Table of contents for Mathematics and its history / John Stillwell.


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1  The Theorem of Pythagoras                                    1
1.1  Arithmetic and Geometry ..........          . . . ....  1
1.2  Pythagorean Triples ..................... .             3
1.3  Rational Points on the Circle ......... . . .....       5
1.4  Right-angled Triangles . ................. ..           8
1.5  Irrational Numbers  . ..................          ... ..  10
1.6  The Definition of Distance .................           12
1.7  Biographical Notes: Pythagoras  . . . . . . . . . .  .  15
2  Greek Geometry                                              17
2.1  The Deductive Method . .   . . . . . . . . . . . . . . . .  17
2.2  The Regular Polyhedra     .................            20
2.3  Ruler and Compass Constructions . . . . .........      25
2.4  Conic Sections ......              .       .28
2.5  Higher-Degree Curves    ..................             31
2.6  Biographical Notes: Euclid ..............          .   35
3  Greek Number Theory                                         37
3.1  The Role of Number Theory ..... . . . . . . . . . . .  37
3.2  Polygonal, Prime, and Perfect Numbers . . . .......    38
3.3  The Euclidean Algorithm   .. . . . ... ...... .41
3.4  Pell's Equation       .......................          43
3.5  The Chord and Tangent Methods . . . ........ ..        48
3.6  Biographical Notes: Diophantus . . . . . . . . ......  49



4  Infinity in Greek Mathematics                              51
4.1  Fear of Infinity  ................    .  . . ....51
4.2  Eudoxus' Theory of Proportions  . . .. . . .. . . .   53
4.3  The Method of Exhaustion ............ ....55
4.4  The Area of a Parabolic Segment .......... ...        61
4.5  Biographical Notes: Archimedes  . . .. . . . . . . . .  64
5  Number Theory in Asia                                      66
5.1  The Euclidean Algorithm   . . . . . . . . . . . . . . .  66
5.2  The Chinese Remainder Theorem  . . . . . .......      68
5.3  Linear Diophantine Equations . . . . . ..........     70
5.4  Pell's Equation in Brahmagupta     .......      ..    72
5.5  Pell's Equation in Bhaskara II ...............        74
5.6  Rational Triangles .  ...........      .........      77
5.7  Biographical Notes: Brahmagupta and Bhaskara .. ...   80
6  Polynomial Equations                                       82
6.1  Algebra  ......     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   82
6.2  Linear Equations and Elimination  . . . . .........   84
6.3  Quadratic Equations ................... .86
6.4  Quadratic Irrationals  ...................            90
6.5  The Solution of the Cubic .. . . . . . . .. ...       91
6.6  Angle Division  . . . . . . . . ......  ......        93
6.7  Higher-Degree Equations    .................          96
6.8  Biographical Notes: Tartaglia, Cardano, and Viete . ..97
7  Analytic Geometry                                         104
7.1  Steps toward Analytic Geometry ... . .. .   ..... 104
7.2  Fermat and Descartes .  .   .......             .    105
7.3  Algebraic Curves ........... .....107
7.4  Newton's Classification of Cubics . . . . . . . . . . 110
7.5  Construction of Equations and B6zout's Theorem  . . . . . 111
7.5  The Arithmetization of Geometry  . . .  . . . . ..... 115
7.6  Biographical Notes: Descartes ...... ...116
8  Projective Geometry                                       120
8.1  Perspective .............          .     .           120
8.2   Anamorphosis ..... ............ ..123
8.3  Desargues' Projective Geometry  ....... ....         125



8.4  The Projective View of Curves .......... ... ..      129
8.5  Homogeneous Coordinates     ..........         . ..  134
8.6  Bezout's Theorem Revisited  ............ ...         137
8.7  Pascal's Theorem  ............                       139
8.8  Biographical Notes: Desargues and Pascal ... .. ..   142
9  Calculus                                                  146
9.1  What Is Calculus? .. . . . . . . . .........         146
9.2  Early Results on Areas and Volumes . . . .......     148
9.3  Maxima, Minima, and Tangents .. . . . . .  ...       150
9.4  The Arithmetica Infinitorum of Wallis . . . . .  . . . .  152
9.5  Newton's Calculus of Series .. . . . . ........      155
9.6  The Calculus of Leibniz ..   ........... ...         159
9.7  Biographical Notes: Wallis, Newton, and Leibniz ..... 160
10 Infinite Series                                           170
10.1 Early Results .........................              170
10.2 Power Series  .........................              173
10.3 An Interpolation on Interpolation ........   ......  176
10.4 Summation of Series .............. .......           177
10.5 Fractional Power Series .................         . 179
10.6 Generating Functions . . . . . . . . . . . . . . .   181
10.7 The Zeta Function .....................              184
10.8 Biographical Notes: Gregory and Euler  .........     186
11 The Number Theory Revival                                 192
11.1 Between Diophantus and Fermat . ............. 192
11.2 Fermat's Little Theorem  .................... 196
11.3 Fermat's Last Theorem  ...................           198
11.4 Rational Right-angled Triangles ........ . . .... 200
11.5 Rational Points on Cubics of Genus 0  . . . . ....... 204
11.6 Rational Points on Cubics of Genus 1 . . . . . . . . . . . 207
11.7 Biographical Notes: Fermat ................ 211
12 Elliptic Functions                                        213
12.1 Elliptic and Circular Functions . . . . . . . . . . .  .  213
12.2 Parameterization of Cubic Curves .............. 214
12.3 Elliptic Integrals   ......................          215
12.4 Doubling the Arc of the Lemniscate ............. 217



12.5 General Addition Theorems  ................. 220
12.6 Elliptic Functions ...................... 222
12.7 A Postscript on the Lemniscate ......   .......... 224
12.8 Biographical Notes: Abel and Jacobi ....   ......... 224
13 Mechanics                                                  231
13.1 Mechanics before Calculus . . . . . . . . . . . . . . . . . 231
13.2 Celestial Mechanics ..................            ..234
13.3 Mechanical Curves ..................... 236
13.4 The Vibrating String  ....................            241
13.5 Hydrodynamics . . . . . ............ . . . .          245
13.6 Biographical Notes: The Bemoullis ............ 248
14 Complex Numbers in Algebra                                 256
14.1 Impossible Numbers .................... 256
14.2 Quadratic Equations  .................. .. .          257
14.3 Cubic Equations ........ ............. .              257
14.4 Wallis' Attempt at Geometric Interpretation ..... . .. 260
14.5 Angle Division  ...........         ...........       262
14.6 The Fundamental Theorem of Algebra ....... .... 266
14.7 The Proofs of d'Alembert and Gauss ......... . ... 268
14.8 Biographical Notes: d'Alembert ........... ... 272
15 Complex Numbers and Curves                                 276
15.1 Roots and Intersections .................... 276
15.2 The Complex Projective Line ................. 279
15.3 Branch Points ......................... 282
15.4 Topology of Complex Projective Curves ........... 285
15.5 Biographical Notes: Riemann ....   ........ ..        288
16 Complex Numbers and Functions                              293
16.1 Complex Functions ............           .........    293
16.2 Conformal Mapping .....................               297
16.3 Cauchy's Theorem   .......... ........            .. 299
16.4 Double Periodicity of Elliptic Functions .. . . . . .  302
16.5 Elliptic Curves ................... ....             305
16.6  Uniformization  ......... .    .  .  .  .  .  ........309
16.7 Biographical Notes: Lagrange and Cauchy .........     310



17 Differential Geometry                                      315
17.1 Transcendental Curves . . . . . . . . . . . . . . . ....  315
17.2 Curvature of Plane Curves ................... 319
17.3 Curvat    ure of Surfaces .................... 322
17.4 Surfaces of Constant Curvature .... . ......     .... 324
17.5 Geodesics ....................... ..326
17.6 The Gauss-Bonnet Theorem   ........ ......... 327
17.7 Biographical Notes: Harriot and Gauss ........... 331
18 Noneuclidean Geometry                                      338
18.1 The Parallel Axiom  ....     ................. 338
18.2 Spherical Geometry  ......      ............       . 341
18.3 Geometry of Bolyai and Lobachevsky ........... 343
18.4 Beltrami's Projective Model . .............. . 344
18.5 Beltrami's Conformal Models  . ............. . 348
18.6 The Complex Interpretations . .............. . 352
18.7 Biographical Notes: Bolyai and Lobachevsky .....   . 357
19 Group Theory                                               361
19.1 The Group Concept ................... .. 361
19.2 Permutations and Theory of Equations ........... 364
19.3 Permutation Groups ......................... 367
19.4 Polyhedral Groups . . . ..................368
19.5 Groups and Geometries ................... 371
19.6 Combinatorial Group Theory . .............. . 373
19.7 Biographical Notes: Galois . ............... . 377
20 Hypercomplex Numbers                                       382
20.1 Complex Numbers in Hindsight     .. . .  . . . . . ... 382
20.2 The Arithmetic of Pairs ...... . . . . . ......     . 383
20.3 Properties of + and x .........        ........ 385
20.4 Arithmetic of Triples and Quadruples  . . . . . ...... 387
20.5 Quaternions, Geometry, and Physics . . . . . ....... 391
20.6  Octonions  ............       . .............        393
20.7 Why C, I, and O Are Special . . . . . . . ........ 396
20.8 Biographical Notes: Hamilton  . . . . . . . ........ 399



21 Algebraic Number Theory                                    404
21.1 Algebraic Numbers ...................... 404
21.2  Gaussian Integers  ......................406
21.3 Algebraic Integers ..................              . 409
21.4 Ideals ................... .........                 412
21.5 Ideal Factorization  .................. .            416
21.6 Sums of Squares Revisited .....418
21.7 Rings and Fields  ................... ... 422
21.8 Biographical Notes: Dedekind, Hilbert, and Noether  . . . 424
22 Topology                                                   431
22.1 Geometry and Topology  ..... . . . . .  . . . . . . .  431
22.2 Polyhedron Formulas of Descartes and Euler  ...432
22.3 The Classification of Surfaces  . . .......434
22.4 Descartes and Gauss-Bonnet ....... . . . . . .. . . 438
22.5 Euler Characteristic and Curvature ..... ...440
22.6 Surfaces and Planes .............                    443
22.7 The Fundamental Group  ............                  448
22.8 Biographical Notes: Poincar6 ....... ....450
23 Sets, Logic, and Computation                               454
23.1 An Explanation ........ . . . . . . . . . . . .  .   454
23.2  Sets  ............455
23.3  Measure  ............ .......459
23.4 Axiom of Choice and Large Cardinals .....462
23.5 The Diagonal Argument ..................464
23.6 Computability ....... ..... ....466
23.7 Logic and Godel's Theorem  ................469
23.8 Provability and Truth ............473
23.9 Biographical Notes: Gdel . . .... . . . . . . .475








Library of Congress subject headings for this publication: Mathematics History