Suppose that in spherical space the physical processes going on at every point are exactly the same as those going on at the antipodal point, so that \PageSep{158} one half of the world is an exact replica of the other half. Let $ABA'B'$ be four points $90°$~apart on a great circle. Let us proceed from~$B'$, \Foreign{via}~$A$, to~$B$; on continuing the journey along~$BA'$ it is impossible to tell that we are not repeating the journey $B'A$~already performed. We should be tempted to think that the arc~$B'A$ was in fact the immediate continuation of~$AB$, $B$~and $B'$ being the same point and only represented as wide apart through some fault in our projective representation---just as in a Mercator Chart we see the same Behring Sea represented at both edges of the map. We may leave to the metaphysicist the question whether two objects can be exactly alike, both intrinsically and in relation to all surroundings, and yet differ in identity; physics has no conception of what is meant by this mysterious differentiation of identity; and in the case supposed, physics would unhesitatingly declare that the observer was re-exploring the same hemisphere. Thus the spherical world in the case considered does not consist of two similar halves, but of a single hemisphere imagined to be repeated twice over for convenience of projective representation. The differential geometry is the same as for a sphere, as given by~\Eq{(67.11)}, but the \emph{connectivity} is different; just as a plane and a cylinder have the same differential geometry but different connectivity. At the limiting circle of any hemisphere there is a cross-connection of opposite ends of the diameters which it is impossible to represent graphically; but that is, of course, no reason against the existence of the cross-connection. This hemisphere which returns on itself by cross-connections is the type of elliptical space. In what follows we shall not need to give separate consideration to elliptical space. It is sufficient to bear in mind that in adopting spherical space we may be representing the physical world in duplicate; for example, the volume $2\pi^{2} R^{3}$ already given may refer to the duplicated world.