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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

  • Open
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