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
