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 1 stitutes a new system . It is easy to see that, with respect to the disk, Euclidean geometry does not properly describe the localization of solid bodies. We have seen that the special theory requires that the bodies in motion undergo a contraction in the direction of their mo- tion. Let us imagine some operators on the disk who try to find the ratio of its circumference to its diameter. Toward this end they take identical rulers and apply them to the circumfer- ence and on a diameter. They read the rulers in both cases and so obtain two numbers, which, divided according to Euclid, should yield as a quotient the number π but they will find it to be greater because they are ignorant of the influence of the gravitational field (pro- duced by the rotation) on the rulers. There are centrifugal forces with respect to the disk that can be considered a gravitational field, whose action on the rulers is neither known nor necessary to know. We know that when the disk is at rest, the measurements are made in the inertial system k and we obtain the results of Euclidean geometry, because the laws of restricted relativity are valid for the system with no field. One understands that the previous measurement must be taken for a same time t of system k. When one puts the disk in mo- tion, the rulers placed on its circumference contract, and, therefore, the number of them has to be greater, while those located on the diameter undergo no change. In this way the ex- perimenter finds that, with respect to the disk, the laws of Euclidean geometry do not work. The same occurs with clocks. If we have two identical ones, one at the center and another at any point of the disk, the latter will run slower than the former. This means that the field acts on the clocks, and, therefore, time cannot be defined by means of identical clocks. If, therefore, Euclidean geometry is not valid, a Cartesian system of coordinates cannot be used, and we find ourselves with the difficulty that on the disk and in any gravitational field an immediate sense cannot be ascribed to the coordinates of space and time. At a first glance it seems impossible for physics to make a description of nature if we do not give, a priori, the meaning of the coordinates and of time, a seemingly insoluble problem, but Gauss had already resolved mathematically a very similar problem, that is, that of the ge- ometry of a surface located in a Euclidean space. What is the geometry of a surface? Let us begin with the plane. The geometry of the plane is Euclidean plane geometry, according to which we locate flat solids on a plane. Let us begin by defining the straight line as a line between two points such that, putting rulers on it, beginning with a point until we reach an- other, the number of necessary rulers be the minimum. To resolve this question there is no need to leave the plane: everything that interests us happens on it. Let us now try to do the same on a surface, and assume that we will not use points outside of the plane, because the surface should suffice to give the law of location upon it. In general, it is not done this way. For example, if we seek a curvature, we use points outside of the plane, and Gauss solved the problem of how to do away with such exterior points. Euclidean geometry does not in general work in locating points on surfaces as an exam- ple, we can take a hemisphere and a small square that we can subdivide into smaller ones, thus obtaining a lattice that cannot be fitted to the hemisphere as it would be to a plane. In view of this, Gauss stated the problem in the following way: it is necessary to give a method for tagging the points of a surface. He resolved it by taking two entirely arbitrary systems of curves, numbering each one of them this way: through each point on the surface k′