The Non-Euclidean, Hyperbolic Plane

I Some Historical Background.- 1. The Parallel-Postulate Problem.- 2. The Lost Centuries.- 3. Saccheri and the “Near Miss”.- 4. The Correct Perspective—Gauss, Bolyai, Lobatchevsky.- 5. Not-Euclidean Geometry, Absolute Geometry.- II Absolute Plane Geometry.- 1. Linear Sets and Linear Order.- 2. Half-Planes, Angles, and Angle Measure.- 3. Triangle Relations, Congruence, Foot in a Set.- 4. Non-intersecting Lines.- 5. Dedekind Cut, Continuity, A Basic Circle Property.- 6. Motions and Symmetries.- III Hyperbolic Plane Geometry.- 1. Hyperbolic Parallels.- 2. Biangles, Hyperparallels.- 3. Saccheri and Lambert Quadrilaterals, Polygon Angle Sums.- 4. Angle of Parallelism Function, Triangle Defect, Distance Variations.- 5. The 3-Point Property, Cycles.- 6. Hyperbolic Compass and Straight Edge Constructions.- 7. Existence Problems; the Method of Associated Right Triangles.- IV A Euclidean Model of the Hyperbolic Plane.- 1. An Overview of the Model.- 2. Circular Inversions in E2.- 3. Angle and Cross Ratio Invariance Under Inversion.- 4. Linear Order and Motions in the Model.- 5. Half-Planes, Angles and Angle Measure in the Model.- 6. Triangle Congruence in the Model, the Consistency of Hyperbolic Geometry.- Appendix Distance Geometrics.- Topic I. Metric Space and Metric Geometry.- Topic II. A Spherical Metric.- Topic III. Elliptic Geometry.- Topic IV. Barbilian Geometries, the Cross Ratio Metric.