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Publisher:

John Wiley

Publication Date:

2010

Number of Pages:

587

Format:

Hardcover

Series:

Pure and Applied Mathematics: A Wiley Series of Texts, Monographs, and Tracts

Price:

115.00

ISBN:

9780470167557

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Art Gittleman

02/24/2011

Hybrid cars combine the advantages of gasoline and electric powered vehicles. Plant and animal breeding produces new hybrids with favorable qualities. *Revolutions of Geometry* is a hybrid of geometry and history with unique advantages. The text is suited for junior and senior mathematics students and especially for prospective teachers who will not only become competent in, but also enjoy, the material.

Geometry here occurs in its historical context interacting with algebra, logic, and calculus rather than as a dry presentation of one sort of modern geometry. Because the focus is on geometry, there is time to include both copious historical background and interesting mathematical derivations with numerous excellent exercises. A history of mathematics covering the whole spectrum of mathematics cannot go into as much mathematical detail which is appropriate if the emphasis is on the history but not if one desires a mathematical emphasis.

The text divides into five parts: Foundations, The Golden Age, Enlightenment, A New Strange World, and New Directions. The foundations go back to the area and volume problems from Egypt and include Babylon and China. The chapter on Thales contains sections on deductive logic and proof writing which will be helpful to the intended audience as will the sections on categorical propositions and categorical syllogisms in the Plato and Aristotle chapter.

The Golden Age comprises Pythagoras, Euclid, and Archimedes. Notably, the method of exhaustion has a careful explanation and treatment of Archimedes’ measurement of a circle and estimate of π as well as of his method applied to calculating volumes. The Enlightenment covers Viète, Descartes, and Desargues. Throughout the text problems come from original sources. For example, the illustration for Descartes’ solution of the four-line Pappus problem does not have the Cartesian coordinates he is noted for today. The idea is there but the form is different.

A Strange New World covers the non-Euclidean geometry of Saccheri, Lambert, Lobachevski and Bolyai. New Directions includes Riemann, Poncelet, and Felix Klein and concludes with Hilbert and the consistency of Euclidean geometry. Prospective teachers and other mathematics students will learn much mathematics in a historical context that will make it much more meaningful.

Art Gittleman (artg@csulb.edu) is Professor of Computer Science at California State University Long Beach.

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.

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