1. Euclid
2. Incidence geometry
3. Axioms for plane geometry
4. Angles
5. Triangles
6. Models of neutral geometry
7. Perpendicular and parallel lines
8. Polygons
9. Quadrilaterals
10. The Euclidean parallel postulate
11. Area
12. Similarity
13. Right triangles
14. Circles
15. Circumference and circular area
16. Compass and straightedge constructions
17. The parallel postulate revisited
18. Introduction to hyperbolic geometry
19. Parallel lines in Hyperbolic geometry
20. Epilogue: Where do we go from here?
Appendices

The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a mode of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom.