• 한국수학사학회지
• /
• 제32권5호
• /
• pp.241-255
• /
• 2019

The proposition that the parallel axiom and the Pythagorean theorem are equivalent in the Hilbert geometry is true when the Archimedean axiom is assumed. In this article, we examine some specific plane geometries to see the existence of the non-archimidean Hilbert geometry in which the Pythagorean theorem holds but the parallel axiom does not. Furthermore we observe that the Pythagorean theorem is equivalent to the fact that the Hilbert geometry is actually a semi-Euclidean geometry.

• 대한수학교육학회지:학교수학
• /
• 제1권1호
• /
• pp.7-17
• /
• 1999

The purpose of this note is to give insights into the role of parallelism to the middle school mathematics. According to NCTM(1998), the middle grades mathematics classroom should be a place in which students regularly engage in thoughtful activity that relates to their emerging capabilities of finding and imposing structure, conjecturing and verifying, and engaging in abstraction and generalization. In this aspect it is desirable to reinvestigate Euclidean Parallel Postulate in the axiomatic point of view in the level of the middle school mathematics. Reconstructing the contents Greenberg(1993), and we explain the statements which are logically equivalent to the Parallel Postulate for mathematics teachers in the middle school.