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Title: Euclidean and Non-Euclidean Geometries: Development and History by Marvin Jay Greenberg ISBN: 0-7167-2446-4 Publisher: W H Freeman & Co. Pub. Date: August, 1993 Format: Hardcover Volumes: 1 List Price(USD): $68.61 |
Average Customer Rating: 4.4 (5 reviews)
Rating: 5
Summary: A real mind stretcher.
Comment: The first edition of this book is the one that I learned Non-Euclidean geometry from and I have always had fond memories of the course. I took it as an independent study, and chose to do all I could on my own, seeking help only when absolutely necessary. It was a time of fascination, as I was often astonished at the results and how they can be applied to the fundamental structure of the universe. The material on the geometry of physical space inspired me to go to the library searching for additional reading material.
This edition is even better than the first, it has many more exercises and projects and the sections on the history of the parallel postulate have been expanded and updated. There is more than enough material for a one-semester course, although you would have to be very selective when culling material, as nearly every page is an element of an essential progression.
I took geometry in high school and found it dull and uninspiring. However, with this book I found my college geometry course to be the most interesting math course that I have ever had, and that is saying a lot. It is an excellent text for learning an essential but often neglected subject.
Published in the recreational mathematics e-mail newsletter, reprinted with permission.
Rating: 4
Summary: Great introduction to a challenging topic
Comment: This is a full-fledged math text that I picked up on discount back when I was working at Bay Tree Bookstore in Santa Cruz. Yes, it's taken me over ten years to finally getting around to reading it. What finally worked for me is the realization that, since I'm not taking it for a class, I don't have to do the problems at the end of each chapter. That finally allowed me to read the book in comfort, as if I were auditing a class.
This book starts with Euclid's first axioms and leads you through the whys and whos of the development of non-Euclidean geometry. First, you get a complete re-introduction to Euclidean geometry itself, which is very handy and leads you directly to later developments. The unprovability of the Parallel Postulate (Euclid's Axiom V) reminded me of the Ultraviolet Catastrophe in physics/chemistry history, and Greenberg shows the motivating effect this had on the mathematics community. Unfortunately, the problem wasn't solved in a matter of decades, as with the Catastrophe, and mathematicians poked at the Parallel Postulate as if it were a sore tooth for hundreds of years before they realized that the REALLY interesting results happened when you discarded the Postulate altogether. In fact, one of the most heartbreaking sections of the book is Greenberg's description of Girolamo Saccheri's work in the 17th century. Saccheri had discovered a type of quadrilateral that seemed able to have acute summit angles and right base angles at the same time. These are perfectly possible in what's now known as hyperbolic geometry, but the only geometry known in Saccheri's time, Euclidean geometry, made no allowances for such a strange creature. Instead of realizing what he was looking at, Saccheri abandoned this line of inquiry in disgust. "It is as if a man had discovered a rare diamond," Greenberg writes, "but, unable to believe what he saw, announced it was glass."
The axioms of hyperbolic geometry are well-presented; I understood them quite well even though it's been 17 years since I took geometry. Klein's and Poincare's models of the hyperbolic plane are presented in an interesting fashion and fleshed out with several excercises and examples. I'm ashamed to say that the book started to pull away from me like an Astin Martin from a Yugo in the final two chapters. Aside from the very advanced nature of the proofs in these chapters, Greenberg's definition of ideal points is not what it could be (sets of rays?), and some of the text relies on results from previous chapters exercises. Someday I might come back to this to do the exercises as well.
Rating: 5
Summary: A Modern Classic
Comment: By a classic I mean to warn people that it isn't very up to date in Non-Euclidean geometry, but it is the best I have been able to find on the subject. I have an even older book by by a Spanish scholar who doesn't do anywhere near as well. If you need this "jive geometry" for your understanding of physics and relativity, this is a very good place to start.
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Title: Statistics and Probability in Modern Life by Joseph Newmark ISBN: 0030063930 Publisher: International Thomson Publishing Pub. Date: November, 1997 List Price(USD): $102.95 |
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Title: Elementary Number Theory and Its Applications (4th Edition) by Kenneth H. Rosen ISBN: 0201870738 Publisher: Pearson Addison Wesley Pub. Date: 06 January, 2000 List Price(USD): $112.00 |
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Title: Linear Algebra and Its Applications (3rd Edition) by David C. Lay ISBN: 0201709708 Publisher: Pearson Addison Wesley Pub. Date: 18 July, 2002 List Price(USD): $108.00 |
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Title: Journey into Mathematics, A: An Introduction to Proofs by Joseph J. Rotman ISBN: 0138423601 Publisher: Prentice Hall Pub. Date: 13 August, 1997 List Price(USD): $76.00 |
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Title: A History of Mathematics by Carl B. Boyer, Isaac Asimov ISBN: 0471543977 Publisher: John Wiley & Sons Pub. Date: 06 March, 1991 List Price(USD): $39.95 |
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