## 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

II | 9 |

III | 10 |

IV | 20 |

V | 29 |

VI | 47 |

VII | 53 |

VIII | 67 |

IX | 68 |

XXXIII | 243 |

XXXV | 244 |

XXXVI | 252 |

XXXVII | 261 |

XXXVIII | 272 |

XXXIX | 282 |

XL | 297 |

XLI | 298 |

X | 75 |

XI | 83 |

XII | 92 |

XIII | 98 |

XIV | 106 |

XV | 114 |

XVI | 119 |

XVII | 120 |

XVIII | 130 |

XIX | 137 |

XX | 142 |

XXI | 150 |

XXII | 160 |

XXIII | 167 |

XXIV | 177 |

XXV | 188 |

XXVI | 197 |

XXVII | 198 |

XXVIII | 207 |

XXIX | 214 |

XXX | 223 |

XXXI | 228 |

XXXII | 233 |

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### Common terms and phrases

algebraic altitudes angle bisectors angle oz Archimedes axiom Cartesian plane circle with center circular inversion congruent convex cross-ratio cube deﬁne deﬁnition dihedral angles dissection draw equal content equation equidecomposable equilateral triangle equivalent Euclid’s Euclid’s Elements example Exercise exists ﬁeld extension ﬁeld F ﬁgure ﬁnd ﬁnite number ﬁrst ﬁve ﬁxed follows Galois group given line Hence Hilbert plane hyperbolic plane icosahedron intersection isomorphic isosceles lemma Let ABC limiting parallel line segments marked ruler meet midpoint neutral geometry non-Euclidean geometry octahedron ordered ﬁeld parallel axiom parallel postulate perpendicular Poincare model polygon polyhedra polyhedron polynomial problem proof Proposition prove radius real Cartesian plane real numbers rectangle reﬂection result right angles right triangle rigid motion rotation ruler and compass Saccheri quadrilateral satisﬁes Section sides splitting ﬁeld square roots steps subﬁeld subgroup suppose tangent tetrahedron theorem triangle ABC unique vertex vertices