Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle.
"This short book, first published in 1897, addresses three geometry puzzles that have been passed down from ancient times. Written for high school students, this book aims to show a younger audience why math should matter and to make the problems found in math intriguing. Klein presents for his readers an investigation of the possibility or impossibility of finding solutions for the following problems in light of mathematics available to him: ¿ duplication of the cube ¿ trisection of an angle ¿ quadrature of the circle Mathematicians and students of the history of math will find this an intriguing work. German mathematician FELIX KLEIN (1849¿1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis and Elementary Mathematics from an Advanced Standpoint: Geometry."
1026051058
Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle.
"This short book, first published in 1897, addresses three geometry puzzles that have been passed down from ancient times. Written for high school students, this book aims to show a younger audience why math should matter and to make the problems found in math intriguing. Klein presents for his readers an investigation of the possibility or impossibility of finding solutions for the following problems in light of mathematics available to him: ¿ duplication of the cube ¿ trisection of an angle ¿ quadrature of the circle Mathematicians and students of the history of math will find this an intriguing work. German mathematician FELIX KLEIN (1849¿1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis and Elementary Mathematics from an Advanced Standpoint: Geometry."
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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.

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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Overview

"This short book, first published in 1897, addresses three geometry puzzles that have been passed down from ancient times. Written for high school students, this book aims to show a younger audience why math should matter and to make the problems found in math intriguing. Klein presents for his readers an investigation of the possibility or impossibility of finding solutions for the following problems in light of mathematics available to him: ¿ duplication of the cube ¿ trisection of an angle ¿ quadrature of the circle Mathematicians and students of the history of math will find this an intriguing work. German mathematician FELIX KLEIN (1849¿1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis and Elementary Mathematics from an Advanced Standpoint: Geometry."

Product Details

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

Table of Contents

Introduction
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
Notes81
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