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Geometry: Euclid and Beyond Reviews

Customer Rating:
Summary: Geometry - anything else you need?
Comment: So this book answers one of the questions I always had. Call me an ignorant if you please, but I had never had a complete reference of the axiomatization of geometry in my hands before. I had read Proffessor Hartshorne's book "Agebraic Geometry" before and I thought he was one of those algebrists that hide themselves inside the name of "Algebraic Geometers". Note that I like Algebraic Geometry myself, but I see it more as an "algebraic" branch of mathematics than a "geometric" one. Anyway, this book proved me wrong yet again. Since I was told some years ago that Geometry could be Axiomatized, I had always hoped to see the structure being constructed. This book finally fulfilled my curiosity. I am indeed grateful with professor Hartshorne just for writting this book.

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Customer Rating:
Summary: Where was this book when I was a student?
Comment: This is a great book, a mature and lively treatment of a familiar subject made new again. If I'd had a text like this as an undergraduate I'd likely still be in math. Most of the serious advances in pre-20th century geometry get subsumed in the typically more topological, or algebraic, but in either case more abstract, treatment one finds today in a typical undergraduate course. Lost in this approach is the intuitive grounding which makes more modern approaches meaningful and not just mere formalism. This book, which would lend itself to self-study as well as to classroom use, goes a long way to restoring that lost grounding. Very highly recommended.

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Customer Rating:
Summary: Bring your copy of Elements!
Comment: I'm still working through this text but I should warn prospective buyers of one thing: The book's early chapters makes heavy references to Euclid's propositions in his books The Elements. I don't just mean references like "Remember that Proposition 43 from Book 2 that says...". No, would that it were so. He'll just give the number and assume you've got your copy of Elements handy.

In that way, it's not really a complete survey of geometry from the start. You'll want to order a copy of Elements with this book. Dover publishes eleven of the books in two volumes.

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Customer Rating:
Summary: a wonderful book by a world famous geometer
Comment: This book reveals the love professor Hartshorne has for geometry and euclid. I became excited about the subject just reading the introduction. The book assumes the student knows high school geometry. which unfortunately eliminates many college students, but I am going to try to use it at least for the second part of my college course.

This is a really well written, expert, wonderfully enthusiastic book, about a great, absolutely classic topic, by a powerful world famous authority in geometry.

The organization assumes the student is reading euclid concurrently. then prof hartshorne explains the difficullties with euclids treatment and shows how to remedy them. e.g. he observes euclids proof of SAS uses a principle of superposition without stating it, then although he adopts the Hilbert option of making this an axiom, he also presents an alternative treatment in which the principle of superposition is an axiom, and SAS is then proved exactly as euclid does. this sort of thing shows very clearly that euclids proofs become correct, merely by clarifying his implicit assumptions.

i love this and think it enhances the subject enormously.

the exercises are so ambitious and far reaching I at first dismissed them as unrealistic, but soon became infected with dr hartshornes enthusiasm for putting the students in touch with their best abilities, and challenging them to reach as deeply as they can.

This book is a remarkable work of scholarship, with far more content than one course can use. The student has here a work that will repay years of study. again the price makes it a bargain compared to far inferior works at double the price.

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Customer Rating:
Summary: Very good textbook but it repeats contemporary fallacies
Comment: The book, like related literature, contends greater rigor in today's geometry or mathematics than in the time of Euclid. I challenge this and give some reasons why.

Examples occur in modern efforts to improve on Euclid's axioms. He, for instance, speaks in his proposition I.4 of applying one of equal figures to another so their parts coincide. This moving of figures is objected to (p.33 in Hartshorne) as not allowed by the axioms. However, equality, congruence, in figures is itself defined by their coinciding if placed upon each other. They are conceptualized accordingly, not requiring an axiom.

Most prominent to me are possibly fallacies connected with non-Euclidean geometries. One of their properties relied on is their understood consistency. This may be traced to past attempts at finding inconsistencies if the controversial parallel postulate is assumed false (see e.g. p.305 in Hartshorne). On finding no inconsistencies, it was inferred that the other geometries are logically valid. The inference commits the fallacy of "denying the antecedent". If inconsistency makes something invalid, it does not follow that consistency makes it valid. For example, inferring from "if A then B" that "if B then A" is not inconsistent, but it is invalid.

A further fallacy permeating non-Euclidean geometry is linguistic equivocation. Beside redefinitions (p.28) of terms like "line", now a straight line, or "curve", now including straight lines, or leaving such terms undefined (p.81), in perhaps confusing the following, concepts like straightness are reinterpreted to include various curvatures (e.g. pp.355-6). Moreover, these procedures are used to assert the parallel postulate unprovable, because they can contradict it. But one cannot make an inference about something, presently the postulate, by changing meanings of words in it. One is then speaking about something else.

Writing more elsewhere, I hope to have adequately touched on the subjects here.

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