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The History of Mathematics: An Introduction

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It's always a pleasure to sit down with a new (to me) textbook on the history of mathematics, and this text by David Burton has much to recommend it. One of the features which sets this book apart from its competitors is the attention that Burton pays to the cultural and historical millieu in which various mathematicians lived and worked through the ages. This is illustrated even in the typical timeline one expects to see inside the front cover; Burton compares mathematical advancements with important historical events such as the engraving of the Rosetta Stone in 195 B.C. (which eventually led to the deciphering of Egyptian hieroglyphics and the understanding of Egyptian mathematics) and the production of paper in Bologna in 1293 (allowing for the cheap production of textbooks using carved wooden blocks and, much later, movable type).

Throughout the book Burton pauses to give lengthy discourses on how the events of the time affected the development of mathematics. We learn about the transition of Christianity from persecuted sect to the official religion of Rome and how that led to a neglect (in the west) of the pagan Greek writings on mathematics. We are given a fascinating description of the Carolingian renaissance of the ninth century and how Charlemagne's system of monastery schools led to both the collection and preservation of libraries and to the design of an easy-to-read Roman script that is the precursor of our modern alphabet. We also are treated to a wonderful story of how the modern American mathematical community came into existence, deeply influenced by the "new" German university system of the nineteenth century (which was itself motivated by the Napoleonic defeat of the Prussian army in 1806). These historical discourses help the student to understand the environment of the time and also to relate the material to topics from their classes in European or world history.

A favorite subject of mine is the history of women in mathematics, and I'm always interested to see how textbooks cover this important topic. Burton includes descriptions of a number of lesser-known female mathematicians, such as Mary Somerville (who wrote many expository versions of 19th century scientific texts, among them Laplace's Mécanique Céleste), Winifred Edgerton (the first female recipient of an American Ph.D. and one of the founders of Barnard College) and Charlotte Scott (who was denied a degree by Cambridge, yet later directed several doctorates as chair of the mathematics department at Bryn Mawr). Many pages are devoted to the life and times of Emmy Noether and to her interactions with fellow mathematicians. I would have like to have seen a similar in-depth coverage of Sophie Germain. Unfortunately, her life and work are described in only a few paragraphs, and while brief mention is made of her interactions with Gauss, Burton left out Gauss' wonderfully chivalrous description of her after his discovery that his mysterious correspondent was a woman.

This sixth edition contains a new section on mathematical developments in China and Arabia. It troubles me to report that (in at least one area) this addition does not live up to the scholarship that one might expect.

Burton notes in this section that the Chinese mathematician Chu Shih-Chieh (a.k.a. Chu Shi-Kie) gave the first eight rows of Pascal's triangle in 1303, yet he fails to mention that this was done earlier by Yang Hui (around 1261), who himself gives credit to Jia Xian (mid 11th century). Two hundred pages later, he does mention Jia Xian (written Chia Hsia), but not Yang Hui, nor does he refer to his earlier discussion of this topic. When discussing the development of Pascal's triangle among Arabic mathematicians, Burton attributes al-Karajî (early 11th century) with the first description of what we now call the binomial coefficients. Unfortunately, Burton completely mangles the formula for generating these numbers, writing it as

Cnk = Cn-1n-k Cn-1k

instead of the correct (not to mention easier-to-read) form of

Cn,k = Cn-1,k-1 + Cn-1,k.

Then, when discussing Pascal's triangle later in the text, Burton ignores al-Karajî entirely and instead credits two other Arab mathematicians.

Would I recommend Burton's text for classroom use? It certainly has many nice features, and each section concludes with an appealing (albeit modest) collection of historically appropriate problems. While the section on Arab and Chinese mathematicians would benefit from some careful editing, I do not see this as being an unsurmountable problem. This book contains so much interesting material, presented in such a lively and entertaining manner, that I feel it would be an excellent reference, and it could also serve well as a primary text. I'll certainly be referring to my copy quite often when teaching my upcoming class on the history of mathematics.

Gregory P. Dresden is Associate Professor of Mathematics at Washington & Lee University in Lexington, VA.


Date Received: 
Monday, February 6, 2006
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David M. Burton
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Gregory P. Dresden
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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 and Monge
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


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Tuesday, July 14, 2009