A method is established to solve Einstein’s equations by successive approximations, uniqueness and compatibility at each step being secured by imposing de Donder’s condition on the field. This turns out to restrict the freedom of choice of the sources in a way that determines their laws of motion, directly from the field quantities, without any use of integration around singularities. If the approximation parameter is the reciprocal velocity of light, the first non trivial approximation results in the constancy of the rest mass, and in Newton’s law of attraction. The next step shows that there is no first-order correction to these laws. The third approximation then gives a small correction to the rest mass, which is found to be increased by the presence of other bodies. There is also a second-order correction to the acceleration (not explicitly evaluated here) which causes the perihelion advance in the two-body problem. Te fourth approximation involves a contribution from the radiation field. It is found that in order to fulfil de Donder’s condition, we must introduce into the field a term which is linear in the sources, although it does not appear in the so-called «linear approximation». This result may cast some doubts on the validity of the latter. In a further paper, we shall solve still higher approximations by means of an improved technique, and find that the fifth-order correction to the acceleration involves a non-conservative term. This last result may be taken as evidence for the reality of gravitational radiation.