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本文原稿是梁绍鸿先生一九七二年为北京师大数学系师资训练班学生所写讲稿的一都分。当时中学课本讲平行线时用等距定义:“同平面内处处距离相等的直线叫做平行线。”代替传统定义:“同平面内不相交的直线叫做平行线。”有人怀疑这两个定义是否等价。梁先生认为是等价的,但是必须另加公理。现在中学课本已经恢复了传统定义,问题已经过去。然而研究几何,由于直观的引诱往往陷入歧途,甚至造成循环论证而不自知。这篇论述在这方面指引得很清楚,很有参考价值。
The original manuscript of this article is the sum of Mr. Liang Shaohong’s speech written in 1972 for a teacher training class in the Beijing Normal University mathematics department. When the middle school textbooks were talking about parallel lines, they were defined with equal distances: “The straight lines at equal distances in the same plane are called parallel lines.” Instead of the traditional definition: “The lines that do not intersect in the same plane are called parallel lines.” Some wonder whether the two definitions are equivalence. Mr. Liang thinks it is equivalent, but it must add axioms. Now that the middle school textbook has been restored to its traditional definition, the problem is over. However, the study geometry, because the intuitive inducement often fall into the wrong direction, and even cause circular argument without knowing. This discussion is very clear in this respect and is of great value.