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chapter 1. euclid's geometry 1. a first look at euclid's elements 2. ruler and compass constructions 3. euclid's axiomatic method 4. construction of the regular pentagon 5. some newer results chapter 2. hilbert's axioms 6. axioms of incidence 7. axioms of betweenness 8. axioms of congruence for line segments 9. axioms of congruence for angles 10. hilbert planes 11. intersection of lines and circles 12. euclidean planes chapter 3. geometry over fields 13. the real cartesian plane 14. abstract fields and incidence 15. ordered fields and betweenness 16. congruence of segments and angles 17. rigid motions and sas 18. non-archimedean geometry chapter 4. segment arithmetic 19. addition and multiplication of line segments 20. similar triangles 21. introduction of coordinates chapter 5. area 22. area in euclid's geometry 23. measure of area functions 24. dissection 25. quadrature circuli 26. euclid's theory of volume 27. hilbert's third problem chapter 6. construction problems and field extensions 28. three famous problems 29. the regular 17-sided polygon 30. constructions with compass and marked ruler 31. cubic and quartic equations 32. appendix: finite field extensions chapter 7. non-euclidean geometry 33. history of the parallel postulate 34. neutral geometry 35. archimedean neutral geometry 36. non-euclidean area 37. circular inversion 38. digression: circles determined by three conditions 39. the poincare model 40. hyperbolic geometry 41. hilbert's arithmetic of ends 42. hyperbolic trigonometry 43. characterization of hilbert planes chapter 8. polyhedra 44. the five regular solids 45. euler's and cauchy's theorems 46. semiregular and face-regular polyhedra 47. symmetry groups of polyhedra appendix: brief euclid notes references list of axioms index of euclid's propositions index