Read This!

The MAA Online book review column

Briefly Noted

March 2001

A theme that resonates through all the essays is that Harriot left us a very large number of manuscript pages, many of which consist of nothing but mathematical calculations. Dealing with these unpublished papers is both challenging and dangerous, because it is all too easy for the historian to impose an interpretation on the material. Harriot's work on algebra is perhaps the best example: there are many handwritten pages on algebra in the Harriot papers, and there is also a posthumously published book. Unfortunately, the papers we have do not include a draft of the book, so that determining the relation between the two becomes an exercise in filling in the blanks. It turns out, for example, that the inequality symbols in Harriot's book (which are famous as the first instance of the use of inequality symbols that are somewhat like the modern ones) are not in Harriot's papers (he uses a pair of different symbols). On the other hand, the papers sometimes seem to suggest that Harriot actually understood things better than what is suggested by the book.

The fact that these lectures were given over a ten-year span by scholars with very different interests and points of view means that we see a wide range of opinions (including some strong disagreement) here. This is therefore not a comprehensive account of Harriot, but it may be an interesting place to begin to learn about him. [Fernando Q. Gouvêa]

Here's a history book in a very different spirit. The historians writing the Harriot lectures strive to put him in the context of his times and rarely interpret his work in modern terms. Narkiewicz, by contrast, is interested in how mathematical ideas evolved at a more technical level. The material is presented roughly in historical sequence, but at each point there are remarks on how other mathematicians dealt with the same problem and on how one might understand (and in some cases how one might justify) the method in modern terms. So, for example, when he is explaining Riemann's famous paper on the zeta function and needs to use the functional equation of the theta function, he gives proofs given by Landsberg in 1893 and by de la Vallée Poussin in 1896 rather than investigating the original work of Jacobi and Cauchy (references are given, however).

When studying the history of relatively recent (and highly technical) mathematics, one needs a guide to the mathematical ideas themselves, and that is what Narkiewicz has provided here. His book does the job admirably, guiding us through the complicated history of a complicated subject. Conversely, this book will also serve as a historically-organized introduction to the analytic theory of the distribution of the primes. There are even exercises at the end of each of the chapters. The result is a valuable book, useful both as a guide to the history and as a guide to the mathematics it discusses. [Fernando Q. Gouvêa]

And here is a history of mathematics book that takes both mathematics and history with deep seriousness. Bos is interested in how the concept of what a "construction" is and what makes a mathematical construction "exact" affected the evolution of mathematics in the early modern period. As at several other points in the evolution of mathematics, this period was marked, among other things, by a "foundational" debate. The question was basically this: when is a curve (or other geometrical object) "known"? The answer, at the time, was the Greek answer: when we can construct it. But that, of course, leads to the question of what counts as a construction. The "Euclidean" answer, that only constructions with lines and circles were acceptable, was clearly not a good answer. But finding an answer to the question was problematic. For example, Omar Khayyam had shown that one could solve a cubic geometrically by intersecting two conic sections. Does this count as a solution of the cubic?

In early modern times, the problem was made more acute by the fact that new curves, related to the study of motion, were being discovered. (From a modern point of view, these curves were solutions of differential equations.) In what sense could these curves be said to be "known"? When is a construction to be considered "exact" (as opposed to approximate)?

Bos argues that these issues had a real impact on early modern mathematics. He also argues that the work of Descartes is a decisive turning point in this story, and therefore he focuses his study on the period from 1590 to 1650, bounded by the publication of Pappus' Collection at one end and by the death of Descartes at the other. A forthcoming book will deal with the rest of the story, up to about 1750.

One interesting point about the question Bos is studying is that it was never solved. Instead, it was discarded, rendered irrelevant by a change in how people understood the existence of mathematical objects. That in itself makes this a fascinating book. Bos tells us in the introduction that he started working on this book in 1977, and it shows: the book is detailed, careful, and most of all interesting. Anyone who wants to do serious historical work on this period will need to read this book, and the rest of us may want to read it too. [Fernando Q. Gouvêa]

Publication Data

Thomas Harriot: An Elizabethan Man of Science, ed. by Robert Fox. Ashgate, 2000. Hardcover, 317 pp, $84.95. ISBN 0754600785 .

The Development of Prime Number Theory, by W. Narkiewicz. Springer-Verlag, 2000. American Mathematical Society, 2000. Hardcover, 448 pp., $94.00. ISBN 3540662898.

Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction, by Henk J. M. Bos. Springer-Verlag, 2000. Hardcover, 470 pp., $99.00. ISBN 0387950907.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

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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 Gouvˆa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.