# The History of Mathematics: An Introduction, 7th Edition

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The History of Mathematics: An Introduction, Seventh Edition, is written for the one- or two-semester math history course taken by juniors or seniors, and covers the history behind the topics typically covered in an undergraduate math curriculum or in elementary schools or high schools. Elegantly written in David Burton’s imitable prose, this classic text provides rich historical context to the mathematics that undergrad math and math education majors encounter every day. Burton illuminates the people, stories, and social context behind mathematics’ greatest historical advances while maintaining appropriate focus on the mathematical concepts themselves. Its wealth of information, mathematical and historical accuracy, and renowned presentation make The History of Mathematics: An Introduction, Seventh Edition a valuable resource that teachers and students will want as part of a permanent library.

# 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

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