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Index
Cover
Title Page
Copyright
Preface
Changes from the Second Edition
Elementary Texts on the History of Mathematics
Part I: What is Mathematics?
Contents of Part I
Chapter 1: Mathematics and its History
1.1 Two Ways to Look at the History of Mathematics
1.2 The Origin of Mathematics
1.3 The Philosophy of Mathematics
1.4 Our Approach to the History of Mathematics
Questions for Reflection
Chapter 2: Proto-mathematics
2.1 Number
2.2 Shape
2.3 Symbols
2.4 Mathematical Reasoning
Problems and Questions
Part II: The Middle East, 2000–1500 BCE
Contents of Part II
Chapter 3: Overview of Mesopotamian Mathematics
3.1 A Sketch of Two Millennia of Mesopotamian History
3.2 Mathematical Cuneiform Tablets
3.3 Systems of Measuring and Counting
3.4 The Mesopotamian Numbering System
Problems and Questions
Chapter 4: Computations in Ancient Mesopotamia
4.1 Arithmetic
4.2 Algebra
Problems and Questions
Chapter 5: Geometry in Mesopotamia
5.1 The Pythagorean Theorem
5.2 Plane Figures
5.3 Volumes
5.4 Plimpton 322
Problems and Questions
Chapter 6: Egyptian Numerals and Arithmetic
6.1 Sources
6.2 The Rhind Papyrus
6.3 Egyptian Arithmetic
6.4 Computation
Problems and Questions
Chapter 7: Algebra and Geometry in Ancient Egypt
7.1 Algebra Problems in the Rhind Papyrus
7.2 Geometry
7.3 Areas
Problems and Questions
Part III: Greek Mathematics From 500 BCE to 500 CE
Contents of Part III
Chapter 8: An Overview of Ancient Greek Mathematics
8.1 Sources
8.2 General Features of Greek Mathematics
8.3 Works and Authors
Questions
Chapter 9: Greek Number Theory
9.1 The Euclidean Algorithm
9.2 The Arithmetica of Nicomachus
9.3 Euclid's Number Theory
9.4 The Arithmetica of Diophantus
Problems and Questions
Chapter 10: Fifth-Century Greek Geometry
10.1 “Pythagorean” Geometry
10.2 Challenge No. 1: Unsolved Problems
10.3 Challenge No. 2: The Paradoxes of Zeno of Elea
10.4 Challenge No. 3: Irrational Numbers and Incommensurable Lines
Problems and Questions
Chapter 11: Athenian Mathematics I: The Classical Problems
11.1 Squaring the Circle
11.2 Doubling the Cube
11.3 Trisecting the Angle
Problems and Questions
Chapter 12: Athenian Mathematics II: Plato and Aristotle
12.1 The Influence of Plato
12.2 Eudoxan Geometry
12.3 Aristotle
Problems and Questions
Chapter 13: Euclid of Alexandria
13.1 The Elements
13.2 The Data
Problems and Questions
Chapter 14: Archimedes of Syracuse
14.1 The Works of Archimedes
14.2 The Surface of a Sphere
14.3 The Archimedes Palimpsest
14.4 Quadrature of the Parabola
Problems and Questions
Chapter 15: Apollonius of Perga
15.1 History of the Conics
15.2 Contents of the Conics
15.3 Foci and the Three-and Four-line Locus
Problems and Questions
Chapter 16: Hellenistic and Roman Geometry
16.1 Zenodorus
16.2 The Parallel Postulate
16.3 Heron
16.4 Roman Civil Engineering
Problems and Questions
Chapter 17: Ptolemy's Geography and Astronomy
17.1 Geography
17.2 Astronomy
17.3 The Almagest
Problems and Questions
Part IV: India, China, and Japan 500 BCE–1700 CE
Contents of Part IV
Chapter 18: Pappus and the Later Commentators
18.1 The Collection of Pappus
18.2 The Later Commentators: Theon and Hypatia
Problems and Questions
Chapter 19: Overview of Mathematics in India
19.1 The Sulva Sutras
19.2 Buddhist and Jain Mathematics
19.3 The Bakshali Manuscript
19.4 The Siddhantas
19.5 Hindu–Arabic Numerals
19.6 Aryabhata I
19.7 Brahmagupta
19.8 Bhaskara II
19.9 Muslim India
19.10 Indian Mathematics in the Colonial Period and After
Questions
Chapter 20: From the Vedas to Aryabhata I
20.1 Problems from the Sulva Sutras
20.2 Aryabhata I: Geometry and Trigonometry
Problems and Questions
Chapter 21: Brahmagupta, the Kuttaka, and Bhaskara II
21.1 Brahmagupta's Plane and Solid Geometry
21.2 Brahmagupta's Number Theory and Algebra
21.3 The Kuttaka
21.4 Algebra in the Works of Bhaskara II
21.5 Geometry in the Works of Bhaskara II
Problems and Questions
Chapter 22: Early Classics of Chinese Mathematics
22.1 Works and Authors
22.2 China's Encounter with Western Mathematics
22.3 The Chinese Number System
22.4 Algebra
22.5 Contents of the Jiu Zhang Suan Shu
22.6 Early Chinese Geometry
Problems and Questions
Chapter 23: Later Chinese Algebra and Geometry
23.1 Algebra
23.2 Later Chinese Geometry
Problems and Questions
Chapter 24: Traditional Japanese Mathematics
24.1 Chinese Influence and Calculating Devices
24.2 Japanese Mathematicians and Their Works
24.3 Japanese Geometry and Algebra
24.4 Sangaku
Problems and Questions
Part V: Islamic Mathematics, 800–1500
Contents of Part V
Chapter 25: Overview of Islamic Mathematics
25.1 A Brief Sketch of the Islamic Civilization
25.2 Islamic Science in General
25.3 Some Muslim Mathematicians and their Works
Questions
Chapter 26: Islamic Number Theory and Algebra
26.1 Number Theory
26.2 Algebra
Problems and Questions
Chapter 27: Islamic Geometry
27.1 The Parallel Postulate
27.2 Thabit ibn-Qurra
27.3 Al-Biruni: Trigonometry
27.4 Al-Kuhi
27.5 Al-Haytham and Ibn-Sahl
27.6 Omar Khayyam
27.7 Nasir al-Din al-Tusi
Problems and Questions
Part VI: European Mathematics, 500–1900
Contents of Part VI
Chapter 28: Medieval and Early Modern Europe
28.1 From the Fall of Rome to the Year 1200
28.2 The High Middle Ages
28.3 The Early Modern Period
28.4 Northern European Advances
Questions
Chapter 29: European Mathematics: 1200–1500
29.1 Leonardo of Pisa (Fibonacci)
29.2 Hindu–Arabic numerals
29.3 Jordanus Nemorarius
29.4 Nicole d'Oresme
29.5 Trigonometry: Regiomontanus and Pitiscus
29.6 A Mathematical Skill: ProsthaphÆresis
29.7 Algebra: Pacioli and Chuquet
Problems and Questions
Chapter 30: Sixteenth-Century Algebra
30.1 Solution of Cubic and Quartic Equations
30.2 Consolidation
30.3 Logarithms
30.4 Hardware: slide rules and calculating machines
Problems and Questions
Chapter 31: Renaissance Art and Geometry
31.1 The Greek Foundations
31.2 The Renaissance Artists and Geometers
31.3 Projective Properties
Problems and Questions
Chapter 32: The Calculus Before Newton and Leibniz
32.1 Analytic Geometry
32.2 Components of the Calculus
Problems and Questions
Chapter 33: Newton and Leibniz
33.1 Isaac Newton
33.2 Gottfried Wilhelm von Leibniz
33.3 The Disciples of Newton and Leibniz
33.4 Philosophical Issues
33.5 The Priority Dispute
33.6 Early Textbooks on Calculus
Problems and Questions
Chapter 34: Consolidation of the Calculus
34.1 Ordinary Differential Equations
34.2 Partial Differential Equations
34.3 Calculus of Variations
34.4 Foundations of the Calculus
Problems and Questions
Part VII: Special Topics
Contents of Part VII
Chapter 35: Women Mathematicians
35.1 Sof'ya Kovalevskaya
35.2 Grace Chisholm Young
35.3 Emmy Noether
Questions
Chapter 36: Probability
36.1 Cardano
36.2 Fermat and Pascal
36.3 Huygens
36.4 Leibniz
36.5 The Ars Conjectandi of James Bernoulli
36.6 De Moivre
36.7 The Petersburg Paradox
36.8 Laplace
36.9 Legendre
36.10 Gauss
36.11 Philosophical Issues
36.12 Large Numbers and Limit Theorems
Problems and Questions
Chapter 37: Algebra from 1600 to 1850
37.1 Theory of Equations
37.2 Euler, D'Alembert, and Lagrange
37.3 The Fundamental Theorem of Algebra and Solution by Radicals
Problems and Questions
Chapter 38: Projective and Algebraic Geometry and Topology
38.1 Projective Geometry
38.2 Algebraic Geometry
38.3 Topology
Problems and Questions
Chapter 39: Differential Geometry
39.1 Plane Curves
39.2 The Eighteenth Century: Surfaces
39.3 Space Curves: The French Geometers
39.4 Gauss: Geodesics and Developable Surfaces
39.5 The French and British Geometers
39.6 Grassmann and Riemann: Manifolds
39.7 Differential Geometry and Physics
39.8 The Italian Geometers
Problems and Questions
Chapter 40: Non-Euclidean Geometry
40.1 Saccheri
40.2 Lambert and Legendre
40.3 Gauss
40.4 The First Treatises
40.5 Lobachevskii's Geometry
40.6 János Bólyai
40.7 The Reception of Non-Euclidean Geometry
40.8 Foundations of Geometry
Problems and Questions
Chapter 41: Complex Analysis
41.1 Imaginary and Complex Numbers
41.2 Analytic Function Theory
41.3 Comparison of the Three Approaches
Problems and Questions
Chapter 42: Real Numbers, Series, and Integrals
42.1 Fourier Series, Functions, and Integrals
42.2 Fourier Series
42.3 Fourier Integrals
42.4 General Trigonometric Series
Problems and Questions
Chapter 43: Foundations of Real Analysis
43.1 What Is a Real Number?
43.2 Completeness of the Real Numbers
43.3 Uniform Convergence and Continuity
43.4 General Integrals and Discontinuous Functions
43.5 The Abstract and the Concrete
43.6 Discontinuity as a Positive Property
Problems and Questions
Chapter 44: Set Theory
44.1 Technical Background
44.2 Cantor's Work on Trigonometric Series
44.3 The Reception of Set Theory
44.4 Existence and the Axiom of Choice
Problems and Questions
Chapter 45: Logic
45.1 From Algebra to Logic
45.2 Symbolic Calculus
45.3 Boole's Mathematical Analysis of Logic
45.4 Boole's Laws of Thought
45.5 Jevons
45.6 Philosophies of Mathematics
45.7 Doubts about Formalized Mathematics: Gödel's Theorems
Problems and Questions
Literature
Name Index
Subject Index
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