Read This!

The MAA Online book review column

Briefly Noted

June 2002

The articles can be divided into three groups. First, there are several methodological articles. These deal with how mathematics is (or can be) communicated across cultures, with what is usually called "ethnomathematics", with philosophical issues about rationality and logic and how they might vary from one culture to another, etc. D'Ambrosio's article, proposing a broad historical framework for understanding the relationship between Western and Non-Western mathematics, is particularly interesting.

The second group of articles discuss the mathematics of cultures that are often discussed in history books: Ancient Iraq (aka Mesopotamia), Ancient Egypt, Medieval Islam, India, China. What is particularly interesting here is the point of view: the authors insist on viewing these mathematical traditions in their own terms, and not in terms of how they anticipated or influenced Western mathematics. The articles by Robson and Ritter are particularly nice.

Third, there are articles on cultures that are discussed much less often: the Hebrew mathematical tradition, the Incas and other Mesoamerican cultures, the Sioux, Pacific cultures, Australia, Subsaharan Africa, Korea, Japan. These are fascinating, and mostly new to me.

Overall, we have here a valuable corrective and supplement to the usual history books. [Fernando Q. Gouvêa]

When the MAA published Sherman Stein's book on Archimedes some years ago, I remember feeling frustrated that we had a useful commentary on the works of Archimedes but that the T. L. Heath translation of those works was no longer in print. Dover has just remedied the situation by bringing their edition of The Works of Archimedes back into print. This actually contains two books: Heath's translation of the surviving works of Archimedes, published in 1897, and a supplement (published in 1912) containing a translation of the recently-discovered "Method".

Heath's edition includes a long introduction (over 170 pages) discussing Archimedes' work. Then come the works themselves, translated into English and into modern mathematical notation. The appendix containing the "Method" has the same structure.

There are two issues surrounding this translation that might bother historians. The first has to do with Heath's decision to modernize the notation. This makes the text easier to read, but sometimes leaves the reader wondering exactly what Archimedes actually wrote. The second has to do with the underlying Greek text: with the recent reappearance of the manuscript containing the "Method" and the increase in knowledge about textual criticism, we are probably at the point where a new critical edition of the Greek text is needed.

Nevertheless, while we wait for scholars to produce a new critical edition and a more literal translation, this edition gives us access to the works of one of the greatest mathematicians of all time. [Fernando Q. Gouvêa]

Let me admit up front that I have only the vaguest sense of what S. Twomey's Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements is about. The main reason to note it here is to call attention to Dover's new series, the "Dover Phoenix Editions". These are hardcover reprints of classics in various scientific and mathematical fields "for which there has been a relatively small but steady demand."

Dover Publications has, of course, been reprinting valuable old mathematics books for many years. Their books range all the way from recreational mathematics aimed at the general public to highly technical books aimed at specialists. Inevitably, the latter don't sell as quickly, and sometimes that makes it difficult to keep them in print. The Phoenix series is an attempt to solve that problem.

The first batch of Phoenix editions includes 20 books, several of which deal with topics in pure or applied mathematics, including well-known classics by Osserman, Hurewicz, and others. Keeping good books in print is a great service to our profession; Dover's efforts are welcome and deserve our support.[Fernando Q. Gouvêa]

Mathematicians tend to identify "probabilistic reasoning" with the mathematical theory of probability. James Franklin's The Science of Conjecture reminds us that there is much more to it than that. The book is a history and analysis of qualitative probabilistic reasoning before the time of Pascal. Thus, though the author is a mathematician, this is not primarily a book about mathematics. Nevertheless, it should be useful to historians of mathematics because it studies the context and tradition from which our quantitative version of probability emerged.

Franklin observes that qualitative probabilistic reasoning still is quite common, particularly in the law. (Think of "beyond a reasonable doubt" or "a preponderance of the evidence.") Much of his book deals with legal reasoning, though it also discusses philosophy, theology, "aleatory contracts" (insurance, annuities, bets), and even some pre-Pascal quantitative analysis of games of chance.

The book is well-written and pleasant (though not easy) to read, and the ideas are provocative. Franklin has an axe to grind: he is out to defend our ability to make rational decisions about truth when the evidence only allows conclusions that are probable or likely. Since this includes essentially all such decisions outside of mathematics, his argument should be of interest to philosophers and others who want to understand how we think and make decisions. [Fernando Q. Gouvêa]

Publication Data

Mathematics Across Cultures: The History of Non-Western Mathematics, ed. by Helaine Selin and Ubiratan D'Ambrosio. Kluwer Academic, 2000. Hardcover, 512pp., $217.00. ISBN 0-7923-6481-3.
This book is also available in a print-on-demand paperback version priced at $55.00. ISBN 1-4020-0260-2.

The Works of Archimedes, ed. by T. L. Heath. Dover Publications, 2002. Paperback, clxxxvi+326+51pp, $24.95. ISBN 0-486-42084-1

Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements, by S. Twomey. Dover Publications, 2002. (Unabridged and corrected edition of the 1977 edition from Elsevier.) Hardcover, 243pp., $32.50. ISBN 0-486-49517-5.

The Science of Conjecture: Evidence and Probability before Pascal, by James Franklin. Johns Hopkins University Press, 2001. Hardcover, xiii+497pp., $59.95. ISBN 0-8018-6569-7.
Paperback edition, September 2002, $22.50. ISBN 0-8018-7109-3.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.