The Non-Euclidean, Hyperbolic Plane Its Structure and Consistency

The discovery of hyperbolic geometry, and the subsequent proof that this geometry is just as logical as Euclid's, had a profound in­ fluence on man's understanding of mathematics and the relation of mathematical geometry to the physical world. It is now possible, due in large part to axioms devised by George Birkhoff, to give an accurate, elementary development of hyperbolic plane geometry. Also, using the Poincare model and inversive geometry, the equiconsistency of hyperbolic plane geometry and euclidean plane geometry can be proved without the use of any advanced mathematics. These two facts provided both the motivation and the two central themes of the present work. Basic hyperbolic plane geometry, and the proof of its equal footing with euclidean plane geometry, is presented here in terms acces­ sible to anyone with a good background in high school mathematics. The development, however, is especially directed to college students who may become secondary teachers. For that reason, the treatment is de­ signed to emphasize those aspects of hyperbolic plane geometry which contribute to the skills, knowledge, and insights needed to teach eucli­ dean geometry with some mastery.

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.