| |

| |

Preface to the Third Edition | |

| |

| |

Preface to the Second Edition | |

| |

| |

Preface to the First Edition | |

| |

| |

| |

The Theorem of Pythagoras | |

| |

| |

| |

Arithmetic and Geometry | |

| |

| |

| |

Pythagorean Triples | |

| |

| |

| |

Rational Points on the Circle | |

| |

| |

| |

Right-Angled Triangles | |

| |

| |

| |

Irrational Numbers | |

| |

| |

| |

The Definition of Distance | |

| |

| |

| |

Biographical Notes: Pythagoras | |

| |

| |

| |

Greek Geometry | |

| |

| |

| |

The Deductive Method | |

| |

| |

| |

The Regular Polyhedra | |

| |

| |

| |

Ruler and Compass Constructions | |

| |

| |

| |

Conic Sections | |

| |

| |

| |

Higher-Degree Curves | |

| |

| |

| |

Biographical Notes: Euclid | |

| |

| |

| |

Greek Number Theory | |

| |

| |

| |

The Role of Number Theory | |

| |

| |

| |

Polygonal, Prime, and Perfect Numbers | |

| |

| |

| |

The Euclidean Algorithm | |

| |

| |

| |

Pell's Equation | |

| |

| |

| |

The Chord and Tangent Methods | |

| |

| |

| |

Biographical Notes: Diophantus | |

| |

| |

| |

Infinity in Greek Mathematics | |

| |

| |

| |

Fear of Infinity | |

| |

| |

| |

Eudoxus's Theory of Proportions | |

| |

| |

| |

The Method of Exhaustion | |

| |

| |

| |

The Area of a Parabolic Segment | |

| |

| |

| |

Biographical Notes: Archimedes | |

| |

| |

| |

Number Theory in Asia | |

| |

| |

| |

The Euclidean Algorithm | |

| |

| |

| |

The Chinese Remainder Theorem | |

| |

| |

| |

Linear Diophantine Equations | |

| |

| |

| |

Pell's Equation in Brahmagupta | |

| |

| |

| |

Pell's Equation in Bhï¿½skara II | |

| |

| |

| |

Rational Triangles | |

| |

| |

| |

Biographical Notes: Brahmagupta and Bhï¿½skara | |

| |

| |

| |

Polynomial Equations | |

| |

| |

| |

Algebra | |

| |

| |

| |

Linear Equations and Elimination | |

| |

| |

| |

Quadratic Equations | |

| |

| |

| |

Quadratic Irrationals | |

| |

| |

| |

The Solution of the Cubic | |

| |

| |

| |

Angle Division | |

| |

| |

| |

Higher-Degree Equations | |

| |

| |

| |

Biographical Notes: Tartaglia, Cardano, and Viï¿½te | |

| |

| |

| |

Analytic Geometry | |

| |

| |

| |

Steps Toward Analytic Geometry | |

| |

| |

| |

Fermat and Descartes | |

| |

| |

| |

Algebraic Curves | |

| |

| |

| |

Newton's Classification of Cubics | |

| |

| |

| |

Construction of Equations, Bï¿½zout's Theorem | |

| |

| |

| |

The Arithmetization of Geometry | |

| |

| |

| |

Biographical Notes: Descartes | |

| |

| |

| |

Projective Geometry | |

| |

| |

| |

Perspective | |

| |

| |

| |

Anamorphosis | |

| |

| |

| |

Desargues's Projective Geometry | |

| |

| |

| |

The Projective View of Curves | |

| |

| |

| |

The Projective Plane | |

| |

| |

| |

The Projective Line | |

| |

| |

| |

Homogeneous Coordinates | |

| |

| |

| |

Pascal's Theorem | |

| |

| |

| |

Biographical Notes: Desargues and Pascal | |

| |

| |

| |

Calculus | |

| |

| |

| |

What Is Calculus? | |

| |

| |

| |

Early Results on Areas and Volumes | |

| |

| |

| |

Maxima, Minima, and Tangents | |

| |

| |

| |

The Arithmetica Infinitorum of Wallis | |

| |

| |

| |

Newton's Calculus of Series | |

| |

| |

| |

The Calculus of Leibniz | |

| |

| |

| |

Biographical Notes: Wallis, Newton, and Leibniz | |

| |

| |

| |

Infinite Series | |

| |

| |

| |

Early Results | |

| |

| |

| |

Power Series | |

| |

| |

| |

An Interpolation on Interpolation | |

| |

| |

| |

Summation of Series | |

| |

| |

| |

Fractional Power Series | |

| |

| |

| |

Generating Functions | |

| |

| |

| |

The Zeta Function | |

| |

| |

| |

Biographical Notes: Gregory and Euler | |

| |

| |

| |

The Number Theory Revival | |

| |

| |

| |

Between Diophantus and Fermat | |

| |

| |

| |

Fermat's Little Theorem | |

| |

| |

| |

Fermat's Last Theorem | |

| |

| |

| |

Rational Right-Angled Triangles | |

| |

| |

| |

Rational Points on Cubics of Genus 0 | |

| |

| |

| |

Rational Points on Cubics of Genus 1 | |

| |

| |

| |

Biographical Notes: Fermat | |

| |

| |

| |

Elliptic Functions | |

| |

| |

| |

Elliptic and Circular Functions | |

| |

| |

| |

Parameterization of Cubic Curves | |

| |

| |

| |

Elliptic Integrals | |

| |

| |

| |

Doubling the Arc of the Lemniscate | |

| |

| |

| |

General Addition Theorems | |

| |

| |

| |

Elliptic Functions | |

| |

| |

| |

A Postscript on the Lemniscate | |

| |

| |

| |

Biographical Notes: Abel and Jacobi | |

| |

| |

| |

Mechanics | |

| |

| |

| |

Mechanics Before Calculus | |

| |

| |

| |

The Fundamental Theorem of Motion | |

| |

| |

| |

Kepler's Laws and the Inverse Square Law | |

| |

| |

| |

Celestial Mechanics | |

| |

| |

| |

Mechanical Curves | |

| |

| |

| |

The Vibrating String | |

| |

| |

| |

Hydrodynamics | |

| |

| |

| |

Biographical Notes: The Bernoullis | |

| |

| |

| |

Complex Numbers in Algebra | |

| |

| |

| |

Impossible Numbers | |

| |

| |

| |

Quadratic Equations | |

| |

| |

| |

Cubic Equations | |

| |

| |

| |

Wallis's Attempt at Geometric Representation | |

| |

| |

| |

Angle Division | |

| |

| |

| |

The Fundamental Theorem of Algebra | |

| |

| |

| |

The Proofs of d' Alembert and Gauss | |

| |

| |

| |

Biographical Notes: d' Alembert | |

| |

| |

| |

Complex Numbers and Curves | |

| |

| |

| |

Roots and Intersections | |

| |

| |

| |

The Complex Projective Line | |

| |

| |

| |

Branch Points | |

| |

| |

| |

Topology of Complex Projective Curves | |

| |

| |

| |

Biographical Notes: Riemann | |

| |

| |

| |

Complex Numbers and Functions | |

| |

| |

| |

Complex Functions | |

| |

| |

| |

Conformal Mapping | |

| |

| |

| |

Cauchy's Theorem | |

| |

| |

| |

Double Periodicity of Elliptic Functions | |

| |

| |

| |

Elliptic Curves | |

| |

| |

| |

Uniformization | |

| |

| |

| |

Biographical Notes: Lagrange and Cauchy | |

| |

| |

| |

Differential Geometry | |

| |

| |

| |

Transcendental Curves | |

| |

| |

| |

Curvature of Plane Curves | |

| |

| |

| |

Curvature of Surfaces | |

| |

| |

| |

Surfaces of Constant Curvature | |

| |

| |

| |

Geodesies | |

| |

| |

| |

The Gauss-Bonnet Theorem | |

| |

| |

| |

Biographical Notes: Harriot and Gauss | |

| |

| |

| |

Non-Euclidean Geometry | |

| |

| |

| |

The Parallel Axiom | |

| |

| |

| |

Spherical Geometry | |

| |

| |

| |

Geometry of Bolyai and Lobachevsky | |

| |

| |

| |

Beltrami's Projective Model | |

| |

| |

| |

Beltrami's Conformal Models | |

| |

| |

| |

The Complex Interpretations | |

| |

| |

| |

Biographical Notes: Bolyai and Lobachevsky | |

| |

| |

| |

Group Theory | |

| |

| |

| |

The Group Concept | |

| |

| |

| |

Subgroups and Quotients | |

| |

| |

| |

Permutations and Theory of Equations | |

| |

| |

| |

Permutation Groups | |

| |

| |

| |

Polyhedral Groups | |

| |

| |

| |

Groups and Geometries | |

| |

| |

| |

Combinatorial Group Theory | |

| |

| |

| |

Finite Simple Groups | |

| |

| |

| |

Biographical Notes: Galois | |

| |

| |

| |

Hypercomplex Numbers | |

| |

| |

| |

Complex Numbers in Hindsight | |

| |

| |

| |

The Arithmetic of Pairs | |

| |

| |

| |

Properties of + and x | |

| |

| |

| |

Arithmetic of Triples and Quadruples | |

| |

| |

| |

Quaternions, Geometry, and Physics | |

| |

| |

| |

Octonions | |

| |

| |

| |

Why C, H, and O Are Special | |

| |

| |

| |

Biographical Notes: Hamilton | |

| |

| |

| |

Algebraic Number Theory | |

| |

| |

| |

Algebraic Numbers | |

| |

| |

| |

Gaussian Integers | |

| |

| |

| |

Algebraic Integers | |

| |

| |

| |

Ideals | |

| |

| |

| |

Ideal Factorization | |

| |

| |

| |

Sums of Squares Revisited | |

| |

| |

| |

Rings and Fields | |

| |

| |

| |

Biographical Notes: Dedekind, Hilbert, and Noether | |

| |

| |

| |

Topology | |

| |

| |

| |

Geometry and Topology | |

| |

| |

| |

Polyhedron Formulas of Descartes and Euler | |

| |

| |

| |

The Classification of Surfaces | |

| |

| |

| |

Descartes and Gauss-Bonnet | |

| |

| |

| |

Euler Characteristic and Curvature | |

| |

| |

| |

Surfaces and Planes | |

| |

| |

| |

The Fundamental Group | |

| |

| |

| |

The Poincarï¿½ Conjecture | |

| |

| |

| |

Biographical Notes: Poincarï¿½ | |

| |

| |

| |

Simple Groups | |

| |

| |

| |

Finite Simple Groups and Finite Fields | |

| |

| |

| |

The Mathieu Groups | |

| |

| |

| |

Continuous Groups | |

| |

| |

| |

Simplicity of SO(3) | |

| |

| |

| |

Simple Lie Groups and Lie Algebras | |

| |

| |

| |

Finite Simple Groups Revisited | |

| |

| |

| |

The Monster | |

| |

| |

| |

Biographical Notes: Lie, Killing, and Cartan | |

| |

| |

| |

Sets, Logic, and Computation | |

| |

| |

| |

Sets | |

| |

| |

| |

Ordinals | |

| |

| |

| |

Measure | |

| |

| |

| |

Axiom of Choice and Large Cardinals | |

| |

| |

| |

The Diagonal Argument | |

| |

| |

| |

Computability | |

| |

| |

| |

Logic and Gï¿½del's Theorem | |

| |

| |

| |

Provability and Truth | |

| |

| |

| |

Biographical Notes: Gï¿½del | |

| |

| |

| |

Combinatorics | |

| |

| |

| |

What Is Combinatorics? | |

| |

| |

| |

The Pigeonhole Principle | |

| |

| |

| |

Analysis and Combinatorics | |

| |

| |

| |

Graph Theory | |

| |

| |

| |

Nonplanar Graphs | |

| |

| |

| |

The Konig Infinity Lemma | |

| |

| |

| |

Ramsey Theory | |

| |

| |

| |

Hard Theorems of Combinatorics | |

| |

| |

| |

Biographical Notes: Erdos | |

| |

| |

Bibliography | |

| |

| |

Index | |