Wilbur Knorr (1945-1997) was one of the most important and influential
scholars of ancient (particularly Greek) and medieval mathematics of the
second half of the twentieth century. Through many books and articles,
Knorr played a major role in the reshaping of our understanding of Greek
mathematics that has marked the years since 1970 or so. His death in 1997
at the age of 52 was a tragic loss for the field. This book is partly the
result of a memorial conference held at the Center for the Study of
Language and Information at Stanford. In addition to the papers presented
at the conference, the book includes other papers written by friends and
colleagues of Knorr. The articles deal with a broad range of topics, as
befits the wide range of Knorr's scholarship. Several of them deal with
Greek mathematics and astronomy. Some of these are specific (e.g., on
Pappus' notes to Euclid's Optics), other deal with important general
issues (e.g., the nature of Greek "analysis"). A few articles pursue
interactions with philosophy (e.g., on Plato's geometrical chemistry). A
couple of articles focus on mathematics and science in Medieval Islam, and
a final article deals with the chronology of Chinese dynasties. Overall,
the book is a fitting tribute to Knorr's life and work, worth a look for
those interested in ancient mathematics and astronomy. My only complaint is
the lack of two items that really should have been part of the package: a
bibliography of Knorr's work (there is a reference to such a bibliography
in a recent issue of Historia Mathematica (vol. 25, 1998,
pp. 123-132), but there was no reason not to reprint it here), and an
overall appreciation of his work and its impact (see the Historia
Mathematica article and the "Éloge" in a recent issue of
Isis). [Fernando Q. Gouvêa]
I first
heard of John Conway's "Surreal Numbers" from Martin Gardner. In one of his
"Mathematical Games" columns, Gardner explained Conway's method for
"creating numbers out of nothing," obtaining, in the process, a bewildering
zoo of infinite and infinitesimal numbers in addition to the usual real
numbers. When, a short time later (in 1977, it was), I saw On Numbers
and Games (originally published in 1976) for sale at a local science
bookstore, I couldn't resist buying a copy.
What a marvelous book it turned out to be! First of all, it was fun to read. (Just look at the names of things: "contorted fractions", "hackenbush unrestrained", "col" and "snort"...) Second, the theory it developed was fascinating. I was only a fledgling mathematician at the time, so many of the details were beyond me, but I could still appreciate and enjoy the creativity and insight on display here.
Here, 25 years later, is a new edition of the book, which has long been out of print. So let's begin by saying that even an unchanged new printing would be a great thing to have: people who missed the chance of buying the book then can now get a copy. The new edition does not include a great number of changes: some corrections have been made, and an Epilogue discusses what progress has been made since 1976 in studying the Surreal Numbers. Coming from Conway, it contains several interesting ideas and even suggested questions for further research.
One of
the more interesting things I learned when I read On Numbers and
Games was that Conway had discovered a connection between numbers and
combinatorial games. In fact, a number turned out to be a certain kind of
game. This theory is developed in the "first" part of ONAG (the
number theory in the zeroth part), but further development was promised in
a forthcoming book. This was Winning Ways, a two-volume book written
by Elwyn Berlekamp, John Conway, and Richard Guy. This was another
marvelous book. It developed an elaborate and powerful theory of
combinatorial games and then applied it to a wide array of different (and
fascinating) games.
The theory introduced in WW has continued to develop, of course. The results of a recent workshop on the subject were published in a book called Games of No Chance, reviewed on MAA Online some years ago. This is to have a sequel in the near future. This book, then, is the entry point to a living mathematical theory.
The original WW appeared in two large volumes. The new edition
splits each of these volumes in two, so that what we have on hand is the
first of four volumes. The new edition is only lightly revised. The authors
have added "Extras" at the ends of the chapters and inserted many
references to more recent work. They also say that they have "corrected
some of the one hundred and sixty-three mistakes." The book is beautifully
produced, with color images where they are helpful. It's great to see this
book back in print. [Fernando Q. Gouvêa]
"Vector analysis," or "vector calculus," as it is sometimes known, is one
of the most fascinating subjects in the undergraduate mathematics
curriculum. It also is one of the subjects that has the largest number of
dramatically different incarnations. Here are two books, both of which
describe themselves as being about "vector analysis." The description is,
in both cases, quite correct, but one might forgive an undergraduate
mathematics major for suspecting that something was wrong, since the books
seem to be about completely different things.
Klaus Jänich's Vector Analysis is about differential manifolds, differential forms, and integration on manifolds. The approach is quite sophisticated, but the author does try to be more helpful to readers than the typical advanced mathematics text. Still, I suspect that most undergraduates would find this far too difficult. They would probably bog down in the definitions of manifolds, tangent spaces, differential forms, and so on, and never really get to the substance of the book.
Chapter 10 of Jänich's book is called "Classical Vector Analysis." In it, the author "translates" the material he has developed back into the classical language of vector fields, obtaining the results that have been described as "Div Grad Curl and all that." (This is by no means the end of the story for Jänich; he goes on to discuss De Rham cohomology, Riemannian manifolds, and more.) By contrast, Applied Vector Analysis, by Matiur Rahman and Isaac Mulolani, is entirely "Div, Grad, Curl", with no hint of the manifolds/forms viewpoint.
In many
ways, this makes sense, since it makes for a straightforward and coherent
development of the material. Still, I can't help but find it a pity. Given
the importance of both points of view, I would have hoped for at least a
small attempt to alert the reader of the existence of an alternative
approach.
The book opens with a short chapter giving some historical background. (As might be expected by anyone who knows me, I wish it had been longer, more detailed, and more careful.) Chapter 2 deals mostly with vector algebra, with partial derivatives thrown in at the end (and looking somewhat out of place.) Then comes the real meat: vector-valued functions, the Del operator, integration (line, surface, and volume integrals), the integral theorems. Finally, a long chapter gives various applications of the material. This chapter is probably the most useful one for someone like me, since it gives me access to material I probably wouldn't have met on my own.
So where do we end up? I first learned this material from Spivak's
beautiful little book, Calculus on Manifolds, but then I learned it
all again when I started to teach vector calculus to engineers. It was
bringing together the two points of view that proved to be really
illuminating. I'm still looking for a book that finds a way to stand in the
middle and connect the two approaches in a serious way. (There is one book
I'd almost certainly use as supplementary reading: Weinreich's Geometrical Vectors.) As for these two books, I'm
afraid Jänich's book is too sophisticated for my students, while
Rahman and Mulolani's isn't sophisticated enough for me! [Fernando
Q. Gouvêa]
Ancient & Medieval Traditions in the Exact Sciences: , ed. by Patrick Suppes, Julius M. Moravcsik, and Henry Mendell. CSLI Publications, 2000. Softcover, 227pp, $24.50, ISBN 1-57586-274-3.
On Numbers and Games, by John Conway. Second Edition. A K Peters, 2001. Hardcover, 242pp., $39.00. ISBN 1-56881-127-6.
Winning Ways, by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. Volume 1, Second Edition. A K Peters, 2001. Paperbound, 276pp., $49.95. ISBN 1-56881-130-6.
Vector Analysis, by Klaus Jänich. Springer-Verlag, 2001. Hardcover, 281pp, $34.95. ISBN 0-387-98649-9.
Applied Vector Analysis, by Matiur Rahman and Isaac Mulolani. CRC Press, 2001. Hardcover, 272pp., $89.95. ISBN 0-8493-1088-1.
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 Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Tue Nov 13 10:18:41 EST 2001