Read This!

The MAA Online book review column

Briefly Noted

May 2004

"... there is the general reader who has not lost interest in the studies of his youth and would wish to know how it came about that a Greek of the name Euclid wrote a textbook which, in an almost literal translation, was used in schools and in the universities of this country as the one recognized basis of intruction in elementary geometry, and on which generations of Senior Wranglers no less than average mortals were brought up, asking nothing better, until some fifty years ago."

(Uff! People don't write sentences like that any more!)

He explains that half the book deals with Euclid and his precursors, and then expresses the hope that readers will want to know more: "And who, having got so far, will not wish to know what heights were scaled by Euclid's successors...?" Indeed!

It's not really a criticism of a book first published in 1931 to say that it is old-fashioned. Of course it is. As the preface makes clear, the first publication of the Rhind papyrus, with its revelations about Egyptian mathematics, happened in the 1920s, after the History was written and before the Manual. Crucial discoveries about the mathematics of ancient Mesopotamia were being made as Heath was writing. And much has happened among scholars of Greek mathematics during the last 70 years.

A particularly noticeable feature of Heath's approach is his exclusive concentration on the content of the major texts. There is very little here about the cultural context in which Greek mathematics was done, very little about the transmission (and reliability!) of the texts, very little about everyday mathematics in ancient cultures.

So while it doesn't count as a criticism, it is still important to say the book is old-fashioned, as a warning to its current readers. By all means use Heath's book, particularly if you would like a quick introduction to the more important mathematical texts. But look for newer books (for example, S. Cuomo's Ancient Mathematics, which despite its title is about Greek, Hellenistic, and Roman mathematics) to fill in the rest of the picture. [Fernando Q. Gouvêa]

In history of mathematics circles, Clifford Truesdell is a name to conjure with (though some historians might want to describe such conjuring as black magic). Edoardo Benvenutto, on the other hand, is mostly unknown. Essays on the History of Mechanics is the proceedings of a conference in honor of both men, who are remembered here as historians of mechanics, and especially of the mechanics of structures. Continuum mechanics was Truesdell's research field before he reinvented himself as a historian, and his contributions to the history of the subject (not least in a book called Essays in the History of Mechanics, published by Springer in 1964) were significant. Benvenutto, whose best known book is An Introduction to the History of Structural Mechanics (Springer, 1991) is the founder of a school of research on the history of mechanics as applied to construction.

The articles are mostly by European scholars, and they are a very mixed bag. Several of them are in "history of history" style. One of them, for example, considers how historians have reacted when they discover errors in the (usually important and respected) works they are studying, with Truesdell as the star example. Other essays are specifically historical (on the history of rose windows, for example, or of timbrel vaults). Two of the essays dealt with early vector-like methods and the "parallelogram of forces". My main regret was that the book does not include bibliographies of the work of either honoree. Not, perhaps, a book to buy, but it'd be nice to find it in a library. [Fernando Q. Gouvêa]

Tables and table-making were, until very recently, a very important part of the practice of mathematics. For mathematicians and (even more so) for users of mathematics, they were a crucial resource: logarithm tables, trigonometric tables, tables of integrals and of transcendental functions. Still in print from Dover, for example, is a Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Academic Press offers a huge book called Table [sic!] of Integrals, Series, and Products, and CRC Press has its Standard Mathematical Tables and Formulae.

Nowadays, of course, tables have for the most part been replaced by calculators and computers, and the modern equivalent table-making is the creation of efficient numerical algorithms. Not that tables are completely gone. On the web, one can still find highly technical tables intended as an aid to "experimental mathematics". I've made some myself.

The History of Mathematical Tables is a collection of essays, mostly by British authors. It isn't really a complete history (the story of mathematical table-making in Medieval Islam is not represented at all, a rather glaring gap given its historical importance), but rather a series of interesting snapshots. The first essay, by Eleanor Robson, is about tables in Ancient Sumer, Babylonia, and Assyria. Following that we take a flying leap into sixteenth century Europe and read about logarithm tables and their construction. There are chapters on actuarial tables, on tabulation of data, and on various attempts at mechanizing the creation of tables. (One of the chapter titles refers to the "unerring certainty of mechanical agency", a concept that may surprise readers living in the time of "computer errors.") The final chapter nominates the spreadsheet as the modern successor of the table and traces its evolution from VisiCalc through Lotus 1-2-3 and its clones to Excel (whose name seems to be on its way to replacing "spreadsheet" as Excel replaces all of its rival programs).

Many things are missing. As mentioned above, there is nothing on Medieval Islam. There is also nothing on non-Western cultures. On the contemporary end, an essay on mathematical software would have been nice. The modern role of such software is much closer to the historical role of tables than the role of spreadsheets seems to be. And it would also have been useful to have something on the role of tables in "experimental mathematics."

Those criticisms, however, amount to simply asking for more of a good thing. As it is, this book is a useful source of information on a subject often neglected by the big historical surveys. It will be of interest not just to historians of mathematics proper, but also to those who are interested in the evolution of computing and of statistics. It might be a bit too expensive for individuals, but it's definitely a must-have for libraries. [Fernando Q. Gouvêa]

Bayes' theorem, sometimes described as the theorem of inverse probabilities for finding the probabilities of the different causes any one of which could have led to an observed effect, is justly famous as one of the key pillars of modern statistics — in particular for the way it allows the incorporation of subjective probabilities into such computations without relying only on frequency-based probabilities. Moreover, the field of Artificial Intelligence has found further valuable uses not envisaged by the Reverend Bayes in the 18th century, and Bayesian Artificial Intelligence has become a rapidly expanding field. But there has been no comprehensive study of the life and works of the author of the theorem. In his preface to Most Honourable Remembrance, Andrew Dale explains that his aim was to "flesh out the shadowy figure after whom one of the major branches of modern statistics is named".

The extensive and careful work which the author has put in to achieve his aim is evident everywhere in the book. It must obviously have involved a lot of labor — evidently a labor of love for the author — to collect all the details he has given about Thomas Bayes' ancestors, his early schooling and life, the town he lived in, the burial ground he was buried in, the inscriptions in the burial ground, the will Bayes left, and so on. It is all there in as much detail as anyone could ask for.

Interesting and useful as all this will be for anyone interested in knowing more about Bayes, this is just part of the riches contained in this book. The seminal essay on The Doctrine of Chances in which Bayes first propounded his views is printed in full, with a an introduction and an illuminating commentary placing the paper in context and relating it to the views of other founders of probability theory such as Bernoulli and Laplace. The commentary also explains how in addition to the famous Bayes theorem, the paper also contains contributions to pure mathematics in relation to issues like the evaluation of the incomplete beta function. Furthermore, Dale provides similar introductions and commentaries for the other lesser-known (one might better say hardly-known) works of Bayes, such as his Introduction to the Doctrine of Fluxions, in which he provided counter-arguments against Bishop Berkeley's criticism of calculus, or his theological paper on Divine Benevolence, and his writings on a miscellany of subjects including electricity and celestial mechanics.

One emerges from the book with two feelings: first, that Bayes was one of the most powerful thinkers of his time and that to think of him only in terms of Bayes' theorem is to do him less than justice; second, that Dale has not just "fleshed out the shadowy figure " of Bayes, he has done much more by his careful and deep analysis of Bayes work as a whole. Beyond doubt this book is a work of the highest quality in terms of the scholarship it displays, and should be regarded as a must for every mathematical library.[Ramachandran Bharath]

Publication Data

A Manual of Greek Mathematics, by Sir Thomas L. Heath. Dover, 2003 (reprinting the 1931 edition). Paperback, 576 pp., $29.95. ISBN 0486432319.

Essays on the History of Mechanics, ed. by Antonio Becchi, Massimo Corradi, Federico Foce, Orietta Pedemonte. Birkhäuser, 2003. Hardcover, 256 pp., $59.95. ISBN 3-7643-1476-1.

A History of Mathematical Tables, ed. by Martin Campbell-Kelly, Mary Croarken, Raymond Flood, and Eleanor Robson. Oxford University Press, 2003. Hardcover, 361 pp., $89.50. ISBN 0-19-850841-7.

Most Honorable Remembrance: The Life and Work of Thomas Bayes, by Andrew I. Dale. Springer-Verlag, 2003. Hardcover, xxiii+677 pp., $99.00. ISBN 0-387-00499-8.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

Ramachandran Bharath is Visiting Professor of Mathematics at Colby College.

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.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.
• October 2002: Algebra from A to Z, Problems in Probability, and a really old trigonometry textbook.
• December, 2002: group representation theory, Irish mathematicians, and integration.
• January, 2003: careers, logic, dueling idiots, and differential geometry.
• March, 2003: the Putnam, the Liber Abaci, and the Millennial Conference.
• April, 2003: analysis, reprints, and Rithmomachia.
• May, 2003: conversation starters, dissections, and Fibonacci numbers.
• July, 2003: mathematical circles, the Scottish Café, and the zeta function.
• September, 2003: history — an encyclopedia, historiography, and algebra.
• October, 2003: selected papers and second editions.
• December, 2003: calculus, relativity, and biology.
• February, 2004: cryptography, generating functions, Wallis, and Stirling.
• March, 2004: differential equations, fundamental groups, reprints, and spherical buildings.
• April, 2004: Ramanujan graphs, time, and Medieval Islam.

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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 Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.