Book Review

A Wealth of NumbersReviewed by Andy R. Magid

A Wealth of NumbersBenjamin WardhaughPrinceton University Press, April 2012Hardcover, 388 pages, US$45.00ISBN-13: 978-0691147758

When it comes to mathematical exposition, wemathematicians are diligent consumers. We buynon-textbook monographs, read Notices articles,and attend departmental colloquia, generally inde-pendent of whether the exposition is high quality(Notices), mixed (monographs), or low (most collo-quia). Perhaps we are motivated by the hope thatideas and methods from other fields will inspireour work in our own specialty, or perhaps wefeel the duty to be informed about the world ofresearch mathematics in general. In any event, wedevote time and effort to consuming mathematicalexposition. Fortunately, some mathematicians arediligent producers of mathematical exposition,although the hard work of producing exposition isusually not rewarded as well, or even at all, com-pared to original research or earning funding fororiginal research. The mathematical world wouldprobably be better off if we were all better citizensin this regard, say, by accepting MathematicalReviews assignments, but it’s also true that thereis some art to exposition, and not everyone whodoes it does it artfully. Those who do it well earn,if not the rewards their employers could offer, therespect of their fellow mathematicians who, afterall, set the standards for what qualifies as excellentexposition.

When it comes to mathematical expositionfor the nonmathematical public, however, math-ematicians are at best interested bystanders.Popularizations of mathematics, like other pub-lishing ventures, succeed or fail in the marketplacethrough coincidences of taste, timeliness, advertis-ing, and fate—which may or may not have anything

Andy R. Magid is George Lynn Cross Research ProfessorEmeritus of Mathematics at the University of Oklahoma. Hisemail address is amagid@math.ou.edu.

DOI: http://dx.doi.org/10.1090/noti959

to do with the quality, or even correctness, of theirmathematical content. For example, consider theAmazon Best Sellers rankings of the following:Constance Reid’s excellent From Zero to Infinity isnumber 907,624, only slightly better than Marilynvos Savant’s execrable The World’s Most FamousMath Problem, number 966,319, while LancelotHogben’s venerable (first published in 1937) andvaluable Mathematics for the Million substantiallyoutranks both at 153,398. For further comparison,John Allen Paulos’s 1980s classic Innumeracy ranks16,221 (all ranks observed 16 November 2012). Thishighly unrepresentative sample of mathematicsbooks for the general reader currently availableconsists of books this reviewer happened to read,either as a general reader himself before becominga mathematician or, for the later ones, out ofcuriosity when the books became newsworthy(Paulos) or notorious (vos Savant). I suspect mostmathematicians have similarly limited exposureto popular mathematics books. “Limited” includeslimited in time: while I wouldn’t be surprised ifmy American coevals among the mathematicalcommunity are familiar with Reid, maybe evenHogben, our millennial successors probably aren’t,and neither would be our prewar predecessors.For all any of us know, the general public is beingoffered, and buying, mathematical exposition thatis erroneous in content, logically opaque, andgracelessly written.

Fortunately, Benjamin Wardhaugh has had thehappy idea to anthologize selections of writingsabout mathematics for the general reader. Ward-haugh, a postdoctoral fellow at All Souls Collegespecializing in the history of mathematics, as-sembled his selections ranging widely over topics(from Mathematical Tricks to Fiction and Humor)and time (1481 to 2004). With approximately onehundred excerpts, each with a brief introduction,comprising eleven chapters, each of which alsohas a short introduction, in some 370 pages, in-cluding index, the selections are necessarily brief.Nonetheless, Wardhaugh has certainly collected arepresentative sample of how people write whenthey are writing in English about mathematicsfor readers whose formal mathematics education

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is only assumed to be “school-level”. As such,this book belongs in every university library orsimilar places where mathematicians and otherswho are curious about the quality of mathematicalpopularizations can dip into it.

Here are some samples of the range of topicsand times: from A Path-way to Knowledge by JohnTapp, London, 1621, instructions about how toreduce fractions, presented as a dialogue betweena teacher, Theodore, and a pupil, Junius; from SirIsaac Newton’s Philosophy Explained for the Use ofLadies by Francesco Algarotti, London, 1739, adescription of Kepler’s second law; from Geometrywithout Axioms by Thomas Perronet Thompson,1833, an eccentric discussion of how to presentEuclidean geometry without axioms or postulates,especially the fifth; from Algebra for the PracticalMan by J. E. Thompson, London, 1931, the cookbookprocedures for solving a cubic equation; and fromthe science fiction novel Blue Mars by Kim StanleyRobinson, London, 1996, a description of a twenty-third century physicist as viewed by an (extremelysuperannuated) scientist from our own era.

But the reader should be cautioned that therepresentative nature of this anthology meansthat it is definitely not a “greatest hits” collection.Wardhaugh warns his readers in the first paragraphof his preface that his collection is designed toanswer the questions, “How did ordinary peoplethink about mathematics in the past? How didthey write about it? How did they learn and teachit?” And although he does later call his selections“a rambling garden of delights” (a point of viewto which this reviewer will take exception), it isa fact that Wardhaugh has alerted us that thisbook is a sourcebook of historical mathematicalpopularization, not an invitation-to-mathematicscollection for the general reader.

Unfortunately, the copywriters who preparedthe cover blurbs for the volume failed to getthe message. They write, “A treasure-house ofpopular mathematics, A Wealth of Numbers willprovide many hours of fun and learning to anyonewho loves math or science.” No, it won’t. Someexamples:

From the selection Napier’s Rules by Alan CliveGardner, 1956 [Gardner is discussing a sphericaltriangle with angles A = 90o, B, C and oppositesides a, b, c]: “Draw a circle and divide it into fivesectors ([figure indicated]). Starting from the rightangle, which is indicated by a heavy line [drawncenter to 12 o’clock], give each sector the nameof one part of the triangle, naming the parts inorder round the circle. The two parts next to theright angle are named normally [clockwise b, c],but the other three parts are replaced by theircomplements [clockwise 90o − B,90o −A,90o −C].

“Two Rules must be memorized in connectionwith the Circular Parts. They are as follows:

(1) Sine Middle Part = Product of Cosines ofOpposite Parts.

(2) Sine Middle Part = Product of Tangents ofAdjacent Parts.”

Examples follow. There is no further discussion ofwhy these rules might be true.

From the selection Making a Star Clock by RoyWorvill, 1974 [this selection is about constructinga device to measure the angle the line containingDubne and Merak (unnamed in the selection)makes with the celestial parallel through Polaris.Worvill has instructed his readers to assemblematerials (cardboard and a paper fastener) andtools (compasses, scissors, and knife) and to cut thecardboard into two 8-inch squares]: “On the othercardboard square draw…five concentric circles…the largest one of radius four inches and theothers, proceeding inwards, of three and a halfinches, three inches, two and a half inches and twoinches. Draw a diameter in the outer circle andmark off angles of thirty degrees so that there aretwelve spaces or sectors. On the outer circle theseare marked with two-hourly divisions from twelveo’clock to midnight throughout the twenty-fourhours, although of course the nocturnal can onlybe used when it is dark enough to see the stars.”

The selection continues in this vein, includinginstructions for use.

From the selection Turtle Fun by Serafim Gas-coigne, 1985 [The first two paragraphs of theselection, in their entirety]: “The world of micro-computers is forever changing. There is alwayssomething new to learn and find out. How can youkeep up? Will you be able to program the com-puters of the future? And what about ‘intelligentmachines’ and robots, especially the LOGO turtle?

“This book will help you to keep up with eventsby introducing you to a very powerful controllanguage of the future, called LOGO. LOGO has infact been developed by computer scientists workingin a new science called Artificial Intelligence. LOGOhas been written for young programmers. It comesfrom another language called LISP, which is oneof the main languages used to control ‘intelligent’computers and robots. Using such commandsas STARTROBOT or REPEAT [FORWARD 50 LEFT120] for example, you can either drive a smallmechanical device called the floor turtle or controlthe screen turtle, a movable cursor on your TVscreen. You can also teach the turtle your owncommands, called ‘procedures’, which can includegraphics text and sound.”

The historical value of these examples is obvi-ous. The common thread of these selections, ofcourse, is their lack of mathematical content, atleast as far as mathematicians would be concerned.

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There are also selections from Euler, Polya, vander Waerden, Sylvester, Feynman, and others, withreal mathematical content. But even in those theselections are so brief that mathematics mustbe interrupted midconcept. This well serves thepurpose of a historical sourcebook, although“anyone who loves math or science” lured by thecover copywriter’s blurb into looking at this bookfor “many hours of learning and fun,” may well bedisappointed.

Does this matter? Readers lured into buyinga book by false advertising can always return itand get their money back. More generally, shouldmathematicians care about books on popularmathematics sold to the general public? I think so.

Consider the greatest years for popular supportfor mathematics (and science) in the United States,arguably from the Sputnik launch to the Mans-field Amendment. That was also the era of twocolossal achievements in the area of mathematicalpopularization: Martin Gardner’s Scientific Amer-ican column “Mathematical Games”, and JamesR. Newman’s anthology The World of Mathematics,published in 1956. To quote Wikipedia, the latteris “a four volume library on the literature of math-ematics from A’h-mos the Scribe to Albert Einstein,presented with commentaries and notes,…[it] cov-ers many branches of mathematics and representsa 15-year effort by Newman to collect what he feltwere the most important essays in the field…froma biography of Srinivasa Ramanujan by Newmanto Bertrand Russell’s Definition of Number. [TheWorld of Mathematics] is often praised as suitablefor any level of mathematical skill.” It is stillavailable new (Amazon rank 553,586).

Whether America’s choice to generously sup-port mathematical research in those years—unparalleled, I believe, before or since—wasencouraged by Gardner and Newman or whetherthe popularity of Gardner and Newman was anartifact of the prestige with which America held itsmathematical community in those years is openfor debate. But we could certainly use figureslike them now, making by example the publiccase for mathematics in the intellectual life ofall Americans. Benjamin Wardhaugh’s anthologyis a useful tool for historians of mathematics,professional and amateur, who would like to knowwhat forms mathematical popularization has taken.But the mathematical community could also use anupdated The World of Mathematics, presenting thepublic with an anthology of the best writing aboutthe best mathematics. It would certainly deliverthe insight and entertainment the cover blurb of AWealth of Numbers falsely promises. Who knows,it might even inspire a resurgence of the liberalfunding of mathematics we once knew.

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