Geometry

A Metric Approach with Models

• Richard S. Millman
• George D. Parker
Textbook

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

1. Front Matter
Pages i-x
2. Richard S. Millman, George D. Parker
Pages 1-14
3. Richard S. Millman, George D. Parker
Pages 15-38
4. Richard S. Millman, George D. Parker
Pages 39-57
5. Richard S. Millman, George D. Parker
Pages 58-82
6. Richard S. Millman, George D. Parker
Pages 83-114
7. Richard S. Millman, George D. Parker
Pages 115-157
8. Richard S. Millman, George D. Parker
Pages 158-184
9. Richard S. Millman, George D. Parker
Pages 185-212
10. Richard S. Millman, George D. Parker
Pages 213-235
11. Richard S. Millman, George D. Parker
Pages 236-271
12. Richard S. Millman, George D. Parker
Pages 272-343
13. Back Matter
Pages 344-358

Introduction

This book is intended as a first rigorous course in geometry. As the title indicates, we have adopted Birkhoff's metric approach (i.e., through use of real numbers) rather than Hilbert's synthetic approach to the subject. Throughout the text we illustrate the various axioms, definitions, and theorems with models ranging from the familiar Cartesian plane to the Poincare upper half plane, the Taxicab plane, and the Moulton plane. We hope that through an intimate acquaintance with examples (and a model is just an example), the reader will obtain a real feeling and intuition for non­ Euclidean (and in particular, hyperbolic) geometry. From a pedagogical viewpoint this approach has the advantage of reducing the reader's tendency to reason from a picture. In addition, our students have found the strange new world of the non-Euclidean geometries both interesting and exciting. Our basic approach is to introduce and develop the various axioms slowly, and then, in a departure from other texts, illustrate major definitions and axioms with two or three models. This has the twin advantages of showing the richness of the concept being discussed and of enabling the reader to picture the idea more clearly. Furthermore, encountering models which do not satisfy the axiom being introduced or the hypothesis of the theorem being proved often sheds more light on the relevant concept than a myriad of cases which do.

Keywords

Cartesian Euclid Geometrie Geometry addition boundary element method concept feeling idea inca model real number theorem value-at-risk will

Authors and affiliations

• Richard S. Millman
• 1
• George D. Parker
• 2
1. 1.Department of Mathematical and Computer ScienceMichigan Technological UniversityHoughtonUSA
2. 2.Department of MathematicsSouthern Illinois UniversityCarbondaleUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-0130-1
• Copyright Information Springer-Verlag New York 1981
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4684-0132-5
• Online ISBN 978-1-4684-0130-1
• Series Print ISSN 0172-6056
• Buy this book on publisher's site