Mathematics and Its History (Third Edition) by John Stillwell
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Mathematics and Its History (Third Edition) written by
John Stillwell , Department of Mathematics, University of San Francisco, San Francisco, CA 94117-1080, USA, [email protected]
Mathematics and Its History (Third Edition) written by
John Stillwell
cover the following topics.
1. The Theorem of Pythagoras
1.1 Arithmetic and Geometry
1.2 Pythagorean Triples
1.3 Rational Points on the Circle
1.4 Right-Angled Triangles.
1.5 Irrational Numbers11
1.6 The Definition of Distance
1.7 Biographical Notes: Pythagoras
2. Greek Geometry
2.1 The Deductive Method.
2.2 The Regular Polyhedra.
2.3 Ruler and Compass Constructions
2.4 Conic Sections.
2.5 Higher-Degree Curves.
2.6 Biographical Notes: Euclid
3. Greek Number Theory
3.1 The Role of Number Theory
3.2 Polygonal, Prime, and Perfect Numbers
3.3 The Euclidean Algorithm
3.4 Pell’s Equation.
3.5 The Chord and Tangent Methods.
3.6 Biographical Notes: Diophantus.
4. Infinity in Greek Mathematics
4.1 Fear of Infinity.
4.2 Eudoxus’s Theory of Proportions.
4.3 The Method of Exhaustion58
4.4 The Area of a Parabolic Segment.
4.5 Biographical Notes: Archimedes.
5. Number Theory in Asia
5.1 The Euclidean Algorithm.
5.2 The Chinese Remainder Theorem
5.3 Linear Diophantine Equations.
5.4 Pell’s Equation in Brahmagupta.
5.5 Pell’s Equation in Bhˆaskara II.
5.6 Rational Triangles
5.7 Biographical Notes: Brahmagupta and Bhˆaskara
6. Polynomial Equations
6.1 Algebra
6.2 Linear Equations and Elimination
6.3 Quadratic Equations
6.4 Quadratic Irrationals.
6.5 The Solution of the Cubic.
6.6 Angle Division
6.7 Higher-Degree Equations
6.8 Biographical Notes: Tartaglia, Cardano, and Vi`ete
7. Analytic Geometry
7.1 Steps Toward Analytic Geometry.
7.2 Fermat and Descartes.
7.3 Algebraic Curves.
7.4 Newton’s Classification of Cubics
7.5 Construction of Equations, B´ezout’s Theorem.
7.6 The Arithmetization of Geometry
7.7 Biographical Notes: Descartes. .
8. Projective Geometry
8.1 Perspective128
8.2 Anamorphosis.131
8.3 Desargues’s Projective Geometry.
8.4 The Projective View of Curves.
8.5 The Projective Plane.
8.6 The Projective Line
8.7 Homogeneous Coordinates
8.8 Pascal’s Theorem.
8.9 Biographical Notes: Desargues and Pascal
9. Calculus
9.1 What Is Calculus?.
9.2 Early Results on Areas and Volumes
9.3 Maxima, Minima, and Tangents.
9.4 The Arithmetica Infinitorum of Wallis.
9.5 Newton’s Calculus of Series.167
9.6 The Calculus of Leibniz.
9.7 Biographical Notes: Wallis, Newton, and Leibniz .
10. Infinite Series
10.1 Early Results.
10.2 Power Series.
10.3 An Interpolation on Interpolation.
10.4 Summation of Series.
10.5 Fractional Power Series.
10.6 Generating Functions.
10.7 The Zeta Function.
10.8 Biographical Notes: Gregory and Euler
11. The Number Theory Revival
11.1 Between Diophantus and Fermat
11.2 Fermat’s Little Theorem
11.3 Fermat’s Last Theorem
11.4 Rational Right-Angled Triangles
11.5 Rational Points on Cubics of Genus 0
11.6 Rational Points on Cubics of Genus 1
11.7 Biographical Notes: Fermat
12. Elliptic Functions
12.1 Elliptic and Circular Functions
12.2 Parameterization of Cubic Curves
12.3 Elliptic Integrals
12.4 Doubling the Arc of the Lemniscate
12.5 General Addition Theorems
12.6 Elliptic Functions
12.7 A Postscript on the Lemniscate.
12.8 Biographical Notes: Abel and Jacobi
13. Mechanics
13.1 Mechanics Before Calculus
13.2 The Fundamental Theorem of Motion.
13.3 Kepler’s Laws and the Inverse Square Law
13.4 Celestial Mechanics
13.5 Mechanical Curves
13.6 The Vibrating String.
13.7 Hydrodynamics.
13.8 Biographical Notes: The Bernoullis
14. Complex Numbers in Algebra
14.1 Impossible Numbers
14.2 Quadratic Equations
14.3 Cubic Equations
14.4 Wallis’s Attempt at Geometric Representation.
14.5 Angle Division
14.6 The Fundamental Theorem of Algebra
14.7 The Proofs of d’Alembert and Gauss
14.8 Biographical Notes: d’Alembert
15. Complex Numbers and Curves
15.1 Roots and Intersections
15.2 The Complex Projective Line
15.3 Branch Points
15.4 Topology of Complex Projective Curves
15.5 Biographical Notes: Riemann
16. Complex Numbers and Functions
16.1 Complex Functions
16.2 Conformal Mapping
16.3 Cauchy’s Theorem
16.4 Double Periodicity of Elliptic Functions
16.5 Elliptic Curves
16.6 Uniformization
16.7 Biographical Notes: Lagrange and Cauchy
17. Differential Geometry
17.1 Transcendental Curves
17.2 Curvature of Plane Curves
17.3 Curvature of Surfaces
17.4 Surfaces of Constant Curvature
17.5 Geodesics
17.6 The Gauss–Bonnet Theorem
17.7 Biographical Notes: Harriot and Gauss
18. Non-Euclidean Geometry
18.1 The Parallel Axiom
18.2 Spherical Geometry
18.3 Geometry of Bolyai and Lobachevsky
18.4 Beltrami’s Projective Model
18.5 Beltrami’s Conformal Models
18.6 The Complex Interpretations
18.7 Biographical Notes: Bolyai and Lobachevsky
19. Group Theory
19.1 The Group Concept
19.2 Subgroups and Quotients
19.3 Permutations and Theory of Equations
19.4 Permutation Groups
19.5 Polyhedral Groups
19.6 Groups and Geometries
19.7 Combinatorial Group Theory
19.8 Finite Simple Groups
19.9 Biographical Notes: Galois
20. Hypercomplex Numbers
20.1 Complex Numbers in Hindsight
20.2 The Arithmetic of Pairs
20.3 Properties of + and ×
20.4 Arithmetic of Triples and Quadruples
20.5 Quaternions, Geometry, and Physics
20.6 Octonions
20.7 Why C, H, and O Are Special
20.8 Biographical Notes: Hamilton
21. Algebraic Number Theory
21.1 Algebraic Numbers
21.2 Gaussian Integers
21.3 Algebraic Integers
21.4 Ideals
21.5 Ideal Factorization
21.6 Sums of Squares Revisited
21.7 Rings and Fields
21.8 Biographical Notes: Dedekind, Hilbert, and Noether
22. Topology
22.1 Geometry and Topology
22.2 Polyhedron Formulas of Descartes and Euler
22.3 The Classification of Surfaces
22.4 Descartes and Gauss–Bonnet
22.5 Euler Characteristic and Curvature
22.6 Surfaces and Planes
22.7 The Fundamental Group
22.8 The Poincar´e Conjecture
22.9 Biographical Notes: Poincar´e
23. Simple Groups
23.1 Finite Simple Groups and Finite Fields
23.2 The Mathieu Groups
23.3 Continuous Groups
23.4 Simplicity of SO(3)
23.5 Simple Lie Groups and Lie Algebras
23.6 Finite Simple Groups Revisited
23.7 The Monster
23.8 Biographical Notes: Lie, Killing, and Cartan
24. Sets, Logic, and Computation
24.1 Sets
24.2 Ordinals
24.3 Measure
24.4 Axiom of Choice and Large Cardinals
24.5 The Diagonal Argument
24.6 Computability
24.7 Logic and G¨odel’s Theorem
24.8 Provability and Truth
24.9 Biographical Notes: G¨odel
25. Combinatorics
25.1 What Is Combinatorics?
25.2 The Pigeonhole Principle
25.3 Analysis and Combinatorics
25.4 Graph Theory
25.5 Nonplanar Graphs
25.6 The K?onig Infinity Lemma
25.7 Ramsey Theory
25.8 Hard Theorems of Combinatorics
25.9 Biographical Notes: Erd?os
Bibliography
Index
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