New Foundation of Euclidean Geometry
New Foundation of Euclidean Geometry
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欧几里得几何新基础
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发表时间：
1931
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通讯作者：
K. Menger
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作者：
K. Menger
我的第二篇关于度量几何的论文 * 包含了n维欧几里德空间在一般半度量空间中的一个特征，即它的点的距离之间的关系。在课程的度量几何在美国大学我已经大大缩短和修改我原来的证明和推广的配方引进的概念同余秩序。下面的文章包含了这些新的证明。在第一部分中，我们证明了每个半度量空间，其中每个n + 3点与n维欧氏空间的n + 3点全等，都与n维欧氏空间的子集全等。这可以通过说n维欧几里得空间的同余阶为n + 3来表示。在第二部分中，我们证明了每个包含n + 3个以上点的半度量空间，其中每个n + 2个点与n维欧氏空间的n + 2个点全等，都与n维欧氏空间的一个子集全等。这一事实表示为说，R。具有拟同余阶n + 2。本文通过系统地研究恰好包含n + 3个点且不与n维欧氏空间中的n + 3个点相对应的集合，而其中的每n + 2个点与n维欧氏空间中的n + 2个点相对应的集合，证明了这一点。这些集合称为伪欧几里德集合。利用这些结果，问题被归结为：在什么条件下n + 2点与R上的n + 2点全等.伪欧几里德（n + 3）元组是用什么距离关系来刻画的。第三部分解决了这些纯代数问题。
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