Read This!

The MAA Online book review column

Briefly Noted

November 2004

Theodore Porter's Karl Pearson: The Scientific Life in a Statistical Age explores the fullness and richness of Pearson's intellectual and emotional life, shows us how "the toil of the years" led to the revolution he wrought in statistics. It details his early intellectual journeys as a philosopher of science, culminating in his then pathbreaking book, The Grammar of Science. It describes his strong and lifelong belief in socialism, and his strong advocacy of education later in life for the working classes and of equal rights for women, and his efforts to help achieve these goals.

All these interests merged when he was about 35 years old, when he had what one might call an epiphany: he realized that the solution to these and other problems required careful and detailed statistical analysis. The application of statistics to science and to human affairs became the ruling passion of his life for the next forty years. There are many strands in this book, and as one reads it, it makes one pause again and again to think about the many fundamental questions which engaged Pearson: the nature of knowledge, the nature of science, the question of equal rights for all sections of the people, the problem of scientific rivalry as in the case of Pearson and R. A. Fisher — and many more such issues. The description of Pearson's sometimes stormy relationships with people, including a few key women in his life, one of them his first wife, is also fascinating and thought provoking.

The book is not merely a picture of a great and revolutionary thinker, it makes the reader reflect on many key questions. As such, I would recommend it as book which would both widen and deepen the reader's understanding of the human condition in general. Most of the book is about these general questions, and only a quarter is about the technical questions of statistics, and those too are explained in a non-technical manner.The book would therefore be a source of both pleasure and profit to any serious reader. [Ramachandran Bharath]

I took great pleasure in reading Across the Board: The Mathematics of Chessboard Problems by John J. Watkins. This book is extremely well writen and is, no doubt, the best exposition of the connection between the chessboard problems and recreational mathematics. The author surveys all the well-known problems about chess and the chessboard: "can a knight follow a path that covers every square once, ending on the starting square?" "How many queens are needed so that every square is targeted or occupied by one of the queens?" The problems are treated in depth from their beginnings through to their status today. Using graph theory, the author gently guides the reader to the forefront of current research in this area of mathematics. Exercises are provided to enhance the reader's involvement.

The book is organized in thirteen chapters. The first chapter introduces the main topics with which we will be concerned: knight's tours, domination, independence, coloring, geometric problems, chessboards on other surfaces, and polyominoes. The next two chapters deal with knight's tours, from the earlier workof De Moivre, Euler, Hamilton up to most recent results known today. Chapter four is devoted to magic squares. The beginning of the chapter includes a delightful presentation of the work of Muhammad Ibn Muhammad, an African Mathematician who discovered a very ingenious idea for constructing magic squares of odd order. This construction was later on rediscovered by Bachetin the early 1600s.

Chapters five and six generalize in different ways from the ordinary chessboard. If we identify the left and the right edges, the chessboard becomes a flat torus and the chess pieces gain considerable freedom of movement since the edges, in effect, disappear. If we identify just one pair of opposite edges, we have a cylindrical board. Finally, we can identify the top and bottom edges of the rectangle just as it was done for the torus, and also identify the left and the right edges, but this time with opposite orientation, so the top and bottom are identified inthe normal way, but the sides get a half-twist before they are identified. A rook in the left-most column moving up and going off the top of the board reappears at the bottom of the board in the same column; but a rook in the bottom row going off the right side of theboard will reappear at the upper left in the top row. This makes the board into a the Klein bottle.

In chapters 7, 8 and 9 the central concept is domination. This is one of the central ideas in graph theory, and is especially important in the application of graph theory to the real world. Of all the chessboard-domination problems, it is that of the queen that continues to hold the most interest among mathematicians. It is a remarkably difficult problem and one that is far from solved even today, although there is much that is known. One nice result from chapter nine is that among chessboards with more than four rows, the 5x12 chessboard is the largest board that can be dominated by four queens. I have a strong feeling that Across the Board will reveal the beauty of mathematics to students, teachers and math lovers. [Mohammed Aassila]

Jackie Stedall is amazing. After giving us a history of English algebra up to 1685 and an annotated translation of Thomas Harriot's manuscripts on algebra, here is her translation of John Wallis's famous Arithmetic of Infinitesimals (Arithmetica Infinitorum, first published in 1656). Thank you, Jackie; please never stop.

Wallis's subtitle gives a good summary of what the book is about: "A New Method of Inquiring into the Quadrature of Curves, and other more difficult mathematical problems". Wallis deploys infinitesimal arguments to solve mathematical problems, in particular problems of "quadrature" (which we would describe as "integration", or perhaps as "computing areas"). Though the book is presented in the classical way, as a sequence of Propositions, it is more exploratory than Euclidean in spirit. Wallis is not afraid to generalize from examples (Stedall correctly calls this "induction," thereby taking the risk of confusing mathematicians unfamiliar with this use of the term). In the latter half of the book the idea of interpolation becomes central as he attempts the quadrature of the circle. The book concludes with what are essentially formulas for π, representing it both as an infinite product and as a continued fraction.

The Arithmetica Infinitorum was written when Newton was 13 and Leibniz was 10 years old, demonstrating once again that much of what we think of as "calculus" was discovered before there was any such subject. (The "Nova Methodus" of Wallis's subtitle is echoed — on purpose, perhaps? — in Leibniz's own "Nova Methodus" article, the first printed account of the new calculus, in 1684.) Stedall's translation gives us access once again to this fascinating book, and her introduction helps us understand its place in history. Not to be missed. [Fernando Q. Gouvêa]

Contemporary Art and the Mathematical Instinct is a beautiful collection of art, published as a catalog to accompany an exhibition of the same name. The exhibition is organized and circulated by the Tweed Museum in Minnesota; it is now at its third stop of its tour — the University of Richmond — until December 12. The catalog features the work of 44 artists, from Arakawa to Zwieg, with commentary; in addition, there are three historical and philosophical essays on mathematics: learning it, doing it, and looking at it. The first of these, by Steven Luecking, gives a brief but very nice overview of the history of mathematical visualization, from string models to sculpture to computer models.

Is this exhibit really mathematics? It is, and it isn't, as the curator Peter Spooner is careful to point out. The ideas that inspire these works are sometimes mathematically shallow (there is a good bit of numerology and gee-whiz fractals), and the descriptions that accompany the works might make some mathematicians itch: "The numerical ratio [22/7]... is of course that of Pi," and " ...the Klein bottle, which like the Moebius strip, is a 3-D form made of one continuous surface."

And yet, the recurring theme in these works feels very mathematical: take an idea (an algorithm, a pattern, a shape), and play with it. What happens in special cases? What happens more generally? Many of the paintings and sculptures in this collection are not only beautiful art, but also beautiful theorems. Among the 44 artists are mathematicians John Simms (a student of the dynamicist Ethan Coven) and Dennis White, along with computer scientists and engineers whose work has taken an artistic turn.

As one of the commentaries notes,

Works like Cartwright's [and that of many other artists in this collection] challenge the assumption that art derived from mathematical information must automatically be sterile, cold, and divorced from the personal.

Publication Data

Across the Board: The Mathematics of Chessboard Problems, by John J. Watkins. Princeton University Press, 2004. Hardcover, 264 pp., $24.95. ISBN 0-691-11503-6.

The Arithmetic of Infinitesimals: John Wallis 1656, by Jacqueline A. Stedall. Sources and Studies in the History of Mathematics and the Physical Sciences. Springer-Verlag, 2004. Hardover, 192 pp., $119.00. ISBN 0-387-20709-0.

Contemporary Art and the Mathematical Instinct, ed. by the Tweed Museum, including articles by Stephen Luecking, John Sims, and Dennis White. Tweed Museum of Art, 2004. Paperback, 88 pp., $19.95. ISBN: 1-889523-27-5.

Mohammed Aassila is a mathematics professor whose research area is analysis. He is interested in mathematics competitions and is the author of two books on the subject: 300 Défis Mathématiques and Olympiades Internationales de Mathématiques.

Ramachandran Bharath teaches at the American University in Bulgaria.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

Annalisa Crannell's primary research is in topological dynamical systems, but she is also active in developing curricular materials for courses on "Mathematics and Art" as well as materials for writing across the curriculum.

Back Issues:

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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.