Read This!
The MAA Online book review column
Briefly Noted
Short Reviews by Fernando Q. Gouvêa
November 1999
There are lots of textbooks on elementary number theory, so when
a new one comes out one has to ask what it has to offer that's new. In the
case of James Tattersall's Elementary Number Theory in Nine
Chapters, the first thing that is immediately apparent is the author's
interest in and deep knowledge of the history of the subject. Beginning
with the title, which alludes to the title of a famous Chinese mathematical
text, the book is full of historical information, historical motivation,
and (best of all) problems derived from these historical sources. The table
of contents includes the standard list of topics for a first course in
number theory, ranging from primes and divisibility through congruences,
quadratic reciprocity, continued fractions, and the usual applications to
cryptology. The book goes on to treat a more unusual topic, the theory of
partitions, including a discussion of generating functions. In contrast to
far too many mathematics books, the book has lots of text surrounding all
the mathematical formulas. It's all a very rich mix, and I suspect
students would enjoy using this book; I also suspect they would find it
quite challenging.
Tattersall's Nine Chapters came out just a week or so too late for
me to select it for my Number Theory course this spring. Next time around,
it'll be high on the list of possible textbooks for the course.
Mathematics at Work is a book that challenges our assumptions about
our subject. This is the fourth edition of a book devoted to
Practical Applications of Arithmetic, Algebra, Geometry, Trigonometry, and
Logarithms to the Step-by-Step Solutions of Mechanical Problems, with
Formulas Commonly Used in Engineering Practice and a Concise Review of
Basic Mathematical Principles.
Browsing through the book reveals an amazing mix of topics, from very
elementary stuff about divisibility to rather sophisticated chapters on
plane and spatial geometry. Many pages contain elaborate tables of results
on trigonometric functions, solving triangles, factoring expressions, and
so on. Many specific construction problems are considered too, such as (to
choose an example at random) how to find the "radius of a circle tangent to
a given circle and to two lines at a given angle." There is a whole section
on approximate formulas which includes a rather sophisticated discussion of
where such formulas come from and how one might decide whether it is OK to
use them. There is a chapter on "gear ratio problems", which turns out to
be about continued fractions. At the back, there are dozens of tables
giving all sorts of interesting and useful functions (one wonders why, in
an age of calculators, one would need tables of common logarithms or
trigonometric functions, but there they are).
I find this a fascinating artifact. On the one hand, it can be an excellent
source of "real world" problems for those of us who teach elementary
mathematics. Many of these problems are quite interesting, and some are
quite difficult; in fact, I'd probably have trouble solving some of them on
my own. On the other hand, it reflects a perspective on what mathematics is
all about that I find deeply alien. This mathematics is about vast
collections of apparently unrelated facts that one looks up as needed. (So,
for example, there are not only tables indicating how the sines and cosines
of complementary and supplementary angles are related, but also similar
tables for tangents, cotangents, secants and cosecants!) Here is
mathematics as a tool, but perhaps also as less than a science.
It is cause for joy and celebration that the "Sources" subseries
of the AMS/LMS "History of Mathematics" series continues to grow. This
book, which is volume 17 in the AMS/LMS history series, contains a small
book by Jacques Hadamard on the connections between non-Euclidean geometry
and the theory of automorphic functions. Hadamard's book was written in the
1920s for publication in Russia, appearing in a series entitled "The
Geometry of Lobachevskii and the Development of Its Ideas". It was
translated into Russian, and the original French text seems to have been
lost; what we have here, accordingly, is a translation from the Russian
edition. It is preceded in this edition by a "Brief History of Automorphic
Function Theory, 1880-1930" by Jeremy Gray. This is short but helpful,
emphasizing the very distinct approaches to the subject used by
Poincaré and by Klein, leading to differing French and German
approaches to the subject. Hadamard's essay belongs firmly in the French
camp, with a strong emphasis on solving differential equations and also
with characteristic foundational weaknesses. This little book should be of
interest both to historians seeking to understand the evolution of the
theory of automorphic functions and to mathematicians working in the area,
and thus it is a valuable addition to the (rather short) list of original
source material available in English translation. Keep them coming,
AMS!
Publication Data
James J. Tattersall,
Elementary Number Theory in Nine Chapters. Cambridge University Press,
1999. Softcover, 407pp, $24.95. ISBN 0-521-58531-7.
Holbrook L. Horton,
Mathematics at Work, edited by Henry H. Ryffel, Edward E. Messal, and
Robert E. Green. Industrial Press, 1999. Softcover, 687pp. $22.95.
ISBN 0-8311-3083-0.
Jacques Hadamard,
Non-Euclidean Geometry in the Theory of Automorphic Functions, edited
by Jeremy J. Gray and Abe Shenitzer (American Mathematical Society and
London Mathematics Society series in the History of Mathematics, volume
17). American Mathematical Society, 1999. Softcover, 95pp, $19.00. ISBN
0-8218-2030-3.
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.
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.
Copyright ©1999 The Mathematical Association of America
MAA Online is edited by Fernando Q. Gouvêa
(fqgouvea@colby.edu).
Last modified: Thu Nov 18 12:51:27 -0500 1999