Geometry: Euclid and Beyond

1. Front Matter

   Pages i-xi

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2. Introduction

   • Robin Hartshorne

   Pages 1-6

3. Euclid’s Geometry

   • Robin Hartshorne

   Pages 7-63

4. Hilbert’s Axioms

   • Robin Hartshorne

   Pages 65-116

5. Geometry over Fields

   • Robin Hartshorne

   Pages 117-163

6. Segment Arithmetic

   • Robin Hartshorne

   Pages 165-194

7. Area

   • Robin Hartshorne

   Pages 195-239

8. Construction Problems and Field Extensions

   • Robin Hartshorne

   Pages 241-294

9. Non-Euclidean Geometry

   • Robin Hartshorne

   Pages 295-433

10. Polyhedra

    • Robin Hartshorne

    Pages 435-480

11. Back Matter

    Pages 481-528

    PDF

In 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.

• Area
• Euclid
• Euclid's Elements
• Geometry
• Non-Euclidean Geometry

• Department of Mathematics, University of California, Berkeley, USA

  Robin Hartshorne

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