Read This!

The MAA Online book review column

Briefly Noted

March 2005

There are three major secondary sources of information about Hamilton (1805-1865): Robert P. Graves' 3-volume Life of Sir William Rowan Hamilton (1882, 1885, 1889), Hankins' volume, and (slightly) more recently, Seán O' Donnell's William Rowan Hamilton: Portrait of a Prodigy (1983). Although Graves does include biographical material, his work consists mainly of selections from Hamilton's poetry, correspondence and other papers, and is thus closer to being a primary source. O'Donnell is emphatic that his work is not a competitor to Hankins', describing his own volume as "an attempt... to further an understanding of the individual rather than what he did."

But it should not be inferred that Hankins ignores Hamilton the man. We see not only Hamilton's work in optics, dynamics, and mathematics but also his philosophy, his poetry, and his professional and personal life. All these are intertwined and interpreted in the context of Victorian science in Ireland. The ten-page bibliographical essay is far richer than an itemized bibliography could ever be.

Mathematicians are most interested in Hamilton's creation (discovery, perhaps?) of quaternions. Once he represented complex numbers as number couples and interpreted them geometrically using perpendicular axes in a plane, the next logical step seemed to be number triples interpreted geometrically using three mutually perpendicular axes in space. Hankins describes in full mathematical detail Hamilton's long and fruitless quest for triples having the properties he desired. It is enthralling to watch each attempt and to see how Hamilton's effort to fix what was "wrong" brought him ever closer to that final step — quaternions. Quaternions, which fulfill all field properties save commutativity under multiplication, were a crucial factor (arguably the crucial factor) in freeing algebra from its ties to arithmetic.

The questions to ask about Hankins' Hamilton 25 years after its original publication are, "Has it been supplanted?" and "Is it worth the challenging read?" The answers are emphatically "No " and "Yes!" respectively. [Patricia R. Allaire]

This book should be on every mathematician's coffee-table!

Pi: A Source Book is a compendium of seventy articles on π (and related things, all cool). These are followed by four appendices, and "A Pamphlet on Pi." The latter is itself filled with arcana: a discussion of the normality question, the hugely fascinating business of Bailey-Borwein-Plouffe formulas which allow for the isolated calculation of individual binary digits of π (and what about other bases, such as 10?). The "Pamphlet" even sports Kaplansky's "A Song about Pi," which, according to Kaplansky's musicological data is a song of non-trivial complexity (only one in five songs, but fully half of the songs in Woody Allen's movies, possess such an exotic refrain-structure...).

The seventy articles comprising the source-book proper range from historical articles and classics by such players as Wallis, Huyghens, Newton, and Euler, to the articles on irrationality and transcendence (with e thrown in for good measure) by Lambert, Hermite, Lindemann, Weierstrass, and Hilbert. There are also the remarkable 17-line proof of π's irrationality given in 1946 (published in 1947) by Ivan Niven, Ramanujan's magical formulas, Mahler on approximation and Baker on linear forms in logarithms, and stuff by Van der Poorten on Apéry and the irrationality of the zeta function at 3. The Chudnovsky brothers weigh in with a discussion of approximations and complex multiplication à la Ramanujan.

Qua numerical analysis there is also a smorgasbord available, from Archimedes' first true algorithm for determining π, through (e.g.) Shanks' 1853 hand-calculation (!) of the first 607 decimal places of π, Brent's 1976 article on multi-precision arithmetic, and various things Borwein (et al), to Kananda's 1988 determination of the first 200 million digits of π in 1988. (Kanada's beyond a trillion now, I am told).

Indeed, Pi: A Source Book is truly an amazing book, irresistible in its own way, and filled with gems. And once it's on your coffee-table, feel free to do more than just browse: it's pretty well-suited for more in-depth study — how could it be otherwise, given the high density of classic stuff in these 800 pages? The authors provide a quintette of points of entry to the major themes represented in the book (caveat: they are arranged chronologically). For example, someone interested in questions of irrationality and transcendence should start with Niven's 1947 coup, go next to Van der Poorten on Apéry's work, and then to Hilbert on π and e. And on from there, of course.

Obviously the book is highly recommended. [Michael Berg]

A great deal of nonsense has been written about the so-called golden ratio,

often called Φ: how it was built into the Great Pyramid, how it determines the prettiest rectangle and where people's navels are situated, and so on. (For anti-nonsense, see George Markowsky's "Misconceptions about the Golden Ratio", College Math. J. 23 (1992) #1, 2-19, his review of Mario Livio's The Golden Ratio in Notices of the AMS 52 (2005) #3, 344-347, or my Numerology, MAA, 1997, Ch. 29.)

Given the human hunger for marvels, fictitious or otherwise, golden numberism might inevitably have arisen, but the person who got the ball rolling was Adolph Zeising (1810-1876) in an 1854 book whose translated title is An Exposition of a New Theory of the Proportions of the Human Body.

Roger Herz-Fischler, author of A Mathematical History of the Golden Number (1987, Dover reprint, 1998) and The Shape of the Great Pyramid (Wilfrid Laurier U. Press, 2000) decided to find out everything he could about Zeising, and this book reports his findings. Zeising was German, taught school for a time, and from 1853 until his death had sufficient means to be an independent scholar. He wrote poems, novels, and plays and works on philosophy, literature, and esthetics. He reviewed books, translated Xenophon, and evidently kept very busy. He is remembered today only as the father of golden numberism.

The book is scholarly and as thorough as is possible, with a complete bibliography of Zeising's known works. It is not for light leisure reading, but it is sure to be the chief source of Zeisingiana from now on. The extensive German quotations are for the most part not translated into English. Other than his works on the golden ratio, Zeising's writings were not much noticed by his contemporaries, nor was he. The book can induce melancholy reflections on the futility of much of human effort. [Underwood Dudley]

Between the 16th and the 18th centuries, Portuguese explorers brought European culture to many parts of the world, including East Asia. This cultural interaction led to the transmission of a good deal of European science and mathematics to the East and to the establishment of many (mostly Jesuit) educational institutions. This collection of historical essays aims to explore the history of this interaction.

As the title indicates, History of Mathematical Sciences: Portugal and East Asia II is the second collection of essays on the subject. The first collection was the proceedings of a conference held in Portugal in 1995. It was published in Portugal by Fundação Oriente and unfortunately seems to be rather hard to obtain. This second collection represents the proceedings of a second conference, this one held in Asia at the University of Macao in 1998. Scholars from both Europe and Asia participated in both conferences.

The range of topics is much wider than the modern term "mathematical sciences" might suggest, covering all sorts of aspects of scientific knowledge of the period, from geography to cosmology. A couple of the essays have an even more general scope, dealing with the movement of all sorts of ideas between Europe and the East. Of particular interest to me were the essay on Jesuit mathematical textbooks and the essays that investigated what, if any, influence ran in the other direction, from Asia to Europe. (They do find some.)

Overall, it is historians of science that will find this book most interesting, particularly those whose research focuses on intercultural issues. At $98 for 182 pages, I suspect that only the bigger libraries will want to have a copy. [Fernando Q. Gouvêa]

Publication Data

Pi: A Source Book, Third Edition, ed. by Lennart Berggren, Jonathan Borwein, and Peter Borwein. Springer-Verlag, 2004. Hardcover, xix + 797 pp., $89.95. ISBN 0-387-20571-3.

Adolph Zeising: The Life and Work of a German Intellectual , by Roger Herz-Fischler. Mzinhigan Publishing, 2005. Hardcover with CD-ROM, 186 pp., $42.00. ISBN 0-9693002-6-3.

History of Mathematical Sciences: Portugal and East Asia II — Scientific Practices and the Portuguese Expansion in Asia (1498-1759), ed. by Luís Saraiva. World Scientific, 2004. Hardcover, 182 pp., $98.00. ISBN 981-256-078-5.

Pat Allaire (pallaire@qcc.cuny.edu) is Associate Professor of Mathematics and Computer Science at Queensborough Community College, CUNY and serves as secretary of the Canadian Society for History and Philosophy of Mathematics.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University.

Underwood Dudley has retired from DePauw University and is now teaching one section of Calculus at Florida State University.

Fernando Q. Gouvêa has too many books on his desk.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
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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.