Book Review

Sources in theDevelopment of

MathematicsReviewed by Tom Archibald

Sources in the Development of MathematicsRanjan RoyCambridge University Press, 2011US$89.10, 994 pagesISBN-13: 978-0521114707

For much of the twentieth century, writing aboutthe history of mathematics emphasized aboveall the development of pure mathematics—andpure mathematics seen in a certain light. Ab-stract mathematics, of both the Bourbaki andnon-Bourbaki varieties, dominated the culturallandscape of twentieth-century mathematics andformed a natural focus for historians studying thatperiod. The creation of abstract structures, newworlds such as those of non-Euclidean geometry,and the apparently progressive evolution of rigorall posed special challenges for historians. Thesedevelopments also had significant philosophicalresonance, as the names Russell and Gödel sufficeto underline, and the importance of the history ofmathematics for philosophy was well illustrated(remains well illustrated) by the work of Lakatos.

The world changes, as does its mathematics, andhistorians looking at the mathematical world nowtend to see a somewhat different picture. Many ofthe same issues remain fundamentally important,but they are no longer seen as providing a completeaccount of all important aspects of mathematicalactivity. The consensus about what mathematicsis, as seen in twentieth-century historical writingsabout mathematics, no longer dominates. Historicalwriters today need to encompass a larger range ofmathematical activity in their studies.

Tom Archibald is professor of mathematics at Simon FraserUniversity. His email address is tarchi@sfu.ca.

DOI: http://dx.doi.org/10.1090/noti1057

One way to achieve such a fresh and broadviewpoint is to seek unifying themes that cutacross contemporary mathematical subfields. Ran-jan Roy’s erudite and broad study does just that,and in so doing provides an alternative threadthrough much historical mathematics of the lastfew centuries. Using series and products, bothfinite and infinite, as a leitmotif, Roy’s forty-onechapters give us just short of one thousand pagesin which mathematics is a concrete entity andmathematicians are working on a wide variety ofproblems. Calculations, formulas, manipulations,and brilliant insights into pattern are massedtogether by Roy to provide the “sources” of mathe-matical development described in the title. Whilemost of the time the path of this thread remainsclose to analysis, the points of contact with manyother major areas of mathematical research areplentiful, and the associations often both stim-ulating and surprising. The book is a personalview based on an evidently long immersion inboth primary and secondary sources. It reads welland easily, and while there are many associationsbetween the chapters, each one can be read asa self-contained vignette. It is broadly accessible,so that while some of the more recent or moretechnical developments described will perhaps beout of the reach of beginners, most people withmathematical training can read a great deal ofthe work and be both informed and entertained.It would be a great book to take on summervacation (well, provided you are not kayaking orbackpacking—it’s a bit heavy).

The treatment is not strictly chronologicalover the course of the book, though chronologyinforms its organization: subjects that “started”later are typically treated later. Indeed, the carewith which the subject has been divided in orderto give both details and a big picture is one of

November 2013 Notices of the AMS 1331

the real services of the work. Given the titles ofchapters, the contents often surprise in a goodway. Let’s take an example: the short (sixteen-page)chapter on inequalities. This begins with a surveysection, discussing the arithmetic-geometric meaninequality and its treatment by Thomas Harriot(before 1631!), Descartes, Newton, Maclaurin, andCauchy; then noting that Cauchy’s method inspiredJensen, tying this to Rogers, Hölder, and Schwarz;and concluding with Hilbert and Riesz. The chaptercontinues with individual sections that provideadditional detail on aspects of the mathematicalwork on this subject by Harriot, Maclaurin, Jensen,and Riesz. For example, the section on Harriotintroduces his own notation, then gives his ownproof of the arithmetic-geometric mean inequality,while the section on Riesz gives his proof of theMinkowski inequality. A concluding section (thereis one of these for every chapter) points to primaryand secondary literature of high quality. Along theway, many other mathematicians are both referredto and quoted; one can savor here, for example,Harald Bohr’s remark that “all analysts spendhalf their time hunting through the literaturefor inequalities which they want to use andcannot prove.” The chapter also has five pages ofexercises, usually with quite direct and explicitlinks to historical literature. This organization istypical of the book as a whole.

Since there are forty-one chapters, making alist of what is treated is not possible here. Royis perhaps most widely known for his book withRichard Askey and George Andrews, so chapters onthe hypergeometric series, q-series, and partitionsare not surprising, nor are many other chaptersthat touch on closely related areas of classicalanalysis involving series and products. There isquite a bit of number theory (L-series, distributionof primes, transcendence), complex analysis (valuedistribution theory, for example), and there are evenchapters on solvability by radicals and finite fields.We can remark on some things that are perhapsquite unexpected features. The first chapter isabout power series in Kerala in the fifteenthcentury, for example—a nice addition. There aremany places where items that are perhaps lesswell known than they should be are brought tolight: I noted with pleasure Harkness and Morley’sproof of the binomial theorem; Zolotarev’s methodfor Lagrange inversion with remainder; Hermiteon approximate integration; Lagrange’s proof ofWilson’s theorem; Bohr, Mollerup, and Artin onthe Γ function. There are many, many more suchinstances. I just spent a semester teaching aseminar for seniors on classical analysis, and thebook was tremendously useful as a resource.

Some of the longer survey chapters are in manycases attractive and motivating introductions to

the subjects they treat. As an example, I’ll mentionthe two chapters on elliptic functions in theeighteenth and nineteenth centuries. The two forma unified historical account at an introductory levelto the basic ideas of the theory, culminating with adiscussion of Eisenstein’s application of the ideasto reciprocity laws. Much of the same materialis covered (more succinctly, with more requiredinput from the reader, and aiming squarely atmodern results) by McKean and Moll in their lovelybook Elliptic Curves. But in Roy’s account thehistorical aspects are treated more fully, the paceis more leisurely, and the connections between thevarious developments are presented in a way thatmakes us more fully aware of the mathematicalcontext of the developments at the time. OftenRoy provides what feels like reliable insight intothe innovations he describes, for example, withEuler’s realization that dt/

√1− t4 could not be

rationalized by substitution, since this would implythat the equation z2 = x4−y4 had integer solutions(as Euler knew from Fermat to be false).

Partly because this is material I find interesting—it is a kind of mathematics I particularly likeand believe to be central to much mathematicalendeavor that grew from these roots—I foundthe book charming. If I make a few minor pointsabout what the work does not provide and indicatethings that I could wish were otherwise, it doesnot detract from my overall view that this isa fine work. As an historian, I would love tohave seen more thorough references, though theforest of citations that would result might havedetracted from readability, and it can be arguedthat in some cases such references can be filledin with relative ease. For example, the statement(p. 83) that “some historians have suggested withsome justification” that personal contact betweenDescartes and Faulhaber could have sent us toKenneth Manders’s paper of 2006, but it didn’t,and the literature survey at the end of the chapterdidn’t either. Sometimes such statements containan implicit historical analysis that may be correctbut isn’t provided: “one can see in the workof Newton, Leibniz, and others that they wereimplicitly aware of the method [of summation byparts]” (p. 195). A fair response to this kind ofcriticism is that “it is not that kind of book,” but Idecided to mention it given the many very welcomereferences to recent historical work that the authordoes bring to the reader’s attention. There are afew typographical errors, even in proper names.Reisz and Frèchet should have been caught by thecopy editors, who no longer seem to exist. Thebook’s title might encapsulate the author’s viewof what he aims to do but unfortunately doesnot describe the work clearly; some variant of thesubtitle using the words “series” and “products”

1332 Notices of the AMS Volume 60, Number 10

would have been better in my view. To counterthese negative remarks, allow me to complimentthe figures and the index, both of which are welldone and useful.

I recommend this book to a wide audience.Undergraduates can learn of the truly vast amountof material that lies alongside some of theirmore standard endeavors, many of which involveonly elementary matters: sums, products, limits,calculus. Graduate students and nonspecialistfaculty can wonder at the ingenuity of theirpredecessors and the connections between nowdisparate areas that are afforded by this veryclassical view. They’ll also get lots of good ideasfor teaching (and they may waste a good dealof time on the problems, as well). Historians,philosophers, and others should read this book,if only for the view of mathematics it propounds.And specialized researchers in the area of specialfunctions and related fields should simply have agood time. All of these readers can benefit fromthe remarkable expository talents of the authorand his careful choice of material. Among personalviews of mathematics that use history as a key tounderstanding, Roy’s book stands out as a model.

Research topic: A three-week summer program for Geometry and graduate students Materials undergraduate students mathematics researchers Education Theme: undergraduate faculty Making Mathematical school teachers Connections math education researchers

IAS/Park City Mathematics Institute (PCMI) June 29 – July 19, 2014 ~ Park City, Utah

Organizers: Mark Bowick, Syracuse University; David Kinderlehrer, Carnegie Mellon University; Govind Menon, Brown University; and Charles Radin, University of Texas Graduate Summer School Lecturers: Michael P. Brenner, Harvard University; Henry Cohn, Microsoft; Veit Elser, Cornell University; Daan Frenkel, University of Cambridge; Richard D. James, University of Minnesota; Robert V. Kohn, Courant Institute; Roman Kotecký, University of Warwick; and Peter Palffy-Muhorray, Kent State University Clay Senior Scholars in Residence: L. Mahadevan, Harvard University; and Felix Otto, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig Other Organizers: Undergraduate Summer School and Undergraduate Faculty Program: Steve Cox, Rice University; and Tom Garrity, Williams College. Summer School Teachers Program: Gail Burrill, Michigan State University; Carol Hattan, Skyview High School, Vancouver, WA; and James King, University of Washington

Applications: pcmi.ias.edu Deadline: January 31, 2014

IAS/Park City Mathematics Institute Institute for Advanced Study, Princeton, NJ 08540

Financial Support Available PCMI receives major financial support from the National Science Foundation

Access MathSciNet, AMS Journals,and eBooks on the go.

Learn more atwww.ams.org/MobilePairing

electronic productselectronic productsAMS

electronic productsare now mobile!

AMERICAN MATHEMATICAL SOCIETY

November 2013 Notices of the AMS 1333