Book Review A Wealth of Numbers Reviewed by Andy R. Magid to do with the quality, or even correctness, of their A Wealth of Numbers mathematical content. For example, consider the Benjamin Wardhaugh Amazon Best Sellers rankings of the following: Princeton University Press, April 2012 Constance Reid’s excellent From Zero to Infinity is Hardcover, 388 pages, US$45.00 number 907,624, only slightly better than Marilyn ISBN-13: 978-0691147758 vos Savant’s execrable The World’s Most Famous Math Problem, number 966,319, while Lancelot When it comes to mathematical exposition, we Hogben’s venerable (first published in 1937) and mathematicians are diligent consumers. We buy valuable Mathematics for the Million substantially non-textbook monographs, read Notices articles, outranks both at 153,398. For further comparison, and attend departmental colloquia, generally inde- John Allen Paulos’s 1980s classic Innumeracy ranks pendent of whether the exposition is high quality 16,221 (all ranks observed 16 November 2012). This (Notices), mixed (monographs), or low (most collo- highly unrepresentative sample of mathematics quia). Perhaps we are motivated by the hope that books for the general reader currently available ideas and methods from other fields will inspire consists of books this reviewer happened to read, our work in our own specialty, or perhaps we either as a general reader himself before becoming feel the duty to be informed about the world of a mathematician or, for the later ones, out of research mathematics in general. In any event, we curiosity when the books became newsworthy devote time and effort to consuming mathematical (Paulos) or notorious (vos Savant). I suspect most exposition. Fortunately, some mathematicians are mathematicians have similarly limited exposure diligent producers of mathematical exposition, to popular mathematics books. “Limited” includes although the hard work of producing exposition is limited in time: while I wouldn’t be surprised if usually not rewarded as well, or even at all, com- my American coevals among the mathematical pared to original research or earning funding for community are familiar with Reid, maybe even original research. The mathematical world would Hogben, our millennial successors probably aren’t, probably be better off if we were all better citizens and neither would be our prewar predecessors. in this regard, say, by accepting Mathematical For all any of us know, the general public is being Reviews assignments, but it’s also true that there offered, and buying, mathematical exposition that is some art to exposition, and not everyone who is erroneous in content, logically opaque, and does it does it artfully. Those who do it well earn, gracelessly written. if not the rewards their employers could offer, the Fortunately, Benjamin Wardhaugh has had the respect of their fellow mathematicians who, after happy idea to anthologize selections of writings all, set the standards for what qualifies as excellent about mathematics for the general reader. Ward- exposition. haugh, a postdoctoral fellow at All Souls College When it comes to mathematical exposition specializing in the history of mathematics, as- for the nonmathematical public, however, math- sembled his selections ranging widely over topics ematicians are at best interested bystanders. (from Mathematical Tricks to Fiction and Humor) Popularizations of mathematics, like other pub- and time (1481 to 2004). With approximately one lishing ventures, succeed or fail in the marketplace hundred excerpts, each with a brief introduction, through coincidences of taste, timeliness, advertis- comprising eleven chapters, each of which also ing, and fate—which may or may not have anything has a short introduction, in some 370 pages, in- cluding index, the selections are necessarily brief. Andy R. Magid is George Lynn Cross Research Professor Nonetheless, Wardhaugh has certainly collected a Emeritus of Mathematics at the University of Oklahoma. His representative sample of how people write when email address is [email protected]. they are writing in English about mathematics DOI: http://dx.doi.org/10.1090/noti959 for readers whose formal mathematics education 326 Notices of the AMS Volume 60, Number 3 is only assumed to be “school-level”. As such, “Two Rules must be memorized in connection this book belongs in every university library or with the Circular Parts. They are as follows: similar places where mathematicians and others (1) Sine Middle Part = Product of Cosines of who are curious about the quality of mathematical Opposite Parts. popularizations can dip into it. (2) Sine Middle Part = Product of Tangents of Here are some samples of the range of topics Adjacent Parts.” and times: from A Path-way to Knowledge by John Examples follow. There is no further discussion of Tapp, London, 1621, instructions about how to why these rules might be true. reduce fractions, presented as a dialogue between From the selection Making a Star Clock by Roy a teacher, Theodore, and a pupil, Junius; from Sir Worvill, 1974 [this selection is about constructing Isaac Newton’s Philosophy Explained for the Use of a device to measure the angle the line containing Ladies by Francesco Algarotti, London, 1739, a Dubne and Merak (unnamed in the selection) description of Kepler’s second law; from Geometry makes with the celestial parallel through Polaris. without Axioms by Thomas Perronet Thompson, Worvill has instructed his readers to assemble 1833, an eccentric discussion of how to present materials (cardboard and a paper fastener) and Euclidean geometry without axioms or postulates, tools (compasses, scissors, and knife) and to cut the especially the fifth; from Algebra for the Practical cardboard into two 8-inch squares]: “On the other Man by J. E. Thompson, London, 1931, the cookbook cardboard square draw…five concentric circles procedures for solving a cubic equation; and from …the largest one of radius four inches and the the science fiction novel Blue Mars by Kim Stanley others, proceeding inwards, of three and a half Robinson, London, 1996, a description of a twenty- inches, three inches, two and a half inches and two third century physicist as viewed by an (extremely inches. Draw a diameter in the outer circle and superannuated) scientist from our own era. mark off angles of thirty degrees so that there are But the reader should be cautioned that the twelve spaces or sectors. On the outer circle these representative nature of this anthology means are marked with two-hourly divisions from twelve that it is definitely not a “greatest hits” collection. o’clock to midnight throughout the twenty-four Wardhaugh warns his readers in the first paragraph hours, although of course the nocturnal can only of his preface that his collection is designed to be used when it is dark enough to see the stars.” answer the questions, “How did ordinary people The selection continues in this vein, including think about mathematics in the past? How did instructions for use. they write about it? How did they learn and teach From the selection Turtle Fun by Serafim Gas- it?” And although he does later call his selections coigne, 1985 [The first two paragraphs of the “a rambling garden of delights” (a point of view selection, in their entirety]: “The world of micro- to which this reviewer will take exception), it is computers is forever changing. There is always a fact that Wardhaugh has alerted us that this something new to learn and find out. How can you book is a sourcebook of historical mathematical keep up? Will you be able to program the com- popularization, not an invitation-to-mathematics puters of the future? And what about ‘intelligent collection for the general reader. machines’ and robots, especially the LOGO turtle? Unfortunately, the copywriters who prepared “This book will help you to keep up with events the cover blurbs for the volume failed to get by introducing you to a very powerful control the message. They write, “A treasure-house of language of the future, called LOGO. LOGO has in popular mathematics, A Wealth of Numbers will fact been developed by computer scientists working in a new science called Artificial Intelligence. LOGO provide many hours of fun and learning to anyone has been written for young programmers. It comes who loves math or science.” No, it won’t. Some from another language called LISP, which is one examples: of the main languages used to control ‘intelligent’ From the selection Napier’s Rules by Alan Clive computers and robots. Using such commands Gardner, 1956 [Gardner is discussing a spherical as STARTROBOT or REPEAT [FORWARD 50 LEFT o triangle with angles A = 90 ; B; C and opposite 120] for example, you can either drive a small sides a; b; c]: “Draw a circle and divide it into five mechanical device called the floor turtle or control sectors ([figure indicated]). Starting from the right the screen turtle, a movable cursor on your TV angle, which is indicated by a heavy line [drawn screen. You can also teach the turtle your own center to 12 o’clock], give each sector the name commands, called ‘procedures’, which can include of one part of the triangle, naming the parts in graphics text and sound.” order round the circle. The two parts next to the The historical value of these examples is obvi- right angle are named normally [clockwise b; c], ous. The common thread of these selections, of but the other three parts are replaced by their course, is their lack of mathematical content, at complements [clockwise 90o − B; 90o − A; 90o − C]. least as far as mathematicians would be concerned. March 2013 Notices of the AMS 327 There are also selections from Euler, Polya, van der Waerden, Sylvester, Feynman, and others, with real mathematical content. But even in those the selections are so brief that mathematics must be interrupted midconcept.