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Rating: 
Summary: A Modern Classic
Review: 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.
Rating: 
Summary: Great for understanding modern axiomatic systems
Review: 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.
Rating: 
Summary: Another good textbook that does not recognize its pitfalls
Review: It is a book that, as fashionable today, accuses Euclid of numerous flaws, while itself replete with them.
One can begin at the beginning of Euclid's Elements. He defined certain basic terms, which, it is now argued, should not be defined if to avoid "infinite regress" (p.11). Ironically, the accordingly undefined terms, and some additions to them, are said to be clarified, filling in gaps in Euclid's proofs (p.13).
"Infinite regress" is alleged because on defining terms by others, we can ask that the others be defined, and so on indefinitely. However, this is not how definition works. We speak and write volumes with barely a definition, because we already understand most of the language. If some of it is uncertain, we can clarify it with familiar words. And what we especially want to understand in discussions are key words as in the above.
Leaving terms undefined is motivated rather by wanting to arbitrarily "interpret" them for subjects like non-Euclidean geometries. This, as I noted elsewhere, commits the fallacy of equivocation. One cannot prove something by changing meanings of words involved.
As to gaps in Euclid's proofs, it is claimed he failed to supply axioms for conditions merely assumed (p.70). But it is axioms that are assumptions. In fact, they are said to (p.10) likewise avoid "infinite regress", by being accepted without proof. But the unstated conditions questioned, like betweenness (p.72), are of fundamental human knowledge gained by observation, the source of certainty even in logical principles, "not conceivable otherwise" and confirmable by diagrams. These conditions are like other words in the proofs tacitly understood, without making them explicit.
More specific to the author are other flaws. He misstates Euclid's 1st postulate, having it say that a unique (straight) line "exists" passing through any two distinct points. The uniqueness (or such as existence) is not stated in Euclid's postulate and is not an issue until Proposition 4.
Or, the author confuses his assertion (p.83) that every segment is congruent (equal) to itself (as usually part of Laws of Thought) with the 4th Common Notion, which states that things which coincide with one another are equal to one another. That Common Notion is referenced in again Proposition 4, and is about placing one shape over another whereupon they coincide and are equal in that sense.
Flaws in the book also enter elementary logic. It is e.g. stated (p.45) that "~[H > C]" ("H does not imply C") is the same as "H & ~C" ("[H and not-C] is true"). The statements are universal, and given the first one, the second can be false in particular cases. In another flaw (p.49) the Law of Excluded Middle is said to state that P implies P or not-P. What the law does state is merely that either P or not-P is true, without positing one of them.
The book contains many other flaws, not coverable within present limitations.
Rating:
Summary: A real mind stretcher.
Review: 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:
Summary: Great introduction to a challenging topic
Review: 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.
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