Preface to the Third Edition | p. vii |

Preface to the Second Edition | p. ix |

Preface to the First Edition | p. xi |

The Theorem of Pythagoras | p. 1 |

Arithmetic and Geometry | p. 2 |

Pythagorean Triples | p. 4 |

Rational Points on the Circle | p. 6 |

Right-Angled Triangles | p. 9 |

Irrational Numbers | p. 11 |

The Definition of Distance | p. 13 |

Biographical Notes: Pythagoras | p. 15 |

Greek Geometry | p. 17 |

The Deductive Method | p. 18 |

The Regular Polyhedra | p. 20 |

Ruler and Compass Constructions | p. 25 |

Conic Sections | p. 28 |

Higher-Degree Curves | p. 31 |

Biographical Notes: Euclid | p. 35 |

Greek Number Theory | p. 37 |

The Role of Number Theory | p. 38 |

Polygonal, Prime, and Perfect Numbers | p. 38 |

The Euclidean Algorithm | p. 41 |

Pell's Equation | p. 44 |

The Chord and Tangent Methods | p. 48 |

Biographical Notes: Diophantus | p. 50 |

Infinity in Greek Mathematics | p. 53 |

Fear of Infinity | p. 54 |

Eudoxus's Theory of Proportions | p. 56 |

The Method of Exhaustion | p. 58 |

The Area of a Parabolic Segment | p. 63 |

Biographical Notes: Archimedes | p. 66 |

Number Theory in Asia | p. 69 |

The Euclidean Algorithm | p. 70 |

The Chinese Remainder Theorem | p. 71 |

Linear Diophantine Equations | p. 74 |

Pell's Equation in Brahmagupta | p. 75 |

Pell's Equation in Bhâskara II | p. 78 |

Rational Triangles | p. 81 |

Biographical Notes: Brahmagupta and Bhâskara | p. 84 |

Polynomial Equations | p. 87 |

Algebra | p. 88 |

Linear Equations and Elimination | p. 89 |

Quadratic Equations | p. 92 |

Quadratic Irrationals | p. 95 |

The Solution of the Cubic | p. 97 |

Angle Division | p. 99 |

Higher-Degree Equations | p. 101 |

Biographical Notes: Tartaglia, Cardano, and Viète | p. 103 |

Analytic Geometry | p. 109 |

Steps Toward Analytic Geometry | p. 110 |

Fermat and Descartes | p. 111 |

Algebraic Curves | p. 112 |

Newton's Classification of Cubics | p. 115 |

Construction of Equations, Bézout's Theorem | p. 118 |

The Arithmetization of Geometry | p. 120 |

Biographical Notes: Descartes | p. 122 |

Projective Geometry | p. 127 |

Perspective | p. 128 |

Anamorphosis | p. 131 |

Desargues's Projective Geometry | p. 132 |

The Projective View of Curves | p. 136 |

The Projective Plane | p. 141 |

The Projective Line | p. 144 |

Homogeneous Coordinates | p. 147 |

Pascal's Theorem | p. 150 |

Biographical Notes: Desargues and Pascal | p. 153 |

Calculus | p. 157 |

What Is Calculus? | p. 158 |

Early Results on Areas and Volumes | p. 159 |

Maxima, Minima, and Tangents | p. 162 |

The Arithmetica Infinitorum of Wallis | p. 164 |

Newton's Calculus of Series | p. 167 |

The Calculus of Leibniz | p. 170 |

Biographical Notes: Wallis, Newton, and Leibniz | p. 172 |

Infinite Series | p. 181 |

Early Results | p. 182 |

Power Series | p. 185 |

An Interpolation on Interpolation | p. 188 |

Summation of Series | p. 189 |

Fractional Power Series | p. 191 |

Generating Functions | p. 192 |

The Zeta Function | p. 195 |

Biographical Notes: Gregory and Euler | p. 197 |

The Number Theory Revival | p. 203 |

Between Diophantus and Fermat | p. 204 |

Fermat's Little Theorem | p. 207 |

Fermat's Last Theorem | p. 210 |

Rational Right-Angled Triangles | p. 211 |

Rational Points on Cubics of Genus 0 | p. 215 |

Rational Points on Cubics of Genus 1 | p. 218 |

Biographical Notes: Fermat | p. 222 |

Elliptic Functions | p. 225 |

Elliptic and Circular Functions | p. 226 |

Parameterization of Cubic Curves | p. 226 |

Elliptic Integrals | p. 228 |

Doubling the Arc of the Lemniscate | p. 230 |

General Addition Theorems | p. 232 |

Elliptic Functions | p. 234 |

A Postscript on the Lemniscate | p. 236 |

Biographical Notes: Abel and Jacobi | p. 237 |

Mechanics | p. 243 |

Mechanics Before Calculus | p. 244 |

The Fundamental Theorem of Motion | p. 246 |

Kepler's Laws and the Inverse Square Law | p. 249 |

Celestial Mechanics | p. 253 |

Mechanical Curves | p. 255 |

The Vibrating String | p. 261 |

Hydrodynamics | p. 265 |

Biographical Notes: The Bernoullis | p. 267 |

Complex Numbers in Algebra | p. 275 |

Impossible Numbers | p. 276 |

Quadratic Equations | p. 276 |

Cubic Equations | p. 277 |

Wallis's Attempt at Geometric Representation | p. 279 |

Angle Division | p. 281 |

The Fundamental Theorem of Algebra | p. 285 |

The Proofs of d' Alembert and Gauss | p. 287 |

Biographical Notes: d' Alembert | p. 291 |

Complex Numbers and Curves | p. 295 |

Roots and Intersections | p. 296 |

The Complex Projective Line | p. 298 |

Branch Points | p. 301 |

Topology of Complex Projective Curves | p. 304 |

Biographical Notes: Riemann | p. 308 |

Complex Numbers and Functions | p. 313 |

Complex Functions | p. 314 |

Conformal Mapping | p. 318 |

Cauchy's Theorem | p. 319 |

Double Periodicity of Elliptic Functions | p. 322 |

Elliptic Curves | p. 325 |

Uniformization | p. 329 |

Biographical Notes: Lagrange and Cauchy | p. 331 |

Differential Geometry | p. 335 |

Transcendental Curves | p. 336 |

Curvature of Plane Curves | p. 340 |

Curvature of Surfaces | p. 343 |

Surfaces of Constant Curvature | p. 344 |

Geodesies | p. 346 |

The Gauss-Bonnet Theorem | p. 348 |

Biographical Notes: Harriot and Gauss | p. 352 |

Non-Euclidean Geometry | p. 359 |

The Parallel Axiom | p. 360 |

Spherical Geometry | p. 363 |

Geometry of Bolyai and Lobachevsky | p. 365 |

Beltrami's Projective Model | p. 366 |

Beltrami's Conformal Models | p. 369 |

The Complex Interpretations | p. 374 |

Biographical Notes: Bolyai and Lobachevsky | p. 378 |

Group Theory | p. 383 |

The Group Concept | p. 384 |

Subgroups and Quotients | p. 387 |

Permutations and Theory of Equations | p. 389 |

Permutation Groups | p. 393 |

Polyhedral Groups | p. 395 |

Groups and Geometries | p. 398 |

Combinatorial Group Theory | p. 401 |

Finite Simple Groups | p. 404 |

Biographical Notes: Galois | p. 409 |

Hypercomplex Numbers | p. 415 |

Complex Numbers in Hindsight | p. 416 |

The Arithmetic of Pairs | p. 417 |

Properties of + and x | p. 419 |

Arithmetic of Triples and Quadruples | p. 421 |

Quaternions, Geometry, and Physics | p. 424 |

Octonions | p. 428 |

Why C, H, and O Are Special | p. 430 |

Biographical Notes: Hamilton | p. 433 |

Algebraic Number Theory | p. 439 |

Algebraic Numbers | p. 440 |

Gaussian Integers | p. 442 |

Algebraic Integers | p. 445 |

Ideals | p. 448 |

Ideal Factorization | p. 452 |

Sums of Squares Revisited | p. 454 |

Rings and Fields | p. 457 |

Biographical Notes: Dedekind, Hilbert, and Noether | p. 459 |

Topology | p. 467 |

Geometry and Topology | p. 468 |

Polyhedron Formulas of Descartes and Euler | p. 469 |

The Classification of Surfaces | p. 471 |

Descartes and Gauss-Bonnet | p. 474 |

Euler Characteristic and Curvature | p. 477 |

Surfaces and Planes | p. 479 |

The Fundamental Group | p. 484 |

The Poincaré Conjecture | p. 486 |

Biographical Notes: Poincaré | p. 492 |

Simple Groups | p. 495 |

Finite Simple Groups and Finite Fields | p. 496 |

The Mathieu Groups | p. 498 |

Continuous Groups | p. 501 |

Simplicity of SO(3) | p. 505 |

Simple Lie Groups and Lie Algebras | p. 509 |

Finite Simple Groups Revisited | p. 513 |

The Monster | p. 515 |

Biographical Notes: Lie, Killing, and Cartan | p. 518 |

Sets, Logic, and Computation | p. 525 |

Sets | p. 526 |

Ordinals | p. 528 |

Measure | p. 531 |

Axiom of Choice and Large Cardinals | p. 534 |

The Diagonal Argument | p. 536 |

Computability | p. 538 |

Logic and Gödel's Theorem | p. 541 |

Provability and Truth | p. 546 |

Biographical Notes: Gödel | p. 549 |

Combinatorics | p. 553 |

What Is Combinatorics? | p. 554 |

The Pigeonhole Principle | p. 557 |

Analysis and Combinatorics | p. 560 |

Graph Theory | p. 563 |

Nonplanar Graphs | p. 567 |

The Konig Infinity Lemma | p. 571 |

Ramsey Theory | p. 575 |

Hard Theorems of Combinatorics | p. 580 |

Biographical Notes: Erdos | p. 584 |

Bibliography | p. 589 |

Index | p. 629 |

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