I.5    Textbooks for a Survey Course in History of Mathematics

The purpose of this page is to provide information that may be helpful to teachers when choosing a textbook for a survey course in the history of mathematics. There are many other textbooks that have been used for other styles of history of mathematics courses, but attention is restricted here to the four textbooks which are currently used in most mathematics history survey courses taught in North America:

• Carl Boyer, A History of Mathematics
  • Reviewed by Campbell, and Jones.
• David M. Burton, The History of Mathematics: An Introduction
  • Reviewed by Calinger, and Wheeler.
• Howard Eves, An Introduction to the History of Mathematics
  • Reviewed by Jones.
• Victor J. Katz, A History of Mathematics: An Introduction
  • Reviewed by Archibald, Grabiner, Rowe, and Tattersall.

  Several new textbooks for survey courses have recently appeared.

Summaries of several reviews of each of these texts are given below (no attempt has been made to list every review of these texts that I am aware of). I have paid special attention to the comments in the reviews that state opinions about what is most desirable in a history of mathematics textbook, as well as what is noteworthy and lacking in each of these. However, no attempt has been made to summarize the contents or to repeat information that appears in several of the reviews. You are encouraged to look up these reviews and to read them in detail. There is a great deal to learn from reading reviews and I a strongly encourage you to develop the habit of reading many of them.

My comments on each of these textbooks are also given.

Carl B. Boyer, A History of Mathematics, New York: John Wiley, second edition, 1989. ISBN 0-471-09763-2.

Reviewed by Paul J. Campbell, Mathematics Magazine, 62 (1989), 284.

This five line review indicates that the first 22 chapters have been left virtually unchanged while those dealing with 19th and 20th century mathematics have been revised and expanded. References and bibliographies have been updated.

Review of the first edition (1968) by Phillip S. Jones, American Mathematical Monthly, 76 (1969), 1163-1165.

"In a historical treatise one looks for scholarship in marshaling the facts, for insight and a broad view in portraying the changing trends, themes, and interrelationships both within mathematics and between mathematics and other aspects of culture. One may also be concerned for the coverage and for the quality of the evaluation of forces directing progress. This book scores high in all of these respects." Recent historical scholarship is taken into account and the bibliographies are "very helpful to students doing individual projects." Jones is concerned that the student will not find this book easy reading for it is "packed with condensed information and interpretation." The student should fill in the details. "The book is remarkably free of errors." "In conclusion, the book rates an excellent score as both a scholarly product which sets a good example for young scholars, and as a textbook in an undergraduate course."

David M. Burton, The History of Mathematics: An Introduction, Boston: Allyn and Bacon, third edition, 1985, ix + 678pp. ISBN 0-697-16089-0.

First edition (1985) reviewed by Ronald Calinger, ISIS, 81 (1990), 545-546.

This solid survey of the history of mathematics is written in a lively style and contains "a rather idiosyncratic store of information." "China, the Hindus, and medieval Islam are largely overlooked," as is Napier's work on logarithms. "The interspersed treatment of the development of the calculus is not adequate." [After reading the sections of Boyer and Katz on the Calculus you should judge for yourself is this treatment is deficient; later editions of Burton are better.]

Reviewed by David Wheeler, The College Mathematics Journal, 17 (1986), 373-375.

"This is a history book in a familiar mold. . . . it pictures mathematics . . . as progressively and inexorably unfolding, brilliantly impelled along its course by a few major characters, becoming the massive edifice of our present inheritance." It is based on up to date secondary sources.

The author leaves out some topics because of their technical difficulty, but should there be no explanation of the Eudoxan theory of proportion, and is the Cantor-Dedekind theory of irrationals "a story best avoided" (p. 577)? "I do not think anyone could find from this book how and when the concept of function surfaced, and how it developed subsequently." Nonetheless, many topics are treated satisfactorily, including Greek geometry and non-Euclidean geometry.

My chief problem with the book, though, is one that Eves' book sets me, too. They both offer a large number of historical facts and anecdotes without making any real attempt to show the reader how a sense of history can deepen understanding of mathematics [Instructors should think hard about how they can do this in their courses. --- VFR], shed light on the roots of mathematical ideas, give a grasp of the role that mathematics plays in human society, and so on. But for what other purposes would one be writing a history of mathematics for undergraduates?

Howard Eves, An Introduction to the History of Mathematics, Saunders Publishing Co., sixth edition, 1990. ISBN 0-03-029558-0.

Review of the first edition (1953) by Phillip S. Jones, The American Mathematical Monthly, 61 (1954), 433-436.

There has long been a need for a book on the history of mathematics which was suitable as a text with undergraduates at the junior-senior level, . . . Eves' book was written expressly for this purpose. It succeeds remarkably well . . . A unique and usable feature of the book is the set of "Problem Studies" at the end of each chapter. . . . Although the reviewer would prefer more emphasis on a topical approach . . . the book is not a mere recital of names and dates, but does well in an attempt to stress the growth of ideas and interrelationships between them for readers who are not too advanced or mature mathematically.

Review of the second edition (1964) by Phillip S. Jones, Scripta Mathematica, 28 (1967), 168.

A readable book but it "lacks pictures and reproductions of portions of original works which the reviewer would like to see."

Victor J. Katz, A History of Mathematics: An Introduction. New York: HarperCollins, 1993, xiv + 786pp. ISBN 0-673-38189-4.

Reviewed by Tom Archibald, Historia Mathematica, 23 (1996), 89-92.

What we want in a history of mathematics textbook

"is a book which is accessible to students who have mastered very little beyond the calculus, which is mathematically respectable, historically very solid, which can serve as a source of problems and projects for the teacher to assign, which can serve as a useful reference, and which can guide the student and the instructor in their further study.

Katz's book succeeds admirably on all these counts, and many more besides." (p. 90)

The combination of clear exposition, superior documentation, and mathematical richness should be actively sought by all who teach such courses." (p. 92)

Archibald notes that he has used this book successfully with junior level math majors, but warns that it would be too much for weaker students [This is definitely true. --- VFR], and would have to be supplemented for more sophisticated students.

Reviewed by Judith Grabiner, The Mathematical Intelligencer, 16:4 (1994), 73-76.

The full-color cover with its detail of The Ambassadors of Hans Holbein the Younger prepares us "for an attractive, well-illustrated book that respects original sources and presents material from them, and that stresses the cultural setting of mathematics." While Katz covers the standard topics the outstanding features of this book is that he closely follows original sources and has an "excellent treatment of mathematics outside the European tradition."

Grabiner has used this book in class and was pleased with the outcome. In a semester one can do the first 12 chapters, up through the calculus of Newton and Leibniz.

The index gives phonetic pronunciation for non-English names, "a feature my students found surprisingly empowering."

"The most serious criticism one can make is that Katz's coverage reflects the limitations of twentieth-century scholarship. The book "is not itself one of path-breaking scholarship. . . . Much remains to be studied." Yet, Katz has "produced an excellent and readable text, based on sound scholarship and attractively presented."

Reviewed by David E. Rowe, ISIS, 85 (1994), 125.

Many history of mathematics courses are taught by "mathematicians with only scant knowledge of history" using "textbooks that betray little knowledge of either primary sources or the scholarly literature concerning them, treat mathematical ideas and problems in Platonic fashion as enjoying an existence largely independent of their cultural content, confine themselves to those aspects of (mostly pure) mathematics that have found a secure place in modern mathematical textbooks, and display a marked Eurocentrism that pays but little attention to the role of mathematics in non-Western cultures." Katz "manages to avoid all of these standard pitfalls." This book is a "highly readable, comprehensive, and exceptionally well-researched work" that "successfully bridges the yawning gap that until now has separated scholarship from teaching in the field of history of mathematics." It is designed for use by prospective secondary teachers and "sets a new standard for historical writing in this genre."

Reviewed by James J. Tattersall, The College Mathematics Journal, 25 (1994), 347-348.

"What really distinguishes this book from any other history of mathematics text is undoubtedly its exercises." (p. 348). They are carefully chosen and based closely on the original texts. The student who does them diligently will become acquainted with many important texts, will come to understand the cultural settings of the text, and will acquire a greater understanding of mathematics. Some exercises in each chapter involve ways of bringing history into the ordinary mathematics classroom. But the problems are hard for most students. They must read the text carefully to learn how to do the problems and also must review much of their mathematical knowledge.

Tattersall's course was for MAT students. They met once a week for two hours for a semester and covered most of the first 13 chapters (through the calculus and Non-Euclidean Geometry). "At the end of the semester, all students agreed that they had learned a lot of mathematics. They felt that Katz's text not only had helped make them better teachers of mathematics but had given them a greater appreciation for the subject and more insight into it." (p. 348).

The students would have liked "time lines using line intervals" as well as more maps. There is plenty of material here for two semesters.

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If you have comments, send email to V. Frederick Rickey at fred-rickey@usma.edu .
Posted December 14, 1996. Last revised July 2005.