The History of Mathematics : A Brief Course

Praise for the Second Edition "An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource . . . essential." ―CHOICE This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed. Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapter-end exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include: Material arranged in a chronological and cultural context Specific parts of the history of mathematics presented as individual lessons New and revised exercises ranging between technical, factual, and integrative Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book) An emphasis on geography, culture, and mathematics In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics. Cover Title Page Copyright Contents Preface Changes from the Second Edition Elementary Texts on the History of Mathematics Part I: What Is Mathematics? Contents of Part I Chapter 1: Mathematics and Its History 1.1. Two Ways to Look at the History of Mathematics 1.1.1. History, but Not Heritage 1.1.2. Our Mathematical Heritage 1.2. The Origin of Mathematics 1.2.1. Number 1.2.2. Space 1.2.3. Are Mathematical Ideas Innate? 1.2.4. Symbolic Notation 1.2.5. Logical Relations 1.2.6. The Components of Mathematics 1.3. The Philosophy of Mathematics 1.3.1. Mathematical Analysis of a Real-world Problem 1.4. Our Approach to the History of Mathematics Chapter 2: Proto-mathematics 2.1. Number 2.1.1. Animals’ Use of Numbers 2.1.2. Young Children’s Use of Numbers 2.1.3. Archaeological Evidence of Counting 2.2. Shape 2.2.1. Perception of Shape by Animals 2.2.2. Children’s Concepts of Space 2.2.3. Geometry in Arts and Crafts 2.3. Symbols 2.4. Mathematical Reasoning 2.4.1. Animal Reasoning 2.4.2. Visual Reasoning Problems and Questions Mathematical Problems Questions for Reflection Part II: The Middle East, 2000–1500 Bce Contents of Part II Chapter 3: Overview of Mesopotamian Mathematics 3.1. A Sketch of Two Millennia of Mesopotamian History 3.2. Mathematical Cuneiform Tablets 3.3. Systems of Measuring and Counting 3.3.1. Counting 3.4. The Mesopotamian Numbering System 3.4.1. Place-value Systems 3.4.2. The Sexagesimal Place-value System 3.4.3. Converting a Decimal Number to Sexagesimal 3.4.4. Irrational Square Roots Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 4: Computations in Ancient Mesopotamia 4.1. Arithmetic 4.1.1. Square Roots 4.2. Algebra 4.2.1. Linear and Quadratic Problems 4.2.2. Higher-degree Problems Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 5: Geometry in Mesopotamia 5.1. The Pythagorean Theorem 5.2. Plane Figures 5.2.1. Mesopotamian Astronomy 5.3. Volumes 5.4. Plimpton 322 5.4.1. The Purpose of Plimpton 322: Some Conjectures Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 6: Egyptian Numerals and Arithmetic 6.1. Sources 6.1.1. Mathematics in Hieroglyphics and Hieratic 6.2. The Rhind Papyrus 6.3. Egyptian Arithmetic 6.4. Computation 6.4.1. Multiplication and Division 6.4.2. “parts” Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 7: Algebra and Geometry in Ancient Egypt 7.1. Algebra Problems in the Rhind Papyrus 7.1.1. Applied Problems: the 7.2. Geometry 7.3. Areas 7.3.1. Rectangles, Triangles, and Trapezoids 7.3.2. Slopes 7.3.3. Circles 7.3.4. The Pythagorean Theorem 7.3.5. Spheres or Cylinders? 7.3.6. Volumes Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Part III: Greek Mathematics from 500 Bce to 500 Ce Contents of Part III Chapter 8: an Overview of Ancient Greek Mathematics 8.1. Sources 8.1.1. Loss and Recovery 8.2. General Features of Greek Mathematics 8.2.1. Pythagoras 8.2.2. Mathematical Aspects of Plato’s Philosophy 8.3. Works and Authors 8.3.1. Euclid 8.3.2. Archimedes 8.3.3. Apollonius 8.3.4. Zenodorus 8.3.5. Heron 8.3.6. Ptolemy 8.3.7. Diophantus 8.3.8. Pappus 8.3.9. Theon and Hypatia Questions Historical Questions Questions for Reflection Chapter 9: Greek Number Theory 9.1. The Euclidean Algorithm 9.2. The Arithmetica of Nicomachus 9.2.1. Factors Vs. Parts. Perfect Numbers 9.2.2. Figurate Numbers 9.3. Euclid’s Number Theory 9.4. The Arithmetica of Diophantus 9.4.1. Algebraic Symbolism 9.4.2. Contents of the Arithmetica 9.4.3. Fermat’s Last Theorem Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 10: Fifth-century Greek Geometry 10.1. “pythagorean” Geometry 10.1.1. Transformation and Application of Areas 10.2. Challenge No. 1: Unsolved Problems 10.3. Challenge No. 2: the Paradoxes of Zeno of Elea 10.4. Challenge No. 3: Irrational Numbers and Incommensurable Lines 10.4.1. The Arithmetical Origin of Irrationals 10.4.2. The Geometric Origin of Irrationals 10.4.3. Consequences of the Discovery Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 11: Athenian Mathematics I: the Classical Problems 11.1. Squaring the Circle 11.2. Doubling the Cube 11.3. Trisecting the Angle 11.3.1. A Mechanical Solution: the Conchoid Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 12: Athenian Mathematics Ii: Plato and Aristotle 12.1. The Influence of Plato 12.2. Eudoxan Geometry 12.2.1. The Eudoxan Definition of Proportion 12.2.2. The Method of Exhaustion 12.2.3. Ratios in Greek Geometry 12.3. Aristotle Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 13: Euclid of Alexandria 13.1. The Elements 13.1.1. Book 1 13.1.2. Book 2 13.1.3. Books 3 and 4 13.1.4. Books 5 and 6 13.1.5. Books 7–9 13.1.6. Book 10 13.1.7. Books 11–13 13.2. The Data Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 14: Archimedes of Syracuse 14.1. The Works of Archimedes 14.2. The Surface of a Sphere 14.3. The Archimedes Palimpsest 14.3.1. The Method 14.4. Quadrature of the Parabola 14.4.1. The Mechanical Quadrature 14.4.2. The Rigorous Quadrature Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 15: Apollonius of Perga 15.1. History of the Conics 15.2. Contents of the Conics 15.2.1. Properties of the Conic Sections 15.3. Foci and the Three and Four-line Locus Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 16: Hellenistic and Roman Geometry 16.1. Zenodorus 16.2. The Parallel Postulate 16.3. Heron 16.4. Roman Civil Engineering Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 17: Ptolemy’s Geography and Astronomy 17.1. Geography 17.2. Astronomy 17.2.1. Epicycles and Eccentrics 17.2.2. The Motion of the Sun 17.3. The Almagest 17.3.1. Trigonometry 17.3.2. Ptolemy’s Table of Chords Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 18: Pappus and the Later Commentators 18.1. The Collection of Pappus 18.1.1. Generalization of the Pythagorean Theorem 18.1.2. the Isoperimetric Problem 18.1.3. Analysis, Locus Problems, and Pappus’ Theorem 18.2. The Later Commentators: Theon and Hypatia 18.2.1. Theon of Alexandria 18.2.2. Hypatia of Alexandria Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Part IV: India, China, and Japan 500 Bce–1700 Ce Contents of Part IV Chapter 19: Overview of Mathematics in India 19.1. The Sulva Sutras 19.2. Buddhist and Jain Mathematics 19.3. The Bakshali Manuscript 19.4. The Siddhantas 19.5. Hindu–arabic Numerals 19.6. Aryabhata I 19.7. Brahmagupta 19.8. Bhaskara II 19.9. Muslim India 19.10. Indian Mathematics in the Colonial Period and After 19.10.1. Srinivasa Ramanujan Questions Historical Questions Questions for Reflection Chapter 20: from the Vedas to Aryabhata I 20.1. Problems from the Sulva Sutras 20.1.1. Arithmetic 20.1.2. Geometry 20.1.3. Square Roots 20.1.4. Jain Mathematics: the Infinite 20.1.5. Jain Mathematics: Combinatorics 20.1.6. The Bakshali Manuscript 20.2. Aryabhata I: Geometry and Trigonometry 20.2.1. Trigonometry 20.2.2. The Kuttaka Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 21: Brahmagupta, the Kuttaka, and Bhaskara II 21.1. Brahmagupta’s Plane and Solid Geometry 21.2. Brahmagupta’s Number Theory and Algebra 21.2.1. Pythagorean Triples 21.2.2. Pell’s Equation 21.3. The Kuttaka 21.4. Algebra in the Works of Bhaskara II 21.4.1. The Vija Ganita (algebra) 21.4.2. Combinatorics 21.5. Geometry in the Works of Bhaskara Ii Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 22: Early Classics of Chinese Mathematics 22.1. Works and Authors 22.1.1. The Zhou Bi Suan Jing 22.1.2. The Jiu Zhang Suan Shu 22.1.3. The Sun Zi Suan Jing 22.1.4. Liu Hui. the Hai Dao Suan Jing 22.1.5. Zu Chongzhi and Zu Geng 22.1.6. Yang Hui 22.1.7. Cheng Dawei 22.2. China’s Encounter with Western Mathematics 22.3. The Chinese Number System 22.3.1. Fractions and Roots 22.4. Algebra 22.5. Contents of the Jiu Zhang Suan Shu 22.6. Early Chinese Geometry 22.6.1. The Zhou Bi Suan Jing 22.6.2. The Jiu Zhang Suan Shu 22.6.3. The Sun Zi Suan Jing Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 23: Later Chinese Algebra and Geometry 23.1. Algebra 23.1.1. Systems of Linear Equations 23.1.2. Quadratic Equations 23.1.3. Cubic Equations 23.1.4. a Digression on the Numerical Solution of Equations 23.2. Later Chinese Geometry 23.2.1. Liu Hui 23.2.2. Zu Chongzhi Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 24: Traditional Japanese Mathematics 24.1. Chinese Influence and Calculating Devices 24.2. Japanese Mathematicians and Their Works 24.2.1. Yoshida Koyu 24.2.2. Seki Kowa and Takebe Kenko 24.2.3. The Modern Era in Japan 24.3. Japanese Geometry and Algebra 24.3.1. Determinants 24.3.2. The Challenge Problems 24.3.3. Beginnings of the Calculus in Japan 24.4. Sangaku 24.4.1. Analysis Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Part V: Islamic Mathematics, 800–1500 Contents of Part V Chapter 25: Overview of Islamic Mathematics 25.1. a Brief Sketch of the Islamic Civilization 25.1.1. The Umayyads 25.1.2. The Abbasids 25.1.3. The Turkish and Mongol Conquests 25.1.4. The Islamic Influence on Science 25.2. Islamic Science in General 25.2.1. Hindu and Hellenistic Influences 25.3. Some Muslim Mathematicians and Their Works 25.3.1. Muhammad Ibn Musa Al-khwarizmi 25.3.2. Thabit Ibn-qurra 25.3.3. Abu Kamil 25.3.4. Al-battani 25.3.5. Abu’l Wafa 25.3.6. Ibn Al-haytham 25.3.7. Al-biruni 25.3.8. Omar Khayyam 25.3.9. Sharaf Al-tusi 25.3.10. Nasir Al-tusi Questions Historical Questions Questions for Reflection Chapter 26: Islamic Number Theory and Algebra 26.1. Number Theory 26.2. Algebra 26.2.1. Al-khwarizmi 26.2.2. Abu Kamil 26.2.3. Omar Khayyam 26.2.4. Sharaf Al-din Al-tusi Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 27: Islamic Geometry 27.1. The Parallel Postulate 27.2. Thabit Ibn-qurra 27.3. Al-biruni: Trigonometry 27.4. Al-kuhi 27.5. Al-haytham and Ibn-sahl 27.6. Omar Khayyam 27.7. Nasir Al-din Al-tusi Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Part VI: European Mathematics, 500–1900 Contents of Part VI Chapter 28: Medieval and Early Modern Europe 28.1. from the Fall of Rome to the Year 1200 28.1.1. Boethius and the Quadrivium 28.1.2. Arithmetic and Geometry 28.1.3. Music and Astronomy 28.1.4. The Carolingian Empire 28.1.5. Gerbert 28.1.6. Early Medieval Geometry 28.1.7. The Translators 28.2. The High Middle Ages 28.2.1. Leonardo of Pisa 28.2.2. Jordanus Nemorarius 28.2.3. Nicole D’oresme 28.2.4. Regiomontanus 28.2.5. Nicolas Chuquet 28.2.6. Luca Pacioli 28.2.7. Leon Battista Alberti 28.3. The Early Modern Period 28.3.1. Scipione Del Ferro 28.3.2. Niccolò Tartaglia 28.3.3. Girolamo Cardano 28.3.4. Ludovico Ferrari 28.3.5. Rafael Bombelli 28.4. Northern European Advances 28.4.1. François Viète 28.4.2. John Napier Questions Historical Questions Questions for Reflection Chapter 29: European Mathematics: 1200–1500 29.1. Leonardo of Pisa (fibonacci) 29.1.1. The Liber Abaci 29.1.2. The Fibonacci Sequence 29.1.3. The Liber Quadratorum 29.1.4. The Flos 29.2. Hindu–arabic Numerals 29.3. Jordanus Nemorarius 29.4. Nicole D’oresme 29.5. Trigonometry: Regiomontanus and Pitiscus 29.5.1. Regiomontanus 29.5.2. Pitiscus 29.6. A Mathematical Skill: Prosthaphæresis 29.7. Algebra: Pacioli and Chuquet 29.7.1. Luca Pacioli 29.7.2. Chuquet Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 30: Sixteenth-century Algebra 30.1. Solution of Cubic and Quartic Equations 30.1.1. Ludovico Ferrari 30.2. Consolidation 30.2.1. François Viète 30.3. Logarithms 30.3.1. Arithmetical Implementation of the Geometric Model 30.4. Hardware: Slide Rules and Calculating Machines 30.4.1. The Slide Rule 30.4.2. Calculating Machines Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 31: Renaissance Art and Geometry 31.1. The Greek Foundations 31.2. The Renaissance Artists and Geometers 31.3. Projective Properties 31.3.1. Girard Desargues 31.3.2. Blaise Pascal Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 32: The Calculus Before Newton and Leibniz 32.1. Analytic Geometry 32.1.1. Pierre De Fermat 32.1.2. René Descartes 32.2. Components of the Calculus 32.2.1. Tangent and Maximum Problems 32.2.2. Lengths, Areas, and Volumes 32.2.3. Bonaventura Cavalieri 32.2.4. Gilles Personne De Roberval 32.2.5. Rectangular Approximations and the Method of Exhaustion 32.2.6. Blaise Pascal 32.2.7. The Relation Between Tangents and Areas 32.2.8. Infinite Series and Products 32.2.9. The Binomial Series Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 33: Newton and Leibniz 33.1. Isaac Newton 33.1.1. Newton’s First Version of the Calculus 33.1.2. Fluxions and Fluents 33.1.3. Later Exposition of the Calculus 33.1.4. Objections 33.2. Gottfried Wilhelm Von Leibniz 33.2.1. Leibniz’ Presentation of the Calculus 33.2.2. Later Reflections on the Calculus 33.3. The Disciples of Newton and Leibniz 33.4. Philosophical Issues 33.4.1. The Debate on the Continent 33.5. The Priority Dispute 33.6. Early Textbooks on Calculus 33.6.1. The State of the Calculus Around 1700 Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 34: Consolidation of the Calculus 34.1. Ordinary Differential Equations 34.1.1. a Digression on Time 34.2. Partial Differential Equations 34.3. Calculus of Variations 34.3.1. Euler 34.3.2. Lagrange 34.3.3. Second-variation Tests for Maxima and Minima 34.3.4. Jacobi: Sufficiency Criteria 34.3.5. Weierstrass and His School 34.4. Foundations of the Calculus 34.4.1. Lagrange’s Algebraic Analysis 34.4.2. Cauchy’s Calculus Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Part VII: Special Topics Contents of Part VII Chapter 35: Women Mathematicians 35.1. Sof’ya Kovalevskaya 35.1.1. Resistance from Conservatives 35.2. Grace Chisholm Young 35.3. Emmy Noether Questions Historical Questions Questions for Reflection Chapter 36: Probability 36.1. Cardano 36.2. Fermat and Pascal 36.3. Huygens 36.4. Leibniz 36.5. The Ars Conjectandi of James Bernoulli 36.5.1. The Law of Large Numbers 36.6. De Moivre 36.7. The Petersburg Paradox 36.8. Laplace 36.9. Legendre 36.10. Gauss 36.11. Philosophical Issues 36.12. Large Numbers and Limit Theorems Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 37: Algebra from 1600 to 1850 37.1. Theory of Equations 37.1.1. Albert Girard 37.1.2. Tschirnhaus Transformations 37.1.3. Newton, Leibniz, and the Bernoullis 37.2. Euler, D’alembert, and Lagrange 37.2.1. Euler 37.2.2. D’alembert 37.2.3. Lagrange 37.3. The Fundamental Theorem of Algebra and Solution by Radicals 37.3.1. Ruffini 37.3.2. Cauchy 37.3.3. Abel 37.3.4. Galois Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 38: Projective and Algebraic Geometry and Topology 38.1. Projective Geometry 38.1.1. Newton’s Degree-preserving Mapping 38.1.2. Brianchon 38.1.3. Monge and His School 38.1.4. Steiner 38.1.5. Möbius 38.2. Algebraic Geometry 38.2.1. Plücker 38.2.2. Cayley 38.3. Topology 38.3.1. Combinatorial Topology 38.3.2. Riemann 38.3.3. Möbius 38.3.4. Poincaré’s Analysis Situs 38.3.5. Point-set Topology Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 39: Differential Geometry 39.1. Plane Curves 39.1.1. Huygens 39.1.2. Newton 39.1.3. Leibniz 39.2. the Eighteenth Century: Surfaces 39.2.1. Euler 39.2.2. Lagrange 39.3. Space Curves: the French Geometers 39.4. Gauss: Geodesics and Developable Surfaces 39.4.1. Further Work by Gauss 39.5. The French and British Geometers 39.6. Grassmann and Riemann: Manifolds 39.6.1. Grassmann 39.6.2. Riemann 39.7. Differential Geometry and Physics 39.8. The Italian Geometers 39.8.1. Ricci’s Absolute Differential Calculus Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 40: Non-euclidean Geometry 40.1. Saccheri 40.2. Lambert and Legendre 40.3. Gauss 40.4. The First Treatises 40.5. Lobachevskii’s Geometry 40.6. János Bólyai 40.7. The Reception of Non-euclidean Geometry 40.8. Foundations of Geometry Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 41: Complex Analysis 41.1. Imaginary and Complex Numbers 41.1.1. Wallis 41.1.2. Wessel 41.1.3. Argand 41.2. Analytic Function Theory 41.2.1. Algebraic Integrals 41.2.2. Legendre, Jacobi, and Abel 41.2.3. Theta Functions 41.2.4. Cauchy 41.2.5. Riemann 41.2.6. Weierstrass 41.3. Comparison of the Three Approaches Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 42: Real Numbers, Series, and Integrals 42.1. Fourier Series, Functions, and Integrals 42.1.1. the Definition of a Function 42.2. Fourier Series 42.2.1. Sturm–liouville Problems 42.3. Fourier Integrals 42.4. General Trigonometric Series Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 43: Foundations of Real Analysis 43.1. What Is a Real Number? 43.1.1. The Arithmetization of the Real Numbers 43.2. Completeness of the Real Numbers 43.3. Uniform Convergence and Continuity 43.4. General Integrals and Discontinuous Functions 43.5. The Abstract and the Concrete 43.5.1. Absolute Continuity 43.5.2. Taming the Abstract 43.6. Discontinuity as a Positive Property Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 44: Set Theory 44.1. Technical Background 44.2. Cantor’s Work on Trigonometric Series 44.2.1. Ordinal Numbers 44.2.2. Cardinal Numbers 44.3. the Reception of Set Theory 44.3.1. Cantor and Kronecker 44.4. Existence and the Axiom of Choice Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Chapter 45: Logic 45.1. from Algebra to Logic 45.2. Symbolic Calculus 45.3. Boole’s Mathematical Analysis of Logic 45.3.1. Logic and Classes 45.4. Boole’s Laws of Thought 45.5. Jevons 45.6. Philosophies of Mathematics 45.6.1. Paradoxes 45.6.2. Formalism 45.6.3. Intuitionism 45.6.4. Mathematical Practice 45.7. Doubts About Formalized Mathematics: Gödel’s Theorems Problems and Questions Mathematical Problems Historical Questions Questions for Reflection Literature Name Index Subject Index