A History of Non-Euclidean Geometry. [Hardcover]

Boris A. Rosenfeld (Author), Abe Shenitzer (Translator), Hardy Grant (Assistant)

Book Description

Publication Date: September 7, 1988 | ISBN-10: 0387964584 | ISBN-13: 978-0387964584 | Edition: 1

This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry.

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Language Notes

Text: English, Russian (translation)

Product Details

• Hardcover: 499 pages
• Publisher: Springer; 1 edition (September 7, 1988)
• Language: English
• ISBN-10: 0387964584
• ISBN-13: 978-0387964584
• Product Dimensions: 9.3 x 6.3 x 1.3 inches
• Shipping Weight: 1.9 pounds (View shipping rates and policies)
• Average Customer Review: 3.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #3,211,305 in Books (See Top 100 in Books)

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1 of 1 people found the following review helpful:

3.0 out of 5 stars A compendium of facts, July 20, 2009

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: A History of Non-Euclidean Geometry. (Hardcover)

This is not a book at all; it is a compendium of facts. Without the slightest hint of a continuous narrative or of relating things to the avowed theme, one work after another is discussed, with the discussion consisting entirely of quotations or mild paraphrases of some highlights of the original works. Thus no one will read this book from cover to cover. It can be used as a reference only. As such, its usefulness is of course directly proportional to its reliability. On this matter I have some doubts.

Let us takes as a random sample the brief discussion of Pascal's poster Essay pour les coniques on p. 140. First of all the publication year of this essay is given as 1604 even though we are informed that Pascal lived 1623-1662 and that he published it at age 16, from which the correct publication year (1639) can be inferred. One cannot help but think that while this misprint was innocent enough, if it is indicative of others then these are bound to be worse. It also makes little sense to remark that "An English translation appeared in 1987" (referring to the book on Desargues by Field & Gray) when an English translation appeared in 1928 (Isis X.1) and was made widely available through inclusion in Smith's famous source book. But of course the mathematics is the most important thing. Here it is claimed of this poster that "in it Pascal proved Pascal's theorem." This is plainly false. Pascal's explicitly states his theorem (in convoluted form) without proof.

Now, on the very next page we read that "Newton shows" (p. 141) his theorem on the projective classification of cubics in the Enumeratio. Again this is plainly false, as again the theorem is stated without proof. Perhaps this can be excused since earlier there is a footnote that says: "Newton's notes on this work, containing the proofs omitted from the printed text, have been published by Derek Thomas Whiteside [602, vol. 2, pp. 10-89]." Fine, but by following this reference we will not find the proof in question, nor in fact anything relating to projective reasoning, as this was unknown to Newton at the time the preliminary treatise referred to was written in the 60s. In fact, Whiteside himself writes right in the introduction to the work referred to (vol. 2, p. 6) that Newton "abandoned completely" the 60s algebraic approach in favour of an approach which "in effect affords a geometric interpretation." Instead of a reference to this "completely abandoned" approach, the correct reference for background notes to Newton's projective classification is vol. VII, pp. 410-433.

In conclusion, in the space of only two pages, Rosenfeld has sent us on two wild goose chases for proofs that do not in fact exist. That is unacceptable for a reference work.