Famous Problems of Elementary Geometry (Dover Phoenix Editions) [Hardcover]

Felix Klein (Author)

Book Description

Series: Dover Phoenix Editions | Publication Date: December 15, 2003

Widely regarded as a classic of modern mathematics, this expanded version of Felix Klein's celebrated 1894 lectures uses contemporary techniques to examine three famous problems of antiquity: doubling the volume of a cube, trisecting an angle, and squaring a circle. Today's students will find this volume of particular interest in its answers to such questions as: Under what circumstances is a geometric construction possible? By what means can a geometric construction be effected? What are transcendental numbers, and how can you prove that e and pi are transcendental? The straightforward treatment requires no higher knowledge of mathematics. Unabridged reprint of the classic 1930 second edition.

Product Details

• Hardcover: 112 pages
• Publisher: Dover Publications; 2 Revised edition (December 15, 2003)
• Language: English
• ISBN-10: 0486495515
• ISBN-13: 978-0486495514
• Product Dimensions: 8.8 x 5.8 x 0.5 inches
• Shipping Weight: 11 ounces
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #4,658,792 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
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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.