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

✍ Scribed by Michael O'Leary

• Publisher: Wiley
• Year: 2010
• Tongue: English
• Leaves: 607
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
CONTENTS
Preface
Acknowledgments
PART I FOUNDATIONS
1 The First Geometers
1.1 Egypt
1.2 Babylon
1.3 China
2 Thales
2.1 The Axiomatic System
2.2 Deductive Logic
2.3 Proof Writing
3 Plato and Aristotle
3.1 Form
3.2 Categorical Propositions
3.3 Categorical Syllogisms
3.4 Figures
PART II THE GOLDEN AGE
4 Pythagoras
4.1 Number Theory
4.2 The Pythagorean Theorem
4.3 Archytas
4.4 The Golden Ratio
5 Euclid
5.1 The Elements
5.2 Constructions
5.3 Triangles
5.4 Parallel Lines
5.5 Circles
5.6 The Pythagorean Theorem Revisited
6 Archimedes
6.1 The Archimedean Library
6.2 The Method of Exhaustion
6.3 The Method
6.4 Preliminaries to the Proof
6.5 The Volume of a Sphere
PART Ill ENLIGHTENMENT
7 François Viète
7.1 The Analytic Art
7.2 Three Problems
7.3 Conic Sections
7.4 The Analytic Art in Two Variables
8 René Descartes
8.1 Compasses
8.2 Method
8.3 Analytic Geometry
9 Gérard Desargues
9.1 Projections
9.2 Points at Infinity
9.3 Theorems of Desargues and Menelaus
9.4 Involutions
PART IV A STRANGE NEW WORLD
10 Giovanni Saccheri
10.1 The Question of Parallels
10.2 The Three Hypotheses
10.3 Conclusions for Two Hypotheses
10.4 Properties of Parallel Lines
10.5 Parallelism Redefined
11 Johann Lambert
11.1 The Three Hypotheses Revisited
11.2 Polygons
11.3 Omega Triangles
11.4 Pure Reason
12 Nicolai Lobachevski and János Bolyai
12.1 Parallel Fundamentals
12.2 Horocycles
12.3 The Surface of a Sphere
12.4 Horospheres
12.5 Evaluating the Pi Function
PART V NEW DIRECTIONS
13 Bernhard Riemann
13.1 Metric Spaces
13.2 Topological Spaces
13.3 Stereographic Projection
13.4 Consistency of Non-Euclidean Geometry
14 Jean-Victor Poncelet
14.1 The Projective Plane
14.2 Duality
14.3 Perspectivity
14.4 Homogeneous Coordinates
15 Felix Klein
15.1 Group Theory
15.2 Transformation Groups
15.3 The Principal Group
15.4 Isometries of the Plane
15.5 Consistency of Euclidean Geometry
References
Index