History of Mathematics, A

Description

A History of Mathematics, Third Edition, provides students with a solid background in the history of mathematics and focuses on the most important topics for today’s elementary, high school, and college curricula. Students will gain a deeper understanding of mathematical concepts in their historical context, and future teachers will find this book a valuable resource in developing lesson plans based on the history of each topic.

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

Features

• The flexible presentation organizes the book by chronological period and then by topic, which gives instructors the option of following a specific theme throughout the course.
• Discussions of the important textbooks of major time periods show students how topics were historically treated, allowing students to draw connections to modern approaches.
• A global perspective integrates non-Western coverage, including contributions from Chinese, Indian, and Islamic mathematicians.  An additional chapter discusses the mathematical achievements of early Africa, America, and Asia.
• Chapter openers include a vignette and quotation to add motivation and human interest.
• Focus essays are boxed features that are set apart from the main narrative of the text for easy reference. Biographies outline the lives and achievements of notable mathematicians. Other essays explore special topics, such as Egyptian influence on Greek mathematics.
• A chronology of major mathematicians at the end of every chapter gives an overview of important individuals and their contribution to the field of mathematics.
• Problems from primary sources enable students to understand how mathematicians were able to solve problems at various times and places.
• Discussion questions promote group work and can be used by future teachers to design lessons for elementary and secondary level math classes.
• An annotated bibliography at the end of each chapter provides easy reference to primary and secondary sources for research and further study.
• A phonetic pronunciation guide is included to aid in the pronunciation of historical names and places.

Author
Victor J. Katz received his PhD in mathematics from Brandeis University in 1968 and has been Professor of Mathematics at the University of the District of Columbia for many years. He has long been interested in the history of mathematics and, in particular, in its use in teaching. He is the editor of The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook (2007). He has edited or co-edited two recent books dealing with this subject, Learn from the Masters (1994) and Using History to Teach Mathematics (2000). Dr. Katz also co-edited a collection of historical articles taken from MAA journals of the past 90 years, Sherlock Holmes in Babylon and other Tales of Mathematical History. He has directed two NSF-sponsored projects to help college teachers learn the history of mathematics and learn to use it in teaching. Dr. Katz has also involved secondary school teachers in writing materials using history in the teaching of various topics in the high school curriculum. These materials, Historical Modules for the Teaching and Learning of Mathematics, have now been published by the MAA. Currently, Dr. Katz is the PI on an NSF grant to the MAA that supports Convergence, an online magazine devoted to the history of mathematics and its use in teaching.