Euclidean and Non-Euclidean Geometries: Development and History

Published in the recreational mathematics e-mail newsletter, reprinted with permission.


A Real Classic   March 17, 2008
Steven G. Krantz
4 out of 4 found this review helpful

This is the fourth edition of a particularly fine text
by Marvin Jay Greenberg. If you want to learn about
Euclidean and non-Euclidean geometries---the great contributions
of Bolyai and Lobachevsky---this is the place to do it. The
book is authoritative but warm and inviting. It is full of
good history and full of good mathematics.

The fourth edition has a good deal of new material. Greenberg
explores some of the subtle logical issues, and also some
of the tricky points of geometry. He makes far-ranging
commentary on how non-Euclidean geometry fits into the modern
flow of mathematical thought. There is even some discussion
of Perelman's proof of the Poincare conjecture.

Even a reader without a strong mathematical background will get
a good deal from dipping into this book. It gives a great
sense of what the mathematical enterprise is all about, written
by a distinguished mathematician (who was also my teacher many
years ago). I consider this work to be one of the treasures on
my bookshelf.


Quintessential Work on Non-Euclidean Geometry   April 26, 2008
Robert Curtis (Cambridge, MA)
3 out of 3 found this review helpful

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.





A wealth of knowledge of geometry   August 8, 2007
George C. Johnson (Gainesville,Georgia)
2 out of 3 found this review helpful

This text provides a wealth of knowledge about geometry. For me, with only a minimum of college level geometry previously studied, it was my first meeting with a rigorous development of any type of geometry, euclidean or noneuclidean. It was very exciting to see how this subject can be so carefully developed. Even though I was exposed to a meticulous construction of real analysis and algebra ,there is quite a difference in the techniques used to develop geometry, which you might anticipate.

Each time I have reviewed Dr.Greenberg's text, I am not only able to retain the material easier, but also to achieve a new level of understanding, which is kind of surprising.

This text is a treasure of knowledge of geometry, but the reader, if not much better prepared than me, needs to understand that digesting this text requires a bit of a committment , but it is well worth the effort. If you are a prior football player, like me, you will probably remember the coach mentioning it will take a 110% effort to win. This is a different way of indicating how tenacious, I feel, you will need to be.

I am really looking forward to reading Dr. Greenberg's most recent edition of this text, which is now available.