Book Review

The Crest of thePeacock:

Non-European Rootsof Mathematics

Reviewed by Clemency Montelle

The Crest of the Peacock: Non-European Rootsof MathematicsGeorge Gheverghese JosephPrinceton University Press, 3rd Edition, 2011US$50.00, 592 pagesISBN-13: 978 0691135267

For many Notices readers, The Crest of the Peacockneeds no introduction. Now in its third edition, thebook has enjoyed widespread popularity for pro-viding an accessible account of the mathematicalculture of non-European peoples. Responding toearlier critiques, Joseph’s revised edition has somenotable changes. For one, the subtitle has beenmodified. In a small but significant alteration, Non-Western Roots of Mathematics has been changed toNon-European Roots of Mathematics. Sections havebeen added and several removed, a reorganizationthat addresses various scholarly objections stem-ming from historical and authenticity concerns.Endnotes have been expanded so as to reflect somecurrent research, and an enlarged bibliographicsection that is now grouped primarily accordingto geographical region provides references forfurther reading. In the third edition it is stated thatthis book is intended to be “an effective resourcefor students and teachers of mathematics whileremaining accessible to general readers.” Indeed,Joseph’s original intention was to appeal to thegeneral public and teachers and students of mathe-matics, and in that aim he has enjoyed success. Thebook has been popular. It covers an assortmentof interesting aspects of the mathematical activity

Clemency Montelle is senior lecturer of mathematics in theDepartment of Mathematics and Statistics at the Univer-sity of Canterbury in New Zealand. Her email address isc.montelle@math.canterbury.ac.nz.

DOI: http://dx.doi.org/10.1090/noti1068

of selected early cultures with a minimum oftechnical details in a humanistic, undemandingway. Joseph aims to summarize the essence oftechnical scholarship that would otherwise be outof reach of this audience.

The third edition opens with a lengthy prefacein which Joseph discusses the underlying rationalefor the book: a response to a field that he believesis overly dominated by the hegemony of a “Westernversion” of mathematics, or Eurocentrism. Suchattitudes, Joseph points out, are epitomized byMorris Kline’s statement that “the mathematics ofEgyptians and Mesopotamians is the scrawling ofchildren just learning to write, as opposed to greatliterature” (p. 175). While the majority of scholarswould have no objection to Joseph’s observation, itis well accepted nowadays that Kline’s attitude, aswell as others that Joseph invokes to support hisargument,1 is outdated. Historiography has altereda lot in the last quarter century. Scholarship in thehistory of mathematics has made an undeniablechange of direction since the time in whichthese attitudes were prevalent. Since then researchculture has moved increasingly away from a narrowmonolithic portrayal of mathematics to a moreinclusive picture that sees as valid and importanta broad range of mathematical activity and isincreasingly sensitive to wider contextual issues.

To be sure, The Crest of the Peacock deserves ashare of credit for injecting momentum into thesegrowing new attitudes, though in fact, these atti-tudes had been present for decades in the work ofmany devoted scholars. Otto Neugebauer broughtserious historical treatment of non-western math-ematics to the forefront beginning in the late1930s and, along with numerous others, published

1Some of the opinions Joseph refers to were made over acentury ago, such as Rouse Ball’s book published in 1908.

December 2013 Notices of the AMS 1459

extensively throughout subsequent decades onvarious cultures of inquiry. Concurrently with theappearance of the first edition of The Crest of thePeacock, Ubiratan D’Ambrosio published a seminalwork on ethnomathematics, a term he had coinedin the late 1970s. This was followed closely byMarcia Ascher writing on the same topic. Themid-1990s saw the rise of books dedicated tobringing non-European mathematical traditionsinto mainstream scholarship. One example fromthe many instances will suffice: the work Mathe-matics across Cultures (2001), edited by HelaineSelin, presented historiographical reflections aswell as expository articles on various mathemati-cal traditions, including Mesopotamian, Egyptian,Islamic, Hebrew, Incan, Mesoamerican, Sioux Tipiand Cone, the Pacific Cultures, Aboriginal, Centraland Southern African, West African, and Yoruba,Chinese, Indian, Japanese, and Korean! Indeed,nowadays a significant number of scholars areactively seeking to be less binary in their ap-proach, and the most current research in thesefields focuses on the interplay of ideas and trendsand eschews a divisive attitude. Scholars are nowreaping the benefits of a measured considerationof the many different cultures of inquiry in thehistory of mathematics; the associated themes oftransmission, comparison, social context, situatedcognition, and the flourishing of mathematicalinquiry in cultures with contrasting epistemicpriorities are proving fruitful, not only for historyof mathematics but for the history and philosophyof science more broadly.

While Joseph passionately denounces the Euro-centric approach throughout his book, at timeshis authorial stance seems to perpetuate the divi-sions caused by these attitudes in the first place.For instance, his lack of a dedicated section onGreece (despite his frequent reference to Greekachievements and their transmission) is restrictive,particularly given the importance of this culture ofinquiry for many others in the book, first and fore-most the Islamic Near East. Indeed, his argumentsin favor of historical inclusiveness would be bettersupported by discussing the relevance of the Greekmathematical culture as well. The polarity between“European” and “non-European” is not mitigatedeither by Joseph’s dismissive references to (mostlyunspecified) “western philosophers” (p. 35 andelsewhere), “western scholars”, and “western his-torians of mathematics”. One must be mindful ofthe fact that European scholars themselves wereintegral to or involved in critical advances in ourknowledge of the so-called non-European cultures.“Western scholars” also deserve much credit fortheir historical work on diverse mathematical cul-tures. Joseph himself relies heavily on them in this

book; an overwhelming number of entries in hisbibliography are in English.

Joseph notes (p. xiii, for instance, and elsewhere)that “the concept of mathematics found outside theGreco-European praxis was very different.” This isan undeniably important observation. One will note,however, that mathematical activity in those so-called Greco-European cultures was not monolithiceither. The tendency for historians to overlookthe various other strands of practice within thesevery cultures has led to regrettable biases. Theemphasis on mathematics associated with elegantdeductive style proposition-proof-type accountsover and above other mathematical activities hasmeant that these latter types of mathematicshave been deemed less important or less relevant,and thus often left out or ignored in historicalaccounts. These activities include recreational,commercial, and utilitarian mathematics, and thewide spectrum of related practices such as teachingand the mathematics applied in other disciplines(such as the broader astral sciences, for instance).Various activities and byproducts in culturesthat formed part of this Greco-European praxis(astrology, horoscopes, numerical tables, roughworking, and so on) are just as important whenit comes to giving a picture of the practice ofmathematical activity in these cultures. Joseph isspot on when he comments that historians need to“confront historical bias, question the social andpolitical values shaping the mathematics (and thewriting of the history of mathematics), and searchfor different ways of ‘knowing’ or establishingmathematical truths in various traditions.” But thissentiment is not to be limited to non-Europeanmathematics. It is applicable to all cultures ofmathematical inquiry.

Critical appraisal of scholarly sources can becomplex, particularly when they are in conflict.One example Joseph confronts is the issue ofthe dating of the Bakhshalı manuscript, a workalmost unique as a surviving physical exemplarof Indian mathematics from before the earlymodern period. Joseph notes (p. 358) that Kaye’sassessment (made in 1933) that the manuscriptbelonged to the twelfth century CE is doubtful.He then instead appears to rely on an assessmentof Hoernle2 made in 1888, which estimates thatthe original was composed sometime in the earlycenturies of the common era. This is in light ofHayashi’s recent analysis (1995), which locates theoriginal text around the seventh century and thesurviving copy as early as the eighth century CEand no later than the twelfth. Hoernle’s dating,generated on the basis of a partial translation,

2Whose name, incidentally, is incorrectly spelled in thereference list.

1460 Notices of the AMS Volume 60, Number 11

relied on assumptions about the content, thesymbolism, and the meter. Hayashi, who producedan authoritative transcription, translation, andcommentary of the entire work, gives compellingreasons that rest on the media of the manuscript,paleographical evidence, and the language (amodified form of classical Sanskrit consistent withmedieval northwest Indian vernaculars at thattime). Given the uniqueness of this text, appealsto content for dating seem questionable, andone must be mindful not to conflate the stylisticfeatures of a copy of an original with what theoriginal may have looked like to establish dating.Choosing between scholarly evaluations of thissort requires a special sort of expertise. One needsfamiliarity with paleographical studies, epigraphy,codicology, as well as detailed knowledge of thetext, to be able to appreciate the reliability of oneset of evidence over another.

So which one does Joseph subscribe to? Theissue of dating the Bakhshalı manuscript appearsin at least three separate places in his survey(pp. 312, 317, and 358). The latter two referencesseem to be contradictory: on page 317 Josephstates, “On the basis of recent evidence, notablythat of Hayashi, the manuscript cannot be datedearlier than the eighth century,” and on page358, “The general consensus supports Hoernle’sdating [to the third century].”3 Despite his apparentagreement with Hayashi at one point in his work,Joseph’s contextualization and analysis proceedon the basis of Hoernle’s dating.4 The situation asJoseph portrays it is thus far from clear for thereader.

The popularity of The Crest of the Peacock restsupon Joseph’s ability to synthesize scholarshipthat would otherwise be unpalatable for a generalaudience. His ample bibliography testifies to thebroad range of sources on which his scholarshiprests. However, all scholarly material includedin the book, even that assumed to be commonknowledge, still ought to be connected back to theoriginal sources. For instance, Joseph’s summaryof Egyptian and Mesopotamian mathematics (pp.181–183) relies point for point on the insights

3The situation is not made much clearer by his summary(p. 367), which says, “The state of Indian mathematics atthe middle of the first millennium AD, as represented by theBakhshalı Manuscript…”4Joseph also comments (p. 358) that there is no clue as towho was the author of the work. This is not entirely true. Infact, some prosopographical details do exist. The Bakhshalımanuscript contains a colophon that reveals that it wascomposed by a Brahman. a who was the son of Chajaka,who wrote it for Hasika the son of Vasis. t.ha and his descen-dants. We cannot be sure, however, whether he was theauthor or the scribe, and nothing more is known about theseindividuals.

in Boyer’s book, which goes unacknowledged byJoseph.5 In another instance, Joseph refers toconnections between Panıni’s grammar and theElements of Euclid (p. 316) without acknowledg-ing the promulgator of this theory, Frits Staal.Frequently invoked throughout the book (p. 103and elsewhere) are Nesselman’s criteria for thedevelopment of algebra (rhetorical, syncopated,symbolic), but Joseph does not indicate their con-nection to Nesselman. Out of these three instances,Boyer does appear in the general bibliography. Thelatter scholars do not. The reviewer provides thereferences here for those interested readers.6

Joseph relays some of the questions that shouldbe addressed when dealing with early mathematicaltexts. He lists these (p. xx):

1. What was the content of the mathematicsknown to that culture?2. How was that mathematics thoughtabout and discussed?3. Who was doing the mathematics?

These are good guidelines for approaching ahistorical document. To endorse this sentimentand complement the work done by Joseph, weoffer some additional remarks to one of hisexamples, the history surrounding the emergenceof triangular tables of binomial coefficients andassociated mathematical relationships. This is tohighlight the complexity of the task of addressingthese questions as well as to emphasize that fullyappreciating a development of this kind requires asynthesis of “western” (Greek) and “non-western”(Islamic, Chinese, Indian) approaches and sourcesrather than focusing exclusively on one group orthe other.

Joseph discusses an instance of triangular tableof binomial coefficients (sometimes known asPascal’s triangle) in his chapter on China, where helinks the first explicit discussion of it to an earlyeleventh-century Chinese mathematician Jia Xian(whose work is lost but is discussed in a work of

5See Carl B. Boyer, A History of Mathematics, 2nd edition,John Wiley & Sons, New York, 1989, pp. 41–42.6Frits Staal, “Euclid and Panıni”, Philosophy East and West,5.2 (1965), 99–116; J. Bronkhorst, “Panıni and Euclid: reflec-tions on Indian geometry”, Journal of Indian Philosophy 29(2001), 43–80; G. H. F. Nesselman, Versuch einer kritischenGeschichte der Algebra, vol 1.: Die Algebra der Griechen,(Reimer Berlin, 1842. Reprint by Minerva, Frankfurt, 1969).Furthermore, Joseph’s preface discusses translation issuesand invokes distinctions between “alienating” or “literal”translations with “user friendly” ones. This distinction refersto the work of Høyrup as contrasted with van der Waerdenand Neugebauer and was discussed in detail by EleanorRobson, The Mathematics of Ancient Iraq: A Social History,Princeton University Press, 2008, pp. 274–284.

December 2013 Notices of the AMS 1461

7 6 5 4 3 2 1 0 1 2 3 4 5 6 7

square cube square square cube square thing unit part part part part part part part

square cube cube square thing square cube square square cube square

cube square cube cube square

cube

128 64 32 16 8 4 2 1 12

14

18

14

14

14

18

18

18

14

14

18

2187 729 243 81 27 9 3 1 13

19

13

19

19

19

13

19

19

19

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13

19

19

19

Table 1. Al-Samaw’al’s table of powers. The factorizations in the last rows are explicitly given byal-Samaw’al also.

Yang Hui around 1261).7 In this context, he argues,the table of binomial coefficients appears to be abyproduct of exploring methods to extract squareand cube roots. Yang Hui tells us that Jia Xianwrote a table of binomial coefficients in a triangleresembling the Pascal triangle up to the sixth row.

However, the exploration of binomial coeffi-cients by early thinkers, their arrangement andmanipulation in triangular arrays, and the ways inwhich they were thought about and used comprisea much more complex and nuanced theme in thehistory of mathematics. For instance, about thesame time, or even a little earlier, the Islamicscholar al-Samaw’al completed a work called AlBahir fı al-Jabr, in which he set out rules in arhetorical form for expanding expressions equiva-lent to (ab)n and (a + b)n for the cases n = 3,4,extending his account to expansions for higherpowers.8 Al-Samaw’al credits this part of his workto his predecessor al-Karajı (953–c. 1029). He alsoincludes in his manuscript a triangular table ofbinomial coefficients as well (see Figure 1).

Joseph’s three questions concerning content,scope, and practice in this case can be properlyappreciated only by considering the wider contextof these achievements and the ways in which othercultures of inquiry had impacted the traditional-Samaw’al and his predecessors were working in.For one, al-Samaw’al is indebted to his Greek pre-decessors in ways that he explicitly acknowledges.However, his work epitomizes the decisive breaksfrom Greek practice that were being advancedamongst Islamic scholars at that time.9 Also of

7It may be relevant to note the activity in India by variousearlier authors. See, for instance, the contributions in RobinWilson and John J. Watkins (eds.), Combinatorics: Ancientand Modern, Oxford University Press.8This passage has been thoroughly analyzed in an upcom-ing publication by S. Bajri, J. Hannah, and C. Montelle.9Joseph does refer to al-T. us. ı’s table, which appeared in1265, and further in the book in an endnote (p. 302, no.8), he notes the appearance “later” in Samarquand (doeshe mean al-T. us. ı?), and on p. 507 and p. 517, no. 35, linksPascal’s triangle to al-Karajı and al-Samaw’al.

Cou

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uti

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Figure 1. The table of binomial coefficients ofal-Samaw’al.

relevance are the new and significant use of di-agrams in the text and an increasingly abstractarticulation of number. All of these features arecritical to understanding the role and functionof this table in this context. Furthermore, thispassage has already attracted much scholarly in-terest because of its relevance to the history ofmathematical induction, a technique that findsantecedents in many different cultures of inquiry.

Al-Samaw’al’s discussion furthermore revealshow rules for laws of indices were to be manipulatedand used in this context. This is contrary toJoseph’s belief that in general “without convenientnotation for indices the laws of indices cannot beformulated precisely” (p. 351). When one consultsal-Samaw’al’s text, one can immediately appreciatethat al-Samaw’al has various rhetorical equivalentsfor his algebraic powers, and he has a clearconception of the relation between successive

1462 Notices of the AMS Volume 60, Number 11

powers (see Figure 1),10 as he presents a table ofthem:

While their recursive relationship is not visiblenumerically (by means of numerical indices orotherwise) the organization of the table reveals asympathy for their mutual relations into succes-sively increasing powers. To the modern eye, thisrelationship is effectively hidden by having thepowers expressed as the appropriate combinationsof the words “square” and “cube”. However, it wasperfectly understood by the actors of the time. Thealignment and arrangement of the table reinforcedthe relations of successive powers. This use of“square” and “cube” is clearly a remnant of theEuclidean geometric tradition and in this contextreveals how transitional al-Samaw’al’s mathemati-cal practice is. Thus by considering how the textwas read and used by those within the traditionand, more generally reflecting on the ways inwhich it resembles as well as contrasts with otherarticulations such as Indian, Chinese, Italian orindeed Blaise Pascal’s account itself, we get a muchricher sense of what it was like for early thinkersto investigate and articulate mathematical resultssuch as these.

No doubt Joseph’s work will continue to enjoyits original popularity; it represents the importanceand impact the history of mathematics can havewhen brought to mainstream audiences. While itis not always a reliable substitute for specialistresearch, the book certainly gives avid readers ataste of the scope and breadth of aspects of earlymathematical cultures of inquiry.

10From S. Ahmad and R. Rashed, Al-Bahir en Algèbre d’As-Samaw’al, Imp. de l’Université de Damas, 1972, p. 21.

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