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

by Felix Klein, Wooster Woodruff Beman (Translator), David Eugene Smith (Translator)

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
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 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