Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle

Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle


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Product Details

ISBN-13: 9781602064171
Publisher: Cosimo
Publication date: 05/01/2007
Pages: 96
Product dimensions: 8.25(w) x 11.00(h) x 0.20(d)

Table of Contents

Practical and Theoretical Constructions2
Statement of the Problem in Algebraic Form3
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 constructed5
5, 6.Normal form of x6
7, 8.Conjugate values7
9.The corresponding equation F(x) = o8
10.Other rational equations f(x) = o8
11, 12.The irreducible equation [phi](x) = o10
13, 14.The degree of the irreducible equation a power of 211
Chapter II.The Delian Problem and the Trisection of the Angle
1.The impossibility of solving the Delian problem with straight edge and compasses13
2.The general equation x[superscript 8] = [lambda]13
3.The impossibility of trisecting an angle with straight edge and compasses14
Chapter III.The Division of the Circle into Equal Parts
1.History of the problem16
2-4.Gauss's prime numbers17
5.The cyclotomic equation19
6.Gauss's Lemma19
7, 8.The irreducibility of the cyclotomic equation21
Chapter IV.The Construction of the Regular Polygon of 17 Sides
1.Algebraic statement of the problem24
2-4.The periods formed from the roots25
5, 6.The quadratic equations satisfied by the periods27
7.Historical account of constructions with straight edge and compasses32
8, 9.Von Staudt's construction of the regular polygon of 17 sides34
Chapter V.General Considerations on Algebraic Constructions
1.Paper folding42
2.The conic sections42
3.The Cissoid of Diocles44
4.The Conchoid of Nicomedes45
5.Mechanical devices47
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 numbers49
2.Arrangement of algebraic numbers according to height50
3.Demonstration of the existence of transcendental numbers53
Chapter II.Historical Survey of the Attempts at the Computation and Construction of [pi]
1.The empirical stage56
2.The Greek mathematicians56
3.Modern analysis from 1670 to 177058
4, 5.Revival of critical rigor since 177059
Chapter III.The Transcendence of the Number e
1.Outline of the demonstration61
2.The symbol h[superscript r] and the function [phi](x)62
3.Hermite's Theorem65
Chapter IV.The Transcendence of the Number [pi]
1.Outline of the demonstration68
2.The function [psi](x)70
3.Lindemann's Theorem73
4.Lindemann's Corollary74
5.The transcendence of [pi]76
6.The transcendence of y = e[superscript x]77
7.The transcendence of y = sin[superscript -1]x77
Chapter V.The Integraph and the Geometric Construction of [pi]
1.The impossibility of the quadrature of the circle with straight edge and compasses78
2.Principle of the integraph78
3.Geometric construction of [pi]79

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