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Which of these two books should I read first? Is one of them more advanced than the other? I'm planning to use them to self-study geometry. Thanks.

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Thanks for your answers. Geometry Revisited it is. I already started to read it. –  becko Dec 30 '10 at 5:28

5 Answers 5

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If you are planning to self study and are just beginning geometry, I would recommend neither at this stage.

Geometry Revisted is from a specific viewpoint: Visiting Elementary Geometry from the viewpoint of transformations.

A snapshot of the backcover of Geometry Revisited:

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So you will miss out on some portions which are not covered by this book.

Introduction to Geometry is an advanced book and presumes fairly high familiarity already.

Note: The books are excellent. I suggest you revisit Geometry Revisited after you have done some amount of study and then perhaps take a shot at Introduction to Geometry.

Having read it at some point myself, I can vouch for Geometry Revisited. It is an excellent book and ought be read at some point.

If your only choices to self learn were Geometry Revisited and Introduction to Geometry, then I would recommend Geometry Revisited.

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Geometry Revisited is much more elementary, aimed at high-school teachers and bright high-school students or college frosh. For example, K. Strubecker wrote in his AMS Math Review

The tenor of the translation of Coxeter's beautiful tome Geometry revisited [Random House, New York, 1967] is in keeping with the objectives of the Klett Textbooks in Mathematics series which are intended to convey to freshmen and teachers of mathematics---via interesting representations---an approach to different aspects of mathematics, especially to geometry, that is kept as concrete as possible and so is applicable in schools. The volume contains six chapters which deal with the following topics: (1) Points and lines connected with a triangle; (2) some properties of circles; (3) collinearity and concurrence; (4) transformations; (5) an introduction to inversive geometry; (6) an introduction to projective geometry; The very lucid presentation takes the reader from elementary problems of plane Euclidean geometry to the fundamental concepts of non-Euclidean geometry, whose metric is briefly illustrated by the conformal model. Starting with simple geometric figures (triangle, lines, circle) and their properties, the volume advances to higher problems and figures in a manner that is convenient for the student and also whets his appetite. The always original developments use very simple tools (theorems of Ceva and Menelaos) and soon proceed to higher configurations (theorems of Pascal and Brianchon on the circle). The conics are obtained from circles as polar figures of the circles. The book is rich in remarkable facts and thereby is very effective in promoting the significance and the value of geometry in mathematical teaching, a promotion which is very necessary in view of today's predominance of set theory, analysis and algebra on the school and university level, and which deserves the skillful hand of distinguished scholars. An advantage in this recruiting endeavor is the high degree of visualizability of geometry, the easy comprehensibility of its problems and interesting theorems, and the challenge emanating from these problems to occupy oneself with their solutions. This purpose is also served by the numerous problems contained in the text whose solutions are listed at the end of the book. Many historical remarks are woven into the text.

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If you are talking about "Introduction to Geometry" by Coxeter and "Geometry Revisited" by Coxeter and Greitzer, the consensus seems to be that both of them are pretty advanced, but "Introduction to Geometry" is significantly more so, while "Geometry Revisited" is closer to something 'right after high school geometry class,' so I guess you should start with the latter (assuming you do have some geometry knowledge already).

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Geometry Revisited has a much narrower domain of content than an Introduction to Geometry. If your goal is to get a sense of what different kinds of problems, techniques, and concepts geometry has evolved to deal with, Introduction to Geometry is a "dated" but somewhat comprehensive choice. (For example, the whole "world" of computational geometry came on the scene after Coexeter's book.) One typically does not read mathematics books from cover to cover and one can dabble in Introduction to Geometry to see some of geometry's many parts. If one has heavy sledding in places than looking at the wiki article of the topic causing trouble might help.

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I'm currently reading Geometry Revisited and working through the solutions to all the problems on a Quora blog with the help of Geometer's Sketchpad and the hints in the back of the book. High school was a LONG time ago for me, and my progression was to be looking for a good next step following reading Euclid's Elements. For me, G-R is great. One of its attributes is to look at modern (i.e. post ancient Greece) results in Euclidean geometry.

I love the style of the book, forcing you to think but giving you everything you need to fill in the gaps, and always doing so succinctly and gracefully. I think you do need to know some of the core propositions in Euclid (or be prepared to look them up), but not a great deal more. I agree with what others have said: Introduction to Geometry is more advanced. While reading G-R I page through I-T-G as a preview of things to come, but for someone with my background G-R is the better place to start. I highly recommend Geometry Revisited!

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