History of the Parallel Postulate Chapter 5

Clavius’ Axiom For any line l and any point P not on l, the equidistant locus to l through P is the set of all the points on a line through P.

Theorem; The following three statements are equivalent for a Hilbert plane:  a) The plane is semi-Euclidean  b) For any line l and any point P not on l the equidistant locus to l through P is the set of all the points on the parallel to l through P obtained by the standard construction.  c) Clavis’ axiom.

Wallis Postulate: Given any triangle  ABC and given any segment DE, there exists a triangle  DEF having DE as one of its sides such that  ABC ~  DEF. (Two triangles are similar ( ~) if their vertices can be put in 1-1 correspon- dence in such a way that corresponding angles are congruent.

Clairaut’s Axiom Rectangles exist.

Proclus Theorem: The Euclidean parallel postulate hold on a Hilbert plane iff the plane is semi-Euclidean (i.e., the angle sum of a triangle is 180°) and Aristotle’s angle unboundedness axiom holds. In particular, the Euclidean parallel postulate holds in an Archimedean semi-Euclidean plane.

Legendre ( with Archimedes Axiom) Theorem: For any acute  A and any point D in the interior of  A, there exists a line through D and not through A that interesects both sides of  A. (The sum of every triangle is 180°) Axiom: For any acute angle and any point in the interior of that angle, there exists a line through that point and not through the angle vertex which intersects both sides of the angle.