1. A set S of undefined elements, for suggestiveness called is called a semi-metrical space provided that for each two elements p, q of S there corresponds a not negative real number, called the distance between the points p and q, such that, denoting this number by pq, we have pq = qp, while pq 0 if and only if the points p anid q are identical. If for two pairs of points p, q; pl, q' we have the relation pq =p'q' we shall say that the two pairs of points are congruent. A mapping of a set S upon a set S' is called a congruent mapping if to each pair of points of S there corresponds a congruent pair of points of S'. Finally, two sets S' and S' are called congruent provided there exists a congruent mapping of one upon the other. Karl Menger has characterized the n-dimensional euclidean space R., among general semi-metrical spaces by means of relations between the distances of its points.* He has shown that the R. has the congruence order n + 3. This means that every semi-metrical space, each n + 3 points of which is congruent with n + 3 points of the R,,, is congruent with a subset of the R1,. It is further shown that each semi-metrical space containing more than n + 3 points, each n + 2 of which is congruenit with iA + 2 points of the R,, is congruent with a subset of the Rn. The notion embodied in) this important result is expressed by saying that the Rn has the quasi-congruence order n + 2. Thus, if a set of points is such that each n + 2 of the points is congruent with n + 2 points of the R,,, while the whole set is not congruent with a subset of the R., the set must consist of exactly n + 3 points. Such sets are called pseudo-euclidean. Let us denote by Sn the n-dimensional sphere; that is, SO is a pair of points, S1 is a circle (the term circle denoting throughout this paper a curve), S2 is the surface of a sphere is three-space, etc. It has been shown that the space Sn has the congruence order n + 3 also; but it has not the quasi-congruence order n + 2. Thus, for' the circle S1 the sets analogous to