## Table of Contents

**Part I. Ancient Mathematics**

**1. Egypt and Mesopotamia**

1.1 Egypt

1.2 Mesopotamia

**2. The Beginnings of Mathematics in Greece**

2.1 The Earliest Greek Mathematics

2.2 The Time of Plato

2.3 Aristotle

**3. Euclid**

3.1 Introduction to the *Elements*

3.2 Book I and the Pythagorean Theorem

3.3 Book II and Geometric Algebra

3.4 Circles and the Pentagon

3.5 Ratio and Proportion

3.6 Number Theory

3.7 Irrational Magnitudes

3.8 Solid Geometry and the Method of Exhaustion

3.9 Euclid’s Data

**4. Archimedes and Apollonius**

4.1 Archimedes and Physics

4.2 Archimedes and Numerical Calculations

4.3 Archimedes and Geometry

4.4 Conic Sections Before Apollonius

4.5 *The Conics* of Apollonius

**5. Mathematical Methods in Hellenistic Times**

5.1 Astronomy Before Ptolemy

5.2 Ptolemy and *The Almagest*

5.3 Practical Mathematics

**6. The Final Chapter of Greek Mathematics**

6.1 Nichomachus and Elementary Number Theory

6.2 Diophantus and Greek Algebra

6.3 Pappus and Analysis

**Part II. Medieval Mathematics**

**7. Ancient and Medieval China**

7.1 Introduction to Mathematics in China

7.2 Calculations

7.3 Geometry

7.4 Solving Equations

7.5 Indeterminate Analysis

7.6 Transmission to and from China

**8. Ancient and Medieval India**

8.1 Introduction to Mathematics in India

8.2 Calculations

8.3 Geometry

8.4 Equation Solving

8.5 Indeterminate Analysis

8.6 Combinatorics

8.7 Trigonometry

8.8 Transmission to and from India

**9. The Mathematics of Islam**

9.1 Introduction to Mathematics in Islam

9.2 Decimal Arithmetic

9.3 Algebra

9.4 Combinatorics

9.5 Geometry

9.6 Trigonometry

9.7 Transmission of Islamic Mathematics

**10. Medieval Europe**

10.1 Introduction to the Mathematics of Medieval Europe

10.2 Geometry and Trigonometry

10.3 Combinatorics

10.4 Medieval Algebra

10.5 The Mathematics of Kinematics

**11. Mathematics Elsewhere**

11.1 Mathematics at the Turn of the Fourteenth Century

11.2 Mathematics in America, Africa, and the Pacific

**Part III. Early Modern Mathematics**

**12. Algebra in the Renaissance**

12.1 The Italian Abacists

12.2 Algebra in France, Germany, England, and Portugal

12.3 The Solution of the Cubic Equation

12.4 Viete, Algebraic Symbolism, and Analysis

12.5 Simon Stevin and Decimal Analysis

**13. Mathematical Methods in the Renaissance**

13.1 Perspective

13.2 Navigation and Geography

13.3 Astronomy and Trigonometry

13.4 Logarithms

13.5 Kinematics

**14. Geometry, Algebra and Probability in the Seventeenth Century**

14.1 The Theory of Equations

14.2 Analytic Geometry

14.3 Elementary Probability

14.4 Number Theory

14.5 Projective Geometry

**15. The Beginnings of Calculus**

15.1 Tangents and Extrema

15.2 Areas and Volumes

15.3 Rectification of Curves and the Fundamental Theorem

**16. Newton and Leibniz**

16.1 Isaac Newton

16.2 Gottfried Wilhelm Leibniz

16.3 First Calculus Texts

**Part IV. Modern Mathematics**

**17. Analysis in the Eighteenth Century**

17.1 Differential Equations

17.2 The Calculus of Several Variables

17.3 Calculus Texts

17.4 The Foundations of Calculus

**18. Probability and Statistics in the Eighteenth Century**

18.1 Theoretical Probability

18.2 Statistical Inference

18.3 Applications of Probability

**19. Algebra and Number Theory in the Eighteenth Century**

19.1 Algebra Texts

19.2 Advances in the Theory of Equations

19.3 Number Theory

19.4 Mathematics in the Americas

**20. Geometry in the Eighteenth Century**

20.1 Clairaut and the *Elements of Geometry*

20.2 The Parallel Postulate

20.3 Analytic and Differential Geometry

20.4 The Beginnings of Topology

20.5 The French Revolution and Mathematics Education

**21. Algebra and Number Theory in the Nineteenth Century**

21.1 Number Theory

21.2 Solving Algebraic Equations

21.3 Symbolic Algebra

21.4 Matrices and Systems of Linear Equations

21.5 Groups and Fields — The Beginning of Structure

**22. Analysis in the Nineteenth Century**

22.1 Rigor in Analysis

22.2 The Arithmetization of Analysis

22.3 Complex Analysis

22.4 Vector Analysis

**23. Probability and Statistics in the Nineteenth Century**

23.1 The Method of Least Squares and Probability Distributions

23.2 Statistics and the Social Sciences

23.3 Statistical Graphs

**24. Geometry in the Nineteenth Century**

24.1 Differential Geometry

24.2 Non-Euclidean Geometry

24.3 Projective Geometry

24.4 Graph Theory and the Four Color Problem

24.5 Geometry in *N* Dimensions

24.6 The Foundations of Geometry

**25. Aspects of the Twentieth Century**

25.1 Set Theory: Problems and Paradoxes

25.2 Topology

25.3 New Ideas in Algebra

25.4 The Statistical Revolution

25.5 Computers and Applications

25.6 Old Questions Answered