Euclidean & Non-Euclidean Geometries: Development and History by Marvin Jay Greenberg35-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 Im 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 isnt 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 Euclids definitions, common notions, and postulates. Its 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 its only 2 days of class, I wanted to have a better understanding of non-Euclidean geometry and of the problems with Euclids 5th postulate.
Enter Marvin Greenbergs 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. Its 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 Im 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 dont know how much information Ill retain, but there were some great quotes and I think Ive got a good, basic grasp of how non-Euclidean geometry was discovered and what its 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.
ISBN 13: 9780716799481
This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry. Convert currency. Add to Basket.
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Freeman and Company , 41 Madison Ave. For much of the last half of the twentieth century, college level mathematics textbooks, particularly calculus texts, have included short, marginal, historical blurbs; a short bio of Brook Taylor in the section on Taylor series, for example. Such inclusions can be interesting for the faculty member who has not had much exposure to the history of mathematics or the student with a pre-existing interest. As a student I found these excerpts tantalizing and they surely whetted my appetite for mathematics history. However, as a professor I have found them frustrating as they rarely say enough about the mathematics itself.
This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry. Marvin J. He received his undergraduate degree from Columbia University, where he was a Ford Scholar.