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“黎曼式非欧几何”发表于数学通报1963年第1期。本文所提的注释就是要严格証明:若采用黎曼几何的关联公理:平面內二直綫恆相交,則不能同时采用欧几里得的关联公理、順序公理、合同公理;也不能同时采用欧氏几何的关联公理、順序公理、連續公理。定理1.利用欧氏几何的关联公理、順序公理、合同公理,可以証明平面上存在着两条不相交的直綫。 証.平面上至少有一条直綫a及a上至少有A,B两点,如果过A,B两点关于a的垂直綫c,d交于一点C的話,那末△ABC的一个外角等于它不相邻的內角,这与合同公理的推論——外角定理矛盾。所以平面上有不相交的两条直綫c,d。
“Riemannian Non-Euclidean Geometry” was published in the 1st issue of the Mathematics Bulletin, 1963. The commentary in this paper is to rigorously prove that if Riemannian geometric axioms are used: the two lines in a plane are constantly intersecting, Euclid’s associated axioms, sequential axioms, and contract axioms cannot be used at the same time; nor can Euclidean be used simultaneously. Geometric axioms, sequential axioms, continuous axioms. Theorem 1. Using the Euclidean geometry axioms, sequential axioms, and contract axioms, it can be proved that there are two disjoint lines in the plane. At least one line a and a on the plane have at least two points A and B. If there are two points A and B on the vertical line c, d, which intersects at point C, then an external angle of △ ABC is equal to Adjacent internal angles, this contradicts the inference of contract axioms - external angle theorem. So there are two lines c, d that do not intersect in the plane.