| |

| |

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-Khowï¿½rizmï¿½ | |

| |

| |

Abï¿½ Kamil and Thï¿½bit 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 Viï¿½ta | |

| |

| |

The Decimal Fractions of Simon Steven | |

| |

| |

Napier's Invention of Logarithms | |

| |

| |

The Astronomical Discoveries of Brahe and Kepler | |

| |

| |

| |

Descartes: The Discours de la Mï¿½thod | |

| |

| |

The Writings of Descartes | |

| |

| |

Inventing Cartesian Geometry | |

| |

| |

The Algebraic Aspect of La Gï¿½ometrie | |

| |

| |

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 Chï¿½telet | |

| |

| |

| |

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 Mï¿½rï¿½'s Problem of Points | |

| |

| |

| |

Pascal's Arithmetic Triangle | |

| |

| |

The Traitï¿½ du Triangle Arithmï¿½tique | |

| |

| |

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 Elï¿½ments de gï¿½ometrie | |

| |

| |

| |

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: Poincarï¿½ | |

| |

| |

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 | |