Lecture 1 (January 24, 2006): Introduction to the course; Axiomatic method of geometry; undefined terms (point, line, lie on, between, congruent); Euclid’s definitions and postulates; the parallel postulate; a preview of what happens when the parallel postulate is negated.
Lecture 2 (January 26): Definitions of segment, circle, ray, angle, opposite rays, supplementary angle, right angle. Logic rules.
Lecture 3 (January 31): More logic rules. Incidence Axioms and propositions that can be deduced from the incidence axioms alone.
Lecture 4 (February 2): Models and interpretations of points and lines. Why the Euclidean parallel property is not deducible from the incidence axioms. The Betweenness Axioms and Proposition 3.1.
Lecture 5 (February 7): Propositions 3.2, 3.3, 3.4. Discussion of Exercise 9 of Chapter1.
Lecture 6 (February 9): Pasch’s theorem; definition of interior of angle and triangle; Propositions 3.5 through 3.9, including Crossbar Theorem.
Lecture 7 (February 14): Congruence Axioms, Propositions 3.10 to 3.12
Lecture 8 (February 16): Propositions 3.13 to 3.18, comparisons and orderings of segments and angles
Lecture 9 (February 21): Propositions 3.19 and 3.23
Lecture 10 (Februray 28): Alternate Interior Angle Theorem
Lecture 11 (March 2): Exterior Angle Theorem and Corollaries
Lecture 12 (March 7): Saccheri-Legendre Theorem
Lecture 13 (March 9): Equivalent Forms of the Parallel Postulate
Lecture 14 (March 21): Angle Sums of Triangles
Lecture 15 (March 23): Attempts by Proclus, Wallis, and Saccheri to prove Euclid’s Parallel Postulate.
Lecture 16 (March 28): Propositions 4.7-4.11 (Equivalent Forms of the Parallel Postulate)
Lecture 17 (March 30): Attempts by Clairaut and Legendre to prove Euclid’s Parallel Postulate.
Lecture 18 (April 4): Hyperbolic Axiom; Universal Hyperbolic Theorem; Similar triangles are congruent.
Lecture 19 (April 11): Parallel lines that admit a common perpendicular (in hyperbolic geometry)
Lecture 20 (April 13): Limiting Parallel rays
Lecture 21 (April 18): More on limiting parallel rays
Lecture 22 (April 20): Consistency of hyperbolic geometry; the Beltrami-Klein model
Lecture 23 (April 25): The Poincare model
Lecture 24 (April 27): Inversion in circles; finding the Poincare line between two points