**Pure and Applied Undergraduate Texts**

Volume: 21;
2013;
469 pp;
Hardcover

MSC: Primary 51;

Print ISBN: 978-0-8218-8478-2

Product Code: AMSTEXT/21

List Price: $75.00

Individual Member Price: $60.00

**Electronic ISBN: 978-1-4704-1436-8
Product Code: AMSTEXT/21.E**

List Price: $75.00

Individual Member Price: $60.00

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#### Supplemental Materials

# Axiomatic Geometry

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*John M. Lee*

The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a model of logical thought.

This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom.

#### Table of Contents

# Table of Contents

## Axiomatic Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface xi12 free
- Euclid 120 free
- Incidence geometry 2342
- Axioms for plane geometry 5372
- Angles 83102
- Triangles 103122
- Models of neutral geometry 123142
- Perpendicular and parallel lines 141160
- Polygons 155174
- Quadrilaterals 175194
- The Euclidean parallel postulate 185204
- Area 199218
- Similarity 213232
- Right triangles 229248
- Circles 247266
- Circumference and circular area 279298
- Compass and straightedge constructions 295314
- The parallel postulate revisited 321340
- Introduction to hyperbolic geometry 337356
- Parallel lines in hyperbolic geometry 355374
- Epilogue: Where do we go from here? 369388
- Hilbert’s axioms 375394
- Birkhoff’s postulates 377396
- The SMSG postulates 379398
- The postulates used in this book 381400
- The language of mathematics 383402
- Proofs 405424
- Sets and functions 423442
- Properties of the real numbers 433452
- Rigid motions: Another approach 441460
- References 451470
- Index 455474 free
- Back Cover Back Cover1489

#### Readership

Undergraduate students interested in geometry and secondary mathematics teaching.

#### Reviews

In the preface, the author announces a textbook for undergraduate students who plan to teach geometry in a North American high-school; this intention is fulfilled perfectly. The author offers, among others, a comprehensive description of the historical development of axiomatic geometry, a careful approach to all arising problems, well-motivated definitions, an analysis of the procedure of proof writing, and plenty of very aesthetical, helpful diagrams. For the reader's convenience, many theorems are named by well-chosen catchwords, thus a very clearly arranged text is reached.

-- Zentralblatt Math

Lee's “Axiomatic Geometry” gives a detailed, rigorous development of plane Euclidean geometry using a set of axioms based on the real numbers. It is suitable for an undergraduate college geometry course, and since it covers most of the topics normally taught in American high school geometry, it would be excellent preparation for future high school teachers. There is a brief treatment of the non-Euclidean hyperbolic plane at the end.

-- Robin Hartshorne, University of California, Berkeley

The goal of Lee's well-written book is to explain the axiomatic method and its role in modern mathematics, and especially in geometry. Beginning with a discussion (and a critique) of Euclid's elements, the author gradually introduces and explains a set of axioms sufficient to provide a rigorous foundation for Euclidean plane geometry.

Because they assume properties of the real numbers, Lee's axioms are fairly intuitive, and this results in a presentation that should be accessible to upper level undergraduate mathematics students. Although the pace is leisurely at first, this book contains a surprising amount of material, some of which can be found among the many exercises. Included are discussions of basic trigonometry, hyperbolic geometry and an extensive treatment of compass and straightedge constructions.

-- I. Martin Isaacs, University of Wisconsin-Madison

Jack Lee's book will be extremely valuable for future high school math teachers. It is perfectly designed for students just learning to write proofs; complete beginners can use the appendices to get started, while more experienced students can jump right in. The axioms, definitions, and theorems are developed meticulously, and the book culminates in several chapters on hyperbolic geometry—a lot of fun, and a nice capstone to a two-quarter course on axiomatic geometry.

-- John H. Palmieri, University of Washington