Revolutions of Geometry (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

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. Cover......Page 1 CONTENTS......Page 9 Preface......Page 13 Acknowledgments......Page 15 PART I FOUNDATIONS......Page 17 1 The First Geometers......Page 19 1.1 Egypt......Page 22 1.2 Babylon......Page 29 1.3 China......Page 36 2 Thales......Page 43 2.1 The Axiomatic System......Page 45 2.2 Deductive Logic......Page 51 2.3 Proof Writing......Page 59 3 Plato and Aristotle......Page 69 3.1 Form......Page 72 3.2 Categorical Propositions......Page 78 3.3 Categorical Syllogisms......Page 88 3.4 Figures......Page 93 PART II THE GOLDEN AGE......Page 101 4 Pythagoras......Page 103 4.1 Number Theory......Page 107 4.2 The Pythagorean Theorem......Page 114 4.3 Archytas......Page 118 4.4 The Golden Ratio......Page 126 5 Euclid......Page 139 5.1 The Elements......Page 140 5.2 Constructions......Page 146 5.3 Triangles......Page 154 5.4 Parallel Lines......Page 163 5.5 Circles......Page 175 5.6 The Pythagorean Theorem Revisited......Page 183 6 Archimedes......Page 189 6.1 The Archimedean Library......Page 190 6.2 The Method of Exhaustion......Page 198 6.3 The Method......Page 209 6.4 Preliminaries to the Proof......Page 220 6.5 The Volume of a Sphere......Page 230 PART Ill ENLIGHTENMENT......Page 241 7 François Viète......Page 243 7.1 The Analytic Art......Page 245 7.2 Three Problems......Page 252 7.3 Conic Sections......Page 262 7.4 The Analytic Art in Two Variables......Page 273 8 René Descartes......Page 283 8.1 Compasses......Page 285 8.2 Method......Page 290 8.3 Analytic Geometry......Page 295 9 Gérard Desargues......Page 309 9.1 Projections......Page 310 9.2 Points at Infinity......Page 314 9.3 Theorems of Desargues and Menelaus......Page 322 9.4 Involutions......Page 328 PART IV A STRANGE NEW WORLD......Page 337 10 Giovanni Saccheri......Page 339 10.1 The Question of Parallels......Page 340 10.2 The Three Hypotheses......Page 346 10.3 Conclusions for Two Hypotheses......Page 353 10.4 Properties of Parallel Lines......Page 356 10.5 Parallelism Redefined......Page 365 11 Johann Lambert......Page 369 11.1 The Three Hypotheses Revisited......Page 371 11.2 Polygons......Page 376 11.3 Omega Triangles......Page 389 11.4 Pure Reason......Page 399 12 Nicolai Lobachevski and János Bolyai......Page 409 12.1 Parallel Fundamentals......Page 413 12.2 Horocycles......Page 420 12.3 The Surface of a Sphere......Page 430 12.4 Horospheres......Page 440 12.5 Evaluating the Pi Function......Page 447 PART V NEW DIRECTIONS......Page 457 13 Bernhard Riemann......Page 459 13.1 Metric Spaces......Page 461 13.2 Topological Spaces......Page 473 13.3 Stereographic Projection......Page 480 13.4 Consistency of Non-Euclidean Geometry......Page 487 14 Jean-Victor Poncelet......Page 499 14.1 The Projective Plane......Page 502 14.2 Duality......Page 508 14.3 Perspectivity......Page 517 14.4 Homogeneous Coordinates......Page 523 15 Felix Klein......Page 535 15.1 Group Theory......Page 536 15.2 Transformation Groups......Page 545 15.3 The Principal Group......Page 551 15.4 Isometries of the Plane......Page 559 15.5 Consistency of Euclidean Geometry......Page 569 References......Page 581 Index......Page 589 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.--jacket. Pt. I. Foundations. 1. The First Geometers -- 2. Thales -- 3. Plato And Aristotle -- Pt. Ii. The Golden Age. 4. Pythagoras -- 5. Euclid -- 6. Archimedes -- Pt. Iii. Enlightenment. 7. François Viète -- 8. René Descartes -- 9. Gérard Desargues -- Pt. Iv. A Strange New World. 10. Giovanni Saccheri -- 11. Johann Lambert -- 12. Nicolai Lobachevski And János Bolyai -- Pt. V. New Directions. 13. Bernhard Riemann -- 14. Jean-victor Poncelet -- 15. Felix Klein. Michael O'leary. Includes Bibliographical References (p. 565-571) And Index. **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.** "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."--BOOK JACKET