The History of Mathematics : A Brief Course

• Record Nr.: UNINA9910795812003321
• Autore: Cooke Roger L
• Pubbl/distr/stampa: New York : , : John Wiley & Sons, Incorporated, , 2012; ©2013
• ISBN: 9781118460290; 9781118217566
• Edizione: [3rd ed.]
• Descrizione fisica: 1 online resource (730 pages)
• Altri autori (Persone): CookeRoger L
• Disciplina: 510/.9
• Soggetti: Mathematics - History; Electronic books.
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Title Page -- Copyright -- 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 -- 1.2 The Origin of Mathematics -- 1.3 The Philosophy of Mathematics -- 1.4 Our Approach to the History of Mathematics -- Questions for Reflection -- Chapter 2: Proto-mathematics -- 2.1 Number -- 2.2 Shape -- 2.3 Symbols -- 2.4 Mathematical Reasoning -- Problems and Questions -- 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 -- 3.4 The Mesopotamian Numbering System -- Problems and Questions -- Chapter 4: Computations in Ancient Mesopotamia -- 4.1 Arithmetic -- 4.2 Algebra -- Problems and Questions -- Chapter 5: Geometry in Mesopotamia -- 5.1 The Pythagorean Theorem -- 5.2 Plane Figures -- 5.3 Volumes -- 5.4 Plimpton 322 -- Problems and Questions -- Chapter 6: Egyptian Numerals and Arithmetic -- 6.1 Sources -- 6.2 The Rhind Papyrus -- 6.3 Egyptian Arithmetic -- 6.4 Computation -- Problems and Questions -- Chapter 7: Algebra and Geometry in Ancient Egypt -- 7.1 ; Algebra Problems in the Rhind Papyrus -- 7.2 Geometry -- 7.3 Areas -- Problems and Questions -- 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 -- 8.2 General Features of Greek Mathematics -- 8.3 Works and Authors -- Questions -- Chapter 9: Greek Number Theory -- 9.1 The Euclidean Algorithm -- 9.2 The Arithmetica of Nicomachus -- 9.3 Euclid's Number Theory -- 9.4 The Arithmetica of Diophantus -- Problems and Questions.; Chapter 10: Fifth-Century Greek Geometry -- 10.1 "Pythagorean" Geometry -- 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 -- Problems and Questions -- Chapter 11: Athenian Mathematics I: The Classical Problems -- 11.1 Squaring the Circle -- 11.2 Doubling the Cube -- 11.3 Trisecting the Angle -- Problems and Questions -- Chapter 12: Athenian Mathematics II: Plato and Aristotle -- 12.1 The Influence of Plato -- 12.2 Eudoxan Geometry -- 12.3 Aristotle -- Problems and Questions -- Chapter 13: Euclid of Alexandria -- 13.1 The Elements -- 13.2 The Data -- Problems and Questions -- Chapter 14: Archimedes of Syracuse -- 14.1 The Works of Archimedes -- 14.2 The Surface of a Sphere -- 14.3 The Archimedes Palimpsest -- 14.4 Quadrature of the Parabola -- Problems and Questions -- Chapter 15: Apollonius of Perga -- 15.1 History of the Conics -- 15.2 Contents of the Conics -- 15.3 Foci and the Three-and Four-line Locus -- Problems and Questions -- 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 -- Chapter 17: Ptolemy's Geography and Astronomy -- 17.1 Geography -- 17.2 Astronomy -- 17.3 The Almagest -- Problems and Questions -- Part IV: India, China, and Japan 500 BCE-1700 CE -- Contents of Part IV -- Chapter 18: Pappus and the Later Commentators -- 18.1 The Collection of Pappus -- 18.2 The Later Commentators: Theon and Hypatia -- Problems and Questions -- 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 -- Questions -- Chapter 20: From the Vedas to Aryabhata I -- 20.1 Problems from the Sulva Sutras -- 20.2 Aryabhata I: Geometry and Trigonometry -- Problems and Questions -- Chapter 21: Brahmagupta, the Kuttaka, and Bhaskara II -- 21.1 Brahmagupta's Plane and Solid Geometry -- 21.2 Brahmagupta's Number Theory and Algebra -- 21.3 The Kuttaka -- 21.4 Algebra in the Works of Bhaskara II -- 21.5 Geometry in the Works of Bhaskara II -- Problems and Questions -- Chapter 22: Early Classics of Chinese Mathematics -- 22.1 Works and Authors -- 22.2 China's Encounter with Western Mathematics -- 22.3 The Chinese Number System -- 22.4 Algebra -- 22.5 Contents of the Jiu Zhang Suan Shu -- 22.6 Early Chinese Geometry -- Problems and Questions -- Chapter 23: Later Chinese Algebra and Geometry -- 23.1 Algebra -- 23.2 Later Chinese Geometry -- Problems and Questions -- Chapter 24: Traditional Japanese Mathematics -- 24.1 Chinese Influence and Calculating Devices -- 24.2 Japanese Mathematicians and Their Works -- 24.3 Japanese Geometry and Algebra -- 24.4 Sangaku -- Problems and Questions -- 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 -- 25.2 Islamic Science in General -- 25.3 Some Muslim Mathematicians and their Works -- ; Questions -- Chapter 26: Islamic Number Theory and Algebra -- 26.1 Number Theory -- 26.2 Algebra -- Problems and Questions -- 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 -- 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 -- 28.2 The High Middle Ages -- 28.3 The Early Modern Period -- 28.4 Northern European Advances -- Questions -- Chapter 29: European Mathematics: 1200-1500 -- 29.1 Leonardo of Pisa (Fibonacci) -- 29.2 Hindu-Arabic numerals -- 29.3 Jordanus Nemorarius -- 29.4 Nicole d'Oresme -- 29.5 Trigonometry: Regiomontanus and Pitiscus -- 29.6 A Mathematical Skill: ProsthaphÆresis -- 29.7 Algebra: Pacioli and Chuquet -- Problems and Questions -- Chapter 30: Sixteenth-Century Algebra -- 30.1 Solution of Cubic and Quartic Equations -- 30.2 Consolidation -- 30.3 Logarithms -- 30.4 Hardware: slide rules and calculating machines -- Problems and Questions -- Chapter 31: Renaissance Art and Geometry -- 31.1 The Greek Foundations -- 31.2 The Renaissance Artists and Geometers -- 31.3 Projective Properties -- Problems and Questions -- Chapter 32: The Calculus Before Newton and Leibniz -- 32.1 Analytic Geometry -- 32.2 Components of the Calculus -- Problems and Questions -- Chapter 33: Newton and Leibniz -- 33.1 Isaac Newton -- 33.2 Gottfried Wilhelm von Leibniz -- 33.3 The Disciples of Newton and Leibniz -- 33.4 Philosophical Issues -- 33.5 The Priority Dispute -- 33.6 Early Textbooks on Calculus -- Problems and Questions -- Chapter 34: Consolidation of the Calculus -- 34.1 Ordinary Differential Equations -- 34.2 Partial Differential Equations -- 34.3 Calculus of Variations -- 34.4 Foundations of the Calculus -- Problems and Questions -- Part VII: Special Topics -- Contents of Part VII -- Chapter 35: Women Mathematicians -- 35.1 Sof'ya Kovalevskaya -- 35.2 Grace Chisholm Young -- 35.3 Emmy Noether -- Questions -- 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 -- 36.6 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 -- Chapter 37: Algebra from 1600 to 1850 -- 37.1 Theory of Equations -- 37.2 Euler, D'Alembert, and Lagrange -- 37.3 The Fundamental Theorem of Algebra and Solution by Radicals -- Problems and Questions -- Chapter 38: Projective and Algebraic Geometry and Topology -- 38.1 Projective Geometry -- 38.2 Algebraic Geometry -- 38.3 Topology -- Problems and Questions -- Chapter 39: Differential Geometry -- 39.1 Plane Curves -- 39.2 The Eighteenth Century: Surfaces -- 39.3 Space Curves: The French Geometers -- 39.4 Gauss: Geodesics and Developable Surfaces -- 39.5 The French and British Geometers -- 39.6 Grassmann and Riemann: Manifolds -- 39.7 Differential Geometry and Physics -- 39.8 The Italian Geometers -- Problems and Questions -- 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 -- Chapter 41: Complex Analysis -- 41.1 Imaginary and Complex Numbers -- 41.2 Analytic Function Theory -- 41.3 Comparison of the Three Approaches -- Problems and Questions -- Chapter 42: Real Numbers, Series, and ; Integrals -- 42.1 Fourier Series, Functions, and Integrals -- 42.2 Fourier Series -- 42.3 Fourier Integrals -- 42.4 General Trigonometric Series -- Problems and Questions -- Chapter 43: Foundations of Real Analysis -- 43.1 What Is a Real Number? -- 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 -- 43.6 Discontinuity as a Positive Property -- Problems and Questions -- Chapter 44: Set Theory.; 44.1 Technical Background.
• Sommario/riassunto: 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.