# Geometry: Euclid and Beyond

• Robin Hartshorne
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xi
2. Robin Hartshorne
Pages 1-6
3. Robin Hartshorne
Pages 7-63
4. Robin Hartshorne
Pages 65-116
5. Robin Hartshorne
Pages 117-163
6. Robin Hartshorne
Pages 165-194
7. Robin Hartshorne
Pages 195-239
8. Robin Hartshorne
Pages 241-294
9. Robin Hartshorne
Pages 295-433
10. Robin Hartshorne
Pages 435-480
11. Back Matter
Pages 481-528

### Introduction

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.

### Keywords

Area Euclid Euclid's Elements Geometry Non-Euclidean Geometry

#### Authors and affiliations

• Robin Hartshorne
• 1
1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-0-387-22676-7
• Copyright Information Springer-Verlag New York 2000
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4419-3145-0
• Online ISBN 978-0-387-22676-7
• Series Print ISSN 0172-6056
• Buy this book on publisher's site