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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