Customer Reviews

Geometry: Euclid and Beyond

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Hartshorne is a famous algebraist and one main contribution of this text is to show fascinating interrelations between classical geometries and modern algebra (of course the book contains lots of pure geometry as well). Example 1: Many texts show the impossibility of the classical problems of constructibility by straightedge and compass (by observing that the coordinates of any point so constructed lie in the smallest extension field of the rationals Q closed under taking square roots of positive numbers). Hartshorne's is the only text that goes further, solving the analogous problem when the straightedge is marked (real roots of cubic and quartic equations must also be allowed); Archimedes observed that any angle can be trisected with these tools. Example 2. Dehn's solution to Hilbert's Third Problem is given, whereby any two polyhedra equivalent under dissection must have equal Dehn invariants, and it shown that a tetrahedron has different invariant than a cube. Example 3. In hyperbolic geometry, Hilbert's arithmetic of ends is developed and applied. Example 4. Pejas' algebraic classification of Hilbert planes is discussed.

Hartshorne's text overlaps mine in correcting Euclid's errors, developing rigorous foundations for Euclidean and Non-Euclidean geometries, and covering much history, presented delightfully. He gives a thorough discussion of area and the open problems in that theory. He concludes with a nice chapter on polyhedra.

Hartshorne is a leading mathematician known for work in rather abstract geometry (see his book ALGEBRAIC GEOMETRY). He takes Euclid's ELEMENTS as great mathematics, no mere genial precursor, and collates it with Hilbert's FOUNDATIONS OF GEOMETRY.

Of course Harshorne proves that Euclid needed the parallel postulate, by exhibiting a non-Euclidean geometry. He gives a very pretty compass and straight-edge Euclidean theory of circles, which then turns into the Poincare plane model for hyperbolic geometry. He also proves that Euclid needed the method of exhaustion for volumes of solids: he gives the agreeably simple Dehn invariant proof that even a cube and a tetrahedron of equal volumes are not decomposable into congruent parts. It is a famous proof, rarely seen, and a beautiful use of the modern algebraic viewpoint in classical geometry. I had always supposed it must be hard but it is not.

Hartshorne also develops the contested "geometric algebra" of Euclid as a modern axiomatic algebra. Many commentators have shown it is wrong to think Euclid was doing "algebra" in the sense of a disguised theory of the roots of quadratic polynomials. But (unless and until Fowler's THE MATHEMATICS OF PLATO'S ACADEMY changes my mind) I think it is reasonable to say Euclid is doing algebra in this sense.

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.

So this book answers one of the questions I always had. I had never had a complete reference of the axiomatization of geometry in my hands before.

I had read Professor Hartshorne's book Algebraic Geometry (Graduate Texts in Mathematics) before and arrived to the conclusion that this branch of mathematics is more an "algebraic" branch of mathematics than a "geometric" one. However, this book gave me the chance to see Professor Hartshorne as a geometer, not an algebrist as I had thought with the previous book. His style is excellent and conveys the geometric insight you want in a Geometry book.

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 writing this book.

The book begins with a quotation from Gauss that suggests the elegance of treating Geometry in the "pure spirit of geometry" i.e. without using real numbers. And Hartshorne follows this principle by developing Euclidean geometry at first from the Elements of Euclid and then (after remarking their weaknesses) by using Hilbert's axioms. However the book is not about the foundations of geometry and much attention is given to the meaning of these axioms in the context of ruler and compass constructions and how this topic is related to analytical methods which lead directly to the theory of field extensions and Galois groups.

I think one of the main purposes of this book is to show how the abstract structure of a Field arise naturally both in Euclidean and Non-euclidean geometry and in this way prove that their typical algebraic models are categorical (that is, they are unique up to isomorphism) which is interesting for its own sake.

So this is not the usual approach to Geometry based on groups of transformations which can be found on other books, but a more "classical" one. But even if the approach is classical, the study of classical problems is always connected with modern algebraic facts, the most striking of them (for me) is the use of algebraic invariants to solve Hilbert's third problem which can be perfectly formulated (but not solved) in elementary terms.

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.

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.

This is without exception the hardest math course I have ever taken. Your understanding of the concepts is pertinent. I had to read the 1st chapter over five times just to understand projective geometry. Hartshorne tries to simplify the material but only so much can be done. It is just a hard course, period. The book does contain many example and logical proofs but be ready to burn the midnight candle on this one.

I told my wife: "If I have to give up all my books but one, then this is the one I'd keep; no question about it." (More comments later.)

I told my wife: "If I have to give up all my books but one, then this is the one I'd keep; no question about it." (More comments later.)