,
as correspond to a three-dimensional continuum of constant positive curvature. We must now investigate whether such an assumption can satisfy the field equations of gravitation.
In order to be able to investigate this, we must first find what differential conditions the three-dimensional manifold of constant curvature satisfies. A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions,[19] is given by the equations
By eliminating
, we get
[19]The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.