The relativistic equations for the deflexion of light, the motion of a particle, and the red shift of spectral lines, in the neighbourhood of a single stationary mass, are rigorously derived on the basra of a strictly Euclidean space and an independent time. Only two ad hoc assumptions are needed, in addition to two very obvious extensions of the special theory; one of these assumptions is already familiar, but the other, involving the mass of a stationary test particle, is believed to be new. The particle equations are derived from a Lagrangian m the usual way. Expressions for the kinetic and potential energies are also readily obtained. It is shown (by what is believed to be a new argument) that matter with an infinite Young's modulus cannot exist, and the fact that actual measuring rods may therefore be affected by tidal forces, even when they are 'unconstrained', is considered. It is shown that in principle observations in the solar system should be made in a tune system which is not Shat in which the clocks of distant observatories are synchronized at present; the difference is below the present errors of the best time signals, but not very much below. A rigorous expression is derived for the numerical value of the radial co-ordinate r, in terms of quantities directly observable by the crew of a space-ship (of negligible mass) moving in a circular orbit at the appropriate circular velocity. Further progress along these lines will depend on their extension to the two-body problem.