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