up these conditions constitute the equations of motion of a particle, at least under the action of non-Newtonian forces. On comparing them with the traditional. Newtonian equations, we see that the g's which characterise any non-Newtonian frame are of the nature of potentials of the non-Newtonian forces introduced by that frame. (26) We now make two assumptions, which are only justified in so far as they work. (a) We assume that it is a universal law of Nature that a particle moves in such a way that the total separation of remote events in its history is stationary, as compared with that of all other possible ways of moving. This is to hold equally whether it be subject only to non-Newtonian or also to Newtonian forces. In that case the equations deduced for the nonNewtonian case become the equations of motion. (b) We assume that in those regions of Nature, regarded as a sum total of events, in which Newtonian forces show themselves, the structure of Nature is not such that the separation can be reduced to the form with constant coefficients. If that be so, the course with the maximum total spatio-temporal separation is not a Euclidean straight line traversed with a constant velocity, as judged by a Newtonian clock. We treat the traditional potentials of the Newtonian forces in any field as first approximations to a set of g's, which satisfy the general equations of motion thus deduced. And we treat the result as the true law of the field. (27) Continuous manifolds of several dimensions, such as Nature has proved itself to be, can be of various intrinsically different kinds. As we might put it, they can be "plane-like," "sphere-like," "egg-like," and so on. Whatever intrinsic spatio-temporal structure Nature may have, there will be an infinite number of different possible frames to be found for placing and dating the events of Nature. Nevertheless, the intrinsic structure of Nature will impose certain conditions on all possible natural frames of reference. These re strictions will take the form of certain very general equations connecting the g's of any possible natural frame. If the structure of Nature be plane-like, the condition is that the unmodified Riemann-Christoffel Tensor shall vanish for the g's of all possible natural frames. If its structure be sphere-like, the condition is that the the Modified Riemann-Christoffel Tensor shall vanish for the g's of all possible frames. The latter is a less rigid condition than the former. (28) If the intrinsic structure of Nature be plane-like, an accurately Newtonian frame will be fitted for dating and placing all the events of Nature; otherwise it will not. (29) If we try to map out a manifold by a frame which is unsuited to its intrinsic structure, we shall only be able to square our measurements with our theory by the assumption of forces which distort our measuring instruments and upset their readings. (30) We cannot find any frame that will transform away gravitational forces always and everywhere, though we can find non-Newtonian frames which will transform them away over sufficiently small regions of space and time. With respect to Newtonian frames all particles are always acted on by gravitational forces, though these may sometimes be negligibly small for practical purposes. It is therefore plausible to suppose that the universality of gravitation with respect to Newtonian frames is a mark of the misfit between this type of frame and the intrinsic structure of Nature. (31) On the other hand (a) gravitation has many analogies to non-Newtonian forces; (b) the traditional law of gravitation, which is certainly very nearly true, can be expressed as a differential equation of the second order, involving the gravitational potential at a place and the co-ordinates of the place with respect to Newtonian axes; and (c) we have already assumed that potentials and the g's of frames are mutually equivalent. (32) The facts mentioned in (31) strongly suggest that the law of gravitation must be some general condition imposed on the g's of all possible natural frames, and expressed as a differential equation of the second order involving these g's. The facts mentioned in (30) suggest that this condition is not that the unmodified Tensor vanishes. For, if this were so, the intrinsic structure of Nature would be such that a Newtonian frame is suited to it, and the necessity of assuming gravitational forces always and everywhere with Newtonian frames strongly suggests that this is not so. (33) It is obvious that the next suggestion to try is to suppose that the law of gravitation is expressed by the vanishing of the Modified Tensor, i.e., that gravitation is the sign of an intrinsically sphere-like structure in Nature. (34) It is found that, if this be the true law of gravitation, the observable effects will in most cases differ so little from those predicted by the traditional law that the difference could not be detected. Hence the very full verification which the traditional law has received is no obstacle to accepting the amended law. (35) On the other hand, there are certain very special cases in which a small observable effect might be expected on the new form of the law and not on the old. In such cases (notably the movement of the perihelion of Mercury and the bending of a ray of light in passing near a very massive body like the sun) the predicted effects have been verified both qualitatively and quantitatively. The following additional works may be consulted with advantage: A. S. EDDINGTON, Report on the Relativity Theory of E. CUNNINGHAM, Relativity, Electron Theory, and Gravitation. H. WEYL, Space, Time, and Matter. VII. Matter and its Appearances; Preliminary Definitions VIII. The Theory of Sensa, and the Critical Scientific Theory IX. The Positions and Shapes of Sensa and of Physical Objects X. The Dates and Durations of Sensa and of Physical Objects XI. Sensible and Physical Motion XII. Sensible and Physical Space-Time XIII. The Physiological Conditions of Sensations, and the Onto- |