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# Monographs on Topics of Modern Mathematics

## Overview

“Of high merit”—*Scientific American*

This stimulating approach to several branches of modern mathematics is geared to those with no background beyond elementary algebra and geometry. Its nine essays by leading mathematicians—including Oswald Veblen, Gilbert Ames Bliss, L. E. Dickson, and David Eugene Smith—cover the foundations of geometry, modern pure geometry and non-Euclidean geometry, fundamental propositions of algebra, algebraic equations, functions, fundamentals of calculus, and number theory. Each essay provides wide coverage, with proofs of important results and descriptions of leading methods. 1911 ed.

## Product Details

ISBN-13: | 9780486438160 |
---|---|

Publisher: | Dover Publications |

Publication date: | 08/25/2004 |

Series: | Dover Phoenix Editions Series |

Pages: | 432 |

Product dimensions: | 5.80(w) x 8.88(h) x 1.06(d) |

## Read an Excerpt

“Of high merit”—*Scientific American*

This stimulating approach to several branches of modern mathematics is geared to those with no background beyond elementary algebra and geometry. Its nine essays by leading mathematicians—including Oswald Veblen, Gilbert Ames Bliss, L. E. Dickson, and David Eugene Smith—cover the foundations of geometry, modern pure geometry and non-Euclidean geometry, fundamental propositions of algebra, algebraic equations, functions, fundamentals of calculus, and number theory. Each essay provides wide coverage, with proofs of important results and descriptions of leading methods. 1911 ed.

## First Chapter

“Of high merit”—*Scientific American*

This stimulating approach to several branches of modern mathematics is geared to those with no background beyond elementary algebra and geometry. Its nine essays by leading mathematicians—including Oswald Veblen, Gilbert Ames Bliss, L. E. Dickson, and David Eugene Smith—cover the foundations of geometry, modern pure geometry and non-Euclidean geometry, fundamental propositions of algebra, algebraic equations, functions, fundamentals of calculus, and number theory. Each essay provides wide coverage, with proofs of important results and descriptions of leading methods. 1911 ed.

## Table of Contents

I. | The Foundations of Geometry | 3 |

Introduction | ||

The Assumption of Order | ||

Order on a Line | ||

The Triangle and the Plane | ||

Regions in a Plane | ||

Congruence of Point Pairs | ||

Congruence of Angles | ||

Intersections of Circles | ||

Parallel Lines | ||

Mensuration | ||

Three-Dimensional Space | ||

Conclusion | ||

II. | Modern Pure Geometry | 55 |

Introduction | ||

Simple Elements in Geometry | ||

The Principle of Duality | ||

Principle of Continuity | ||

Points at Infinity | ||

Fundamental Theorem | ||

Metric Properties | ||

Anharmonic Ratios | ||

Elementary Geometric Forms | ||

Correlation of Elementary Forms | ||

Curves and Sheaves of Rays of the Second Order | ||

Pascal's and Brianchon's Theorems | ||

Pole and Polar Theory | ||

Conclusion | ||

III. | Non-Euclidean Geometry | 93 |

Introduction | ||

Parallel Lines | ||

The Euclidean Assumption | ||

The Lobachevskian Assumption | ||

The Riemannian Assumption | ||

The Sum of the Angles of a Triangle | ||

Areas | ||

Non-Euclidean Trigonometry | ||

Non-Euclidean Analytic Geometry | ||

Representation of the Lobachevskian Geometry on a Euclidean Plane | ||

Relation between Projective and Non-Euclidean Geometry | ||

The Element of Arc | ||

IV. | The Fundamental Propositions of Algebra | 151 |

Introduction | ||

The Addition of Angles and the Multiplication of Distances | ||

The Abstract Theory of these Operations | ||

Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane | ||

The Abstract Theory of the Algebra of Complex Quantities | ||

Appendix | Other Examples of the Algebra of Complex Quantities | |

Geometric Proof that Every Algebraic Equation has a Root | ||

V. | The Algebraic Equation | 211 |

General Introduction | ||

Historical Sketch and Definitions | ||

Equations with One Unknown and with Literal Coefficients | ||

Equations with One Unknown and with Numerical Coefficients | ||

Simultaneous Equations | ||

A Few References | ||

VI. | The Function Concept and the Fundamental Notions of the Calculus | 263 |

Introduction | ||

Variables and Functions | ||

The Fundamental Notions of the Calculus | ||

VII. | The Theory of Numbers | 307 |

Introduction | ||

Factors | ||

Diophantine Equations | ||

Congruences | ||

Binomial Congruences | ||

Quadratic Congruences | ||

Bibliography | ||

VIII. | Constructions with Ruler and Compasses; Regular Polygons | 353 |

Introduction | ||

Analytic Criterion for Constructibility | ||

Graphical Solution of a Quadratic Equation | ||

Domain of Rationality | ||

Functions Involving no Irrationalities other than Square Root | ||

Reducible and Irreducible Functions | ||

Fundamental Theorem; Duplication of the Cube; Trisection of an Angle; Quadrature of the Circle | ||

Connection between Regular Polygons and Roots of Unity | ||

De Moivre's Theorem | ||

Regular Pentagon and Decagon | ||

Regular Polygon of 17 Sides | ||

Construction of the Regular Polygon of 17 Sides | ||

Gauss's Theory of Regular Polygons | ||

Primitive Roots of Unity | ||

Gauss's Lemma | ||

Irreducibility of the Cyclotomic Equation | ||

Proofs of Theorems Cited Earlier | ||

References | ||

IX. | The History and Transcendence of [pi] | 389 |

The Nature of the Problem | ||

The History of the Problem | ||

The Transcendence of e | ||

The Transcendence of [pi] |

## Reading Group Guide

I. | The Foundations of Geometry | 3 |

Introduction | ||

The Assumption of Order | ||

Order on a Line | ||

The Triangle and the Plane | ||

Regions in a Plane | ||

Congruence of Point Pairs | ||

Congruence of Angles | ||

Intersections of Circles | ||

Parallel Lines | ||

Mensuration | ||

Three-Dimensional Space | ||

Conclusion | ||

II. | Modern Pure Geometry | 55 |

Introduction | ||

Simple Elements in Geometry | ||

The Principle of Duality | ||

Principle of Continuity | ||

Points at Infinity | ||

Fundamental Theorem | ||

Metric Properties | ||

Anharmonic Ratios | ||

Elementary Geometric Forms | ||

Correlation of Elementary Forms | ||

Curves and Sheaves of Rays of the Second Order | ||

Pascal's and Brianchon's Theorems | ||

Pole and Polar Theory | ||

Conclusion | ||

III. | Non-Euclidean Geometry | 93 |

Introduction | ||

Parallel Lines | ||

The Euclidean Assumption | ||

The Lobachevskian Assumption | ||

The Riemannian Assumption | ||

The Sum of the Angles of a Triangle | ||

Areas | ||

Non-Euclidean Trigonometry | ||

Non-Euclidean Analytic Geometry | ||

Representation of the Lobachevskian Geometry on a Euclidean Plane | ||

Relation between Projective and Non-Euclidean Geometry | ||

The Element of Arc | ||

IV. | The Fundamental Propositions of Algebra | 151 |

Introduction | ||

The Addition of Angles and the Multiplication of Distances | ||

The Abstract Theory of these Operations | ||

Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane | ||

The Abstract Theory of the Algebra of Complex Quantities | ||

Appendix | Other Examples of the Algebra of Complex Quantities | |

Geometric Proof that Every Algebraic Equation has a Root | ||

V. | The Algebraic Equation | 211 |

General Introduction | ||

Historical Sketch and Definitions | ||

Equations with One Unknown and with Literal Coefficients | ||

Equations with One Unknown and with Numerical Coefficients | ||

Simultaneous Equations | ||

A Few References | ||

VI. | The Function Concept and the Fundamental Notions of the Calculus | 263 |

Introduction | ||

Variables and Functions | ||

The Fundamental Notions of the Calculus | ||

VII. | The Theory of Numbers | 307 |

Introduction | ||

Factors | ||

Diophantine Equations | ||

Congruences | ||

Binomial Congruences | ||

Quadratic Congruences | ||

Bibliography | ||

VIII. | Constructions with Ruler and Compasses; Regular Polygons | 353 |

Introduction | ||

Analytic Criterion for Constructibility | ||

Graphical Solution of a Quadratic Equation | ||

Domain of Rationality | ||

Functions Involving no Irrationalities other than Square Root | ||

Reducible and Irreducible Functions | ||

Fundamental Theorem; Duplication of the Cube; Trisection of an Angle; Quadrature of the Circle | ||

Connection between Regular Polygons and Roots of Unity | ||

De Moivre's Theorem | ||

Regular Pentagon and Decagon | ||

Regular Polygon of 17 Sides | ||

Construction of the Regular Polygon of 17 Sides | ||

Gauss's Theory of Regular Polygons | ||

Primitive Roots of Unity | ||

Gauss's Lemma | ||

Irreducibility of the Cyclotomic Equation | ||

Proofs of Theorems Cited Earlier | ||

References | ||

IX. | The History and Transcendence of [pi] | 389 |

The Nature of the Problem | ||

The History of the Problem | ||

The Transcendence of e | ||

The Transcendence of [pi] |

## Interviews

I. | The Foundations of Geometry | 3 |

Introduction | ||

The Assumption of Order | ||

Order on a Line | ||

The Triangle and the Plane | ||

Regions in a Plane | ||

Congruence of Point Pairs | ||

Congruence of Angles | ||

Intersections of Circles | ||

Parallel Lines | ||

Mensuration | ||

Three-Dimensional Space | ||

Conclusion | ||

II. | Modern Pure Geometry | 55 |

Introduction | ||

Simple Elements in Geometry | ||

The Principle of Duality | ||

Principle of Continuity | ||

Points at Infinity | ||

Fundamental Theorem | ||

Metric Properties | ||

Anharmonic Ratios | ||

Elementary Geometric Forms | ||

Correlation of Elementary Forms | ||

Curves and Sheaves of Rays of the Second Order | ||

Pascal's and Brianchon's Theorems | ||

Pole and Polar Theory | ||

Conclusion | ||

III. | Non-Euclidean Geometry | 93 |

Introduction | ||

Parallel Lines | ||

The Euclidean Assumption | ||

The Lobachevskian Assumption | ||

The Riemannian Assumption | ||

The Sum of the Angles of a Triangle | ||

Areas | ||

Non-Euclidean Trigonometry | ||

Non-Euclidean Analytic Geometry | ||

Representation of the Lobachevskian Geometry on a Euclidean Plane | ||

Relation between Projective and Non-Euclidean Geometry | ||

The Element of Arc | ||

IV. | The Fundamental Propositions of Algebra | 151 |

Introduction | ||

The Addition of Angles and the Multiplication of Distances | ||

The Abstract Theory of these Operations | ||

Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane | ||

The Abstract Theory of the Algebra of Complex Quantities | ||

Appendix | Other Examples of the Algebra of Complex Quantities | |

Geometric Proof that Every Algebraic Equation has a Root | ||

V. | The Algebraic Equation | 211 |

General Introduction | ||

Historical Sketch and Definitions | ||

Equations with One Unknown and with Literal Coefficients | ||

Equations with One Unknown and with Numerical Coefficients | ||

Simultaneous Equations | ||

A Few References | ||

VI. | The Function Concept and the Fundamental Notions of the Calculus | 263 |

Introduction | ||

Variables and Functions | ||

The Fundamental Notions of the Calculus | ||

VII. | The Theory of Numbers | 307 |

Introduction | ||

Factors | ||

Diophantine Equations | ||

Congruences | ||

Binomial Congruences | ||

Quadratic Congruences | ||

Bibliography | ||

VIII. | Constructions with Ruler and Compasses; Regular Polygons | 353 |

Introduction | ||

Analytic Criterion for Constructibility | ||

Graphical Solution of a Quadratic Equation | ||

Domain of Rationality | ||

Functions Involving no Irrationalities other than Square Root | ||

Reducible and Irreducible Functions | ||

Fundamental Theorem; Duplication of the Cube; Trisection of an Angle; Quadrature of the Circle | ||

Connection between Regular Polygons and Roots of Unity | ||

De Moivre's Theorem | ||

Regular Pentagon and Decagon | ||

Regular Polygon of 17 Sides | ||

Construction of the Regular Polygon of 17 Sides | ||

Gauss's Theory of Regular Polygons | ||

Primitive Roots of Unity | ||

Gauss's Lemma | ||

Irreducibility of the Cyclotomic Equation | ||

Proofs of Theorems Cited Earlier | ||

References | ||

IX. | The History and Transcendence of [pi] | 389 |

The Nature of the Problem | ||

The History of the Problem | ||

The Transcendence of e | ||

The Transcendence of [pi] |

## Recipe

I. | The Foundations of Geometry | 3 |

Introduction | ||

The Assumption of Order | ||

Order on a Line | ||

The Triangle and the Plane | ||

Regions in a Plane | ||

Congruence of Point Pairs | ||

Congruence of Angles | ||

Intersections of Circles | ||

Parallel Lines | ||

Mensuration | ||

Three-Dimensional Space | ||

Conclusion | ||

II. | Modern Pure Geometry | 55 |

Introduction | ||

Simple Elements in Geometry | ||

The Principle of Duality | ||

Principle of Continuity | ||

Points at Infinity | ||

Fundamental Theorem | ||

Metric Properties | ||

Anharmonic Ratios | ||

Elementary Geometric Forms | ||

Correlation of Elementary Forms | ||

Curves and Sheaves of Rays of the Second Order | ||

Pascal's and Brianchon's Theorems | ||

Pole and Polar Theory | ||

Conclusion | ||

III. | Non-Euclidean Geometry | 93 |

Introduction | ||

Parallel Lines | ||

The Euclidean Assumption | ||

The Lobachevskian Assumption | ||

The Riemannian Assumption | ||

The Sum of the Angles of a Triangle | ||

Areas | ||

Non-Euclidean Trigonometry | ||

Non-Euclidean Analytic Geometry | ||

Representation of the Lobachevskian Geometry on a Euclidean Plane | ||

Relation between Projective and Non-Euclidean Geometry | ||

The Element of Arc | ||

IV. | The Fundamental Propositions of Algebra | 151 |

Introduction | ||

The Addition of Angles and the Multiplication of Distances | ||

The Abstract Theory of these Operations | ||

Geometric Example of the Algebra of Complex Quantities: The System of Points in the Plane | ||

The Abstract Theory of the Algebra of Complex Quantities | ||

Appendix | Other Examples of the Algebra of Complex Quantities | |

Geometric Proof that Every Algebraic Equation has a Root | ||

V. | The Algebraic Equation | 211 |

General Introduction | ||

Historical Sketch and Definitions | ||

Equations with One Unknown and with Literal Coefficients | ||

Equations with One Unknown and with Numerical Coefficients | ||

Simultaneous Equations | ||

A Few References | ||

VI. | The Function Concept and the Fundamental Notions of the Calculus | 263 |

Introduction | ||

Variables and Functions | ||

The Fundamental Notions of the Calculus | ||

VII. | The Theory of Numbers | 307 |

Introduction | ||

Factors | ||

Diophantine Equations | ||

Congruences | ||

Binomial Congruences | ||

Quadratic Congruences | ||

Bibliography | ||

VIII. | Constructions with Ruler and Compasses; Regular Polygons | 353 |

Introduction | ||

Analytic Criterion for Constructibility | ||

Graphical Solution of a Quadratic Equation | ||

Domain of Rationality | ||

Functions Involving no Irrationalities other than Square Root | ||

Reducible and Irreducible Functions | ||

Fundamental Theorem; Duplication of the Cube; Trisection of an Angle; Quadrature of the Circle | ||

Connection between Regular Polygons and Roots of Unity | ||

De Moivre's Theorem | ||

Regular Pentagon and Decagon | ||

Regular Polygon of 17 Sides | ||

Construction of the Regular Polygon of 17 Sides | ||

Gauss's Theory of Regular Polygons | ||

Primitive Roots of Unity | ||

Gauss's Lemma | ||

Irreducibility of the Cyclotomic Equation | ||

Proofs of Theorems Cited Earlier | ||

References | ||

IX. | The History and Transcendence of [pi] | 389 |

The Nature of the Problem | ||

The History of the Problem | ||

The Transcendence of e | ||

The Transcendence of [pi] |