Famous Problems of Elementary Geometry

Overview


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,...
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FAMOUS PROBLEMS OF ELEMENTARY GEOMETRY

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Overview


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.
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Product Details

  • ISBN-13: 9780486495514
  • Publisher: Dover Publications
  • Publication date: 12/15/2003
  • Series: Dover Phoenix Editions Ser.
  • Format: Library Binding
  • Edition number: 2
  • Pages: 112
  • Product dimensions: 5.72 (w) x 8.68 (h) x 0.53 (d)

Table of Contents

Introduction
Practical and Theoretical Constructions 2
Statement of the Problem in Algebraic Form 3
Part I. The Possibility of the Construction of Algebraic Expressions
Chapter I. Algebraic Equations Solvable by Square Roots
1-4. Structure of the expression x to be constructed 5
5, 6. Normal form of x 6
7, 8. Conjugate values 7
9. The corresponding equation F(x) = o 8
10. Other rational equations f(x) = o 8
11, 12. The irreducible equation [phi](x) = o 10
13, 14. The degree of the irreducible equation a power of 2 11
Chapter II. The Delian Problem and the Trisection of the Angle
1. The impossibility of solving the Delian problem with straight edge and compasses 13
2. The general equation x[superscript 8] = [lambda] 13
3. The impossibility of trisecting an angle with straight edge and compasses 14
Chapter III. The Division of the Circle into Equal Parts
1. History of the problem 16
2-4. Gauss's prime numbers 17
5. The cyclotomic equation 19
6. Gauss's Lemma 19
7, 8. The irreducibility of the cyclotomic equation 21
Chapter IV. The Construction of the Regular Polygon of 17 Sides
1. Algebraic statement of the problem 24
2-4. The periods formed from the roots 25
5, 6. The quadratic equations satisfied by the periods 27
7. Historical account of constructions with straight edge and compasses 32
8, 9. Von Staudt's construction of the regular polygon of 17 sides 34
Chapter V. General Considerations on Algebraic Constructions
1. Paper folding 42
2. The conic sections 42
3. The Cissoid of Diocles 44
4. The Conchoid of Nicomedes 45
5. Mechanical devices 47
Part II. Transcendental Numbers and the Quadrature of the Circle
Chapter I. Cantor's Demonstration of the Existence of Transcendental Numbers
1. Definition of algebraic and of transcendental numbers 49
2. Arrangement of algebraic numbers according to height 50
3. Demonstration of the existence of transcendental numbers 53
Chapter II. Historical Survey of the Attempts at the Computation and Construction of [pi]
1. The empirical stage 56
2. The Greek mathematicians 56
3. Modern analysis from 1670 to 1770 58
4, 5. Revival of critical rigor since 1770 59
Chapter III. The Transcendence of the Number e
1. Outline of the demonstration 61
2. The symbol h[superscript r] and the function [phi](x) 62
3. Hermite's Theorem 65
Chapter IV. The Transcendence of the Number [pi]
1. Outline of the demonstration 68
2. The function [psi](x) 70
3. Lindemann's Theorem 73
4. Lindemann's Corollary 74
5. The transcendence of [pi] 76
6. The transcendence of y = e[superscript x] 77
7. The transcendence of y = sin[superscript -1]x 77
Chapter V. The Integraph and the Geometric Construction of [pi]
1. The impossibility of the quadrature of the circle with straight edge and compasses 78
2. Principle of the integraph 78
3. Geometric construction of [pi] 79
Notes 81
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