Preface | |

Early Number Systems and Symbols | |

Primitive Counting | |

A Sense of Number | |

Notches as Tally Marks | |

The Peruvian Quipus: Knots as Numbers | |

Number Recording of the Egyptians and Greeks | |

The History of Herodotus | |

Hieroglyphic Representation of Numbers | |

Egyptian Hieratic Numeration | |

The Greek Alphabetic Numeral System | |

Number Recording of the Babylonians | |

Babylonian Cuneiform Script | |

Deciphering Cuneiform: Grotefend and Rawlinson | |

The Babylonian Positional Number System | |

Writing in Ancient China | |

Mathematics in Early Civilizations | |

The Rhind Papyrus | |

Egyptian Mathematical Papyri | |

A Key to Deciphering: The Rosetta Stone | |

Egyptian Arithmetic | |

Early Egyptian Multiplication | |

The Unit Fraction Table | |

Representing Rational Numbers | |

Four Problems from the Rhind Papyrus | |

The Method of False Position | |

A Curious Problem | |

Egyptian Mathematics as Applied Arithmetic | |

Egyptian Geometry | |

Approximating the Area of a Circle | |

The Volume of a Truncated Pyramid | |

Speculations About the Great Pyramid | |

Babylonian Mathematics | |

A Tablet of Reciprocals | |

The Babylonian Treatment of Quadratic Equations | |

Two Characteristic Babylonian Problems | |

Plimpton | |

A Tablet Concerning Number Triples | |

Babylonian Use of the Pythagorean Theorem | |

The Cairo Mathematical Papyrus | |

The Beginnings of Greek Mathematics | |

The Geometric Discoveries of Thales | |

Greece and the Aegean Area | |

The Dawn of Demonstrative Geometry: Thales of Miletos | |

Measurements Using Geometry | |

Pythagorean Mathematics | |

Pythagoras and His Followers | |

Nichomachus' Introductio Arithmeticae | |

The Theory of Figurative Numbers | |

Zeno's Paradox | |

The Pythagorean Problem | |

Geometric Proofs of the Pythagorean Theorem | |

Early Solutions of the Pythagorean Equation | |

The Crisis of Incommensurable Quantities | |

Theon's Side and Diagonal Numbers | |

Eudoxus of Cnidos | |

Three Construction Problems of Antiquity | |

Hippocrates and the Quadrature of the Circle | |

The Duplication of the Cube | |

The Trisection of an Angle | |

The Quadratrix of Hippias | |

Rise of the Sophists | |

Hippias of Elis | |

The Grove of Academia: Plato's Academy | |

The Alexandrian School: Euclid | |

Euclid and the Elements | |

A Center of Learning: The Museum | |

Euclid's Life and Writings | |

Euclidean Geometry | |

Euclid's Foundation for Geometry | |

Book I of the Elements | |

Euclid's Proof of the Pythagorean Theorem | |

Book II on Geometric Algebra | |

Construction of the Regular Pentagon | |

Euclid's Number Theory | |

Euclidean Divisibility Properties | |

The Algorithm of Euclid | |

The Fundamental Theorem of Arithmetic | |

An Infinity of Primes | |

Eratosthenes, the Wise Man of Alexandria | |

The Sieve of Eratosthenes | |

Measurement of the Earth | |

The Almagest of Claudius Ptolemy | |

Ptolemy's Geographical Dictionary | |

Archimedes | |

The Ancient World's Genius | |

Estimating the Value of p | |

The Sand-Reckoner | |

Quadrature of a Parabolic Segment | |

Apollonius of Perga: The Conics | |

The Twilight of Greek Mathematics: Diophantus | |

The Decline of Alexandrian Mathematics | |

The Waning of the Golden Age | |

The Spread of Christianity | |

Constantinople, A Refuge for Greek Learning | |

The Arithmetica | |

Diophantus's Number Theory | |

Problems from the Arithmetica | |

Diophantine Equations in Greece, India, and China | |

The Cattle Problem of Archimedes | |

Early Mathematics in India | |

The Chinese Hundred Fowls Problem | |

The Later Commentators | |

The Mathematical Collection of Pappus | |

Hypatia, the First Woman Mathematician | |

Roman Mathematics: Boethius and Cassiodorus | |

Mathematics in the Near and Far East | |

The Algebra of al-Khowârizmî | |

Abû Kamil and Thâbit ibn Qurra | |

Omar Khayyam | |

The Astronomers al-Tusi and al-Karashi | |

The Ancient Chinese Nine Chapters | |

Later Chinese Mathematical Works | |

The First Awakening: Fibonacci | |

The Decline and Revival of Learning | |

The Carolingian Pre-Renaissance | |

Transmission of Arabic Learning to the West | |

The Pioneer Translators: Gerard and Adelard | |

The Liber Abaci and Liber Quadratorum | |

The Hindu-Arabic Numerals | |

Fibonacci's Liver Quadratorum | |

The Works of Jordanus de Nemore | |

The Fibonacci Sequence | |

The Liber Abaci's Rabbit Problem | |

Some Properties of Fibonacci Numbers | |

Fibonacci and the Pythagorean Problem | |

Pythagorean Number Triples | |

Fibonacci's Tournament Problem | |

The Renaissance of Mathematics: Cardan and Tartaglia | |

Europe in the Fourteenth and Fifteenth Centuries | |

The Italian Renaissance | |

Artificial Writing: The Invention of Printing | |

Founding of the Great Universities | |

A Thirst for Classical Learning | |

The Battle of the Scholars | |

Restoring the Algebraic Tradition: Robert Recorde | |

The Italian Algebraists: Pacioli, del Ferro and Tartaglia | |

Cardan, A Scoundrel Mathematician | |

Cardan's Ars Magna | |

Cardan's Solution of the Cubic Equation | |

Bombelli and Imaginary Roots of the Cubic | |

Ferrari's Solution of the Quartic Equation | |

The Resolvant Cubic | |

The Story of the Quintic Equation: Ruffini, Abel and Galois | |

The Mechanical World: Descartes and Newton | |

The Dawn of Modern Mathematics | |

The Seventeenth Century Spread of Knowledge | |

Galileo's Telescopic Observations | |

The Beginning of Modern Notation: Francois Vièta | |

The Decimal Fractions of Simon Steven | |

Napier's Invention of Logarithms | |

The Astronomical Discoveries of Brahe and Kepler | |

Descartes: The Discours de la Méthod | |

The Writings of Descartes | |

Inventing Cartesian Geometry | |

The Algebraic Aspect of La Géometrie | |

Descartes' Principia Philosophia | |

Perspective Geometry: Desargues and Poncelet | |

Newton: The Principia Mathematica | |

The Textbooks of Oughtred and Harriot | |

Wallis' Arithmetica Infinitorum | |

The Lucasian Professorship: Barrow and Newton | |

Newton's Golden Years | |

The Laws of Motion | |

Later Years: Appointment to the Mint | |

Gottfried Leibniz: The Calculus Controversy | |

The Early Work of Leibniz | |

Leibniz's Creation of the Calculus | |

Newton's Fluxional Calculus | |

The Dispute over Priority | |

Maria Agnesi and Emilie du Châtelet | |

The Development of Probability Theory: Pascal, Bernoulli, and Laplace | |

The Origins of Probability Theory | |

Graunt's Bills of Mortality | |

Games of Chance: Dice and Cards | |

The Precocity of the Young Pascal | |

Pascal and the Cycloid | |

De Méré's Problem of Points | |

Pascal's Arithmetic Triangle | |

The Traité du Triangle Arithmétique | |

Mathematical Induction | |

Francesco Maurolico's Use of Induction | |

The Bernoullis and Laplace | |

Christiaan Huygens's Pamphlet on Probability | |

The Bernoulli Brothers: John and James | |

De Moivre's Doctrine of Chances | |

The Mathematics of Celestial Phenomena: Laplace | |

Mary Fairfax Somerville | |

Laplace's Research on Probability Theory | |

Daniel Bernoulli, Poisson, and Chebyshev | |

The Revival of Number Theory: Fermat, Euler, and Gauss | |

Martin Mersenne and the Search for Perfect Numbers | |

Scientific Societies | |

Marin Mersenne's Mathematical Gathering | |

Numbers, Perfect and Not So Perfect | |

From Fermat to Euler | |

Fermat's Arithmetica | |

The Famous Last Theorem of Fermat | |

The Eighteenth-Century Enlightenment | |

Maclaurin's Treatise on Fluxions | |

Euler's Life and Contributions | |

The Prince of Mathematicians: Carl Friedrich Gauss | |

The Period of the French Revolution: Lagrange, Monge, and Carnot | |

Gauss's Disquisitiones Arithmeticae | |

The Legacy of Gauss: Congruence Theory | |

Dirichlet and Jacobi | |

Nineteenth-Century Contributions: Lobachevsky to Hilbert | |

Attempts to Prove the Parallel Postulate | |

The Efforts of Proclus, Playfair, and Wallis | |

Saccheri Quadrilaterals | |

The Accomplishments of Legendre | |

Legendre's Eléments de géometrie | |

The Founders of Non-Euclidean Geometry | |

Gauss's Attempt at a New Geometry | |

The Struggle of John Bolyai | |

Creation of Non-Euclidean Geometry: Lobachevsky | |

Models of the New Geometry: Riemann, Beltrami, and Klein | |

Grace Chisholm Young | |

The Age of Rigor | |

D'Alembert and Cauchy on Limits | |

Fourier's Series | |

The Father of Modern Analysis, Weierstrass | |

Sonya Kovalevsky | |

The Axiomatic Movement: Pasch and Hilbert | |

Arithmetic Generalized | |

Babbage and the Analytical Engine | |

Peacock's Treatise on Algebra | |

The Representations of Complex Numbers | |

Hamilton's Discovery of Quaternions | |

Matrix Algebra: Cayley and Sylvester | |

Boole's Algebra of Logic | |

Transition to the Twenthieth Century: Cantor and Kronecker | |

The Emergence of American Mathematics | |

Ascendency of the German Universities | |

American Mathematics Takes Root: 1800-1900 | |

The Twentieth Century Consolidation | |

Counting the Infinite | |

The Last Universalist: Poincaré | |

Cantor's Theory of Infinite Sets | |

Kronecker's View of Set Theory | |

Countable and Uncountable Sets | |

Transcendental Numbers | |

The Continuum Hypothesis | |

The Paradoxes of Set Theory | |

The Early Paradoxes | |

Zermelo and the Axiom of Choice | |

The Logistic School: Frege, Peano and Russell | |

Hilbert's Formalistic Approach | |

Brouwer's Intuitionism | |

Extensions and Generalizations: Hardy, Hausdorff, and Noether | |

Hardy and Ramanujan | |

The Tripos Examination | |

The Rejuvenation of English Mathematics | |

A Unique Collaboration: Hardy and Littlewood | |

India's Prodigy, Ramanujan | |

The Beginnings of Point-Set Topology | |

Frechet's Metric Spaces | |

The Neighborhood Spaces of Hausdorff | |

Banach and Normed Linear Spaces | |

Some Twentieth-Century Developments | |

Emmy Noether's Theory of Rings | |

Von Neumann and the Computer | |

Women in Modern Mathematics | |

A Few Recent Advances | |

General Bibliography | |

Additional Reading | |

The Greek Alphabet | |

Solutions to Selected Problems | |

Index | |

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