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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