Contents

Foreword to the Second Edition

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Chapter 1: Traces

Concepts and Relationships

Early Number Bases

Number Language and Counting

Spatial Relationships

Chapter 2: Ancient Egypt

The Era and the Sources

Numbers and Fractions

Arithmetic Operations

“Heap” Problems

Geometric Problems

Slope Problems

Arithmetic Pragmatism

Chapter 3: Mesopotamia

The Era and the Sources

Cuneiform Writing

Numbers and Fractions: Sexagesimals

Positional Numeration

Sexagesimal Fractions

Approximations

Tables

Equations

Measurements: Pythagorean Triads

Polygonal Areas

Geometry as Applied Arithmetic

Chapter 4: Hellenic Traditions

The Era and the Sources

Thales and Pythagoras

Numeration

Arithmetic and Logistic

Fifth-Century Athens

Three Classical Problems

Quadrature of Lunes

Hippias of Elis

Philolaus and Archytas of Tarentum

Incommensurability

Paradoxes of Zeno

Deductive Reasoning

Democritus of Abdera

Mathematics and the Liberal Arts

The Academy

Aristotle

Chapter 5: Euclid of Alexandria

Alexandria

Lost Works

Extant Works

The Elements

Chapter 6: Archimedes of Syracuse

The Siege of Syracuse

On the Equilibriums of Planes

On Floating Bodies

The Sand-Reckoner

Measurement of the Circle

On Spirals

Quadrature of the Parabola

On Conoids and Spheroids

On the Sphere and Cylinder

Book of Lemmas

Semiregular Solids and Trigonometry

The Method

Chapter 7: Apollonius of Perge

Works and Tradition

Lost Works

Cycles and Epicycles

The Conics

Chapter 8: Crosscurrents

Changing Trends

Eratosthenes

Angles and Chords

Ptolemy’s Almagest

Heron of Alexandria

The Decline of Greek Mathematics

Nicomachus of Gerasa

Diophantus of Alexandria

Pappus of Alexandria

The End of Alexandrian Dominance

Proclus of Alexandria

Boethius

Athenian Fragments

Byzantine Mathematicians

Chapter 9: Ancient and Medieval China

The Oldest Known Texts

The Nine Chapters

Rod Numerals

The Abacus and Decimal Fractions

Values of Pi

Thirteenth-Century Mathematics

Chapter 10: Ancient and Medieval India

Early Mathematics in India

The Sulbasutras

The Siddhantas

Aryabhata

Numerals

Trigonometry

Multiplication

Long Division

Brahmagupta

Indeterminate Equations

Bhaskara

Madhava and the Keralese School

Chapter 11: The Islamic Hegemony

Arabic Conquests

The House of Wisdom

Al-Khwarizmi

‘Abd Al-Hamid ibn-Turk

Thabit ibn-Qurra

Numerals

Trigonometry

Tenth- and Eleventh-Century Highlights

Omar Khayyam

The Parallel Postulate

Nasir al-Din al-Tusi

Al-Kashi

Chapter 12: The Latin West

Introduction

Compendia of the Dark Ages

Gerbert

The Century of Translation

Abacists and Algorists

Fibonacci

Jordanus Nemorarius

Campanus of Novara

Learning in the Thirteenth Century

Archimedes Revived

Medieval Kinematics

Thomas Bradwardine

Nicole Oresme

The Latitude of Forms

Infinite Series

Levi ben Gerson

Nicholas of Cusa

The Decline of Medieval Learning

Chapter 13: The European Renaissance

Overview

Regiomontanus

Nicolas Chuquet’s Triparty

Luca Pacioli’s Summa

German Algebras and Arithmetics

Cardan’s Ars Magna

Rafael Bombelli

Robert Recorde

Trigonometry

Geometry

Renaissance Trends

François Viète

Chapter 14: Early Modern Problem Solvers

Accessibility of Computation

Decimal Fractions

Notation

Logarithms

Mathematical Instruments

Infinitesimal Methods: Stevin

Johannes Kepler

Chapter 15: Analysis, Synthesis, the Infinite, and Numbers

Galileo’s Two New Sciences

Bonaventura Cavalieri

Evangelista Torricelli

Mersenne’s Communicants

René Descartes

Fermat’s Loci

Gregory of St. Vincent

The Theory of Numbers

Gilles Persone de Roberval

Girard Desargues and Projective Geometry

Blaise Pascal

Philippe de Lahire

Georg Mohr

Pietro Mengoli

Frans van Schooten

Jan de Witt

Johann Hudde

René François de Sluse

Christiaan Huygens

Chapter 16: British Techniques and Continental Methods

John Wallis

James Gregory

Nicolaus Mercator and William Brouncker

Barrow’s Method of Tangents

Newton

Abraham De Moivre

Roger Cotes

James Stirling

Colin Maclaurin

Textbooks

Rigor and Progress

Leibniz

The Bernoulli Family

Tschirnhaus Transformations

Solid Analytic Geometry

Michel Rolle and Pierre Varignon

The Clairauts

Mathematics in Italy

The Parallel Postulate

Divergent Series

Chapter 17: Euler

The Life of Euler

Notation

Foundation of Analysis

Logarithms and the Euler Identities

Differential Equations

Probability

The Theory of Numbers

Textbooks

Analytic Geometry

The Parallel Postulate: Lambert

Chapter 18: Pre- to Postrevolutionary France

Men and Institutions

The Committee on Weights and Measures

D’Alembert

Bézout

Condorcet

Lagrange

Monge

Carnot

Laplace

Legendre

Aspects of Abstraction

Paris in the 1820s

Fourier

Cauchy

Diffusion

Chapter 19: Gauss

Nineteenth-Century Overview

Gauss: Early Work

Number Theory

Reception of the Disquisitiones Arithmeticae

Astronomy

Gauss’s Middle Years

Differential Geometry

Gauss’s Later Work

Gauss’s Influence

Chapter 20: Geometry

The School of Monge

Projective Geometry: Poncelet and Chasles

Synthetic Metric Geometry: Steiner

Synthetic Nonmetric Geometry: von Staudt

Analytic Geometry

Non-Euclidean Geometry

Riemannian Geometry

Spaces of Higher Dimensions

Felix Klein

Post-Riemannian Algebraic Geometry

Chapter 21: Algebra

Introduction

British Algebra and the Operational Calculus of Functions

Boole and the Algebra of Logic

Augustus De Morgan

William Rowan Hamilton

Grassmann and Ausdehnungslehre

Cayley and Sylvester

Linear Associative Algebras

Algebraic Geometry

Algebraic and Arithmetic Integers

Axioms of Arithmetic

Chapter 22: Analysis

Berlin and Göttingen at Midcentury

Riemann in Göttingen

Mathematical Physics in Germany

Mathematical Physics in English-Speaking Countries

Weierstrass and Students

The Arithmetization of Analysis

Dedekind

Cantor and Kronecker

Analysis in France

Chapter 23: Twentieth-Century Legacies

Overview

Henri Poincaré

David Hilbert

Integration and Measure

Functional Analysis and General Topology

Algebra

Differential Geometry and Tensor Analysis

Probability

Bounds and Approximations

The 1930s and World War II

Nicolas Bourbaki

Homological Algebra and Category Theory

Algebraic Geometry

Logic and Computing

The Fields Medals

Chapter 24: Recent Trends

Overview

The Four-Color Conjecture

Classification of Finite Simple Groups

Fermat’s Last Theorem

Poincaré’s Query

Future Outlook

References

General Bibliography

Index