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Index
Cover Contents Title Copyright Dedication 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
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