Book Review

The MαTH βOOKReviewed by Sarah J. Greenwald andJill E. Thomley

The MαTH βOOK: From Pythagoras to the 57thDimension, 250 Milestones in the History ofMathematicsClifford A. PickoverSterling Milestones, 2009, 2012528 pagesUS$22.00 (hardcover), US$13.00 (paperback)ISBN-13: 978-1-4027-8829-1

The Neumann Prize from the British Society forthe History of Mathematics is awarded to a bookwritten for a general audience that incorporatesmathematical history. The book under review, byClifford Pickover, won this prize in 2011. Onereason is surely the visual beauty of The MαTHβOOK, starting with its striking cover, which looksgreat on a coffee table. In fact, it has been sittingon our own coffee tables, and, while undersizedcompared to our other books, it holds its own.Catching the light from a certain angle, the coverreveals numbers and objects in what resemblesa colorful foil-stamped effect. Look closely tosee spheres and Möbius bands, as well as thenumbers from 1 to 999. Certain numbers arehighlighted—they are the primes from 2 to 997.

Sarah J. Greenwald is professor of mathematics and aWomen’s Studies core faculty member at Appalachian StateUniversity. Her email address is greenwaldsj@appstate.edu.

Jill E. Thomley is professor of statistics at Appalachian StateUniversity. Her email address is thomleyje@appstate.edu.

DOI: http://dx.doi.org/10.1090/noti1238

Why is the number 1 also highlighted? We guessthat was a mistake on the part of the layout artist.Nevertheless, we love the cover effects and feelthey would definitely appeal to those browsing theshelves in a bookstore or a library.

The book is a collection of two-page articles,in which the first page is devoted to text and thefacing page to an accompanying image. So eachturn of a page opens to a new and wonderfulpicture. Some of the images successfully makeconnections and extend the mathematics of thearticles they are paired with. For instance, anapproximation of a hyperbolic dodecahedron byPaul Nylander accompanies the article on Platonicsolids. However, other images, though still visuallyappealing, are less mathematically effective, suchas the cicada picture accompanying an article onprime number life cycles.

The organization of the book is chronological,with each article focusing on one “milestone.” Thefirst is “Ant Odometer,” which is dated circa 150million BC, and the last is “Mathematical UniverseHypothesis,” dated 2007. In each article the datecorresponding to the milestone is prominentlydisplayed in large type. This is attractive graphicdesign, but in some cases, assigning a single daterather than a range of dates is awkward or evendownright misleading. For example, the articleon non-Euclidean geometry carries the date of1829, as that is the year of publication of NikolaiLobachevsky’s paper “On the principles of geom-etry.” The implication is that the field started in1829, whereas in fact it evolved slowly as mathe-maticians through the centuries grappled with the

384 Notices of the AMS Volume 62, Number 4

fifth postulate (see, e.g., [3]). To be fair, the articledoes highlight some other contributors, includingEuclid of Alexandria, János Bolyai, Bernhard Rie-mann, and Albert Einstein; Carl Friedrich Gauss,however, is notably absent. For some articles, thebook uses not publication dates but the date ofa significant event to indicate the timing of themathematical development being described. Forexample, the article on Bayes’s Theorem gives theyear of Thomas Bayes’s death as the milestone,even though the theorem that bears his name wasnot officially published until three years later, in1764 [1].

In fact, the use of the term “milestones” createsits own awkwardness. For example, the Klein bottleis discussed a number times in the book and evenhas its own article, as do several other surfaces. Tocall the Klein bottle a “milestone” makes little sense.A more fitting milestone might have been FelixKlein’s Erlanger Programm, which had profoundeffects on geometry research but is not mentionedat all in the book. We also wouldn’t have includedcertain other topics in the book as mathematical“milestones,” such as the Rubik’s Cube or thetelevision show NUMB3RS. While they certainlyhelped popularize mathematics and are of interestto a general audience, we do not consider them tobe a true “milestone” in the history of mathematics.Perhaps a different term than “milestones” wouldhave worked better for the articles.

Despite the chronological ordering of the articles,the real focus of the book is on people and places,starting with the book’s subtitle From Pythagorasto the 57th Dimension. Pythagoras is mentioned inseveral articles, including one called “PythagorasFounds Mathematical Brotherhood.” The 57thdimension is found in the article “The Quest for LieGroup E8,” which highlights the contributions ofMarius Sophus Lie and Wilhelm Karl Joseph Killing.We like the way Pickover has included contributionsnot just of the historical stars of mathematics butalso of many different people in many differentcultures. There are some missteps: the name of theHausa people of Africa is misspelled as “Hasua,”and the statement that “Abaci are still used in. . .parts of the Soviet Union” needs “former” before“Soviet.” Nevertheless, the fact that contributionsfrom so many peoples are included is a realstrength of the book.

Sometimes the effort to dramatize the contri-butions of diverse mathematicians pulls the focusaway from the mathematics. For example, theinside front cover of the book poses the ques-tion, “Why was the first woman mathematicianmurdered?” Virtually the entire article on Hypatiaof Alexandria focuses on the circumstances of herdeath, in rather gruesome detail, rather than herlife or mathematical work; the latter receives only

a scant three sentences. Even worse, from a mathe-matics history standpoint, this article asserts asfact many speculative details about Hypatia, suchas those from a fictionalized account of her lifeand work [4]. The article also explicitly mentionsHypatia’s attractiveness even though there is nohistorical evidence indicating what she looked like[2]. For example, she might have resembled Egyp-tian women of the day rather than the blond-haired,nude image of Hypatia that was imagined by anineteenth-century artist and that accompaniesthe article. The author’s treatment of other femalemathematicians seems more consistent with theauthor’s presentations of male mathematicians,where he provides some interesting historical andpersonal details but focuses on their mathematicalcontributions. At times the exposition is a bitsensationalist and exploitive, as in the use of theword “sinister” when describing the quipu as anInca death calculator, but, overall, the writing,like in Pickover’s other books, is creative andengaging and connects mathematics to society,human thought, and applications.

The book successfully highlights many topicsthat would be of interest to a general audience. Itmakes for great recreational reading for students ofscience and mathematics and interested laypeople.It can serve as a starting point for those who wantto learn more, as further readings are included inthe notes at the end of the book. Although certaintopics like puzzles and geometric spaces areoverrepresented, the book successfully showcasesthe beauty, utility, and breadth of mathematics. Ifyou are seeking a scholarly work on the history ofmathematics, look elsewhere. However, if insteadyou want a handsome, lively, and entertainingbook, we highly recommend this one.

References[1] Thomas Bayes, An essay towards solving a problem

in the doctrine of chances, Philosophical Transactionsof the Royal Society of London 53 (1764), 370.

[2] Michael Deakin, Hypatia of Alexandria: Mathemati-cian and Martyr, Prometheus Books, Amherst, NewYork, 2007.

[3] John O’Connor and Edmund Robertson, MacTutorHistory of Mathematics Archive: Non-Euclidean Ge-ometry, http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html

[4] Elbert Hubbard, Little Journeys to the Homes of GreatTeachers, The Roycrofters, East Aurora, New York,1908.

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