L E C T U R E S A T T H E U N I V E R S I T Y O F M A D R I D 8 8 5 appeared, and physics of the late nineteenth century had already done away with such a concept, which the qualities of the phenomena did not permit, either. At that time he thought that inertia is only relative. In the general theory of relativity, the physical properties of space depend on matter, for geometry, which gives the rules for localizing bodies, is influenced by the gravitational field, which, in turn, depends on matter. Therefore, space does not have absolute properties. Then, if this theory of relativity or reciprocity of causes really exists, the properties of space, of metrics, of inertia, or of gravitation depend on bodies. That is, space does not ex- ist there is only matter. We have found the gravitational field equations, and we can rely on them in our calcu- lations. E.g., the problem of the planetary system has been studied by supposing that Eu- clidean geometry prevails in the part of space where the Sun and the planets can be found if it does not, they would not exist. The equations obtained in this way are such that those hypotheses can be extended to the whole universe, and we admit that at infinite distance from celestial bodies special relativity can be applied. Now, the “unpleasant point” in Mach also exists in the theory of relativity, which is not permitted by its very essence. In the first place, space should have properties influenced by matter, and this introduces an unacceptable dualism. In the second place, when the laws of gravity are calculated with greater accuracy, one should find the influence of masses in mo- tion on others, finding things in agreement with the ideas of Mach. For example, if we ac- celerate all the masses of the universe except for one, this one should also turn out to be accelerated. If we only accelerate a portion, there would appear a kind of force of induction acting in the same direction. Besides, if all masses are in rotation, the whole system should rotate to avoid relative accelerations, and thus it is necessary to believe that if we seek a system of local coordinates that behaves in a Galilean way, the system should also take on a slight rotation. For exam- ple, a gyroscope should be influenced by the mass of the other bodies. All these conse- quences of Mach’s ideas are confirmed by the gravitational equations, only that these effects are too weak to be observed. Thus the hypothesis that the world be almost Euclidean in the infinite is unnatural, and it is much more logical to admit that all the laws of space are a consequence of matter. There is a more direct cause that prevents us to think that space is Euclidean in the infinite. We can form an idea of the structure of the universe by imagining that there is matter every- where whose mean density is not zero—, something we could not know from experience because it is very hard to believe that our stellar system constitutes a kind of island in space. The first hypothesis contradicts both Newton’s theory and that of relativity. If we admit that there are stars everywhere and that space is not totally empty, it turns out that there is an imperfection in the law of gravity, and it is easy to see that the postulate of relativity permits us to modify the form of the equations a bit, adding to its second term another term which will not change the character of the equations, and thus one sees that a world with a non- zero mean density exists which, insofar as space is concerned, is closed, and Mach’s pos- tulate is satisfied, because if the additional term is not zero, one cannot have a static empty