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Publisher:

Dover Publications

Publication Date:

1983

Number of Pages:

321

Format:

Paperback

Price:

11.95

ISBN:

9780486634159

Category:

Textbook

[Reviewed by , on ]

William J. Satzer

01/10/2012

First published in 1955, this book has become a sort of touchstone for college geometry courses, particularly those that are aimed in part at prospective high school teachers. It carries some signs of its age, more in perspective than in treatment of its subject. The author suggests that the course is most appropriate for students with the equivalent of one or preferably two years of college mathematics. One of those years would include something like a college algebra course of the sort no longer commonly taught. No calculus background is assumed.

The subject is mostly classical geometry, broadly considered. Euclidean geometry has a place, but it doesn’t show up until more than halfway through the book. The author chooses to start — after a short discussion of preliminaries — with synthetic projective geometry, followed by analytic projective geometry. A chapter on coordinate systems is sandwiched in between. Then the author takes up affine geometry. This is followed by treatments of Euclidean and non-Euclidean geometry that are accompanied by a historical perspective on their development. The book then concludes with a brief tour of elementary topics in topology.

The author views his subject as a hierarchy. It begins with topology at the top with projective geometry immediately below, bifurcating into affine geometry on one side, and non-Euclidean geometry on the other. Euclidean geometry sits at the bottom, being just one kind of affine geometry. With the exception of topology, the organization of the book follows the same pattern. This seems odd to me because, from most students’ perspective, Euclidean geometry is the most familiar, and thus perhaps the natural place to start exploring other geometries. While the author’s approach might well appeal to stronger students, the less able are more likely to struggle if they have to start with a formal deductive system based on the unfamiliar postulates of projective geometry.

The other notable aspect of the book is its level of exposition. While it might have been written with prospective high school teachers in mind, it makes few concessions to the less sophisticated. It is not an easy book to read, and few of the exercises are straightforward. However it might have been used in the past, it is hard to imagine using this now in a class for future high school teachers. There is wonderful geometry here, though better suited to the more mathematically mature.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

CHAPTER 1. FOUNDATIONS OF GEOMETRY | |||||||

1-1 Logical systems | |||||||

1-2 Logical notations | |||||||

1-3 Inductive and deductive reasoning | |||||||

1-4 Postulates | |||||||

1-5 Independent postulates | |||||||

1-6 Categorical sets of postulates | |||||||

1-7 A geometry of number triples | |||||||

1-8 Geometric invariants | |||||||

CHAPTER 2. SYNTHETIC PROJECTIVE GEOMETRY | |||||||

2-1 Postulates of incidence and existence | |||||||

2-2 Properties of a projective plane | |||||||

2-3 Figures | |||||||

2-4 Duality | |||||||

2-5 Perspective figures | |||||||

2-6 Projective transformations | |||||||

2-7 Postulate of Projectivity | |||||||

2-8 Quadrangles | |||||||

2-9 Complete and simple n-points | |||||||

2-10 Theorem of Desargues | |||||||

2-11 Theorem of Pappus | |||||||

2-12 Conics | |||||||

2-13 Theorem of Pascal | |||||||

2-14 Survey | |||||||

CHAPTER 3. COORDINATE SYSTEMS | |||||||

3-1 Quadrangular sets | |||||||

3-2 Properties of quadrangular sets | |||||||

3-3 Harmonic sets | |||||||

3-4 Postulates of Separation | |||||||

3-5 Nets of rationality | |||||||

3-6 Real projective geometry | |||||||

3-7 Nonhomogeneous coordinates | |||||||

3-8 Homogeneous coordinates | |||||||

3-9 Survey | |||||||

CHAPTER 4. ANALYTIC PROJECTIVE GEOMETRY | |||||||

4-1 Representations in space | |||||||

4-2 Representations on a plane | |||||||

4-3 Representations on a line | |||||||

4-4 Matrices | |||||||

4-5 Cross ratio | |||||||

4-6 Analytic and synthetic geometries | |||||||

4-7 Groups | |||||||

4-8 Classification of projective transformations | |||||||

4-9 Polarities and conics | |||||||

4-10 Conics | |||||||

4-11 Involutions on a line | |||||||

4-12 Survey | |||||||

CHAPTER 5. AFFINE GEOMETRY | |||||||

5-1 Ideal points | |||||||

5-2 Parallels | |||||||

5-3 Mid-point | |||||||

5-4 Classification of conics | |||||||

5-5 Affine transformations | |||||||

5-6 Homothetic transformations | |||||||

5-7 Translations | |||||||

5-8 Dilations | |||||||

5-9 Line reflections | |||||||

5-10 Equiaffine and equiareal transformations | |||||||

5-11 Survey | |||||||

CHAPTER 6. EUCLIDEAN PLANE GEOMETRY | |||||||

6-1 Perpendicluar lines | |||||||

6-2 Similarity transformations | |||||||

6-3 Orthogonal line reflections | |||||||

6-4 Euclidean transformations | |||||||

6-5 Distances | |||||||

6-6 Directed angles | |||||||

6-7 Angles | |||||||

6-8 Common figures | |||||||

6-9 Survey | |||||||

CHAPTER 7. THE EVOLUTION OF GEOMETRY | |||||||

7-1 Early measurements | |||||||

7-2 Early Greek influence | |||||||

7-3 Euclid | |||||||

7-4 Early euclidean geometry | |||||||

7-5 The awakening in Europe | |||||||

7-6 Constructions | |||||||

7-7 Descriptive geometry | |||||||

7-8 Seventeenth ce | |||||||

7-9 Eighteenth century | |||||||

7-10 Euclid's fifth postulate | |||||||

7-11 Nineteenth and twentieth centuries | |||||||

7-12 Survey | |||||||

CHAPTER 8. NONEUCLIDEAN GEOMETRY | |||||||

8-1 The absolute polarity | |||||||

8-2 Points and lines | |||||||

8-3 Hyperbolic geometry | |||||||

8-4 Elliptic and spherical geometries | |||||||

8-5 Comparisons | |||||||

CHAPTER 9. TOPOLOGY | |||||||

9-1 Topology | |||||||

9-2 Homeomorphic figures | |||||||

9-3 Jordan Curve Theorem | |||||||

9-4 Surfaces | |||||||

9-5 Euler's Formula | |||||||

9-6 Tranversable networks | |||||||

9-7 Four-color problem | |||||||

9-8 Fixed-point theorems | |||||||

9-9 Moebius strip | |||||||

9-10 Survey | |||||||

BIBLIOGRAPHY | |||||||

INDEX |

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