Euclidean and Non-Euclidean Geometries: Development and History [Hardcover]

Marvin J. Greenberg (Author)

Book Description

Publication Date: September 28, 2007 | ISBN-10: 0716799480 | ISBN-13: 978-0716799481 | Edition: 4th

This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.

Product Details

• Hardcover: 637 pages
• Publisher: W. H. Freeman; 4th edition (September 28, 2007)
• Language: English
• ISBN-10: 0716799480
• ISBN-13: 978-0716799481
• Product Dimensions: 9.3 x 6.4 x 1.4 inches
• Shipping Weight: 2.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.4 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #119,871 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

44 of 46 people found the following review helpful:

4.0 out of 5 stars Great introduction to a challenging topic, April 13, 2002

By 

B. McGovney (Redondo Beach, CA USA) - See all my reviews
(REAL NAME)   

This review is from: Euclidean and Non-Euclidean Geometries: Development and History (Hardcover)

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.

30 of 34 people found the following review helpful:

4.0 out of 5 stars Detailed, but frustrating at times, April 24, 1999

By A Customer

This review is from: Euclidean and Non-Euclidean Geometries: Development and History (Hardcover)

This was the textbook used in an upper division college level geometry class I recently took. For the most part, I found it to be very detailed and well-written. I also liked the fact that it presented the history of Euclidean and non-Euclidean geometry and the philosophical implications of the discovery of non-Euclidean geometry in addition to the mathematics. This made for a more well-rounded course and gave me a greater appreciation for the mathematics involved. That having been said, I must warn those who might think about using this book outside of a classroom setting to make sure they are well-versed in basic Euclidean geometry. This book gets into some very advanced topics, and at times can be very annoying in that the author makes statements like "It should be obvious that..." which immediately provokes me to think "Well maybe to you, Mr PhD!" Overall, though, this book is well thought out, and really teaches one to appreciate the beauty of building a mathematical system from a set of basic axioms. This book would be good in conjunction with some other books on the topic.

11 of 11 people found the following review helpful:

5.0 out of 5 stars Quintessential Work on Non-Euclidean Geometry, April 26, 2008

By 

Robert Curtis "Robert Curtis" (Cambridge, MA) - See all my reviews
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This review is from: Euclidean and Non-Euclidean Geometries: Development and History (Hardcover)

I had the pleasure of reading and studying the Second Edition of this text while in college. This course with this text was my favorite course during all of my undergraduate math courses.

Being a fan of the subject, I was eager to see the new Fourth Edition of the text. The Fourth Edition is quite expanded from earlier editions, going past the wonderful main story of the Parallel Postulate - told better by Greenberg than any other author, IMHO - and diving into the different non-Euclidean geometries that "open one's eyes" by setting aside the "obvious axiom of a unique parallel". The last chapters are greatly enhanced, with a superb presentation of the issue of straightedge and compass constructions in the Hyperbolic plane.

This presentation of Non-Euclidean geometry is more serious than the "popularized" books on advanced mathematical topics. If you're looking for a "light, fun" reading of this topic, this is not the book for you.

I feel that the real power of the story of the maturing of intellectual thought, so brilliantly portrayed in the story of the Parallel Postulate, must be experienced, through the effort (and often hard work) of actually **doing** geometry, rather than just reading lightly about it. If you want to dive in and actual experience geometry (and the consequent rewards), then this is the book for you. The explanations are magnificent, the problems are wonderful (and, at times, very challenging), all culminating in the "wow!" of modifying the Euclidean way of thinking to a new and beautiful alternate geometrical universe.

As other reviewers have noted, this text reads like a great novel - a drama involving geometry. If PBS/Nova ever make a "What does Parallel mean anyway?" show, this text will be the basis for that show.

I believe this Fourth Edition can be considered the quintessential text on this topic, on which all future discussion of the topics can be based, including both the introductory materials, as well as moving to the forefront of research on many topics in Hyperbolic geometry.

For a university course, weaker students will find this text quite challenging, and possibly too hard. For average students, this text will provide sufficient challenge and interest, and ample areas in the text that will not overwhelm. For advanced students, this text will certainly challenge in many different directions and interests, both in the later chapter discussions, and various problems throughout.

Greenberg's writing is meticulous - you will never find an error, a comma out of place, nor a sentence that is not perfect.