Geometry: Euclid and Beyond [Hardcover]

Robin Hartshorne (Author)

Book Description

ISBN-10: 0387986502 | ISBN-13: 978-0387986500 | Publication Date: June 8, 2000

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

Product Details

• Hardcover: 544 pages
• Publisher: Springer (June 8, 2000)
• Language: English
• ISBN-10: 0387986502
• ISBN-13: 978-0387986500
• Product Dimensions: 10.2 x 7.2 x 1.5 inches
• Shipping Weight: 2 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (11 customer reviews)
• Amazon Best Sellers Rank: #437,108 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

73 of 75 people found the following review helpful:

5.0 out of 5 stars Well beyond Euclid, December 3, 2000

By 

Marvin J. Greenberg (Berkeley, CA USA) - See all my reviews
(REAL NAME)   

This review is from: Geometry: Euclid and Beyond (Hardcover)

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.

28 of 28 people found the following review helpful:

5.0 out of 5 stars A stunning book, July 14, 2001

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

Amazon Verified Purchase

This review is from: Geometry: Euclid and Beyond (Hardcover)

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.

15 of 16 people found the following review helpful:

5.0 out of 5 stars a wonderful book by a world famous geometer, May 22, 2007

By 

mathwonk - See all my reviews

This review is from: Geometry: Euclid and Beyond (Hardcover)

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.