We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, _i.e._ by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be _nil_. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX. If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, _i.e._ the universe will be quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection[25] between the space-expanse of the universe and the average density of matter in it. [25] For the radius _R_ of the universe we obtain the equation image037 The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027; ρ is the average density of the matter and _k_ is a constant connected with the Newtonian constant of gravitation. APPENDICES APPENDIX I SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION (SUPPLEMENTARY TO SECTION XI) For the relative orientation of the co-ordinate systems indicated in Fig. 2, the _x_-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the _x_-axis. Any such event is represented with respect to the co-ordinate system _K_ by the abscissa _x_ and the time _t_, and with respect to the system _K′_ by the abscissa _x′_ and the time _t′_. We require to find _x′_ and _t′_ when _x_ and _t_ are given. A light-signal, which is proceeding along the positive axis of _x_, is transmitted according to the equation _x_ = _ct_ or _x_ – _ct_ = 0 . . . . . (1).