Euclidean and Non-Euclidean Geometries: Development and History

by Marvin Jay Greenberg

Book Overview

Euclidean and Non-Euclidean Geometries presents the discovery of non-Euclidean geometry and the reformulation of the foundations of Euclidean geometry. This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 0716711036
• ISBN13: 9780716711032
• Release Date: January 1980
• Publisher: W.H. Freeman & Company
• Length: 400 Pages
• Weight: 1.57 lbs.
• Dimensions: 1.1" x 6.3" x 9.5"

Customer Reviews

Quintessential Work on Non-Euclidean Geometry

I had the pleasure of reading and studying the Second Edition of this text while in college. This course with this text was my favorite course during all of my undergraduate math courses. Being a fan of the subject, I was eager to see the new Fourth Edition of the text. The Fourth Edition is quite expanded from earlier editions, going past the wonderful main story of the Parallel Postulate - told better by Greenberg than any other author, IMHO - and diving into the different non-Euclidean geometries that "open one's eyes" by setting aside the "obvious axiom of a unique parallel". The last chapters are greatly enhanced, with a superb presentation of the issue of straightedge and compass constructions in the Hyperbolic plane. This presentation of Non-Euclidean geometry is more serious than the "popularized" books on advanced mathematical topics. If you're looking for a "light, fun" reading of this topic, this is not the book for you. I feel that the real power of the story of the maturing of intellectual thought, so brilliantly portrayed in the story of the Parallel Postulate, must be experienced, through the effort (and often hard work) of actually **doing** geometry, rather than just reading lightly about it. If you want to dive in and actual experience geometry (and the consequent rewards), then this is the book for you. The explanations are magnificent, the problems are wonderful (and, at times, very challenging), all culminating in the "wow!" of modifying the Euclidean way of thinking to a new and beautiful alternate geometrical universe. As other reviewers have noted, this text reads like a great novel - a drama involving geometry. If PBS/Nova ever make a "What does Parallel mean anyway?" show, this text will be the basis for that show. I believe this Fourth Edition can be considered the quintessential text on this topic, on which all future discussion of the topics can be based, including both the introductory materials, as well as moving to the forefront of research on many topics in Hyperbolic geometry. For a university course, weaker students will find this text quite challenging, and possibly too hard. For average students, this text will provide sufficient challenge and interest, and ample areas in the text that will not overwhelm. For advanced students, this text will certainly challenge in many different directions and interests, both in the later chapter discussions, and various problems throughout. Greenberg's writing is meticulous - you will never find an error, a comma out of place, nor a sentence that is not perfect.

Euclidean and Non-Euclidean Geometries, Fourth Edition, by Marvin Jay Greenberg

The Fourth Edition of M.J. Greenberg's textbook is a wonderful addition to the geometry textbook literature. No praise could be higher than to say that it is even better--indeed, a good deal better--than the highly regarded earlier editions. There are important revisions to each of the chapters and appendices, some of them extensive. As Greenberg aptly notes: "this book is a resource for a wide variety of students, from the naive to the sophisticated, from the non-mathematical-but-educated to the mathematical wizards." In this reviewer's opinion, Greenberg's fourth edition along with the Robin Hartshorne's mathematically more technical Geometry: Euclid and Beyond (2000)--a text to which Greenberg repeatedly makes reference--are far and away the most informed, up-to-date, and historically and philosophically sensitive geometry texts on the market today. No one with an interest in the foundations of geometry can afford to be without copies of these two great works.

A Real Classic

This is the fourth edition of a particularly fine text by Marvin Jay Greenberg. If you want to learn about Euclidean and non-Euclidean geometries---the great contributions of Bolyai and Lobachevsky---this is the place to do it. The book is authoritative but warm and inviting. It is full of good history and full of good mathematics. The fourth edition has a good deal of new material. Greenberg explores some of the subtle logical issues, and also some of the tricky points of geometry. He makes far-ranging commentary on how non-Euclidean geometry fits into the modern flow of mathematical thought. There is even some discussion of Perelman's proof of the Poincare conjecture. Even a reader without a strong mathematical background will get a good deal from dipping into this book. It gives a great sense of what the mathematical enterprise is all about, written by a distinguished mathematician (who was also my teacher many years ago). I consider this work to be one of the treasures on my bookshelf.

Excellent Book

This book is written like a mystery, and I thoroughly enjoyed the way it led me into an understanding of non-Euclidean geometry. It builds the foundation - neutral geometry, while keeping you into suspense as to whether the parallel postulate can be proved. It includes just enough history of the mathematicians who spent their lives trying to prove the parallel postulate, with excellent referencing for further study. I hate to give away the high point of the mystery, but it has to do with the parallel postulate being independent of neutral geometry! (Read the book if you don't realize the significance of that!) The book then goes into detail on hyperbolic geometric models, such as those of Poincare and Klein. The referencing is complete and thorough. It is just a well written book, as fun to read as a math book can ever be. A classic. I highly recommend it for students and anyone interested in geometry.

A real mind stretcher.

The first edition of this book is the one that I learned Non-Euclidean geometry from and I have always had fond memories of the course. I took it as an independent study, and chose to do all I could on my own, seeking help only when absolutely necessary. It was a time of fascination, as I was often astonished at the results and how they can be applied to the fundamental structure of the universe. The material on the geometry of physical space inspired me to go to the library searching for additional reading material. This edition is even better than the first, it has many more exercises and projects and the sections on the history of the parallel postulate have been expanded and updated. There is more than enough material for a one-semester course, although you would have to be very selective when culling material, as nearly every page is an element of an essential progression. I took geometry in high school and found it dull and uninspiring. However, with this book I found my college geometry course to be the most interesting math course that I have ever had, and that is saying a lot. It is an excellent text for learning an essential but often neglected subject.Published in the recreational mathematics e-mail newsletter, reprinted with permission.