## Geometry: Euclid and BeyondIn recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks. |

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### Contents

7 | |

Hilberts Axioms | 65 |

Geometry over Fields | 117 |

Congruence of Segments and Angles | 140 |

Rigid Motions and SAS | 148 |

NonArchimedean Geometry | 158 |

Segment Arithmetic | 165 |

Similar Triangles | 175 |

Finite Field Extensions | 280 |

NonEuclidean Geometry | 295 |

History of the Parallel Postulate | 296 |

Neutral Geometry | 304 |

Archimedean Neutral Geometry | 319 |

NonEuclidean Area | 326 |

Circular Inversion | 334 |

Circles Determined by Three Conditions | 346 |

Introduction of Coordinates | 186 |

Area | 195 |

Area in Euclids Geometry | 196 |

Measure of Area Functions | 205 |

Dissection | 212 |

Quadratura Circuli | 221 |

Euclids Theory of Volume | 226 |

Hilberts Third Problem | 231 |

Construction Problems and Field Extensions | 241 |

Three Famous Problems | 242 |

The Regular 17Sided Polygon | 250 |

Constructions with Compass and Marked Ruler | 259 |

Cubic and Quartic Equations | 270 |

The Poincaré Model | 355 |

Hyperbolic Geometry | 373 |

Hilberts Arithmetic of Ends | 388 |

Hyperbolic Trigonometry | 403 |

Characterization of Hilbert Planes | 415 |

Polyhedra | 435 |

The Five Regular Solids | 436 |

Eulers and Cauchys Theorems | 448 |

Brief Euclid | 481 |

495 | |

499 | |

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algebraic altitudes angle bisectors Archimedes area function axiom Cartesian plane circle with center circular inversion congruent construction convex cross-ratio cube define definition dihedral angles dissection draw edges equal content equation equidecomposable equilateral triangles equivalent Euclid's Elements Euclidean plane example Exercise exists faces field extension field F figure finite number follows Galois group given group G Hence Hilbert's axioms hyperbolic plane icosahedron inscribed intersection isomorphic lemma Let ABC limiting parallel line segments marked ruler meet midpoint non-Euclidean geometry obtain octahedron ordered field orthogonal P-line parallel postulate perpendicular Poincaré model polygon polyhedra polyhedron polynomial problem proof Proposition prove radius real numbers rectangle result right angles right triangle rigid motion rotation ruler and compass Saccheri quadrilateral Section segment arithmetic sides splitting field square roots steps subgroup suppose symmetry tangent tetrahedron theorem triangle ABC unique vertex vertices