L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 5 7 Instead of forming the grid with the bars, one could, with their help, draw a grid of lines (coordinates). When it is possible to draw grids with lines U and V in this way (Figure 1), which cross one another at right angles and such that they form squares, the surface is Eu- clidean. In that case, the distance ds between points P and P’, whose coordinates differ by du and dv, is expressed by the equation: (1) Clearly, other lines could be used, for example a grid of concentric circles with a group of straight lines emanating from the center of these. In this case, assuming a Euclidean sur- face, even when the expression of the distance between two points differs in form from the previous one, it nonetheless remains defined by the coordinates. This is characteristic of Euclidean continua. For example, given the coordinates of three points, one can calculate the distance separating them, the angle formed by the lines connecting them, and the area these lines enclose. Gaussian Coordinates We have seen—said Dr. Einstein—that in the case of the rotating disk, it is impossible to achieve Euclidean geometrical constructions by using solid bars that is, its surface is not a Euclidean continuum. We must therefore abandon the Cartesian method of coordinates and replace it with a different one that does not assume that possibility. Gauss, the sage continued, has developed the geometry of surfaces without using the space in which it is located, so that it is applicable regardless of the nature of that space. Points not contained on the surface itself are excluded from his considerations. Gauss, he added, imagines a system of “u” and “v” curves drawn on the surfaces and numbers them (Figure 2). (We should draw them using one or more standard rulers.) Only those in one “family” cross those of the other. Each point on the surface corresponds to a carefully determined value for “u” and another for “v”: these two values are their coordinates. ds2 du2 dv2 +=