Revolutions of Geometry / Edition 1

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Guides readers through the development of geometry and basic proof writing using a historical approach to the topic

In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfully equipped with the necessary logic to develop a full understanding of geometric theorems.

Following a presentation of the geometry of ancient Egypt, Babylon, and China, the author addresses mathematical philosophy and logic within the context of works by Thales, Plato, and Aristotle. Next, the mathematics of the classical Greeks is discussed, incorporating the teachings of Pythagoras and his followers along with an overview of lower-level geometry using Euclid's Elements. Subsequent chapters explore the work of Archimedes, Viete's revolutionary contributions to algebra, Descartes' merging of algebra and geometry to solve the Pappus problem, and Desargues' development of projective geometry. The author also supplies an excursion into non-Euclidean geometry, including the three hypotheses of Saccheri and Lambert and the near simultaneous discoveries of Lobachevski and Bolyai. Finally, modern geometry is addressed within the study of manifolds and elliptic geometry inspired by Riemann's work, Poncelet's return to projective geometry, and Klein's use of group theory to characterize different geometries.

The book promotes the belief that in order to learn how to write proofs, one needs to read finished proofs, studying both their logic and grammar. Each chapter features a concise introduction to the presented topic, and chapter sections conclude with exercises that are designed to reinforce the material and provide readers with ample practice in writing proofs. In addition, the overall presentation of topics in the book is in chronological order, helping readers appreciate the relevance of geometry within the historical development of mathematics.

Well organized and clearly written, Revolutions of Geometry is a valuable book for courses on modern geometry and the history of mathematics at the upper-undergraduate level. It is also a valuable reference for educators in the field of mathematics.

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Editorial Reviews

From the Publisher

"An excellent supplemental resource or main textbook for an overview of mathematics course for upper-level undergraduate and graduate students." (Choice, October 2010).

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Product Details

Meet the Author

Michael O'Leary, PhD, is Professor of Mathematics at the College of DuPage. He received his PhD in mathematics from the University of California, Irvine in 1994.

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Table of Contents

Preface xi

Acknowledgments xiii

Part I Foundations

1 The First Geometers 3

1.1 Egypt 6

1.2 Babylon 13

1.3 China 20

2 Thales 27

2.1 The Axiomatic System 29

2.2 Deductive Logic 35

2.3 Proof Writing 43

3 Plato and Aristotle 53

3.1 Form 56

3.2 Categorical Propositions 62

3.3 Categorical Syllogisms 72

3.4 Figures 77

Part II The Golden Age

4 Pythagoras 87

4.1 Number Theory 91

4.2 The Pythagorean Theorem 98

4.3 Archytas 102

4.4 The Golden Ratio 110

5 Euclid 123

5.1 The Elements 124

5.2 Constructions 130

5.3 Triangles 138

5.4 Parallel Lines 147

5.5 Circles 159

5.6 The Pythagorean Theorem Revisited 167

6 Archimedes 173

6.1 The Archimedean Library 174

6.2 The Method of Exhaustion 182

6.3 The Method 193

6.4 Preliminaries to the Proof 204

6.5 The Volume of a Sphere 214

Part III Enlightenment

7 François Viète 227

7.1 The Analytic Art 229

7.2 Three Problems 236

7.3 Conic Sections 246

7.4 The Analytic Art in Two Variables 257

8 René Descartes 267

8.1 Compasses 269

8.2 Method 274

8.3 Analytic Geometry 279

9 Gérard Desargues 293

9.1 Projections 294

9.2 Points at Infinity 298

9.3 Theorems of Desargues and Menelaus 306

9.4 Involutions 312

Part IV A Strange New World

10 Giovanni Saccheri 323

10.1 The Question of Parallels 324

10.2 The Three Hypotheses 330

10.3 Conclusions for Two Hypotheses 337

10.4 Properties of Parallel Lines 340

10.5 Parallelism Redefined 349

11 Johann Lambert 353

11.1 The Three Hypotheses Revisited 355

11.2 Polygons 360

11.3 Omega Triangles 373

11.4 Pure Reason 383

12 Nicolai Lobachevski János Bolyai 393

12.1 Parallel Fundamentals 397

12.2 Horocycles 404

12.3 The Surface of a Sphere 414

12.4 Horospheres 424

12.5 Evaluating the Pi Function 431

Part V New Directions

13 Bernhard Riemann 443

13.1 Metric Spaces 445

13.2 Topological Spaces 457

13.3 Stereographic Projection 464

13.4 Consistency of Non-Euclidean Geometry 471

14 Jean-Victor Poncelet 483

14.1 The Projective Plane 486

14.2 Duality 492

14.3 Perspectivity 501

14.4 Homogeneous Coordinates 507

15 Felix Klein 519

15.1 Group Theory 520

15.2 Transformation Groups 529

15.3 The Principal Group 535

15.4 Isometries of the Plane 543

15.5 Consistency of Euclidean Geometry 553

References 565

Index 573

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