You are here

A History of Mathematics: An Introduction

Victor J. Katz
Publication Date: 
Number of Pages: 
[Reviewed by
Jacqueline Stedall
, on

Victor Katz’s A History of Mathematics: An Introduction is already well known as a comprehensive textbook in the history of mathematics, courageously covering material from ‘ancient civilizations’ to ‘computers and their applications’ in just under 900 pages. In content the third edition (2009) remains essentially the same as the second edition (1998) but it has also been revised and updated. What is new?

Most significantly, since working on the previous edition Katz has edited The Mathematics of Egypt, Mesopotamia, China, India, and Islam, and this has led to substantial changes to A History of Mathematics, with new sections on each of those five regions. Thus new scholarship on the ancient and medieval world has been rapidly introduced into a widely used textbook. In the better known field of Greek mathematics, Euclid’s Elements, so essential to an understanding of later European mathematics, now benefits from a longer treatment than before, and there is also new discussion of the Archimedes palimpsest and the discoveries that have arisen from it.

For later periods of history, where the focus moves to western Europe, there is are helpful separations of material so that, for example, Viète and Stevin, who were so very different in motivation and output, now each have their own subsections; similarly, the mature calculus of Newton and Leibniz is treated separately from the earlier seventeenth-century ‘beginnings of calculus’.

Katz’s revised discussion of eighteenth-century calculus is also differently arranged, with a new section on translating the methods of Newton’s Principia into differential calculus. For the eighteenth century, probability, algebra, and geometry now have a whole chapter each, and there is also a new chapter on probability in the nineteenth century, so that it is now possible to follow the distinct threads of algebra, analysis, probability, and geometry through the second half of the book. The volume ends with a new discussion of twentieth-century solutions to some old problems: Fermat’s last theorem, the four-colour problem, and the Poincaré conjecture.

Despite the welcome updating, some of the problems of earlier editions remain. One is the translation of historical mathematics into modern notation. Of course this is a useful, sometimes necessary, thing to do to aid understanding, but to do it without ever returning to the original texts obscures historical reality. Katz’s account of Newton’s discovery of the general binomial theorem, for example (pp. 547–548), claims that Newton did so by means of an elegant modern formula. This bears little relation to the manuscript evidence of Newton pursuing a lengthy process of trial and error and empirical observation, with a formula of sorts emerging only at the end.

A second problem, which compounds the first, is the lack of references, making it very difficult for readers to return to original sources for themselves. Thus, in the context just discussed of Newton’s binomial theorem, Katz tells us (p. 550) that the infinite series for arcsin appeared for the first time in Europe in De Analysi. A library search for a book of this name, however, will reveal nothing. Newton’s ‘De analysi’, written in 1669 was unpublished for many years (and which, therefore, in keeping with a widely used convention, I write in quotes rather than italic). When it was finally printed, in 1711, it was under the title of Analysis per Quantitatum, Series, Fluxiones, ac Differentias. This, like many other historical mathematical texts, is now available online, a development of inestimable value for historians of mathematics, but in order to identify and find such texts the student needs accurate dates and titles, which Katz all too rarely gives.

Lists of references are provided at the end of each chapter, but some are now a little dated. The reference list for algebra in the eighteenth century, for instance (pp. 684–685), lists as ‘recent’ three publications from 1973, 1984, and 1985, and cites only the second edition (why not the first?) of Maclaurin’s A Treatise of Algebra (1748).

Updating a book of this length is, of course, a major undertaking, but it is a pity that some of these shortcomings of earlier editions could not have been addressed in the latest round of revisions. Nevertheless, Katz’s A History of Mathematics remains, as it has been for some years, the most comprehensive textbook available in the history of mathematics, and for this reason alone is a valuable resource for students and teacher alike.

Jacqueline Stedall is lecturer in the history of mathematics and a Fellow of The Queen’s College, Oxford.



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