A Long Way from Euclid (Dover Books on Mathematics) by Constance Reid (PDF)

Description

Mathematics has come a long way indeed in the last 2,000 years, and this guide to modern mathematics traces the fascinating path from Euclid’s Elements to contemporary concepts. No background beyond elementary algebra and plane geometry is necessary to understand and appreciate author Constance Reid’s simple, direct explanations of the arithmetic of the infinite, the paradoxes of point sets, the “knotty” problems of topology, and “truth tables” of symbolic logic. Reid illustrates the ways in which the quandaries that arose from unsolvable problems promoted new ideas. Numerical concepts expanded to accommodate such concepts as zero, irrational numbers, negative numbers, imaginary numbers, and infinite numbers.Geometry advanced into the widening territories of projective geometry, non-Euclidean geometries, the geometry of n-dimensions, and topology or “rubber sheet” geometry. More than 80 drawings, integrated with the text, assist in cultivating a grasp of the abstract foundations of modern mathematics, the search for truly consistent assumptions, the recognition that absolute consistency is unattainable, and the realization that some problems can never be solved.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I am always enthralled by Constance Reid’s books.

⭐thanxs

⭐Constance Reid selected a fascinating series of mathematical topics to present to lay readers, tied together with the unifying thread of what Euclid pioneered. Her writing is lively, much of it is informative, and I would guess that this book, originally published in 1963 and now reprinted by Dover, has been widely read.It being understood that a popular book must necessarily slough over technicalities in order to convey general ideas, I am nevertheless shocked that Reid, with all her mathematical contacts, including her famous sister and brother-in-law (alive when she wrote this), did not have professional mathematicians check the correctness of what she wrote.Here are a few inaccuracies I found:p.60. “Their [Greek] geometry was based on an axiom which stated in essence that parallel lines never meet …” (By definition, parallel lines are lines in the same plane that do not meet. No axiom is needed to guarantee that!)p.152. “… the fifth postulate, which makes a statement very roughly equivalent to our common statement that parallel lines never meet.”p.136 Reid states falsely that Gauss was the first person in the history of mathematics to question the age-old assumption that the four classical constuction problems could be solved using straightedge and compass alone. Descartes, for one, preceded him.p.157 “The true surface of hyperbolic geometry … is what is called the pseudosphere, a world of two unending trumpets.”p.255 She states incorrectly that the domain of arithmetic presented by Hilbert in his Grundlagen is that of the constructible numbers. It is only a subfield of that.p. 278 She states that elementary algebra is a decidable theory according to Tarski, without specifying (as she did for “elementary geometry”) what that is. Ditto her statement that elementary arithmetic is decidable.I was also disappointed that Reid hardly gave any references and has no bibliography. Surely many readers became interested in topics she presented and would wish to read more about them.Reid still can fix these defects in a subsequent edition.

⭐It was very hard to get a copy of this book, and it cost me a ton of money. I do not regret this purchase in the least. If you have any interest in math (or if you don’t you should read it and maybe you’ll become interested) this book is incredible. I have reread it at least 10 times. It tries for nothing less than a story of the major advances in geometry from the greeks to the present. It is now a little out of date, (it says that Fermat’s theorem is unproven) Constance Reid is such a good writer, that it does not matter. You should without a doubt try to get a copy of this book. Chapters 7, 9, 12 are exceptional.

⭐I think that I shall never see A constant lovelier than e,Whose digits are too great too state,They’re 2.71828…. . .But only Euler could make an e.Of all the constants you will need,There’s only one that you should Reid.

⭐Excellent book.

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