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Introduction to Non-Euclidean Geometry

Dover Publications
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The synthetic approach to teaching non-Euclidean geometry has fallen out of fashion. There are some good reasons for this — students can get a good feel for the axiomatic method from Euclid’s Elements and the results of non-Euclidean geometry can be more efficiently obtained using transformational or model-based methods. Comparison to planar geometry usually comes after the model is established.

We must not lose sight, however, of the major role non-Euclidean geometry played in the development of modern mathematical thought. The story of mathematicians trying to prove Euclid’s parallel postulate from the other four is a compelling one, featuring interesting characters and a twist ending: it can’t be done, because hyperbolic and spherical geometry work just fine. What an axiomatic approach to non-Euclidean geometry brings pedagogically is forced reliance on the axioms, since intuition will always lead the uninitiated astray, just as it did the mathematicians who attempted to prove the parallel postulate.

Harold E. Wolfe’s Introduction to Non-Euclidean Geometry is an excellent text that takes the axiomatic approach. The attention to history is a delight. The first three chapters are consumed with the discovery of non-Euclidean geometry. He begins with a careful study of Euclid’s postulates and common notions, discussing some of its gaps in rigor that led to Hilbert’s revisiting them. Wolfe then works through a variety of non-proofs of the parallel postulate. This was wonderful to read, as we get to see the greats fall into the same traps we warn our students about. It also gives an effective working understanding of how these guys were doing mathematics. We then proceed to the discovery of non-Euclidean geometry, wherein we learn that the parallel postulate really is independent and that Gauss was a pretty smart guy.

Theorems about hyperbolic geometry begin in Chapter 4. Wolfe derives essential results about parallel lines, triangles, and Saccheri quadrilaterals in a way that is convincing but not oppressively formal. Diagrams are abundant (I defy anyone in the know to work through this book without instinctively redrawing everything in the Poincaré disk). We move to hyperbolic trigonometry in the next chapter, where things take a more analytic turn, and then a chapter on calculus-based results where students will have the opportunity to think more deeply about how calculus is used to calculate arc length and area. Hyperbolic geometry is the star of the show, of course, but elliptic geometry gets a chapter as well.

The Poincaré disk does make an appearance in the final chapter, though not by name and not to prove more theorems in hyperbolic geometry. This chapter is about consistency of the axioms, which Wolfe proves by building a consistent model (the model for elliptic geometry — the sphere — is covered briefly and dismissed as fairly obvious). Wolfe thus ends around where contemporary studies of this topic often begin.

The book includes several exercise sets (without solutions), an index, and a couple of short appendices on background material.

The course for which this is the right textbook is not taught much anymore. It would make a fine broadly-accessible text for a special topics class. The material could be used in various ways to supplement almost any kind of geometry or history of mathematics class and it would be appropriate for self-study or independent projects. Everything is developed from elementary geometry, so prerequisites are light. It is a classic text every academic library and many personal libraries (especially at the always-reasonable Dover reprint price) should have.

Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa. He has written enough glowing reviews of Dover reprints that he feels obligated to confirm that no money has changed hands.

Date Received: 
Wednesday, October 24, 2012
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Harold E. Wolfe
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Bill Wood


I. The Foundation of Euclidean Geometry
II. The Fifth Postulate
III. The Discovery of Non-Euclidean Geometry
IV. Hyperbolic Plane Geometry
V. Hyperbolic Plane Trigonometry
VI. Applications of Calculus to the Solutions of Some Problems in Hyperbolic Geometry
VII. Elliptic Plane Geometry and Trigonometry
VIII. The Consistency of the Non-Euclidean Geometries
I. The Foundation of Euclidean Geometry
II. Circular and Hyperbolic Functions
III. The Theory of Orthogonal Circles and Allied Topics
IV. The Elements of Inversion


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Wednesday, October 24, 2012