For instance, “parallels” are said to be two straight lines, being in the same plane, which can never meet. The geometry which we learned in our boyhood says: “Through a given point there can be only one straight line parallel to a given straight line.” This is said to be Euclid’s postulate. Riemann, however, does not admit this and wishes to replace it by: “Through a given point there cannot be any straight line parallel to a given straight line”—that is to say, any line which never meets it. Upon this Riemann founds a quite consistent system of geometry. Who will venture to say that Euclid’s geometry is true and that of Riemann false? As theoretical ideal constructions they are both equally true. * * * * * A question that we may legitimately ask is: Does the real universe correspond to the classical geometry of Euclid or to that of Riemann? It was long believed that it corresponded to Euclid’s geometry. Poincaré himself, speaking of Euclid’s system, said: “It is, and will remain, the most convenient, (1) because it is the simplest; (2) because it agrees very well with the properties of natural solids, the bodies with which our limbs and our eyes are concerned, and out of which we make our measuring instruments.” When people used to say in earlier ages that the earth is flat, they argued pretty much as Poincaré does: “This theory is the most convenient, (1) because it is the simplest; (2) because it agrees very well with the properties of the natural objects with which we are in contact.” But when men came into touch with more remote objects, when navigators and astronomers multiplied these remote objects, the idea of a flat earth ceased to be the most convenient, the simplest, and the best suited to the facts of experience. Then appeared the idea that the earth is round, and this was found infinitely more convenient, simpler, and better adapted to the material universe. “Convenience,” which Poincaré makes a criterion of scientific truth, is a contingent and elastic thing. A point of view may be convenient in London and not in Bedford. A theory may be convenient in an area of a hundred yards and no longer convenient for an area of a hundred million miles. The hypothesis of a flat earth has been replaced by the theory of the earth’s rotundity. The stationary earth has been replaced by a revolving globe. In the same way, it seems that in our time Euclid’s geometry must give way to another as a _convenient_ representation of the real world. Can there be, in our universe, our space, a parallel to a straight line? That is to say, is it true that two straight lines being in the same plane will never meet? The real meaning of the question is: Is it impossible for two luminous rays, travelling in empty space and being in what (for each fraction of the rays) we will call the same plane, ever to meet? _The answer to this question is in the negative._