1 0 0 4 A P P E N D I X N An observer would describe the new phenomenon in the same way as the first case. Thus, the existence or non-existence of a gravitational field depends on the state of mo- tion of a reference system. We imagine still, drawing from the same example, that the material point referred to above will now be connected by a thread to the roof of the box. In the first case, the tension one would verify for the thread would be induced by the inert mass dragging in the box’s accelerated motion. In the second case, the thread’s tension, of the same intensity as in the previous case, would be induced by the action of the gravitational field on the heavy mass. You see, thus, that you would have the same description by the observer. The two phe- nomena: that of the motion of the inert material point in an accelerated reference system, or the material point subject to gravitation in a reference system at rest or in uniform motion. One can already see the influence of gravitation on the laws of physical phenomena. The gravitational field laws could then be logically deduced by a simple transformation of coordinates. Equally, any accelerated system could be substituted for a suitable gravita- tional field. For accelerated systems and, therefore, for the gravitational field, Euclidean geometry is no longer valid. In fact, let’s consider the case of a rotating disc in relation to an inertial system. The measurements of its circumference and of its diameter lead to a relation, differing from the classic value of N. (pi). In fact, in a given time of the motion, the different constit- uent elements of the circumference would have their length reduced—observed from a ref- erence system at rest—due to the longitudinal contraction, as was already demonstrated in the restricted theory of relativity. However, for the diameter, one would not observe this contraction, given that at any in- stant in time its direction of motion is normal and the diameter itself is not contracting. An analogous statement would apply to time, where a slowing influence caused by mo- tion varies according to the distance to the center of the disk. Look at it thus, that the metric depends on gravitation and, consequently, matter. Matter, gravitation, and metrics, then, condition each other mutually. Gauss had to solve an analogous problem in the study of surfaces. He had to define points on a surface, introducing systems of coordinates characterized by two interdepen- dent systems of curves in which the concept of measurement yields to the concept of order, where Euclidean geometry would be valid only for an infinitely small region. It is possible then to constitute geometry without an immediate physical signification. With the help of Gauss’s coordinates one can interpret the general laws of physics, which would be indepen- dent of the chosen system. Turning to the new gravitational theory, its equations represent a second approximation, making Newton’s law the first approximation. It is curious to observe that the two theories, so profoundly different in their principles, lead to results that differ extremely little.