It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.
H.L. SKALA
2. MENGER'S AXIOMS
We first list the axioms, nine in all, given by Menger et al. for the hyperbolic plane. (Points will be denoted by capital letters and lines by lower-case letters.)
- Any two distinct points are on exactly one line.
2. Each line is on at least one point.
- There exist three collinear points and three noncollinear points. 4. If the point P is on neither of two intersecting lines l and m, then P is on a line that meets l but not m. 5. Of three collinear points, at least one has the property that every line through it intersects at least one of each pair of intersecting lines through the other two. 6. If P is not on l, then there exist two distinct lines on P not meeting l and such that each line meeting l meets at least one of those two lines.
From these six axioms one can derive all the usual properties of linear and planar order ([3], [4], E5], E11]). The definition of betweenness is as follows: of three distinct collinear points A, B and C, the point B is said to be between A and C, and we write *(A, B, C), if every line through B intersects at least one line of each pair of intersecting lines which pass through A and C, respectively. Note that this definition would be useless for Euclidean geometry since, in the Euclidean plane, it would imply that any point is between any other two points collinear with it.
Using the concept of betweenness one can define ray and segment in the usual manner. Two rays r and s with endpoints R and S, respectively, are said to be parallel if r and s are noncollinear and if every line that meets one of the rays meets either the other ray or the segment RS. Two lines, or a ray and a line, are said to be parallel if they contain rays that are parallel. Note that this definition describes strict parallelism. That is to say, there are exactly two lines through a given point which are parallel to a given line. A further postulate is assumed:
7. Any two noncollinear rays have a common parallel line.
From these seven axioms, all the usual properties of betweenness and parallelism can be derived ([3], E4], E5], Eli]).
In order to state the next two postulates, we make the following definitions. A pair (a, b) of parallel lines is called a rimpoint and is said to lie on a line I if l is parallel to a and to b and there exists a line which intersects a, b and I. A