Euclidean & Non-Euclidean Geometries: Development and History

by Marvin Jay Greenberg

This classic text provides overview of both classic and hyperbolic geometries, placing the work of key mathematicians/ philosophers in historical context. Coverage includes geometric transformations, models of the hyperbolic planes, and pseudospheres.

Genres: Mathematics, Geometry, Textbooks, Nonfiction, History, Stem, Reference

512 pages, Hardcover

First published September 28, 1974

Reviews

[Rick Davis · Rating 5 out of 5]

35-year-old-Rick-from-January-2018: Well, I just finished reading a book about the history and development of Non-Euclidean Geometry.
15-year-old-Rick-from-January 1998: Wait, are you me from the future? How did you get here?
35yo-Rick: It would take too long to explain. Just ask Gödel.
15yo-Rick: Okay, but why did you just read a book about geometry? Surely I'm still not in school 20 years from now!
35yo-Rick: I read it for fun.
15yo-Rick: Fun?! You think Geometry is fun? Oh no. Please tell me this isn't who I grow up to become.
-------------------
Okay, this sounds crazy to my 15 year old self, and probably crazy to a lot of other people, but I have found that some of the most calming things to read are math books. Something about the order and elegance of a good proof, something I also appreciate in formal logic. I picked this book to read particularly because one of the classes I teach to high schoolers covers Euclid's definitions, common notions, and postulates. It's not a math class, but we quickly cover Euclid for philosophical purposes. Mainly we talk about his axiomatic method and how it informed Descartes later on. However, even though it's only 2 days of class, I wanted to have a better understanding of non-Euclidean geometry and of the problems with Euclid's 5th postulate.

Enter Marvin Greenberg's excellent book. There are 10 chapters and 2 appendices that intend to take the reader on a journey through the history of geometry while rigorously inculcating the principles of geometric proofs. It's kind of an all-in-one program, and Greenberg offers ideas on how to teach the book for various classes in the introduction. There are chapters to work through with a math class of moderate skill (Chapters 1-6 and the beginning of 7 [minus all the major exercises]). There are chapters to work through with a class of liberal arts students (Chapters 1, 2, 5, parts of 6 and 7, and 8). There are chapters to work through with a math class of advanced students (Chapters 1-7 with all exercises).

Being a glutton for punishment, I decided to work through all the chapters and do the review exercises (but not the major exercises, because I'm not that crazy). I found that I was able to follow the discussion well through the first 6 chapters, and I made it part of the way through chapter 7 before I was completely over my head. Chapter 8 was a philosophical overview of the implications of non-Euclidean geometry for philosophy of mathematics; that was a great chapter. Chapters 9 and 10 made my brain hurt, and would have required far more time than I wanted to spend in order to fully grasp. I don't know how much information I'll retain, but there were some great quotes and I think I've got a good, basic grasp of how non-Euclidean geometry was discovered and what it's all about. (Hint: It has nothing to do with Cthulhu. Thanks for confusing me, H. P. Lovecraft!)

Another thing this book reinforced for me was my long-held belief that math is the closest thing to actual magic that exists in the world. If there were really a school for witchcraft and wizardry, it would look a lot less like Hogwarts and a lot more like a school full of people doing abstract mathematics and pure analytic geometry.

[Lewis · Rating 4 out of 5]

This book has increased my love of geometry proofs, and proofs in general. They used to be just another part of math to me, now they are exciting and elegant.

[Benjamin · Rating 5 out of 5]

This book is packed with information. I found some initial, minor organizational choices odd, but I think this is really a symptom of just how much content Greenberg puts into the text. The exercises I did were always rewarding and interesting. The book certainly deserves far more time than I had available to devote to it.

my favorite quote: "Only impractical dreamers spent two thousand years wondering about proving Euclid's parallel postulate, and if they hadn't done so, there would be no spaceships exploring the galaxy today."

[Brian · Rating 5 out of 5]

This is a full-fledged math text that I picked up on discount back when I was working at Bay Tree Bookstore in Santa Cruz. Yes, it's taken me over ten years to finally getting around to reading it. What finally worked for me is the realization that, since I'm not taking it for a class, I don't have to do the problems at the end of each chapter. That finally allowed me to read the book in comfort, as if I were auditing a class.

This book starts with Euclid's first axioms and leads you through the whys and whos of the development of non-Euclidean geometry. First, you get a complete re-introduction to Euclidean geometry itself, which is very handy and leads you directly to later developments. The unprovability of the Parallel Postulate (Euclid's Axiom V) reminded me of the Ultraviolet Catastrophe in physics/chemistry history, and Greenberg shows the motivating effect this had on the mathematics community. Unfortunately, the problem wasn't solved in a matter of decades, as with the Catastrophe, and mathematicians poked at the Parallel Postulate as if it were a sore tooth for hundreds of years before they realized that the REALLY interesting results happened when you discarded the Postulate altogether. In fact, one of the most heartbreaking sections of the book is Greenberg's description of Girolamo Saccheri's work in the 17th century. Saccheri had discovered a type of quadrilateral that seemed able to have acute summit angles and right base angles at the same time. These are perfectly possible in what's now known as hyperbolic geometry, but the only geometry known in Saccheri's time, Euclidean geometry, made no allowances for such a strange creature. Instead of realizing what he was looking at, Saccheri abandoned this line of inquiry in disgust. "It is as if a man had discovered a rare diamond," Greenberg writes, "but, unable to believe what he saw, announced it was glass."

The axioms of hyperbolic geometry are well-presented; I understood them quite well even though it's been 17 years since I took geometry. Klein's and Poincare's models of the hyperbolic plane are presented in an interesting fashion and fleshed out with several excercises and examples. I'm ashamed to say that the book started to pull away from me like an Astin Martin from a Yugo in the final two chapters. Aside from the very advanced nature of the proofs in these chapters, Greenberg's definition of ideal points is not what it could be (sets of rays?), and some of the text relies on results from previous chapters exercises. Someday I might come back to this to do the exercises as well.

[Whitney Evans · Rating 4 out of 5]

It started out awful... It's hard to unlearn the Euclidean Geometry we are so used to, but by the end, I had learned so much about what it means to truly use a definition to define every decision and characteristic of something rather than trusting your eyes, intuition, or prior knowledge. It's a completely different way of thinking about diagrams and using models. It stretched my brain and made me a better mathematician! I couldn't have done it without a teacher to help me make sense of it all, though.

[Shabnam]

ترجمه:م.ه.شفيعيها

[Dominick Joo]

Skimmed the last n chapters, tl;dr hyperbolic geometry has interesting features:

- The area of a triangle in a given hyperbolic plane is bounded
- There's an absolute unit of length (?)
- Different definitions exist for rectangle-like quadrilaterals

[Am]

Ok

[همس المشاعر]

Thins

[Jace Davies · Rating 5 out of 5]

Absolutely delicious. An essential read post-Elements.