THE HISTORY OF MATHEMATICS BRIEF VERSION VICTOR J* KATZ. University of the District of Columbia. SUB Gottingen A 3130

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1 THE HISTORY OF MATHEMATICS BRIEF VERSION VICTOR J* KATZ University of the District of Columbia SUB Gottingen A 3130 PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal

2 PREFACE CHAPTER ONE Egypt and Mesopotamia Egypt Introduction Number Systems and Computations Linear Equations and Proportional Reasoning Geometry Mesopotamia Introduction Methods of Computation Geometry Square Roots and the Pythagorean Theorem Solving Equations Conclusion 25 Exercises 25 References 27 CHAPTER TWO Greek Mathematics to the Time of Euclid The Earliest Greek Mathematics Thales, Pythagoras, and the Pythagoreans Geometric Problem Solving and the Need for Proof Euclid and His Elements The Pythagorean Theorem and Its Proof Geometric Algebra The Pentagon Construction Ratio, Proportion, and Incommensurability Number Theory Incommensurability, Solid Geometry, and the Method of Exhaustion 59 xtii

3 CHAPTER THREE CHAPTER FOUR CHAPTER FIVE Exercises 62 References 65 Greek Mathematics from Archimedes to Ptolemy Archimedes The Determination of n Archimedes' Method of Discovery Sums of Series Analysis Apollonius and the Conic Sections Conic Sections before Apollonius Definitions and Basic Properties of the Conies Asymptotes, Tangents, and Foci Problem Solving Using Conies Ptolemy and Greek Astronomy Astronomy before Ptolemy ;> Apollonius and Hipparchus Ptolemy and His Chord Table Solving Plane Triangles Solving Spherical Triangles 96 Exercises 99 References 103 Greek Mathematics from Diophantus to Hypatia Diophantus and the Arithmetica Linear and Quadratic Equations Higher-Degree Equations The Method of False Position \ Pappus and Analysis Hypatia 113 Exercises 114 References 115 Ancient and Medieval China Calculating with Numbers Geometry The Pythagorean Theorem and Surveying Areas and Volumes Solving Equations Systems of Linear Equations Polynomial Equations 128

4 CHAPTER SIX CHAPTER SEVEN CHAPTER EIGHT 5.4 The Chinese Remainder Theorem Transmission to and from China 134 Exercises 134 References Ancient and Medieval India Indian Number Systems and Calculations Geometry Algebra Combinatorics Trigonometry Transmission to and from India 156 Exercises 157 References 159 Mathematics in the Islamic World Arithmetic!\ Algebra The Algebra of al-khwarizmi The Algebra of Abu Kamil The Algebra of Polynomials Induction, Sums of Powers, and the Pascal Triangle The Solution of Cubic Equations Combinatorics Counting Combinations Deriving the Combinatorial Formulas Geometry The Parallel Postulate Volumes and the Method of Exhaustion Trigonometry The Trigonometric Functions Spherical Trigonometry Values of Trigonometric Functions Transmission of Islamic Mathematics 187 Exercises 188 References 191 Mathematics in Medieval Europe Geometry Abraham bar Hiyya's Treatise on Mensuration Leonardo of Pisa's Practica geometriae 196

5 CHAPTER NINE CHAPTER TEN 8.2 Combinatorics The Work of Abraham ibn Ezra Levi ben Gerson and Induction Medieval Algebra Leonardo of Pisa's Liber abbaci The Work of Jordanus de Nemore The Mathematics of Kinematics 206 Exercises 210 References 211 Mathematics in the Renaissance Algebra The Abacists Algebra in Northern Europe The Solution of the Cubic Equation Bombelli and Complex Numbers Viete, Algebraic Symbolism, and Analysis Geometry and Trigonometry Art and Perspective The Conic Sections Regiomontanus and Trigonometry Numerical Calculations Simon Stevin and Decimal Fractions Logarithms Astronomy and Physics Copernicus and the Heliocentric Universe Johannes Kepler and Elliptical Orbits Galileo and Kinematics 247 Exercises 252 References 255 Precalculus in the Seventeenth Century Algebraic Symbolism and the Theory of Equations William Oughtred and Thomas Harriot Albert Girard and the Fundamental Theorem of Algebra Analytic Geometry Fermat and the Introduction to Plane and Solid Loci Descartes and the Geometry The Work of Jan de Witt Elementary Probability Blaise Pascal and the Beginnings of the Theory of Probability Christian Huygens and the Earliest Probability Text 274

6 viii Contents 10.4 Number Theory 276 Exercises 278 References 280 CHAPTER ELEVEN Calculus in the Seventeenth Century Tangents and Extrema Fermat's Method of Finding Extrema Descartes and the Method of Normals Hudde's Algorithm Areas and Volumes Infinitesimals and Indivisibles Torricelli and the Infinitely Long Solid Fermat and the Area under Parabolas and Hyperbolas Wallis and Fractional Exponents The Area under the Sine Curve and the Rectangular Hyperbola Rectification of Curves and the Fundamental Theorem Van Heuraet and the Rectification of Curves Gregory and the Fundamental Theorem Barrow and the Fundamental Theorem Isaac Newton Power Series Algorithms for Calculating Fluxions and Fluents The Synthetic Method of Fluxions and Newton's Physics Gottfried Wilhelm Leibniz Sums and Differences The Differential Triangle and the Transmutation Theorem The Calculus of Differentials The Fundamental Theorem and Differential Equations. 321 Exercises 324 References 327 CHAPTER TWELVE Analysis in the Eighteenth Century Differential Equations The Brachistochrone Problem Translating Newton's Synthetic Method of Fluxions into the Method of Differentials Differential Equations and the Trigonometric Functions The Calculus of Several Variables The Differential Calculus of Functions of Two Variables Multiple Integration Partial Differential Equations: The Wave Equation 343

7 12.3 The Textbook Organization of the Calculus Textbooks in Fluxions Textbooks in the Differential Calculus Euler's Textbooks The Foundations of the Calculus George Berkeley's Criticisms and Maclaurin's Response Euler and d'alembert Lagrange and Power Series 357 Exercises 360 References 362 CHAPTER THIRTEEN Probability and Statistics in the Eighteenth Century Probability Jakob Bernoulli and the Ars Conjectandi De Moivre and The Doctrine of Chances Applications of Probability to Statistics Errors in Observations De Moivre and Annuities Bayes and Statistical Inference The Calculations of Laplace 374 Exercises 375 References 377 CHAPTER FOURTEEN Algebra and Number Theory in the Eighteenth Century Systems of Linear Equations Polynomial Equations Number Theory Fermat's Last Theorem Residues 385 Exercises 386 References 387 CHAPTER FIFTEEN Geometry in the Eighteenth Century The Parallel Postulate Saccheri and the Parallel Postulate Lambert and the Parallel Postulate Differential Geometry of Curves and Surfaces Euler and Space Curves and Surfaces The Work of Monge 395

8 x Contents 15.3 Euler and the Beginnings of Topology 396 Exercises 398 References 399 CHAPTER SIXTEEN Algebra and Number Theory in the Nineteenth Century Number Theory Gauss and Congruences Fermat's Last Theorem and Unique Factorization Solving Algebraic Equations Cyclotomic Equations The Theory of Permutations The Unsolvability of the Quintic The Work of Galois Jordan and the Theory of Groups of Substitutions Groups and Fields The Beginning of Structure Gauss and Quadratic Forms Kronecker and the Structure of Abelian Groups Groups of Transformations Axiomatization of the Group Concept The Concept of a Field Matrices and Systems of Linear Equations Basic Ideas of Matrices Eigenvalues and Eigenvectors Solutions of Systems of Equations Systems of Linear Inequalities 426 Exercises ; 427 References ; 429 CHAPTER SEVENTEEN Analysis in the Nineteenth Century Rigor in Analysis Limits Continuity Convergence Derivatives Integrals Fourier Series and the Notion of a Function The Riemann Integral Uniform Convergence The Arithmetization of Analysis Dedekind Cuts 446

9 Cantor and Fundamental Sequences The Theory of Sets Dedekind and Axioms for the Natural Numbers Complex Analysis Geometrical Representation of Complex Numbers Complex Functions The Riemann Zeta Function Vector Analysis Surface Integrals and the Divergence Theorem Stokes's Theorem 457 Exercises 459 References 461 CHAPTER EIGHTEEN Statistics in the Nineteenth Century The Method of Least Squares., The Work of Legendre Gauss and the Derivation of the Method of Least Squares Statistics and the Social Sciences Statistical Graphs 469 Exercises 474 References 474 CHAPTER NINETEEN Geometry in the Nineteenth Century Non-Euclidean Geometry Taurinus and Log-Spherical Geometry The Non-Euclidean Geometry of Lobachevsky and Bolyai Models of Non-Euclidean Geometry Geometry in n Dimensions Grassmann and the Ausdehnungslehre Vector Spaces Graph Theory and the Four-Color Problem 488 Exercises 492 References 493 CHAPTER TWENTY Aspects of the Twentieth Century The Growth of Abstraction The Axiomatization of Vector Spaces The Theory of Rings The Axiomatization of Set Theory 497

10 xii Contents 20.2 Major Questions Answered The Proof of Fermat's Last Theorem The Classification of the Finite Simple Groups The Proof of the Four-Color Theorem Growth of New Fields of Mathematics The Statistical Revolution Linear Programming Computers and Mathematics The Prehistory of Computers Turing and Computability Von Neumann's Computer 516 Exercises 518 References 519 APPENDIX Using This Textbook in Teaching Mathematics 521 Courses and Topics f Sample Lesson Ideas for Incorporating History 525 Time Line 528 ANSWERS TO SELECTED PROBLEMS 536 GENERAL REFERENCES IN THE HISTORY OF MATHEMATICS 542 INDEX 544