Geometry: Euclid and Beyond

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.


A stunning book   July 14, 2001
Colin McLarty (Chardon, OH USA)
23 out of 23 found this review helpful

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.


a wonderful book by a world famous geometer   May 23, 2007
mathwonk
8 out of 9 found this review helpful

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.


Geometry - anything else you need?   April 15, 2008
Rodrigo Hernandez (Cuautitlan Izcalli, Mexico Mexico)
4 out of 5 found this review helpful

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. After reading it, I found out that Proffessor Hartshorne is really good explaining geometry.

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.


Where was this book when I was a student?   October 10, 2007
swimjay (Berkeley, CA)
4 out of 6 found this review helpful

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.