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 30.

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 * between the space-expanse of the universe and the average density of matter in it.

  Notes

*) For the radius R of the universe we obtain the equation

eq. 28:  file eq28.gif

The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27; p is the average density of the matter and k is a constant connected with the Newtonian constant of gravitation.

APPENDIX I

Simple derivation of the Lorentz transformation
(supplementary to section 11)

For the relative orientation of the co-ordinate systems indicated in Fig. 2, the x-axes of both systems pernumently 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 K1 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).

Since the same light-signal has to be transmitted relative to K1 with the velocity c, the propagation relative to the system K1 will be represented by the analogous formula

x’ — ct’ = O . . . (2)

Those space-time points (events) which satisfy (x) must also satisfy (2).  Obviously this will be the case when the relation