目录
目 录 PREFACE CHAPTER ONE Egypt and Mesopotamia 1.1 Egypt 1.1.1 Introduction 1.1.2 Number Systems and Computations 1.1.3 Linear Equations and Proportional Reasoning 1.1.4 Geometry 1.2 Mesopotamia 1.2.1 Introduction 1.2.2 Methods of Computation 1.2.3 Geometry 1.2.4 Square Roots and the Pythagorean Theorem 1.2.5 Solving Equations 1.3 Conclusion Exercises References CHAPTER TWO Greek Mathematics to the Time of Euclid 2.1 The Earliest Greek Mathematics 2.1.1 Thales, Pythagoras, and the Pythagoreans 2.1.2 Geometric Problem Solving and the Need for Proof 2.2 Euclid and His Elements 2.2.1 The Pythagorean Theorem and Its Proof 2.2.2 Geometric Algebra 2.2.3 The Pentagon Construction 2.2.4 Ratio, Proportion, and Incommensurability 2.2.5 Number Theory 2.2.6 Incommensurability, Solid Geometry, and the Method of Exhaustion Exercises References CHAPTER THREE Greek Mathematics from Archimedes to Ptolemy 3.1 Archimedes 3.1.1 The Determination ofrr 3.1.2 Archimedes'''' Method of Discovery 3.1.3 Sums of Series 3.1.4 Analysis 3.2 Apollonius and the Conic Sections 3.2.1 Conic Sections before Apollonius 3.2.2 Definitions and Basic Properties of the Conics 3.2.3 Asymptotes, Tangents, and Foci 3.2.4 Problem Solving Using Conics 3.3 Ptolemy and Greek Astronomy 3.3.1 Astronomy before Ptolemy 3.3.2 Apollonius and Hipparchus 3.3.3 Ptolemy and His Chord Table 3.3.4 Solving Plane Triangles 3.3.5 Solving Spherical Triangles Exercises References CHAPTER FOUR Greek Mathematics from Diophantus to Hypatia 4.1 Diophantus and the Arithrnetica 4.1.1 Linear and Quadratic Equations 4.1.2 Higher-Degree Equations 4.1.3 The Method of False Position 4.2 Pappus and Analysis 4.3 Hypatia Exercises References CHAPTER FIVE Ancient and Medieval China 5.1 Calculating with Numbers 5.2 Geometry 5.2.1 The Pythagorean Theorem and Surveying 5.2.2 Areas and Volumes 5.3 Solving Equations 5.3.1 Systems of Linear Equations 5.3.2 Polynomial Equations 5.4 The Chinese Remainder Theorem 5.5 Transmission to and from China Exercises References CHAPTER SIX Ancient and Medieval India 6.1 Indian Number Systems and Calculations 6.2 Geometry 6.3 Algebra 6.4 Combinatorics 6.5 Trigonometry 6.6 Transmission to and from India Exercises References CHAPTER SEVEN Mathematics in the Islamic World 7.1 Arithmetic 7.2 Algebra 7.2.1 The Algebra of al-Khwarizmi 7.2.2 The Algebra of Aba Kamil 7.2.3 The Algebra of Polynomials 7.2.4 Induction, Sums of Powers, and the Pascal Triangle 7.2.5 The Solution of Cubic Equations 7.3 Combinatorics 7.3.1 Counting Combinations 7.3.2 Deriving the Combinatorial Formulas 7.4 Geometry 7.4.1 The Parallel Postulate 7.4.2 Volumes and the Method of Exhaustion 7.5 Trigonometry 7.5.1 The Trigonometric Functions 7.5.2 Spherical Trigonometry 7.5.3 Values of Trigonometric Functions 7.6 Transmission of Islamic Mathematics Exercises References CHAPTER EIGHT Mathematics in Medieval Europe 8.1 Geometry 8.1.1 Abraham bar .Hiyya''''s Treatise on Mensuration 8.1.2 Leonardo of Pisa''''s Practica geometriae 8.2 Combinatorics 8.2.1 The Work of Abraham ibn Ezra 8.2.2 Leviben Gerson and Induction 8.3 Medieval Algebra 8.3.1 Leonardo of Pisa''''s Liber abbaci 8.3.2 The Work of Jordanus de Nemore 8.4 The Mathematics of Kinematics Exercises References CHAPTER NINE Mathematics in the Renaissance 9.1 Algebra 9.1.1 The Abacists 9.1.2 Algebra in Northern Europe 9.1.3 The Solution of the Cubic Equation 9.1.4 Bombelli and Complex Numbers 9.1.5 Viete, Algebraic Symbolism, and Analysis 9.2 Geometry and Trigonometry 9.2.1 Art and Perspective 9.2.2 The Conic Sections 9.2.3 Regiomontanus and Trigonometry 9.3 Numerical Calculations 9.3.1 Simon Stevin and Decimal Fractions 9.3.2 Logarithms 9.4 Astronomy and Physigs 9.4.1 Copernicus and the Heliocentric Universe 9.4.2 Johannes Kepler and Elliptical Orbits 9.4.3 Galileo and Kinematics Exercises References CHAPTER TEN Pre. calculus in the Seventeenth Century 10.1 Algebraic Symbolism and the Theory of Equations 10.1.1 William Oughtred and Thomas Harriot 10.1.2 Albert Girard and the Fundamental Theorem of Algebra 10.2 Analytic Geometry 10.2.1 Fermat and the Introduction to Plane and Solid Loci 10.2.2 Descartes and the Geometry 10.2.3 The Work of Jan de Witt 10.3 Elementary Probability 10.3.1 Blaise Pascal and the Beginnings of the Theory of Probability 10.3.2 Christian Huygens and the Earliest Probability Text 10.4 Number Theory Exercises References CHAPTER ELEVEN Calculus in the Seventeenth Century 11.1 Tangents and Extrema 11.1.1 Fermat''''s Method of Finding Extrema 11.1.2 Descartes and the Method of Normals 11.1.3 Hudde''''s Algorithm 11.2 Areas and Volumes 11.2.1 Infinitesimals and Indivisibles 11.2.2 Torricelli and the Infinitely Long Solid 11.2.3 Fermat and the Area under Parabolas and Hyperbolas 11.2.4 Wallis and Fractional Exponents 11.2.5 The Area under the Sine Curve and the Rectangular Hyperbola 11.3 Rectification of Curves and the Fundamental Theorem 11.3.1 Van Heuraet and the Rectification of Curves 11.3.2 Gregory and the Fundamental Theorem 11.3.3 Barrow and the Fundamental Theorem 11.4 Isaac Newton 11.4.1 Power Series 11.4.2 Algorithms for Calculating Fluxions and Fluents 11.4.3 The Synthetic Method of Fluxions and Newton''''s Physics 11.5 Gottfried Wilhelm Leibniz 11.5.1 Sums and Differences 11.5.2 The Differential Triangle and the Transmutation Theorem 11.5.3 The Calculus of Differentials 11.5.4 The Fundamental Theorem and Differential Equations Exercises References CHAPTER TWELVE Analysis in the Eighteenth Century 12.1 Differential Equations 12.1.1 The Brachistochrone Problem 12.1.2 Translating Newton''''s Synthetic Method of Fluxions into the Method of Differentials 12.1.3 Differential Equations and the Trigonometric Functions 12.2 The Calculus of Several Variables 12.2.1 The Differential Calculus of Functions of Two Variables 12.2.2 Multiple Integration 12.2.3 Partial Differential Equations: The Wave Equation 12.3 The Textbook Organization of the Calculus 12.3.1 Textbooks in Fluxions 12.3.2 Textbooks in the Differential Calculus 12.3.3 Euler'''' s Textbooks 12.4 The Foundations of the Calculus 12.4.1 George Berkeley''''s Criticisms and Maclaurin''''s Response 12.4.2 Euler and d''''Alembert 12.4.3 Lagrange and Power Series Exercises References CHAPTER THIRTEEN Probability and Statistics in the Eighteenth Century 13.1 Probability 13.1.1 Jakob Bernoulli and the Ars Conjectandi 13.1.2 De Moivre and The Doctrine of Chances 13.2 Applications of Probability to Statistics 13.2.1 Errors in Observations 13.2.2 De Moivre and Annuities 13.2.3 Bayes and Statistical Inference 13.2.4 The Calculations of Laplace Exercises References CHAPTER FOURTEEN Algebra and Number Theory in the Eighteenth Century 14.1 Systems of Linear Equations 14.2 Polynomial Equations 14.3 Number Theory 14.3.1 Fermat''''s Last Theorem 14.3.2 Residues Exercises References CHAPTER FIFTEEN Geometry in the Eighteenth Century 15.1 The Parallel Postulate 15.1.1 Saccheri and the Parallel Postulate 15.1.2 Lambert and the Parallel Postulate 15.2 Differential Geometry of Curves and Surfaces 15.2.1 Euler and Space Curves and Surfaces 15.2.2 The Work of Monge 15.3 Euler and the Beginnings of Topology Exercises References CHAPTER SIXTEEN Algebra and Number Theory in the Nineteenth Century 16.1 Number Theory 16.1.1 Gauss and Congruences 16.1.2 Fermat''''s Last Theorem and Unique Factorization 16.2 Solving Algebraic Equations 16.2.1 Cyclotomic Equations 16.2.2 The Theory of Permutations 16.2.3 The Unsolvability of the Quintic 16.2.4 The Work of Galois 16.2.5 Jordan and the Theory of Groups of Substitutions 16.3 Groups and Fields -- The Beginning of Structure 16.3.1 Gauss and Quadratic Forms 16.3.2 Kronecker and the Structure of Abelian Groups 16.3.3 Groups of Transformations 16.3.4 Axiomatizafion of the Group Concept 16.3.5 The Concept of a Field 16.4 Matrices and Systems of Linear Equations 16.4.1 Basic Ideas of Matrices 16.4.2 Eigenvalues and Eigenvectors 16.4.3 Solutions of Systems of Equations 16.4.4 Systems of Linear Inequalities Exercises References CHAPTER SEVENTEEN Analysis in the Nineteenth Century 17.1 Rigor in Analysis 17.1.1 Limits 17.1.2 Continuity 17.1.3 Convergence 17.1.4 Derivatives 17.1.5 Integrals 17.1.6 Fourier Series and the Notion of a Function 17.1.7 The Riemann Integral 17.1.8 Uniform Convergence 17.2 The Arithmetization of Analysis 17.2.1 Dedekind Cuts 17.2.2 Cantor and Fundamental Sequences 17.2.3 The Theory of Sets 17.2.4 Dedekind and Axioms for the Natural Numbers 17.3 Complex Analysis 17.3.1 Geometrical Representation of Complex Numbers 17.3.2 Complex Functions 17.3.3 The Riemann Zeta Function 17.4 Vector Analysis 17.4.1 Surface Integrals and the Divergence Theorem 17.4.2 Stokes''''s Theorem Exercises References CHAPTER EIGHTEEN Statistics in the Nineteenth Century 18.1 The Method of Least Squares 18.1.1 The Work of Legendre 18.1.2 Gauss and the Derivation of the Method of Least Squares 18.2 Statistics and the Social Sciences 18.3 Statistical Graphs Exercises References CHAPTER NINETEEN Geometry in the Nineteenth Century 19.1 Non-Euclidean Geometry 19.1.1 Taurinus and Log-Spherical Geometry 19.1.2 The Non-Euclidean Geometry of Lobachevsky and Bolyai 19.1.3 Models of Non-Euclidean Geometry 19.2 Geometry in n Dimensions 19.2.1 Grassmann and the Ausdehnungslehre 19.2.2 Vector Spaces 19.3 Graph Theory and the Four-Color Problem Exercises References CHAPTER TWENTY Aspects of the Twentieth Century 20.1 The Growth of Abstraction 20.1.1 The Axiomatization of Vector Spaces 20.1.2 The Theory of Rings 20.1.3 The Axiomatization of Set Theory 20.2 Major Questions Answered 20.2.1 The Proof of Fermat''''s Last Theorem 20.2.2 The Classification of the Finite Simple Groups 20.2.3 The Proof of the Four-Color Theorem 20.3 Growth of New Fields of Mathematics 20.3.1 The Statistical Revolution 20.3.2 Linear Programming 20.4 Computers and Mathematics 20.4.1 The Prehistory of Computers 20.4.2 Turing and Computability 20.4.3 Von Neumann''''s Computer Exercises References APPENDIX Using This Textbook in Teaching Mathematics Courses and Topics Sample Lesson Ideas for Incorporating History Time Line ANSWERS TO SELECTED PROBLEMS GENERAL REFERENCES IN THE HISTORY OF MATHEMATICS INDEX
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