9 7 0 A P P E N D I X G If we establish that the segment of a ruler contained between two marks is of invariable length, geometry acquires an experimental character it ceases to be—he added—a system of words and phrases and becomes instead a system of assertions that may be true or false, since one might inquire if the rules deduced from it correspond to real objects that is, whether the world is Euclidean or not. That materialization of length has already been used in the Theory of Special Relativity by utilizing a solid body to define coordinates and give them physical meaning. Finite Space as Determined by the Theory of General Relativity In the Theory of General Relativity, Euclidean geometry does not allow for a description of real objects, or, stated differently, the laws by which one can locate bodies differ from those deduced from that geometry. If we trace a circle around the sun, the eminent sage said, and if we measure the length of its circumference and diameter respectively and its quotient is formed, we obtain a number that is not equal to N [π], which derives from the fact that the physical qualities of space depend on the distribution of matter. This, he added, simul- taneously indicates the gravitational field as well as the properties that lead to finding the solid bodies. That is, gravitation and metrics are indicated by a single physical reality. Naturally, Dr. Einstein continued, if the distribution of matter influences the metrical qualities of space, is it necessary to ask what the form of geometric space is. Gauss had al- ready considered the possibility that the geometrical space of the material world might be finite. According to the Theory of General Relativity, this is what appears to be the case. The conception of such a thing clashes, of course, with our mental habits, which come from the representations we accept as images of experience. Thus, a man, unaccustomed to such matters, when told that our space is finite, will reply with this amusing question: And what's on the other side? My purpose, the sage went on, is to show, through the use of a simple example, that there is nothing paradoxical about saying that a space can be both finite and limitless. If we imag- ine, he continued, illustrating with circles, beings endowed with intelligence like ours but incapable of perceiving more than two dimensions (two being the number of dimensions of a circle), living on a spherical surface, they will consider the world they live in to be flat or indefinite, even when we know that it is finite and limitless limitless, since nowhere do we encounter the impossibility of moving to a neighboring point, no matter which direction is considered. If, on the other hand, these beings were in possession of the principles of Euclidean ge- ometry, they would deduce, for example, that the [ratio] between the length of the circum- ference and the diameter was equal to N[π]. But, if by using a materialized straight line segment they were to measure the length of a circle drawn in their world and that of the diameter they imagine to be a straight line, they would be surprised to find a different value. They would thus arrive at the certainty that the two-dimensional Euclidean geometry they had constructed does not apply to real objects. If they were to introduce a third dimension, which escapes their conceptual capabilities, they would be able to re-establish harmony