By Akira Yamamoto; Stephen A Gourlay; Arnaud Devred; Lawrence Berkeley National Laboratory.; United States. Dept. of Energy. Office of Scientific and Technical Information.; All authors

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**Sample text**

This relation is independent of the 5th postulate, belongs to the realm of absolute geometry, and has the reflexivity, symmetry, and transitivity properties: A 2 A; if A 2 B, then B 2 A; if A 2 B and B 2 C , then A 2 C. If a relation has the above properties, then it is called an equivalence relation. It is well known that any equivalence relation in a The Revolution of Ja'nos Bolyai Figure 3. Corresponding points set gives rise to a subdivision of the set into disjoint subsets. These are called equivalence classes.

Earlier we mentioned that the lines have directions too, let us designate them as M , N (see Figure 3). Assume that the angle M A B is equal to the angle NBA. Then the points A, B are called isogonal corresponding, or briefly corresponding points (Gauss' terminology) and the fact is expressed by the relation A 2 B (notation of JBnos Bolyai). This relation is independent of the 5th postulate, belongs to the realm of absolute geometry, and has the reflexivity, symmetry, and transitivity properties: A 2 A; if A 2 B, then B 2 A; if A 2 B and B 2 C , then A 2 C.

This approach has been followed for a long time. Euclid's fifth postulate, the starting-point of various geometrical systems, is J&nos Bolyai's llthaxiom. Over the years Elements has had several editions, some of which of have been supplemented. In this article, the terms 'fifth postulate' and 'eleventh axiom', will be used interchangeably. In addition to the axioms in Elements, there are also definitions and theorems. Euclid defines the concepts of 'point' and 'straight line' as follows: A 'point' is that which has no parts and a 'straight line' is a length without width.