New Foundation of Euclidean Geometry

My second paper on metrical geometry * contains a characterisation of the n-dimensional euclidean space among general semi-metrical spaces in terms of relations between the distances of its points. In courses on metrical geometry at American universities I have considerably shortened and revised my original proofs and generalized the formulations by introducing the concept of congruence order. The following paper contains these new proofs. In the first part we prove that every semi-metrical space, each n + 3 points of which are congruent with n + 3 points of the n-dimensional euclidean space, is congruent with a subset of the n-dimensional euclidean space. This is expressed by saying that the n-dimensional euclidean space has the congruence order n + 3. In the second part we prove that each semi-metrical space containing more than n + 3 points each n + 2 points of which are congruent with n + 2 points of the n-dimensional euclidean space, is congruent with a subset of the n-dimensional euclidean space. This fact is expressed by saying that the R. has the quasi-congruence order n + 2. It is proved by a systematic study of those sets which contain exactly n + 3 points and are not corLgruent with n + 3 points of the n-dimensional euclidean space whereas each n + 2 of them are congruent with n + 2 points of the n-dimensional euclidean space. These sets are called pseudo-euclidean sets. By means of these results the problem is reduced to the question: under what conditions are n + 2 points congruent with n + 2 points of the R. and by what distance relations are the pseudo-euclidean (n + 3)-tuples characterized. These purely algebraic problems are solved in the third part.t