Finite geometries: Pappus and Desargues.
a. Axioms for the finite geometry of Pappus.
1. There exits at least one line.
2. Every line has exactly three points.
3. Not all points are on the same line.
4. Given a line ,m, and a point, P, not on m there exists a unique line parallel to m passing through P.
5. If P is a point not on a line, m, there exists a unique point on P, Q, such no line contains both P and Q.
6. Except for the case described in Axiom 5, given any two distinct points there exists a single line through them.
2. Axioms for the finite geometry of Desargues.
1. There exits at least one point.
2. Each point has at least one polar.
3. Every line has at most one pole.
4. Each two distinct points are on at most one line.
5. Every line has exactly three distinct points on it.
6. If a line does not contain a certain point, then there is a point on both the line and any polar of the point.