Revolutions of Geometry
Published on 10. March 2010
608 pages
978-0-470-59179-6 (ISBN)
Description
Guides readers through the development of geometry and basicproof writing using a historical approach to the topic
In an effort to fully appreciate the logic and structure ofgeometric proofs, Revolutions of Geometry places proofs intothe context of geometry's history, helping readers to understandthat proof writing is crucial to the job of a mathematician.Written for students and educators of mathematics alike, the bookguides readers through the rich history and influential works, fromancient times to the present, behind the development of geometry.As a result, readers are successfully equipped with the necessarylogic to develop a full understanding of geometric theorems.
Following a presentation of the geometry of ancient Egypt,Babylon, and China, the author addresses mathematical philosophyand logic within the context of works by Thales, Plato, andAristotle. Next, the mathematics of the classical Greeks isdiscussed, incorporating the teachings of Pythagoras and hisfollowers along with an overview of lower-level geometry usingEuclid's Elements. Subsequent chapters explore the work ofArchimedes, Viete's revolutionary contributions to algebra,Descartes' merging of algebra and geometry to solve the Pappusproblem, and Desargues' development of projective geometry. Theauthor also supplies an excursion into non-Euclidean geometry,including the three hypotheses of Saccheri and Lambert and the nearsimultaneous discoveries of Lobachevski and Bolyai. Finally, moderngeometry is addressed within the study of manifolds and ellipticgeometry inspired by Riemann's work, Poncelet's return toprojective geometry, and Klein's use of group theory tocharacterize different geometries.
The book promotes the belief that in order to learn how to writeproofs, one needs to read finished proofs, studying both theirlogic and grammar. Each chapter features a concise introduction tothe presented topic, and chapter sections conclude with exercisesthat are designed to reinforce the material and provide readerswith ample practice in writing proofs. In addition, the overallpresentation of topics in the book is in chronological order,helping readers appreciate the relevance of geometry within thehistorical development of mathematics.
Well organized and clearly written, Revolutions ofGeometry is a valuable book for courses on modern geometry andthe history of mathematics at the upper-undergraduate level. It isalso a valuable reference for educators in the field ofmathematics.
Reviews / Votes
"An excellent supplemental resource or main textbook for an overview of mathematics course for upper-level undergraduate and graduate students." (Choice, October 2010).More details
Other editions
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Person
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.
Content
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 III 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.
