Mathematics and its history / John Stillwell.
Series Undergraduate texts in mathematicsEditor: New York : Springer, c2002Edición: 2nd edDescripción: xviii, 542 p. : il. ; 24 cmISBN: 0387953361Tema(s): Mathematics -- HistoryOtra clasificación: 01-01 (00A05) Recursos en línea: Table of contents | Google Book SearchContents 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] 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 Bézout’s Theorem [111] 7.5 The Arithmetization of Geometry [115] 7.6 Biographical Notes: Descartes [116] 8 Projective Geometry 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 Bézout’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 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 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 Curvature 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 4- 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] 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 Gödel’s Theorem [469] 23.8 Provability and Truth [473] 23.9 Biographical Notes: Gödel [475] Bibliography [479] Index [514]
| Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | 01 St857 (Browse shelf) | Available | A-8375 |
Incluye referencias bibliográficas (p. 479-513) e índice.
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] --
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 Bézout’s Theorem [111] --
7.5 The Arithmetization of Geometry [115] --
7.6 Biographical Notes: Descartes [116] --
8 Projective Geometry --
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 Bézout’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 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 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 Curvature 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 4- 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] --
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 Gödel’s Theorem [469] --
23.8 Provability and Truth [473] --
23.9 Biographical Notes: Gödel [475] --
Bibliography [479] --
Index [514] --
MR, 2002i:01001
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