Famous Problems Of Elementary Geometry: The Duplication Of The Cube, The Trisection Of An Angle, The Quadrature Of The Circle, : An Authorized Translation Of F. Klein's Vorträge [Paperback]

Felix Klein (Author)

Book Description

Publication Date: November 26, 2007

This book is a facsimile reprint and may contain imperfections such as marks, notations, marginalia and flawed pages.

--This text refers to an alternate Paperback edition.

Product Details

• Paperback: 92 pages
• Publisher: Hard Press (November 26, 2007)
• Language: English
• ISBN-10: 1407602764
• ISBN-13: 978-1407602769
• Product Dimensions: 8.8 x 5.9 x 0.4 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,364,185 in Books (See Top 100 in Books)

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3 of 3 people found the following review helpful:

5.0 out of 5 stars An interesting "unpretending little book", April 18, 2005

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: Famous Problems of Elementary Geometry (Dover Phoenix Editions) (Hardcover)

The back cover claims that this book is "widely regarded as a classic of modern mathematics", but I think one should rather listen to Klein himself: in his preface, Klein calls it an "unpretending little book", apparently based on a few (unpretending?) little lectures, typed up into a book by some (unpretending?) little student.

Circumventing Galois Theory, Klein proves the impossibility of doubling a cube and trisecting an angle in the first ten pages. We then look at the solvable cases of circle division, especially the constructability of the regular 17-gon. Gauss proved algebraically that p-division of the circle is possible for Fermat primes p=2^2^k+1. We work through Gauss's proof in the special case p=17. Giving up on a fully geometric proof, "there remains, then, only the geometric translation of the individual algebraic steps". In the second part of the book, to prove that it is impossible to square the circle we prove the transcendence of pi, and in the process we must also show the transcendence of e (and then go through Euler's formula e^i*pi=-1). These proofs use only elementary mathematics but unfortunately they are still rather technical. Perhaps the history of these proofs--they are the results of successive reworkings of much more advanced analytic proofs--vouches for their artificiality.