Read This!

The MAA Online book review column

Briefly Noted

June 2000

Most mathematicians, if they have heard of Pappus at all, know him as the author of a famous theorem about solids and surfaces of revolution. People interested in the history of mathematics may have heard a little more about him, and particularly about his role as a recorder and transmitter of the Greek mathematical tradition. Most historians don't give the man much credit. Cuomo quotes Alexander Jones, for example, who says that during Pappus's time (the fourth century AD) Greek mathematics had "experienced a deep and permanent decline" and describes Pappus as an "author in this degenerate tradition" whose reputation is high only because of the large volume of his work, the large percentage of that work which is still extant, and the fact that he preserves important bits of earlier tradition. Cuomo sets out to change this impression, not so much by arguing for Pappus as a creative mathematician as by trying to see him in the context of his time. The book is too short for the author to investigate Pappus's work in depth (there is a lot of it), but she does a good job of describing how mathematics fit into late Hellenistic culture and how Pappus can be seen as a great mathematician in this context. Cuomo does a few close readings of material from Pappus, and these are the most interesting parts of the book. The result is readable, interesting, and an eye-opener for those who have only met Greek mathematics at a superficial level. This little book in the "Cambridge Classical Series" is one that mathematicians interested in the history of their subject will want to read. (Fernando Q. Gouvêa)

Abraham Pais, who has written first-class biographies of Einstein and Bohr, presents us here with a "portrait gallery of twentieth century physicists." The portraits (short articles, many of which have appeared as independent pieces before) are arranged in alphabetical order, from Niels Bohr and Max Born to Viktor Weisskopf and Eugene Wigner. Rather than focusing exclusively on their scientific work, Pais strives to bring out each scientist's personality and to relate the person and the work. To do this, he relies on the fact that he knew each of these people personally. The chapters contain great stories, interesting quotes, and perceptive summaries of the scientific work of some of the century's most influential physicists. There's only one mathematician in the bunch (John Von Neumann), but these physicists knew their mathematics and often had a real influence on the development of mathematics in our time. Great scientists are fascinating, and great science is even more fascinating: Pais gives us both, and the result is a valuable book. (Fernando Q. Gouvêa)

We typically learn (and teach!) the law of quadratic reciprocity in courses on Elementary Number Theory. In that context, it seems like something of a miracle. Why should the question of whether p is a square modulo q have any relation to the question of whether q is a square modulo p? After all, the modulo p world and the modulo q world seem completely independent of each other. (Isn't that what the Chinese Remainder Theorem says?) The proofs in the elementary textbooks don't help much. They prove the theorem all right, but they do not really tell us why the theorem is true. So it all seems rather mysterious. On top of that, the books often tell us that Gauss gave a whole bunch of proofs of this theorem, and that hundreds of proofs are now known... and we are left with a feeling that we are missing something. What we are missing is what Franz Lemmermeyer's book is about. Right at the beginning, he makes the point that even the quadratic reciprocity law should be understood in terms of algebraic number theory, and from then on he leads us on a wild ride through some very deep mathematics indeed as he surveys the attempts to understand and to extend the reciprocity law. Most of the book deals with the many "higher reciprocity laws" which were a central theme in nineteenth century number theory. As the introduction suggests, in the twentieth century this theme developed into what is now known as "Class Field Theory," and the only unfortunate thing about this book is that it doesn't follow the thread all the way to the end. But never fear: the author promises us a second volume, to pick up where this one leaves off and lead us all the way to the Artin Reciprocity Law and (I hope!) beyond. For now, this is a very good expository account of some difficult, deep, and beautiful mathematics. (Fernando Q. Gouvêa)

Publication Data

Pappus of Alexandria and the Mathematics of Late Antiquity, by S. Cuomo. Cambridge University Press, 2000. Hardcover, 234pp., $59.95. ISBN 0521642116.

The Genius of Science: a Portrait Gallery, by Abraham Pais. Oxford University Press, 2000. Hardcover, 368pp., $30.00. ISBN 0198506147.

Reciprocity Laws : From Euler to Eisenstein, by Franz Lemmermeyer. Springer-Verlag, 2000. Hardcover, 487pp., $79.95. ISBN 3540669574.

Fernando Gouvêa (fqgouvea@colby.edu) is the editor of 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.

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.

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