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History of the Parallel Postulate

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:

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History of the Parallel Postulate

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  1. History of the Parallel Postulate Chapter 5

  2. 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.

  3. 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.

  4. 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.

  5. Clairaut’s Axiom • Rectangles exist.

  6. 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.

  7. 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.

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