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American Mathematical Monthly Vita Mathematica: Historical Research and Integration with Teachingby Ronald Calinger
Vita Mathematica: Historical Research and Integration with Teachingby Ronald Calinger
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Cilt:
104
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english
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The American Mathematical Monthly
DOI:
10.2307/2974755
Date:
May, 1997
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Review Author(s): Hardy Grant Review by: Hardy Grant Source: The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 471478 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2974755 Accessed: 13122015 18:31 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions moneyconcerns,some have to do odd jobs, some have children.Of course, all this interfereswith their study. It would be much better for them to have started to solve nontrivialproblemsyears ago. This inconsistencyis especially noticeable when one reads the initial Chapter Zero, intended for students of ages 1011. C4Theproblems of this chapter have virtuallyno mathematicalcontent,"naively claim the authors.Actually the problems of this chapterrequirethe most fundamentalabilitythe abilityfor abstract thinking.This ability should by no means be taken for granted;it develops as a result of careful and wellthoughtschooling.Let us rememberthat most teachers of circles in Russia were not professional teachers. In most cases they were universitystudents,inspiredbut unexperienced.Their teachingwas successfuldue to the sound preparationprovidedby the nationaleducationalsystem.In myyoung years the Russianeducationaladministrationseemed vegr stupid to me, but now I see ; how efficient actually it was. Don't ask me how does this square with the tyrannicalSoviet rule and the ailing Russian economybecause I don't have all the answers. Some people ask whether there are competent enthusiasts in Anlerica who could and would teach classes of creativeproblemsolving.The answeris C4yes,of course,"but this is not the rightquestion to ask. The right questionis whether the educational system can teach the basics of mathematicsso that childrenwill be able to attend such classes. REFERENCES 1. Boris A. KordemskySTheMoscowPazzles,359 Mathematical Recreations.Dover PublicationsS 1992. 2. H. St&inhausSMathematical Snapshots.New YorkS Oxford Universit7 Press, 1969. DeparTment of Maffiematics Uniuersity of the IncamateWord 4301 Broadway SanAntonio, IX 78209 toom@thecollege.iwct .edu VitaMathematica: HistoricalResearchand Integrationwith Teaching.Edited by Ronald Calinger.WashingtonSD.C., MathematicalAssociation of America, 1996, 350 pp., $34.95. Reviewedby Hardy Grant That mathematicshas an 4'imageproblem"with large segments of the general public has been obvious for a long time. The stock indictmentis all too familiar, for it has echoed repeatedlyin the cultural creations that reflect and shape our collective values. Since Aristophanes, mathematics and its devotees have been lampooned,gently or savagely,as narrow,austere, mechanical,passionless excessively cerebral,cold Examples abound. Jane Austen'sEmma (1816) considers at one point that the romantic events unfolding around her must impinge on "the coldest heart and the steadiest brain,"on a linguist, a grammarian,4even [t] a mathematician." The "Master Mathematician"in Oscar Wilde's TheHappyPrince (1888) "frowned and looked very severe, for he did not approve of children dreaming."Have perceptions changed in our own time? A few months ago, the 1997] REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions 471 parentsin the enjoyablecomic strip SallyForth,weighingpossible summerdestinations for their 9yearold,threatenedher with "AlgebraCamp"("justoutside Geek City")as the absolute antithesisof a place where a kid could hope to have some fun. Readers of the MONTHLY could multiplythese instances at will. The problem runs very deep, and does not look like going awayany time soon. Is there in fact a cure? Countlessadultsblame their distastefor mathematicson bad teaching,so the schools would seem the naturalplace to start. But what to do there? For many the best hope remains the use of history in the mathematics classroomas a "humanizing"corrective.The Danish educatorTorkil Heiede goes so far as to say, in his contributionto the volume under review, that to teach mathematics"without its history"is to teach it "as if it were dead."The book's origin reflects the now widespreadsharingand institutionalizationof such sentiments. Many of the papers assembledhere began as talks given in the summerof 1992 at either the QuadrennialMeeting of the InternationalStudy Group on the Relationsbetween Historyand Pedagogyof Mathematics("HPM"),at Toronto,or the Seventh International Congress on MathematicalEducation ("ICME"), at Quebec City.Accordingto Ronald Calinger,the volume'seditor, other papers are the fruits of a call sent out by him to "historiansof mathematics."The book's subtitle,'4HistoricalResearchand Integrationwith Teaching,"captureswell enough such focus and unity as this wildly diverse collection may be said to possess. Calinger identifies the primary audience as "mathematics teachers, research mathematicians,historiansof mathematics,and historiansof science"(p. vii). Predictably,these thirty articles vary enormouslyin subject matter, in level of presentation, and in putative audience. Someone, presumablythe editor, has gamelytried to partitionthem accordingto theme, but the subsets'boundariesare rather slippery,and indeed I would argue that perhapshalf a dozen contributions have ended up in the wrongboxes. I shall adopt a somewhatdifferentgroupingin what follows. I should add that any attempt to summarizeso many papers in so brief a compass must be lamentablyselective and superficial,and due apology is hereby extended both to their authorsand to potential readers. A preliminaryword about the book's"productionvalues"may be in order. The index is good, and in many articles the bibliographyalone is worth the price of admission. Many marvellous illustrations have been culled from inaccessible sources. The writingis mostlyvery competent, if occasionallyclumsy and seldom inspired.Surprisinglyor not, some of the authorsfor whom English is presumably a learned language write it better than do some presumablynative users. Alas, these pages will not reassure those readers (I am one) who fear that in our time professionalstandardsof editing and proofreadingare in steep decline. Thus (to offer a samplingthat is very far from exhaustive)"data"is stubbornlytreated as singular(pp. 92, 335), and "phenomena"too for good measure(p. 332); two verbs on p. 94 disagree with their respective subjects;several references of a paper to itself have been left as "p. ???";and computergremlinshave at severalplaces been allowedto substitutecommasfor occurrencesof "e."(One mighthave thoughtthat "Dieudonn,"(pp. 9, 10) would rouse the sleepiest proofreader;but apparentlynot.) It is only fair to add that the passages involvingtechnical mathematicsseem to have been treated with commendablecare. Not much in these pages will jolt readersout of their seats, or sparkrevolutions. Few seeds of potential controversyare here sown. But one paper, David Rowe's fine surveyof trends in the historiographyof mathematics,deals at some length with old disagreementsthat still smolder; I shall come back to that paper, and those issues, towardthe end of the review.The author here whom I should most 472 REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions [May enjoy debatingface to face is UbiratanD'Ambrosio,who offers a spiriteddefence of an "ethnomathematics" conceivedso broadlyas to coincide, so far as I can see, with all of culturalanthropology("the study of techniquesdeveloped in different cultures for explaining,understanding,and coping with their physical and socioculturalenvironments").Regrettably,a definitionso vague and idiosyncraticseems likely to take the paper out of the mainstreamdiscussionof its ostensible subject. One of the book's great delights is its severalventures along relativelyuntrodden bywaysof our mathematicalheritage. Complete with charmingcartoons (by the author?),BeatriceLumpkin'sarticlecleverlyinterweavesthe historyof the rule of false position (which she traces to ancient Egypt) and the career of the outstandingmathematicseducator Benjamin Banneker (d. 1806). Peggy Kidwell tells the absorbingstory of the various objects, especially geometric models, that saw service as teaching aids in l9thcenturyschools and universities.Kidwelllinks these longago educators'enthusiasmfor such objects to their belief that "it was more useful to show pupils the propertiesof surfacesand solids using models than to offer formalproofs of those properties"a notion that has not lost its validity. KarenDee Michalowiczrescues from obscurityMaryEverestBoole (the mountain is named for her uncle), wife of George Boole and a distinguishedwriter on, and teacher of, mathematicsin her own right. Those who still doubt that women have "come a long way"in our subjectmay like to ponder a despairingremarkmade by Mary'sfather in 1842, when she was ten. If she could go to university,he said (whichshe could not), she "would carryeverythingbefore her . . . But what could a girl do learning mathematics?"This article includes twoplus pages of short quotationsfrom MaryBoole'swritingson the art of teaching,a little treasurythat I wouldn'ttrade for ten years'worth of "aimsand objectives"frommy local Ministry of Education. VitaMathematicacontains a number of papers on "straight"history of mathematics,without overt or obviouspedagogicalapplication.Jens H0yruptraces from old Babyloniato the Renaissancethe career of a single problem to find the side of a square from the sum of its perimeter and area; this, he says, turns out to belong to a nonscholarlytraditionof practicalgeometry,whose interactionwith the contemporary"literate"mathematicalculture he discusses at length. Wilbur Knorr'stheme is the ancient practice of the "method of indivisibles"and its influence in the age of Cavalieri.Knorr argues that an "indivisibilistheuristic" predatedArchimedes,and that earlymodernversionsrepresent"a reconstruction of the lost heuristicby geometerswho were impatientover the demandsof formal demonstration." These two papers are substantialmonographs,with the detail and rigor of argumentand documentationthat we expect from their authors.Shorter and slighter, but perhaps on that account better suited to "integration with teaching,"are an overviewof traditionalChinese mathematicsby Frank Swetz, a discussionof combinatoricsand inductionin medievalHebrew and Islamicmathematics by Victor Katz, and BarnabasHughes' descriptionof the earliest correct algebraicsolution of cubic equations,by Master Dardi of Pisa (c. 1350).All three of these essays are solid and useful. ZarkoDadic well summarizesthe work of the Croatianscientist Marin Getaldic (15681626). Getaldic'wrote interalia a treatise De resolutioneet compositionemathematica which is to say, the use in mathematics of the method of "analysisand synthesis."Some day that volume, and Dadic's paper here, will be soughtout by the authorof one of the great unwrittenbooks of the western intellectual tradition,a history of analysisand synthesisin their two millennia of life, not just in mathematicsbut in philosophy and earlymodern science as well. 1997] REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions 473 take the reader much The other purely historical essays in VitaMathematica nearer to our own time. Roger Cooke gives an admirable account of Sof'ya Kovalevskaya's"mathematicallegacy,"in particularher "discoveryof a physical configurationfor which the equationsof motion of a rigidbody about a fixed point under the influence of gravitycan be integratedin closed analyticform."William Aspray,Andrew Goldstein, and BernardWilliamsentertaininglyrecount the rise of theoretical computer science and engineering, in both their intellectual and social aspects but with emphasison the latter, specificallythe supportingrole of the National Science Foundation. Other paperstreat of the histog7of mathematicseducation.Hans Niels Jahnke, in one of the book's best essays, relates the complicated l9thcenturyhistoxyof "algebraicanalysis" the most familiar manifestationis probablyLagrange'srecasting of the calculus in terms of power series and explainshow its vogue had the remarkableeffect of excluding infinitesimal methods from the curriculaof German gymnasiafor several decades. Jahnke argues that pedagogicallythis was no bad thing, for it entailed a stress on "concrete"problemsolvingas opposed to blind applicationof algorithms.Susann Hensel's topic is the mathematicaleducation of engineersin late19thcenturyGermany.As she says, some of the questions then under debate are hardy perennials: how "pure" and rigorous should the mathematicsin "service"coursesbe, and should they be taughtonly by mathematicians? llonald Calinger focusses on the early years (1861 ff.) of the famous mathematics seminar at Berlin; the midwives at its birth were Kummer and Weierstrass.Calingerweaves into his tale a good deal of the mathematicsof these two giants and of their contemporaries. The papers in yet another group get closer to the volume'sprofessed goal of integratinghistory with teaching. Fred llickey makes the case in general terms, describes particulartechniques in his own practice, and calls attention to such resourcesas the HPM newsletterand the MAA'sthrivingemail discussiongroup on the histoiy of mathematics.(Rickey is too modest to mention here that the latter is his own creation.)TorkilHeiede relaysthe welcome news that the Danish the use of the historyof mathematicsin "uppergrades." governmenthas mandated One longs to know more: who had the ears of the bureaucrats, and what argumentswere used that might be transplantedto less progressivejurisdictions? A repeated theme in these pages is the value of histoxyin promotinga view of mathematicsas a process,rather than merely a product, of human strivingand discovery.This contrastexplicitlyguides EvelynBarbin'spersuasiveadvocacy,with historical examples, of a problemsorientedapproachto teaching. The benefits, she says, include a better understandingand tolerance of pupils'errors. Similarly ManfredKronfeller,after providinga short histow of the function concept, draws pedagogicallessons;that include the realizationthat student errors may actually mimic those of the great masters.Indeed, Kronfellersays, in teaching any set of ideas one should follow as much as possible their historicalevolution,for one can "assume"that aspects elucidated earlier in history should be easier for pupils to grasp.Some empiricalevidencein the same directionis offered by Peter Bero, who conducted in Slovakiaa survey of young people's perceptions of the continuum. (The nodding proofreadershere provide the funniest of the book's innumerable typos:the age range of Bero'srespondentsis said (p. 304) to be "1 to 18."So what does the playpen set make of Zeno?) Bero reports that these elementaryschool and gymnasiumstudents conceive the continuumin ways "often similar to those displayedby ancient Greek mathematicians."One thinks of the biologists'catchy 474 REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions [May old sayingthat "ontogenyrecapitulatesphylogeny"the individual'sdevelopment retracesthe evolution of her species. Other authorsseek to base teaching on the direct use or creativereworkingof primarysources. RichardLaubenbacherand David Pengelley describe an upperlevel honors course that presents students with "mathematicalmasterpieces," rangingin time from Archimedesto John Conway,and asks them to function as "critics"in the sense in which this term is used in the arts. Israel Kleineroutlines a course, for teachers, built around the use of quotations about mathematics, supplementedby a "very concise"chronology.Very instructivehere are a number of pairs of mutuallyopposingquotations,which give a vivid sense of the complexity of issues and of their inherent drama. This potential for drama is taken to its natural conclusion by Gavin Hitchcock, who has written stageable dialogue in which figures from the historyof mathematicsdebate their respectivestances. In the first of these playletsSimon Stevin touts irrationalnumbersagainstthe qualms of Michael Stifel; in the second, set in 1827, George Peacock and Augustus de Morgantry to make WilliamFrend (himself a mathematician)accept multiple and negative roots of equations.Given the right actors and audience, these exchanges could "come off the page"with real theatricaleffectiveness. Several authors present case studies that extract pedagogical morals from specific historical episodes. John Fauvel's point of departure is a wonderfully curious "tree"diagram,from a book of 1808, which shows graphicallythe various factors that led to the abolition of the slave trade. Fauvel's agenda here is threefold. He argues that (i) "graphicalrepresentationand modelling"have been neglected by historians of mathematics,(ii) they have also been "devalued"by teachers,in comparisonwith "prose text or algebraicsymbolism,"and (iii) they can and should be used to "empower"students, that is, to encouragein students the "knowledgeand belief that they can use and create mathematicsto influence their way in the world."Marie Franc,oiseJozeau and Michele Gregoire describe the meridianmeasurementsin France (179299) that led to the first definition of the meter, and tell also (a bit cursorily,to my regret) of an imaginativereenactment, near Paris, of part of that labor by a group of modern high school students. Jim Tattersallsketches the historyof attemptsto estimate the total numberof people who ever lived, and proposes a classroom exercise to the same end. This piece revives an ancient wheeze which is the best joke in a book not brimmingwith humor.A class of studentswas asked how they knew they were going to die, and one replied brightlythat "it was because so far most people have." Several other papers (in addition to Tattersall's)draw their case studies from the history of the calculus. Judy Grabiner summarizesthe sharply contrasting approaches taken to the calculus by Maclaurin (geometric) and by Lagrange (algebraic),respecti;vely.That is a familiarstory,but Grabinergoes on to trace the differenceto differingculturalinfluences,and sets out the lesson for teachers:the tendency of students to adopt diverse approaches to mathematicalproblems is both natural and legitimate, especially where "nontraditional"backgroundsare involved.MartinFlashmansuggeststhat instructorsgive an extra historicaldimension to the textbook account of the FundamentalTheorem of the Calculus by presenting in detail the wholly geometric version proved by Isaac Barrow just before the watershed work of Newton and Leibniz. ManKeung Siu considers "integrationin finite terms"fromits first rigoroustreatmentby Liouville(1830s)to the modernclassroom.He gives splendidexpositionsboth of the mathematicsitself which is far from elementary and of its history, and in both aspects he 1997] REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions 475 maintains an exemplary balance between superficialityon the one hand and excessive detail on the other. And Siu is equally good on the pedagogicalissues. He has a fine sense of the kind of questionthat better students are likely to raise, and he specifically addresses the impact of computers on the teaching of his chosen topic. For my money his paper is another of the volume'shighlights. I come finallyto the excellent essay, alreadycited above, in which David Rowe depicts "new trends and old images"in the historiographyof mathematics.In particularhe discussesthe notoriousdebate that climaxedin the late 70s over the "geometricalalgebra"commonlycredited to the Greeks. Many results stated and proved in geometricallanguageby Euclid can easily be "translated"into elementary algebraicidentities;but does such restatementdistort the Greeks'own point of view? So argued the historian Sabetai Unguru, whose views then evoked rebuttalsfrom mathematiciansof the stature of Andre Weil, Hans Freudenthal, and B. L. van der Waerden.Rowe expoundsthis particulardispute with applaudable fairness, and then sets it in a larger context, centering his discussionaround some provocativeviews voiced by the same Andre Weil in a famous lecture in 1978. Weil declaredon that occasion that for mathematicians the "firstuse"of the subject's history is "to put or keep before our eyes 'illustrious examples' of firstrate mathematicalwork."Therefore "the craft of mathematicalhistory can best be practicedby those of us who are or have been active mathematiciansor at least who are in contact with active mathematicians."Moreover,Weil urged, it is appropriate,indeed necessary,to interpretthe mathematicalideas of past cultures in modern terms. For example, "it is impossible for us to analyze properly the contents of Books V and VII of Euclid [on ratios of magnitudesand on number theory, respectively]without the concept of group and even that of groups of operators, since the ratios of magnitudes are treated as a multiplicativegroup operatingon the additivegroup of the magnitudesthemselves." The tendency exhibited in this last quotation is probably more seductive in mathematicsthan in any other subject.For in mathematics,as the same quotation shows, many ancient ideas can be fitted with perilous ease into modern abstract frameworksto which they are wholly foreign in conception and in spirit. Perhaps then it was no great surpriseto read recently a hint that mathematicsmay have acquired among outsiders a certain reputation for this kind of thing. In the obituary of Joseph Needham that she wrote for the June 1996 issue of Isis, Francesca Bray tossed out the aside that "projecting modern meanings onto ancientterms [is] a practicethat often passes unnoticed,I am told, in the historyof Westernmathematics.""Often,"indeed; for the temptationis chronic. But not, in fact, "unnoticed."It was just this tendency that Sabetai Unguru protested back in the 70s, and now David Rowe takes up the cause. Still addressinghimselftspecificallyto the views of Andre Weil, quoted above, Rowe explainswith great civilitywhat "disturbs"historiansabout viewing ancient ideas through modern lenses. He associates Weil's vision of history with a Platonist philosophy,which makes mathematicaltruths independentof time and of historical milieu. As he says, this orientationdisposes one to "present the development of mathematical ideas as a steadily unfolding search for Platonic truths that transcendthe particularculturalcontexts in which these ideas arose"(p. 10). But he raises at once, in the completion of the sentence just quoted, the obvious objection, that one can write history in this way "only by discountingthe rich varietyof meaningsthat accompanied"those ideas in their actualconcrete settings. With this objectionI agree entirely but I would push the argumenta bit further. There is a facet of the issue that most discussionsmention scarcelyif at all. 476 REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions [May I suppose that those who credit the ancientswith this or that modernidea think that the ascription amounts to generous praise. For all their limitations, the argumentgoes, these forerunnershad the Right Stuff, they were spirituallyso close to us that they seem (in Littlewood'smemorable phrase) like "fellows of another college";and wouldn'tthey rejoice in our good opinion of them? Well, maybeso. It mightjust be, however,that the mathematiciansof bygoneages would also, or even rather,wish to be studied and judged for whattheywere,in their own proud individualityand distinctiveness.In the writing of historywithin the traditional academicdisciplineof that name, the goal of an empatheticportrayalof the past on its owntermsis now a commonplace.This is not at all to say, of course, that anticipationsof, influenceson, the present are not worth notice. But the aim is the polar opposite of the deliberate backwardprojectionof modern modes of thought and feeling. Rather, the quest is for what WilliamBlake would call the "minuteparticulars,"the specific and defininguniqueness?of each time and place and people of the past. Historiographyso conceivedhas at its very heart a moralimperativeS a scrupulous respectfor, and honoringof, those who went before. At its highest,it is dare one say? like an act of love, which would know and possess yet ultimatelyleave singularand autonomous.That makesits practiceat once a noble and a formidably demanding enterprise. George Steiner once wrote, in a different but parallel context, that "there can be no other thanks [to the creators of our cultural heritage]than extreme precision,than the patient, provisional,alwaysinadequate attemptto get each case right."Thatchallengeis present no less in the historiography of mathematicsthan elsewhere, despite the (for some) supposedly eternal characterof the truths unveiled. Indeed7in the last analysisthis timelessnessis a red herringafor the most passionate Platonist must concede that mathematical discovery is the work of individualminds in historicallyconditionedsettings. So who should write the historyof that discovery?Where the object of study is the technical progress made in the 20th century, Andre Weil's claim that only mathematiciansneed apply seems incontestable.In most fields of currentmathematics, the dynamicof the developmentis so overwhelmingly4'internal,77 and the barriers of specialist knowledge so forbidding, that the prospective historian cannot hope to succeed without something very close to the active researcher's intimate grasp of, and 44feel"for, the subject.The scarcityof people qualified in this sense mustbe the primereason for the deplorableunderrepresentationof the 20th centuryin the journalsand conferenceprogramsdevoted to serious historyof mathematics. But if the historiographyof 20thcentuiy mathematics must be left to the practitioners,the pasteven the relatively recently past is a different case. Roger Cooke says acutely in this volume (p. 177) that, thanks above all to the massivemoderntrend towardabstractionand generalization, To reconstructprecisely the present state of any nineteenthcenturymathematical topic is in a sense impossible. No mathematicalproblem is understood exactlyas it was understood . . . a centuryago. Then how muchwider still the gulf between us and the still more distantpastt The fartherback in time, the more foreign and elusive must be the vCmindset^' that the historianwould seek to know. Moreover,and cruciallySincreasingremotenessfrom the present increases also the role o£ 'sexternal factors in mathematical activity and so diminishes,in proportionSthe place of purelymathematicalskills 1997] REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions 477 0 X g X g f , g g g Wo . / _. _ ;K<o, _ _ . . _.. g .._ > _. t . . W _ 2_ _ 9. . ; __ ffi .._ _ _. R _ > .__._. w _ . ._ and knowledgein the aspiringhistorian'sstock in trade. Where ancientmathematics is in question, the historianneeds little if any technical grasp beyond the very elementarylevels (quadraticequations or whatever) achieved by the civilization under her gaze. The resources that must sustain her are in other directions entirely. The importance of the relevant linguistic competence is obvious. The whole of the surroundingcultural and social matrix is germane and must be mastered.Perhapsthe insightsof modernanthropologycan be broughtto bear, as by Geoffrey Lloyd in his superb studies of ancient Greek science. Above all, the successful explorer of mathematics'distant past must bring the precious gifts of imaginationand of empathythat alone give any hope of access to alien minds. In this terribly difficult undertakingthe research mathematicianas such has absolutely no privilegedstatus, no claim whateverto special authority. The good things in the book under review,and there are many,make their own valuablecontributionsto the historyof mathematicsand to the creativeuse of that history in our classrooms.Thus they may serve also toward fulfilling the hope, articulatedhere by Fred Rickey, of educating"a general populationwith a much better feel for what mathematiciansdo and why it is important."In that urgenttask VitaMathematicawill not set the world on fire, but it should light some candles; and everybit helps. 539 HighlandAvenue Ottawa,OntarioK2A 2J8 Canada hgrant@freenet. carleton.ca Contributedby Russ Hood, Rio Linda,CA 478 REVIEWS This content downloaded from 131.111.164.128 on Sun, 13 Dec 2015 18:31:12 UTC All use subject to JSTOR Terms and Conditions [May