Customer Reviews

Euclidean and Non-Euclidean Geometries: Development and History

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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.

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.

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.

I have taught from this book since 1992, and my review consists of two parts:

(1) His treatment of Hilbert's axioms (restricted to two dimensions) is excellent for students. Without such a detailed study students will not understand the nature of modern axiomatic systems. I wish he included the proof of the Crossbar theorem, though!

(2) His treatment of the early history of geometry is very poor. The modern translation of Euclid's postulates and his explanation of them is badly misinformed. His history of attempts to prove the parallels postulate is exceedingly weak for antiquity and the medieval period (especially for medieval Islam). This is in part excusable, since his first edition was written before historians had a good grasp of the nature of pre-modern geometry, but it should be updated. His treatment of the modern period is much better.

The Fourth Edition of M.J. Greenberg's textbook is a wonderful addition to the geometry textbook literature. No praise could be higher than to say that it is even better--indeed, a good deal better--than the highly regarded earlier editions. There are important revisions to each of the chapters and appendices, some of them extensive. As Greenberg aptly notes: "this book is a resource for a wide variety of students, from the naive to the sophisticated, from the non-mathematical-but-educated to the mathematical wizards." In this reviewer's opinion, Greenberg's fourth edition along with the Robin Hartshorne's mathematically more technical Geometry: Euclid and Beyond (2000)--a text to which Greenberg repeatedly makes reference--are far and away the most informed, up-to-date, and historically and philosophically sensitive geometry texts on the market today. No one with an interest in the foundations of geometry can afford to be without copies of these two great works.

This book is written like a mystery, and I thoroughly enjoyed the way it led me into an understanding of non-Euclidean geometry. It builds the foundation - neutral geometry, while keeping you into suspense as to whether the parallel postulate can be proved. It includes just enough history of the mathematicians who spent their lives trying to prove the parallel postulate, with excellent referencing for further study. I hate to give away the high point of the mystery, but it has to do with the parallel postulate being independent of neutral geometry! (Read the book if you don't realize the significance of that!) The book then goes into detail on hyperbolic geometric models, such as those of Poincare and Klein. The referencing is complete and thorough. It is just a well written book, as fun to read as a math book can ever be. A classic. I highly recommend it for students and anyone interested in geometry.

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.

First of all, I must point out that i am reviewing the second edition of this book. I'm sure the third edition is different, but i think the main points of my review will still hold.

I bought this book because i needed to brush up on my geometry for the California Subject Examination for Teachers (CSET) in mathematics. While it is certainly a well written book (I found the historical aspects of it particularly interesting), its major flaw is that there are no answers to the end-of-chapter excercises! This makes the book virtually useless to anyone not in school wanting to learn geometry in their own time (i.e. not for a class). Whilst i managed to do most of the exercises at the end of the first chapter (at least i think i did), it seemed pointless to attempt subsequent problems as they were quite in depth and there would be no way for me to know whether they were right or not! A big improvement would be if the number of problems were cut down (seriously, it would take years for someone to finish all of the end-of-chapter problems!) and something resembling answers was in the back of the book.

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.

This is a good text book. It has lots of clear worked out proofs of propostitions. Many of the exercises at the end of each lesson require that you find proofs of propostions mentioned in the text but were not proven. This book also has good drawings of shapes that explain what the propositions, theorems,and axioms mean. Since I am a student, I personally think the exercises at the end of the chapters are somewhat difficulty to figure out, but for others they might not be. If your professor doesn't explain the material well, you can at least try to understand by yourself using this text. Overall, this is a good textbook.