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Preface to the Third Edition | |
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Preface to the Second Edition | |
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Preface to the First Edition | |
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The Theorem of Pythagoras | |
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Arithmetic and Geometry | |
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Pythagorean Triples | |
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Rational Points on the Circle | |
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Right-Angled Triangles | |
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Irrational Numbers | |
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The Definition of Distance | |
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Biographical Notes: Pythagoras | |
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Greek Geometry | |
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The Deductive Method | |
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The Regular Polyhedra | |
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Ruler and Compass Constructions | |
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Conic Sections | |
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Higher-Degree Curves | |
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Biographical Notes: Euclid | |
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Greek Number Theory | |
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The Role of Number Theory | |
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Polygonal, Prime, and Perfect Numbers | |
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The Euclidean Algorithm | |
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Pell's Equation | |
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The Chord and Tangent Methods | |
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Biographical Notes: Diophantus | |
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Infinity in Greek Mathematics | |
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Fear of Infinity | |
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Eudoxus's Theory of Proportions | |
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The Method of Exhaustion | |
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The Area of a Parabolic Segment | |
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Biographical Notes: Archimedes | |
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Number Theory in Asia | |
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The Euclidean Algorithm | |
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The Chinese Remainder Theorem | |
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Linear Diophantine Equations | |
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Pell's Equation in Brahmagupta | |
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Pell's Equation in Bh�skara II | |
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Rational Triangles | |
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Biographical Notes: Brahmagupta and Bh�skara | |
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Polynomial Equations | |
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Algebra | |
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Linear Equations and Elimination | |
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Quadratic Equations | |
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Quadratic Irrationals | |
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The Solution of the Cubic | |
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Angle Division | |
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Higher-Degree Equations | |
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Biographical Notes: Tartaglia, Cardano, and Vi�te | |
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Analytic Geometry | |
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Steps Toward Analytic Geometry | |
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Fermat and Descartes | |
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Algebraic Curves | |
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Newton's Classification of Cubics | |
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Construction of Equations, B�zout's Theorem | |
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The Arithmetization of Geometry | |
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Biographical Notes: Descartes | |
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Projective Geometry | |
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Perspective | |
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Anamorphosis | |
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Desargues's Projective Geometry | |
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The Projective View of Curves | |
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The Projective Plane | |
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The Projective Line | |
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Homogeneous Coordinates | |
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Pascal's Theorem | |
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Biographical Notes: Desargues and Pascal | |
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Calculus | |
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What Is Calculus? | |
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Early Results on Areas and Volumes | |
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Maxima, Minima, and Tangents | |
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The Arithmetica Infinitorum of Wallis | |
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Newton's Calculus of Series | |
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The Calculus of Leibniz | |
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Biographical Notes: Wallis, Newton, and Leibniz | |
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Infinite Series | |
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Early Results | |
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Power Series | |
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An Interpolation on Interpolation | |
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Summation of Series | |
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Fractional Power Series | |
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Generating Functions | |
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The Zeta Function | |
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Biographical Notes: Gregory and Euler | |
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The Number Theory Revival | |
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Between Diophantus and Fermat | |
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Fermat's Little Theorem | |
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Fermat's Last Theorem | |
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Rational Right-Angled Triangles | |
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Rational Points on Cubics of Genus 0 | |
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Rational Points on Cubics of Genus 1 | |
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Biographical Notes: Fermat | |
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Elliptic Functions | |
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Elliptic and Circular Functions | |
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Parameterization of Cubic Curves | |
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Elliptic Integrals | |
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Doubling the Arc of the Lemniscate | |
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General Addition Theorems | |
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Elliptic Functions | |
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A Postscript on the Lemniscate | |
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Biographical Notes: Abel and Jacobi | |
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Mechanics | |
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Mechanics Before Calculus | |
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The Fundamental Theorem of Motion | |
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Kepler's Laws and the Inverse Square Law | |
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Celestial Mechanics | |
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Mechanical Curves | |
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The Vibrating String | |
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Hydrodynamics | |
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Biographical Notes: The Bernoullis | |
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Complex Numbers in Algebra | |
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Impossible Numbers | |
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Quadratic Equations | |
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Cubic Equations | |
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Wallis's Attempt at Geometric Representation | |
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Angle Division | |
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The Fundamental Theorem of Algebra | |
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The Proofs of d' Alembert and Gauss | |
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Biographical Notes: d' Alembert | |
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Complex Numbers and Curves | |
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Roots and Intersections | |
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The Complex Projective Line | |
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Branch Points | |
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Topology of Complex Projective Curves | |
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Biographical Notes: Riemann | |
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Complex Numbers and Functions | |
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Complex Functions | |
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Conformal Mapping | |
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Cauchy's Theorem | |
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Double Periodicity of Elliptic Functions | |
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Elliptic Curves | |
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Uniformization | |
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Biographical Notes: Lagrange and Cauchy | |
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Differential Geometry | |
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Transcendental Curves | |
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Curvature of Plane Curves | |
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Curvature of Surfaces | |
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Surfaces of Constant Curvature | |
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Geodesies | |
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The Gauss-Bonnet Theorem | |
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Biographical Notes: Harriot and Gauss | |
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Non-Euclidean Geometry | |
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The Parallel Axiom | |
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Spherical Geometry | |
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Geometry of Bolyai and Lobachevsky | |
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Beltrami's Projective Model | |
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Beltrami's Conformal Models | |
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The Complex Interpretations | |
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Biographical Notes: Bolyai and Lobachevsky | |
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Group Theory | |
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The Group Concept | |
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Subgroups and Quotients | |
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Permutations and Theory of Equations | |
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Permutation Groups | |
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Polyhedral Groups | |
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Groups and Geometries | |
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Combinatorial Group Theory | |
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Finite Simple Groups | |
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Biographical Notes: Galois | |
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Hypercomplex Numbers | |
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Complex Numbers in Hindsight | |
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The Arithmetic of Pairs | |
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Properties of + and x | |
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Arithmetic of Triples and Quadruples | |
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Quaternions, Geometry, and Physics | |
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Octonions | |
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Why C, H, and O Are Special | |
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Biographical Notes: Hamilton | |
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Algebraic Number Theory | |
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Algebraic Numbers | |
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Gaussian Integers | |
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Algebraic Integers | |
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Ideals | |
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Ideal Factorization | |
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Sums of Squares Revisited | |
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Rings and Fields | |
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Biographical Notes: Dedekind, Hilbert, and Noether | |
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Topology | |
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Geometry and Topology | |
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Polyhedron Formulas of Descartes and Euler | |
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The Classification of Surfaces | |
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Descartes and Gauss-Bonnet | |
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Euler Characteristic and Curvature | |
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Surfaces and Planes | |
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The Fundamental Group | |
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The Poincar� Conjecture | |
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Biographical Notes: Poincar� | |
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Simple Groups | |
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Finite Simple Groups and Finite Fields | |
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The Mathieu Groups | |
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Continuous Groups | |
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Simplicity of SO(3) | |
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Simple Lie Groups and Lie Algebras | |
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Finite Simple Groups Revisited | |
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The Monster | |
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Biographical Notes: Lie, Killing, and Cartan | |
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Sets, Logic, and Computation | |
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Sets | |
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Ordinals | |
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Measure | |
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Axiom of Choice and Large Cardinals | |
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The Diagonal Argument | |
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Computability | |
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Logic and G�del's Theorem | |
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Provability and Truth | |
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Biographical Notes: G�del | |
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Combinatorics | |
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What Is Combinatorics? | |
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The Pigeonhole Principle | |
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Analysis and Combinatorics | |
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Graph Theory | |
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Nonplanar Graphs | |
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The Konig Infinity Lemma | |
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Ramsey Theory | |
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Hard Theorems of Combinatorics | |
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Biographical Notes: Erdos | |
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Bibliography | |
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Index | |