He has written enough glowing reviews of Dover reprints that he feels obligated to confirm that no money has changed hands. It is a classic text every academic library and many personal libraries (especially at the always-reasonable Dover reprint price) should have.īill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa. Everything is developed from elementary geometry, so prerequisites are light. 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. It would make a fine broadly-accessible text for a special topics class. The course for which this is the right textbook is not taught much anymore. The book includes several exercise sets (without solutions), an index, and a couple of short appendices on background material.
Wolfe thus ends around where contemporary studies of this topic often begin. 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).
The Poincaré disk does make an appearance in the final chapter, though not by name and not to prove more theorems in hyperbolic geometry. Hyperbolic geometry is the star of the show, of course, but elliptic geometry gets a chapter as well. 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. Diagrams are abundant (I defy anyone in the know to work through this book without instinctively redrawing everything in the Poincaré disk). Wolfe derives essential results about parallel lines, triangles, and Saccheri quadrilaterals in a way that is convincing but not oppressively formal. Theorems about hyperbolic geometry begin in Chapter 4. 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. It also gives an effective working understanding of how these guys were doing mathematics. This was wonderful to read, as we get to see the greats fall into the same traps we warn our students about. Wolfe then works through a variety of non-proofs of the parallel postulate. 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. The first three chapters are consumed with the discovery of non-Euclidean geometry.
Wolfe’s Introduction to Non-Euclidean Geometry is an excellent text that takes the axiomatic approach. 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. 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. We must not lose sight, however, of the major role non-Euclidean geometry played in the development of modern mathematical thought. Comparison to planar geometry usually comes after the model is established. 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. Those numbered 834-866 are open to undergraduate students who have completed 45 semester hours of credit and to graduate students undergraduates are awarded upper division credit graduate students are awarded graduate credit.The synthetic approach to teaching non-Euclidean geometry has fallen out of fashion. Workshops numbered 800-833 are open to all undergraduate and graduate students and are awarded lower division credit. 600-level courses are open to graduate students only. To earn graduate credit additional course requirements must be met.
3 2.2 Propositions Which Are True for Restricted Figures. 1 INTRODUCTION 1 2 PANGEOMETRY 3 2.1 Propositions Depending Only on the Principle of Superposition. Those who teach Geometry should have some knowledge of this subject. 500-level classes are advanced undergraduate classes. Non-Euclidean Geometry is now recognized as an important branch of Mathe-matics. Courses numbered 300-599 are designated as senior college (upper division) courses if completed at a regionally accredited four-year institution. Courses numbered 100-299 are designated as junior college (lower division) courses. Courses numbered 000-099 are classified as developmental courses (unless a lab section which corresponds with a 100-599 lecture course).
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