The History of Mathematics: An Introduction, 7e by David M. Burton

Preface

1 Early Number Systems and Symbols

1.1

Primitive Counting

A Sense of Number

Notches as Tally Marks

The Peruvian Quipus: Knots as Numbers

1.2

Number Recording of the Egyptians and Greeks

The History of Herodotus

Hieroglyphic Representation of Numbers

Egyptian Hieratic Numeration

The Greek Alphabetic Numeral System

1.3

Number Recording of the Babylonians

Babylonian Cuneiform Script

Deciphering Cuneiform: Grotefend and Rawlinson

The Babylonian Positional Number System

Writing in Ancient China

2 Mathematics in Early Civilizations

2.1

The Rhind Papyrus

Egyptian Mathematical Papyri

A Key to Deciphering: The Rosetta Stone

2.2

Egyptian Arithmetic

Early Egyptian Multiplication

The Unit Fraction Table

Representing Rational Numbers

2.3

Four Problems from the Rhind Papyrus

The Method of False Position

A Curious Problem

Egyptian Mathematics as Applied Arithmetic

2.4

Egyptian Geometry

Approximating the Area of a Circle

The Volume of a Truncated Pyramid

Speculations About the Great Pyramid

2.5

Babylonian Mathematics

A Tablet of Reciprocals

The Babylonian Treatment of Quadratic Equations

Two Characteristic Babylonian Problems

2.6

Plimpton

A Tablet Concerning Number Triples

Babylonian Use of the Pythagorean Theorem

The Cairo Mathematical Papyrus

3 The Beginnings of Greek Mathematics

3.1

The Geometric Discoveries of Thales

Greece and the Aegean Area

The Dawn of Demonstrative Geometry: Thales of Miletos

Measurements Using Geometry

3.2

Pythagorean Mathematics

Pythagoras and His Followers

Nichomachus' Introductio Arithmeticae

The Theory of Figurative Numbers

Zeno's Paradox

3.3

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

3.4

Three Construction Problems of Antiquity

Hippocrates and the Quadrature of the Circle

The Duplication of the Cube

The Trisection of an Angle

3.5

The Quadratrix of Hippias

Rise of the Sophists

Hippias of Elis

The Grove of Academia: Plato's Academy

4 The Alexandrian School: Euclid

4.1

Euclid and the Elements

A Center of Learning: The Museum

Euclid's Life and Writings

4.2

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

4.3

Euclid's Number Theory

Euclidean Divisibility Properties

The Algorithm of Euclid

The Fundamental Theorem of Arithmetic

An Infinity of Primes

4.4

Eratosthenes, the Wise Man of Alexandria

The Sieve of Eratosthenes

Measurement of the Earth

The Almagest of Claudius Ptolemy

Ptolemy's Geographical Dictionary

4.5

Archimedes

The Ancient World's Genius

Estimating the Value of π

The Sand-Reckoner

Quadrature of a Parabolic Segment

Apollonius of Perga: The Conics

5 The Twilight of Greek Mathematics: Diophantus

5.1

The Decline of Alexandrian Mathematics

The Waning of the Golden Age

The Spread of Christianity

Constantinople, A Refuge for Greek Learning

5.2

The Arithmetica

Diophantus's Number Theory

Problems from the Arithmetica

5.3

Diophantine Equations in Greece, India, and China

The Cattle Problem of Archimedes

Early Mathematics in India

The Chinese Hundred Fowls Problem

5.4

The Later Commentators

The Mathematical Collection of Pappus

Hypatia, the First Woman Mathematician

Roman Mathematics: Boethius and Cassiodorus

5.5

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

6 The First Awakening: Fibonacci

6.1

The Decline and Revival of Learning

The Carolingian Pre-Renaissance

Transmission of Arabic Learning to the West

The Pioneer Translators: Gerard and Adelard

6.2

The Liber Abaci and Liber Quadratorum

The Hindu-Arabic Numerals

Fibonacci's Liver Quadratorum

The Works of Jordanus de Nemore

6.3

The Fibonacci Sequence

The Liber Abaci's Rabbit Problem

Some Properties of Fibonacci Numbers

6.4

Fibonacci and the Pythagorean Problem

Pythagorean Number Triples

Fibonacci's Tournament Problem

7 The Renaissance of Mathematics: Cardan and Tartaglia

7.1

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

7.2

The Battle of the Scholars

Restoring the Algebraic Tradition: Robert Recorde

The Italian Algebraists: Pacioli, del Ferro and Tartaglia

Cardan, A Scoundrel Mathematician

7.3

Cardan's Ars Magna

Cardan's Solution of the Cubic Equation

Bombelli and Imaginary Roots of the Cubic

7.4

Ferrari's Solution of the Quartic Equation

The Resolvant Cubic

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

8 The Mechanical World: Descartes and Newton

8.1

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

8.2

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

8.3

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

8.4

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

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

9.1

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

9.2

Pascal's Arithmetic Triangle

The Traité du Triangle Arithmétique

Mathematical Induction

Francesco Maurolico's Use of Induction

9.3

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

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

10.1

Martin Mersenne and the Search for Perfect Numbers

Scientific Societies

Marin Mersenne's Mathematical Gathering

Numbers, Perfect and Not So Perfect

10.2

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

10.3

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

11 Nineteenth-Century Contributions: Lobachevsky to Hilbert

11.1

Attempts to Prove the Parallel Postulate

The Efforts of Proclus, Playfair, and Wallis

Saccheri Quadrilaterals

The Accomplishments of Legendre

Legendre's Eléments de géometrie

11.2

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

11.3

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

11.4

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

12 Transition to the Twenthieth Century: Cantor and Kronecker

12.1

The Emergence of American Mathematics

Ascendency of the German Universities

American Mathematics Takes Root: 1800-1900

The Twentieth Century Consolidation

12.2

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

12.3

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

13 Extensions and Generalizations: Hardy, Hausdorff, and Noether

13.1

Hardy and Ramanujan

The Tripos Examination

The Rejuvenation of English Mathematics

A Unique Collaboration: Hardy and Littlewood

India's Prodigy, Ramanujan

13.2

The Beginnings of Point-Set Topology

Frechet's Metric Spaces

The Neighborhood Spaces of Hausdorff

Banach and Normed Linear Spaces

13.3

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