then, once again, that absolute shape, size and straightness escape us in every case, and we cannot even say that Reimann’s straight lines are curved with respect to Euclidean ones. All that is relevant is the mutual behaviour of bodies, their laws of disposition.