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Revolutions of Geometry

Michael L. O'Leary

ISBN: 978-0-470-16755-7 February 2010 608 Pages

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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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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.


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.


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.


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.


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.



"An excellent supplemental resource or main textbook for an overview of mathematics course for upper-level undergraduate and graduate students." (Choice, October 2010).
  • Highlights from the history of geometry are intertwined with explanations on how to read and write proofs
  • This is the first book to present the works of Euclid and Hilbert in addition to other geometers in chronological order, all in an effort to show how the subject matter developed over time
  • Hints and both partial and complete solutions are included at the end of the book as an aid for selected exercises
  • An important contribution to the teaching of geometry, this book proves that learning to read and write proofs is a crucial aspect of the subject
  • An Instructor's Solutions Manual is available upon request to the Wiley editorial department