Mathematics and its history:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2002
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 479 - 513 |
| Beschreibung: | XVIII, 542 S. Ill., graph. Darst. |
| ISBN: | 0387953361 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804129012983791616 |
|---|---|
| adam_text | Contents
Preface to the Second Edition vii
Preface to the First Edition ix
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
xiii
xjv Contents
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
s 6.4 Quadratic Irrationals 90
I 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 Bezout s Theorem Ill
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
Contents xv
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
xvi Contents
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 Bernoullis 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 Branchpoints 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
Contents xvii
17 Differential Geometry 315
17.1 Transcendental Curves 315
17.2 Curvature of Plane Curves 319
17.3 Curvature of Surfaces 322
17.4 Surfaces of Constant Curvature 324
17.5 Geodesies 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, H, and O Are Special 396
20.8 Biographical Notes: Hamilton 399
xviii Contents
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: Poincare 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: Godel 475
Bibliography 479
Index 514
|
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| discipline | Mathematik Philosophie |
| edition | 2. ed. |
| era | Geschichte gnd |
| era_facet | Geschichte |
| format | Book |
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| isbn | 0387953361 |
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| spelling | Stillwell, John 1942- Verfasser (DE-588)128427264 aut Mathematics and its history John Stillwell 2. ed. New York [u.a.] Springer 2002 XVIII, 542 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Literaturverz. S. 479 - 513 Geschichte gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Mathematik (DE-588)4037944-9 s Geschichte (DE-588)4020517-4 s DE-604 Geschichte z 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009690116&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
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| title | Mathematics and its history |
| title_auth | Mathematics and its history |
| title_exact_search | Mathematics and its history |
| title_full | Mathematics and its history John Stillwell |
| title_fullStr | Mathematics and its history John Stillwell |
| title_full_unstemmed | Mathematics and its history John Stillwell |
| title_short | Mathematics and its history |
| title_sort | mathematics and its history |
| topic | Mathematik (DE-588)4037944-9 gnd Geschichte (DE-588)4020517-4 gnd |
| topic_facet | Mathematik Geschichte |
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