Read This!

The MAA Online book review column

Briefly Noted

February 2004

Cryptography: An Introduction is a translation from the Russian by Sergei Lando. Each chapter is, for the most part, self-contained and written by a different author. I would recommend this book for faculty, and undergraduates that have an understanding of Number Theory and/or Abstract Algebra. Although the books jacket recommends this book for advanced high school students, and undergraduates in my opinion this is not the "Idiots Guide to Cryptography."

Students and faculty alike will find this book to be an excellent guide/reference to cryptography. The concepts are explained clearly and many references are given for further study. The chapter I found most interesting was chapter 4, which covers the RSA cryptosystem, how to construct large prime numbers, and related issues. The book also contains some interesting examples of cryptography in literature from A. Conan Doyle's The Dancing Men, J. Verne's Journey to the Center of the Earth, and others.

The old adage "too many cooks spoil the broth" doesn't really hold for this book, but one must keep in mind the way the book is written. Each chapter is self-contained and covers a different topic in cryptography. Although the book can be read straight through, that wouldn't be what I recommend. After reading the first chapter I suggest skipping to the last chapter to work on some of the exercises (answers and hints are provided for those of us who are a little impatient). This will give you an idea of which types of problems in cryptography interest you and which chapters you'd like to read. This book is a smorgasbord of cryptography and covers all the major concepts. So read what you like and enjoy![Kevin Anderson]

Lando's Lectures on Generating Functions is distinguished by two closely intertwined features: It is driven by very, very interesting problems and examples; and it takes off in directions which are perhaps a little unusual in an introductory text on generating functions, fitted in the larger framework of combinatorics. The former feature is already abundantly evident from the first sentence of the book's preface: "After multiplying by (2n-1)! the coefficient of [the n-th term] in the power series expansion of the function tan(x) becomes a positive integer..." Thus, without naming names — and thereby giving away the tantalizing secret — Lando begins the book, even before the first chapter, with a veiled allusion to a candidate for the most famously named numbers in number theory. And he can hardly contain his enthusiasm (which is indeed quickly infectious) as he goes on to the second paragraph of the preface: "Mathematicians of the 18th and 19th centuries knew functions 'personally'..." Surely this is an irresistible challenge to join him in the recovery of a lost art!

Regarding the second property, the author's particular choice of topics, it's really all an immense amount of fun. After dealing very effectively with the requisite background concerning fundamental properties of elementary generating functions, Lando hits, for example, the Fibonacci numbers, the Catalan numbers, and formal grammars. Then, quickly introducing more and more (beautiful) machinery as he goes along, he gets to Pascal's triangle, the Dyck triangle, the Bernoulli-Euler triangle, and the Euler numbers, all in the context of generating functions of several variables. Very good stuff! And the last chapters of the book deal with such gems as the theory of partitions, continued fractions, and enumeration problems for embedded graphs.

The book is very well suited to self study or use in a seminar for a hand-picked audience. It is beautifully written, even if the exposition is a bit on the terse side, and it is certainly indicated that the reader or student should possess a fair amount of "mathematical maturity." This having been said, reading Lando's book is a trip well worth taking for any one interested in the topic of generating functions and the arithmetically well-endowed numbers or related objects, such as certain ordinary differential equations (see paragraph 5.8) which they describe.[Michael Berg]

Publication of important historical source material is always cause for rejoicing. This is particularly so in the case of the first volume of the Correspondence of John Wallis (1616–1703). I think Wallis has received less attention from the general histories of mathematics than he deserves. Partly, this is because he had the bad luck to have made substantial contributions to analysis and to physics that were later viewed mainly as pointing towards calculus and Newtonian mechanics. (Being a predecessor can be a formula for obscurity...) It may also have something to do with the fact that Wallis worked in so many fields (he did significant work in theology and cryptography, for example, in addition to mathematics). The fact that he had a rather prickly personality and was involved in many controversies (the most famous one being the one with Thomas Hobbes) probably has something to do with it too.

There is no doubt, however, that Wallis was one of the most important mathematicians of his time, and that his works contain a lot of fascinating material. In particular, he was attuned to the history of mathematics, and several of his books contain portions in which he gives historical accounts of the subject. These are notoriously partisan and polemical, but I find them fascinating.

This volume is the first in a series collecting all of Wallis' letters (in their original languages). The letters have been collected from archives spread throughout Europe, and are presented here with careful textual notes. There are no translations or commentary except for a useful introduction summarizing Wallis' life and interests. So this is a book for scholars, and its price makes it a book that only libraries (perhaps only the bigger libraries, in fact) are likely to buy. Still, it is good news that the letters are being published. I hope it'll lead to more work being done on Wallis and his mathematics.

Also important is Ian Tweddle's new edition of James Stirling's Methodus Differentialis. This is an annotated translation of the original text, and hence perhaps a little more friendly to non-historians than the Beeley-Scriba edition of Wallis' letters. Tweddle has translated the Latin and added many notes. Much of Stirling's book deals with numerical methods for finding the sums of series and other function values; Stirling gives no error estimates for his methods, so Tweddle has given these in his notes.

The original title of Stirling's book (originally published in 1730) can be translated as "The Method of Differentials, or, a Treatise on Summation and Interpolation of Infinite Series." That is a good description of what is in the book. Stirling's Formula is here, as are the original appearances of the Stirling numbers and of Stirling's interpolation formula. There is also lots of stuff on transformations of series (for example, to accelerate their convergence). The blurb on the back cover describes the book as a classic of numerical analysis, and that seems exactly right.

This is another expensive book, a bit too expensive in fact, which is a pity. At a lower price, it might have reached a bigger public. (Working through Stirling might make a very interesting — and very challenging — undergraduate project, one that would make sense to students interested in "getting a number out" of all the mathematical theory.) As it is, I hope that all serious libraries will consider buying a copy. [Fernando Q. Gouvêa]

Publication Data

Cryptography: An Introduction, edited by V. V. Yaschenko. Student Mathematical Library, vol. 18. American Mathematical Society, 2002. Paperback, 229 pp., $39.00 ($31.00 to AMS members). ISBN 0-8218-2986-6.

Lectures on Generating Functions, by S. K. Lando. Student Mathematical Library, vol. 23. American Mathematical Society, 2003. Paperback, 128 pp., $29.00 ($23.00 to AMS members). ISBN 0-8218-3481-9.

Correspondence of John Wallis (1616–1703), Volume I (1641–1659), ed. by Philip Beeley and Christoph J. Scriba. Oxford University Press, 2003. Hardcover, 651 pp., $225.00.. ISBN 0-19-851066-7.

James Stirling's Methodus Differentialis: An Annotated Translation of Stirling's Text, by Ian Tweddle. Springer-Verlag, 2003. Hardcover, 295 pp., $129.00. ISBN 1-85233-723-0.

Kevin Anderson (andersk@mwsc.edu) is assistant professor of mathematics at Missouri Western State College.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME. He is interested in number theory, the history of mathematics, Christian theology, poetry, science fiction, comic books, politics, classics, and football (the real thing, not the American version).

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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.

MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu).
Last modified: Sun Mar 14 13:49:15 Eastern Standard Time 2004