Chapter
Chapter 1: Mathematics and Its History
1.1. Two Ways to Look at the History of Mathematics
1.1.1. History, but Not Heritage
1.1.2. Our Mathematical Heritage
1.2. The Origin of Mathematics
1.2.3. Are Mathematical Ideas Innate?
1.2.6. The Components of Mathematics
1.3. The Philosophy of Mathematics
1.3.1. Mathematical Analysis of a Real-world Problem
1.4. Our Approach to the History of Mathematics
Chapter 2: Proto-mathematics
2.1.1. Animals’ Use of Numbers
2.1.2. Young Children’s Use of Numbers
2.1.3. Archaeological Evidence of Counting
2.2.1. Perception of Shape by Animals
2.2.2. Children’s Concepts of Space
2.2.3. Geometry in Arts and Crafts
2.4. Mathematical Reasoning
Part II: The Middle East, 2000–1500 Bce
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
3.4.1. Place-value Systems
3.4.2. The Sexagesimal Place-value System
3.4.3. Converting a Decimal Number to Sexagesimal
3.4.4. Irrational Square Roots
Chapter 4: Computations in Ancient Mesopotamia
4.2.1. Linear and Quadratic Problems
4.2.2. Higher-degree Problems
Chapter 5: Geometry in Mesopotamia
5.1. The Pythagorean Theorem
5.2.1. Mesopotamian Astronomy
5.4.1. The Purpose of Plimpton 322: Some Conjectures
Chapter 6: Egyptian Numerals and Arithmetic
6.1.1. Mathematics in Hieroglyphics and Hieratic
6.4.1. Multiplication and Division
Chapter 7: Algebra and Geometry in Ancient Egypt
7.1. Algebra Problems in the Rhind Papyrus
7.1.1. Applied Problems: the
7.3.1. Rectangles, Triangles, and Trapezoids
7.3.4. The Pythagorean Theorem
7.3.5. Spheres or Cylinders?
Part III: Greek Mathematics from 500 Bce to 500 Ce
Chapter 8: an Overview of Ancient Greek Mathematics
8.2. General Features of Greek Mathematics
8.2.2. Mathematical Aspects of Plato’s Philosophy
Chapter 9: Greek Number Theory
9.1. The Euclidean Algorithm
9.2. The Arithmetica of Nicomachus
9.2.1. Factors Vs. Parts. Perfect Numbers
9.3. Euclid’s Number Theory
9.4. The Arithmetica of Diophantus
9.4.1. Algebraic Symbolism
9.4.2. Contents of the Arithmetica
9.4.3. Fermat’s Last Theorem
Chapter 10: Fifth-century Greek Geometry
10.1. “pythagorean” Geometry
10.1.1. Transformation and Application of Areas
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
10.4.1. The Arithmetical Origin of Irrationals
10.4.2. The Geometric Origin of Irrationals
10.4.3. Consequences of the Discovery
Chapter 11: Athenian Mathematics I: the Classical Problems
11.1. Squaring the Circle
11.3. Trisecting the Angle
11.3.1. A Mechanical Solution: the Conchoid
Chapter 12: Athenian Mathematics Ii: Plato and Aristotle
12.1. The Influence of Plato
12.2.1. The Eudoxan Definition of Proportion
12.2.2. The Method of Exhaustion
12.2.3. Ratios in Greek Geometry
Chapter 13: Euclid of Alexandria
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
14.4.1. The Mechanical Quadrature
14.4.2. The Rigorous Quadrature
Chapter 15: Apollonius of Perga
15.1. History of the Conics
15.2. Contents of the Conics
15.2.1. Properties of the Conic Sections
15.3. Foci and the Three and Four-line Locus
Chapter 16: Hellenistic and Roman Geometry
16.2. The Parallel Postulate
16.4. Roman Civil Engineering
Chapter 17: Ptolemy’s Geography and Astronomy
17.2.1. Epicycles and Eccentrics
17.2.2. The Motion of the Sun
17.3.2. Ptolemy’s Table of Chords
Chapter 18: Pappus and the Later Commentators
18.1. The Collection of Pappus
18.1.1. Generalization of the Pythagorean Theorem
18.1.2. the Isoperimetric Problem
18.1.3. Analysis, Locus Problems, and Pappus’ Theorem
18.2. The Later Commentators: Theon and Hypatia
18.2.1. Theon of Alexandria
18.2.2. Hypatia of Alexandria
Part IV: India, China, and Japan 500 Bce–1700 Ce
Chapter 19: Overview of Mathematics in India
19.2. Buddhist and Jain Mathematics
19.3. The Bakshali Manuscript
19.5. Hindu–arabic Numerals
19.10. Indian Mathematics in the Colonial Period and After
19.10.1. Srinivasa Ramanujan
Chapter 20: from the Vedas to Aryabhata I
20.1. Problems from the Sulva Sutras
20.1.4. Jain Mathematics: the Infinite
20.1.5. Jain Mathematics: Combinatorics
20.1.6. The Bakshali Manuscript
20.2. Aryabhata I: Geometry and Trigonometry
Chapter 21: Brahmagupta, the Kuttaka, and Bhaskara II
21.1. Brahmagupta’s Plane and Solid Geometry
21.2. Brahmagupta’s Number Theory and Algebra
21.2.1. Pythagorean Triples
21.4. Algebra in the Works of Bhaskara II
21.4.1. The Vija Ganita (algebra)
21.5. Geometry in the Works of Bhaskara Ii
Chapter 22: Early Classics of Chinese Mathematics
22.1.1. The Zhou Bi Suan Jing
22.1.2. The Jiu Zhang Suan Shu
22.1.3. The Sun Zi Suan Jing
22.1.4. Liu Hui. the Hai Dao Suan Jing
22.1.5. Zu Chongzhi and Zu Geng
22.2. China’s Encounter with Western Mathematics
22.3. The Chinese Number System
22.3.1. Fractions and Roots
22.5. Contents of the Jiu Zhang Suan Shu
22.6. Early Chinese Geometry
22.6.1. The Zhou Bi Suan Jing
22.6.2. The Jiu Zhang Suan Shu
22.6.3. The Sun Zi Suan Jing
Chapter 23: Later Chinese Algebra and Geometry
23.1.1. Systems of Linear Equations
23.1.2. Quadratic Equations
23.1.4. a Digression on the Numerical Solution of Equations
23.2. Later Chinese Geometry
Chapter 24: Traditional Japanese Mathematics
24.1. Chinese Influence and Calculating Devices
24.2. Japanese Mathematicians and Their Works
24.2.2. Seki Kowa and Takebe Kenko
24.2.3. The Modern Era in Japan
24.3. Japanese Geometry and Algebra
24.3.2. The Challenge Problems
24.3.3. Beginnings of the Calculus in Japan
Part V: Islamic Mathematics, 800–1500
Chapter 25: Overview of Islamic Mathematics
25.1. a Brief Sketch of the Islamic Civilization
25.1.3. The Turkish and Mongol Conquests
25.1.4. The Islamic Influence on Science
25.2. Islamic Science in General
25.2.1. Hindu and Hellenistic Influences
25.3. Some Muslim Mathematicians and Their Works
25.3.1. Muhammad Ibn Musa Al-khwarizmi
Chapter 26: Islamic Number Theory and Algebra
26.2.4. Sharaf Al-din Al-tusi
Chapter 27: Islamic Geometry
27.1. The Parallel Postulate
27.3. Al-biruni: Trigonometry
27.5. Al-haytham and Ibn-sahl
27.7. Nasir Al-din Al-tusi
Part VI: European Mathematics, 500–1900
Chapter 28: Medieval and Early Modern Europe
28.1. from the Fall of Rome to the Year 1200
28.1.1. Boethius and the Quadrivium
28.1.2. Arithmetic and Geometry
28.1.3. Music and Astronomy
28.1.4. The Carolingian Empire
28.1.6. Early Medieval Geometry
28.2. The High Middle Ages
28.2.2. Jordanus Nemorarius
28.2.7. Leon Battista Alberti
28.3. The Early Modern Period
28.3.1. Scipione Del Ferro
28.3.2. Niccolò Tartaglia
28.4. Northern European Advances
Chapter 29: European Mathematics: 1200–1500
29.1. Leonardo of Pisa (fibonacci)
29.1.2. The Fibonacci Sequence
29.1.3. The Liber Quadratorum
29.2. Hindu–arabic Numerals
29.3. Jordanus Nemorarius
29.5. Trigonometry: Regiomontanus and Pitiscus
29.6. A Mathematical Skill: Prosthaphæresis
29.7. Algebra: Pacioli and Chuquet
Chapter 30: Sixteenth-century Algebra
30.1. Solution of Cubic and Quartic Equations
30.3.1. Arithmetical Implementation of the Geometric Model
30.4. Hardware: Slide Rules and Calculating Machines
30.4.2. Calculating Machines
Chapter 31: Renaissance Art and Geometry
31.1. The Greek Foundations
31.2. The Renaissance Artists and Geometers
31.3. Projective Properties
Chapter 32: The Calculus Before Newton and Leibniz
32.2. Components of the Calculus
32.2.1. Tangent and Maximum Problems
32.2.2. Lengths, Areas, and Volumes
32.2.3. Bonaventura Cavalieri
32.2.4. Gilles Personne De Roberval
32.2.5. Rectangular Approximations and the Method of Exhaustion
32.2.7. The Relation Between Tangents and Areas
32.2.8. Infinite Series and Products
32.2.9. The Binomial Series
Chapter 33: Newton and Leibniz
33.1.1. Newton’s First Version of the Calculus
33.1.2. Fluxions and Fluents
33.1.3. Later Exposition of the Calculus
33.2. Gottfried Wilhelm Von Leibniz
33.2.1. Leibniz’ Presentation of the Calculus
33.2.2. Later Reflections on the Calculus
33.3. The Disciples of Newton and Leibniz
33.4. Philosophical Issues
33.4.1. The Debate on the Continent
33.5. The Priority Dispute
33.6. Early Textbooks on Calculus
33.6.1. The State of the Calculus Around 1700
Chapter 34: Consolidation of the Calculus
34.1. Ordinary Differential Equations
34.1.1. a Digression on Time
34.2. Partial Differential Equations
34.3. Calculus of Variations
34.3.3. Second-variation Tests for Maxima and Minima
34.3.4. Jacobi: Sufficiency Criteria
34.3.5. Weierstrass and His School
34.4. Foundations of the Calculus
34.4.1. Lagrange’s Algebraic Analysis
34.4.2. Cauchy’s Calculus
Chapter 35: Women Mathematicians
35.1. Sof’ya Kovalevskaya
35.1.1. Resistance from Conservatives
35.2. Grace Chisholm Young
36.5. The Ars Conjectandi of James Bernoulli
36.5.1. The Law of Large Numbers
36.7. The Petersburg Paradox
36.11. Philosophical Issues
36.12. Large Numbers and Limit Theorems
Chapter 37: Algebra from 1600 to 1850
37.1. Theory of Equations
37.1.2. Tschirnhaus Transformations
37.1.3. Newton, Leibniz, and the Bernoullis
37.2. Euler, D’alembert, and Lagrange
37.3. The Fundamental Theorem of Algebra and Solution by Radicals
Chapter 38: Projective and Algebraic Geometry and Topology
38.1. Projective Geometry
38.1.1. Newton’s Degree-preserving Mapping
38.1.3. Monge and His School
38.3.1. Combinatorial Topology
38.3.4. Poincaré’s Analysis Situs
38.3.5. Point-set Topology
Chapter 39: Differential Geometry
39.2. the Eighteenth Century: Surfaces
39.3. Space Curves: the French Geometers
39.4. Gauss: Geodesics and Developable Surfaces
39.4.1. Further Work by Gauss
39.5. The French and British Geometers
39.6. Grassmann and Riemann: Manifolds
39.7. Differential Geometry and Physics
39.8. The Italian Geometers
39.8.1. Ricci’s Absolute Differential Calculus
Chapter 40: Non-euclidean Geometry
40.2. Lambert and Legendre
40.4. The First Treatises
40.5. Lobachevskii’s Geometry
40.7. The Reception of Non-euclidean Geometry
40.8. Foundations of Geometry
Chapter 41: Complex Analysis
41.1. Imaginary and Complex Numbers
41.2. Analytic Function Theory
41.2.1. Algebraic Integrals
41.2.2. Legendre, Jacobi, and Abel
41.3. Comparison of the Three Approaches
Chapter 42: Real Numbers, Series, and Integrals
42.1. Fourier Series, Functions, and Integrals
42.1.1. the Definition of a Function
42.2.1. Sturm–liouville Problems
42.4. General Trigonometric Series
Chapter 43: Foundations of Real Analysis
43.1. What Is a Real Number?
43.1.1. The Arithmetization of the Real Numbers
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
43.5.1. Absolute Continuity
43.5.2. Taming the Abstract
43.6. Discontinuity as a Positive Property
44.1. Technical Background
44.2. Cantor’s Work on Trigonometric Series
44.3. the Reception of Set Theory
44.3.1. Cantor and Kronecker
44.4. Existence and the Axiom of Choice
45.1. from Algebra to Logic
45.3. Boole’s Mathematical Analysis of Logic
45.3.1. Logic and Classes
45.4. Boole’s Laws of Thought
45.6. Philosophies of Mathematics
45.6.4. Mathematical Practice
45.7. Doubts About Formalized Mathematics: Gödel’s Theorems
The History of Mathematics
：A Brief Course
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