- World Part 1. The World of Mathematics and the Mathematics of the
- Chapter 1. The Origin and Prehistory of Mathematics
- Numbers
- 1.1. Animals'use of numbers
- 1.2. Young children's use of numbers
- 1.3. Archaeological evidence of counting
- Numbers
- Continuous magnitudes
- 2.1. Perception of shape by animals
- 2.2. Children's concepts of space
- 2.3. Geometry in arts and crafts
- Symbols
- Mathematical inference
- 4.1. Visual reasoning
- 4.2. Chance and probability
- Questions and problems
- Continuous magnitudes
- Chapter 2. Mathematical Cultures I
- The motives for creating mathematics
- 1.1. Pure versus applied mathematics
- The motives for creating mathematics
- India
- 2.1. The Sulva Sutras
- 2.2. Buddhist and Jaina mathematics
- 2.3. The Bakshali Manuscript
- 2.4. The siddhantas
- 2.5. Aryabhata I
- 2.6. Brahmagupta
- 2.7. Bhaskara II
- 2.8. Muslim India
- 2.9. Indian mathematics in the colonial period and after
- China
- 3.1. Works and authors
- 3.2. China's encounter with Western mathematics
- China
- Ancient Egypt
- Mesopotamia
- The Maya
- 6.1. The Dresden Codex
- Questions and problems vi CONTENTS
- Chapter 1. The Origin and Prehistory of Mathematics
- Chapter 3. Mathematical Cultures II
- Greek and Roman mathematics
- 1.1. Sources
- 1.2. General features of Greek mathematics
- 1.3. Works and authors
- Greek and Roman mathematics
- Japan
- 2.1. Chinese influence and calculating devices
- 2.2. Japanese mathematicians and their works
- Japan
- The Muslims
- 3.1. Islamic science in general
- 3.2. Some Muslim mathematicians and their works
- The Muslims
- Europe
- 4.1. Monasteries, schools, and universities
- 4.2. The high Middle Ages
- 4.3. Authors and works
- Europe
- North America
- 5.1. The United States and Canada before
- 5.2. The Canadian Federation and post Civil War United States
- 5.3. Mexico
- 5.1. The United States and Canada before
- North America
- Australia and New Zealand
- 6.1. Colonial mathematics
- Australia and New Zealand
- The modern era
- 7.1. Educational institutions
- 7.2. Mathematical societies
- 7.3. Journals
- The modern era
- Questions and problems
- Chapter 4. Women Mathematicians
- Individual achievements and obstacles to achievement
- 1.1. Obstacles to mathematical careers for women
- Individual achievements and obstacles to achievement
- Ancient women mathematicians
- Modern European women
- 3.1. Continental mathematicians
- 3.2. Nineteenth-century British women
- 3.3. Four modern pioneers
- American women
- The situation today
- Questions and problems
- Modern European women
- Part 2. Numbers
- Chapter 5. Counting
- Number words
- Bases for counting
- 2.1. Decimal systems
- 2.2. Nondecimal systems
- Bases for counting
- Counting around the world
- 3.1. Egypt
- 3.2. Mesopotamia CONTENTS vii
- 3.3. India
- 3.4. China
- 3.5. Greece and Rome
- 3.6. The Maya
- Counting around the world
- What was counted?
- 4.1. Calendars
- 4.2. Weeks
- What was counted?
- Questions and problems
- Chapter 6. Calculation
- Egypt
- 1.1. Multiplication and division
- 1.2. "Parts"
- 1.3. Practical problems
- Egypt
- China
- 2.1. Fractions and roots
- 2.2. The Jiu Zhang Suanshu
- China
- India
- Mesopotamia
- The ancient Greeks
- The Islamic world
- Europe
- The value of calculation
- Mechanical methods of computation
- 9.1. Software: prosthaphaeresis and logarithms
- 9.2. Hardware: slide rules and calculating machines
- 9.3. The effects of computing power
- Mechanical methods of computation
- Questions and problems
- Chapter 7. Ancient Number Theory
- Plimpton
- Ancient Greek number theory
- 2.1. The Arithmetica of Nicomachus
- 2.2. Euclid's number theory
- 2.3. The Arithmetica of Diophantus
- China
- India
- 4.1. Varahamihira's mystical square
- 4.2. Aryabhata I
- 4.3. Brahmagupta
- 4.4. Bhaskara II
- India
- Ancient Greek number theory
- The Muslims
- Japan
- Medieval Europe
- Questions and problems
- Chapter 8. Numbers and Number Theory in Modern Mathematics
- Modern number theory
- 1.1. Fermat
- 1.2. Euler viii CONTENTS
- 1.3. Lagrange
- 1.4. Legendre
- 1.5. Gauss
- 1.6. Dirichlet
- 1.7. Riemann
- 1.8. Fermat's last theorem
- 1.9. The prime number theorem
- Modern number theory
- Number systems
- 2.1. Negative numbers and zero
- 2.2. Irrational and imaginary numbers
- 2.3. Imaginary and complex numbers
- 2.4. Infinite numbers
- Combinatorics
- 3.1. Summation rules
- Combinatorics
- Questions and problems
- Number systems
- Part 3. Color Plates
- Part 4. Space
- Chapter 9. Measurement
- Egypt
- 1.1. Areas
- 1.2. Volumes
- Egypt
- Mesopotamia
- 2.1. The Pythagorean theorem
- 2.2. Plane figures
- 2.3. Volumes
- Mesopotamia
- China
- 3.1. The Zhou Bi Suan Jing
- 3.2. The Jiu Zhang Suanshu
- 3.3. The Sun Zi Suan Jing
- 3.4. Liu Hui
- 3.5. Zu Chongzhi
- Japan
- 4.1. The challenge problems
- 4.2. Beginnings of the calculus in Japan
- Japan
- India
- 5.1. Aryabhata I
- 5.2. Brahmagupta
- India
- Questions and problems
- China
- Chapter 10. Euclidean Geometry
- The earliest Greek geometry
- 1.1. Thales
- 1.2. Pythagoras and the Pythagoreans
- 1.3. Pythagorean geometry
- 1.4. Challenges to Pythagoreanism: unsolved problems
- 1.6. Challenges to Pythagoreanism: incommensurables CONTENTS ix
- 1.7. The influence of Plato
- 1.8. Eudoxan geometry
- 1.9. Aristotle
- The earliest Greek geometry
- Euclid
- 2.1. The Elements
- 2.2. The Data
- Euclid
- Archimedes
- 3.1. The area of a sphere
- 3.2. The Method
- Archimedes
- Apollonius
- 4.1. History of the Conies
- 4.2. Contents of the Conies
- 4.3. Apollonius' definition of the conic sections
- 4.4. Foci and the three- and four-line locus
- Apollonius
- Questions and problems
- Chapter 11. Post-Euclidean Geometry
- Hellenistic geometry
- 1.1. Zenodorus
- 1.2. The parallel postulate
- 1.3. Heron
- 1.4. Pappus
- Hellenistic geometry
- Roman geometry
- 2.1. Roman civil engineering
- Roman geometry
- Medieval geometry
- 3.1. Late Medieval and Renaissance geometry
- Medieval geometry
- Geometry in the Muslim world
- 4.1. The parallel postulate
- 4.2. Thabit ibn-Qurra
- 4.3. Al-Kuhi
- 4.4. Al-Haytham
- 4.5. Omar Khayyam
- 4.6. Nasir al-Din al-Tusi
- Non-Euclidean geometry
- 5.1. Girolamo Saccheri
- 5.2. Lambert and Legendre
- 5.3. Gauss
- 5.4. Lobachevskii and Janos Bolyai
- 5.5. The reception of non-Euclidean geometry
- 5.6. Foundations of geometry
- Non-Euclidean geometry
- 4.1. The parallel postulate
- Questions and problems
- Geometry in the Muslim world
- Chapter 12. Modern Geometries
- Analytic and algebraic geometry
- 1.1. Fermat
- 1.2. Descartes
- 1.3. Newton's classification of curves
- 1.5. Challenges to Pythagoreanism: the paradoxes of Zeno of Elea
- 1.4. Algebraic geometry x CONTENTS
- Analytic and algebraic geometry
- Projective and descriptive geometry
- 2.1. Projective properties
- 2.2. The Renaissance artists
- 2.3. Girard Desargues
- 2.4. Blaise Pascal
- 2.5. Newton's degree-preserving mappings
- 2.6. Charles Brianchon
- 2.7. Monge and his school
- 2.8. Jacob Steiner
- 2.9. August Ferdinand Mobius
- 2.10. Julius Plucker
- 2.11. Arthur Cayley
- Projective and descriptive geometry
- Differential geometry
- 3.1. Huygens
- 3.2. Newton
- 3.3. Leibniz
- 3.4. The eighteenth century
- 3.5. Gauss
- 3.6. The French and British geometers
- 3.7. Riemann
- 3.8. The Italian geometers
- Differential geometry
- Topology
- 4.1. Early combinatorial topology
- 4.2. Riemann
- 4.3. Mobius
- 4.4. Poincare's Analysis situs
- 4.5. Point-set topology
- Topology
- Questions and problems
- Part 5. Algebra
- Chapter 13. Problems Leading to Algebra
- Egypt
- Mesopotamia
- 2.1. Linear and quadratic problems
- 2.2. Higher-degree problems
- India
- 3.1. Jaina algebra
- 3.2. The Bakshali Manuscript
- India
- Mesopotamia
- China
- 4.1. The Jiu Zhang Suanshu
- 4.2. The Suanshu Shu
- 4.3. The Sun Zi Suan Jing
- 4.4. Zhang Qiujian
- Questions and problems
- China
- Chapter 14. Equations and Algorithms
- The Arithmetica of Diophantus
- 1.1. Diophantine equations CONTENTS xi
- 1.2. General characteristics of the Arithmetica
- 1.3. Determinate problems
- 1.4. The significance of the Arithmetica
- 1.5. The view of Jacob Klein
- The Arithmetica of Diophantus
- China
- 2.1. Linear equations
- 2.2. Quadratic equations
- 2.3. Cubic equations
- 2.4. The numerical solution of equations
- 2.1. Linear equations
- China
- Japan
- 3.1. SekiK5wa
- Japan
- Hindu algebra
- 4.1. Brahmagupta
- 4.2. Bhaskara II
- 4.1. Brahmagupta
- Hindu algebra
- The Muslims
- 5.1. Al-Khwarizmi
- 5.2. Abu Kamil
- 5.3. Omar Khayyam
- 5.4. Sharaf al-Din al-Muzaffar al-Tusi
- The Muslims
- Europe
- 6.1. Leonardo of Pisa (Fibonacci)
- 6.2. Jordanus Nemorarius
- 6.3. The fourteenth and fifteenth centuries
- 6.4. Chuquet
- 6.5. Solution of cubic and quartic equations
- 6.6. Consolidation
- Europe
- Questions and problems
- Chapter 15. Modern Algebra
- Theory of equations
- 1.1. Albert Girard
- 1.2. Tschirnhaus transformations
- 1.3. Newton, Leibniz, and the Bernoullis
- 1.4. Euler, d'Alembert, and Lagrange
- 1.5. Gauss and the fundamental theorem of algebra
- 1.6. Ruffini
- 1.7. Cauchy
- 1.8. Abel
- 1.9. Galois
- Theory of equations
- Algebraic structures
- 2.1. Fields, rings, and algebras
- 2.2. Abstract groups
- 2.3. Number systems
- Questions and problems
- Algebraic structures
- Part 6. Analysis
- Chapter 16. The Calculus
- Prelude to the calculus xii CONTENTS
- 1.1. Tangent and maximum problems
- 1.2. Lengths, areas, and volumes
- 1.3. The relation between tangents and areas
- 1.4. Infinite series and products
- Prelude to the calculus xii CONTENTS
- Newton and Leibniz
- 2.1. Isaac Newton
- 2.2. Gottfried Wilhelm von Leibniz
- 2.3. The disciples of Newton and Leibniz
- Newton and Leibniz
- Branches and roots of the calculus
- 3.1. Ordinary differential equations
- 3.2. Partial differential equations
- 3.3. Calculus of variations
- 3.4. Foundations of the calculus
- Branches and roots of the calculus
- Questions and problems
- Chapter 17. Real and Complex Analysis
- Complex analysis
- 1.1. Algebraic integrals
- 1.2. Cauchy
- 1.3. Riemann
- 1.4. Weierstrass
- Complex analysis
- Real analysis
- 2.1. Fourier series, functions, and integrals
- 2.2. Completeness of the real numbers
- 2.3. Uniform convergence and continuity
- 2.4. General integrals and discontinuous functions
- 2.5. The abstract and the concrete
- 2.G. Discontinuity as a positive property
- Real analysis
- Questions and problems
- Chapter 16. The Calculus
- Part 7. Mathematical Inferences
- Chapter 18. Probability and Statistics
- Probability
- 1.1. Cardano
- 1.2. Fermat and Pascal
- 1.3. Huygens
- 1.4. Leibniz
- 1.5. The Ars Conjectandi of Jakob Bernoulli
- 1.6. De Moivre
- 1.7. Laplace
- 1.8. Legendre
- 1.9. Gauss
- 1.10. Philosophical issues
- 1.11. Large numbers and limit theorems
- Probability
- Statistics
- 2.1. Quetelet
- 2.2. Statistics in physics
- 2.3. The metaphysics of probability and statistics CONTENTS xi"
- 2.4. Correlations and statistical inference
- Statistics
- Questions and problems
- Chapter 19. Logic and Set Theory
- Logic
- 1.1. From algebra to logic
- 1.2. Symbolic calculus
- 1.3. Boole's Mathematical Analysis of Logic
- 1.4. Boole's Laws of Thought
- 1.5. Venn
- 1.6. Jevons
- Logic
- Set theory
- 2.1. Technical background
- 2.2. Cantor's work on trigonometric series
- 2.3. The reception of set theory
- 2.4. Existence and the axiom of choice
- 2.5. Doubts about set theory
- Philosophies of mathematics
- 3.1. Paradoxes
- 3.2. Formalism
- 3.3. Intuitionism
- 3.4. Mathematical practice
- Philosophies of mathematics
- Questions and problems
- Literature
- Subject Index
- Name Index
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