The History of Mathematics: An Introduction
David Burton
The History of Mathematics: An Introduction, 7e by David M. Burton Preface 1 Early Number Systems and Symbols 1.1 Primitive Counting A Sense Notches as Tally Marks Peruvian Quipus: Knots Numbers 1.2 Recording the Egyptians Greeks Herodotus Hieroglyphic Representation Egyptian Hieratic Numeration Greek Alphabetic Numeral System 1.3 Babylonians Babylonian Cuneiform Script Deciphering Cuneiform: Grotefend Rawlinson Positional Writing in Ancient China 2 Mathematics Civilizations 2.1 Rhind Papyrus Mathematical Papyri Key to Deciphering: Rosetta Stone 2.2 Arithmetic Multiplication Unit Fraction Table Representing Rational 2.3 Four Problems from Method False Position Curious Problem Applied 2.4 Geometry Approximating Area a Circle Volume Truncated Pyramid Speculations About Great 2.5 Tablet Reciprocals Treatment Quadratic Equations Two Characteristic 2.6 Plimpton Concerning Triples Use Pythagorean Theorem Cairo 3 Beginnings 3.1 Geometric Discoveries Thales Greece Aegean Dawn Demonstrative Geometry: Miletos Measurements Using 3.2 Pythagoras His Followers Nichomachus' Introductio Arithmeticae Theory Figurative Zeno's Paradox 3.3 Proofs Solutions Equation Crisis Incommensurable Quantities Theon's Side Diagonal Eudoxus Cnidos 3.4 Three Construction Antiquity Hippocrates Quadrature Duplication Cube Trisection an Angle 3.5 Quadratrix Hippias Rise Sophists Elis Grove Academia: Plato's Academy 4 Alexandrian School: Euclid 4.1 Elements Center Learning: Museum Euclid's Life Writings 4.2 Euclidean Foundation for Book I Proof II on Algebra Regular Pentagon 4.3 Divisibility Properties Algorithm Fundamental Infinity Primes 4.4 Eratosthenes, Wise Man Alexandria Sieve Eratosthenes Measurement Earth Almagest Claudius Ptolemy Ptolemy's Geographical Dictionary 4.5 Archimedes World's Genius Estimating Value pi Sand-Reckoner Parabolic Segment Apollonius Perga: Conics 5 Twilight Diophantus 5.1 Decline Waning Golden Age Spread Christianity Constantinople, Refuge Learning 5.2 Arithmetica Diophantus's 5.3 Diophantine Greece, India, Cattle India Chinese Hundred Fowls 5.4 Later Commentators Collection Pappus Hypatia, First Woman Mathematician Roman Boethius Cassiodorus 5.5 Near Far East al-Khowarizmi Abu Kamil Thabit ibn Qurra Omar Khayyam Astronomers al-Tusi al-Karashi Nine Chapters Works 6 Awakening: Fibonacci 6.1 Revival Carolingian Pre-Renaissance Transmission Arabic West Pioneer Translators: Gerard Adelard 6.2 Liber Abaci Quadratorum Hindu-Arabic Numerals Fibonacci's Liver Jordanus de Nemore 6.3 Sequence Abaci's Rabbit Some 6.4 Tournament 7 Renaissance Cardan Tartaglia 7.1 Europe Fourteenth Fifteenth Centuries Italian Artificial Writing: Invention Printing Founding Universities Thirst Classical 7.2 Battle Scholars Restoring Algebraic Tradition: Robert Recorde Algebraists: Pacioli, del Ferro Cardan, Scoundrel 7.3 Cardan's Ars Magna Solution Cubic Bombelli Imaginary Roots 7.4 Ferrari's Quartic Resolvant Story Quintic Equation: Ruffini, Abel Galois 8 Mechanical World: Descartes Newton 8.1 Modern Seventeenth Century Knowledge Galileo's Telescopic Observations Beginning Notation: Francois Vieta Decimal Fractions Simon Steven Napier's Logarithms Astronomical Brahe Kepler 8.2 Descartes: Discours la Inventing Cartesian Aspect La Geometrie Descartes' Principia Philosophia Perspective Desargues Poncelet 8.3 Newton: Mathematica Textbooks Oughtred Harriot Wallis' Infinitorum Lucasian Professorship: Barrow Newton's Years Laws Motion Years: Appointment Mint 8.4 Gottfried Leibniz: Calculus Controversy Work Leibniz Leibniz's Creation Fluxional Dispute over Priority Maria Agnesi Emilie du Chatelet 9 Development Probability Theory: Pascal, Bernoulli, Laplace 9.1 Origins Graunt's Bills Mortality Games Chance: Dice Cards Precocity Young Pascal Cycloid De Mere's Points 9.2 Pascal's Triangle Traite Arithmetique Induction Francesco Maurolico's 9.3 Bernoullis Christiaan Huygens's Pamphlet Bernoulli Brothers: John James Moivre's Doctrine Chances Celestial Phenomena: Mary Fairfax Somerville Laplace's Research Daniel Poisson, Chebyshev 10 Fermat, Euler, Gauss 10.1 Martin Mersenne Search Perfect Scientific Societies Marin Mersenne's Gathering Numbers, Pe rfect Not So 10.2 From Fermat Euler Fermat's Famous Last Eighteenth-Century Enlightenment Maclaurin's Treatise Fluxions Euler's Contributions 10.3 Prince Mathematicians: Carl Friedrich Period French Revolution: Lagrange, Monge, Carnot Gauss's Disquisitiones Legacy Gauss: Congruence Dirichlet Jacobi 11 Nineteenth-Century Contributions: Lobachevsky Hilbert 11.1 Attempts Prove Parallel Postulate Efforts Proclus, Playfair, Wallis Saccheri Quadrilaterals Accomplishments Legendre Legendre's geometrie 11.2 Founders Non-Euclidean Attempt at New Struggle Bolyai Models Riemann, Beltrami, Klein Grace Chisholm 11.3 Rigor D'Alembert Cauchy Limits Fourier's Series Father Analysis, Weierstrass Sonya Kovalevsky Axiomatic Movement: Pasch 11.4 Generalized Babbage Analytical Engine Peacock's Representations Complex Hamilton's Discovery Quaternions Matrix Algebra: Cayley Sylvester Boole's Logic 12 Transition Twenthieth Century: Cantor Kronecker 12.1 Emergence American Ascendency German Takes Root: 1800-1900 Twentieth Consolidation 12.2 Infinite Universalist: Poincare Cantor's Sets Kronecker's View Set Countable Uncountable Transcendental Continuum Hypothesis 12.3 Paradoxes Zermelo Axiom Choice Logistic Frege, Peano Russell Hilbert's Formalistic Approach Brouwer's Intuitionism 13 Extensions Generalizations: Hardy, Hausdorff, Noether 13.1 Hardy Ramanujan Tripos Examination Rejuvenation English Unique Collaboration: Littlewood India's Prodigy, 13.2 Point-Set Topology Frechet's Metric Spaces Neighborhood Hausdorff Banach Normed Linear 13.3 Twentieth-Century Developments Emmy Noether's Rings Von Neumann Computer Women Few Recent Advances General Bibliography Additional Reading Alphabet Selected Index
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