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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 | |