“Agem....WhatMozartwastomusicandEinsteinwasto

physics,Ramanujanwastomath....”

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“ENLIGHTENING. . . . a magic, tragic ugly-duckling fable. . . .Ramanujan’sremarkablestorycomesthrough....”

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Contents

Prologue

One/INTHETEMPLE’SCOOLNESS/1887to19031.DakshinGange2.SarangapaniSannidhiStreet3.ABrahminBoyhood4.Off-scale5.TheGoddessofNamakkal

Two/RANGINGWITHDELIGHT/1903to19081.TheBookofCarr2.TheCambridgeofSouthIndia3.Flight4.AnotherTry5.TheNotebooks6.AThoughtofGod7.EnoughisEnough

Three/THESEARCHFORPATRONS/1908to19131.Janaki2.Door-to-Door3.“Leisure”inMadras4.JacobBernoulliandHisNumbers5.ThePortTrust6.TheBritishRaj7.TheLetter

Four/HARDY/G.H.Hardyto19131.ForeverYoung2.HorseshoeLane3.FlintandStone

4.AFellowofTrinity5.“TheMagicAir”6.TheHardySchool

Five/“IBEGTOINTRODUCEMYSELF...”/1913to19141.TheLetter2.“IHaveFoundinYouaFriend...”3.“DoesRamanujanKnowPolish?”4.ADreamatNamakkal5.AttheDock

Six/RAMANUJAN’SSPRING/1914to19161.OutofIndia2.Together3.TheFlamesofLouvain4.TheZeroesoftheZetaFunction5.S.Ramanujan,B.A.

Seven/THEENGLISHCHILL/1916to19181.HighTable2.AnIndianinEngland3.“ASingularlyHappyCollaboration”4.DeepeningtheHole5.“AllUsBigSteamers”6.TheDanishPhenomenon7.TroubleBackHome8.TheNelsonMonument9.Ramanujan,Mathematics,andGod

10.SingularitiesatX=111.SlippedfromMemory

Eight/“INSOMEWHATINDIFFERENTHEALTH”/From19181.“AlltheWorldSeemedYoungAgain”2.ReturntotheCauvery3.TheFinalProblem4.ASonofIndia5.RamanujanReborn

6.BetterBlastFurnaces?7.Svayambhu

EpiloguePhotographsAboutRobertKanigelNotesSelectedBibliographyAuthor’sNoteandAcknowledgmentsIndex

ForMomandDadwithloveandthanks

Prologue

One day in the summer of 1913, a twenty-year-old Bengali from an old andprosperous Calcutta family stood in the chapel of King’s College in the medievaluniversity town of Cambridge, England. A glorious, grandly proportioned place,morecathedralthanchapel,itwastheworkofthreekingsofEnglandgoingbackto1446.Lightstreamedinthroughstainedglasspanelsrangingacrossthesouthwall.Greatflutedcolumnsreachedheavenward,flaringoutintothemassivesplayedvaultoftheroof.

PrasanthaChandraMahalanobiswassmitten.ScarcelyofftheboatfromIndiaandplanningtostudyinLondon,hehadcomeuponthetrainforthedaytosightsee.Butnow, having missed the last train back to London and staying with friends, hecouldn’t stop talking about the chapel and its splendors, how moved he’d been,how...

Perhaps,proposedafriend,heshouldforgetLondonandcometoKing’sinstead.ThatwasallMahalanobisneededtohear.Thenextdayhemetwiththeprovost,andsoon,tohisastonishmentanddelight,hewasastudentatKing’sCollege,Cambridge.

He had been at Cambridge for about six months when his mathematics tutoraskedhim,“HaveyoumetyourwonderfulcountrymanRamanujan?”

Hehadnotyetmethim,buthehadheardofhim.Ramanujanwasaself-taughtmathematicalprodigyfromatownoutsideMadras,inSouthIndia,athousandmilesfrom the sophisticatedCalcutta thatMahalanobis knew best, a world as differentfromhis own asMahalanobis’swas fromEngland.TheSouth, as educatedNorthIndianswerewonttoseeit,wasbackwardandsuperstitious,scarcelybrushedbytheenlightened rationality ofBombay andCalcutta.And yet, somehow, out of such aplace,fromapoorfamily,cameamathematiciansoalivewithgeniusthattheEnglishhad practically hand-delivered him toCambridge, there to share his giftswith thescholarsofTrinityCollegeandlearnwhatevertheycouldteachhim.

Among the colleges of Cambridge University, Trinity was the largest, with themost lustrous heritage, home to kings, poets, geniuses. IsaacNewton himself hadstudiedthere;since1755,hismarblelikeness,holdingtheprismhe’dusedtoexplorethe polychromatic nature of light, stood in its chapel. Lord Byron had gone toTrinity. So had Tennyson, Thackeray, and Fitzgerald. So had the historianMacaulay, and thephysicistRutherford, and the philosopherBertrandRussell. SohadfiveEnglishprimeministers.

Andnow,RamanujanwasatTrinity,too.Soon Mahalanobis did meet him, and the two became friends; on Sunday

mornings, after breakfast, they’d go for long walks, talk about life, philosophy,mathematics. Later, looking back, Mahalanobis would date the flowering of theirfriendshiptoonedayinthefallfollowingRamanujan’sarrival.He’dgonetoseehimathisplaceinWhewell’sCourt,athree-storystonewarrenofroomsbuiltaroundagrassy quadrangle laced with arched Gothic windows and pierced at intervals bystaircases leadingtorooms.Onesuchportal ledtoRamanujan’ssmallsuite,onthegroundfloor,asteportwooffthecourt.

IthadturnedcoldinCambridgeandasMahalanobiscamein,hesawRamanujan,withhisfleshy,pockmarkedface,sittinghuddledbythefire.HerewastheprideofIndia, the man whom the English had moved heaven and earth to bring toCambridge.Butwell-laidplanshadgoneawry.Itwastheshamefulyearof1914,andEurope had gone to war. The graceful arched cloisters of Nevile’s Court, SirChristopher Wren’s eternal stamp on Trinity, had become an open-air hospital.Thousandshadalreadyleftforthefront.Cambridgewasdeserted.Andcold.

Areyouwarmatnight?askedMahalanobis,seeingRamanujanbesidethefire.No,repliedthemathematicianfromalways-warmMadras,hesleptwithhisovercoaton,wrappedinashawl.Figuringhisfriendhadn’tenoughblankets,Mahalanobissteppedbackintothelittlesleepingalcoveontheothersideofthefireplacefromthesittingroom.Thebedspreadwasloose,asifRamanujanhadjustgottenup.Yettheblanketslayperfectlyundisturbed,tuckedneatlyunderthemattress.

Yes,Ramanujanhadenoughblankets;hejustdidn’tknowwhattodowiththem.Gently,patiently,Mahalanobisshowedhimhowyoupeeledthemback,madealittlehollowforyourself,slippedinside...

•••

Forfiveyears,walledofffromIndiabythewar,Ramanujanwouldremaininstrange,cold, distant England, fashioning, through twenty-one major papers, an enduringmathematicallegacy.Then,hewouldgohometoIndia,toahero’swelcome,anddie.

“Srinivasa Ramanujan,” an Englishman would later say of him, “was amathematician so great that his name transcends jealousies, the one superlativelygreatmathematicianwhomIndiahasproducedinthelastthousandyears.”Hisleapsofintuitionconfoundmathematicianseventoday,sevendecadesafterhisdeath.Hispapersarestillplumbedfortheirsecrets.Histheoremsarebeingappliedinareas—polymerchemistry,computers,even(ithasrecentlybeensuggested)cancer—scarcely

imaginable during his lifetime.And always the nagging question:Whatmight havebeen,hadhebeendiscoveredafewyearsearlier,orlivedafewyearslonger?

Ramanujanwas a simpleman.Hisneedswere simple.Sowerehismanners,hishumor.Hewas no idiot savant; he was intelligent in realms outsidemathematics,persistent,hardworking, and even, in his ownway, charming. But by the lights ofCambridgeor,forthatmatter,ofCalcuttaorBombay,hewassupremelynarrowandnaive.SomethingsosmallasMahalanobis’slessonintheartofblanketingcouldleavehim“extremelytouched.”Hewasshamedbythemostinsignificantslight.Hisletters,outsidetheirmathematicalcontent,arebarrenofgraceorsubtlety.

In this book I propose to tell Ramanujan’s story, the story of an inscrutableintellectandasimpleheart.

It is a story of the clash of cultures between India and theWest—between theworld of Sarangapani Sannidhi Street in Kumbakonam in South India, whereRamanujan grew up, and the glittering world of Cambridge; between the pristineproofsoftheWesternmathematicaltraditionandthemysteriouspowersofintuitionwithwhichRamanujandazzledEastandWestalike.

Itisastoryofonemanandhisstubbornfaithinhisownabilities.Butitisnotastorythatconcludes,Geniuswillout—thoughRamanujan’s,inthemain,did.Becausesonearlydideventsturnoutotherwisethatweneednoimaginationtoseehowtheleast bit less persistence, or the least bit less luck, might have consigned him toobscurity.Inaway,then,thisisalsoastoryaboutsocialandeducationalsystems,andabouthowtheymatter,andhowtheycansometimesnurturetalentandsometimescrush it. How many Ramanujans, his life begs us to ask, dwell in India today,unknownandunrecognized?AndhowmanyinAmericaandBritain,lockedawayinracialoreconomicghettos,scarcelyawareofworldsoutsidetheirown?

This isastory, too,aboutwhatyoudowithgeniusonceyoufind it.Ramanujanwas brought to Cambridge by an English mathematician of aristocratic mien andpeerlessacademiccredentials,G.H.Hardy,towhomhehadwrittenforhelp.Hardysaw that Ramanujan was a rare flower, one not apt to tolerate being stuffedmethodicallyfullofall themathematicalknowledgehe’dneveracquiredinIndia.“Iwasafraid,”hewrote,“thatifI insistedundulyonmatterswhichRamanujanfoundirksome,Imightdestroyhisconfidenceandbreakthespellofhisinspiration.”

Ramanujanwasamanwhogrewupprayingtostonedeities;whoformostofhislife took counsel from a family goddess, declaring it was she to whom hismathematical insights were owed; whose theorems would, at intellectuallybackbreaking cost, be proved true—yet leave mathematicians baffled that anyonecoulddivinetheminthefirstplace.This isalsoabook, then,aboutanuncommon

andindividualmind,andwhatitsquirksmaysuggestaboutcreativity,intuition,andintelligence.

•••

Likemostbooks, thisone startedwithan idea.Sadly, itwasnotmine,but thatofBarbara Grossman, then senior editor at Crown Publishers, now publisher atScribners. Barbara first encountered the name of Ramanujan in late 1987, a timewhenmagazinesandnewspapersintheUnitedStates,India,andBritainwerefullofarticlesmarkingthehundredthanniversaryofhisbirth.LikeMahalanobisinKing’sCollegeChapel,Barbarawassmitten.First,withthesheerromanceofhis life—thestoryinit.Butalsowithhowtoday,yearsafterhisdeathandlongintothecomputerage, some of his theorems were, as she put it later, being “snatched back fromhistory.”

“Ramanujanwho?”Isaidwhenmyagent,VickyBijur,toldmeofBarbara’sinterestinabiographyofhim.Thoughskeptical,Ididsomepreliminaryresearchintohislife,asrecordedbyhisIndianbiographers.AndthemoreIlearned,themoreI,too,cameunderRamanujan’sspell.Hiswasarags-to-intellectual-richesstory.Partsofit,wrotean English mathematician, B. M. Wilson, in the 1930s, “might be lifted almostunchangedbyascenario-writerforthetalkies.”Mydoubtsfellaway.Myexcitementmountedattheprospectofdelvingintothelifeofthisstrangegenius.

Earlyon,IviewedadocumentaryaboutRamanujan’slifebytheBritishfilmmakerChristopherSykes.ReleasedbytheBBCthepreviousyearasLettersfromanIndianClerk,Sykes’sfilmsuperblydistilled,intoasinglehour,somethingoftheromanceofRamanujan’s life.Butwatching it, I grewbeguiledbyG.H.Hardy, too.Hardy, itturnedout,wasthethirdEnglishmathematiciantowhomRamanujanhadappealed;theothertwodeclinedtohelp.AndHardydidnotjustrecognizeRamanujan’sgifts;hewenttogreatlengthstobringhimtoEngland,schoolhiminthemathematicshehadmissed,andbringhimtotheattentionoftheworld.

WhyHardy?Wasitsheermathematicalacumen?Probablynot;theothertwomathematicians

wereequallydistinguished.Theremusthavebeenother,lesspurelyintellectualtraitsdemandedofhim—aspecialopenness,perhaps,awillingnesstodisrupthislifeandstakehisreputationonsomeonehe’dneverseen.

Hardy,I learned,wasabizarreandfascinatingcharacter—acricketaficionado,amasterfulprosestylist,amanblessedwithgorgeousgoodlookswhotohisowneyeswas so repulsively ugly he couldn’t look at himself in themirror. And this enfant

terribleofEnglishmathematicswas,atthetimeheheardfromRamanujan,workingarevolutiononhisfieldthatwouldbefeltforgenerationstocome.

Oneis,ofcourse,movedtopraiseHardy’sabilitytoseegeniusinthetatteredgarbinwhichitwasclothed,andtoagreethattheworldwasenrichedasaresult.But,itstruckme,Hardy was enriched, too. His whole life was shaped by his time withRamanujan, which he called “the one romantic incident in my life.” The story ofRamanujan,then,isastoryabouttwomen,andwhattheymeanttoeachother.

Imustsayonethingmore,thoughindoingsoIriskalienatingafewreaders.ToAmericans,IndiaandCambridgeare,indeed,foreigncountries.AndasL.P.Hartleyhaswritten,thepastisalienterritory,too:“Theydothingsdifferentlythere.”Thus,theyearsaroundtheturnof thecenturywhenourstorybegins,a timewhenIndiawas still British and Victoria still queen, represent a second foreign country toexplore.Now,tothesetwoworldsremoteinplaceandtime,Imustaddathird—themathematicsthatRamanujanandHardydid,aloneandtogether,astheirlife’swork.

It is tempting to concentrate exclusively on the exotic and flavorful elements ofRamanujan’slifeandskipthemathematicsaltogether.Indeed,virtuallyallwhohavetakenRamanujanastheirsubjecthaveseveredhisworkfromhislife.Biographiesasdoexisteitherignorethemathematics,orbanishittothebackofthebook.Similarly,scholarly papers devoted to Ramanujan’s mathematics normally limit to a fewparagraphstheirattentiontohislife.

Andyet, canweunderstandRamanujan’s lifewithout someappreciation for themathematicsthathelivedforandloved?Whichistosay,canweunderstandanartistwithoutgainingafeelforhisart?Aphilosopherwithoutsomeglimpseintowhathebelieved?

Mathematics,Iammindful,presentsaspecialproblemtothegeneralreader(andwriter).Art,atleast,youcansee.Philosophyandliterature,too,havetheadvantagethat, however recondite, they can at least be rendered into English.Mathematics,however,ismiredinalanguageofsymbolsforeigntomostofus,exploresregionsofthe infinitesimally small and the infinitely large that elude words, much lessunderstanding. So specialized is mathematics today, I am told, that mostmathematicalpapersappearinginmostmathematicsjournalsareindecipherableeventomostmathematicians.PennsylvaniaState’sGeorgeAndrews,whorediscoveredalong-forgotten Ramanujan manuscript at Trinity College, says it took someonealreadyexpertinthenarrowareaofmathematicswithwhichitdealttorecognizeit—that merely being a professional mathematician with a Ph.D. would not havesufficed.

Whathope,then,hasthegeneralreaderfacedwithRamanujan’swork?

Little,certainly,ifwesetasthetasktofollowoneofRamanujan’sproofsthroughtwenty pages of hieroglyphics in a mathematics journal—especially in the case ofRamanujan,who routinely telescoped adozen steps into two, leavinghis reader tofindtheconnections.Buttocomeawaywithsomeflavorofhiswork, thepathsbywhich he got there, its historical roots? These pose no insuperable problem—certainlynomorethanfollowingaphilosophicalargument,orachallenging literaryexegesis.

Inonesense,atleast,Ramanujan’smathematicsismoreaccessiblethansomeotherfields; much of it comes under the heading of number theory, which seeks outpropertiesof, andpatternsamong, theordinarynumberswithwhichwedealeveryday; and 8s, 19s, and 376s are surely more familiar than quarks, quasars, andphosphocreatine. While the mathematical tools Ramanujan used were subtle andpowerful, the problems to which he applied them were often surprisingly easy toformulate.

•••

InanoteattheendofthebookImentionbynamemanywhohavehelpedmealongtheway.ButhereIwishtoespeciallythankthepeopleofSouthIndiaasawholeformakingmytimetheresopersonallysatisfying.

IspentfiveweeksintheSouth,travelingtoplacesthatfiguredinRamanujan’slife.Irodetrainsandbuses,touredtemples,atewithmyhandsoffbananaleaves.IwasbuttedinthebehindbyacowonthestreetsofKumbakonam,sharedaroomwithalizardinKodumudi.IsawErode,thetownwhereRamanujanwasborn.Itouredthehouse in which he grew up, participated in opening exercises at his alma mater,wandered through the grounds of the temple inNamakkal towhich he came at aturningpointinhislife,andsawtheroominwhichhediedinMadras.ThepeopleofSouthIndiatookmeintotheirhomes.Theybestoweduponmeeverykindness.Autorickshawandbicyclerickshawdriversoftenwenttoextraordinarylengthstogetmeto out-of-the-way places, forever struggling to understand what to them was myatrociouspronunciationofSouthIndianplace-names.Theystaredatthewhitenessofmyskinbutalwaystreatedmewithgentlenessandgoodwill.

Spendany lengthof timeamongthepeopleofSouthIndiaand it ishardnot tocome away with a heightened sense of spirituality, a deepened respect for hiddenrealms, that implicitly questionsWestern values andways of life. In theWest, allthrough the centuries, artists have sought to give expression to religious feeling,creatingBach fugues andGothic cathedrals in thanks and tribute to their gods. In

South India today, such religious feeling hangs heavy in the air, and to discern aspiritual resonance inRamanujan’smathematics seemsmore natural by far than itdoesinthesecularWest.

Ramanujan’s champion,Hardy,was a confirmed atheist.Yetwhen he died, onemournerspokeofhis

profound conviction that the truths of mathematics described a bright and clear universe, exquisite andbeautifulinitsstructure,incomparisonwithwhichthephysicalworldwasturbidandconfused.Itwasthiswhichmade his friends . . . think that in his attitude to mathematics there was something which, beingessentiallyspiritual,wasneartoreligion.

Thesame,butmoreemphatically,goesforRamanujan,whoallhislifebelievedintheHindugodsandmadethelandscapeoftheInfinite,inrealmsbothmathematicalandspiritual, his home. “An equation forme has nomeaning,” he once said, “unless itexpressesathoughtofGod.”

BaltimoreMay1990

CHAPTERONE

IntheTemple’sCoolness

[1887to1903]

1.DAKSHINGANGE

Heheard itallhis life—theslow,measured thwap . . . thwap . . . thwap . . .ofwetclothes being pounded clean on rocks jutting up from the waters of the CauveryRiver.Bornalmostwithinsightof the river,Ramanujanheard it evenasan infant.Growingup,heheard it ashe fetchedwater from theCauvery, orbathed in it,orplayed on its sandy banks after school. Later, back in India after years abroad,fevered, sick,andclose todeath,hewouldhear that rhythmic slappingsoundoncemore.

TheCauverywasafamiliar,recurringconstantofRamanujan’slife.Atsomeplacesalong its length, palm trees, their trunks heavywith fruit, leaned over the river atrakishangles.Atothers, leafy trees formedacanopyofgreenover it, theirgnarled,knottedrootssnakingalongtheriverbank.Duringthemonsoon,itswatersmightriseten,fifteen,twentyfeet,sometimesdrowningcattleallowedtograzetoolongbesideit.Come thedry season, the torrent became amemory, the riverbankswide sandybeaches,andtheCauveryitselfbutafeebletrickletracingthedeepestchannelsoftheriverbed.

But always it was there. Drawing its waters from the Coorg Mountains fivehundredmilestothewest,branchingandrebranchingacrossthepeninsula,itsflowchanneledbydamsandcanals someofwhichwentback fifteenhundredyears, theCauverypainted the surrounding countryside an intense, unforgettable green.Andthatsinglefact,morethananyother,madeRamanujan’sworldwhatitwas.

Kumbakonam,hishometown, flankedby theCauvery andoneof its tributaries,layintheheartlandofstaunchlytraditionalSouthIndia,160milessouthofMadras,inthedistrictthenknownasTanjore.Halfthedistrict’sthirty-sevenhundredsquaremiles, an area the size of the state ofDelaware, waswatered directly by the river,whichfellgently,threefeetpermile,tothesea,spreadingitsrichalluvialsoilacrossthedelta.

The Cauvery conferred almost unalloyed blessing. Even back in 1853, when itflooded,coveringthedeltawithwaterandcausing immensedamage, few liveswerelost.More typically, thegreat rivermade the surrounding land immune toyear-to-yearvariationinthemonsoon,uponwhosecapricesmostoftherestofIndiahung.In1877, in thewake of two straight years of failedmonsoons, South India had beenvisitedbydrought, leaving thousandsdead.ButTanjoreDistrict,nourishedby theunfailing Cauvery, had been scarcely touched; indeed, the rise in grain pricesaccompanyingthefaminehadbroughtthedeltaunprecedentedprosperity.

NowonderthattheCauvery,liketheGangesathousandmilesnorth,wasoneofIndia’s sacred rivers. India’s legendary puranas told of amortal known asKavera-muniwhoadoptedoneofBrahma’sdaughters.Infilialdevotiontohim,sheturnedherselfintoariverwhosewaterwouldpurify fromallsin.EventheholyGanges, itwassaid,periodicallyjoinedtheCauverythroughsomehiddenundergroundlink,soastopurgeitselfofpollutionborneofsinnersbathinginitswaters.

DakshinGange, theCauverywas called—theGangesof theSouth.And itmadethedelta themostdenselypopulated and richest region in all ofSouth India.Thewholeedificeoftheregion’slife,itswealthaswellastherichspiritualandintellectuallivesitswealthencouraged,alldependedonitswaters.TheCauverywasaplaceforspiritual cleansing; for agricultural surfeit; for drawing water and bathing eachmorning;forcattle,ledintoitsshallowwatersbymeninwhitedhotisandturbans,todrink;andalways,forwomen,standingknee-deepin itswaters,to lettheirsnakingribbonsofcottonorsilkdriftoutbehindthemintothegentlecurrent, thengatherthemupintosoddenclumpsofclothandslapthemslowly,relentlessly,against thewater-wornrocks.

2.SARANGAPANISANNIDHISTREET

InSeptember 1887, twomonths before her childwas due to be born, a nineteen-year-oldKumbakonam girl namedKomalatammal traveled to Erode, her parentalhome, 150miles upriver, to prepare for the birth of the child she carried.That awomanreturnedtohernativehomeforthebirthofherfirstchildwasatraditionsowidelyobservedthatofficialschargedwithmonitoringvitalstatisticsmadeapointofallowingforit.

Erode, a county seat home to about fifteen thousand people,was located at theconfluenceof theCauvery and one of its tributaries, theBhavani, about 250milessouthwest of Madras. At Erode—the word means “wet skull,” recalling a HindulegendinwhichanenragedSivatearsoffoneofBrahma’sfiveheads—theCauveryisbroad, itsstreambed litteredwithgreatslabsofprotrudingrock.Not far fromthe

river,in“thefort,”asthetown’soriginaltradingareawasknown,wasthelittlehouse,onTeppukulamStreet,thatbelongedtoKomalatammal’sfather.

ItwasherethatasonwasborntoherandherhusbandSrinivasa,justaftersunsetontheninthdayoftheIndianmonthofMargasirsha—orThursday,December22,1887.Onhiseleventhdayof life, again inaccordancewith tradition, thechildwasformallynamed, and a year almost to theday afterhis birth,SrinivasaRamanujanIyengarandhismother returned toKumbakonam,wherehewould spendmostofthenexttwentyyearsofhislife.

“Srinivasa”—its initial syllable pronounced shri—was just his father’s name,automaticallybestowedandrarelyused;indeed,onformaldocuments,andwhenhesignedhisname,itusuallyatrophiedintoaninitial“S.”“Iyengar,”meanwhile,wasacastename,referringtotheparticularbranchofSouthIndianBrahminstowhichheandhisfamilybelonged.Thus,withonenamethatofhisfatherandanotherthatofhiscaste,only“Ramanujan”washisalone.AshewouldlaterexplaintoaWesterner,“I have no proper surname.”Hismother often called himChinnaswami, or “littlelord.”Butotherwisehewas,simply,Ramanujan.

Hegotthename,bysomeaccounts,becausetheVaishnavitesaintRamanuja,wholivedaroundA.D.1100andwhosetheologicaldoctrinesinjectednewspiritualvitalityintoawitheredHinduism,wasalsobornonaThursdayandsharedwithhimotherastrological likenesses. “Ramanujan”—pronounced Rah-MAH-na-jun, with onlylight stress on the second syllable, and the last syllable sometimes closer to jum—meansyoungerbrother(anuja)ofRama,thatmodelofIndianmanhoodwhosestoryhasbeenhandeddownfromgenerationtogenerationthroughtheRamayana,India’snationalepic.

•••

Ramanujan’smother,Komalatammal,sangbhajans,ordevotionalsongs,atanearbytemple.Half the proceeds from her group’s performanceswent to the temple, theother half to the singers.Withher husband earning only about twenty rupees permonth, the five or ten she earned this way mattered; never would she miss arehearsal.

Yetnow,inDecember1889,shewasmissingthem,fourorfiveinarow.Sooneday, the head of the singing group showed up at Komalatammal’s house toinvestigate.

Thereshefound,pilednearthefrontdoor,leavesofthemargosatree;someone,itwas plain to her, had smallpox. Stepping inside, she saw a small, dark figure lying

atopabedofmargosaleaves.Hismother,chantingallthewhile,dippedtheleavesinwaterlacedwithgroundturmeric,andgentlyscouredtwo-year-oldRamanujan’spox-ridden body—both to relieve the infernal itching and, South Indian herbalistsbelieved,subduethefever.

Ramanujan would bear the scars of his childhood smallpox all his life. But herecovered,andinthatwasfortunate.ForinTanjoreDistrict,aroundthetimehewasgrowingup, a bad year for smallpoxmeant four thousand deaths. Fewer than oneperson in fivewas vaccinated.A cholera epidemicwhenRamanujanwas ten killedfifteenthousandpeople.Threeorfourchildrenineverytendiedbeforethey’dlivedayear.

Ramanujan’sfamilywasacasestudyinthedamningstatistics.Whenhewasayearandahalf,hismotherborea son,Sadagopan.Threemonths later,Sadagopanwasdead.

WhenRamanujanwas almost four, inNovember 1891, a girlwas born.By thefollowingFebruary,she,too,wasdead.

WhenRamanujanwassixandahalf,hismothergavebirthtoyetanotherchild,Seshan—whoalsodiedbeforetheyearwasout.

Muchlater,twobrothersdidsurvive—LakshmiNarasimhan,bornin1898,whenRamanujanwasten,andTirunarayanan,bornwhenhewasseventeen.Butthedeathofhisinfantbrothersandsisterduringthoseearlyyearsmeantthathegrewupwiththesolicitousregardandcentralpositionofanonlychild.

After the death of his paternal grandfather, who had suffered from leprosy,Ramanujan,sevenatthetime,brokeoutinabadcaseofitchingandboils.Butthiswasnotthefirsthintofatemperamentinclinedtoextremeandunexpectedreactionstostress. Indeed,Ramanujanwasa sensitive, stubborn,and—ifawordmoreoftenreserved for adults in their prime can be applied to a little boy—eccentric child.While yet an infant back inErode, hewouldn’t eat except at the temple.Later, inKumbakonam,he’dtakeallthebrassandcoppervesselsinthehouseandlinethemupfromonewalltotheother.Ifhedidn’tgetwhathewantedtoeat,hewasknowntorollinthemudinfrustration.

For Ramanujan’s first three years, he scarcely spoke. Perhaps, it is tempting tothink,becausehesimplydidn’tchooseto;hewasanenormouslyself-willedchild.Itwascommon in thosedays forayoungwife to shuttlebackand forthbetweenherhusband’shouseand thatofherparents, andKomalatammal,worriedbyher son’smuteness, took Ramanujan to see her father, then living in Kanchipuram, nearMadras.There, at theurgingof anelderly friendofher father’s,Ramanujanbeganthe ritual practice of Akshara Abhyasam: his hand, held and guided by his

grandfather, wasmade to trace outTamil characters in a thick bed of rice spreadacrossthefloor,aseachcharacterwasspokenaloud.

SoonfearsofRamanujan’sdumbnessweredispelledandhebegantolearnthe12vowels, 18 consonants, and 216 combined consonant-vowel forms of the Tamilalphabet. On October 1, 1892, the traditional opening day of school, known asVijayathasami,hewasenrolled,totheaccompanimentofancientVedicchants,inthelocalpialschool.ApialisthelittleporchinfrontofmostSouthIndianhouses;apialschoolwasjustateachermeetingtherewithhalfadozenorsopupils.

Butfive-year-oldRamanujan,dislikingtheteacher,bristledatattending.Evenasachild, he was so self-directed that, it was fair to say, unless he was ready to dosomething on his own, in his own time, hewas scarcely capable of doing it at all;schoolforhimoftenmeantnotkeystoknowledgebutshacklestothrowoff.

Quietandcontemplative,Ramanujanwasfondofaskingquestionslike,Whowasthe firstman in the world?Or,How far is it between clouds?He liked to be byhimself, a tendency abetted by parentswho, when friends called, discouraged himfromgoingouttoplay;sohe’dtalktothemfromthewindowoverlookingthestreet.Helackedallinterestinsports.Andinaworldwhereobesitywasvirtuallyunknown,wherebonesprotrudedfromhumansandanimalsalike,hewas, firstasachildandthen for most of his life, fat. He used to say—whether as boast, joke, or lamentremainsunclear—thatifhegotintoafightwithanotherboyhehadonlytofallonhimtocrushhimtopieces.

For about two years, Ramanujan was shuffled between schools. Beginning inMarch 1894, while still at hismother’s parents’ house inKanchipuram, he brieflyattendedaschoolinwhichthelanguageofinstructionwasnothisnativeTamilbuttherelatedbutdistinctTelugu.There,sometimespunishedbyhavingtositwithhisarmsfoldedinfrontofhimandonefingerturneduptohislipsinsilence,hewouldattimesstalkoutofclassinahuff.

In a dispute over a loan, his grandfather quit his job and left Kanchipuram.Ramanujan and his mother returned to Kumbakonam, where he enrolled in theKangayan Primary School. But when his other grandfather died, Ramanujan wasbouncedbacktohismaternalgrandparents,whobynowwereinMadras.Therehesofiercelyfoughtattendingschoolthatthefamilyenlistedalocalconstabletoscarehimbacktoclass.

Bymid-1895,afteranunhappysixmonthsinMadras,RamanujanwasoncemorebackinKumbakonam.

•••

Kumbakonam was flanked by the Cauvery and the Arasalar, its tributary. Moststreets ran parallel to these rivers or else marched straight down to their banks,perpendicular to the first set, making for a surprisingly regular grid system. Andthere,nearthemiddleof this compactgrid,onSarangapaniSannidhiStreet, adirtroadaboutthirtyfeetwidewithsquatlittlebuildingsclosepackedoneitherside,wasRamanujan’shouse.

Theone-storystructure,thatchedwithpalmleaves,stoodbackabouttenfeetfromthestreet,insulated,asitwere,byitstwo-tiered,coveredpial:itwasasteportwoupfrom the dusty (ormuddy) street, another few up to the little porch. The stuccohouse faced the streetwitha twelve-foot-widewallbrokenbyawindowto the leftandadoortotheright.Abystanderinthestreet,peeringthroughtheopendoorandintothegloomoftheinterior,couldsightallthewaythroughtotheback,wherehisgazewouldbearrestedbyasplashofsunlightfromtheopenrearcourt.Therestofthehouse,meanwhile,wasoffsettotheleft,behindthefrontwindow.Herewasthemainlivingareaand,behindit,asmallkitchen,redolentwithyearsofcookingsmells.

SouthIndiawasnotalwayshot;butitwasnevercold.Atalatitudeofaboutelevendegreesnorth.Kumbakonam laycomfortablywithin the tropics; evenona Januarywinter’snight, the thermometerdropped,onaverage,only to seventydegrees.Andthat climatological fact established an architectural fact, for it gave South Indianhomesakindofpermeability;theirinteriorsalwayssavoredalittleoftheoutside(afeeling familiar toAmericanswith screened-in porches).Windows therewere, buttheseweremerely cutouts in thewall, perhapswith bars or shutters, never space-sealingpanesofglassthatleftyouconsciousofbeingononesideoranother.Inthemiddleofmosthouseswas a small courtyard, themuttam, open to the sky—like askylightbutagainwithouttheglass—thatbroughtrainintothecenterofthehouse,whereitwasfunneledtoadrainthatledbackoutside.InRamanujan’shouse,smellsfrom outside wafted inside. Lizards crawled,mosquitos flew unimpeded. The softSouthIndianair,fragrantwithroses,withincense,withcowdungburnedasafuel,waftedovereverything.

Justoutside thedoor layKumbakonam,anancient capitalof theCholaEmpire.TheCholashad reached their zenitharoundA.D. 1000,whenEuropewallowed intheDarkAges, and had ruled, alongwith northernCeylon,most ofwhat, duringRamanujan’s day, was known as the Madras Presidency (which, with those ofBombay and Calcutta, constituted the chief administrative and political units ofBritish-ruled India).Thedozenor somajor templesdating from thisperiodmadeKumbakonamamagnettopilgrimsfromthroughoutSouthIndia.

Everytwelveyears,aroundFebruaryorMarch,theycamefortheMahammakhamfestival, commemorating a legendary post-Deluge event in which the seeds ofcreation,driftingupon thewaters in a sacredpot—orkumba, sourceof the town’sname—waspiercedbythegodSiva’sarrow.Thenectarthusfreed,itwassaid,hadcollectedintheMahammakham“tank,”theoutdoorpoolforritualbathingthatwaspartofeverytemple.Atsuchtimes—asin1897,whenRamanujanwasnine—three-quarters of a million pilgrims might descend on the town. And the great tank,surroundedbypicturesquemandapams,orhalls,andcoveringanexpanseoftwentyacres, would be so filled with pilgrims that its water level was said to rise severalinches.

When not in use, temple tanks could seem anything but spiritually uplifting.Open, stone-lined reservoirs, sometimes stocked with fish, frequently green withalgae,theyoftenservedasbreedinggroundsformalarialmosquitos.Situatedonlowground between two rivers, Kumbakonam was notorious for its bad water, itsmosquitos,anditsfilarialelephantiasis,amosquito-bornediseasethatleftitsvictimswith grotesquely deformed limbs, sometimes with scrota the size of basketballs.When Ramanujan was six, the town completed a drainage system. But this wasdesignedtocarryoffonly surfacewater,not sewage,andmostof the town’shealthproblemscontinuedunabated.

Kumbakonam,aday’strainridefromMadras,whichwasalmost200milesnorthand thenearest realmetropolis, had a seventy-two-bed hospital. It had four policestations, two lower secondaryEnglishschools, three conducting classes inTamil, ahighschoolofexcellentreputation,andacollege. Indeed,withapopulationduringRamanujan’s day ofmore than fifty thousand, it was nomere village, but amajortown,sixthlargestintheMadrasPresidency.

Just outside town and all through the surrounding districts ranged some of therichest cropland in all of India. Two-thirds of the population—including wholecastes given over to agricultural labor alone, like the Paraiyans and the Pallans—workedthelandforafewannasaday.SiltcarrieddowntothedeltabytheCauverymade the use of expensivemanure as fertilizer unnecessary.Narrow strips of landbesidetheriver,annuallysubmergedinthesilt-ladenwaterofmonsoon-bornefloods,were especially valued and used to raise bamboo, tobacco, or banana.Meanwhile,most of the rest of the delta’s arable cropland,more than three-quarters of it,wasgivenovertorice.

InmanypartsofSouthIndia,thelandwas,formuchoftheyear,ableakbrown.Buthere,midstthericefieldsoftheCauvery,thelandscapesuddenlythickenedwithlushgreeneryinarichpaletteofshadesandtextures.Farmersnurseddelicateinfant

rice seedlings in small, specially watered plots whose rich velvety green stood outagainstneighboringfields.When,afterthirtyorfortydays,theplantswerehealthyand strong, laborers individually scooped them out with their root pods andtransplantedthemto large, flooded fields; thesemade fora softergreen.There theplantsgrewuntilayellowerhuesignaledtheywerereadyforharvest.

Almosteverysquarefootofthedeltawasundertheplow,andhadbeensincetimeimmemorial.Cattleandsheepfoundlittleroominwhichtograze;thelandwasjusttoovaluable.Forestswerefew,justisolatedcoconut,banyan,orfigtreesand,towardthecoast,palmyrapalmsandAlexandrianlaurel.The342squaremilesofthetaluk,orcounty,ofwhichKumbakonamwaschief town,comprised justabout that samenumberofvillages.Mostwere littlemorethantinyinhabitedislandsmidsta seaofwavingcrops—acoupleofdozenthatched-roofhutsandafewhundredinhabitants,half-hidden by coconut palms, sitting on cramped little sites a few feet above theneighboringricefields.

Andyet,whatevertheirdebttotheland,Ramanujan’sfamilywasnotoftheland.Theyweretownspeople.Theywerepoor,buttheywereurbanpoor;theyinhabitednotjustthegroundonwhichtheylivedbutawiderworldofthemindandspirit.TheCauvery freed the town fromunduepreoccupationwith theday’sweather and theseason’scrops,bestowinguponitameasureofwealth.AndRamanujan’sfamilywasamongthemanywho,indirectly,livedoffitsbounty.

Like theAmerican cityofDesMoines,with its similar relationship to the corn-rich countryside of Iowa, Kumbakonam was more cosmopolitan than itssurroundings,wasacenterfortheworkofeye,hand,andbrain,whichneedsadegreeof leisure to pursue.A census taken around the timeRamanujan was growing upfound ithad ahigherproportionofprofessionals than anywhere in thepresidencybut Madras itself. The crafts were especially strong. One specialty was finemetalwork; Kumbakonam craftsmen, six hundred of them, it was estimated, keptEuropeanmarkets stockedwithdeitiesof theHindupantheonexecuted incopper,silver,andbrass.

Anotherspecialtywassilksaris,theproductoftwothousandsmallloomsmannedbythreethousandpeople.NoplaceinSouthIndiawasbetterknownforitsfinesilksaris,dazzlinginbrightcolors,embroideredinsilverstripes,fringedwithgold,thanKumbakonamandneighboringTanjore.SariswoveninKumbakonamcouldcostasmuchasahundredrupees—ayear’sincometomanypoorfamilies.

Bountiful harvests made the delta home to many wealthy farmers, and themarriageofoneoftheirdaughtersmightmeanthepurchaseofadozenormoresaris.Beforethewedding,thewholefamilywouldtroopintotown,maketheirselections,

onlylatertobebilledforwhattheytookaway;themerchantswerehappytoextendcredit to such well-heeled customers. Otherwise, though, it was normally thehusbands who did the buying, worried lest their wives, as one Kumbakonam sariweaverandshopownertoldanEnglishvisitoraroundthistime,spenttoomuch.

Srinivasa Iyengar, twenty-four athis sonRamanujan’sbirth andabout five yearsolderthanhiswife,wasaclerkinonesuchshop,justashisownfather,Kupuswamy,hadbeen.Normally, such a clerk remained one all his life—waiting on customers,takingorders,performingroutinepaperwork,perhapstravelingtonearbyvillagestocollectbills.Occasionallyaclerkmightbetakenintothebusinessorwouldgoofftostarthisown.Butthatrequiredsomespecialdriveorentrepreneurialtemperament.ApparentlySrinivasawasgoodatappraisingfabrics,askilluponwhichhisemployerrelied;butbeyondthat,whateverittooktosteptoabetterjobhecouldnevermuster.

ClerkslikeSrinivasareportedtoworkateightorsointhemorninganddidn’tgethometilllongafterdark(which,soclosetotheequator,variedlittleacrosstheyearfrom about 6:00 P.M.). Sometimes they would return home at midday for lunch,thoughmorelikelytheirwivespackedfoodinsmallmetalcannistersforthemtoeatat the shop. Because certain months were deemed unpropitious, weddings wouldoften stack up in months reckoned as lucky, making business quite seasonal andleaving clerks to sit idle for longperiods.At such times,Srinivasamightwell havebeenfoundasleepintheshopinthemiddleoftheday.

Dayafterday,yearafteryear,hewasattheshop,largelyabsentfromRamanujan’searly life. Indian society generally left the father little role to play at home, castinghim as an aloof, physically undemonstrative, even unwelcoming figure whoserelationshipwithhischildrenwas largely formal.Srinivasawasalmost invisible,hisname largely absent from family accounts. “Very quiet,” a boyhood friend ofRamanujancalledhim.Someoneelsewouldresorttotheword“weightless.”Butevenhadhebeenotherwise,hecouldscarcelyhavecompetedwithKomalatammalasaninfluenceontheirson.

Years later, while away in England, and with at least one letter to his fatherconfined to reminders to keep up the house and not let the gutter run over,RamanujanwrotehismotheraboutthetitanicstruggleunleashedinEuropewiththeonsetoftheGreatWar,downtodetailsofthenumberofmenfighting,thewidthofthebattlefronts,theuseofairplanesincombat,andthecontributionofIndianrajastotheBritishwareffort.

He must have known such an account would interest her. He and his motherunderstood each other. They talked the same language, enjoyed one another’sintelligentcompany, shared the same intensityof feeling.Whenhewasyoung, the

twoofthemdueledatGoatsandTigers,playedwithpebbles,onagridresemblingaperspectiveviewof railroad tracks receding to thehorizon, crossedbyother tracksperpendiculartothem.Three“tigers”soughttokillfifteen“goats”byjumpingthem,as incheckers,while the goats tried to encircle the tigers, immobilizing them.Thegamedemandedlogic,strategy,andfierce,chesslikeconcentration.Thetwoofthemreveledinit.

Komalatammal,whomRamanujanresembledphysically,was,inthewordsofoneaccount,“ashrewdandculturedlady.”HerfamilycouldclaimSanskritscholarsinitsline, scholars uponwhom local kings had bestowed gifts. Shewas the daughter ofNarayana Iyengar, well known in Erode as amin, an official in the district courtcharged with calling witnesses, taking court notes, and conferring with lawyers.WhenRamanujanwasaboutfour,herfatheroffendedsomehigher-upand losthisjob. It was then that he and his wife, Rangammal, moved to Kanchipuram, thetemple city near Madras. There he managed a choultry, a temple annex wheremarriagesareheldandpilgrimsputup.

A picture of Komalatammal survives, probably taken in her forties or fifties. Itshows a woman whose corpulence even nine yards of sari cannot hide. Only herhands,restinglightlyoverthearmsofherchair,suggestease.Thewholerestofherbodyconveysrawintensity:headcockedtooneside,eyesalive,almostglaring,mouthset, leaninga little forward inthechair,only theballsofherbare feet touchingthefloor,poised as if ready to spring.The overall impression is one of great personalforceonlybarelycontainedwithinherbody.

Shewasanintense,evenobsessivewoman,nevershyaboutthrustingherpowerfulpersonalityontoobjectsofherinterest.Andherprimaryobjectalltheyearshewasgrowingupwasherson,Chinnaswami.InIndia,strongtiesbetweensonandmotherarelegendary;closeindeedmusthavebeentherelationshipbetweenRamanujanandhismotherthatevenhisIndianbiographersinvariablysawfittocommentuponit.

Komalatammalfedhimhisyogurtandrice,hisspicy,pickledfruitsandvegetables,hislentilsoup.Shecombedhishairandcoileditintothetraditionaltuft,sometimesplacing in it a flower.She tiedhisdhoti (or, as itwasknown inTamil,veshti), thelongpieceofclothwrappedaroundthewaistandpulledupbetweenthelegsthatallbut themostWesternizedmen wore. She applied the namam, the powdery castemark, to his forehead. Shewalked him to school; before going,Ramanujan wouldtouchher feet in the traditional Indiansignof respectandsecureherblessing.Shemonitored his friends and his time, made his decisions. Later, when Ramanujandidn’t get the treatment at school she thought he deserved, she stormed into theprincipal’sofficeandprotested.Andwhenshedecidedheoughttomarry,shefound

him a wife and arranged for the wedding—all without bothering to consult herhusband.

Shepouredprodigiousenergyintoherspirituallife.InHindufamilies,thewomenwereapttobemorepious,andmorescrupulousaboutobservingtradition,thanthemen. So it had been in her own family; her mother was said to have gone intohypnotic trances that placed her in communion with the gods. And so it was inRamanujan’sfamily.Komalatammalwasfiercelydevout,heldprayermeetingsatherhome,sangatthetemple,pursuedastrologyandpalmistry.Always,thenameoftheirfamilydeity,theGoddessNamagiriofNamakkal,wasonherlips.“Anexceptionallygiftedladywithpsychicpowersandaremarkableimagination”washowonefriendofthe family described her. She had “’a remarkable repertoire ofmythological storiesandusedtotellmestoriesfrom[the]ancientMahabharataandRamayanato[the]later Vikramaditya legends.” Any pause in the telling was cause for yet anothermurmuredappealtoNamagiri.

Fromhismother,Ramanujanabsorbedtradition,masteredthedoctrinesofcaste,learnedthepuranas.Helearnedtosingreligioussongs,toattendpujas,ordevotions,atthetemple,toeattherightfoodsandforswearthewrongones—learned,inshort,whathemustdo,andwhathemustneverdo,inordertobeagoodBrahminboy.

3.ABRAHMINBOYHOOD

For thousands of years Brahmins had been the learned men, teachers, andinterpretersofHindulife.Brahminswithheadssoshavedinfrontthattheylookedprematurely bald, prominent caste marks of dried, colored paste upon theirforeheads, locks of hair in the back like little ponytails, and thin, white, knottedthreadsworndiagonallyacrosstheirbarechests,wereaneverydaysightonthestreetsof Kumbakonam and within its temples. Kumbakonam was a bedrock ofBrahminism,thetraditionalHinduismassociatedwithitshighest,priestlycaste.

Four percent of the South Indian population, Brahmins were to most Hindusobjects of veneration and respect; in pre-British India, at least, wealthy patronsacquired religious merit and washed away sins by giving them land, houses, gold.Brahminswerethetemplepriests,theastrologers,thegurus,thepanditsspecializingin sacred law and Vedic exegesis, indispensable at every wedding and funeral,occupyingthemostexaltednicheintheIndiancastesystem.

BooksaboutIndiabyBritishwritersaroundthistimeseemedtodelightinregalingtheirreaderswiththehorrorsofthecastesystem—ofmenandwomenpunishedforsins committed in past lives by being consigned in this one to low and pitiablestations.Therewerefourcastes,theseaccountsrecorded:Brahmins,atthetopofthe

heap;Kshatriyas, or warriors; Vaisyas, or merchants and traders; and Sudras, ormenials.Afifthgroup,theuntouchables,layproperlyoutsidethecastesystem.Thefirstthreecasteswereentitledtowearthesacredthreadthataffirmedthem“twice-born.”TheSudrascouldnot,butcouldenterthetemples.Theuntouchablescouldnotevendothat.Norcouldtheydrawwaterfromthevillagewell.Nor,traditionally,couldeventheirshadowscrossthepathofaBrahminwithouthishavingtoundergoapurificationritual.

Even this rudimentary breakdown, based on caste law first set down in theInstitutesofManu,aSanskritworkdatingtothethirdcentury,didn’tquiteapplyinSouthIndia;fortheKshatriyaswentlargelyunrepresentedintheSouth.Butmore,itomitted the reality of India’s several thousand self-governing subcastes, each withrules as to who could eat with whom, and whom one could marry. Most wereoriginally,andoftenstill,rootedinoccupationalcategories.Thus,therewerecastesofagriculturalworkers,barbers,weavers,carpenters.Itwasthesesubcastes,or jatis,towhich one really belonged.One simplywas aVanniar, or aChettiar.Or, as inRamanujan’scase,aVaishnaviteBrahmin;hisveryname,Iyengar,labeledhimone.

FromtheHindupantheonofBrahma,Vishnu,andSiva,Vaishnavites—aboutoneBrahmin in four—singled out Vishnu as object of special devotion. Furthertheologicalnuances—forexample,over justhowmuchhumaneffortwasneededtosecuredivinegrace—layinthesplitbetweenitsTengalaiandVedagalai,ornorthernand southern, branches.Suchdistinctionswerenotunlike thosemarkingoff, first,ChristiansfromJews,thenProtestantsfromRomanCatholics,andfinally,LutheransfromMethodists.AndliketheirWesterncounterparts,thedifferenceswereoftenasmuchmatters of style, tone, ritual emphasis, and historical accident as theologicaldoctrine.

AllHindus believed in reincarnation and karma, heard the same tellings of thegreat Indian epics, shared certain sensibilities, values, and beliefs. ButVaishnaviteBrahmins,asarule,simplydidnotmarryShaiviteBrahmins,thosedevotedtoSiva.Each group had its own temples, shrines, and centers of religious teaching.Ramanujan wore a caste mark on his forehead—the namam, a broad white “U”intersectedbya redvertical slash—whollydistinct fromthe threewhitehorizontalstripeswornbyShaivites.

Castebarriersrosehighestatmealtime.ABrahminateonlywithotherBrahmins,could be served only by other Brahmins. In the cities, restaurants and hotelsemployingBrahmin chefs prominently advertised that fact. A Brahmin away fromhomewenttoelaboratepainstoverifythesourceoffoodheate.Brahminfamilieson

pilgrimages to distant shrines would pull over to the side of the road to eat whatthey’dbroughtwiththemratherthanchancefoodpreparedbywho-knew-whom.

Mostoften,itwasaBrahminmale’swifewhopreparedandservedhismeals.Buthe never ate with her—another example of heathen ways the English cited asrepugnanttoproperChristians:womenpreparedthemealsofthemenandchildrenof thehousehold, serving them fromvesselsof silver, copper, andbrass (not china,whichwasdeemedinsufficientlyclean),andhoveringoverthemduringmealtimetodispense fresh helpings. The men would eat, largely oblivious to them, then risetogetheratmeal’send.Onlythen,oncehavingcleanedup,wouldthewomenretreattothekitchenandeatwhateverremained.

Ramanujanatewhileseatedonthefloor,fromaroundmetaltrayor,moreoften,banana leaves setbeforehimand laterdiscarded, likepaperplates.Heatewithhishands.Thisdidnotmeanusingbreadtoscooporsopupfood.ThestaplefoodupNorth was wheat, that of the South rice; bread played little role in its diet. SoRamanujanatepreciselyaseveryWesterntoddlerlearnsnottoeat—withhisfingers.

Into the centerof thebanana leafwouldbe ladledahelpingof rice.Toward theperipheryoftheleaf—aboutthesizeofaplacematinaWesternhouseholdandstillgreen and fresh, with a thick, muscular rib running down the middle—would godollopsofsharplypickledfruitsorvegetables,likemangos,onions,ororanges;spicedfruit chutneys; sambhar, a thick lentil soup stocked with potatoes; and yogurt.Sometimes justa fewselections, sometimes, fora festivemeal, asmanyasadozen.Withthefingersofhisrighthand(andonlyhisrighthand),Ramanujanwouldmixricewithoneorseveralotherfoods.Then,withfourfingersandthumbformedintoapincer,he’d shape someof the loosemixture into apastyball andplop it ontohistongue.

South Indian cuisinewas tasty andnutritious, if not always subtle. Itwas neverbland;thecurrieddishesweresharpandspicy,theothersalmostmaddeninglysweet.Rice andyogurt, beyond theirnutritive value, softened andblunted thebite of thespices themselves.Coconuts and bananas (or actually plantains, a shorter, stubbiervariety,tastingmuchthesame)werethemainfruits,alongwithmangoandguava.

That Ramanujan never ate meat, then, was no act of painful self-denial. Likevirtually all Brahmins, he was a strict vegetarian. And yet to say meat was“prohibited”tohimsubtlymissesthepoint.Itscarcelyneededtobeprohibited,andforthesamesimple,invisiblereasonanorthodoxJeworMuslimneedn’tbetoldnottoeatpork:youjustdidn’tdoit.Othersatemeat;hedidn’t.Hewouldhavegaggedatthe thought. Some of his friends even avoided ingredients, like beetroot, that gavefoodareddishcastreminiscentofblood.

Ramanujanabsorbedsuchdosanddon’tsofBrahminlifeasnaturallyashelearnedto walk and talk. “As the child learned to accept responsibility for its own bodilycleanliness, it was also taught the importance of avoiding the invisible pollutionconferred by the touch of members of the lowest castes,” is how one scholar, G.Morris Carstairs, would later depict the Indian socializing process at work. “Themother or grandmother would call him in and make him bathe and change hisclothes if this should happen, until his repugnance for a low caste person’s touchbecameasinvoluntaryashisdisgustforthesmellandtouchoffeces.”

Every morning a Hindu male underwent an elaborate cleansing ritual. Hedefecated, using his left hand only to clean himself with water. Then he bathed,preferably in a holy river like theCauvery, but always paying special heed to ears,eyes,andnostrils.Indrinking,heneverbroughtacup tohis lipsbut rather spilledwater from it into his mouth. After a meal, he got up, left the eating area, andceremoniously poured water over his hands. For all the dirt and lack of modernsanitaryfacilitieswhichsobotheredEnglishvisitors,therewasafastidiousnessaboutHindulifethatnooneobservedmorescrupulouslythanorthodoxBrahmins.

Though sometimes scorned as haughty, Brahmins felt pride that, in their ownestimation,even thepoorest among themwere cleaner andpurer thanothers; thattheleasteducatedBrahminknewsomeSanskrit,theancientlanguageofHinduism’ssacredtexts;thatnormallytheywereaccordeddeferenceandrespectbyothers;thateducationallyandprofessionally,theyexcelled.Allthiscontributedtoasensealmostuniversal among them—and nothing suggests Ramanujan failed to share it—thatBrahminswere,inarealsense,chosen.

4.OFF-SCALE

AmongBrahmins, traditionally, a sanyasi, or itinerantbeggarwhogaveupworldlyinterests for spiritual, was not deemed a failure. An ascetic streak ran throughBrahminculture.As Sanskrit scholarDaniel Ingalls has written in an essay, “TheBrahminTradition,”“Asceticismandmysticismhavebeen,formanycenturiesnow,totherespectableIndianclasseswhatarthasbeenforthelastcenturyandahalftothe bourgeoisie ofWestern Europe”—something to which, whether aspiring to itthemselvesornot,theyatleastgavelipservice,andrespected.

This tradition lifted an eyebrow toward any too-fevered a rush toward worldlysuccess, lauded a life rich inmind and spirit, bereft though itmightbe of physicalcomfort. Even wealthy Brahmin families often kept homes that, both byWesternstandardsand thoseofotherwell-off Indians,were conspicuousby their simplicityandspartangrace,withbarefloors,themeanestoffurniture.“Simplelivingandhigh

thinking,” is how one South Indian Brahmin would, years later, characterize thetradition.

ButintheyearsRamanujanwasgrowingup,thingswerechanging.Brahminswerestillthepriestsandgurus,thelogiciansandpoets,theSanskritscholarsandsanyasisofHindulife.Butnowtheoldcontemplativebentwastakingnewform;thespiritualwasbeingtransmutedintothesecular.LikeJewsinEuropeandAmericaataboutthesame time (with whom South Indian Brahmins would, almost a century later,routinelycomparethemselves),theywerebecomingprofessionals.

Thecensus followingRamanujan’sbirthnotedthatofSouthIndia’ssixhundredthousandmaleBrahmins,some15percent—anextraordinarilyhighnumber—heldpositions in thecivil service, the learnedprofessions, andminorprofessional fields.Theyalreadydominatedtheranksofthecollegeeducated,andwithinageneration,by1914,of650graduatesoftheUniversityofMadrasnofewerthan452wouldbeBrahmins—morethantentimestheirproportionofthepopulation.Theoldmiddleclassoftradersandbarristershad traditionallybeendrawn fromtheirowndistinctcastes. But the British had helped build a new middle class of brokers, agents,teachers, civil servants, journalists, writers, and government clerks. And thesepositionsBrahminsnowbegantofill.

In Brahminically steepedKumbakonam, one in five adultmales could read andwrite, more than anywhere else in South India with the possible exception ofTanjore,thedistrictseat,andMadrasitself.KumbakonamBrahminshadatasteforphilosophical and intellectual inquiry, a delight in mental exercise, that led oneEnglish observer to pronounce them “proverbial for ability and subtlety.”Ramanujan’sparents,whennotmired inoutrightpoverty, clung to thenethermostreachesofthemiddleclassandwere illiterate inEnglish,thoughnot intheirnativeTamil;hisfriends,however,mostlycamefrombetter-offfamiliesandwereboundforpositionsaslawyers,engineers,andgovernmentofficials.

Indoingso,theytrodcareerpathswithonethingincommon:thewaywasalwaysmarkedinEnglish.

Ramanujan’s native languagewasTamil, one of a family ofDravidian languagesthat includes Malayalam, Canarese, and the musical-sounding Telugu. Europeanscholars acclaimedTamil for its clear-cut logic; “a language made by lawyers andgrammarians,” someoneonce called it.Spoken from just northofMadraswithin abroad,kidney-shapedregionwesttotheNilgiriHillsandsouthtoCapeComorinatthetipofthesubcontinent,aswellasinnorthernCeylon,Tamilrepresentednoout-of-the-waylinguisticoutpost.Ithaditsownrichliterature,distinctfromtheHindi

ofthenorth,goingbacktothefifthcenturyB.C.,boastedaverseformreminiscentofancientGreek,andwasspokenbyalmosttwentymillionpeople.

Butintheearly1900s,asnow,EnglishwasascendantinIndia.Itwasthelanguageof the country’s rulers. It fueled the machinery of government. It was the linguafranca towhich Indians, who spokemore than a dozen distinct languages, turnedwhentheydidnototherwiseunderstandoneanother.AmongIndiansasawhole,tobe sure, the proportionwho spokeEnglishwas small. Even among relativelywell-educatedTamilBrahminmales,onlyabout11percentwere(in1911) literate in it.So, those who did speak and read it were, in obedience to the law of supply anddemand,propelledontothefasttrack.Asaclerk,evenasmatteringofitgotyouanextrafewrupees’pay.Itwastheticketofadmissiontotheprofessions.

•••

While a pupil at Kangayan Primary School, Ramanujan studied English from anearlyage,andinNovember1897,justshyoften,hepassedhisprimaryexaminations—in English, Tamil, arithmetic, and geography—scoring first in the district. ThefollowingJanuary,heenrolledintheEnglishlanguagehighschool,TownHigh.

TownHighSchoolhad its origins in1864 in twohousesonBigStreet, amainthoroughfare near the heart of town. When, some years later, the local collegedroppeditslowerclasses,agroupofpublic-spiritedcitizensrushedtofillthevacantacademic niche from below, through an expanded Town High. They would teardown the old buildings, erect a new one on the existing site . . .No, pronouncedThambuswami Mudaliar, a magnificently mustachioed eminence on the school’smanaging committee, better to start afresh. And for the school’s new campus, heoffered seven prime acres then harboring a banana orchard. There, he personallysupervisedconstructionofthefirstbuildings.

Today, Town High’s cluster of handsome white buildings occupies an oasis oftropical charm insulated fromthenoisy streetout frontbya sandy field shadedbytall margosa trees. At the time Ramanujan attended, however, the first block ofclassrooms,withitsroofofdenselylayeredredclaytilesandporchoverhangsofpalmleafthatching,hadgoneupjustafewyearsbefore.Itsclassroomswere laidend-to-end,makingforabuildingoneroomwide,withwindowsonbothsidestocatchanyhintofbreeze.

The windows would have caught any adolescent clamor, too, but there wasprobably little to carry. Years later an alumnus would recall the long coats andturbans of the teachers and the respect they commanded among the students.

Headmaster during Ramanujan’s time, and for twenty-two years in all, was S.KrishnaswamiIyer,asevere-facedmanpartialtoimpromptustrollsbetweenclasses.The tapping of his walking stick would alert both teachers and students to hiscoming. Sometimes he’d step into a class, take over from the teacher, questionstudents,andteachtherestof theclass—withenoughflair, it seems, thatwhenhetaughtGrey’s “EtonCollege” one student imagined littleTownHigh as Eton, theirrigationditchcrossingthecampusastheThames.

Theschool,whichstoodaboutafive-minutewalkfromRamanujan’shouse,drewthecreamofKumbakonamyouthandlaunchedthemintocollegeandcareer.Alumniwould later recall it with genuine fondness. And it nourished Ramanujan for sixyears,bringinghimascloseashe’devercometoasatisfyingacademicexperience.

Evenallowingfortheretrospectivehalothatseesineveryschoolboyexploitofthefamous a harbinger of future greatness, it’s plain that Ramanujan’s gifts becameapparent early. Ramanujan entered Town High’s first form at the age of ten,correspondingtoaboutanAmericanseventhgrade.Andalreadywhilehewasinthesecondform,hisclassmateswerecomingtohimforhelpwithmathematicsproblems.

Soon, certainly by the third form,hewas challenginghis teachers.Oneday, themathteacherpointedoutthatanythingdividedbyitselfwasone:Dividethreefruitsamongthreepeople,hewassaying,andeachwouldgetone.Divideathousandfruitsamongathousandpeople,andeachwouldgetone.SoRamanujanpipedup:“Butiszerodividedbyzeroalsoone?Ifnofruitsaredividedamongnoone,willeachstillgetone?”

Ramanujan’sfamily,alwaysstrappedforcash,oftentookinboarders.Aroundthetimehewaseleven,thereweretwoofthem,Brahminboys,onefromtheneighboringdistrictofTrichinopoly,onefromTirunelvelifartothesouth,studyingatthenearbyGovernmentCollege.NoticingRamanujan’sinterestinmathematics,theyfeditwithwhatever they knew. Within months he had exhausted their knowledge and waspesteringthemformathtextsfromthecollegelibrary.Amongthosetheybroughttohim was an 1893 English textbook popular in South Indian colleges and Englishpreparatory schools, S. L. Loney’s Trigonometry, which actually ranged into moreadvancedrealms.BythetimeRamanujanwasthirteen,hehadmasteredit.

Ramanujanlearned fromanolderboyhowtosolvecubicequations.Hecame tounderstandtrigonometricfunctionsnotastheratiosofthesidesinarighttriangle,asusually taught in school, but as far more sophisticated concepts involving infiniteseries. He’d rattle off the numerical values of π and e, “transcendental” numbersappearingfrequentlyinhighermathematics,toanynumberofdecimalplaces.He’d

take exams and finish in half the allotted time.Classmates two years aheadwouldhandhimproblemstheythoughtdifficult,onlytowatchhimsolvethemataglance.

Occasionally, his powers were put to good use. Some twelve hundred studentsattended the schoolandeachhad tobeassigned toclassrooms,and to the school’sthree dozen or so teachers, while satisfying any special circumstances peculiar toparticularstudents.AtTownHigh,theseniormathteacher,GanapathiSubbier,wasregularlyshackledwiththemaddeningjob—andhewouldgiveittoRamanujan.

By the timehewas fourteenand in the fourth form, someofhis classmateshadbegun towriteRamanujanoff as someoneoff in thecloudswithwhomtheycouldscarcely hope to communicate. “We, including teachers, rarely understood him,”rememberedoneofhiscontemporarieshalfacenturylater.Someofhisteachersmayalready have felt uncomfortable in the face of his powers. Butmost of the schoolapparentlystoodinsomethinglikerespectfulaweofhim,whethertheyknewwhathewastalkingaboutornot.

Hebecamesomethingofaminorcelebrity.Allthroughhisschoolyears,hewalkedoffwithmeritcertificatesandvolumesofEnglishpoetryasscholasticprizes.Finally,ataceremonyin1904,whenRamanujanwasbeingawardedtheK.RanganathaRaoprize for mathematics, headmaster Krishnaswami Iyer introduced him to theaudience as a student who, were it possible, deserved higher than the maximumpossiblemarks.

AnA-plus, or100percent,wouldn’t do to ratehim.Ramanujan,hewas saying,wasoff-scale.

Still, during most of his time in school, Ramanujan’s life remained in roughbalance.Atgraduation,hewashismother’sson,motivatedandsuccessfulinschool,gettingsettoenrollthefollowingyear,withascholarship,intheGovernmentCollegeat the other end of town, looking ahead to academic achievement, a career,marriage...

But soon, very soon, that uneasy balance would be destroyed, and Ramanujanwould be led out into a new,mentally unsettling realm of intellectual passion andfierce,unbendingintensitythatwouldruletherestofhislife.

Forbesidethereasoned,rationalsideofRamanujanlayanintuitive,evenirrationalstreak that most of his Western friends later could never understand—but withwhichhewasatease,andtowhichhehappilysurrenderedhimself.

5.THEGODDESSOFNAMAKKAL

It would take a fewminutes for his eyes to adjust to the shadows. There, in theSarangapanitemple’souterhall,itseemedgloomyafterthebrightsunoutside.What

lighttherewassweptinfromtheside,softlymodelingtheintricatesculptedshapes,thelionsandgeometricallycutstone,ofthehall’scloselyspacedcolumns.

Away further from the light, nestled among the columns,were areas favored bybats for nesting. SometimesRamanujan could hear the quick, nervous swatting oftheirwings.Orevenseethemhangingfromtheceiling,chirpingaway,thenabruptlyflutteringintoflight.

UnlikeWesternchurcheswhich,architecturally,drewyouhigherandhigher,herethedevoutwerepulled,asitwere,innerandinner.Withinthehighstonewallsofthetemple complex stood a broad court, open to the sky and,within that, the roofedcolumnedarea. In further yet, you came to the great chariot, its enormouswheels,several feet in diameter, drawn by sculpted horses and elephants. Within thebuilding-within-a-building thatwas the chariot stood, in a dark stone cellwhere alampburnednightandday,thesanctumsanctorum,theprimarydeityhimself—thegreat god Vishnu, rising up from his slumber beside the many-headed serpentrepresentingEternity.

Always the temple stirred with little bright devotional fires, the chanting ofmantras,thesmellofincenseinsmallshrinesanddarknichesdevotedtosecondarydeities.Thecloseroneapproachedto thecentral shrine itself, thedarker itgrew—moremysterious,moreintimatelyscaled,progressivelysmaller,tighter,closer.Whatfromthenoisystreetbeyondthetemplewallsmighthaveseemedafitsiteforgreatpublic spectacles,here, inside,within stone grottosblackenedby centuriesof ritualfirepresidedoverbybare-chestedBrahminpriests,wasaplaceforonemanandhisgods.

From the outside, the gopuram, or entrance tower, of this great temple built byNayak kings sometime before A.D. 1350 was a massive twelve-story trapezoid ofintricately sculpted figures,90 feetacrossat itsbaseandrising146 into the sky. Itwas so high you could scarcely discern the images at the top,much less the facialexpressions uponwhich their sculptors had lavished attention. There were figuresclothed and naked, figures sitting and standing, with human shapes and animal,realisticandutterlyfantastic.Therewerefiguresdancing,onhorseback,makinglove,strumminginstruments—afullpanoplyofhumanactivity,denselyrealizedinstone.

To Ramanujan, growing up within sight of the temple, these were not neutralimages.Eachrepresented legendsontowhich, sincehisearliest childhood, layersofimageryandsignificancehadbeenheapedup—scenesandstorieshehadheardathismother’s knee, stories from the great epics, the Ramayana and the Mahabharata,storiesmeant to edify, or amuse. Every Hindu child learned of mischievous littleKrishna—a child now, not yet a god—coming upon a group of women bathing,

stealingtheirsaris,andescapingupatreewiththem,thewomenfranticallyimploringhim for their return.Here,Ramanujan had only to lift his gaze to thewall of thegopuramtoseeKrishnaperchedinthelegendarytree.

Allhislife,forfestivals,ordevotions,orjusttopassthetime,withhisfamilyorbyhimself, Ramanujan came to the temple. He’d grown up virtually in its shadow.Steppingoutofhislittlehouse,hehadbuttoturnhisheadtosee,attheheadofthestreet, close enough that he couldmake out the larger figures, the great gopuram.Indeed,theverystreetonwhichhelivedborethetemple’sname.ItwasSarangapaniSannidhiStreet;sannidhimeantentranceorprocessionway.

There was no special premium on silence within the temple; it was natural forRamanujan to strike up conversations there. But the prevailing feelingwas that ofquiet and calm, a stone oasis of serenity, while outside all India clamored withboisterouslife.

Here, to the sheltered columned coolness, Ramanujanwould come.Here, awayfromthe family,protected fromthehighhot sunoutside,hewouldsometimes fallasleepinthemiddleoftheday,hisnotebook,withitspagesofmathematicalscrawl,tucked beneath his arm, the stone slabs of the floor around him blanketed withequationsinscribedinchalk.

More than a dozen major temples studded the town and nearby villages, somedevotedtoLordSiva, sometoLordVishnu.Eachhad itsprominentgopurams, itscolumned halls, its dark inner sanctums, its tanks, or large, ritual purifying pools.The town fairly exuded spirituality. That once every twelve years the greatMahammakham tank received water from the Ganges—from which geographybooksshowedithopelesslyremote—was, inKumbakonam,a truthstatednotwithapology to secular sensibilities, or qualifiers like “traditionhas it,” or “according tolegend,”butsimply,baldly,asfact.

Itwas aworld inwhich the spiritual, themystical, and themetaphysicalweren’tconsigned to the fringesof life, but laynear its center.Ramanujanhadbut to stepoutsidehishouse,wanderalongthestreet,orlollaboutthetemple,tofindsomeoneeager to listen to a monologue on the traits of this or that deity, or the mysticqualitiesofthenumber7,orman’sdutiesassetforthintheBhagavad-Gita.

Not that practical matters were dismissed in the high-caste Brahmin world inwhichRamanujangrewup;money,comfort,andsecurityhadtheirplace.ButsodidVishnu and his incarnations, and what they meant, and how they might bepropitiated,andupcomingfestivals,andtheproperformfordevotions.Thesewerenotmere distractions or diversions from the business of everyday life. They were

integraltoit,ascentraltomostSouthIndiansasafternoonteaandcricketweretoupper-classEnglishmen,orfreeenterpriseandtheirautomobilestoAmericans.

Yearslater,afterhewasdead,someofhisWesternfriendswhothoughttheyknewhim would say that Ramanujan was not really religious, that his mind wasindistinguishable from any brilliant Westerner’s, that he was a Hindu only bymechanicalobservance,orforform’ssakealone.

Theywerewrong.Alltheyearshewasgrowingup,helivedthelifeofatraditionalHinduBrahmin.

He wore the kutumi, the topknot. His forehead was shaved. He was rigidlyvegetarian.Hefrequentedlocaltemples.Heparticipatedinceremoniesandritualsathome.He traveled all over South India for pilgrimages.He regularly invoked thenameofhisfamilydeity,thegoddessNamagiriofNamakkal,andbasedhisactionsonwhathe tooktobeherwishes.Heattributedto thegodshisability tonavigatethrough the shoals of mathematical texts written in foreign languages. He couldrecite from the Vedas, the Upanishads, and other Hindu scriptures. He had apenchantforinterpretingdreams,atasteforoccultphenomena,andamysticalbentuponwhichhisIndianfriendsunfailinglycommented.

OnceayearduringtheyearshewasgrowingupinKumbakonam,hewouldsetoutalong the road heading east past the railroad station. Outside of town, the mudhouseswiththeirthatchedroofshuggingthesideoftheroadthinnedout.Hecouldseebullockstiedtostakesbesidetheroad,goatswanderinginandoutofhouses,littleroadside shrines, trails leading off the road and into the flat, green countryside.Aboutfourmiles fromKumbakonam,he’dreachabroadloopingcurve intheroadwherethetownofThirunageswarambegan,andwheretheancientUppiliapanKoiltemplestood.HereRamanujancameeveryyear,atthetimeofthefullmoon,inthemonthofSravana(aroundAugust)torenewhissacredthread.

Whenhewasfiveyearsold,participatinginatime-honoredceremonyoffireandchantingthattypicallylastedfourdays,Ramanujanhadbeeninvestedwiththesacredthread—three intertwined strands of cotton thread draped across the bare chest,from the left shoulder diagonally down to the right hip, like a bandolier. Theupanayanam ceremony solemnized his “twice-born” status as a Brahmin; the firstbirth,saidtheancientlawgiverManu,isfromthemother,thesecondfromthetakingofthesacredthread.Thenceforth,hecouldreadthesacredVedasandperformtheritesofhis caste.AndeachyearduringSravanam,midst foodofferings and sacredfiresandworship,hereneweditinthecompanyofotherBrahminsattheUppiliapantemple.

Onetime,afriendrecalledlater,heandRamanujanwalkedthroughthemoonlightthesixmilestothenearbytownofNachiarkovil,siteofaVishnutemple,towitnessareligiousfestival.Allthewhile,RamanujanrecitedpassagesfromtheVedasandtheShastras,ancientSanskrittomes,andgaverunningcommentariesontheirmeaning.

Anothertime,whenhewastwenty-one,heshowedupatthehouseofateacher,gotdrawn into conversation, and soonwas expatiating on the ties he sawbetweenGod,zero,andinfinity—keepingeveryonespellboundtilltwointhemorning.Itwasthat way often for Ramanujan. Losing himself in philosophical and mysticalmonologues,he’dmakebizarre,fancifulleapsoftheimaginationthathisfriendsdidnotunderstandbutfoundfascinatinganyway.Soabsorbedwouldtheybecomethatlateralltheycouldrecallwasthepenetratingsetofhiseyes.

“Immensely devout,” R.Radhakrishna Iyer, a classmate of his, would later termhim.“Atruemystic...intenselyreligious,”recalledR.Srinivasan,aformerprofessorofmathematics.Towardtheendofhislife,influencedbytheWest,Ramanujanmayhaveedgedtowardmoresecular,narrowlyrationalvalues.Butthatcamemuchlater.AndgrowingupmidstthedenseandubiquitousspiritualityofSouthIndia,hecouldscarcelyhavecomeawayuntouchedbyit—evenifonlyinrebellion.

Ramanujanneverdidrebel.Hedidnotdenytheunseenrealmofspirit,norevenholditatarm’slength;rather,heembracedit.HiswasnotalifesetintensionwiththeSouthIndiafromwhichhecame,butratheroneresonatingtoitsrhythms.

•••

SouthIndiawasaworldapart.AllacrossIndia’snorthernplains,thecenturieshadbroughtinvasion,war,turmoil,andchange.Around1500B.C.light-skinnedAryansswept in throughmountain passes from the north. For eight centuries, Buddhismcompeted with traditional Brahminism, before at last being overpowered by it.Beginning in the tenth century, it was the Muslims who invaded, ultimatelyestablishing theirownMoghulEmpire.Oneempiregaveway toanother, the racesmingled,religionscompeted,menfought.

Andyetbyallthis,theSouthremainedlargelyuntouched,safebehinditsshieldofmountains,rivers,andmiles.

North ofwhatwould become themodern city ofBombay, stretching across thewesternedge of the subcontinent at roughly the latitude of theTropic ofCancer,loomedtheVindhyamountains,abrokenchainofruggedhillsrisingashighasthreethousand feet and reaching inland almost seven hundred miles. At their base layfurther obstacles to movement south into the tapering Indian peninsula—the

Narbada and Tapti rivers, flowing west into the Arabian Sea, and theMahanadiRiver,flowingintotheBayofBengalontheeast.These,togetherwithsheerdistance,exhaustedmostinvadersbeforetheyreachedveryfarsouth.

Thus, spared both the fury of the North and the fresh cultural winds foreversweepingthroughit, theSouthremainedaplaceunto itself,remarkably“pure.”Nopartof Indiawasmore homogenous.Racially, theSouthwas populatedmostly byindigenousDravidian peoples with black, often curly hair, broad noses, and skinalmost as dark as nativeAfricans; even the Brahmins, thought to be derived fromAryanstock,werenotsolight-skinnedasthoseseenupNorth.Linguistically,NorthandSouthweredivided,too.TamilandtheotherDravidianlanguagesborefewtiestoHindiandtheotherSanskrit-basedlanguagesoftheNorth.Religiously,theSouthwas more purely Hindu than any other part of India; nine in ten of those inRamanujan’s Tanjore District, for example, were Hindu, only about 5 percentMuslim.SospecialanddistinctwastheSouthinthemindsofitsinhabitantsthatinwritingoverseastheywereapttomake“SouthIndia”partofthereturnaddress.Onthepoliticalmap,nosuchplaceexisted;yetitexpressedaprofoundculturaltruth.

InSouthIndiaanundilutedspiritualityhadhadachancetoblossom.IftheNorthwaslikeEuropeduringtheEnlightenment,theSouthwas,religiously,stillrootedintheMiddle Ages. If Bombay was known for commerce, and Calcutta for politics,Madraswasthemostsingle-mindedlyreligious.Itwasaplacewheretherewasless,asitwere,todistractyou—justricefields,temples,andhiddengods.

Here,inthissetting,withthesecularworldheldatbay,withinatraditionalculturealwayswillingtoseemysticalandmagical forcesatwork,Ramanujan’sbelief intheunseenworkingsofgodsandgoddesses,hissupremecomfortwithamentaluniversetiedtogetherbyinvisiblethreads,cameasnaturallyasbreathitself.

•••

AllthroughSouthIndia,everyvillageofafewdozenhutshadashrinetoMariammaorIyenar,SeliammaorAngalamma—godsandgoddesseswhoseoriginswentbacktotheverydawnofagriculturalcommunities.Thesedeitiesrepresentedpowerswhichvillagers hoped to propitiate, like smallpox, cholera, and cattle plague. Most werereckonedasfemale.Afewwererecent,incorporatingthespiritsofmurdervictimsorwomenwhohaddiedinchildbirth.In1904,someboysthoughttheyheardtrumpetscomingfromananthill,andsoonthedeityoftheanthillwasattractingthousandsofpeople from nearby villages, who would lie “prostrate on their faces, rapt inadoration.”

Grama devata, or village gods, these deities were called, and they had virtuallynothingtodowiththeformalBrahminicHinduismastudentofcomparativereligionmightlearnaboutincollege.ThevillagersmightgivelipservicetoVishnuandSiva,thetwopillarsoforthodoxworship.Butat timeofpestilenceor famine, theywereapt to turnback to their little shrines—perhaps abrickbuilding threeor four feethigh,orasmallenclosurewithafewrudestonesinthemiddle—whereguardianshipoftheirvillagelay.

Mereidolworship?Nomorethanaprimitive,aboriginalanimism?Sosomecriticsof Hinduism argued. And to the extent that these Dravidian gods were part ofHinduism,onecouldargue,thecriticsweren’tfaroff.

But theHinduismofwhichKumbakonamwas such a stronghold, and inwhichRamanujanwas steeped, was a world apart from all this. InTanjoreDistrict, oneEnglishobserverwouldnote,“BrahminicalHinduismisherealivingrealityandnottheneglectedcult,shoulderedoutbytheworshipofaboriginalgodlings,demonsanddevilswhichitsooftenisinotherdistricts.”

The great temples of the South fairly shouted out the difference. Temples inKumbakonam, inKanchipuram, inTanjore,Madurai, andRameswaram, were, asone authority put it, as superior to more famous ones in the North, say, “asWestminster Abbey and St. Paul’s are to the other churches of London.”One atRameswaram,towhichRamanujanandhisfather,mother,andbabybrotherwentonapilgrimagein1901,builtoverahundred-yearspanduringtheseventeenthcenturyonan islandoff thecoastoppositeCeylon,was1000 feet longand650 feetacross,builtwithgopurams100 feethighoneach face,withalmost4000 feetofcorridorsrichinextravagantlysculpteddetail.

A Western observer to such a temple might still be brought up short by thebewildering variety of deities he’d find there—sculpted figures, statues large andsmall, inwood and stone, sometimes garlandedwith flowers, even dressed in rudeclothing. But inmainstreamHinduism, these could all be seen as part of a grandedificeofbeliefvastlymoresophisticatedthanthereligionofthevillages.

The threechiefdeities in theHindupantheon,Brahma,Siva, andVishnu,weretraditionally represented as, respectively, the universe’s creative, destructive, andpreserving forces. In practice, however, Brahma, once having fashioned the world,wasseenascoldandaloof,andtendedtobe ignored.SothetwogreatbranchesofBrahminicHinduismbecameShaivismandVaishnavism.

Shaivism had a kind of demonic streak, a fierceness, a malignity, a raw sexualenergy embodied in the stylized phallic symbol known as a lingam that was the

centerpiece of every Shaivite temple. Think of sweeping change, of cataclysmicdestruction,andyouinvokedLordSiva.

Vaishnavism,befittingitsidentificationwiththeconservinggodVishnu,hadmoreplacid connotations.One contemporaryEnglish account likened it to theSpirit ofMan—a distinctly gentler idea. Figuring largely in Vaishnavism were Rama andKrishna,heroesof Indian legend,and twoof the incarnations,or avatars, inwhichVishnuappears.

InHindulore,eachofthethreeprimalgodsappearedinmanyforms.SivacouldbeParmeswara.Vishnu could beNarasimha orVenkatarama. They had consortsandrelatives,eachofwhomthemselveshad,overthecenturies,becometheobjectsofworship,thecentersoftheirowncults.Vishnu,forexample,wasworshippedintheformofhisconsortLakshmi,andasthemonkeygod,Hanuman.Eachwasendowedwithdistinctpersonalities;eachgaineditsownadherents.

Someworshippers, certainly, construed those stone figures literally, viewed themasgods,pureandsimple,inawaynotsodifferentfromthegramadevataworshipofthe villages. Indeed, onehistoryofSouth India spokeof a “fusionofvillagedeitiesandVedicBrahminicaldeities”goingbacktoaroundthebeginningoftheChristianerathathadbroughtacominglingofdifferentformsofworship.

But sophisticatedHindus, at least, understood that these stone “deities” merelyrepresented forms or facets of a single godhead; in contemplating them, you werereawakened to the Oneness of all things. For those whose worship remainedprimitive,meanwhile, the garish stone figures could be seen as hooks bywhich tosnare the spirituallyunsophisticatedanddirect them toward somethinghigher andfiner.

The genius of Hinduism, then, was that it left room for everyone. It was aprofoundly tolerant religion. It deniedno other faiths. It set out no single path. Itprescribednoonecanonofworshipandbelief.Itembracedeverythingandeveryone.Whateveryour personality there was a god or goddess, an incarnation, a figure, adeity,withwhichtoidentify,fromwhichtodrawcomfort,torouseyoutoahigherordeeperspirituality.Thereweregodsforeverypurpose,tosuitanyframeofmind,anymood, any psyche, any stage or station of life. In taking on different forms, Godbecameformless;indifferentnames,nameless.

Among the thousands of deities, most South Indian families tended to investspecial powers in a particular one—which became as much part of the family’sheritage as stories passed down through the generations, or its treasured jewelry.This kula devata became the focal point of the family’s supplications in time oftrouble,much as some Roman Catholics invoke a particular patron saint. Things

wouldgowrong,andyou’dpropitiateyourfamilydeitybeforeyouwouldanyother.InSouthIndia,manyawell-traveledBrahminwithwideknowledgeoftheworld—perhapsascholar,aprofessional,fluentinEnglish,well-readinSanskrit,whocouldintelligently discuss Indian nationalism, Tamil poetry, or mathematics—routinelyandardentlyprayedbeforetheshrineofhisfamilydeity.

InRamanujan’sfamily,thefamilydeitywasthegoddessNamagiri,consortofthelion-god Narasimha. Her shrine at Namakkal was about a hundred miles fromKumbakonam, about three-quarters of the way to Erode, near whereKomalatammal’sfamilycamefrom.ItwasNamagiriwhosenamewasalwaysonhismother’s lips,whowastheobjectof those firstdevotions,whoseassumedviewsonmattersgreatandsmallweretakenwiththeutmostseriousness.

IthadbeenNamagiritowhomRamanujan’smotherandfather,childlessforsomeyearsaftertheymarried,hadprayedforachild.Ramanujan’smaternalgrandmother,Rangammal,wasadevoteeofNamagiriandwassaidtoenteratrancetospeaktoher.One time, a vision of Namagiri warned her of a bizarre murder plot involvingteachers at the local school.Another time,many years earlier, beforeRamanujan’sbirth,Namagiri revealed toher that thegoddesswouldonedayspeak throughherdaughter’s son. Ramanujan grew up hearing this story. And he, too, would utterNamagiri’s name all his life, invokeher blessings, seekher counsel. Itwas goddessNamagiri,hewouldtellfriends,towhomheowedhismathematicalgifts.Namagiriwould write the equations on his tongue. Namagiri would bestow mathematicalinsightsinhisdreams.

Sohetoldhisfriends.Didhebelieveit?Hisgrandmotherdid,andsodidhismother.Herson’sbirth,afterlongprayerto

Namagiri, had only intensified her devotion,made hermore fervent in her belief.That’showKomalatammalwas:Why,shehadpracticallywilledherselfachild.Thewholeforceofherpersonality,herferociouswill,surgedthroughallshedid.

Ramanujanabsorbedthatfromher;sheneverhadtoteachittohim,becauseitwasimprinted in theexampleofher life.He learned fromher toheed thevoicewithinhimselfandtoexertthewill toacton it.His fatherwasmired intheday-to-day,aslavetoitsroutines,preoccupiedwithrupeesandannas;hewouldwantRamanujanmarried off, bringingmoney into the family,well settled. ButKomalatammal gaveherselfover to deeper forces, dwelt in a rich, innerworld—andpulledRamanujanintoitwithher.

SothatwhenapowerfulnewinfluenceonRamanujan’syounglifecamealong,hehadhismother’s sanction to embrace it, togivehis lifeover to it, to follow itwithabandon.

CHAPTERTWO

RangingwithDelight

[1903to1908]

1.THEBOOKOFCARR

ItfirstcameintohishandsafewmonthsbeforeheleftTownHighSchool,sometimein1903.Probably,collegestudentsstayingwithRamanujan’sfamilyshowedhimthebook.Inanycase,itstitleborenohintoftheholditwouldhaveonhim:ASynopsisofElementaryResultsinPureandAppliedMathematics.

Inessence, thebookwasa compilationof five thousandor soequations,writtenout one after the other—theorems, formulas, geometric diagrams, and othermathematicalfacts,marchingdownthepage,tiedtogetherbytopic,withbig,bold-faced numbers beside each for cross-reference. Algebra, trigonometry, calculus,analytic geometry, differential equations—great chunks of mathematics as it wasknowninthelatenineteenthcentury,rangednotoverawholeshelfoftextbooks,butcompressedwithin twomodest volumes (the secondofwhichRamanujanmaynothaveseenuntillater).

“The book is not in any sense a great one,” someonewould later say of it, “butRamanujanhasmadeitfamous.”

•••

TheSynopsiswas a product of the genius ofGeorgeShoobridgeCarr.Except thatCarr was no genius.Hewas amathematician of distinctlymiddling rankwho foryearstutoredprivatelyinLondon;thebookwasadistillationofhiscoachingnotes.

Mathematics students in England during the late nineteenth century werepreoccupied to the point of obsession with a notoriously difficult examination,knownas theTripos, one’s rankingonwhich largelydeterminedone’s career.TheTripos system encouraged what educators today might deride as “teaching to thetest,”andsoonmathematicianswouldclamorforitsreform.Butbackinthe1860s,intheperiodgivingrisetothebook,itsholdwentunchallenged.Notsurprisingly,given

the exam’s importance, armiesofprivate tutorshadarisen to coach students for it.Carrwasoneofthem.

Carrhimselfhadapeculiaracademichistory.Bornin1837inTeignmouth,nearwhere the Pilgrims sailed for the New World, he attended school in Jersey, aChannelIslandofftheFrenchcoast,andlaterUniversityCollegeSchoolinLondon.Atleastby1866,andperhapsearlier,hestartedtutoring.Apparentlyhetookitquiteseriously and was forever updating his notes, refining his teaching methods,developingmnemonicstohelphispupilscoverthevastrangeofmaterial theyweresupposedtomaster.

Then,atthirty-eight,moreinthemodernstylethanwascommonatthetime,hedecided togoback to school.Admitted toGonville andCaiusCollege,CambridgeUniversity,hereceivedhisB.A.in1880,andthen—fouryearsshyoffifty—hisM.A.

He was no star student. In the Tripos, he was classed among the “SeniorOptimes,”not the higher-ranking “Wranglers,” and only twelfth among them.HeknewhewasnotthebrightestlightinthefirmamentofEnglishmathematics.IntheprefacetotheSynopsishesuggestedthat“ablerhandsthanmine”mighthavedoneabetter job with it, but that “abler hands might also, perhaps, be more usefullyemployed”—presumably inmaking the originalmathematical discoveries to whichhisintellectortemperamentfailedtosuithim.

ButwhileCarrasamathematicianwasnomorethannormallybright,hehadtheenthusiasmandloveofsubjecttoteachittothoseablerthanhimself.Inanycase,itwasjustaboutthetimehewasgrantedhisCambridgeB.A.that,onMay23,1880,fromhisdesk inHadley,outsideLondon,heput the finishing toucheson the firstvolume—asecondappearedin1886—oftheSynopsiswhichwouldlinkhisnametoRamanujan’sforever.

•••

OnestrengthofCarr’sbookwasamovement,a flow totheformulasseeminglylaiddownoneaftertheotherinartlessprofusion,thatgavethebookasly,seductivelogicofitsown.

Take,forexample,thefirststatementonthefirstpage:

a2−b2=(a−b)(a+b)

Thisis,firstofall,anequation.Itsays—anyequationsays—thatwhateverisontheleft-handsideoftheequalssignisequivalenttowhat’sontheright,asin2+2=4.Onlyinthiscase,it’snotnumbers,butsymbols—thelettersaandb—thatfigurein

theequation.Thattheyaresymbolschangesnothing.Someequationsaretrueonlywhentheirvariablestakeoncertainvalues;thejob,then,isto“solve”theequation,todeterminethosevalues—x=3,say,orz=−8.2—thatmakeitvalid.Butthisone,an“identity,”isalwaystrue;whateveryoumakeaandb,thestatementstillholds.

So,tryit:Leta=11,say,andb=6.Whathappens?Well,a+bisjust11+6,whichis17.Anda−bis11−6,or5.Now,tosetoffquantitiesinparentheses,astheyareinCarr’sequation,meansjust

tomultiplythem—(a+b)and(a−b)—together.Inthiscase,(17)(5)isjust17×5,or85.That’stheright-handsideoftheequation.

Nowfortheleft.a2,ofcourse,isjustatimesa,whichis11×11,or121.b2is36.a2−b2,then,isjust121−36,or85.Whichisjustwhattheright-handsideoftheequationcomesto.Sureenough,thetwosidesmatch.Theequationholds.

Youcouldkeepondoingthisforever—verifyingthattheequationholds,withbignumbersand littlenumbers,positivenumbersandnegative, fractionsanddecimals.Youcoulddothat,butwho’dwantto?

Moresensibleistodowhatmathematiciansdo—provetheidentityholdsgenerally,for any a and any b. To do that, you dispense with particular numbers andmanipulateinsteadthesymbolsthemselves.Youaddandsubtractthelettersaandb,multiplyanddividethem,justasyouwouldnumbers.

Inthiscase,theequationtellsustomultiply(a−b)times(a+b).Doing that isaboutassimpleasitlooks.Ifyoumade$10anhour,butthengotapaycutof$1perhour, you couldmultiply the number of hours youworked by 10, then by 1, andsubtractoneproduct fromtheother.Oryoucouldsimplymultiplythetotalhoursworkedby9.Samething.Inthiscase,youcanmultiplythewholesecondterm,(a+b),bya,thenbyb,andthensubtractoneproductfromtheother.Or,symbolically,

(a−b)(a+b)=a(a+b)−b(a+b)

Whatnow?Well,a(a+b) is justa2+ab.Andb(a+b) is justba+ b2. But ba(whichmeansb×a)isjustthesameasab.Soweget:

(a−b)(a+b)=(a2+ab)−(ab+b2)

Inmanipulating their equations,mathematicians often get caught in a clutter ofnumbers,letters,andsymbols.Andforthesamereasonsyoudoaroundthehouse,theyperiodicallytaketimetotidyup—sothey’renotforeversteppingovermoundsofmathematicaldebris,andsoanyattractivequalitiesoftheirmathematicalhabitatare shown off to best advantage. “Grouping like terms” is one form housecleaning

takes; you cluster mathematical entities in their appropriate categories. You placedirtyclothesinthelaundrybin,freshlylaunderednapkinsinthelinendrawer,cerealbackinthepantry.Youputthingswheretheybelong.

Inthiscase,webringeverythingout frombehindtheparentheses,addupthea2

terms,andtheb2 terms,andtheab terms.Andwhenwedo,something interestinghappens.Theabtermscanceleachother;they“dropout.”The+abandthe−abadduptoagrandtotalofzero,sothatthequantityabjustdisappearsfromtheequation.Whichleavesuswitha2−b2,whichisjustwhat’sontheleft-handsideoftheoriginalequation—andjustwhatwe’resupposedtoprove.

What this simple exercise demonstrates is a “proof” of sorts, though amathematicianmightshudderattheclaim.Butatleastoncasualinspection,itseemsthatforanyaandanyb,thetwosidesofCarr’sequationarethesame.Wedon’thavetochecka=735andb=.0231.Weknowitwillworkbecauseweprovedthegeneralcase.

SomuchforCarr’sfirstequation.Hissecondisthis:

a3−b3=(a−b)(a2+ab+b2)

Proving this differs little from proving the first. Working with the symbols, youmultiply,add,andsubtract,lineupliketerms—addapplestoapples,andorangestooranges, but never apples and oranges together—hope something cancels out, andsoonareleftwitheithersideoftheequalssignthesame.Why,it’shardlyworththetroubletogothroughit....

And right there, in the normal, natural—and appropriate—impulse to say it’shardlyworth the trouble, we gain a clue toCarr’s pedagogical wisdom (and to howmathematicians, generally, think).The second equation, though different from thefirst,resemblesit,seemsanextensionornaturalprogressionfromit:Infollowingonewiththeother,Carrwasgoingsomewhere.Therewasadirection,adevelopment,notwithin the mathematical statements he set down but implicit within the order inwhichhesetthemdown.Thefirstequationdealtwithaandb“raisedtothesecondpower,”intheformofa2−b2;thesecondwithaandbtothethirdpower,asa3−b3.What,onemightnowwonder,wouldbe theequation fora4−b4?Havingworkedout the first two, you’d suspect you couldwork it out easily, following the earlierexamples.Andyou’dberight;theanswerholdsnosurprises.

So Carr didn’t set it down at all. That would have been tedious, and trivial.Instead,hegeneralized:

an−bn=(a−b)(an−1+an−2b+...bn−1)

This is thedecisive step, for in taking it, the last ordinarynumber in the equationdisappears. It’snot the secondpower towhicha andb are raised this time, or thethird,ortheeighth,butthenth.

Abruptly,weareinanewworld.It’sstillsimplealgebra,butbydaringtoreplacethose safe 2s and3s by themoremysteriousn, the equation short-circuits routinemathematicalmanipulations.Now,yougivemeanumberandIcanjustwriteouttheequation, merely by substituting for the general n. The ellipsis appearing midwaythroughtheequation,thethreelittledots,justmeansyoucontinueinthepatternthefirsttwotermsestablish.O.K.,son=8?Fine,plug it intoCarr’sgeneralequation,andtheequationwritesitself.WhereCarr’sequationsaysn,youwrite8.Whereitsaysn−1,youwrite7,andsoon.

Asmathematiciansmightsay, theequationwiththen’s ismoregeneral than theprevioustwo.Or,putanotherway,thefirsttwoequationsweremerelyspecialcasesof the third.WereRamanujan not already familiarwith it—and it’s inconceivablethathewasn’t—hecouldhaveconfirmeditataglance.Still,itsuggestshowhewasguidedthroughmathematicalrealmsnewtohim;itwasn’t justthestatementsCarrmadethatcounted,butthepathhenudgedthestudentalonginmakingthem.

Andthewayinwhichhesetaboutprovingthem.Or,rather,notprovingthem.In fact, Carr didn’t prove much in his book, certainly not as mathematicians

normallydo, andnotevenaswehavehere.Then,asnow, the typicalmathematicstext methodically worked through a subject, setting out a theorem, then goingthrough the steps of its proof.The studentwas expected to dutifully follow alongbehind theauthor, trackinghis logic,perhaps filling in small gaps inhis reasoning.“Oh,yes,thatfollows...”thestudentthinks.“Yes,Isee...”

Butmathematics isnotbest learnedpassively;youdon’tsopituplikearomancenovel.You’ve got to goout to it, aggressive and alert, like a chessmasterpursuingcheckmate. And mechanically following a proof laid out by another hardlyencouragesthat, leavesscantopportunitytobringmuchofyourselfto it.Whateverits othermerits, the trigonometry text by S. L. Loney that Ramanujan had sailedthroughafewyearsbeforehadclungtothemold; itwasatextyoufollowedratherthanonewhichdemandedyoucutyourownpath.

Carr’swasdifferent.Therewasnoroomfordetailedproofs in theSynopsis.Manyresultswere stated

without so much as a word of explanation. Sometimes, a little note would beappendedtotheresult.Theorem245,forexample,simplynotes,“by(243),(244).”Thatis,onecanarriveattheconclusionofno.245byextendingthelogicof243and244.Theorem2912notes:“Proof—Bychangingxintoπxin(2911).”Inotherwords

—mathematicians use this trick all the time—by an astute change of variable, theresultassumesaclearer,morerevealingform.Inanycase,Carrofferednoelaboratedemonstrations,nostep-by-stepproofs,justagentlepointingoftheway.

ScholarswouldonedayprobeCarr’sbook,searchingfortheelusivemathematicalsophistication that might have inspired Ramanujan. Some would point to how itcovered,orfailedtocover,thisorthatmathematicaltopic.Somewouldpointtoitsunusuallyhelpfulindex,otherstoitsbroadcompass.

But in fact it’s hard to imagine a book more apt to influence a mathematicallyprecocious sixteen-year-old, at least one like Ramanujan. For in baldly stating itsresultsitalmostdaredyoutojumpinandprovethemforyourself.ToRamanujan,eachtheoremwasitsownlittleresearchproject.Orlikeacrosswordpuzzle,withitsemptygridbeggingtobefilledin.Oroneofthoseirresistiblelittlequizzesinpopularmagazinesthatinviteyoutorateyourcreativityoryoursexappeal.

Norwasallthisjustanaccident,oraby-productoftheconcisionanycompendiummightdemand;Carrhadit inmindallalong,andsaidso inthepreface. “Ihave,inmanycases,”heexplained,

merely indicated the salient points of a demonstration, or merely referred to the theorems by which theproposition is proved. . . . The difference in the effect upon the mind between reading a mathematicaldemonstration, and originating one wholly or partly, is very great. It may be compared to the differencebetweenthepleasureexperienced,andinterestaroused,whenintheonecaseatravellerispassivelyconductedthroughtheroadsofanovelandunexploredcountry,andintheothercasehediscoverstheroadsforhimselfwiththeassistanceofamap.

Butitwasn’tevenamapCarrsupplied;rather,advicelike,Onceoutoftown,turnleft.AWesternmathematicianwhoknewRamanujan’sworkwellwouldlaterobserve

thattheSynopsishadgivenhimdirection,buthad“nothingtodowithhismethods,the most important of which were completely original.” In fact, there were nomethods,atleastnotdetailedones,inCarr’sbook.SoRamanujan,chargingintothedensemathematicalthicketof itsfivethousandtheorems,hadlargelytofashionhisown.That’swhathenowabandonedhimselftodoing.“Throughthenewworldthusopenedtohim,”twoofhisIndianbiographerslaterwrote,“Ramanujanwentrangingwithdelight.”

2.THECAMBRIDGEOFSOUTHINDIA

In 1904, soon after discoveringCarr, Ramanujan graduated from high school andentered Kumbakonam’s Government College with a scholarship awarded on thestrengthofhishighschoolwork.HewasanF.A.student,forFirstArts,acourseof

studythat,byyearsinschool,mighttodaycorrespondtoanassociatesdegreebutinIndia,then,countedforconsiderablymore.

Fromthecenteroftown,thecollegewasaboutatwenty-minutewalk—alongthestreetthatranbyTownHigh,downtotheCauvery’sedge,thenright,alongtherivertoa point opposite the college.The bridge today spanning the river dates only to1944;before that, a little boat ferried you across.Or else, you’d swim—a feat lessdauntinginMarchandApril,whentheriverhaddriedtoatrickle.

GovernmentCollegewas small, its faculty consistingofbarely adozen lecturers.AndthebestlocalstudentshadbeguntoforsakeitforlargerschoolsinMadras.Still,for its time and place, it was pretty good—good enough, at any rate, to earn themoniker “the Cambridge of South India.” Its link to the great English universityrestedinpartonthecampus’sproximitytotheCauvery,whichflowedbesideitliketheRiverCaminCambridge.ButalsoplayingarolewasthereputeofitsgraduatesandthepositionsmanyofthemheldinSouthIndianlife.

The year 1854 saw the college’s founding on land given by the maharani ofTanjore; you could still see the steps, leading down from the dressing cabin, thatroyal princesses took down to the river to bathe. Beginning in 1871, existingbuildingswererepairedandenlarged,newonesbuilt.Inthe1880sitslastsecondaryclassesweredropped,anditbecameafull-blowncollege.Itsgroundswereenlargedand landscaped.Agymnasiumwasbuilt.WhileRamanujanwas there, ahostel forseventy-two students was going up, complete with separate dining facilities forBrahmins.

Thecollegeoccupiedasiteofconsiderablenaturalbeauty.Theriverstreamedby.Birds chirped. Groves of trees afforded shelter from the high, hot sun. Luxuriantvinescrawledeverywhere,foreverthreateningtooverrunthecollegebuildings.Evenwiththenewconstructionsincethemaharani’stime,thecollegedidnotdominateitssitebutratherclungthere,atnature’ssufferance.Theplacewaslovely,idyllic,serene.

AndthesceneofRamanujan’sfirstacademicdebacle.

•••

OnecanonlyguessattheeffectsofabooklikeCarr’sSynopsisonamediocre,orevennormallybrightstudent.ButinRamanujan,ithadignitedaburstoffiercelysingle-minded intellectual activity.Until then, he’d keptmathematics in balancewith therestofhislife,hadbeenproperlyattentivetootherclaimsonhisenergyandtime.Butnow, ensnaredby puremathematics, he lost interest in everything else.Hewas allmath. He couldn’t get enough of it. “College regulations could secure his bodily

presence at a lecture on history or physiology,” E. H. Neville, an EnglishmathematicianwholaterbefriendedRamanujan,wouldwrite,“buthismindwasfree,or,shallwesay,wastheslaveofhisgenius.”

AshisprofessorintonedaboutRomanhistory,Ramanujanwouldsitmanipulatingmathematicalformulas.“Hewasquiteunmindfulofwhatwasgoingonaroundhim,”recalledone classmate,N.HariRao. “Hehadno inclinationwhatsoever for eitherfollowing the class lessons or taking an interest in any subject other thanmathematics.”He showedHariRaohow to construct “magic squares”—tic-tac-toegrids stuffedwithnumberswhich, in everydirection, addup to the samequantity.He worked problems in algebra, trigonometry, calculus. He played with primenumbers,thebuildingblocksofthenumbersystem,andexploredthemforpatterns.Hegothishandsonthefewforeign-languagemathtextsinthelibraryandmadehiswaythroughatleastsomeofthem;mathematicalsymbols,ofcourse,aresimilarinalllanguages.

Onemathprofessor,P.V.SeshuIyer,sometimeslefthimtodoashepleasedinclass, even encouraging him to tackle problems appearing inmathematics journalsliketheLondonMathematicalGazette.OnedayRamanujanshowedhimhisworkinanareaofmathematicsknownasinfiniteseries;“ingeniousandoriginal,”SeshuIyerjudgedit.Butattentionlikethatwasrare,andRamanujan’sintellectualeccentricitieswere, on the whole, little indulged. More typical was the professor from whomRamanujan borrowed a calculus book who, once he saw how it interfered withRamanujan’sotherschoolwork,demandeditsreturn.EvenSeshuIyermaynothavebeenassolicitiousashelaterremembered;Ramanujancomplainedtoonefriendthathewas“indifferent”tohim.

Meanwhile,heignoredthephysiology,theEnglish,theGreekandRomanhistoryhewassupposedtobestudying;hewasnolonger,ifhe’deverbeen,“well-rounded.”Back in 1897, his high standing on the Primary Examination had depended onexcelling inmany subjects, including English. Letters known to bewritten by himlater, while showing no special grace, were competent enough, as were hismathematics notebooks when he used words, rather than symbols, to explainsomething.Yetnow,atGovernmentCollege,hefailedEnglishcomposition.“Tothecollege authorities,” E.H.Neville observed later, “he was just a student who wasneglectingflagrantlyallbutoneofthesubjectshewassupposedtobestudying.Thepenaltywasinevitable:hisscholarshipwastakenaway.”

Hismother,ofcourse,wasincensedandwenttoseetheprincipal.Howcouldherefusehersonascholarship?Hewasunequaledinmathematics.Theyhadneverseen

hislike.Theprincipalwaspolite,butfirm.Ruleswererules.HersonhadfailedtheEnglishcompositionpaper,andmiserablyso.Thatwasthat.

Ramanujan’s scholarship was no matter of mere prestige to him. Tuition wasthirty-tworupeesperterm—asmuchashisfathermadeinamonthandahalf.Thescholarshipinsulatedhimfromit;itenabledhimtoattend.Heneededit.

Still,hemanagedtohangonforafewmonths,showingupforclassenoughtoearnacertificateinJuly1905attestingtohisattendance.Theeffortmusthavetaxedhim.He’d lost the scholarship, and everybody knew it.His parentswere under a heavyfinancialburden;heknewthat,too.Hefeltpressuretodowellinhisothersubjects,yethedidn’twanttolaymathematicsasidefortheirsake.Hewastornandmiserable.

Heenduredthesituationuntilhecouldendureitnolonger.InearlyAugust1905,Ramanujan,seventeenyearsold,ranawayfromhome.

3.FLIGHT

Asthehotbreezepouredthroughtheopenwindowsoftherailwaycar,RamanujanwatchedtheSouthIndiancountrysideslipbyattwenty-fivemilesanhour.Villagesofthatchedroofsweatheredtoadullbarn-gray;intensepinkflowerspokingoutfrombushes and trees; palm trees, like exclamation points, punctuating the rice fieldflatness.Fromadistance,themeninthefieldsbesidethetrackswerelittlemorethanbrownsticks, theirdhotisand turbanswhitecottonpuffs.Thewomenwerebrightsplashesofcolor,theirorangeandredsarissetoffagainstthestartlinggreenofthericefields.

A snapshot might have recorded the scene as a charming bucolic tableau, butRamanujan saw people everywhere engaged in purposeful activity. Men tendingcattle.Women,stoopedoverinthefields,nursingthecrops.Sometimestheyworkedalone,sometimestogetheringroupsofadozenormore,basketsperchedatoptheirheads, fetching water from streams. Occasionally, a child with its mother wouldglanceupfromthesurroundingfieldsandwaveatthetrainbearingRamanujannorthtoVizagapatnam.

•••

Foreons,transportationinIndia,bybullockcartortheone-horsevehicleknownasajutka,hadbeenpainfullyslow.Roadswereterrible.EvenbyRamanujan’stime,onlyabout an eighth of Tanjore District’s seventeen hundred miles of road were“metalled,”orpavedwith limestoneorotherrock.Thedifferencewasconsiderable.Cartdrivers forced to travel on bumpydirt roads thickly covered by dust ormud,

ratherthanametalledone,normallyplannedoncarryingtwo-thirdstheload,attwo-thirdsthespeed.Twenty-fivemileswasagoodday’sjourney.

The coming of the railroad had changed Indian life. It was the crowningengineeringachievementoftheBritishRaj,emerginginthemidnineteenthcenturytoknitthefar-flungcountrytogether.IntheSouth,thefirstlineshadbeenlaidin1853,and in 1874 they began pushing south from Madras. In 1892, with the line toVizagapatnamstillunfinished,togettherefromKumbakonamcouldstilltakethreeweeksbytrain,bullockcart,andcanalboat.Bythefollowingyear,constructionnowcomplete,thetriptookoneday.

The railroads were the great leveler; everyone used them, irrespective of caste.“When you get to the third-class railway carriage you override even such a toughobstacle as caste,” an English writer from this period noted. “Into it are bundledBrahminandPariah; they sit on the same seat; they rub shoulderswhomightnotmingleshadows.‘Youmustdropyourcaste,’saystherailway,‘ifyouwanttotravelatafarthingamile’;anditisdropped—toberesumedagainoutsidethestation.”

Ramanujanhadgrownupwiththerailways—aswhen,achild,he’dbeenshuttledamongschools inKumbakonam,Kanchipuram,andMadras.Andnow, in1905, inthe wake of losing his scholarship at Government College, the rails facilitated hisflight.

Madras was 194 miles up the tracks of the South Indian Railways fromKumbakonam.AndVizagapatnam,followingthemainlinealongthecoast,was484milesbeyondthat.Atownofabout fortythousand, it lay inanangleof theBayofBengalformedbyapromontoryknownas the “Dolphin’sNose.”BoastingtheonlynaturalharboronIndia’seastcoast,itwasaflourishingseaport;throughit,yarnandpiecegoodsenteredIndiaandmanganeseoreandrawsugar left.Anew lighthousehadjustbeenbuiltneartheanchorage.Nowtheengineerswereplanningtodredgethebackwaterandriverandbuildnewdocks.Vizagapatnamwasonthemove.

AnditwasforthislargelyTelugu-speakingtownhalfwayupthecoasttoCalcuttathatRamanujan, informingnoone, setout.Fragmentary accounts from theperiodvariously give as reasons the influence of a friend, thepursuit of a scholarship, thewishtofindapatron,or—underpressurefromhisfather—ajob.Butinvariably,theyalsouselanguagelike“owingtodisappointment,”“ranaway,”and“toosensitivetoaskhis parents for help,” and it’s plain that whatever Ramanujanmay have sought inVizagapatnam,hewasrunningfromsomething,too.

There is evidence that the family, distraught over their son’s disappearance,advertisedinnewspapersforhim;thathisfatherwenthousetohouseinMadrasandTrichinopoly, looking for him.Otherwise, details ofRamanujan’s impetuous flight

arescanty.Exceptthatsoon,probablybySeptember,hisparentshadhimsafelybackinKumbakonam.

•••

ItwasthefirstoftheGreatDisappearances,thefirstofnumeroussuchoccasionsonwhich Ramanujan would abruptly vanish, and about which little subsequentlybecameknown.Butitwasnotthefirsttimehe’dtakenabruptandheedlessactioninthewakeofwhathedeemedanintolerableblowtohisself-esteem.

Back in 1897, aged nine, when he took his primary exam at Town Hall inKumbakonam,hehadscoreda42outof45onthearithmeticportion,whileafriend,K.Sarangapani Iyengar, got a43.Hurt andangry,Ramanujan refused to speak tohim.Sarangapaniwasmystified;whatwas the big deal?Trying tomollify him, hepointedoutthatintheothersubjectsRamanujanhadscoredhigher.Didn’tmatter,grumbledRamanujan—inarithmetichealwaysscoredhighest.Thistimehehadnot,andeveryoneknewit.Itwasalltoomuchtobear—whereuponheranhomecryingtohismother.

Later, in high school, Ramanujan saw how trigonometric functions could beexpressedinaformunrelatedtotherighttrianglesinwhich,superficially,theywererooted. It was a stunning discovery. But it turned out that the great SwissmathematicianLeonhardEuler had anticipated it by 150 years.WhenRamanujanfoundout,hewassomortifiedthathesecretedthepapersonwhichhehadrecordedtheresultsintheroofofhishouse.

Adolescent behavior quirks, irrelevant in the broad sweep of a genius’s life?Perhaps. But together, and coupled with many other such instances later, theysuggest an almost pathological sensitivity to the slightest breath of publichumiliation.When,yearslater,Ramanujanstoppedgettinglettersfromaonce-closefriend,hewrotetothefriend’sbrotherthatperhaps“heistoosorryforhisfailureintheExamtowritetome.”Plainly,itwasbehaviortowhichhewaskeenlysensitive.

Shameiswhatpsychologistscallthissensitivitytopublicdisgrace,somethingquitedistinct from “guilt.” Guilt, roughly speaking, comes from doing wrong, shame atbeingdiscovered,orattheprospectofbeingdiscovered,insomefailureorvice;you’recaught masturbating, say, or with your hand in the till. “An obligatory aspect ofshameistherolediscoveryplays,”writesLeonWurmser,aUniversityofMarylandpsychiatrist,inTheMaskofShame.“Itisusuallyamoreorlesssuddenexposure,andexposure that abruptly brings to light the discrepancy between expectation andfailure.”Thefeelingisthatofsudden,sharp,inescapablehumiliation—ofayawning

gapbetweenwhoyousayyouareandwhoyourfailuresrevealyoutobe,ofanuglystainuponyourpublicface.

It is not necessary to actually be discovered,Wurmser points out; one can feelshamebeforeoneself,atthemerethoughtofdiscovery.“Wemaywinceatourselvesinthemirroranddespiseanddegradeourselvesforthedishonorwefeelwithin....Nooneelsehastoseethisstain—theshameremains.”

The single most reliable marker of the shame syndrome is the impulse to flee.WritesWurmser:“Hidingisintrinsictoandinseparablefromtheconceptofshame.”One experiences “the wish to hide, to flee, to ‘cover one’s face,’ to ‘sink into theground.’ ” And that’s just what Ramanujan did when faced with the ignominy ofscoringonlysecondinthearithmeticexam;inhidingevidencethathisdiscoverywasinfactrediscovery;andinrunningofftoVizagapatnam.OneaccounthasRamanujansuffering a “mental aberration” during this period. Another calls it “a temporaryunsoundnessofmind.”Whateveritwas,acutelyfeltshamemayhavetriggeredit.

Years later, the memory of his school failure would make Ramanujan seekassurancethatascholarshiphehadbeenofferedwouldnot leavehimwithanotherexaminationtopass.TheGovernmentCollegefiascohumiliatedhim,apparentlytothepointofpsychic trauma.His impulse, as itwouldbe allhis life,was to escape.AndinfleeingtoVizagapatnam,heyieldedtoit.

Norwasitincongruousthatonewho,asmathematician,wouldprovesofreefromtheintellectualblindersofthecrowdshouldcaresodeeplyhowthecrowdperceivedhim. Ramanujan was supremely self-assured about his mathematical gifts. Yetsocially, he was a thoroughgoing conformist. If he cared not at all to followmathematicalpathsothershadtrod,hecareddeeplyhowothersesteemedthepathhehadchosen.

Later, while in England and learning of a mathematics prize, he breathlesslyinquired whether hemight apply for it; formal, outward acknowledgment was nomatterofindifferencetohim,andheneverpretendedotherwise.Similarly,whentheBritishawardedhimahighhonor,hisletteracknowledgingwordofitfairlybubbledoverwithexcitement.

Was he respected as a mathematician? Was he deemed a dutiful son, a goodBrahmin?Didheholdanimportantscholarship?Hadhewonaprize?Theanswers,asoutwardmarkersofacceptanceor success, counted—andcertainlynevermoresothannow,asateenager,atanageofexquisitesensitivitytotheopinionsofothers.

Tales ofRamanujan’s youth reveal a boy content to campout on thepial of hishouse and work at mathematics, outwardly oblivious to the raucous play of hisfriends out on the street.Often,wrappedup inmathematics, hewas oblivious.At

other times, though, he must have wanted to be part of it. His thirst for publicacknowledgment of his gifts, his pain when denied it, and his sensitivity to socialslight,showhowdeeply,atanotherlevel,hereallycared.

4.ANOTHERTRY

Pachaiyappa Mudaliar, born in 1784 of a destitute rural family, was a dubash, amasterof two languages, who thereby served as a vital link in commercewith theBritish.Bythetimehewastwenty-onehehadamassedafortune.Athisdeath,agedforty-six,heleftgreatheapsofittocharity.Thecollegebearinghisname,foundedin1889 and open only toHindus, was by 1906 a respectable institution. Surely thebuilding in which it was housed did nothing to sully its reputation—a greatcolumnedstructuremodeledontheTempleofTheseusinAthens,locatedonwhatwasthenknownasChinaBazaarRoadinthebusyGeorgetownsectionofMadras.

It was to Pachaiyappa’s—pronounced Pa-shay-a-pas—College that Ramanujanwasboundwhen,onedayearlyin1906,hearrivedatEgmoreStationinMadras,sotiredanddisorientedthathefellasleepinthewaitingroom.Amanwokehim,tookhimbacktohishouse,fedhim,gavehimdirections,andsenthimonhiswaytothecollege.

In India a college degree was no mere prerequisite for a good job; it virtuallyguaranteed you one, and a good start in your career. You earned a degree not bytaking so many courses, or accumulating so many credits, but by passing anexamination administered by the University of Madras; the “university” was notteachersandstudents,butmerelyanexaminingbody.“Toappearandsucceedattheuniversityexaminationshasbeentheambitionofeveryyouthofpromise,”anEnglishwriterfromtheperiodnoted.SomeofRamanujan’scontemporariesatthecollegeinKumbakonamtransferred toPresidencyCollege inMadras, the crown jewelof theSouthIndianeducationalsystem,inhopesofbetterpreparingfortheall-importantexamination.

For most who sought a degree, though, it was all in vain. Of those taking thematriculationexam—equivalenttoahighschooldiplomabutmoreeagerlysought—halffailed.Asimilarproportionfelloutateachdegreestepalongtheway;failuresofPachaiyappa’sstudentsontheF.A.examranto80percent.In1904,fewerthanfivethousandboys—andjustforty-ninegirls—wereenrolledinthepresidency’scollegesandprofessional schools.Andamongall its forty-threemillionpeople, thenumberearninganF.A.degreeeachyearcametobarelyathousand.

Ramanujan, eighteen years old now, aimed to be one of them. A year after hisfailureinKumbakonam,hewasgivingcollegeanothertryinMadras.

Foratime,he liveda fewblocksawayfromPachaiyappa’s inasmall laneoff thefruitbazaaronBroadwayinhisgrandmother’shouse.Itwasdingyanddark.Andtheairseemedtohang,staticandclose.Butatleasthewasbackinschool.

Ramanujan’s newmath teacher, shown his notebooks, came away so impressedthathe introducedhim to the principal—who, on the spot, awardedhim a partialscholarship.ThoughinterruptedbyabadboutofdysenterythatbroughthimbacktoKumbakonamforthreemonths,Ramanujan’searlydaysatPachaiyappa’sCollegeseemedfilledwithnewpromise.

N. Ramanujachariar, the math teacher, would take two sliding blackboards towork out a problem in algebra or trigonometry, reaching the solution in a dozenscrawledmathematicalsteps;Ramanujanwouldgetupandshowhowtosolve it inthree or four. “Uh,whatwas that?”Ramanujachariar,whowas a littledeaf,wouldhave to ask. SoRamanujanwould obligingly run through it again. Sometimes theteacherwouldinterruptthelecture,turntoRamanujan,andask,“Andwhatdoyouthink, Ramanujan?” The prodigy from Kumbakonam tended to jump around theproblem,workingoutkeystepsinhisheadbutomittingthemfromhisexposition—leavinghisclassmatesthoroughlyconfused.

Sometimes he’d get together with the college’s senior math professor, P.SingaraveluMudaliar.Singaravelu—somethingofacatch forPachaiyappa’s,havingformerly been an assistant professor of mathematics at the more prestigiousPresidencyCollegeacrosstown—wasstruckbyRamanujan’sgifts.Togetherthetwoof themwould tackle problems appearing inmathematical journals. If Ramanujancouldn’t crack one of them, he’d give it to Singaravelu to work on overnight;invariablytheprofessorcouldn’tsolveit,either.

EveryonewasstruckbyRamanujan’sgifts;buttherewasnothingnewinthat.Norwas there anything new in that nothing tangible came of it. For his experience inKumbakonamnowrepeated itselfatPachaiyappa’s.AtGovernmentCollege, itwasEnglish that had been his undoing. Now, among other subjects remote frommathematicshehadtomaster,therewasphysiology.Andthishefoundnotmerelyboring,butrepellent.

The text was a small book, Physiology for Beginners, written by two Cambridgedons,MichaelFosterandLewisE.Shore,publishedin1894,andconsistingmostlyofthekindofflatdescriptiveaccountsthatpassedforscienceinthelatenineteenthcentury: “At the upper left-hand part of the stomach is the opening into it of theesophagus,atubewhichpassesfromthemouthdowntheneck,throughthethorax,andpiercing the diaphragm, enters the stomach.” Itwas full of elaborate drawingsshowing a rabbit with its skin peeled back, its internal organs revealed in graphic

detail; a sheep’s heart fillingmost of one page, a cutaway of a humanmouth andtongueonanother.

This was as far from the abstract heights of mathematics as you could get; ifmathematicswasartdeco,withitscoolgeometricelegance,physiologywasakindofartnouveau, fluidandsumptuous. Itwasaworld forwhichRamanujan,asa strictvegetarian, could scarcely have hadmuch taste: “Procure a rabbit which has beenrecentlykilled,butnotskinned,”chapter3ofthetextbegan.“Fastentherabbitonitsbackbyitsfourlimbstoaboard,andthen,withasmallsharpandpointedknifeandapairofscissors...”

Ramanujanreactedtoallthiswithaskittish—anduncharacteristic—sarcasm.Theprofessorwoulddissectabig,anesthetizedfrog,earnestlypointingoutphysiologicalsimilarities to humans, only to have Ramanujan pipe up with, And where is theserpent in this frog?—apparently a reference to the nade, or serpent power, thatHindutraditionascribes tohumannature.Another time,onan examcovering thedigestivesystem,Ramanujansimplywroteafewlinesintheanswerbookandhandeditbackunsigned:“Sir,thisismyundigestedproductoftheDigestionchapter.”Theprofessorhadnotroublefiguringwhoseitwas.

Ramanujan, it need hardly be said, flunked physiology. Except formath he didpoorlyinallhis subjects,but inphysiologyhereachedparticularly impressive lows,oftenscoringlessthan10percentonexams.He’dtakethethree-hourmathexamandfinishitinthirtyminutes.Butthatgothimexactlynowhere.InDecember1906,heappeared again for theF.A. examination and failed.The following year, he took itagain.Andfailedagain.

GovernmentCollege,Kumbakonam, 1904 and 1905 . . . Pachaiyappa’sCollege,Madras,1906and1907...Inthefirstdecadeofthetwentiethcentury,therewasnoroomforSrinivasaRamanujan in the higher education systemof South India.Hewasgifted,andeveryoneknewit.Butthathardlysufficedtokeephiminschoolorgethimadegree.

TheSystemwouldn’tbudge.

•••

Describing the obsession with college degrees among ambitious young Indiansaroundthis time, anEnglishwriter,HerbertCompton,notedhow “the loaves andfishesfallfarshortofthemultitude,andtheresultisthecreationofarmiesofhungry‘hopefuls’—thename isa literal translationof thevernaculargeneric termomedwarused in describing them—who pass their lives in absolute idleness,waiting on the

skirts of chance, or gravitate to courses entirely opposed to thosewhich educationintended.”Ramanujan,itmighthaveseemedin1908,wasjustsuchanomedwar.Outofschool,withoutajob,hehungaroundthehouseinKumbakonam.

Times were hard. One day back at Pachaiyappa’s, the wind had blown offRamanujan’s cap as he boarded the electric train for school, and Ramanujan’sSanskrit teacher,who insisted that boyswear their traditional tufts covered, askedhim to step back out to the market and buy one. Ramanujan apologized that helacked even the few annas it cost. (His classmates, who’d observed his often-threadbaredress,chippedintobuyitforhim.)

Ramanujan’sfathernevermademorethanabouttwentyrupeesamonth;arupeeboughtabouttwenty-fivepoundsofrice.Agriculturalworkersinsurroundingvillagesearnedfourorfiveannas,oraboutaquarterrupee,perday;somanyfamilieswerefarworse off than Ramanujan’s. But by the standards of the Brahmin professionalcommunityinwhichRamanujanmoved,itwasclosetopenury.

Thefamilytookinboarders;thatbroughtinanothertenrupeespermonth.AndKomalatammal sang at the temple, bringing in a few more. Still, Ramanujanoccasionally went hungry. Sometimes, an old woman in the neighborhood wouldinvitehim in foramiddaymeal.Another family, thatofRamanujan’s friendS.M.Subramanian,wouldalsotakehimin,feedinghimdosai,thelentilpancakesthatareastapleofSouthIndiancooking.Onetimein1908,Ramanujan’smotherstoppedbytheSubramanianhouselamentingthatshehadnorice.Theboy’smotherfedherandsentheryoungerson,Anantharaman,tofindRamanujan.Anantharamanledhimtothehouseofhisaunt,whofilledhimuponriceandbutter.

Tobringinmoney,Ramanujanapproachedfriendsofthefamily;perhapstheyhadaccounts topost, orbooks to reconcile?Or a son to tutor?One student, for sevenrupees amonth,wasViswanatha Sastri, son of aGovernmentCollege philosophyprofessor. Early each morning, Ramanujan would walk to the boy’s house onSolaiappaMudaliStreet,attheotherendoftown,tocoachhiminalgebra,geometry,and trigonometry. The only trouble was, he couldn’t stick to the coursematerial.He’d teach the standard method today and then, if Viswanatha forgot it, wouldimproviseawhollynewone tomorrow.Soonhe’dbe lost inareas theboy’s regularteachernevertouched.

Sometimeshewouldflyoffontophilosophicaltangents.They’dbediscussingtheheightof awall,perhaps for a trigonometryproblem, andRamanujanwould insistthatitsheightwas,ofcourse,onlyrelative:whocouldsayhowhighitseemedtoanantorabuffalo?One timeheaskedhowtheworldwould lookwhen first created,beforetherewasanyonetoviewit.Hetookdelight,too,inposingslylittleproblems:

Ifyoutakeabelt,heaskedViswanathaandhisfather,andcinchittightaroundtheearth’s twenty-five-thousand-mile-long equator, then let it out just 2π feet—abouttwoyards—howfarofftheearth’ssurfacewoulditstand?Sometinyfractionofaninch?Nope,onefoot.

ViswanathaSastri foundRamanujan inspiring;otherstudents,however,didnot.Oneclassmatefromhighschool,N.GovindarajaIyengar,askedRamanujantohelphimwithdifferentialcalculus forhisB.A.exam.Thearrangement lastedallof twoweeks.Youcanthinkofcalculusasasetofpowerfulmathematicaltools;that’showmoststudentslearnitandwhatmostexamsrequire.Orelseyoucanappreciateitforthe subtle questions it poses about the nature of the infinitesimally small and theinfinitely large. Ramanujan, either unmindful of his students’ practical needs orunwilling to cater to them, stressed the latter. “Hewould talk only of infinity andinfinitesimals,”wroteGovindaraja,whowasnoslouchintellectuallyandwoundupaschairmanofIndia’spublicservicecommission.“Ifeltthathistuition[teaching]mightnotbeofrealusetomeintheexamination,andsoIgaveitup.”

Ramanujanhadlostallhisscholarships.Hehadfailedinschool.Evenasatutorofthesubjecthelovedmost,he’dbeenfoundwanting.

Hehadnothing.Andyet,viewedalittledifferently,hehadeverything.Fornowtherewasnothing

todistracthimfromhisnotebooks—notebooks,crammedwiththeorems,thateachday,eachweek,bulgedwider.

5.THENOTEBOOKS

“In proving one formula, he discovered many others, and he began to compile anotebook”torecordhisresults.That’showRamanujan’sfriendNevilleputitmanyyearslater,anditremainsasconciseadistillationasanyofhowhisnotebookscametobe.Certainly, itwas inworking throughCarr’sSynopsis, as he tottered throughcollegeduringtheyears from1904to1907, thathebegankeeping them inearliestform.

After Ramanujan’s death, his brother prepared a succession of handwrittenaccountsoftherawfacts,data,anddatesofhislife.Andpreserved intheiroriginalformastheyare, theyremindusofaworldbeforecomputersandwordprocessorsmade revisioneasyand routine:we see rude scrawlsgrowingneater,moredigestedandrefined,astheyarecopiedandrecopiedthroughsuccessiveversions.

SuchwasthelikelygenesisofRamanujan’snotebooks.The first of the published Notebooks that come down to us today, which

Ramanujan may have prepared around the time he left Pachaiyappa’s College in

1907,waswritteninwhatsomeonelatercalled“apeculiargreenink,”itsmorethantwohundred largepagesstuffedwithformulasonhypergeometricseries,continuedfractions,singularmoduli...

But this “first”notebook,whichwas later expandedandrevised intoa second, ismuchmorethanmereoddnotes.Brokenintodiscretechaptersdevotedtoparticulartopics,itstheoremsnumberedconsecutively,itsuggestsRamanujanlookingbackonwhathehasdoneandprettying itup for formalpresentation,perhaps tohelphimfind a job. It is, in other words, edited. It contains few outright errors; mostly,Ramanujancaught themearlier.Andmostof itscontents,arrayedacross fifteenortwentylinesperpage,areentirelylegible;oneneedn’tsquinttomakeoutwhattheysay.No,this isno imprompturecord,nopileofsketchesorsnapshots;rather, it islikeamuseumretrospective, theviewerbeingguidedthroughwell-markedgallerieslinedwiththeartist’swork.

Orsotheywereintended.Atfirst,Ramanujanproceededmethodically, inneatlyorganizedchapters,writingonlyontheright-handsideofthepage.Butultimately,itseems,hisresolvebrokedown.Hebegantousethereversesidesofsomepages forscratchwork,orforresultshe’dnotyetcategorized.Mathematicaljottingspiledup,nowinamoreimpetuoushand,withsomeofitstruckout,andsometimeswithscriptmarchingupanddownthepageratherthanacrossit.OnecanimagineRamanujanvowing that, yes, this time he is going to keep his notebook pristine . . . when,workingonanideaandfindingneitherscratchpapernorslateathand,heabruptlyreachesforthenotebookwithitsbeckoningblanksheets—theresultcomingdowntoustodayasflurriesofthoughttransmutedintopaperandink.

Inthose flurries,wecan imagine theveryearliestnotebooks, thosepredating thepublishedones,comingintobeing.RamanujanhadsetouttoprovethetheoremsinCarr’s book but soon left his remote mentor behind. Experimenting, he saw newtheorems,wentwhereCarr had never—or, inmany cases, no one had ever—gonebefore.Atsomepoint,ashisminddailyspunoffnewtheorems,hethoughttorecordthem. Only over the course of years, and subsequent editions, did those early,haphazard scribblings evolve into the published Notebooks that today sustain averitablecottageindustryofmathematiciansdevotedtotheirstudy.

•••

“Twomonkeyshavingrobbedanorchardof3timesasmanyplantainsasguavas,areabout tobegin their feastwhen they espy the injuredownerof the fruits stealthilyapproachingwithastick.Theycalculatethatitwilltakehim21/4minutestoreach

them.Onemonkeywho can eat 10 guavas perminute finishes them in2/3 of thetime,and thenhelps theother toeat theplantains.They just finish in time. If thefirstmonkeyeatsplantainstwiceasfastasguavas,howfastcanthesecondmonkeyeatplantains?”

ThischarminglittleproblemhadappearedsomeyearsbeforeRamanujan’stimeinan Indianmathematical textbook. Exotic as itmight seem at first, one has but tochange themonkeys to foxes, and the guavas to grapes, to recognize one of thoseexercises, beloved of some educators, supposed to inject life and color intomathematics’ presumably airless tracts.Needless to say, this sort of trifle, howevertrickytosolve,bearsnokinshiptothebrandofmathematicsthatfilledRamanujan’snotebooks.

Ramanujanneedednovisionofmonkeyschompingonguavastospurhisinterest.Forhim,itwasn’twhathisequationstoodforthatmattered,buttheequationitself,aspatternandform.Andhispleasurelaynotinfindinginitanumericalanswer,butfromturningitupsidedownandinsideout,seeinginitnewpossibilities,playingwithitasthepoetdoeswordsandimages,theartistcolorandline,thephilosopherideas.

Ramanujan’s world was one in which numbers had properties built into them.Chemistrystudentslearnthepropertiesofthevariouselements,thepositionsintheperiodic table they occupy, the classes to which they belong, and just how theirchemicalpropertiesarisefromtheiratomicstructure.Numbers,too,havepropertieswhichplacethemindistinctclassesandcategories.

Forstarters,thereareevennumbers,like2,4,and6;andoddnumbers,like1,3,and5.

Therearetheintegers—wholenumbers,like2,3,and17;andnonintegers,like171/4and3.778.

Numberslike4,9,16,and25aretheproductofmultiplyingtheintegers2,3,4,and5bythemselves;theyare“squares,”whereas3,10,and24,forexample,arenot.

A6differsfundamentallyfroma5,inthatyoucangetitbymultiplyingtwoothernumbers,2and3;whereasa5istheproductonlyofitselfand1.Mathematicianscall5andnumbers like it (2,3,7,and11,butnot9) “prime.”Meanwhile,6andothernumbersbuiltupfromprimesaretermed“composite.”

Then, there are “irrational”numbers,which can’tbe expressedas integersor theratioofintegers,like ,whichisapproximately1.414...,butwhich,howevermanydecimalplacesyoutaketoexpressit,remainsapproximate.Numberslike3,1/2,and911/16,ontheotherhand,are“rational.”

Andwhataboutnumbers, likethesquarerootof−1,whichseemimpossibleorabsurd? A negative number times a negative number, after all, by mathematical

conventionispositive;sohowcananynumbermultipliedbyitselfgiveyouanegativenumber?Noordinarynumber,ofcourse,can;thosesodefinedarecalled“imaginary,”and assigned the label i; . Such numbers, it turns out, can bemanipulated like any other and find wide use in such fields as aerodynamics andelectronics.

Thathappensofteninmathematics;anotionatfirstglancearbitrary,ortrivial,orparadoxicalturnsouttobemathematicallyprofound,orevenofpracticalvalue.Afteraninnocentchildhoodofordinarynumberslike1,2,and7,one’sinitialexposuretonegativenumbers,like−1or−11,canbeunsettling.Here,itdoesn’trequiremucharm-twisting to accept the idea: If t represents a temperature rise, but thetemperaturedrops 6 degrees, you certainly couldn’t assign the same t= 6 that youwouldforanequivalenttemperaturerise;someothernumber,−6,seemsdemanded.Somewhat analogously, imaginary numbers—as well as many other seeminglyarbitraryordownrightbizarremathematicalconcepts—turnouttomakesolidsense.

Ramanujan’snotebooksrangedovervast terrain.Butthis terrainwasvirtuallyall“pure”mathematics.Whateverusetowhichitmightonedaybeput,Ramanujangaveno thought to its practical applications.Hemight have laughed out loud over themonkey and the guava problem, but he thought not at all, it is safe to say, aboutraising the yield of South Indian rice. Or improving the water system. Or evenmakinganimpactontheoreticalphysics;that,too,was“applied.”

Rather,hedid it just todo it.Ramanujanwasanartist.Andnumbers—andthemathematicallanguageexpressingtheirrelationships—werehismedium.

•••

Ramanujan’snotebooksformedadistinctlyidiosyncraticrecord.Inthemevenwidelystandardized terms sometimes acquired new meaning. Thus, an “example”—normally, as in everyday usage, an illustration of a general principle—was forRamanujanoftenawhollynewtheorem.A“corollary”—atheoremflowingnaturallyfromanothertheoremandsorequiringnoseparateproof—wasforhimsometimesageneralization,whichdidrequireitsownproof.Asforhismathematicalnotation,itsometimesborescantresemblancetoanyoneelse’s.

Inmathematics,theassignmentofx’sandy’sneedconformtonoparticularrule;whileanequationmayrevealprofoundmathematicaltruths,justhowitiscouched—the letters and symbols assigned to its various entities, for example—is quitearbitrary.Still, in amature field, oneor very fewnotational systemsnormally takehold.AmathematicianlayingopenanewfieldpickstheGreekletterπ,say,tostand

foracertainvariable;soon,throughhistoricalaccidentorforceofhabit,it’sbecomeenshrinedinthemathematicalliterature.

Topickanexamplefamiliarfromhighschoolalgebra,thetworootsofaquadraticequation(whichdescribesthegeometricfigureknownasaparabola)aregivenby

where a, b, and c are constants, x a variable. So entrenched is this form of theequationthatit’shardtoimagineanythingelse.Andyet,there’snoreasonwhytheconstantscouldn’tbep,q, andr.Orm1,m2, andm3.And the quantitywithin thesquare root signcouldbe seenas thedifferenceof twosquaresandbrokenup intotwoterms.Andthesquarerootitselfcouldbeexpressedasafractionalpower.Andeachofthetworootscouldgetitsownequation.Theresultwouldbe:

The mathematical gymnastics don’t matter here, only that this is identical to themore canonical version—and yet, on its face, unrecognizable. Someone coming upwiththeresultonhisown,andexpressingitinthisaliennotationbecausehedidnotknow the established one, would face extra roadblocks to being understood andmightbewrittenoffasunorthodoxorstrange.

Which is just how Ramanujan’s notebooks would tend to be regarded bymathematicians,bothofhisowndayandofourown.Intheareaofellipticfunctions,where everybodyusedk for themodulus, an important constant,Ramanujan usedtheGreekletter or .Sometimesnwas,inhisnotebooks,acontinuousvariable,which for professional mathematicians it never was. As for the quantity π(x), bywhicheveryoneelsemeantthenumberofprimenumbersamongthefirstxintegers,itneverappearedatall.

Therewasnothing“wrong”inwhatRamanujandid;itwasjustweird.Ramanujanwas not in contact with other mathematicians. He hadn’t read last month’sProceedings of the London Mathematical Society. He was not a member of themathematicalcommunity.Sothattoday,scholarscitinghisworkmustinvariablysay,“InRamanujan’snotation,”or“ExpressingRamanujan’sideainstandardnotation,”orusesimilarsuchlanguage.

Hewaslikeaspeciesthathadbranchedofffromthemainevolutionarylineand,likeanAustralian echidna orGalápagos tortoise, had come to occupy a biologicalnicheallhisown.

•••

Ifoffbeattoothermathematicians,theparadeofsymbolsinRamanujan’snotebooksamountedtoaforeignlanguagetomostlaypeople.Andyet,asarcanealanguageasitwas,theconceptsitexpressedoftenturnedouttobesurprisinglystraightforward.

Take,forexample,thef(x)’sandotherexamplesof“functionalnotation”thatlitterRamanujan’snotebooks.Here,f(x),read“efofex,”doesn’tmeanftimesx,butrathersomeunspecifiedfunctionofx; something, inotherwords,dependsonx.Withoutdefining the function we don’t know how it depends. Later, we may specify, forexample, that f(x)=3x+1.Thenwedo knowhow itdependsonx; thealgebraicformulatellsus,describingitsmathematicalbehavior:Inthiscase,whenx=1,f(x)=4;whenx=2,f(x)=7;andsoon.Butoften,themathematiciandoesn’twanttogetdowntospecifics.Functionalnotationletshimworkinthemoreabstractrealmsheprefers,freefromslaverytoparticularcases.

In functional notation, φ (a,b), read “phi of ay and bee,” just means someunspecifiedfunction,φ, thatdependsonthevariablesaandb.And f(3) justmeansf(x) evaluatedwhenx= 3.And g(−x) justmeans g(x)with− 1 plugged into theequationwheneverx=1,−2whenx=2,andsoon.Withsuchbroadbrushstrokes,sometimesneverstoopingtoparticular functionsatall, themathematician fashionshisworld.

Orsometimeshedoesmake f(x) a specific function, thengoeson todiscover itsodd or revealing properties. On page 75 of the first notebook, for example,Ramanujanwrites

φ(x)+φ(−x)=1/2φ(−x2)

foraparticularfunctiondefinedpreviously.Evaluateφ(x)at,say,x=1/2,thenatx=−1/2.Addupthetworesults.Andthatwillequalhalfofwhatyougetifyouevaluatethefunctionatx=−1/4.ButRamanujan’sequationsays itmoregenerally, revealsthefunction’smathematicalidiosyncrasies.Andsaysitwithoutsomanywords.

WhichisonereasonwhyRamanujan,andallmathematicians,usetheirseeminglyalien language in the first place—as a standin for long-winded verbiage.When onpage 86 of the first notebook, and inmany other places,RamanujanwritesΣ, theGreeklettersigma,hemeans,simply,“thesumof...”Anotationalfragmentlike

maybereadas“thesumofallthetermsoftheformxtothekthpowerdividedbyk,whenkgoesfrom1toinfinity.”Thatmeans,“wheneveryouseeak,replaceitby1,andnoteit;thenbya2,andaddittothefirst....Continueinthiswayforever.”Andthatisequivalentto

Mathematics is full of similarly simple ideas lurking behind alien terminology.Wanttospecifyaseriesoftermsthatalternatebetweenpositiveandnegative?That’seasy:Justincludethefragment(−1)k.Askmarchesupthroughtheintegersonebyone,thesignalternatesautomaticallybetweenplusandminus,becauseminus-times-minusisplusandminus-times-plusisminus.Ormaybeyouwishtospecifyonlyoddnumbers, like1,3,5,7,andsoon?Theexpression2n+1churnsthemout.Fornequals0,2n+1equals1.Forn=1,2n+1=3.Forn=2,2n+1=5...

Short,sweet,concise.

•••

Ifyoudon’tknowEnglish,youcan’twriteajobapplication,andyoucan’twriteKingLear.But justknowingEnglish isn’t enough towriteShakespeare’splay.Thesameapplies to Ramanujan’s notebooks: its pages of mathematical scrawl were, toprofessionalmathematicians,whatwasleastdifficultaboutthem.AswiththeEnglishofLear,itwaswhattheysaidthattookallthework.

Andwork itwas—inexpressingmathematicalentities,performingoperationsonthem,tryingspecialcases,applyingexistingtheoremstonewrealms.Butsomeofthework, too, was numerical computation. “Every rational integer was his personalfriend,”someoneoncesaidofRamanujan;aswithfriends,helikednumbers,enjoyedbeingintheircompany.

Even in the published notebooks, you can see Ramanujan giving concretenumerical form to what others might have left abstract—plugging in numbers,gettingafeelforhowfunctions“behaved.”Somepages,withtheirdearthofΣ’sandf(x)’s,andtheirprofusionof61sand3533s,looklesslikemathematicaltreatise,morelike the homework assignment of a fourth-grader.Numerical elbow grease it was.Andheput inplentyof it.OneRamanujan scholar,B.M.Wilson, later toldhow

Ramanujan’s research into number theory was often “preceded by a table ofnumericalresults,carriedusuallytoalengthfromwhichmostofuswouldshrink.”

Fromwhichmostofuswouldshrink.There’sadmirationthere,butmaybeawispofderision,too—asifinwonderthatRamanujan,ofallpeople,couldstoopsowillinglyto the realmof themerelyarithmetical.Andyet,Ramanujanwasdoingwhatgreatartists always do—diving into his material. He was building an intimacy withnumbers,forthesamereasonthatthepainterlingersoverthemixingofhispaints,orthemusicianendlesslypracticeshisscales.

And his insight profited.He was like the biological researcher who sees thingsothersmissbecausehe’sthere inthe labeverynighttoseethem.Hisfriendsmightlaterchoosetorecallhowhemadeshortworkofschoolproblems,couldseeinstantlyintothosetheyfoundmostdifficult.ButtheproblemsRamanujantookupwereastoughsloggingtohimasschoolproblemsweretothem.Hissuccessesdidnotcomeentirelythroughflashesofinspiration.Itwashardwork.Itwasfulloffalsestarts.Ittooktime.

Andthatwastheirony:inthewakeofhisfailureatschool,timewasonethinghehadplentyof.

6.ATHOUGHTOFGOD

In1807,ahundredyearsbeforeRamanujanwastofailhisF.A.examforthelasttimeand experience India’s educational system in all its oppressive rigidity, WilliamThackeray, an Englishman with experience in India as translator, judge, and civilservant, concluded his Report on Canara, Malabar, and Ceded Districts. In it, hewrote:

It is very proper that in England, a good share of the produce of the earth should be appropriated tosupportcertainfamiliesinaffluence,toproducesenators,sagesandheroesfortheserviceanddefenseofthestate;orinotherwords,thatagreatpartoftherentshouldgotoopulentnobilityandgentry,whoaretoservetheircountryinParliament, inthearmy, inthenavy, in thedepartmentsofscienceand liberalprofessions.Theleisure,independenceandhighidealswhichtheenjoymentofthisrentaffordshasenabledthemtoraiseBritaintopinnaclesofglory.Longmaytheyenjoyit.ButinIndiathathaughtyspirit,independenceanddeepthought which the possession of great wealth sometimes gives ought to be suppressed. They are directlyaverse to our power and interest.The nature of things, the past experience of all governments, renders itunnecessary to enlarge on this subject. We do not want generals, statesmen and legislators; we wantindustrioushusbandmen.IfwewantedrestlessandambitiousspiritsthereareenoughoftheminMalabartosupplythewholepeninsula.

IfThackeray’s sentimentsmirroredBritish educationalpolicy in India,nobetterevidenceforitcouldbefoundthanRamanujan,forwhomcollegemighthaveseemedaimed at suppressing “haughty spirit, independence, and deep thought.” Indeed,

Indianhighereducation’sfailuretonurtureoneofsuchundoubted,butidiosyncratic,giftscouldserveastextbookexampleofhowbureaucraticsystems,policies,andrulesreallydomatter.People, as individuals, appreciated and respectedRamanujan; butthe System failed to find a place for him. It was designed, after all, to churn outbright, well-rounded young men who could help their British masters run thecountry,notthe“restlessandambitiousspirits”Thackeraywarnedagainst.

Viewed one way, then, for at least the five years between 1904 and 1909,Ramanujan floundered—mostly out of school, without a degree, without a job,withoutcontactwithothermathematicians.

Andyet,wasthecuphalf-empty—orhalf-full?Thegreatnineteenth-centurymathematicianJacobibelieved,asE.T.Bellputitin

Men ofMathematics, that youngmathematicians ought to be pitched “into the icywater to learn to swimordrownby themselves.Many studentsputoff attemptinganything on their own account till they havemastered everything relating to theirproblem that has been done by others.The result is that but few ever acquire theknackofindependentwork.”

Ramanujantossedaloneintheicywatersforyears.Thehardshipandintellectualisolationwoulddohimgood?Theywouldspurhis independentthinkingandhonehistalents?NooneinIndia,surely,thoughtanythingofthekind.Andyet,thatwastheeffect.Hisacademic failure forcedhimtodevelopunconventionally, freeof thesocialstraitjacketthatmighthaveconstrainedhisprogresstowell-wornpaths.

Forfivesolidyears,Ramanujanwasleftalonetopursuemathematics.Hereceivedno guidance, no stimulation, no money beyond the few rupees he made fromtutoring.But forall theeconomicdeadweightherepresented,his familyapparentlydiscouragedhimlittle—notenough,inanycase,tostophim.India,itmightbesaid,left room for the solitary genius in him as it would for the sage, the mystic, thesanyasi.Hisfriends,hismother,andevenhisfathertoleratedhim,madenoundulyurgentdemandsthathefindworkandmakesomethingofhimself.Indeed,inlookingback to Ramanujan’s early years, Neville would refer to “the carefree days before1909.”And,inasense,theywere.Insomeways,theywerethemostproductiveofhislife.Ramanujanhadfoundahomeinmathematics,onesothoroughlycomfortablehescarcely ever wished to leave it. It satisfied him intellectually, aesthetically,emotionally.

And,theevidencesuggests,spiritually,aswell.Countlessstorieswouldlaterattestto how, in Ramanujan, the mathematical and the metaphysical lay side by side,inextricablyintertwined.Once,whileastudentatPachaiyappa’sCollege,heissaidtohavewarnedtheparentsofasickchildtomovehimaway;“thedeathofaperson,”he

toldthem,“canoccuronlyinacertainspace-timejunctionpoint.”Anothertime,inadream,hesawahandwriteacrossa screenmaderedby flowingblood, tracingoutellipticintegrals.

OneideaRamanujanbruitedaboutdealtwiththequantity2n−1.That,afriendremembered him explaining, stood for “the primordial God and several divinities.When n is zero the expression denotes zero, there is nothing; when n is 1 theexpression denotes unity, the Infinite God. When n is 2, the expression denotesTrinity;whennis3,theexpressiondenotes7,theSapthaRishis,andsoon.”

Ramanujan was unfailingly congenial to metaphysical speculation. InKumbakonam, there was a gymnastics teacher, Satyapriya Rao, whose feveredoutpourings even tolerant South Indians dismissed.Hewould stand there, by theCauvery,staringintothesun,raving;sometimeshe’dhavetobechainedupwhenhegot too hysterical. Most people ignored him. But not Ramanujan, who wouldsometimescollectfoodforhim;somethoughthemustbemadtoindulgehimso.Yes,Ramanujan explained, he knew themanhad visions, saw tiny creatures. But in anearlierbirth,hewassure,Satyapriyahadearnedgreatmerit.Whatotherswroteoffastheravingsofamadmanwasactuallyahighlyevolvedvisionofthecosmos.

Later, in England,Ramanujanwould build a theory of reality aroundZero andInfinity, thoughhis friendsneverquite figuredoutwhathewasgettingat.Zero, itseemed,representedAbsoluteReality.Infinity,or∞,wasthemyriadmanifestationsof thatReality.Theirmathematical product,∞× 0, was not one number, but allnumbers,eachofwhichcorrespondedtoindividualactsofcreation.Tophilosophers,perhaps—and tomathematicians, certainly—the ideamighthave seemed silly.ButRamanujan found meaning in it. One friend, P. C. Mahalanobis—the man whodiscovered him shivering in his Cambridge room—later wrote how Ramanujan“spokewithsuchenthusiasmaboutthephilosophicalquestionsthatsometimesIfelthewouldhavebeenbetterpleasedtohavesucceededinestablishinghisphilosophicaltheoriesthaninsupplyingrigorousproofsofhismathematicalconjectures.”

IntheWest,therewasanolddebateastowhethermathematicalrealitywasmadeby mathematicians or, existing independently, was merely discovered by them.Ramanujanwassquarelyinthelattercamp;forhim,numbersandtheirmathematicalrelationships fairly threw off clues to how the universe fit together. Each newtheoremwasonemorepieceoftheInfiniteunfathomed.Sohewasn’tbeingsilly,orsly,orcutewhenlaterhetoldafriend,“AnequationformehasnomeaningunlessitexpressesathoughtofGod.”

7.ENOUGHISENOUGH

Attheageoftwenty,Ramanujanwas,ashe’dbeenmostofhislife,fat.Hewasshortand squat, with a full nose set onto a fleshy, lightly pockmarked face, bare ofmustache or beard. His shaved forehead, with its prominent red and white castemark,andtherestofhis fullblackhairgatheredbehindhishead intoa tuft,madehimseemevenrounder,fleshier,andfullerthanhewas.

Buttherewasnothick,lumberingsluggishnesstoRamanujan’sbulk;ifanythingitwasmore like that of a sumowrestler, or a Buddha,with a lightness to it, even adelicacy.Hewalkedwithheaderect,asprightlinessinhisgait,bodypitchedforwardonto his toes. He had long arms, hands of a surprising, velvety smoothness, andslender,taperingfingersforeverinmotionashetalked.

Whenhegrewanimated,thewordstumbledout.Eveneating,whichhedidwithgusto, rarely staunched the flow; he’d go on with an idea or a joke even with hismouthfull.Andalways,hisdarkeyesglowed;therestofhimcouldsometimesseemtofallaway,leavingonlythelightinhiseyes.

Occasionallyhe’ddropbythecollegethathadflunkedhim,toborrowabook,orsee a professor, or hear a lecture.Or he’dwander over to the temple. Butmostly,Ramanujan would sit working on the pial of his house on Sarangapani SannidhiStreet,legspulledintohisbody,alargeslatespreadacrosshislap,madlyscribbling,seeminglyobliviousto thesqueakof thehardslatepencilupon it.Forall thenoisyactivity of the street, the procession of cattle, of sari-garbedwomen, of half-nakedmenpullingcarts,heinhabitedanislandofserenity.Humanactivitypassedcloseby,yetlefthimalone,andfree,unperturbedbyexamshehadnowishtotake,orsubjectshehadnowishtostudy.

TheHindu,SouthIndia’spremierEnglishlanguagenewspaper,hadobservedinan1889 editorial that “the Indian character has seldom been wanting in examples ofwhat may be called passive virtues. Patience, personal attachment, gentleness andsuch like have always been prominent. But for ages together, India has not hadamongsthersonsonelikeGordon,Garibaldi,orWashington....Inalldepartmentsoflife,” itwenton,“theHindusrequireavigorousindividuality,adeterminationtosucceed and to sacrifice everything in the attempt.” And despite his seemingindifferencetoworldlysuccess,Ramanujan, inwardly,wasamodelofall theHindueditorialwritercouldhavewanted.

Adeterminationtosucceedandtosacrificeeverythingintheattempt.Thatcouldbeaprescriptionforanunhappylife;certainlyforalifeoutofbalance,sneeringattimidityandrestraint.Sometimes,asRamanujansatorsquattedonthepial,he’dlookuptowatch the childrenplaying in the streetwithwhatoneneighbor rememberedas “ablankandvacantlook.”Butinside,hewasonfire.

Whenhe thought hard, his face scrunched up, his eyes narrowed into a squint.When he figured something out, he sometimes seemed to talk to himself, smile,shakehisheadwithpleasure.Whenhemadeamistake, too impatientto laydownhisslatepencil,hetwistedhisforearmtowardhisbodyinasinglefluidmotionandusedhiselbow,nowaimedattheslate,asaneraser.

Ramanujan’swasnocool,steadyIntelligence,solemnlyappliedtotheproblemathand;hewasallenergy,animation,force.

Hewasalsoayoungmanwhohungaroundthehouse,whohadflunkedoutoftwocolleges,whohadnojob,whoindulgedinmysticaldisquisitionsthatfewunderstood,andinmathematicsthatnoonedid.Whatvaluewashisworktoanyone?Maybehewas a genius,maybe a crank.But in any case,whywaste one’s time and energy inactivitysodivorcedfromthecommonpurposesoflife?Didn’thisfather,workingasalowlyclerkinasilkshop,dotheworldandhimselfmoregoodthanhe?

Foralongtimehisparentsputupwithhim.Butintheendtheytooreachedtheirlimits, grew irritated and impatient. Enough is enough, his mother decided. Andsometime probably late in 1908 she moved decisively to invoke what the Indianpsychologist AshisNandy has called “that time-tested Indian psychotherapy”—anarrangedmarriage.

CHAPTERTHREE

TheSearchforPatrons

[1908to1913]

1.JANAKI

One day late in 1908, Ramanujan’s mother was visiting friends in the village ofRajendram,about sixtymileswestofKumbakonam.There she spiedabright-eyedwispofagirl,Janaki,daughterofadistantrelative.Sheaskedforthegirl’shoroscope—the first step in virtually every arranged marriage in India—drew her son’shoroscopeon thewall of thehouse, compared it to thatof thegirl, and concludedthat yes, this wouldmake a goodmatch.Negotiations ensued for themarriage ofRamanujanandJanaki,thenaboutnineyearsold.

Itwas inmanywaysanaptmatch,betweentwopersonsofequallymeagersocialstanding.Janakiwasanunassuming,onlyordinarilyprettygirlfromavillagesotinyitappearsonnonebutthemostdetailedmaps.Thefamilyhadoncebeenbetteroff;herfatherdealtinjewelry-makingsuppliesandhadonceownedalittleproperty.Butnow, fallen on harder times, they could offer only amodest dowry, perhaps a fewpolished copper vessels. They could not afford to be choosy about a husband,especiallysinceJanakiwasbutoneoffivedaughters(alongwithoneson).Mostofall,theysoughtafamilyfortheirdaughterapttotreatherkindlyduringthoseearlyyearswhen, stillwithout children, she toiled under the imperious eye and unquestionedauthorityofhermother-in-law.

Ramanujan, meanwhile, was no great catch. Outwardly, he was a total failure,lackingdegree,job,orprospects.Janakiknewnothingofthemanwhowastobecomeherhusband;shewouldnotsomuchasglimpsehisfaceuntiltheirwedding.Hewasan ordinary young man from an ordinary family. Maybe, she thought later, herparentshadheardKomalatammaltouthersonasamathematicalgenius;ifso,she’dknownnothingofit.

SofarasKomalatammalwasconcerneditwasallset,whichinRamanujan’sfamilymeant itwas.ButwhenhusbandSrinivasa learnedof the arrangements,he fumed.Theboycandobetterthanthat,heprotested.ManyfamiliesinKumbakonamwould

beproudtocounthimasson-in-law;infact,twoyearsbefore,whenRamanujanwasoff at Pachaiyappa’s College, a family inKanchipuram had come forward with anoffer,andonlyadeathinthebride’sfamilyhadgottenintheway.WhatreallyriledSrinivasa,though,wasthathe’dhadnosayintheplans.Nothingmollifiedhim.Sothe following July, when it came time to travel to Rajendram for the wedding, hestayedhome.

Intheseevents,Srinivasa’sexclusionwasunusual.Arrangedmarriages,withoutasay for the bride or groom,were virtually universal, the institution of child-bridesalmostasmuchso;mostgirlsmarriedbeforepuberty,thoughtheydidn’tactuallylivewith their husbands, consummating the marriage, until later. The practice wasrepugnant to most Europeans, but the British, ever sensitive to local custom, didnothing to change it. In 1894, the state of Mysore had passed a law barring themarriageofgirlsyoungerthaneight;asimilarprovisioninMadrashadfailed.

Extendingoverfourorfivedays,anIndianweddingwasagloryofcolorandtinsel,musicandceremony.Thewholeeconomywasinfluencedbythescaleandexpenseofthese grand affairs, on which six months’ income might be blown with scarcely athought.Eventhepoorestfamiliesunblinkinglyassumedeveryburden—savedeveryspare rupee, indebted themselves to local usurers—to provide their daughters’dowries,tobuynewsaris,andtopayforthemealsandmusicoftheweddingitself.

Ramanujan’s was a double wedding, Janaki’s sister Vijayalakshmi being set tomarrythesameday.(ByDecember,shewouldbedead,preytoaseverefever.)Theotherbridegroomshoweduponschedule.Butlongintothedaybeforethewedding,andthen intothenight,Ramanujanandhis family failedtoappear. Janaki’s father,Rangaswamy,hadneverbeenentirelywonoverbytheprospectofthematch.Now,Janakiheardhimsay,ifRamanujandidn’tshowupsoon,they’dmarryheroff,thenandthere,tosomeoneelse,maybehisnephew....

The train from Kumbakonam rolled into Kulittalai, the station nearestRajendram,hourslate.AndsoitwaslongpastmidnightbeforeRamanujanandhismother, on a bullock cart from the station, reached the village. Rangaswamy, hisnervesstretchedtothebreakingpoint, railed, spokeofcallingoff thewedding.ButKomalatammal,marshalingherconsiderablepersuasivepowers,wonderedout loudwhetherafatheroffivedaughtersoughttohesitatewhenopportunityknocked....

Asusual,Komalatammalhadherway.Thebridegroom’sreceptiontookplaceatoneo’clockinthemorning.Thencamethekasiyatra,inwhichthebridegroommakesashowofrenouncingdomesticpleasures,evenstartsoffforBenares,thesacredcityof the North, to become a sanyasi; he gets maybe a hundred yards before beingheadedoffby thebride’s family,whowashhis feet in supplication andbeghim to

return.Finally,onJuly14,1909Janakitookthesaptapadi,orsevensteps,thatmadethemarriageirrevocable.

Inauspicious incidents, however, marred the wedding. While Ramanujan andJanaki, inthe finest silk sariher familycouldafford, sat togetheron the traditionalswingbeingserenadedbysingers,thescreamsofaretardedgirlfromtownshatteredthemoment’sharmony.At anotherpoint, a garland Janaki sought toplace aroundRamanujan’sneckfelltotheground.Finally,asdrummersandmusiciansentertainedthem,afirebrokeoutinacornerofthechoultrywheretheweddingwasbeingheld.Thoughquicklyextinguished,itwasdeemedanillomen.

Throughitall, thedoughtyKomalatammalremainedcheerful,herunflappabilitywinninghersympathyandnotalittlewonder.

•••

Atfirst,Ramanujan’smarriagechangednothing,atleastoutwardly.Janakiwouldn’tactually join him for three years, until after she’d reached puberty. Rather, after abrief spell with his family in Kumbakonam, she would return to her own inRajendram, there toworkwith hermother around the kitchen, learn cooking anddomesticchores,andbefurtherschooledintheartsofobedienceandrespectforherparents-in-lawandhusband.

But thoughoutward circumstanceshad changed little,Ramanujanhadenteredanew stage of life. Hindu thinking sees life passing through four stages. Asbrahmacharya, you are a student, learning the spiritual and intellectual ropes. Asgrihasta,occupyingthe longestspan,youareahouseholder,withresponsibilitiestohomeandfamily.Asvanaprastha,or“inhabitantoftheforest,”youbegintothrowoffthebustleof family lifeandseeksolitude, introspectivecalm.Finally,assanyasi,yourelinquisheverything—family,possessions,attachments—inpursuitofspiritualfulfillment.Athiswedding,inheadingoffforBenares,Ramanujanhadrituallyoptedforthislaststage.Butinfact,hewasnowagrihasta.Hehadresponsibilitiesnow.Hehadawife.Hisfatherwaspushingfifty.Nolongerwasheafreespirit,left“rangingwithdelight”throughmathematics,happilyonhisown.Itwastimethatheassumethemantleofadulthood.

Butnowamedicalproblemintervened.Someaccountslaterfounditmoredelicateto refer vaguely to “kidney trouble,” but in fact Ramanujan had developed ahydrocele,anabnormalswellingofthescrotalsac.

“Hydrocele” is a physical finding, not some particular illness. A subtle, andotherwiseharmless, imbalanceintherateofabsorptionofscrotalfluidcancauseit.

So can filariasis, endemic in South India, an infection of the lymph system bymosquito-borneparasites.Socanotherinfections,amongthemtubercular.Usually,there are no symptoms, not even sexual; men sometimes carry a small hydrocelearoundwiththemforyears.Onlywhenonereachesthesizeof,say,atennisball,doessheer mechanical inconvenience make it a problem and demand surgery. Theoperationissimple;anincisionismadeinthescrotalsactoreleasetheblockedfluid.Because thearea is sorich inbloodvessels,healing isnormallyrapid,and infectionrare,evenunderpoorsanitaryconditions.

There was one problem; the family had no money for the operation.Komalatammalaskedfriendsforhelp,butnonewasforthcoming.Finally,inJanuary1910, a certain Dr. Kuppuswami volunteered to do the surgery for free. As thechloroform was being administered, a friend later recalled in wonder, Ramanujannotedtheorderinwhichhisfivesenseswereblocked.

For a time,Ramanujanwas leftprostrate.Oneday,hurryingontohis legs againtoosoon, hewalkedwith his friendAnantharaman to a village a fewmiles out oftown; thewoundbegan to bleed.But soon, recovered, fueled by the new resolve alongrestcanbring,Ramanujanbegantogoouttothewiderworldbeyondthepial.

Since discovering Carr, he had turned his back on anything—school, family,friends—that tookhimaway frommathematics.During thatperiod,he’dprobablyneededtobeleftalone,undistracted,freetofollowhismathematicalmuse.Butnow,aftersixyears,maybeitwastimetostitchhimselfintothebroadersocialfabricagain.Itistempting,ofcourse,toseethehandofhismotherinallthis—that,consciouslyornot,she’drealizedthatifhersonwastoachieveanything,hehadtoreachouttotheworld,andthathismarriagewouldforcehimtodothat.Inanyevent,that’swhathappened. Ramanujan was a grihasta now, and even if inwardly kicking andscreaming,hegaveupthesocialwildernessthathadlongbeenhishome.

2.DOOR-TO-DOOR

Ramanujan sought now not a scholarship, nor even the chance to be amathematician,butjustajob,achanceatafuture,anewlife.Forthenexttwoyears,thesheerdesperationofhislotsenthimacrossSouthIndia,firstfromKumbakonamashisbaseandthen,increasingly,Madras.

Once again, he took to the rails, though hewould often have to depend for histicket on friends and well-wishers. To the English, even first-class seats were abargain.ButforRamanujan,round-triptoMadrasataquarter-annaorsopermileforthecrowdedthird-classcarriagewasworthmorethanaweek’spaytohisfather,theequivalentofmorethanahundredpoundsofrice.

Early during this period, at least, he had no real home, but camped out withfriends.Atonepoint,he showedupat thehouseofa friendbegging foraplace tostayandwasdirectedtoquartershemight sharewithanoldmonk.Forawhile in1910,hestayedwithViswanathaSastri,whomhehadtutoredinKumbakonamandwhowas now a student atPresidencyCollege inMadras.Viswanatha lived at theVictoriaStudentHostelnearthecollege,alargeredandblackbrickstructurewhoseturrets and three stories of brick-columned arches looked as if they had beentransplanted intact from England. Ramanujan joined him there, heading out eachmorninginsearchofstudentstotutor.

But apparentlyhis reputation as a tutorofunworldlybentprecededhim, forhedrewfewstudents.Atnight,ViswanathaSastrirecalledlater,

heusedtobemoanhiswretchedconditioninlife.WhenIencouragedhimbysayingthatbeingendowedwithavaluablegiftheneednotbesorrybutonlyhadtowaitforrecognition,hewouldreplythatmanyagreatmanlikeGalileodiedininquisitionandhislotwouldbetodieinpoverty.ButIcontinuedtoencouragehimthatGod,whoisgreat,wouldsurelyhelphimandheoughtnottogivewaytosorrow.

It was an emotionally fragile period; even so thin a ray of pleasure as the thin,pepperysoup,orrasam,thehostelserved,wouldloomlargeinRamanujan’smemoryacrosstheyears.

Laterin1910andonintothefollowingyear,RamanujanlivedonVenkatanarayanLane,inaneighborhoodcalledParkTownneartheredbuildingsofCentralRailwayStation.This timehe lived on the sufferance of two oldKumbakonam friends,K.NarasimhaIyengarandhisbrother,K.SarangapaniIyengar(whomhe’dapparentlyforgivenforscoringhigherthanheonthearithmeticexamtenyearsbefore).BackinKumbakonam, the brothers had sometimes footed the bill for his clothes and railfares.Now,theywerehelpinghimagain.

Narasimhawas a student atMadrasChristianCollege, a school runbyScottishmissionaries, and Ramanujan tutored him in math. As the F.A. examinationapproached,Narasimha,whowasnomathematician,becamenervousanddepressed,evenweighedskippingitaltogether.Onthedayoftheexam,RamanujanwalkedthefourmilesfromParkTowntoPresidencyCollege,whereitwasbeingheld.There,helocated his friend, convinced him to take the exam, gave him a little pep talk, andsuppliedafewlast-minutetips.Whateverhesaidworked,ifonlybarely:Narasimhasqueakedbywiththelowestpassingscore.

OnedayprobablysoonafterthisRamanujanappeared,deliveredbyhorsecart,atthedoorstepofhis friend fromPachaiyappa’s days,R.Radhakrishna Iyer.Hewassick again, perhaps suffering the effects of his operation earlier that year.Radhakrishna took him in, saw that he was properly fed, and called in a doctor.

Ramanujan,advisedDr.Narayanaswami,neededconstantnursing.SoRadhakrishnatookRamanujantoBeachStation,neartheharbor,andputhimonthetrainbacktohis family in Kumbakonam. But before he left, in a moment that Radhakrishnawouldrememberalways,Ramanujan turned tohimand said, “If Idie,pleasehandtheseovertoProfessorSingaraveluMudaliar[fromPachaiyappa’s]ortotheBritishprofessor,EdwardB.Ross,oftheMadrasChristianCollege,”towhomhe’drecentlybeenintroduced.

And with that, Ramanujan handed him two large notebooks stuffed withmathematics.

•••

Ramanujan’s notebooks were no longer for him merely a private record of hismathematicalthought.Astheprecedingincidentsuggests,theywerehislegacy.Andtheywere a sellingdocument, his ticket to a job—“evidence,” as hisEnglish friendNevillewouldlaterputit,“thathewasnottheincorrigibleidlerhisfailuresseemedtoimply.” Propelled by necessity, he had begun calling on influential men who, hethought, couldgivehima job.Andslungunderhis armashe calledwere—just asphotographershavetheirportfolios,orsalesmentheirdisplaycases—hisnotebooks.Ramanujan had become, in the year and a half since hismarriage, a door-to-doorsalesman.Hisproductwashimself.

In India more than elsewhere, it wasn’t elaborate correspondence and formalapplicationtoanonymousbureaucratsthatgotyouahearingorlandedyouajob,butpersonal connections to someone at the top. Armed with an introduction from afriend of the family, or the family of a friend, you’d plunk yourself down at hisdoorstep. The physical blurring of the line between inside and outside in SouthIndianhomeswasmatchedby the permeability ofSouth Indian social life; privateand public realms were not so rigidly walled off as in the West. Often, you’d beadmittedintotheGreatMan’spresence.Nodaywassocrowded,itseemed,notimesosqueezed,thatitcouldn’taccommodateonemorejob-seeker.

Ramanujan’srefrainwasalwaysthesame—thathisparentshadmadehimmarry,that now he needed a job, that he had no degree but that he’d been conductingmathematicalresearchesonhisown.Andhere...well,whydidn’tthegoodsirjustexaminehisnotebooks.

Hisnotebookswerehissolecredential inasocietywhere,evenmorethan intheWest,credentialsmattered;whereacademicdegreesusuallyappearedonletterheadsandwerementionedaspartofany introduction;where,whentheywerenot,you’d

take care to slip them into the conversation. “Like regimentswe have to carry ourdrums,andtambourinageisasessentialathingtothemarchofourcareersasitistothemarchofsoldiersintheWest,”IndiannovelistandcriticNiradC.Chaudhurihaswrittenofhiscountrymen’sbentforself-promotion.“Inoursociety,amanisalwayswhathisdesignationmakeshim.”Ramanujan’s onlydesignationswereunemployed,and flunk-out. Without his B.A., one prominent mathematics professor told himstraightout,hewouldsimplyneveramounttoanything.

Ramanujan,then,hadthetoughestsortofsellingjob.Buthebroughttoitqualitiesthat, as he hawked his wares across South India, brought him a warm reception.Peoplelikedhim.

•••

“Hewassofriendlyandgregarious,”onewhoknewhimlaterinMadraswouldsayofhim.Hewas“alwayssofulloffun,everpunningonTamilandEnglishwords,tellingjokes,sometimeslongstories,andgoingintofitsoflaughterwhenrelatingthem.Histuftwouldcomeundoneandhewouldtrytoknotitbackashecontinuedtotellthestory.” Sometimes he’d start laughing before reaching the punch line, garble itstelling,andhavetorepeatit.“Hewassofulloflifeandhiseyesweremischievousandsparkling....Hecouldtalkonanysubject.Itwashardnottolikehim.”

NotthatRamanujanwasthehail-fellow-well-mettype.Moreoftenheseemedshy,his geniality emerging more among a few friends than in a crowd. Nor was heparticularly attuned to interpersonal nuance. More than one otherwise fondreminiscenceislikethatofN.HariRao,acollegeclassmatefromKumbakonam,whovisitedRamanujaninMadrasduringthisperiod.“Hewouldopenhisnotebooksandexplain tome intricate theoremsand formulaewithout in the least suspecting thatthey were beyondmy understanding or knowledge.”He just didn’t pick up. OnceRamanujanwaslostinmathematics,theotherpersonwasasgoodasgone.

Andyet,ironically,thissamewantofsocialsensitivityconferredonhimaspeciesof charm.For its flip sidewas an innocence, a sincerity,uponwhich allwhoknewhiminvariablyremarked.

“Ramanujan was such a simple soul that one could never be unfriendly towardhim,” recalled N. Raghunathan, a high school classmate who himself went on tobecomeamathematicsprofessor.Hishumorrantowardtheobvious.Hispunswerecrude.Hisideaofentertainmentwaspuppetshows,orbommalattam;orelsesimplestreet dramas, terrukutu, that ran all night during village festivals and to whichRamanujanwouldgowith friends, cracking jokesand telling stories along theway.

Ramanujan wore his spirits on his sleeve. There was something so direct, sounassuming,sotransparentabouthimthatitmelteddistrust,madeyouwanttolikehim,madeyouwanttohelphim.

InKumbakonam a few years before, an oldwoman from the neighborhoodhadtaken Ramanujan under her wing, often inviting him in for midday snacks. “Sheknewnothingofmathematics,”oneofRamanujan’sIndianbiographerswouldnote.Butitwas“thegleamintheeyesofRamanujanandhistotalabsorptioninsomething—itis these thathadendearedRamanujantoher.”Andthatunstudiedabsorptiondrewotherstohim,too.

Peopledidn’ttaketoRamanujanbecausehewassensitivetothem,orbecausehewasespeciallyconsiderate.Theymightnotunderstandhismathematics.Theymightevenflirtwiththeideathathewasacrank.Andtheymight,intheend,beunabletohelphim.But,somehow,theycouldn’thelpbutlikehim.

•••

Sometimelatein1910,RamanujanboardedanorthboundtrainfromKumbakonamand,abouthalfwaytoMadras,gotoffatVillupuram, justwestofPondicherry, thecoastal city then still in French hands. At Villupuram, he changed trains for thetwenty-mile trip west to Tirukoilur, a town of about nine thousand that washeadquarters of its district. In Tirukoilur, V. Ramaswami Iyer held the midlevelgovernmentpostofdeputycollector.(Iyer,alsospelledAiyar,wasthecastenameofBrahminswhoworshippedSiva,andwasubiquitousinSouthIndia.)

What made Ramaswami especially worth traveling to see was that he was amathematician; in particular, he had recently founded the Indian MathematicalSociety. Everyone called him “Professor,” though he held no academic post. Backwhile a student at Presidency College, it seems, he had contributedmathematicalarticles to theEducationalTimes inEngland. Its editors, assuminghewas a collegeprofessor,addressedhimassuch,andthenamestuck.

Now,asever,Ramanujancamearmedwithhisnotebook.TheProfessorlookedatit. He was a geometer, and the mathematics he saw before him was mostlyunfamiliar.Still,at least in theglowofmemory, “Iwas struckby theextraordinarymathematicalresultscontainedinit.”DidthatmeanhewouldgiveRamanujanajobinthetalukoffice?Hardly.“Ihadnomindtosmotherhisgeniusbyanappointmentinthelowestrungsoftherevenuedepartment,”hewrotelater.Sohesenthimonhisway,withnotesofintroduction,tomathematicalfriendsinMadras.

Oneofthem,achartermemberoftheMathematicalSociety,wasP.V.SeshuIyer,a pinch-faced man with glasses who’d been one of Ramanujan’s professors atGovernment College. Since about 1906, they’d not seen one another. Now, fouryears later,SeshuIyerhadmoveduptoPresidencyCollege inMadras.Ramanujanmet him there, notebooks in hand, but also this time with Ramaswami Iyer’srecommendation.Heleftwithleadsandyetothernotesofintroduction.

He went to see S. Balakrishna Iyer, then himself just starting his career as amathematicslectureratTeachers’CollegeintheMadrassuburbofSaidapet.Wouldhe,Ramanujanasked,recommendhimtohisEnglishboss,acertainDodwell,forajobasaclerk? Itdidn’tmatterhowpoorly itpaid; anythingwoulddo.Balakrishnaserved him coffee, looked at his notebooks, which he didn’t understand, and laterwenttoseeDodwellthreeorfourtimesonRamanujan’sbehalf.Nothingcameofit.“Iwasnotbigenough,”apologizedBalakrishnaIyerlater—notimportantenoughtoexertanyinfluence.

InDecember,RamanujanwenttoseeR.RamachandraRao,whowasindeed“bigenough.”EducatedatMadras’sPresidencyCollege,hehadjoinedtheprovincialcivilservice in1890, at the ageofnineteen, and in time rose tobecome registrar of thecity’sCooperativeCreditSocieties.NowhewasdistrictcollectorofNellore,atownof about thirty-five thousand, a hundred miles up the East Coast Railway fromMadras. Earlier in the year, he had been named “Dewan Bahadur,” which wassomethinglikeaBritishknight.Allthis,andhewasamathematician,too,servingassecretary of the Indian Mathematical Society, the group Ramaswami Iyer hadfounded four years earlier, and even sometimes contributing solutions toproblemsposed in itsJournal. Intelligent,wealthy, andwell connected,R.RamachandraRaowas just thekindofpaternal figure, at thehead of a retinue of family and friends,throughwhoseofficesonegotthingsdoneinIndia.

JusthowRamanujangotanaudiencewithhimisunclear, thoughaccountsagreethat Ramachandra Rao’s nephew, R. Krishna Rao, was the final intermediary.Ramanujan’sfriend,RadhakrishnaIyer,towhomhe’dearliergivenhisnotebooksforsafekeeping, saidhewrote his father-in-law, an engineer inNellore, to arrange themeeting. Seshu Iyer said later that he paved theway.Ramanujan’s English friend,Neville, laterspeculatedthatSeshuIyerdidindeedsupplyRamanujanwitha letterofintroductiontoRamachandraRao—butthatRamanujanwas“tootimid”touseit.Hemay,then,haveneededsomeextrapushtogoandmeetthispowerfulman.Ifso,hegotitfromC.V.Rajagopalachari.

RajagopalachariwasjustafewmonthsolderthanRamanujan,hadgrownupinthesame town, frequented the same temple, attended Town High with him. One

afternoonbackin1902,duringrecess,anolderstudent,saidtobethesmartestinhisclass, handed him amath problem.Ramanujanwas so smart?Well, then, let himsolvethis:

At firstglance fallingunder the familiarheadingof “twosimultaneousequations intwounknowns,”theproblemactuallyconfrontedRamanujanwithadifficultfourth-degree equation and meant recalling a theorem applicable to a particular class ofthem.To any ordinarily smart fourteen-year-old, itwould be exceedingly difficult.“To my astonishment,” Rajagopalachari remembered later, “Ramanujan worked itoutinhalfaminuteandarrivedattheanswerbytwosteps.”

Infact,heprobablydidn’t“workitout”atall,butsimplylookedatit,guessedtheanswermightbeonewhere eachwas a square, tried a coupleofpossibilities inhishead,andsawthesolution,x=9andy=4,jumpoutathim;inotherwords,itwasapiece of fancy footwork, nothing mathematically profound. Still, it impressedRajagopalachari,andheandRamanujanbecamefriends.

Over the years,Rajagopalachari had followed a straight career trajectory towardbecomingalawyer,whileRamanujanfloundered.Thetwolostcontact.Butnow,in1910, almost a decade later, they met again by chance in Madras. Despondent,RamanujantoldRajagopalachariabouthisschoolfailure.Hehadnofuture,hesaid.No one appreciated him. He’d written a famous mathematician in Bombay,ProfessorSaldhana,withsamplesofhiswork.He’dwrittentheIndianMathematicalSociety.Nothinghadcomeofanyofit.So,thankstoafriendwhowassupplyingtheticket,hewastakingatrainbacktoKumbakonamthatverynight.

Don’tgo,saidRajagopalachari.Ramanujanmayhavementionedhehadaletterofintroduction to Ramachandra Rao, but had not yet acted on it. In any case,Rajagopalachari said that he would take him to meet Ramachandra Rao. WhenRamanujanprotestedthathehadnomoneytoremaininMadras,hisfriendsaidhe’dfoottheexpenses.

Themeetingoccurred.RamachandraRaowroteaboutitlater,inthesewords:

Severalyearsago,anephewofmine,perfectlyinnocentofmathematics,spoketome,“Uncle,Ihaveavisitorwhotalksofmathematics;Idonotunderstandhim;canyouseeifthereisanythingtohistalk?”Andintheplenitudeofmymathematicalwisdom, I condescended topermitRamanujan towalk intomypresence.Ashortuncouthfigure,stout,unshaved,notoverclean,withoneconspicuousfeature—shiningeyes—walkedin,withafrayednotebookunderhisarm.

Three times, according toRajagopalachari, Ramanujanmetwith the greatman.Thefirsttime,RamachandraRaoaskedtokeepRamanujan’spapersafewdays.Thesecondtime,havingperusedthem,hesaidhe’dneverseenanythinglikeRamanujan’stheorems,butsincehecouldmakenothingofthem,hehopedtheywouldnottroublehimagain.Theydid,ofcourse,sonow,onthisthirdoccasion,RamachandraRaoputthingsmore plainly. Perhaps Ramanujan was sincere, he allowed; but if nomoralfraud,hewasmorethan likelyan intellectualone.Inotherwords,hedoubtedthatRamanujanknewwhathewastalkingabout.

As the two friends left, Ramanujan mentioned that with him he had hiscorrespondence with Professor Saldhana, the eminent Bombay mathematician.Saldhana, too,hadconcluded thathecouldn’thelphim.ButmanyofRamanujan’sformulas,he’dwritteninthemarginsofthesheetofpaperRamanujanhadsenthim,seemed intriguing indeed. Itwas just thathe couldhardly throw theweight of hisreputationbehindsomeoneworkinginareassounfamiliartohim.

Thiswashardlyaringingendorsement;indeeditdifferedonlyslightlyfromwhatRamanujan would hear all through his early years—that his work was not wellenough understood to classify as either the fulminations of a crank or theoutpouringsofagenius.RamachandraRaohimself,insomanywords,hadsaidthat;dubious,he’derredonthesideofcaution,anddecidednottotakeupRamanujan’scase.ButSaldhana,erringevenfurtheronthesideofcaution,hadatleastmadeclearthat,whateverelsehewas,Ramanujanwasnocrank.

That was enough for the tenacious Rajagopalachari, who saw in Saldhana’scommentsawaytoallayRamachandraRao’sdoubts.Backtheywent—onsofineaknife edgedidRamanujan’s fate hinge—a fourth time.At first,RamachandraRaowasangry.Hereagain?Justafewminuteslater?ButthenhewasshowntheSaldhanacorrespondence, as well as some of Ramanujan’s easier, more accessible results.“These,”hewrotelater,“transcendedexistingbooksandIhadnodoubtthathewasaremarkable man. Then, step by step he led me to elliptic integrals, andhypergeometricseries.Atlast,histheoryofdivergentseries,notyetannouncedtotheworld,convertedme.Iaskedhimwhathewanted.”

Whathewanted,Ramanujan replied,was apittanceonwhich to live andwork.Or,asRamachandraRaolaterputit,“Hewantedleisure,inotherwords,simplefoodtobeprovidedtohimwithoutexertiononhispart,andthatheshouldbeallowedtodreamon.”

3.“LEISURE”INMADRAS

Hewantedleisure...

ThewordleisurehasundergoneashiftsincethetimeRamachandraRaouseditinthiscontext.Today,inphraseslikeleisureactivityorleisuresuit,itimpliesrecreationor play. But the word actually goes back to the Middle English leisour, meaningfreedom or opportunity. And as the Oxford English Dictionary makes clear, it’sfreedomnotfrombut“todosomethingspecifiedorimplied”[emphasisadded].Thus,E.T.Bellwritesofa famousseventeenth-centuryFrenchmathematician,PierredeFermat, thathe found in theKing’s service “plenty of leisure”—leisure, that is, formathematics.

So it was with Ramanujan. It was not self-indulgence that fueled his quest forleisure; rather, he sought freedom to employ his gifts. In his Report on Canara,MalabarandCededDistricts,Thackerayspokeofthe“leisure,independenceandhighideals”thathadpropelledBritaintoitsculturalheights.TheEuropean“gentlemanofleisure,”freefromtheneedtoearnalivelihood,presumablychanneledhistimeandenergyintohighermoralandintellectualrealms.Ramanujandidnotbelongtosuchanaristocracyofbirth,butheclaimedmembershipinanaristocracyoftheintellect.Inseeking“leisure,”hesoughtnothingmorethanwhatthousandsborntoelitestatusaroundtheworldtookastheirdue.

And remarkably—ina testament tohis stubbornness asmuchashisbrains—hefoundit.

ThathewasaBrahminprobablyhelped.Ramanujanwaspoor,fromafamilythatsometimeslackedenoughtoeat.But inIndia,economicclasscounted for less thancaste.Being aBrahmin gave him access to circles otherwise closed to him. In fact,virtually all those whom Ramanujan met during these years were Brahmins.RamaswamiIyerwasaBrahmin.SowasSeshuIyer.SowasRamachandraRao.HadRamanujanbeenofanothercaste,hemightlikewisehavesought,andreceived,helpfrom wealthy and influential castemen. But in no other caste did prestige,connections,andatasteforthelifeofthemindmergesonaturallyastheydidamongBrahmins.

AsaBrahmin,Ramanujanmayalsohavefeltfreertoseekthesortofconstructiveidlenesshethoughtheneeded—andperhapseven,insomemeasure,conceivedashisdue.Traditionally,Brahminswererecipientsofalmsandtemplesacrifices;earningalivelihood was for them never quite the high and urgent calling it was for others.Uncharitably, it might be said that Ramanujan exhibited a prima donna-like self-importancethatlefthimunwillingtostudywhathehadnowishtostudy,ortoworkforanyreasonbuttosupporthismathematics.Lessharshly—and,onbalance,withgreaterjustice—hewasasecularsanyasi.

•••

RamachandraRaosentRamanujanbacktoSeshuIyer,sayingitwouldbecrueltolethimrotinabackwaterlikeNellore.No,hewouldnotgivehimajobinthelocaltalukofficebutratherwouldseekforhimsomescholarshiptowhich,despitehispenchantforfailingexaminations,hemightbeeligible.Meanwhile,lethimstayinMadras;he,RamachandraRao,wouldpayhisway.

Monthly,fromthenon,Ramanujanbeganreceivingamoneyorderfortwenty-fiverupees. It wasn’t much. But it was enough to free him from economic cares. Lifeopenedup forhim.Now,moredecisively thanbefore,he left theKumbakonamofhisyouthbehindand,fromearly1911andforthenextthreeyears,steppedintothewiderworldofSouthIndia’scapital,Madras.

•••

Itwas the fifth-largest city in theBritishEmpire and, afterCalcutta andBombay,third-largestonthesubcontinent.Sometraceditsnametothelegendofafishermannamed Madarasen; others to a corruption of Mandarajya, meaning realm of thestupid, or even Madre de Dios, Portuguese for mother of God. The city itself,however,wasaninventionofBritishcolonialpolicy.TheBritishEastIndiaCompanybought land at themouth of the CooumRiver, and Fort St. George, which theyconstructedtherein1642,becametheadministrativehuboftheBritishpresenceinSouthIndia.

Madraswasnotacompactcity.The550,000peoplewhoinhabiteditin1910werespreadup and down along theBay of Bengal formiles, dispersed in quite distinctpopulationcenters—Georgetown,Triplicane,Mylapore,Chepauk,andothers.Manyof these placeswent back hundreds or thousands of years.Three and a halfmilessouthof the ragged centerof town, for example,wasMylapore, site of the reveredKapalaswaraTemple. There, St. Thomas the Apostle, patron saint of India, hadsettled in the first centuryA.D.But the areawasknown to the ancientGreeks andRomans,asaport,longbeforethat.

ThemoderncityofMadrasslunglowovertheland,onlytheoccasionalgopuramofathousand-year-oldtemplepunctuatingtheflatness;nopartofthecityrosemorethanfiftyfeetabovesealevel.Spreadallacrossit,especiallyatthesitesofoldvillages,were clusters of “hutments,” one-room dwellings of mud and thatch, tens ofthousands of them. But even the more substantial structures with red tile roofsalmostneverrosehigherthanthesecondfloor.Overtheyears,thecityhadexpandedhorizontally,notvertically;you’daddanextensiontothefrontorbackofthehouse

rather thanbuild another story.Madras, then,wasmore like a leisurely, sprawlingPhoenixorSanDiegothanarestless,denselypackedNewYork.

Therewerestilllargeruraltractswithinthecity,withpalmtreesandpaddyfields,buffalo and washermen in rivers and lagoons, fishermen’s thatched huts andcatamarans idledon thebeach.Save fora fewmorecrowdeddistricts, thecrushofpeople squeezed onto every square inch that theWesterner today associates withIndiancitieswasstillinthefuture.Thecityretainedaneasygoingvillageslowness.

Itwaspossibletogazedownfromthetopofthelighthouseoverlookingtheharborandnote,asoneEnglishvisitordidaroundtheturnofthecentury,that

Madrasismorelostingreenthanthegreenestcityfurthernorth.UnderyourfeettheredhuddledroofsoftheBlackTown[theadjacentnativequarter]areonlyaspeck.Ononesideisthebosomoftheturquoisesea,thewhite lineofsurf, the leaguesofbroad,empty,yellowbeach;on theother, the forestofEuropeanMadras,dense,round-polledgreenrollingawaysouthwardsandinlandtillyoucanhardlyseewhereitpassesintothepalergreenofthefields.

ThatwasaEuropeanperspective,ofcourse.ButamongIndians,too,Madraswasregardedasslowerandmorecongenial,greenerandmorespaciousthanaCalcuttaorBombay. It was hard being poor anywhere in India. But it was a little easier inMadras.Therewasneverthecoldtobear.AndbeingsoremovedfromtheNorth,somucharegionalcapital,somuchSouthIndian,thecityfeltcomfortableandfamiliarto the thousands who, like Ramanujan, had moved to it from towns and villagesacrosstheSouth.

•••

In May 1911, Ramanujan left the place he shared on Venkatanarayan Lane andmoved toa little alleyboardinghouse, onSwamiPillaiStreet, bearing the inflatedname“SummerHouse.”Therehe lived for the rest of the year andmuchof 1912with close to a dozen others, mostly students, who frequented a Brahmin-runrestaurant on Pycroft’s Road, the main street of a neighborhood known asTriplicane.

Afewminutes’walkdownPycroft’s,rightbesidePresidencyCollege,laythebeach.EventhenitwasaMadraslandmark,aplaceanyonewho’dvisitedthecityforevenafewdaysalwaysremembered.Itwasnotjustabeach,butafreakofnature,asweepofsand piled up by the roaring surf over the eons, that then had been refined,manicured, and developed by an otherwise obscure Madras governor, oneMountstuartGeorgeGrant-Duff,backinthe1880s.Attheendofthelongsloping

sand, the breakers rumbled. Yet so deep was the beach that, having once steppedontoit,itwasasifyoustillhadagreatdeserttocrosstoreachthem.

It was here that Ramanujan would come, letting his mathematical ruminationspercolate as he strolled along by the sea. Or else, come the cooler hours of theevening,hewouldcomewithhisfriends,plunkinghimselfdownonthelightbrownsand,fleckedwithtinyfragmentsofseashell,andspinoccultstoriesuntil longafterdark.

Therewasanopennessouthere,awayfromthehot,dustystreetsofTriplicane,adelightfulcoolness.Lookinginland,RamanujancouldseethedomedclocktowerofPresidencyCollege,madegoldbythesettingsun.Lookingouttosea,hecouldspymerchantvessels—distantgray shapes, andothers, closer to shore, all cargoboomsandbrightpaint—plyingtheirwayupthecoastfromColombo,inCeylon,andfromaroundthesoutherntipofIndia,boundforMadras.

Lightened by the load Ramachandra Rao’s generosity had lifted from hisshoulders,Ramanujanwashappy,orsomethingclosetoit.Now,afterthatanxious,gropingtwoyearsfollowinghismarriage,hewassurroundedbyfriends,doingwhathelikedtodo,carefreeandcheerful.C.R.KrishnaswamiIyer,who’dknownhimatPachaiyappa’s and now shared a room with him in SummerHouse, rememberedhowonceRamanujanstayedupexclaimingonastronomicalwonderstilllateintothenight. Finally, Krishnaswami’s cousin, his sleep shattered by Ramanujan’smonologue, poured a pot ofwater over him; that would cool his fevered brain, hesaid. But Ramanujan took it all in stride. Ah, yes, a refreshing Gangasnanam—apurgingbathintheRiverGanges;couldhehaveanother?

1911wasagoodandhopefulyear.ItwastheyearthecapitalofIndiawasshifted,with great pomp and ceremony, from Calcutta to Delhi. The year a new sewersystem, complete with underground conduits, sand filters, and pumps, was beinginstalledinMadras.Theyearitsoil-litstreetsbegantogivewaytoelectricity.AnditwastheyearSrinivasaRamanujan’sfirstpaperappeared,intheJournaloftheIndianMathematical Society—representing his initial step onto the stage of Indianmathematics,andtheworld’s.

4.JACOBBERNOULLIANDHISNUMBERS

Five years before, in late 1906, several dozen professors at colleges in Madras,Mysore, Coimbatore, and elsewhere in South India received a letter from V.Ramaswami Iyer, in which he proposed the formation of a mathematical society.Behind the idea lay simplewant. Just asRamanujanhad sodependedonwhateverfewmathematicalbookshadcomehisway, sodidIndianmathematiciansgenerally

suffer a lack of books and journals from Europe and America. The society, inRamaswami’sconception,wouldsubscribetojournalsandbuybooks,thencirculatethem to members. Twenty-five rupees per year from even half a dozen memberswouldbeenoughtogetthesocietyofftheground.

Hewoundupwith20foundingmembers,allhungryformathematicalfellowship,andwhat was known first as the Analytical Club, then the Indian MathematicalSociety,was born. Soon itwas publishing a journal of its own. Just a dozen yearslater, at its second conference in Bombay, it would claim 197 members and becirculating35EuropeanandAmericanjournals.

Theseeventsawaitedmoderntimes.ButathousandyearsbeforetheBritishcame,Indiansweredoingmathematics.Beforetheseventhcentury,whiletheWestwasstillmired in awkward Roman numerals, India had introduced the numerals we usetoday.Thezero,asymbolexpressingnothingness,representedaparticulartriumph;itmaygobacktoasearlyasthesecondcenturyB.C.butdefinitelyappearedinabookinthethirdcenturyandonthewallofatemplenearGwalior,incentralIndia,intheninth(whereithelpedspecifyaflowergardenas270unitslong).

ManyofIndia’scontributionstomathematicswerespurredbytheneedtoknow,based on astronomical factors, the correct times for Vedic ceremonies. Algebra,geometry,andtrigonometrywerealltherebyenriched.FigureslikeAryabhata,bornin A.D. 476, who established one of the earliest and best values for π, andBrahmagupta,150yearslater,lefttheoremsevennowassociatedwiththeirnames.

Itwasa rich tradition,butonequitedifferent fromthatofGreece, thecradleofWestern mathematics.Whereas the Greeks, especially Euclid, emphasized formalproof,asinthestep-by-stepprocesshighschoolstudentsfirstencounteringeometry,Indianmathematicsstressedtheresultsthemselves,howeverobtained.Andwithoutthat winnowing out of mathematical dross that formal proof achieved, Indianmathematicswaswildlyuneven;someofitwasjustplainwrong.OneMuslimwriternotedinabookaboutIndiathatHindumathematicswas“amixtureofpearlshellsandsourdates...ofcostlycrystalandcommonpebbles.”

By the twentieth century, thepearl shells andcrystalhad long lainburied in thedustoftime.Forcenturies,Indiahadstooditsmathematicalgroundagainsttherestof the world. But now, that was ancient history; of late it had added little to theworld’smathematicaltreasure.OnlyalineofbrilliantmathematiciansinKerala,onthe subcontinent’s southwest tip,broke thegloomthatotherwise extendedback tothe great Bhaskara of the twelfth century. The birth of theMathematical Societycould not ensure a rebirth. But its founders—hungry to connect with the West,

proudoftheircountry’sheritageyetsoberlyawarethatreverenceforthepastwasnosubstituteforpresentachievement—surelyhopeditdid.

It was into this nascent newworld thatRamanujan “came out,” as it were, as amathematician in 1911. He had met Ramaswami Iyer, the society’s founder, theprevious year when, in search of a job, he had traveled to Tirukoilur. NowRamanujan’sworkwas appearing in volume 3 ofRamaswami Iyer’s new Journal—which, like most mathematics publications, opened its pages to provocative orentertainingproblemsfromitsreaders.

OneoftwoproblemsRamanujanposed,asquestion289,simplyaskedthereadertoevaluate

Seeminglystraightforwardarithmetic,withnotsomuchasanxorytocomplicateit?Well, three issues of the Journal came and went—six months—with no solutionoffered; in the end, Ramanujan supplied it himself. The problem was those threelittledots, indicatingthatthenestingofsquareroots,andthesequenceofnumbersbegun,wastocontinueadinfinitum.

Ramanujan had generated the problem years before in the form of an exampleillustrating amore general theorem. The fourth equation down in chapter 12, onpage105ofhisfirstnotebook,read:

Break any number into three components,x,n, anda, the equation said, and youcouldrepresentitintheformofthoseendlesslynestingsquareroots.Forexample,3couldbeimaginedas(x+n+a)inwhichx=2,n=1,anda=0.PlugthosevaluesintotheequationandyouwindupwithjustwhatRamanujanhad,inquestion289,askedtohaveevaluated.Theanswer,inotherwords,wasjustplain3.Ofcourse,howyou’dfigureitoutwithoutRamanujan’sequationwasscarcelyobvious.

In its small way, Ramanujan’s deceptively difficult problem rose up frommathematical terrain that had fascinated him for as long as he had worked inmathematics.Asuperficiallysimilarproblemmighthaveaskedthereadertotakethenestedsquarerootsonlysofar,outto,say,10,or100,or1000.Thatway,youcouldpluginthenumbers,runthroughthecomputation,andbefinished.Butthiswasn’ttheproblemRamanujanposed.Heasked,whathappenedifyouwereneverfinished?Whatifthenumberofnestedsquarerootswasinfinite?

This was, in a sense, a contradiction in terms: how could any number “equal”infinity? Infinitywasnoplaceyoucould reach,noquantityyoucouldplug intoanequation; there was no “last number.” So to understand how a mathematicalexpressionbehaved“at”infinitywastoexploreanelusiveandmysteriousterrainoutbeyondallseeing.

Andnooneexploredthisterrainmoreardently,orknewitmoreintimately,thanRamanujan.

Many kinds ofmathematical processes can be ordered to proceed ad infinitum.Ramanujan’sproblemintheMathematicalSocietyJournalwasbuiltaround“nestedradicals”—squarerootsofsquarerootsofsquarerootsof . . . ,anarealittlestudiedthenornow.ButRamanujanalsostudiedcontinuedfractions—fractionsoffractionsoffractionsof...Mostofall,heexploredinfiniteseries,whichappearedonvirtuallyeverypageofhisnotebooksandwould fairly litterhis first substantialpaper in theJournal later in 1911. “Infinite series,” one mathematician has written, “wereRamanujan’sfirstlove.”

Thetelltalethreedotsheraldingtheirpresenceshowedupearlyinhisnotebooks,thoughmoreoftenhesimplywrote“&c,”whichmeantthesamething:numbersoralgebraicterms,followingsomeparticularpattern,weretobeaddedtooneanotherforever.Thus,

1+2+3+...

clearlysuggeststhatthenexttermistobe4,then5,andsoon.Of course, this isnot very interestingbecause,here, an infinitenumberof terms

justaddsuptoinfinity.Whatmakesinfiniteseriessointriguing,sovaluable,andtheobjectofsomuchstudy,iswhentheydon’tbuildupwithoutbound,whentheyaddup to something finite. As mathematicians put it, the series “converges” to aparticularvalue.Forexample,

1+1/2+1/4+1/8+...

Here,thenexttermis1/16,thenext1/32,andsoon.Andthecuriousthing,knowneventotheGreeks,isthateventhoughyouaddtermsforever,eachtermdiminishesso rapidly from the preceding one that the sum, even after an infinite number ofterms,isaquitemanageable2—thevaluetowhichtheseriesissaidtoconverge.Themoretermsyouadd,thecloseryougetto2.

But just because each successive term in a series is smaller than the one beforedoesn’tmeanitconverges.Forexample,

1+1/2+1/3+1/4+...

issuperficiallysimilartotheearlierconvergentseries,butdoesn’tconverge;goas faroutintotheseriesasyoulike,butsosoonasyouthinkit’saddinguptosomething,moretermsalwaystakeyoubeyondit.Forexample,isthesumofthisseriesperhaps2,asitwasforthepreviousone?No—fourtermsareenoughtoexceedit.Maybe3?Here it takes a little longer, but already by the eleventh term, you’ve passed it.Perhaps10?Twelvethousand,threehundredninetytermsadduptomorethan10.Itturnsout—andcanbeproven—thatwhatevernumberyoupick,thesameholds:Thesumoftheseriesisinfinite;itdoesn’tconverge.

Theseriesofinteresttomathematicians,then,arethosethatdoconverge,orthatconvergeundercertaincircumstances.Andit’swhattheyconvergetothatisoftenthewonderfulthingaboutthem.

Take, for example, the trigonometric functions thatmany remember from highschool,whichteacherspracticallyalwaysfirstintroduceastheratiosofthelegsofaright triangle. These functions—named sine, cosine, tangent, and so on—are allcomputedbytakingthelengthofoneoranothersideofarighttriangleanddividingitby the lengthofanother.Thebacksof trigonometrybooksare stuffedwith longlistsofangleswiththecorrespondingvaluesoftheirtrigfunctions.Givemeanangle,thetablessay,andI’llgiveyouthevalue,forexample,ofthesineofthatangle.Theangleis30degrees?Itssineis.5000.Andsoon.Suchtablesinhand,navigatorscrossoceans,engineersdesignmachines.

Andyet these same trigonometric functions,historically rooted in right trianglesand ratios, can be evaluated in away seemingly unrelated—as the sums of infiniteseries.If,say,theangleθ(theta,theGreekletter)isexpressednotindegreesbutinanothermeasureofangularity thatmathematicians findmoreconvenient (radians),then:

(Here,5!,read“factorialfive,”justmeans5×4×3×2×1=120.)Wantthesineof30degrees?Justplugintotheequationitsradianequivalent(π/6,

orabout.5236),andaddupasmanytermsasyouwanttogetavalueasaccurateasyouwant.Here,eventhreetermsareenoughtogetyouto.500002—quiteclosetothecorrect.500000;theseriesconvergesrapidly.

Thus, this alternating infinite series—addingabithere, subtracting something alittle smaller there, and so on through an infinite number of terms—inexplicably

equalsjustwhatyougetfromdividingonelegofatrianglebyanother.Just this sort of seemingly unexpected connection shows up all the time with

infinite series, which is what has made them so attractive to mathematicians,Ramanujanmost particularly.Bernoulli numbers, the subject of his first publishedpaper, were defined in terms of infinite series. And every page of his paper wasriddledwithmoreofthem.

Jacob Bernoulli was among the first in a line of eminent seventeenth-andeighteenth-century mathematicians derived from a merchant family that had fledanti-ProtestantmassacresinAntwerpandsettledinSwitzerland.Hehelpedextendcalculus,thepowerfulsetofmathematicaltoolsfordealingwithcontinuouslyvaryingquantities, beyond the point that Germany’s Gottfried von Leibniz, along withEngland’s Sir IsaacNewton, had taken it two decades before. Along the way, hederivedthenumbersthathavesincebornehisname.

Bernoullinumbersareintimatelytiedtothequantityewhich,likeπ,isanumberwhosespecialpropertiesmakeitubiquitousinmathematics.Itisdefinedas

Now,whenaparticularalgebraicexpressioninvolvingeisexpressedasaninfiniteseries,thecoefficientsofeachtermturnouttohavespecialsignificance.(Coefficientsare just the ordinary numbers by which the algebraic parts are multiplied; in theequation3x+1/2x

2=12,3and1/2arethecoefficients.)ThesecoefficientsweretheBernoullinumbers,whichfirstappearedinhisbookArsConjectandi,publishedafterhis death in 1713. Notational inconsistencies confuse matters some, but in onesystemthefirstfewBernoullinumbers,genericallyBn,areB1=−1/2,B2=1/6,B4=−1/30,B6=1/42.(Theodd-numberedones,exceptforthefirst,areallzero.)

RamanujanhadstumbledonBernoullinumbersforthefirsttimeabouteightyearsbefore, though probably without having ever heard of them as such. The secondvolume of Carr’s Synopsis contained references to them in various guises, butRamanujanmaynothaveseenituntil1904,whenhewasatGovernmentCollege—ayear after he apparently beganworkingwith them. In any case, he’d worked withthem ever since, using them repeatedly, through the Euler-Maclaurin summationformula, to approximate the values of mathematical entities known as “definiteintegrals”(unrelatedto“integers”)incalculus.Pages30and31ofthefirstnotebookcitedthem.Sodidmuchofchapter5ofhissecond.

NowRamanujanwasmakingthemthesubjectofhisfirst formalcontributiontothemathematical literature.Children take theSalkor theSabinvaccine toprotect

them against polio. Supersonic aircraft exceedMach 1, the speed of sound in themeasurement system named for Austrian physicist Ernst Mach. In science andmedicine, immortality ishaving something—a treatment, aunitofmeasurement, atheory—namedafteryou.So,too,inmathematics.Bernoullinumbersbearhisnamebecausetheyappearagainandagain inawidevarietyofmathematicalapplications.Theyweren’t justflukesofmathematicalnature,meaninglesschainsofdigits; therewererelationshipsamongthem,andRamanujanhaddiscovered—or,insomecases,rediscovered—whatsomeofthemwere.

“SomePropertiesofBernoulli’sNumbers,”hecalledhispaper,and itwasanapttitle. The physical properties of a metal, like its melting point or specific gravity,appear in any chemical handbook; Ramanujan was discovering mathematicalproperties of these numbers. Bernoulli numbers were expressed as fractions; forexample, . Well, Ramanujan found that the denominators (thebottomparts)ofthosefractionswerealwaysdivisibleby6.HefoundalternativewaysofcalculatingBnbasedonearlierBernoullinumbers.Thesixthofeighteennumberedsectionsbegan:

6.Itwillbeobservedthatifnisevenbutnotequaltozero,

(i)Bnisafractionandthenumeratorof initslowesttermsisaprimenumber,

(ii)thedenominatorofBncontainseachofthefactors2and3onceandonlyonce,

(iii) isanintegerandconsequently2(2n−1)Bnisanoddinteger.

OnandonRamanujan’spaperwentlikethat,fillingseventeenpagesoftheJournal.By one reckoning, it stated eight theorems, offering proofs, of a sort, for three ofthem; two were stated as corollaries of two other theorems, three more as mereconjectures.

Ramanujan’smanuscripthadproblemswhenitfirstreachedtheeditor’sdesk.“Mr.Ramanujan’s methods were so terse and novel and his presentation so lacking inclearnessandprecision,”itwouldlaterbeobserved,“thattheordinary[mathematical]reader, unaccustomed to such intellectual gymnastics, couldhardly followhim.” Inother words, his paper was a mess. And this was written by a champion ofRamanujan’swork.M.T.Narayana Iyengar, amath professor atCentralCollege,Bangalore and the Journal’s editor during the early years, confessed later “that theeditor’sworkinconnectionwithRamanujan’scontributionswasbynomeanslight,”andthatthemanuscriptwentbackandforthbetweenhimanditsauthorthreetimes.

Inthisfirstpaper,asallthroughhiswork,Ramanujanfoundconnectionsbetweenthings that seemedunconnected.Othermathematicianswould later provemost of

themtrue;Ramanujan,though,eithercouldn’tbebothered,ordidn’tseetheneedto.Whatproofshedidofferweresketchyorincomplete.

A testament to the influence on him of George Shoobridge Carr? That’s whatmostscholarslaterconcluded.Carr,writingasynopsisofresultsratherthanmakingoriginalcontributions,hadgivenproofs,wherehedidsoatall,onlyinbareoutline.NowRamanujan,whowasmakingoriginal contributions, clung to thepattern.Hehadasserted,forexample,thatthenumeratorofthenthBernoullinumberdividedbyn was always a prime number. Proof? Not a shred. Another mathematician laterobserved,“Hetakesthenumericalevidenceassufficient,andthereisnotraceofanysuggestionthatthereisneedofotherproof.”WhetherRamanujancaredaboutproofis debatable; that normally he didn’t furnish it, sometimes offering the mostprovocativeresultswithoutascintillaofevidencetosupportthem,isnot.

In this particular case, as it happens, Ramanujan was wrong. For example, thenumeratorofB20/20=174611,whichisnotaprimenumberatall,asheclaimed,butequalto283×617.

This,though,wastheexceptiontoprovetherule;muchmoreoften,Ramanujan’strust in himself was wholly justified. In his paper on Bernoulli numbers, in thenotebooks, inhismathematical correspondence, inhis otherpublishedpapers—hewas,withastoundingconsistency,right.

5.THEPORTTRUST

AppearingintheJournalof theIndianMathematicalSociety,Ramanujanwason theworld’s mathematical map at last, if tucked into an obscure corner of it. He wasstartingtobenoticed.

Earlythefollowingyear,K.S.Srinivasan,astudentatMadrasChristianCollegewho’dknownRamanujanbackinKumbakonam,droppedbytoseehimatSummerHouse.

“Ramanju,”hesaid,“theycallyouagenius.”Hardly a genius, repliedRamanujan, “Look atmy elbow.Thatwill tell you the

story.” It was rough, dirty, and black.Working from his large slate, he found thequickflipbetweenwritinghandanderasingelbowa lotfaster,whenhewascaughtupinthethroesofhiswork,thanreachingforarag.“Myelbowismakingageniusofme,”hesaid.

Why, Srinivasan asked, didn’t he just use paper? Can’t afford it, repliedRamanujan.Hewasgettingmoney fromRamachandraRao.But thatonlywent sofar.Paper?He’dneedfourreamsofitamonth.

Another friend from the Summer House days, N. Ramaswami Iyer [norelationshiptothe“Professor”]alsorecalledRamanujan’s“hugeappetite”forpaper.Ramaswami pictured him lying on a mat, his shirt torn, “his long hair carelesslyboundupwithapieceofthinstring,”workingfeverishly,notebooksandloosesheetsofplainwhitepaperpileduparoundhim.AfriendfromPachaiyappa’swhomethimin Madras a little later, T. Srinivasacharya, recalled that, for want of paper,Ramanujanwouldsometimeswriteinredinkonpaperalreadywrittenupon.

It was during this period that, apparently worried something might happen toRamanujan’s notebook, Ramachandra Rao prevailed on him to copy it over.Ramanujan did so, though not without revising and expanding it as he did,incorporating thenotes appended to it into appropriate sectionsof thenewone—whichcomesdowntoustodayasthe“second”Notebook.

Half a century after Ramanujan was dead, in one of themanymemorial bookshonoringhim,onesponsorwouldbeamanufacturerofwritingandprintingpapersinErode,Ramanujan’splaceofbirth.“Paper,TheGreatImmortalizer,”itsone-pageadwas headed. “GoodPaper,” itwent on, “has helped preserve and propagate thegreatthoughtsofMan.”Itwouldbeafittingtribute.

•••

For about a year, Ramanujan lived on Ramachandra Rao’s generosity. He wasmathematically productive, peppering the Mathematical Society Journal with oneinterestingnewproblemafteranother,andcompletingasecondpaper.Buthewas,afterall,unemployed,andthisgrewtobotherhim.Notlongbefore,throughoneofhis patrons, Ramanujan got a temporary job in theMadras AccountantGeneral’sOffice,makingtwentyrupeespermonth,buthelditonlyafewweeks.Now,earlyin1912, Ramachandra Rao had turned to others among his influential friends, andRamanujanwasapplyingforanewjob:

Sir,Iunderstandthereisaclerkshipvacantinyouroffice,andIbegtoapplyforthesame.Ihavepassedthe

MatriculationExaminationandstudieduptotheF.A.butwaspreventedfrompursuingmystudiesfurtherowing to several untoward circumstances. I have, however, beendevoting allmy time toMathematics anddevelopingthesubject.IcansayIamquiteconfidentIcando justice tomywork if Iamappointedtothepost.Ithereforebegtorequestthatyouwillbegoodenoughtoconfertheappointmentonme.

Ibegtoremain,Sir,

Yourmostobedientservant,S.Ramanujan

Ramanujan’s letterwaswritten in aneat, schoolboyhandunremarkable in everywaysavefort’swhosehorizontalarmsrarelyintersectedtheverticalstemstheyweremeant to cross, but floated off instead to their right.Appended to it was a hand-copied version of a recommendation by a mathematics professor at PresidencyCollege,E.W.Middlemast,whodescribedhimas“ayoungmanofquiteexceptionalcapacity inMathematics.”Infact,recommendationand letterwereprobablybothamatter of form, the job for which he applied doubtless his all along, thanks toRamachandraRao.

Theletterwasdated9thFebruary1912.ItlistedRamanujan’sreturnaddressas7,SummerHouse,Triplicane.ItwasaddressedtoTheChiefAccountant,MadrasPortTrust.

•••

Fromitsbeginnings,Madrashadbeenatradingsettlementandport,thoughnaturehadequippeditpoorlyforthejob.Anuncommonlyroughsurfcrashedrelentlessly.Therewere insidiously trickyoceancurrentsandpeculiar sandbuildups. Ithadnonatural harbor atwhich cargo vesselsmightunload. Instead, shipswould anchor aquartermileoffshore,andmasulaboats—flat-bottomedcraftabouttwenty-fivefeetlongmadefromthinplanksstitchedtogetherwithcoconutfiber—pilotedbydaringmenexpertinreadingthesurf,wouldrowouttotheships,loadcargoafewtonsatatime,andreturnwith it to thebeach.Ninetypercentofall lossessufferedbyshipscallingatMadrascameinthattreacherouslastquartermile.

Yetsomehowthey’dmadeaportoutofit.In1796,alighthousehadbeenbuilt;itburned coconut oil in lamps visible from seventeen miles away. In 1861, a pierextendingelevenhundredfeetintotheBayofBengalwascompleted,and1876sawconstructionbeginonarectangularartificialharbor,twelvehundredyardsonaside,builtupfromtwenty-seven-tonconcreteblocks.Thenewharbor improvedmatterslittle.Unloadinglossesremainedhigh,andtheentrancerapidlybegansiltingup,thehigh-waterlineonthesouthsideoftheharboradvancingseventyfeetayearintothebay.

The port of Madras carried more than 60 percent of the Madras Presidency’simportsandexportstoBritain.Eachyear,twelvehundredshipscalledthere,bringingin iron and steel,machinery, and railway equipment, and leavingwith hides, piecegoods, indigo, and raw cotton. Still, by early in the century, it was a troubledoperationneedingmajorchanges.PlacedinchargeofthePortTrustin1904,andasitschiefengineerchargedwithoverseeingthosechanges,wasSirFrancisSpring.

Abaldmanwith sleepy eyes, whitemustache, and goatee, known inMadras asamong the first in South India to own his own motorcar, Sir Francis was in hissecondcareer.BorninIrelandin1849,agraduateofTrinityCollege inDublin,hehad joined the India Government engineering service in 1870 and for more thanthirtyyearshadplayedakeyroleinthedevelopmentoftheSouthIndianRailwaysSystem,where he had, among other feats, spanned theGodavariRiverwith a bigrailroadbridge.Fortheseaccomplishments,hehadbeennamedknightcommanderoftheIndianEmpirein1911.Sevenyearsbefore,he’dcometothePortTrust,andwithhimhe’dbroughtS.NarayanaIyer.

Narayana Iyer was not an engineer by training; the British had set up Indiancolleges to trainbright clerks toadminister thebureaucracy,not equip themtogetalongwithoutEuropeantechnicalexpertise.ThesonofaBrahminpriest,heheldanM.A.fromSt. Joseph’sCollege, inTrichinopoly,wherehe’dstayedontobecomealecturerinmathematics.Therehe’dmetSirFrancis.At thePortTrust,hewas,asofficemanagerandthenaschiefaccountant,thehighest-rankingIndian.SirFrancisreliedonhimheavily.

NarayanaIyerneversuccumbedtoWesterndressbutworethetraditionaldhotiand turban until his death in 1937; years later, his family would point to that astestimonytohispersonalstrength.Allduringtheseyears,windsofchangefromtheWestinfluencedevenmattersofdress.While someridiculedIndianswhoadoptedEuropean trousers, coat, collared tie, and boots or shoes, more common was theattitudereflectedinaHindueditorialinthelate1890s:“TherecanbenodoubtthatbootsandtrouserswiththeEuropeancoatconstitutethemostconvenientdressformovingaboutquickly.Theorientaldressissuitedtoalifeofleisure,indolence,andslowlocomotion,whereastheWesterncostumeindicatesanactiveandself-confidentlife.”Bythe1910s,educated,upwardlymobileIndianshadgottenthemessage.Inaformal photo taken at an Indian Mathematical Society conference in 1919,Ramachandra Rao, for example, wore Western garb, as did more than half theothers.NarayanaIyer,alsointhephoto,sittingonthegroundinthefirstrow,infullturbanandflowingrobes,didnot.

Hisfamilywouldtellofascrupulouslyhonest,restrained,anddignifiedman,notalittle forbidding in aspect, who championed Indian independence—but quietly.Who, as the patriarch of a house where more than two dozen cousins, brothers,sisters,andassortedhangers-ondependedonhim,alwayshelpedthosewhocametohim for money—but quietly. Who harbored a searchingly independent mind—behindoutwardbehaviorconformingineveryrespecttotraditionalHinduism.

Thesetwomen,SirFrancisSpringandNarayanaIyer,weretoplayanimportantrole inRamanujan’s life over the coming years.But now, onMarch1, 1912, threeweeksafter he’d applied for the job, Ramanujan knew only that he worked underthemasaClassIII,GradeIVclerkintheaccountssection,earningthirtyrupeespermonth.

•••

Duringallthistime,Janakihadbeenfarfromherhusband’sside,shuttlingbackandforth between her parents in Rajendram and her mother-in-law’s house inKumbakonamfor“training” inthewifelyarts.Now, late in1912,pastpubertyandwithRamanujaninasteadyjob,thetwofinallybecamemanandwifeinsomethingmorethanname.

SummerHouseinTriplicanewasaboutthreemilesfromRamanujan’snewjobattheofficesofthePortTrustoppositetheharborcomplexnorthofFortSt.George.Soafewmonthsafterstartingthe job,Ramanujanhadmovedmuchcloser, joininghisgrandmotherinalittlehouseonSaivaMuthiahMudaliStreet,offBroadway,inthe district known as Georgetown. And it was there that, three years after theirmarriage,Janaki—alongwithRamanujan’smother,Komalatammal—joinedhim.

UntilthevisittoMadrasin1906ofthePrinceofWales,thefutureKingGeorgeV,Georgetownwas still BlackTown, the original area set aside for the native, or“black,” population; the area within the Fort, set aside for Europeans, wasWhiteTown.Itsteemingstreetsheldathirdofthecity’spopulationon9percentofitsland.Cowsandbullocks, chickens andgoats, roamed freely. Inone street,metalworkerswouldsquatinfrontoftheirtinystalls,hammeringoutshapes,ortossingscrapsoftinintolittlebucketstobemelted.Thenextstreetwouldbecloggedwithbullocks,shouldering huge sacks of grain. Then streets of jewelry stalls, of textile shops,oilmongers, basketweavers, fruit and vegetable wholesalers . . . And everywhere,drivingitall,wasmusclepower,black-hairedmen,shoelessandshirtless,cladonlyintheir dhotis, ribs and muscles pushing out against glistening dark brown skin,straining as they pulled carts, or bent low under heavy loads upon their backs, orwhippingtheiranimalsthroughthedustystreets.

On Saiva Muthiah Mudali Street, in the tiny house for which they paid threerupeespermonth,Ramanujanandhisfamilywererightontopofeachother.YetheandJanaki,notevena teenageryet,had littlecontact.Theyscarcelyspoke.Duringtheday,hemightaskhertofetchhimsoaporanarticleofclothing.Atnights,she’drecall, she mostly slept beside Komalatammal—at her mother-in-law’s insistence.

They were never alone. If Komalatammal had to go to Kumbakonam, hisgrandmother,Rangammal,remainedbehind,monitoringtheircontact.

None of this was uncommon. Until she had children of her own, a new wife’spositioninthefamilyofherhusbandvergedonthatofaslave.Shewasthereonlytoserve,todohermother-in-law’sbidding.

Later,RamachandraRaowouldsayhehelpedRamanujanget“asinecurepost”atthe Port Trust. And sinecure it was, though whether intended as such from thebeginning or becoming that later is not clear. Still, even just putting in the hoursmade for a lifemorehectic thanRamanujanwasused to. “Iused to seehimmanytimes running to his office via the Beach Road,” recalled a friend from SummerHouse days, referring to the road that ran right up beside the PortTrust offices.“Withhiscoat,tailandall,flyinginthebreeze,andhislonghaircomingundone,abright namam [his trident-shaped caste mark] adorning his forehead, the younggeniushadnotimetowaste;hewasalwaysinahurry.”

Janakiwould laterrecallhowbeforegoingtoworkinthemorningheworkedonmathematics;andhowwhenhecamehomeheworkedonmathematics.Sometimes,he’d stay up till six the next morning, then sleep for two or three hours beforeheading in towork.At theoffice,his jobprobably included verifying accounts andestablishingcashbalances.Atonepoint, somemonthsafterhestarted,hereplacedanotherclerk,onleaveforamonth,as“pilotagefundclerk.”Inanyevent,theworkwashardlytaxing,andsoonhewasbeing leftalonetoworkonmathematics,beingtoleratedinthis,ifnotexplicitlyencouraged,bybothNarayanaIyerandSirFrancis.

Once,thestorygoes,afriendfoundhimaroundthedocksduringworkinghours,prowlingforpackingpaperonwhichtoworkcalculations.Anothertime,SirFranciscalledNarayanaIyer intohisoffice.How,hedemandedtoknow,sternly regardinghis aide, had these pages ofmathematical results gottenmixed into this importantfile?Was he, perhaps, using office time to dabble inmathematics?Narayana Iyerpleadedinnocent,claimedthemathwasn’t inhishandwritingatall, thatperhaps itwasRamanujan’swork.SirFrancislaughed.OfcourseitwasRamanujan’swork.He’dknownasmuchallalong.

NarayanaIyer,amemberoftheMathematicalSocietyandlongitstreasurer,wasnot just Ramanujan’s immediate superior, but his colleague. In the evenings, theywouldretiretotheelderman’shouseonPycroft’sRoadinTriplicane.There,they’dsitout on theporchupstairs overlooking the street, slatesproppedon theirknees,sometimes until midnight, the interminable scraping of their slate-pencils oftenkeepingothersup.Sometimes,aftertheyhadgonetosleep,Ramanujanwouldwake

and,inthefeeblelightofahurricanelamp,recordsomethingthathadcometohim,he’dexplain,inadream.

Narayana Iyerwas nomeanmathematician. But inworkingwith himhe foundthatRamanujan’spenchant for collapsingmany steps intoone lefthimas lost as adazedWatsoninthewakeofarunofSherlockHolmeslogic.How,NarayanaIyerwouldask,couldheexpectotherstounderstandandaccepthim?“Youmustdescendtomy level of understanding andwrite at least ten steps between the two steps ofyours.” What for? Ramanujan would ask. Wasn’t it obvious? No, Narayana Iyerwouldreply,itwasnotobvious.Patiently,hewouldpersist,cajolinghim,intheendsometimesgettinghimtoexpandalittleonhisthinking.

Itwasn’t longbeforeNarayana Iyerwasnot just aboss toRamanujan,nor evenjustacolleague,butadvisor,mentor,andfriend.“Somepeople,”Janakilaterrecalledhimsaying, “lookuponhim[Ramanujan]asordinaryglass,but theywill remaintosee him soon to be a diamond.”He brought Sir Francis around to his view, too,makinghimRamanujan’schampionaswell.

And itwas in coming to the attentionofSirFrancis and to thewebof contactsradiating out from him that, sometime around the middle of 1912, RamanujansteppedintoBritishIndia.HehadgrownupandlivedalmosthisentirelifewithonlythebarestcontactwiththeBritish.Nowthatwasabouttochange.

6.THEBRITISHRAJ

WestoftheMadrasPresidency,highintheNilgiriHills,wasOotacomund,knownas “Ooty,” the Presidency’s summer home, where Englishmen and their families,moresteeledtofrigidwindsblowinginofftheNorthSeathantothetropicalheat,fledtoescapethelowlandsummer.Onthefarsideofthehillslayanarrownorth-to-southstripofwet, rainycountry, theMalabarcoast,densewith tropicalvegetationand rich with the pepper, nutmeg, and other spices that had drawn the eyes ofEurope to India in the first place. In nearby Mysore, thick forests were home toexotic sandal and rosewood, and animals like the tiger and elephant that had soinflamedtheEnglishimagination.

ABritishpresenceinIndiawentbackto1600,whentheEastIndiaCompanywasformed.FortwocenturiesBritainclashedwiththeFrench,Dutch,Portuguese,andothersforcontrolofthesubcontinentand, in1876,madeIndiapartoftheempire.Now,in1912,morethanhalfacenturyhadelapsedsincetheSepoyMutiny—Indiannationalists called it the First War of Independence—had last challenged Britishrule.Theyearbefore,KingGeorgehadpassedthroughtheGatewayofIndiaerectedtohishonorinBombay,thentraveledtothenewcapital,Delhi,toassumehisthrone.

The very notion of India without Britain was, in the words of Lord Curzon, theviceroy, “treason to our trust.”Who could imagine that India, led byGandhi andNehru, would wrest independence from the Crown in 1947? And that the fair-skinnedfewnowinpowerwerepresidingoverthelastdaysoftheBritishRaj?

The marvel of British rule was that so few administered it. The Indian CivilService,thelegendaryICS,numberedbarelyathousandmen.ItwasIndia’scentralnervous system, quietly controlling mechanical arms wielded by Indian clerks andBritishengineers,physicians,andpolice.The viceroy and the governors ofMadrasand the other presidencies were not ICS, but everyone else who mattered was.EvolvingoutoftheEastIndiaCompany’sstaffofcommercialagents,theICSwasin1853thrownopenbycompetitiveexamtoIndians.Buttheyneverheldmorethanafewdozenpositionsthroughoutthecountry.Andtheirsalarieswerelimitedtotwo-thirdsthoseoftheBritish.

OnlylexicalaccidentlinkstheICStothebureaucraticinefficiencyandmediocritytodayconjuredupby“civilservice.”Infact,theIndianCivilServiceattractedmanyofBritain’sbest. Itsmemberswereculled fromtheupperclassesand intellectualelite.Theywereproductsofthefinestpublicschools,graduatesofCambridgeandOxford.Theyhadpassed arduous examinations. In their spare time, they translatedworksfrom Sanskrit, deciphered temple inscriptions, wrote grammars, compileddictionaries.Theyweremenwho,asoneaccountlaterhadit,“werefondofthinkingof themselves as Plato’s ideal rulers.” Reared with patrician values, imbuedwith asenseofresponsibilityandpublictrust, theyestablishedareputationfordedicationandfair-mindedness.

Buttherewasanothersidetothem—aninsufferablesmugness,atoweringsenseofmoralsuperiority.“Themembersofthisservice,”wrotearetiredmemberofitaboutthistime,

have generally shown the capacitywhich is awakenedby responsibility inmen ofBritish race:with amplesalariestheyhavehardlybeentemptedbydishonesty,andtheirdetachedimpartialityhasnotbeendisturbedbytheimportunityofrelationsorfriends.Tothecreditoftheirnationtheyhaveestablishedandmaintainedagovernment,which,foritsresources,isexceedinglyefficient.

At the district level, the government representative was called a collector, and hewielded the power of a prince. But, as one late nineteenth-century account noted,“ThecollectorandhisEnglishstaffhardlyeverknowthevernacular.Bythenativesthey are regarded with awe, not affection.” Observed another: “The collector isseparatedbyanimpassablegulffromthepeopleofthecountry....Totheeyesofanative, the English official is an incomprehensible being, inaccessible, selfish,overbearing,irresistible.”

The central fact of the British presence in India, then, was distance. Prints andengravings todayonviewatFortSt.George inMadrasshowBritish lifewalledofffrom India,marked by awell-ordered calm reminiscent of nothing somuch as anEnglishgarden.OneviewshowsturbanedIndianssurroundingasnakecharmer,asan Englishman, attended by a native servant, looks on from the safe remove of asecond-floorporch.Inthesescenes,nativeswork—bearingpalanquins,orbalancingloadsatoptheirheads,orurgingmasulaboatsoutintotheangrysurf.TheEnglish,invariablyattheirease,standshieldedfromthesununderanumbrellaor,intophat,leanlanguidlyagainstacolumn.

AnEnglishman typicallyhadhis ownwasherman,whogot fouror five rupees amonth,livedonhispremises,andfunctionedassomethinglikehisprivateproperty.AfterlongenoughinIndia,theEnglishmanforgothowtosomuchasbrushandfoldhis clothes. When he finally took the steamer home, he’d be taken aback as anEnglish steward stooped to serve him tea. (Indeed, years later, asked as part of asurvey what most struck them about England, students from former Asian andAfricancoloniesinvariablymentionedthesightofwhitemendoingmanuallabor.)

That therewas an ineradicable splitbetweenEnglishmanand Indian theBritishthemselveswereeagertoacknowledge.“EastisEastandWestisWestandneverthetwain shallmeet,” wrote Kipling. Indians might work, even live with you in yourbungalow,notedoneoldIndiahand,HerbertCompton;butintheend,“thereisnoassimilation between black and white. They are, and always must remain, racesforeign to one another in sentiment, sympathies, feelings, andhabits.Between youand anative friend there is a great gulfwhichno intimacy canbridge—the gulf ofcasteandcustom.Amalgamationisutterlyimpossibleinanybutthemostsuperficialsense,andaffinityoutofthequestion.”

It was this “great gulf” that, in the succeeding months and years, Ramanujanwould,ofnecessity,confront.HehadgrownupduringthereignofQueenVictoria.Coins inhispocketbore likenessesof theBritish sovereign;until1902, they’d saidVICTORIAEMPRESS,whileafterherdeath,theyboretheprofileofEdwardVII,KING

ANDEMPEROR.Inhighschool,Ramanujan’sscholasticprizesincludedananthologyof patrioticEnglish verse, a collection of LordMacaulay’s essays, and a volume ofWordsworth’s poetry—certainly nothing Indian. Later, while he attendedGovernmentCollege inKumbakonam, the sixty thousandrupees thatconstructionofanewstudenthostelrequiredwereraisedinmemorynotofsomeIndiannotablebutofQueenVictoria.

Yet all this had left England for Ramanujan no more than symbol, image, andabstraction; English people he had scarcely known. Now, however, that was

changing. His friend Narasimha had introduced him to E. B. Ross, of MadrasChristianCollege.HehadmetE.W.MiddlemastofPresidencyCollegeandsecuredfromhimarecommendation.Now,atthePortTrust,hehadmetSirFrancis;soonhewouldmeetSpring’sfriends.

Whatever prejudicesmay have distorted theBritish view ofRamanujan did notlikely extend to his intellect.Of the Indian, one English writer noted around thistime,“heiscunningandcontentiousinargument,andhisintellectualpowers,wheneducated,arecapableof considerabledevelopment. . . . In this respect,heputs theEnglishmantoshame,andwereallpostsintheIndianGovernmentthrownopentoexaminationinIndia,weshouldprobablyseetheadministrationfilledwithBengaliBaboosandMahrattaBrahmins.”

But that’s as far as British esteem for the Indian temperament went. HerbertCompton,whohadrunaplantationforyears,observedinabookpublishedin1904that “whilst you can polish the Hindu intellect to a very high pitch, you cannottempertheHinducharacterwiththosemoralandmanlyqualitiesthatareessentialforthepositionsheseeksto fill.”Aretiredmemberof theIndianCivilService,SirBampfylde Fuller, marveled at how Hindu boys could flock to classrooms andlibraries,andpursueWestern literatureandscience,yetunaccountablycling to . . .well,Indianways.Indians,hesaid,wereundulysentimental,wildlyinconsistent.“AnEnglishmanisconstantlydisconcertedbytheextraordinarycontradictionswhichheobserves between the words and the actions of an educated Indian, who seemsuntouchedby inconsistencieswhich tohimappear scandalous. . . .Theygiveeagerintellectualassentto[European]ideals,yetlivetheirlivesunchanged.”

Other laudable traits the British discerned in the Indian personality verged ondamningwith faintpraise.Of theSouth’sDravidian stock, oneBritonwrote, theywere“hardworking,docileandenduring.Theyaremoresober,self-denyingandlessbrutish in their habits than Europeans.They show greater respect for animal life,they havemore natural courtesy ofmanner, and, as servants, attach themselves tothosewhotreatthemwellwithfargreateraffectionthanEnglishservants.”Howeverseemingly admirable, these were hardly traits that left Indians the equals of theirBritishmasters.

Itwaswithsomesuchblendofdueregardforhisintellect,coupledwithalingeringdubietyabouthischaracterandtemperament,allseenacrossavastsocialdivide,thattheBritishwouldtendtoviewRamanujan.

7.THELETTER

Through Narayana Iyer, Ramachandra Rao, Presidency College mathematicsprofessorMiddlemast,andotherstestifyingtohismathematicalgifts,ithadbecomecleararoundPortTrustofficesbylate1912thatRamanujanwassomethingspecial.Thequestion facingSirFrancisandotherBritishofficialswashow special?And injustwhatwayspecial?Andwhat,inanycase,weretheytodowithhim?Werehisgiftstrivial,orprofound?Weretheythegiftsofthegeniusor,intellectuallyspeaking,theshaman?WasRamanujanaminorodditywhocouldbesafelydismissed,oraprodigydemandingnurtureandguidance?

No one would go out on a limb. The harsh explanation is that, despite firmopinions,theywereafraidto,onthechancethateventsmightprovethemwrongandhistory judge them harshly. The kinder and simpler explanation—and the morelikely—isthattheyjustdidn’tknow,andknewtheydidn’tknow.

NarayanaIyer,ofcourse,thoughthedidknow;heworkedwithRamanujaneverydayand sawhis abilitiesupclose. IfRamanujanwas toachievehispromise, itwasclear,heneededtheBritishsolidlyinhiscorner.NarayanaIyerlobbiedonhisbehalfwithSirFrancis.

Drawing on his connections, Ramachandra Rao also tried to gain Spring’s ear.“Dear Sir Francis,” wrote C. L. T. Griffith, a forty-year-old civil engineeringprofessor at Madras Engineering College, on November 12, 1912, apparently atRamachandra Rao’s behest. “You have in your office an Accountant, on Rs 25, ayoungmannamedS.Ramanujan,whoisamostremarkablemathematician.Hemaybeaverypooraccountant,but Ihopeyouwill see thathe is lefthappilyemployeduntilsomethingcanbedonetomakeuseofhisextraordinarygifts.”Sincefewcouldfollow, much less meaningfully critique, Ramanujan’s work, he went on, he waswriting anothermathematician (M. J.M.Hill, inLondon) for advice, and sendinghimsomeofRamanujan’spapers.“Ifthereisanyrealgeniusinhim,”wroteGriffith,“hewillhavetobeprovidedwithmoneyforbooksandwithleisure,butuntilIhearfrom home,” he added, hedging his bets, “I don’t feel sure that it is worthwhilespendingmuchtimeormoneyonhim.”

Among those to whom Spring turned for advice was A. G. Bourne, Madras’sdirectorofpublicinstruction,whoadvisedthatRamanujanbesenttoseeoneorbothoftwoMadrasmathematiciansheindicatedbyname.Then,headded:“Ifhisgeniusis so elusive or mysterious that good mathematicians, possessed besides of muchcommonsense,cannotrecognizeandappreciateitevenifitcarriesthembeyondtheirscope,Ishoulddoubtitsexistence.”

Two weeks later Ramanujan went to see W. Graham, Madras’s accountantgeneral,oneofthoseBournehadsuggested.“Whether[Ramanujan]hasthestuffof

greatmathematiciansor not I do not know,”Grahamwrote after seeing him. “Hegivesmetheimpressionofhavingbrains.”Givesmetheimpression...Withsuchcaredidhewordhisassessment,noonecouldpossiblyfaulthimshouldheprovewrong.Confusingmattersmore,hesuggestedthat“itispossiblehisbrainsareakintothoseofthecalculatingboy.”

Grahamwasreferringtothosefreaksofnature,someofthemtodaydescribedasidiotsavants,whothoughlackingrealunderstandingofhighermathematics,possessapeculiarabilitytoperformextremelyrapidcalculations—tounerringlymultiplyanddivide long strings of ten-digit numbers, or give the day of the week on which athousand-year-oldbattleoccurred,orperformsimilarlytrivialcomputingtasks.

Infact,justassomeartistsofsurpassingbrilliancearenogoodatdrawingstraightlinesorrepresentingthehumanfigure,sodoesmerefacilityinarithmetic—whetherextractingsquareroots,orbalancingbooks,orworkingouttrickywordproblems—havenothing todowith realmathematics.Amathematicianmay be adept at suchskills,justastheartistmaybeadeptatroutinedraftsmanshiporfiguredrawing.Butpossessionofsuchskillsdoesnotpredictmathematicaltalent.

Ramanujanwasmorethanordinarilygoodinarithmeticcalculation;ontheotherhand,hiswasn’taskilldevelopedtofreakishproportions.Andcertainlyinnootherway did he resemble the “calculating boy” model. Nonetheless, it was one moreamongvariouspossibilitiesas,duringlate1912,BritishofficialdominMadrasgropedwiththequestionofwhattodowithRamanujan.

Griffith,towhomGrahamhadalsowritten,wroteSpringthenextday:“IthinkIwasrightinwritingtoProf.Hill,”saidhe,“andwemustwaithisopinion.”

Micaiah JohnMullerHill was Griffith’s professor from twenty years before, atUniversityCollegeinLondon,andateacherknownmoreforthepatienceandcarehelavished on his students than for his mathematical researches. Around mid-DecemberGriffith heard fromhim at last.He couldnot look through allGriffithhad sent him just now, Hill apologized, but a glance was sufficient to show thatRamanujan had fallen into some pitfalls; some of his results were simply absurd.Shouldhewant toovercomehis evident lacks,Bromwich’sTheoryof InfiniteSerieswas the text to consult. If still interested in publication, he ought to write thesecretaryoftheLondonMathematicalSociety.But,Hillwarned,“Heshouldbeverycarefulwithhis [manuscripts.They] shouldbe very clearlywritten, and shouldbefreefromerrors;andheshouldnotusesymbolswhichhedoesnotexplain”—ashehadinthepublishedpaperonBernoullinumbersGriffithhadsenthim.

But Hill’s letter didn’t answer the question: Had Ramanujan somethingextraordinary to offer theworld?Whatwas thenature and extent of his genius, if

geniusitwas?“Whatyousayabouthimpersonallyisveryinteresting”—presumablyareferencetohisunusualintellectualhistory—“andIhopesomethingmaycomeofhiswork,”wasaboutallHillwouldadd.

Afewdayslater,Hillwrotehisformerstudentagain.Itwasacuriousletter,stillnotdefinitive,butthistimemoreencouraging.Ontheonehand,Ramanujan’spaperon Bernoulli numbers, he said, was riddled with holes. “He has in fact observedcertainpropertiesof theearlierBernoullinumbersandassumedthemtobe trueofthem all without proof. For [this and other] reasons, I feel sure that the LondonMathematicalSocietywouldnothaveacceptedthepaperfortheirProceedings.”Onthe other hand, he said, “Mr. Ramanujan is evidently a man with a taste formathematics, andwith some ability.”His educational deficitwas hurting him.HeneededtogetthatBromwichbook,hesaidagain,thistimecitingthespecificchapterthatwouldclearupRamanujan’smisunderstandings.

And then, in a personal aside,Hill said perhaps themost revealing thing of all.“When I was a student in Cambridge, 1876–9, these things were not properlyunderstood,” he wrote, referring to the subtle but crucial points underminingRamanujan’swork,“andthemoderntheoryhasonlyrecentlybeenestablishedonafirm basis. Many illustrious mathematicians of earlier days stumbled over thesedifficulties, so it is not surprising that Mr. Ramanujan, working by himself, hasobtainederroneousresults.Ihopehewillnotbediscouraged.”

1876–9.Hill’sCambridge years, as it happened, coincided exactlywith those ofGeorge Shoobridge Carr, author of the book so important to Ramanujan. Here,then,wasthe firsthintof thepriceRamanujanhadpaid in findingnomorerecentinspiration: he had missed out on all that had been learned in the mathematicalcapitalsofEuropeoverthepastfortyyears.Ramanujan’smathematics,ineffect,wastrappedinatimewarp.Nowonderhehadgoneastray.

Hill, who scarcely remembered his old student Griffith, had been sufficientlyintriguedbyRamanujan’sworktowritetwolonglettersinresponsetoit.Ofcourse,itcametonothingmorethanthat.Hewasnotofferingtotakehimonasastudent.Why,ifanything,hehadjudgedRamanujan’sfirstpaperunfitforpublication.Still,though he did not fully understand all Ramanujan had done, his reply probablycontainedmoreserious,reasoned,professionaladvicethanRamanujanhadgottenallhislife.AnditwasencouragingenoughtoquellmostlingeringsuspicionamongtheBritishinMadrasthatmaybeRamanujanwasmorecrankthangenius.

Forsometimenow,manyhadadvisedRamanujanthatnoone inIndiaproperlyunderstoodhim,thathe’dnotbeabletofindtheretheexpertiseandencouragementheneeded,thatheshouldinsteadwritetoCambridge,orelsewhereintheWest,for

help. One who did was Singaravelu Mudaliar, his old professor at Pachaiyappa’sCollege, to whom he had drawn close during his brief time there. Another wasBhavaniswamiRao,oneofRamanujan’sprofessorsatKumbakonamCollege.AthirdwashisfriendNarasimha,withwhomhehadlivedinParkTownacoupleofyearsbefore.Morerecently,NarayanaIyerprobablygavehimsimilaradvice.

IfRamanujan needed convincing to look toward theWest, now, in thewake ofHill’sletterfromEngland,heneedednomore.Eventshadconspiredtotellhimthathewas,ineffect,toobigforIndianmathematics,andthathewasapttogetamoresympathetichearingfromEuropeanmathematicians.

Indiawas aquarterof theway around the globe fromEurope,but themailwascheap,reliable,and—longbeforeairmailshranktheworld—surprisinglyfast;peoplegrumbledif letterstoEnglandtookas longastwoweeks.Andso, in late1912andearly1913,itwastotheinternationalmailsthatRamanujanturned.InlettersdraftedwiththehelpofNarayanaIyer,SirFrancisSpring,andperhapsP.V.SeshuIyer,hebegantowrite leadingmathematiciansatCambridgeUniversity, includingwithhisletterssamplesofhiswork.

He wrote to H. F. Baker, who held a long string of high honors as amathematician,includingafellowshipoftheRoyalSociety,andhadbeenpresidentoftheLondonMathematicalSocietyuntiltwoyearsbefore.CouldBakerofferhimhelporadvice?

Either through the kind of formulaic letter of polite discouragement thatimportant men learn to write, or by returning his unsolicited material withoutcomment,orbyignoringhisletteraltogether,Bakersaidno.

RamanujanwrotetoE.W.Hobson,anequallydistinguishedmathematician,alsoaFellowof theRoyal Society, and holder ofCambridge’s Sadleirian chair in puremathematics.

Hobson,too,saidno.On January 16, 1913, Ramanujan wrote to still another Cambridge

mathematician,G.H.Hardy,whoatthirty-five,agenerationyoungerthantheothermen,wasalreadysettingthemathematicalworldofEnglandonitsear.CouldHardyhelphim?

AndHardysaidyes.

CHAPTERFOUR

Hardy

[G.H.Hardyto1913]

1.FOREVERYOUNG

Hewasastudyinperpetualyouth.One day in the spring of 1901, Hardy took his friend Lytton Strachey to the

privategreenbehindTrinityCollege, towhichasa fellowof thecollegeheenjoyedaccess, for a game of bowls. “He is the mathematical genius,” Strachey wrote hismother, “and looks a babe of three.” Even into his thirties,Hardy was sometimesrefused beer and at least once, while at lunch with other Trinity dons, he wasmistakenforanundergraduate.

Hehad ice-cleareyes,a finelychiseled face,and in1913, straight,close-trimmedhair.Hewasbeautiful.Hedidn’tthinkso,ofcourse,andcouldscarcelybeartolookathimself.Hiscollegeroomshadnomirrors,andinahotelroomhewouldcoveranywithtowels,shavingbytouch.Buthealonewasdeceived.Evenwhenpast fifty,hislookswere arresting.His skin, wrote a friend from those years, the novelistC. P.Snow,was tanned to “a kind ofRed Indian bronze.His facewas beautiful—withhigh cheek bones, thin nose, spiritual and austere. . . . [Cambridge] was full ofunusual and distinguished faces, but even thereHardy’s could not help but standout.” He was not, by every yardstick, handsome, at least not “ruggedly” so; hisfeaturesweretoodelicateforthat.Andpursed,ungenerouslythinlips,turneddownalittleatthecorners,hintedatajudgmentalstreakinhim.

Hardy was forever judging, weighing, comparing.He ratedmathematicians, thework they did, the books and papers they wrote. He held firm opinions oneverything, and expressed them.When a Cambridge club to which he’d belongedmoved to change its official colors, Hardy took six pages to attack the plan. Hefaulted a sacrosanct academic tradition of almost two centuries’ standing, andcondemnedit,unrelentingly,formorethantwentyyears.Allhisenthusiasms,peeves,and idiosyncrasies were like that—sharp, unwavering, vehement. He hated war,politiciansasaclass,andtheEnglishclimate.Helovedthesun.Helovedcats,hated

dogs.Hehatedwatchesand fountainpens, lovedTheTimes ofLondon crosswordpuzzles.

InTheCase of the Philosophers’Ring, a SherlockHolmesmysterywritten half acenturyafter thedeathofArthurConanDoyle, the characters includeRamanujanandHardy.In it,authorRandallCollinspicturesHardyasasortofWhiteRabbithoppingaroundtheFellowsGardenatTrinityinwhiteflannelsandcap,cricketbatinhand, frantically searching forhis cricketgloves, crying, “There’samatchdue tobegin,andIcan’tfindthem.I’mlate,I’mlate!”Inaprefatorynote,Collinsabjuresallclaimtohistoricalaccuracy.ButinHardy,he’sclosetothemark.

Hardywas a cricket aficionado of almost pathological proportion.He played it,watched it, studied it, lived it. He analyzed its tactics, rated its champions. Heincludedcricketmetaphorsinhismathpapers.“Theproblemismosteasilygraspedin the language of cricket,” he would write in a Swedish mathematical journal;foreigners failed to grasp it at all.His highest accoladewas to rate amathematicalproof, say, as being “in the Hobbs class”—leaving the benighted to imagine thephilosopherThomasHobbes,notthelegendarySurreycricketerJackHobbs.Hardywouldplaythegameintohissixties.Hissisterwouldbereadingtohimaboutcricketwhenhedied.

HardyjudgedGod,andfoundHimwanting.Hewasnotjustanatheist;hewasadevoutone.Asanundergraduate,hewastoldthattobeexcusedfromchapelhehadto informhis devout parents; he agonized over what to do—but ultimately wrotethemwiththecrushingnews.God,itwouldbesaidofhim,washispersonalenemy.Yet his friends included clerics, and some of his infidel posturing was just that—anotheroftheharmlessgameshenevertiredofplaying.“It’sratherunfortunate,”heoncegrumbledtoafriendasachurch’ssixo’clockchimessoundedtheendofasunnydayof cricketatFenner’s cricket ground inCambridge, “that someof thehappiesthoursofmylifeshouldhavebeenspentwithinsoundofaRomanCatholicChurch.”

Shyandself-conscious,hedislikedsmalltalk;cricket,ofcourse,wasnotsmalltalk.Heabhorredformalintroductions,wouldnotshakehands,wouldwalk, facedown,alongthestreet,ignoringthosewhomightexpecthimtoexchangehow-do-you-dos.Hewas“oneofthemoststrangeandcharmingofmen,”wroteLeonardWoolf,whoknew him at Cambridge long before Woolf married Virginia Stephen and, withStracheyandothers,launchedtheBloomsburyliterarymovement.Hiseccentricitieswouldossifywithage,becomecaricaturesof themselves, the stuff of story.Buthispersonality, temperament, and values were already largely formed when he heardfromRamanujan.

RamanujanknewnothingofthissideofHardy,ofcourse.Heknewhimonlyasamathematician.And in1913,at theageof thirty-five,Hardywasalreadya famousone.Hehadappearedinthemathematicalliteratureforfifteenyears,countedmorethanahundredpapers, and threebooks, tohis credit.Hewas aFellowofTrinityCollege,themeccaofCambridgemathematics,andhenceEnglishmathematics.HehadbeennamedtotheRoyalSociety,Britain’smostelitebodyofscientists,in1910.Indeed, more than sniping at God, or delighting in cricket, or fashioning slyconversational gambits over dinner, Hardy cared about discovering mathematicaltruth. A brilliant mathematician, he was also a major influence on othermathematicians.Awholeschoolhadbeguntoformaroundhim.HehadservedontheCounciloftheLondonMathematicalSocietyforthreeyears,wouldlateroccupynumerous other posts within the mathematical community. “My devotion tomathematicsisindeedofthemostextravagantandfanaticalkind,”hewouldwrite.“Ibelieve in it, and love it, and should be utterly miserable without it.” Hismathematicalresearch,hewouldsay,was“theonegreatpermanenthappinessofmylife.”

Hardyspokebeautifully.Hebattedoutsparklingbonsmotsthewayhedidcricketballs from the popping crease—provoking, challenging, asserting. He wasscrupulouslyhonest,fastidiousaboutgivingotherstheirdue,onceevenadmittedthatthepro-Godpositioninadebatehadbeenbetterargued.Hewasendlesslyamusing—butitwasalllikethegauzysilkenshimmerofawoman’sdress,meanttodistractanddisguisemorethanreveal.Conversation,oneofhisresearchstudentswouldsay,wastohim“oneofthegameswhichhelovedtoplay,anditwasnotalwayseasy tomakeoutwhathisrealopinionswere.”

C. P. Snow once reported that the longer you spent in Einstein’s company, themoreextraordinaryheseemed;whereasSnowfoundthatthelongeryouspentwithHardy,themorefamiliarafigureheseemedtobecome—morelikemostpeople,only“moredelicate,lesspadded,finer-nerved”;thathisformidablewallofcharmandwitshieldedanimmenselyfragileego;thatwithinlaysomeonesimple,caring,andkind.

There is a picture of Hardy from middle age that shows him slouched in anupholsteredwicker chair, one leg crossed over the other, right hand cocked at thewrist, lightlygrippingacigarette, leftarmsuspendedatanunlikelyangleacrosstheback of the chair. Awisp of hair slips down over his forehead.He does not lookrelaxed; innophotographofHardydoeshe ever look relaxed.Always there’s thathauntedlookinhiseyes,like“aslightlystartledfawn,”asLeonardWoolfoncesaidofhim. There he sits, brows knitted, lips pursed, peering out over the tops of hisreading glasses, imperious and forbidding. Someone spying this picture once said,

“To sit that way you have to have been educated in a public school” (or what toAmericansisaprivateboardingschool).

HardywastheproductofthefinestBritishpublicschooleducation.Buthehadn’tcomebyitintheusualway.TherewerenoviscountsintheHardyline,nocountrysquires.Hisfamilywasneitherrichnorwellborn—wasofhumblerlineage,inasense,than Ramanujan’s: in caste-bound India, Ramanujan was a Brahmin, while inEngland,wheresocialclasscounted,Hardycamefromschoolteacherstock.Indeed,Hardy would later be offered as an “example of how far the English educationalsystemcanbringoutthepersonalpowersandcapabilitiesofaman.”Hisintellectwasso luminous, hewasmarked from the start.His success implied a blurring of thetraditional British class system, a filtering down into the middle classes ofopportunitiesoncelargelylimitedtoathinstratumofsocietyatthetop.

2.HORSESHOELANE

In1896,whenHardyandhisnewclassmates took turns signing thegreat leather-bound book, with quarter-inch-thick covers, that had been used since 1882 toregister each new class at Trinity College, they noted the schools they’d attendedpreviously—Eton,Harrow,Marlborough, and inHardy’s case,Winchester. But afewstudents,oneortwoperpageoftwenty-sixnames,wroteinnoschoolatallbutonly“privatetutor,”orjust“private,”inthespaceprovided.Itwasinthissense,then,thatEton,Harrow,andWinchesterwere“public”schools,notprivate.Thoughmanywent back centuries, public schools as a powerful social institution had blossomedonlyinthewakeofreformsachievedbyThomasArnoldattheRugbySchoolintheearly part of the century. By Hardy’s time they were the de rigueur means forfashioningthebodies,minds,characters(andaccents)ofupper-classboys.

But England, all during Queen Victoria’s sixty-four-year reign, was changing.NewlyprosperousfarmersandtradesmencouldhardlysendtheirsonstoEtonandHarrowyetwerenotcontent,either,withthebare-bonesgrammarschooleducationafforded the poor. The 1850s and 1860s saw much debate about how best toaccommodate thisnewmiddle class and the establishment of new schools to servethem.

OnewasCranleighSchool,locatedinthecountysouthwestofLondonknownasSurrey.“Whiletheupperclass[enjoy]thegreatpublicschools,andmuchhasbeendoneandstillcontinuestobedonetoimprovetheeducationofthelowerorders,theprovision for the sons of farmers, and others engaged in commercial pursuits is soinadequate that the labourer’s sonoftenreceivesabettereducation than the sonofhisemployer.”Soadvisedtheschool’sprospectusatitsfoundingin1863.Thenstill

calledSurreyCountySchool,itaimedtoredressthisimbalanceand,initsearlyyears,did:of113boysenteringaround1880,forexample,55werethesonsoftradesmen,20ofclerks,14offarmers.

In1871,takingupthepostofassistantmaster,teachinggeographyanddrawing,wastwenty-nine-year-oldIsaacHardy,whohadearliertaughtatagrammarschoolinLincolnshire.Three years later, the school awarded him an extra fifty pounds peryear—probably more than half again as much as he then earned; he was gettingmarried,andwouldbelivingoffschoolpremises.InJanuary1875,SophiaHall,threeyears younger thanhe, and then seniormistress at theLincolnDiocesanTrainingCollege, became his wife. Littlemore than a year later, she was pregnant, and onFebruary7,1877shegavebirthtoGodfreyHaroldHardy.Twoyearslater,hewasjoined by a sister,GertrudeEdith.There, across the road from the school, on theoutskirtsofCranleigh,avillageoftwothousandsouls,theygrewup.

Surrey’s northern border was formed by the RiverThames which, further east,meanderedthroughLondon.Inthe1840s,therailroadhadbegunpushingoutfromthe great metropolis and in forty years had doubled the county’s population, to342,000; by the yardstick of the centuries, that was breakneck growth. But inCranleigh, at the other end of the county, things changedmore slowly. True, therailroad had in 1865 leftCranleigh just a quick fortymiles, by theGuildford andHorshambranchoftheBrightonRailway,fromLondon.Andrichindustrialistshadbeguntobuyfarmshereandbuildnewhomesonthem.ButduringHardy’syouth,the rolling countryside around Cranleigh remained largely unspoiled, a peacefultableauofdirtroads,windmills,oldmanors,andthatched-roofcottages.

Hardy’sparentslivedmidsttheoldSurreycharm,butwerenotofit,havingmovedtherefromtheothersideofLondon,150milesaway.Theyhadhadnomoneyforauniversity education. Isaac Hardy’s father had been a laborer and foundryman.Sophia’s,onceaturnkeyatthecounty jail,wasabakeratthetimeofhermarriage.Butbothwerebright, sought somethingbetter in life—and, as schoolteachers,hadfoundanicheinthehumblerreachesoftheacademicworld.

Isaac Hardy, as C. P. Snow pictured him, was “a gentle, indulgent, somewhatineffectualmanwithmore than a touchof theWhiteKnight abouthim.”Hewasprobablyahappymanaswell,witharefinedaestheticsenseandabuoyantattitudetoward life.Hewasthe leadingtenor intheschoolchoirand,soonaftercomingtoCranleigh,wasgiving twice-a-weeksinging lessons.Heedited the schoolmagazine,played football (soccer, toAmericans), was active in fraternal organizations, was amemberof theRoyalGeographicSociety.Whenhedied, at age fifty-nine, hewasearnestly and sincerelymourned, the shops alongHigh Street closed, their blinds

drawn.Hewas“oneofthoserareandprecioussouls,”theheadmasterwouldsay,who“neverutteredanunkindword,neverpainedanylivingbeing,neverhadanenemy.”YoumightchalkituptoinflatedVictoriansentimentalismexceptforaphotographofhim, takenwhenHaroldwasa child. It showsamanwith sunnyeyes, thinninghair, and the kind of thick beard that canmake aman seem forbidding; on IsaacHardy,though,itlooksaswelcomingasitdoesonSantaClaus.

ThegrimformalityVictorianphotographersmorenormallyfoundintheirsubjectsreturns,however,inaportraitofhiswifetakeninCranleighataboutthesametime.Init,SophiaHallHardywearsanornate,embroidereddress,herhaircombedandpiled back onto her head. The small lips, turned down a little at the corners, thestartled look in the eyes, would reappear in her son. She was, like Ramanujan’smother, a pious woman. Sundays, she would dragHarold and sister Gertrude tochurchtwoorthreetimes.WhenshelefttheLincolnDiocesanCollegeinDecember1874 to get married, the school thanked her for the “high religious tone, andconsistentChristian conduct”withwhich shehad influencedher studentsover thepastfouryears;forthegoodperformanceofherstudentsinarithmetic;andfor“thewisecombinationoffirmnessandkindnessdisplayed . . . inthemanagementoftheStudents, togetherwiththequietnesswithwhichshehasmaintainedherauthorityover them.” Altogether, it is hard not to think that Sophia was a stern, upright,surpassinglycompetentwoman.

Like her husband, she had a taste for cultural pursuits. She taught piano. Sheattendedconcerts—likeaperformanceofHandel’sMessiah,heldattheschool,whenHaroldwasone.Asschoolteachers,embracingaworldofArtandLearning,bothsheandIsaachadtranscendedtheirroots.Andnowasparents,proudofthestation inlifethey’dreached,theyapproachedtheirownchildren’seducationwiththeutmostearnestnessandcare.

Andthesensibilitiestheyhopedtobequeath,atleasttheintellectualones,took.

•••

InMay1891,GertrudewouldenterSt.Catherine’sSchool,Cranleigh’ssisterschool,aboutfivemilesuptheroadinBramley.There,sheexcelled,earningprizecertificatesindrawingandLatin.In1903,afterearningaB.A.fromtheUniversityofLondon,shereturnedtoheralmamaterasartmistress,remainingtheremostoftherestofherlife.In1926,shebecameeditoroftheschoolmagazine,whichshefilledwithpoetry,stories,andessaysbetterbyfarthananythingthatsmallhadarighttoexpect.Hereditorials were graceful. Her own poems, many of them rich with literary and

scholarly allusion, were suffused with an extravagant comfort in, and love of, thewholeworldof learning.In thisuncharacteristicattemptatdoggerel, she swipedattheself-satisfiedignoranceshefoundamongsomestudents:

ThereisagirlIcan’tabide.Hername?I’llbediscreet.

IfeelI’dneedsomesavoirdireShouldIherparentsmeet!

...

Shesays“InevercoulddoMaths.WhenDaddywasatschool

Hecouldnotadd!”I’dlovetosay“ThenDaddywasafool!”

“IndictéeIgotminustwo;There’snotaverbIknow;

IalwayswritethefuturetenseOf‘rego,’‘regĕbo.’

“ButthenmyMothercannotwriteOrspeakaforeigntongue.”

Sweetmaid,howmuchtheworldhadgainedIftheyhadbothdiedyoung!”

WhateverMr.andMrs.Hardylaidontheirchildren,itmusthavecomeinmassivedoses,relentlesslyadministered, tohavesiredapoemlike that. Indeed, theHardysheld firm theories about education, quite definite ideas about raising children thatmovedSnowtocallthem“alittleobsessive.”Haroldandhissisterhadnogoverness;anursetaughtthemtoreadandwrite.Theyhadrelativelyfewbooks,butthosetheydid have were always “good” ones; as a boy,Harold would readDonQuixote andGulliver’sTravelstohissister.Theywereneverallowedtoplaywithbrokentoys.Itwas as if therewere an invisible standardof integrity and excellence towhich theywere invited to compare everything—and to sternlydiscardor reject anything thatfailedtomeasureup.

WhenHardywastwo,hewaswritingdownnumbersintothemillions,acommonmarkerofmathematicaltalent.Atchurch,he’dbusyhimselfseekingthefactorsinthehymnnumbers:Hymn84?That’s2×2×3×7.Atschool,youngHardyapparently

neversatinaregularmathclass,butwasrathercoachedprivatelybyEustaceThomasClarke, who presided over the school’s mathematics instruction. Clarke came toCranleigh fresh from St. John’s College, Cambridge, where, as a “Wrangler,” he’drankedamongthe topmathematicsstudents.Coupledtohismathematicalabilitieswasuncommonenergyanddrive,andhewonareputationforinstillingthosetraitsinhisstudents.

ItwasnotjustmathematicsinwhichHardywasgifted,butalltowhichhetookhishand.Yet a fragility, a diffidence, sometimes undermined it.Hewas painfully shyandself-consciousandcouldscarcelybeargoingupbeforethewholeschooltoaccepta prize. Sometimes he’d give wrong answers to ensure he wouldn’t have to do so.“Over-delicate,”his friendSnowdescribedhim later. “He seems tohave beenbornwiththreeskinstoofew.”

Sowashissister,whomoneofherstudentswoulddescribeas“shyanddiffident.”WhensheenteredSt.Catherine’sSchool,Gertrudewouldrecall,“Iwasveryshy,andtheatmosphere inwhichIhadbeenbroughtupwasapoorpreparation foragirls’schoolofthatage.”Whatevermadeforthat“atmosphere,”itgrippedbothchildren.Neither evermarried.Both spent their lives in academic settings.Both emerged asdelicate,enchantedby intellect,and—apparentlyreactingtotheirmother’szealotry—contemptuousofreligion.

Hardywasanatheistevenasaboy.Once,asheandaclergymanwalkedinthefog,they sawaboywitha stringand stick.Theclergyman likenedGod’spresence toakite,feltbutunseen.Inthefog,hetoldyoungHardy,“youcannotseethekiteflying,butyoufeelthepullonthestring.”Butinfog,Hardythought,thereisnowindandnokitecanfly.Gertrudefeltmuchthesame.Once,asanoldwomanconfinedtoanursinghome, shewasaskedher religiouspreference.Shereplied “Mohammedan,”bewailedthewantofamosquecloseby,evensetabouttryingtolocateaprayerrugtoenhancethedeception.

Wheninthe1920saphotowastakenofthefacultyatSt.Catherine’sSchool,alltwentyorsostaredstraightaheadintothecamera.All, that is,savefor“Gertie,”asshewasknown,whogazedoffcameratoherleft,leavingheralmostinprofile,herleftsidehidden.Gertrudelosthereyeasachild,whenHarold,playingcarelesslywithacricket bat, struck her; she had to wear a glass eye for the rest of her life. Theincident,however,didnothingtodisrupttheirsiblingcloseness,andmayevenhaveenhanced it.Theywere devoted to one another all their lives, kept in close touchalmostastwinsaresaidto,andformanyyearssharedanapartmentinLondon.

•••

In 1880, when Hardy was three, the board of Cranleigh School approved takingtwenty-fouroftheyoungeststudentsandboardingthemseparatelyinwhathadbeenthe sick house across the road, thus freeing up places in the main school andincreasing its income. Running it would be Mr. and Mrs. Hardy, helped by agoverness.This preparatory school (whichmay have existed in some form earlier)wasadistinctoperation,keptseparateintheschool’saccountbooks.In1881,Mrs.Hardy was paid 281 pounds, 12 shillings, from which, presumably, she met theHouse’soperatingexpenses.

The House, as the preparatory school came to be known, was a sprawling,barrackslikeaffairwithadouble-gabledroofthatstooduptheslopefromHorseshoeLaneopposite the school.At least around1881,whenHardywas four, the familylived a few steps down from it toward the road, in a small, two-story bricksemidetachedhouse,trimmedwiththeblackandredscallopedclaytilingubiquitousinSurreysincetheseventeenthcentury.“Mt.Pleasant”the littlehousewasgrandlycalled. It had two small bedrooms on the second floor, and a low-ceilingedsittingroomandkitchen,dominatedbyabigbrickfireplace,onthefirst.Fromthesittingroom, you could see the clock tower of the school across the road and the rosewindowofitschapel.

Around1881,censusrecordstell,thehousewashomenotjusttoHaroldandhismother,father,andsister,butalsotoElizaDenton,thirtyyearsold,whohelpedMrs.HardyattheHouse;twenty-two-year-oldCatherineMaynard,whomayhavebeenthe children’s nurse; Alice Lee, an eighteen-year-old servant; and another servant,LauraChandler, awidowed thirty-eight-year-old—eightpeople, in all, packed intowhat amounted to a cottage. By today’s standards, it must have made for a tightsqueeze.Butontheothersideofthefireplace,inamirrorimageoftheHardyhouse,liveda still larger family—anagricultural laborerandhiswife,her father, twosonsalreadyatworkaslaborerandploughboy,andsixotherchildrenrangingfromelevendowntofourmonths.

Indeed, with only two children, the housemight have seemed toMr. andMrs.Hardyscandalouslyempty—makingitnatural,withthechildrenstillyoungandthefamily’sfinancesprecarious,torentoutsleepingspace,probablyinthebasement,tolocal servants, some of whom may have worked at the school. During the day,presumably,theservantswouldscattertotheirjobs,Mr.Hardytoschool,andMrs.HardytoministertoherflockattheHouse,leavingHaroldandhissisterwithMissMaynard.

TheHardyswerebetteroffthanmanyotherswholivedalongHorseshoeLane—farmersand laborerswith fiveor sixchildren,mostofwhomhad lived theirwhole

lives in Cranleigh or nearby towns like Alfold, Woking, or Dorking. But Isaac’smodest position left them by nomeans flush.The school’s headmaster (principal)madeathousandpoundsayear.Butatatimewhenworkmenmadesixtyorseventypoundsayearandanupper-middle-classpersonthreehundred,theschool’ssecondmastermadeonlyahundred.AssistantmasterHardy,itissafetosay,madeevenless.

TheHardychildren,someonelaterwrote,werebroughtupin“atypicalVictoriannursery”; typical is thekeyword.Whenhewas six,Haroldwasphotographed inastandard-issueVictoriansailorsuit,withabowtie.Hegrewuponaroadthatafewyearsbeforehadbeen littlemore than amud track.His hometown, spreading outhaphazardly from its little High Street of gabled, half-timbered shops, wasunprepossessing.Meanwhile,theschooltowhichhisfamilyhadsuchclosetiessentakindred message: its modest red brick buildings, situated at the end of a slopingdriveway up fromHorseshoe Lane, were handsome enough, but bore none of theweightofhistoryandtraditionboastedbyolderpublicschools.

Indeed,theatmosphereatCranleighSchooldifferedfromanythingHardywouldmeet later. It was not there to groom England’s elite. It was ordinary andunpretentious,shot throughwithakindofyouthful freshness.Onevisitor in1875noted that “Cranleigh boys may wander where they please, and this freedom ischaracteristic of the establishment throughout.” Teachers played on school teamsuntil 1888.Absentwere the rigid hierarchies of the older public schools, the tightrestrictionsonstudentbehavior.Boyswentoffontheirown,smokedpipes,hungoutbehindthegym.

Snowwrote thatHardy enjoyed a fortunate childhood, “enlightened, cultivated,highlyliterate....Heknewwhatprivilegemeant,andheknewthathehadpossessedit.”Indeed,athome,hegrewupwithanemphasisonintellectandlearningthat,inagespast, only the aristocracy enjoyed.But at school,his teachersweremostlynotpublic school boys and everyone was aware of being not Eton, not Harrow, notWinchester.Andso,if“privilege”itwas,itwasofarareandperhapsidealsort—incircumstances at once economically modest and culturally enriched, like theimmigrantJewsoftwogenerationsago,say,ortheimmigrantAsiansoftoday.AndifHardygrewup,ashedid, insistenton intellectualexcellence,yet sensitive to thosesociallyscornedoreconomicallyunlucky,Cranleighmayhavehelpedmakehimthatway.

•••

InJuly1889,J.T.Ward,afellowofSt.John’sCollege,Cambridge,reportedtotheheadmaster at Cranleigh the results of his yearly examinations of students in theschool’supperforms.StudentsatCranleighadvancedthrougheach“form,”orgrade,basedlargelyonmerit,notage;indeed,oneyearamongtheschool’s347boyswasasixth-formerwhowas twenty.Hardyreached the sixth format twelve.But thoughfiveyearsyoungerthanmostotherboys,reportedWard,hesurpassedmostoftheminalgebra,geometry,andtrigonometry.Amucholderboycameinfirstthatyear.ButHardy came next, “far ahead of the third boy, although he has read [studied] noConicSections yet, and a very littleMechanics [the science, not the trade]; takinginto account his present age, I am confident,” wrote Ward, “that he ought todistinguishhimselfgreatlyinthefuture.”

Distinguishhimselfhedid.Cranleighhadnourishedhimbutnowofferedhim,atage twelve, nothing more. His parents, who had the highest aspirations for theirprodigiouslyintelligentson,kepttabsonscholarshipsforwhichhemightbeeligible.Around this time, about the scarcest and most coveted scholarship of all was toWinchester,oneofEngland’smosthallowedpublicschoolsandatraditionalprovinggroundforthemathematicallygifted.Itsscholarshipexamwasamongthetoughestof its kind; one year in the 1860s, 137 boys competed for seven scholarships. Toprepare for it, you normally went to a coach or special school. If youwon it, youturneddownanythingelseyoumightalsohavewon.Itwastheultimatefeatherinaboy’sacademiccap.

Hardyappliedforitandin1890,amongafieldof102candidates,placedfirst.By the early 1890s, Cranleigh School was beginning to shed the easygoing

opennessofitsearlyyears,Hardy’sfatherhadbeennamedbursar,ortreasurer,andthefamilywaslivinginwhatwasprobablyamorespacioushome,“Connel.”Butbythen,G.H.HardywasgonefromCranleigh,headedofftothewiderworld.

3.FLINTANDSTONE

If originally endowed, as were many venerable public schools, to educate poorscholars,Winchesterhadlongagoevolvedintoapreserveforthegentry.AfewhoursbyrailfromLondon’sWaterlooStation,inthecathedralcitybearingthesamename,Winchesterwastherealthing—thesortofplaceyouconjuredup,alongwithEton,Harrow,andRugby,whenyouthoughtofpublicschool,andwhichoccupiedheightstowhichCranleigh’sfounderscouldonlydreamofaspiring.

Overtheyears,theschoolgraduatedmorethanitsshareofthosewhomoneschoolhistorian could aptly class as “gentlemanly rebels and intellectual reformers.” Andtwenty years later, itwouldbe fair to describeHardy as one of them.Winchester

didn’ttryto,ofcourse;itsmoreusualproductswerereserved,patrician,conservativesocial and political leaders. But those who did rebel often became distinguishedrebels.

At Winchester, Hardy found much against which to rebel. From outside theancientcomplex,heconfrontedtheoriginalcollegewall,allflintandstone,piercedbytiny slotted windows; these formidable architectural details stemmed from thePeasants’Revolt,andfrombloodytown-gownbattlesatOxford,bothstillofrecentmemory in the fourteenth century when the school was founded. Scholarshipstudents like Hardy lived in a sort of intellectual ghetto within the college, afortresslikecomplexofmedieval,graystonebuildings,worldsapart fromthesunnyopennessofCranleigh.

As a new student,Hardywas grilled on “notions”—a vast lexicon of jargon andslang peculiar to Winchester. Some of them went back to the Latin, some weresubmergedinthemistsoftheschool’smedievalpast.Collectively,theydefinedgoodformandbad, asWinchester, across the spanof centuries, had come to see them.“Tugs”wasstalenews.To“brock”wastobully.A“remedy,”derivedfromtheLatinremedium,meant“aholiday.”A“tunding”wasafloggingatthehandsofaprefect,orsenior student officer. Learning your notions made for no idle study. There werethousandsofwords,withwholepublishedglossariesgivenovertothem,somegracedwithexquisitedrawingsandilluminatedcapitals.Allhadtobememorized.

Makingsuretheylearnedtheirnotionsandotherwiseconformedweretheprefects.It was Student Power run amok. Even afterMatthewArnold’s reforms, expresslyaimedatcurbingthepoweroftheolderboysatEnglishpublicschools,Winchesterlayinthegripoftheseprefects,wholordeditovertheyoungerandweakerstudents.Conditions were better than a quarter century before, when a particularly cruelincident of tunding—thirty strokes, with a ground-ash stick, across the back andshoulders of a student who bristled at taking his notions exam—had outraged aparentandbroughtinthepress.ButWinchesterwasstillreckonedamongthemostbrutalofthepublicschools.Beatingsweretherule,nottheexception.Andstudentprefects were still left free by school officials to exercise a Lord of the Flies–likesavagery.Theplacewasavastadolescenthierarchy,a tribalsocietybuiltonpower,privilege,andforce.

And tradition. If anything, the current headmaster, W. A. Fearon, wasstrengthening tradition’s grip on the place. While Hardy was there, he began“MorningHills,” a twice-yearly schoolwide trek to the top of St. Catherine’sHillculminating in prayers and a calling of the school roll.He also revived the ancientprocessionsaroundtheCloisters,wherestudentssangthemorninghymn, lam lucis

orto sidere, as they solemnly circled through the stone-arched walkways. After thesweet,softairofCranleigh,Winchesterwas likeawork farm.TheamiablewifeofHardy’smathematicsteacher,SarahRichardson,knownbyeveryoneas“Mrs.Dick,”whoheldopenhouseonSundaywithcakesandbackissuesoftheIllustratedLondonNews, could scarcely do much to ease the grayness. Neither could Fearon’shumanizinglittletripstoLondon.Neither,amazingly,couldcricket.

•••

BackwhenHardywaseight,eventhenthebuddingwriter,heputouthisownlittlenewspaper, complete with editorial, advertisements, a speech by Prime MinisterGladstone—andafullreportofacricketmatch.

AsachildgrowingupinCranleigh,allthroughhisWinchesteryears,andbeyond,Hardy’sworldresonatedtothesoundofcricketbatsandthewhirlofwhiteflannelunder the summer sun. In Britain’s more gentlemanly circles, cricket was asubiquitousandimportantasbasketballisinAmericaninner-cityneighborhoods,orasbaseballoncewasinBrooklyn.ItwasthegoldenageofcountycricketinSurrey,andasayouthHardywouldgototheKenningtonOvalinLondontoseecricketgreatsRichardsonandAbel,thenintheirprime.Backatschool,practicesweresacrosanct;even detention didn’t interfere. The school magazine was heavy with accounts ofmatches—likethisonebetweenschoolandvillagein1888,whenHardywaseleven:

Played at Cranleigh, July 21st, on a very slow wicket. The village batted first, and were opposed byRobinsonandBlaker.Robinsonbowledextremelywell,andnoonebutStreetcouldofferanyresistance,thewholeofthesidebeingdismissedfor53.IntheSchool’sinnings,Douglaswasbowledinthefirstoverandthiswasonlythecommencementofaseriesofdisasters,ourtotalonlyreaching45,Warnerbeingtheonlyonetoobtaindoublefigures.Inthe2ndinnings,thevillagedidmuchbetter....

In cricket, one man hurls a ball, another bats it, and fielders rush to catch it.Soundslikebaseball,asdoesitspenchantforstatistics.Butcricketisamoreleisurelygame,oneinspiredbyquiteadifferentspirit.Theballisdelivered—“bowled”—withastraightarm,attheendofacuriouslopingrun,normallyfirststrikingthegroundinfrontofthebatter.The“batsman,”asheisproperlycalled,usesapaddlelikewoodenimplement to “defend” the “wicket,” a two-foot-high tridentlike affair, stuck in theground, atopwhich lie lightly balanced rods. If the bowler dislodges these rods orinduces the batsman to do so, thewicket is lost.The batsman’s object is to get asmanyrunsaspossiblebeforeheis“dismissed.”

Suchanaccountscarcelydoesjusticetoagamewhoseroots,byHardy’stime,wentbacksixhundredyears,thatinorganizedformhaddominatedEnglishsummersfor

twohundred,thathaditsownrichlore, itsownetiquette, itsownarcanelanguage.(A“stickywicket,”forexample,meansthatthegroundinfrontofthebatsmaniswet,makingtheball’sbouncemoredifficulttoestimate.)Cricket,oneconnoisseurofthegame,NevilleCardus,oncewrote,“isathingofpersonalartandskill;itdependsnotmainlyonresults,butontheamountofgeniusandcharacterwhich isput into it.”Geniusandcharacter:theEnglishtookthegameseriously.

And noEnglishmanmore so thanHardy, towhose days cricket gave almost asmuchmeaning as did mathematics. He devotedly studied his Wisden, the cricketannualcrammedwithbowlingaverages,test-matchresults,andotherarcaniaofthegame. In 1910, the minutes of a Cambridge club would alliteratively cite hiscommand of “theUniversity Constitution, themethods of Canvassing, Clarendontype, and professional cricket.” As a young Fellow of Trinity College, he’d play abastardizedformof it inhisrooms,withwalkingstickandtennisball.He’dplay itmore seriously into his sixties. He reveled in the batsman’s backswing, in tacticalnuance,inthelogicbehindchangingbowlersorpositioningfielders...

ButnowatWinchester, evencricketgavehim littlepleasure.WinchesterwasasobsessedbythegameasCranleigh.Indeed,somehadgrownalarmedbytherampantathleticismembodiedincricket’sholdonstudents.Comesummer,onewriterfortheschoolpaperlamentedinJune1893,threeyearsintoHardy’stenure,“weproceedasiflifewereonelonggameofcricket.”

ButthoughHardywasanaturalathlete,hiscricketskillsatrophiedatWinchester.Hewas, as someone later describedhim, “small, taut, andwiry,” and played othersports well, especially soccer. “Hardy made an excellent rush . . . Hardy movedneatly . . . Hardy was magnificent,” accounts from the period record. But as forcricket,heneverplayedontheteam,andleftWinchesterfeelingslighted.Deniedthecoachingtowhichhefelthistalentsentitledhim,Snowremembershimgrumbling,his defects as a batsman persisted—thus frustrating, at least in his imaginings, abrilliantcricketcareer.

ThatwasonegrudgeHardyboreagainstWinchester;givenhisfixationwith thegame, thatwas probably enough for him to forever hate the place.But therewereothers.Poorteachingcrushedanyartisticabilityheinheritedfromhisfather.Hewasshy and sometimes sickly, in a setting with scant tolerance for such frailties. Onewinter he got so sick he almost died. Later he would feel a shock of envy for thehappier experience of a friend who attended school from home, as a day student.AfterHardy leftWinchester, he couldn’t eatmutton, which was served there, bystatute,fivedaysaweek.Heneverreturnedtovisit.Heneverattendedareunion.

At Winchester, in those days, the classics were still lopsidedly represented. Oftwenty-sixclasshourseachweek,fiveeachwenttoGreek,Latin,andhistory,threetoFrench,twotodivinity,twotoscience,andfourtomathematics.ButHardyprobablyneverattendedmathematics classesas such, insteadworkingalonewith the secondmaster, George (“Dick”) Richardson. Something of an anomaly at Winchester,RichardsonhadattendedCambridge,rankinghighinthemathematicalTriposexamthere,butwasnothimselfapublicschoolgraduate.Hewasnomodelteacher,hardlybotheringwithanybutthebetterstudents.These,ofcourse,includedHardy,whoin1893walkedoffwiththeschool’sDuncanPrizeinmathematics.

ButHardywasnotjustinterestedinmathematics.Hestudiedphysicsonhisown,read Tyndall and Huxley, was a devotee of Ruskin, enjoyed headmaster Fearon’sbrandofhistory.SomidwaythroughhissixyearsatWinchester,itwasnotyetclearwherehisfuturelay.

•••

Duringlaterlife,Hardyhadaweaknessfordetectivestories,andoverthecourseofoneboringLondonweekendreputedlyconsumedseveraldozenofthem;sowhileheenjoyedgoodliterature,hewasnotaboveescapistfare.Onedaywhenhewasaboutfifteen, he came upon a bookwritten a few years before under the nameAlan St.Aubyn, a pseudonym for a certain Mrs. Frances Marshall who, displaying ablejudgment,chosenottoassociatehernamewithit.ThebookwascalledAFellowofTrinity.ItwasaboutuniversitylifeatTrinityCollege,Cambridge.Anditwasawful,its characters silly, its prose insipid. Yet, somehow, it touched a chord in youngHardy, especially its concluding scene, which showsHerbert Flowers, the earnesthero,rejoicinginacademicvictory.

HerbertwasbackagainatTrinity.HeworealongB.A.gownnow,withribbonsonit,thathemadenoattempttohide,andafurhoodoverhissurplice,andsatingreatdignityintheBachelors’seatsinchapelandattheFellows’tableinHall.

HewasaFellowofTrinity!Hesatatthehightablenow,andthegraveportraitsofthefoundersandtheillustriousdeadlookeddown

uponhimapprovingly.Theardours,thesorrows,thestrugglesoftherace,wereallover;onlythebrilliantachievementremained.

ThegreatcloudofwitnessesthatlookeddownfromthoseoldraftersoverheaduponthosewhofeastedtherehadneverapprovedamorenoblyearnedsuccessintherichintellectualhistoryofthepastofTrinity.

Heworehishonoursashehadwornhismisfortunes,withbecomingmodesty,andwaswarmlywelcomedbythegrave,scholarlyoldFellowswhosataroundthegreathorseshoetableintheCombinationRoom.

Perhapsheneverquiterealizeduntilhesatthere,onthatfirstnightofhisFellowshipafterHall,muteandwondering,enjoyingthewalnutsandthewine—andallthatthewalnutsandthewineroundthathorseshoetablerepresentedofscholarlyandphilosophicallearningandculture—howgreathadbeenhissuccess!

Hardywas bewitched. “Flowerswas a decent enough fellow (so far as ‘Alan St.Aubyn’ could draw one),” Hardy reminisced later, “but even my unsophisticatedmind refused to accepthim as clever. If he coulddo these things,whynot I?”He,Hardy,wouldbecomeaFellowofTrinity,andmathematicswouldbehistickettoit.

OmittedfromHardy’saccountwasthatTrinitywas,afterall,Cambridge’scrownjewel, enjoying the richest mathematical tradition of any Cambridge or Oxfordcollege. Still, for a Wykehamist, as Winchester men were known, New College,Oxford,wasthemoreorthodoxchoice.BothWinchesterandNewCollegehadbeenfoundedbyWilliamofWykeham,bishopofWinchesterandtwicelordchancellor,whoconceivedthetwoinstitutionsasone,encompassingalltheyearsofschoolanduniversity.Soofthetwogreatuniversities,WykehamistsnormallypickedOxford,inparticular New College. In Hardy’s year, 1896, forty-six Wykehamists went tovariousOxfordcolleges, fifteenofthemtoNewCollege.Onlyeightwenttoall theCambridgecollegescombined.

Hardywasoneofthem,andonascholarshiptoboot.“CongratulationsareduetoSmyth and Hardy for their brilliant success,” noted TheWykehamist, the schoolnewspaper,withuncharacteristicfervor.“Itisanuniquehonortogaintwoofthefourbestscholarshipsoftheyear;Winchesterhascertainlyneveraccomplisheditbefore,andwedoubtifanyotherschoolcanboastsuchaperformance.”

Back inCranleigh, itwas a fine time tobe aHardy. IsaacHardy’s studentshaddonesowell inthebigartexamination,gushedheadmasterRev.G.C.Allenattheschool’sannualSpeechDay,thattheexaminerhadcomparedthemfavorablytothebestinthecountry.Hearhear,andwarmapplauseeruptedfromtheaudience.Andnow,Allenwenton,Mr.Hardy’ssonhadwonaMajorScholarshiptoTrinity—anachievement which, the local paper wrote, “was one of the highest things evenWinchestercouldexpecttoget.”

4.AFELLOWOFTRINITY

ItwouldbeimpossibletoexaggeratetheroleinBritishlifeofCambridgeandOxford.When Havelock Ellis in 1904 prepared his Study of British Genius, based on athousandorsoeminentBritons,ofthehalfwhohadbeentouniversity,74percenthadbeentooneortheotherofthem.

Cambridge, like Oxford eighty miles to the southwest, was first a town—amidsizedtradingsettlementthatgrewoutofaRomanencampmentonahillnorthoftheRiverCam.Beginninginthethirteenthcentury,scholarscongregatedthereandformed colleges. What ultimately emerged was a federation of colleges more likeAmerica under theArticles ofConfederation—states unto themselves exercising a

jealous grip on their rights and powers—than America under the strong centralgovernmentoftheConstitution;youwerefirstastudentofyourcollege,onlythenofthe university. Until the Senate House was built in 1720, the university had nocentralmeetingplace.

Eachcollegehaditsownhistory,itsownendowment,itsownfacultywithitsownacademicstrengths, itsownpredominantarchitectural style.Eachattracted itsownmixofstudents,whoworetheirowndistinctiveundergraduategowns.Eachhaditsown roster of distinguished graduates. There was Peterhouse, oldest of theCambridgecolleges,establishedin1282.AndKing’sCollege,withitsstoriedchapel—Wordsworth’s“immenseandgloriousworkoffineintelligence!”AndMagdalene,pronounced“maudlin,”atthenorthendoftownacrosstheRiverCam,foundedbyBenedictinemonksin1428andwhoseSamuelPepysLibrarycontainedhis famousDiary.TherewereSt. John’s, Jesus,Gonville andCaius,Pembroke . . . byHardy’stimeclose to twenty inall.Somehad justoneor twohundredundergraduatesandfellows;some,likeTrinity,closertoathousand.

Architecturally, the colleges could be imagined as variations on a powerful andpersistenttheme:facingthestreetwasamoreorlesscontinuouswalloftwo-orthree-storybuildings in stone or brick.Entering the college through a great arched gate,youfoundyourselfinadarkpassage,toonesideofwhichstoodthePorter’sLodge.Thereamaninabowlerhat,whoseaccentinstantlymarkedhimasporterandnotstudent, saw to the daily workings of the college—admitting visitors, deliveringmessages,keepingtrackofkeys.PastthePorter’sLodgewastheenclosedcourtofthecollege,whosesmartlycoiffedgrassplotwould,evenonthegrayestofEnglishdays,leapat youwith green.Around the court, typically on all four sides,werewalls ofbuildings. These included the chapel, the library, and the dining hall, typically agrand, high-ceilinged affair, richly paneled in wood, and studded with framedportraitsofcollegeluminaries.

A student might attend university lectures, but he mostly learned through“supervisions,”attheelbowofacollegetutor,inhisroomswithinthecollege,twoorthree students at a time. He also had a tutor who watched over his “moral”development,andadirectorofstudies,whomonitoredhisscholarlyprogress;these,too,weremembersof thecollege,notdrawn fromtheuniversityat large.Likelyasnot,theundergraduatelived“incollege”—inasetofroomsoffacourtofthecollegereachedthroughoneofthestaircasesspacedeverydozenorsopacesaroundit.

“Rooms” at Cambridge did not mean the bare furnished room conjured up byAmericanstodayin“roomforrent.”Infact,theywereoftensmallapartmentsoftheirown, some of them well furnished and ornate, with fine wallpaper, rugs, framed

paintings,sometimesevenformaldining-roomtables.Cambridgemenweredeemedwhollyunabletocook,clean,orotherwisecareforthemselves.Theywereservedbybedmakersand “gyps,”whobrought coalup to feed the fireplace, took in themail,fetched lunch, set out bed linen, towels, and tea cloths, served tea. Students wereinvariably addressed as Mister and treated with no little respect. But perks atCambridgedidnotextendtothemodernplumbingandconveniencesthatordinarymiddle-class Americans were enjoying by this time. The few baths were clusteredtogether, often across a windswept court from the student’s room. Hot and coldrunningwaterwereunknown.Thegyp’s first job,uponshowingupatseven inthemorning,wastofillatinsaucerwithcoldwaterforwashing.

Trinity was among the oldest of the Cambridge colleges; it was certainly thelargest,andthemostfamous,formedfromtwosmallercollegeseachofwhichwentback to the fourteenth century. It had taken its present form underHenry VIII,whose plump, strutting sculpted figure facedHardy each time hewalked past thebicyclesmassedoutsidethegatetoGreatCourt.

Dominatedbyapicturesquedomedfountainthatwentbackto1602,GreatCourtwas the largest among Trinity’s five courts and, indeed, the largest enclosedquadrangleinEurope,morethanfourhundredfeetdiagonallyacross;youcoulddropabaseballdiamondwithinitanditswallswouldmakeforrespectablydistantoutfieldfences.

Hardy was assigned a room in Whewell’s Court, across cobblestoned TrinityStreetfromGreatCourt,inacomplexofbuildingscompletedjustthirtyyearsearlier.More intimately scaled than most of the rest of Trinity, and a little isolated,Whewell’sCourtwasitsownquietpreserveofstoneandlawn.FromhisroomonthesecondfloorofStaircaseM,HardycouldlookoutthrougharchedGothicwindowsto thebusysidewalk,or to the stone rampartsofSidneySussexCollegeacross thestreet. Yet within the inner sanctum of the court itself, the sound of bicyclesclattering down Bridge Street or of excited conversation among students in JesusLanedidnotpenetrate.

After Winchester, of course, Trinity was not primarily an aesthetic delight forHardy. Winchester was even older than Trinity, or almost anyplace else atCambridge for that matter, its chapel as stately, its courts as tradition bound, itscloisters as imposing. So for nineteen-year-old Hardy, as he signed the TrinityAdmissions Book in 1896, it was what Cambridge and Trinity represented thatmattered.Andwhatitrepresentedwasbeingatthetopoftheintellectualheap.

Hardy, however, had a shock coming—a disillusioning that would lead him toweigh dropping mathematics. It came in the form of a venerable Cambridge

institutioncalledtheTripos.

•••

Theword ispronounced try-poss, and it originally referred to a three-legged stool.Onit, inoldendaysatCambridge,satamanwhose jobwastodispute,sometimeshumorously, sometimes aggressively, the candidate for the degree in mathematics.The word subsequently came to refer generally to the examinations mathematicalcandidates took to earn their degrees. Still later, the word was extended toexaminationsinotherfields—aClassicalTripos,aNaturalSciencesTripos,andsoon.

The mathematical Tripos was impossibly arduous. You sat for four days ofproblems,often late into theevening, tookaweek’sbreak, thencameback for fourmore days. The first half, which stressed quickness and counted even merearithmetical facility, covered the easy stuff; showing even modest aptitude wasenough to earn a degree.The second halfweighed doubly and encompassedmoredifficultmaterial.Here,sometimesonproblemafterproblem,thebrighteststudents,destined for distinguished mathematical careers, would not even know where tobegin. Itwas a frightful ordeal.Recollections invariably lapse into awedhyperbole.The Tripos, wrote one English-born mathematician years later, “became far andaway the most difficult mathematical test that the world has ever known, one towhichnouniversityofthepresentdaycanshowanyparallel.”

ButtheTriposwasmorethananexamination.Itwasaninstitution.BythetimeHardyarrivedinCambridgein1896,itcomprisedtheacademicritualssurroundingit, the esteem accorded it, the system built to support it, even the style ofmathematics it encouraged, all rolled into one. The Tripos went back, in earliestform, to 1730 and had always been difficult. But over the years, as it grew moredemanding andmore important, it also, in inimitableEnglish fashion, tookon theluster—anddeadweight—ofTradition.

Tripos candidates were ranked on the basis of their performance, and a wholeritual surrounded the reading of theOrder ofMerit at the SenateHouse. Thoseranking in the first of three classeswere deemed “Wranglers,” the topmost amongthembeingnamedSeniorWrangler; intheearlydaysoftheTripos,thedisputants“wrangled” over points of logic. To learn who the SeniorWrangler was, everyoneflocked to the Senate House. Even women, normally banished to the fringes ofCambridgelife,showedup.Mingledreverenceandsexappeal,ifitcanbeimagined,accruedtotheWranglers,who“usuallyexpected,”aCambridgevice-chancellorwas

advisedback in1751, “that all the youngLadies of theirAcquaintance . . . shouldwishthemJoyoftheirHonour.”

If anything, the Joy and Honour were greater by Hardy’s time, the SeniorWrangler and those just behind him in the order of merit earning applause andhurrahsfromfriendsandcollegemates.Atgraduation,thevice-chancellorsatonthedaisatoneendoftheSenateHouse,andcollegetutorspresentedtohimthosetakingtheir degrees. The recipient kneeled before him, and the vice-chancellor took hishandbetweenhisownandrepeatedinLatintheawardingofthedegree.

StandingbelowtheWranglersweretheSeniorOptimes,orsecondclass,andtheJunior Optimes, or third. When the lowest scoring of the Junior Optimes—thebottom-rankingman in the class—went to receive his degree, his friends solemnlylowered from the SenateHouse gallery theWooden Spoon. In fact itwas a hugemaltingshovel,aslargeasthemanreceivingit,inscribedinGreek,andflamboyantlydecorated.Upon rising tohis feet,he’d take theungainly thing and, shouldering ittriumphantly,stridefromthehallwithhisfriends.

Of course, the Wooden Spoon was strictly a consolation prize. The honoraccordedtheSeniorWrangler,ontheotherhand,wasnolaughingmatter,butclungtohim, like an aura, for a lifetime. “If onepersonwere consensusAll-American, aRhodesscholar,andBacheloroftheYear,”observedoneaccountaimedatAmericanreaders,“hewouldnotcomeclosetocommandingthelastingdistinctionthatcametotheSeniorWrangler.”Toalesserextent,Wranglersamongthefirsttenorsosharedsomeofthegloryandwerevirtuallyguaranteeddistinguishedcareers.ObituariesofEnglishmathematicianshalfacenturylaterinvariablynotedthatyouhadoncebeenSeniorWrangler, or SecondWrangler, or Fourth.As a history of theCambridgePhilosophical Society accurately noted, “the SeniorWranglerwas not invariably agreatmathematician.. .[but]itwasvirtuallycertainthathecould,ifhewished,beaninfluentialone.”

AroundCambridge and to an extent elsewhere in Britain, the SeniorWranglerwasacelebrity.PeoplerelatedtohimmuchastheydothewinneroftheKentuckyDerby—withoutknowingthefirstthingabouthorseflesh.Hewasastar.TheTimesof London invariably recorded his elevation. Picture postcards bearing hisphotographic likeness were sold around town. One from around the turn of thecentury shows the victor sitting outside, posed on an ornate chair, polished shoesgleaming, hands demurely clasped together—the Derby winner shown off to thepress.

All this was very nice in its way, very British. No one questioned that Tripossuccess reflected, more or less, mathematical ability. Nor that, if anyone was to

become a great mathematician, it would be the Wrangler and not the WoodenSpoon.On theotherhand,precise ranking in theOrderofMerit, everyoneknew,meantfarlessthanitseemed.Indeed,itwasanopensecretthatSecondWranglersseemed todistinguish themselvesmore thanSeniorWranglers.Maxwell, the greatmathematical physicist who united electricity and magnetism, had been SecondWrangler. So had J. J. Thomson, discoverer of the electron. Then there wasthermodynamicist Lord Kelvin, then still William Thomson, surely the bestmathematicianofhisyear.Everyone,includinghimself,thoughthewasashoo-inforSeniorWrangler.“Oh,justrundowntotheSenateHouse,willyou,andseewhoisSecondWrangler,”he askedhis servant.The servant returnedand said, “You, sir.”Someone else, his name today forgotten, had proved better able tomasterTriposmathematics.

Andthatwastheproblem:therewassuchathingas“Triposmathematics,”anditbore little kinship to anything of interest to serious,workingmathematicians.TheTriposwas trickyandchallenging, andcertainly separated theWranglers fromtheWoodenSpoon; in1881, for example, theSeniorWrangler got a score of 16,368“marks,”outofapossible33,541,theWoodenSpoon247.Buttherewasadatednessto the problems, a preoccupation with Euclid, and Newton, and exercises inmathematicalphysics—aspherespinningonacylinderwiththecandidateaskedtoestablishtheequationsgoverningitsmotion,oraproblembasedonCarnot’sCycleinthermodynamics,andsoon.Theydemandedaccuracyandspeedinthemanipulationofmathematicalformulas,ashallowcleverness,perhaps,butnotrealinsight.

And not even stubborn persistence; a proof demanded by a Tripos questioncouldn’tbetoo longor too involved;soyou learnedto lookfor thathiddenTripostwist. During one Tripos exam, a top student—that year’s Senior Wrangler—observed a less capable candidatemaking short work of a problem over which heagonized.Must be a trick, he realized—and went back and found it himself. ThepersonalqualitiesencouragedbytheTripos,J.J.Thomsonwouldmakesoboldastosuggest,madeitexcellenttraining—forthebar.

InhomagetotheTripos,analternativeeducationalsystemhadsprungup.Inthenineteenthcentury,universitylecturersatOxfordandCambridgeinhabitedaworldof their ownwith scant contact with students. Teaching was left to tutors in theindividual colleges, who were hardly up to the task of preparing students for theTripos.SoaThirdForcehademergedtofillthevacuum—privatecoaches.

These coaches were not out to teach youmathematics for its own sake, but toschoolyou,forhandsomefees,inTriposmathematics.Youweretrained,onefutureSeniorWrangler wrote, like a racehorse. In groups of four or five, you and your

coachwentoverproblems.Maybethreetimesaweek,helecturedtoyou.Heporedoveroldexaminations,wrotelittleprivatetracts,codifyingmathematicalknowledgeinto neat bundles. Only rarely did a coach himself produce important newmathematicalresults.But“produce”hedid—SeniorWranglers.Atonepoint,coachE.J.Routhchurnedoutsometwodozenoftheminasmanyyears.

Forstudents,theworkloadwasprodigious.Onedistinguishedmathematician,J.E.Littlewood,wrotethat“tobeintherunningforSeniorWrangleronehadtospendtwo-thirds of the time practicing how to solve difficult problems against time.”Attending lectures became a luxury. Of his professors in the late 1870s, A. R.Forsythhaswrittenthatthey

didnot teach us;we did not give them the chance.Wedid not read theirwork: itwas asserted, andwasbelieved,tobeofnohelpintheTripos.Probably,manyofthestudentsdidnotknowtheprofessorsbysight.Suchanoddsituation,formathematicalstudentsinaUniversityfamedformathematics,wasduemainly,ifnotentirely,totheTriposanditssurroundingswhich,asundefinedasistheBritishconstitution,hadsettledintoapositionbeyondthepaleofaccessiblecriticism.

ThatwashowthingsstoodinForsyth’stime,andthat’showtheyremainedduringHardy’s. The Tripos system discouraged exploration of any area of mathematics,however personally satisfying, not apt to show up on the examination. It grantedprofessionalsuccess—afellowshipatagoodcollege,say—tothosedoingwellontheexam,notthosedemonstratingabentforresearch,orboldnessinpursuingit.Tripossuccess became, like marriage for the prototypical Southern belle, not a happypreludetoone’slifebutitsculmination.

Hardywasdrawnupintothesystem,inhisfirsttermbeinghandedovertoR.R.Webb, the day’s foremost producer of Senior Wranglers. At Cambridge, theacademic year consisted of three terms of seven or eight weeks. Typically, for tenstraighttermsthecoachesdidtheirwork,andtheirstudentsdidtheirs.Untilfinally,onacoldJanuaryintheunheatedSenateHouse,thestudentwouldsitdowntobegintheTriposordeal.ThiswaswhatHardyhadtolookforwardto.

HewasnotaloneamongthemathematicallygiftedrevoltedbythebanalityoftheTripos.BertrandRussell,whorankedasSeventhWranglerwhenhetooktheTriposin1893,andwhowouldmakenumerouscontributionstomathematicalphilosophyintheyearsahead,wrotelaterhowpreparingforit“ledmetothinkofmathematicsasconsistingofartfuldodgesandingeniousdevicesandasaltogethertoomuchlikeacrosswordpuzzle.”TheTriposover,hesworehe’dneverlookatmathematicsagain,andatonepointsoldallhismathbooks.

ButRusselldid go through it.Andsodidmostothers.Hardy’s futurecolleague,Littlewood,laterconfidedthathe,too,hadseenTriposmathematicsasempty.But

he simply gritted his teeth and, as Hardy told the story years later, “decideddeliberatelytopostponehismathematicaleducation,andtodevotetwoyearstotheacquisitionof a completemasteryof all theTripos technique, resuminghis studieslater with the Senior Wranglership to his credit and, he hoped, without seriousprejudice to his career.” And that, Hardy declared “in hopeless admiration,” waspreciselywhathedid.

ButtheyoungHardy,newtoCambridgeanddeeplydisenchanted,begantothinkhe just couldn’t go through with it. It was all too stupid. Maybe he would quitmathematicsaltogether.

•••

“Icannotremembereverhavingwantedtobeanythingbutamathematician,”Hardywouldlaterwrite.“Isupposethatitwasalwaysclearthatmyspecificabilitieslaythatway,anditneveroccurredtometoquestiontheverdictofmyelders.”Modestly—and falsely—hewoulddisclaimall artistic ability.As a philosopher, hedecided, hewouldhavebeeninsufficientlyoriginal.Onlyasajournalistmighthehavemadeagoofit.Buttheseruminationslayalmosthalfacenturyinthefuture.Atthetime,itwasnoneofthesefieldsthatheweighedentering,buthistory...

Inabout1064,twoyearsbeforebeingkilledbyanarrowintheeyeattheBattleofHastings,Harold, son of Godwin, swore to PrinceWilliam ofNormandy not tocontest theEnglish throne, and in returnwas crownedEarl ofWessex.The scenewasdepictedinthefamousBayeuxtapestries,andasaboy,HaroldHardyhadtakenthatmasterpieceashismodelforabrightlycoloredilluminationofsurpassinggrace,ontowhichhedutifullycalligraphedthecaption“YeCrowningeofYeEarlHarold.”Later,WinstonChurchill,amongothers,wouldalsotellthestoryofHarold,inTheBirth of Britain, the first of his epic four-volume History of the English-SpeakingPeoples.AnditwasacomprehensivehistorysomethinglikethatwhichHardy,attheage of about ten, dared undertake. Unfortunately, he approached it in overmuchdetailandsonevergotmuchpastthegoodEarlHarold.

AnhistoricalessayHardyhadwrittenonenteringTrinityimpressedhisexaminerssomuchhe could aswell have gotten his scholarship in history as inmath.Now,recallingHeadmasterFearon’s treatmentof the subjectasabright spotof thedrabWinchesteryears, he flirted with changing fields.Hemight well have gone aheadwith it. But in themidst of his confusion, hewent to his director of studies,whobrought another influence to bear on his malleable young mind in the person ofAugustusEdwardHoughLove.

Athirty-three-year-oldmanwithahuge,bushymustache,mutton-chopsideburns,andavastbaldovalofahead,Lovehadbeennamed,afewyearsbefore,aFellowoftheRoyalSociety,Britain’smostdistinguishedscientificbody.In1893,he’dfinishedhistwo-volumeTreatiseontheMathematicalTheoryofElasticity, summarizingwhatwas thenknownofhowmaterialsdeformunder impact, twisting, andheavy loads.ButLovedidnotpushHardyintohisownfield.Thoughanappliedmathematician,hehadabentforfundamentals,basicprinciples,abstractformulations.Once,talkingwithafriendwhowasexplainingsomethinggeometrically,Loveshookhishead,andsaidhedidn’tfollow.“Youseeitisallx,y,zforme,andnotyourpicturesatall.”Andso,perhapsattunedtoHardy’snaturalbent,hesuggestedHardyreadamathematicaltext alien to the libraries ofmost appliedmathematicians, the FrenchmanCamilleJordan’sCoursd’analysedel’EcolePolytechnique.

WhenJordandiedalmost threedecades later, itwouldbeHardywhowrotehisobituaryinProceedingsoftheRoyalSocietyandtooktheopportunitytocommentonthebook that changedhis life: “Tohave read it andmastered it is amathematicaleducationinitself,”Hardywrote,“anditishardlypossibletooverstatetheinfluencewhichithashadonthosewho,comingtoitasIdidfromtheelaboratefutilitiesof‘Tripos’ mathematics, have found themselves at last in [the] presence of the realthing.”

Jordan and other Continental mathematicians were taking seemingly obviousmathematical concepts and subjecting them to the most searching scrutiny. Forexample, mathematicians spoke of “continuous functions”—relationships betweenvariables unmarked byweird and abrupt lurches.And in just such vague, intuitivewaysdidtheytendtothinkofthem.Butwhat,exactly,wasafunction?Andwhatdiditmeantosayitwascontinuous?Among“analysts,”asthisbreedofmathematicianwasknown,justsuchquestionsweretheirmeat.

Sayyoudrawa circleonapieceofpaper;obviously, the circledivides thepaperintotworegions—withinthecircle,andoutsideit.Now,sayyou’vegottwopointsonthe paper, both of which lie outside the circle: surely you can connect them (notnecessarilywithastraightline)withoutcuttingthecircle,right?Andjustassurely,ifonepointliesoutsidethecircleandtheotherinside,thenanycontinuouslinelinkingthemhastocutthecircle...

“Surely?”“Justassurely?”ForEnglishmathematiciansuntouchedbytheprecisionoftheContinent,suchnotionswereobvious,scarcelyworthyofanotherthought.ButJordanactuallystatedtheseseeminglyself-evidenttruthsastheoremsandsetabouttryingtoprovethemrigorously.Infact,hecouldn’tdoit,oratleastnotcompletely;hisproofswere lacedwith flaws, andhis successorshad later to correct them.But

theyinvokedjustthekindofclose,sophisticatedreasoningthatHardy,cominguponJordannow,attheageofbarelytwenty,foundbeguiling.“Ishallneverforget,”Hardylater wrote of Jordan’s book, whose second, much-improved edition had justappeared in 1896, “the astonishmentwithwhich I read that remarkablework, thefirstinspirationforsomanymathematiciansofmygeneration,andlearntforthefirsttimeasIreaditwhatmathematicsreallymeant.”

Ofcoursenoneof thishadanything inthe least todowiththeTripos,which—nowamathematicianagainmore thanever—Hardywouldhave to take.And that,realistically speaking, meant studying with a coach. So Hardy—yes, his friendLittlewood later seemed to gloat, “even the rebel Hardy”—surrendered. It was abitter pill; “real” mathematics was put aside, Tripos mathematics reluctantlyembraced. “When I look back upon those two years of intensive study,” hewouldrecall later, “itseemstomealmost incrediblethatanyonenotdestituteofabilityorenthusiasm shouldhave found it possible to take somuch trouble and to learnnomore.”

Inaugurating something of a trend among the better students, Hardy took theTriposattheendofhissecondyear,ratherthanhisthird,thusfreeinghimselffromitsgrip thatmuch sooner. Still hemanaged to come in fourth—by anymeasure asuperb performance. “Hurrah,” one friend, the future historian G. M. Trevelyan,wrotewhenHardynotifiedhim.“Itisagreattriumph,notonlyforyoubutthegoodcauseoftakingthetriposesinthesecondyear.”AndyetforallhiscontemptfortheTripos system,Hardy told his friend Snow later, it rankled that hewasn’t SeniorWrangler. “He was enough of a natural competitor,” wrote Snow, “to feel that,thoughtheracewasridiculous,heoughttohavewonit.”

From then on, though, Hardy’s stature among his generation of Englishmathematicians steadily rose. In 1898, when he was twenty-one, his firstmathematical publication appeared; like Ramanujan’s, it was in the form of aquestion,followedthreeissueslaterbyitssolution.Hardygraduatedin1899andthefollowingyeartookPartIIoftheTripos;PartIwastheoneforwhichthecoachesgroomedtheirstudentsandwhichdeterminedWranglerstanding,PartIIthemoreprovocativeandmorechallenging.Onit,HardyscoredfirstandwaspromptlynamedaFellowofTrinity.

5.“THEMAGICAIR”

Atitstwo-hundred-fiftiethmeeting,onMay23,1901,theCambridgeShakespeareSocietyheldareadingofthefirstthreeactsofTwelfthNight.Hardywasthere—notto performbut, as his friends deemed his role, as “critic.”With himwas his close

friend R. K. Gaye, a classical scholar, who played Malvolio. Others lendingretrospective luster to the evening were Lytton Strachey, playing Maria, LeonardWoolf inthedualroleofValentineandtheCaptain,andJ.T.Stephen,thefutureVirginiaWoolf’sbrother,whoplayedSirTobyBelch. JamesHopwoodJeans,whocameuptoTrinitythesameyearasHardyandwouldenjoyalmostasdistinguishedacareer in applied mathematics as Hardy would in pure, took photographs bymagnesiumlight.Whatapitytheydon’tsurvive!Themensportedlavishcostumes,sparingnothinginthedetailtheyexpendedontheladies’gownssomeofthemwore.Gayeworetheyellowstockings,cross-garteredofShakespeare’stext.Stephenworeastomacher,anelaboratelyembroideredchestpiece.Woolfworehisoldbowlerhat.Itwas a madcap evening. The reading went well enough but, as the secretary wasdelighted to record, “elementary stage rules, such as facing the audience, wereneglectedbythosewhoshouldhaveknownbetter.Letushopeitwasmodesty!”

A fewmonthsbefore,on January22,QueenVictoriahaddied; thecentury thatborehername,withitsstraitlacedproprietiesanditsmoralfervor,wasover,andthefirststirringsofthenew,morerelaxederaheraldedbyEdward’sreigncouldbefelt.

OnSeptember3,HardywasinCranleighforthefuneralofhisfather,who’ddiedfourdaysbefore,attheageoffifty-nine.Therehejoinedhissisterandmotherintheprocessionwhichborehisfather’sbody,inawheeledbiercoveredwithwreathsandlacedwithivy,fromtheHouse,totheschoolchapel,andthencethroughthevillagetotheparishchurch,intheshadowofwhosesquat,stonetoweritwaslaidtorest.

BackatTrinity,wherehenowoccupiedroomsinGreatCourt,twenty-four-year-oldHardytookuphisold lifeagain.Shyandsensitive thoughhewas,Hardyhad,sincetakingtheTripos in1898,becomesomethingofasocialanimal.Hewasnowpart of a circle to which the sons of schoolteachers normally did not belong andwhose values his family might scarcely have been able to imagine. He attendedShakespeareSocietymeetings.HebelongedtoDecemviriandMagpie&Stump,twoCambridge debating societies, which took up such questions as “An enlightenedselfishnessisthehighestvirtue,”and“ThecompletedegradationoftheLatinracesisfinal and irrevocable.” But he also belonged to another group, which neitheradvertised its meetings, nor openly recruited members, nor even proclaimed itsexistence.

WoolfandStracheywerebothpresentor futuremembersofasecret intellectualsocietyknownastheApostles.SoweretwoothersintheTwelfthNightcast.AndsowasHardy.

Ithadbeenstartedin1820astheConversazioneSociety,oneofmanyCambridgestudentgroupsdevotedtodebateandgoodfellowship.Foundedasitwasbytwelve

men,itsoonbecameknownastheApostles,andintimeassimplytheSociety.BythetimeHardyjoined,itsrankshadincludedsomeofthemostbrilliantmenCambridgehad ever produced. There was Tennyson, the poet; Whitehead, the philosopher;JamesClerkMaxwell,thephysicist;BertrandRussell;andmanyotherswhosenamesare less recognizable today.Of the tenwhomLeonardWoolfwould later countasthenucleusof“OldBloomsbury,”theiconoclasticintellectualcommunitythatwouldreachfullflowerinthe1920sand1930s,nofewerthansevenwereApostles.

Ascientist,asoneApostlenotedroundthistime,waselectedonlyif“hewasaverynice scientist.”Hardy, apparently,was nice enough; his crystalline intellect, his slycharm,his aesthetic sensibilities,his good looks,his loveof goodconversation—allthesewouldhaveendearedhimtosuchmen.Actually,forreasonsunknown,hefirstdeclinedmembership.Butfinally,inagala,well-attendedmeetingoftheSocietyontheeveningofFebruary19,1898,hewasinducted.

His“father,”themanputtinghimupformembership,wasG.E.Moore.LeonardWoolf would later write of Moore that he pursued truth “with the tenacity of abulldog and the integrity of a saint.”AnApostle since 1894, hewas a charismaticphilosopherwhobrookednoimprecisioninthought,feeling,orexpression;alwayshewasasking,Whatdoyouexactlymean to say?Hisphilosophy, setout inPrincipiaEthica in1903,coupledwithwhatWoolfwouldcall“hispeculiarpassionfortruth,for clarity and common sense,” made him the dominant influence on his wholegenerationofCambridgeintellectuals,andultimatelyonBloomsburyasawhole.

EachSaturdaynighttheApostlesmet. (Thenormal farewas“whales,” thenamegiven to sardines or anchovies on toast.) One of them read a paper on someintellectual topic, such as, “IsAny EventNecessary?”Or, “CanMoral PhilosophyProvideAnyAntidote forUnhappiness?”Or, “DoesYouthApproveofAge?”Thequestionwould thenbedebatedandput to the floor for a vote. (On that lastone,Hardyvotedno.)Thegroupwantedfornothinginarroganceandpreciousness,asitspeculiarjargonsuggested:An“embryo”wasacandidateformembership,a“birth”hisinduction ceremony. “Phenomena”was the rest of theworld—anything or anyonenotanApostle;“reality”wastheSociety,itsmembersandactivities.

“There were to be no taboos, no limitations, nothing considered shocking, nobarriers to absolute freedom of speculation,” wrote Bertrand Russell, who wasinductedsixyearsbeforeHardy.“Wediscussedallmannerofthings,nodoubtwithacertainimmaturity,butwithadetachmentandinterestscarcelypossibleinlaterlife.”Meetingslastedtillaboutoneinthemorning,followedbyinformaldiscussionupanddown the cloisters ofNevile’sCourt, atTrinity,which alongwithKing’s suppliedmostofthegroup’smembersduringthisperiod.“Thesoulofthething,asIfeltit,”

wroteGoldsworthyLowesDickinson,whohadjoinedin1885,“isincommunicable.Whenyoungmenaregrowinginmindandsoul,whenspeculationisapassion,whendiscussion is made profound by love, there happens something [that would beunbelievable]toanybutthosewho...breathethemagicair.”

A classical scholar byway ofKing’s, a short, vigorousman of formidable charmwho had briefly swerved into medicine and later would become a peace activist,Dickinsonwas also a homosexual.Not all theApostleswere homosexual, perhapsnot evenmost; and yet a homosexual current ran through the Society and, in theyears afterHardy joined, intensified. In 1901, E.M. Forster,whowas towriteAPassagetoIndia,joined;hewashomosexual.In1902,LyttonStracheyjoined,andin1903 JohnMaynardKeynes, the economist (who once advisedHardy that hadhefollowedstocksasavidlyashedidcricket,hewouldhavebecome rich).Bothwerehomosexual, too, and their election, as one chronicler of the Society later put it,transmuted “its naughty verbal mannerisms and Walt Whitmanesque feelings ofcomradeship into overt full-blooded—almost aggressive—homosexuality.”Homosexualitywaselevatedalmosttothestatusofanartformoraestheticdoctrine.“TheHigherSodomy,”Apostlestermedit—theassertionthatthelovebetweenmanand man could be higher and finer than that of man for woman, thus raisinghomosexualrelationshipstoanalmostspiritualplane.

At the timeHardy joined, theApostleshadnotyet reached thepointwhere, asDuncanGrant would put it, “even the womanisers pretend to be sods, lest theyshouldn’t be thought respectable.” But while he formally “took wings” from theSocietyjustbeforeStracheyandKeynesjoined,Hardymaintainedfriendshipswith“brethren,”manyofthemhomosexual,foryearsthereafter.

ThatHardyhimselfwasatleastofhomosexualdispositionisscarcelyindoubt.Nowoman,asidefromhismotherandsister,playedtheslightestsubstantiveroleinhislife.Andhehadnumerousmalefriendsofwhomhewaspassionatelyfond.

In 1903, for example, he shared a double suite of rooms at Trinity with R.K.Gaye,whowasalsoafellowofthecollege.Gaye,whohadjoinedintheTwelfthNightrevelry two years before, entered Trinity the same year as Hardy and showedevidence of being as brilliant a classical scholar as Hardy was a mathematician.Already,attwenty-five,hehadreceivedahostofmedalsandprizes,soontoincludetheHarePrizeforhisPlatonicConceptionofImmortalityandItsConnexionwith theTheoryofIdeas.Muchlater,hewouldcommitsuicide.Butfornow,LeonardWoolfwouldwrite,heandHardy“wereabsolutelyinseparable;theywereneverseenapartand rarely talked to other people.” Woolf records them in their rooms sitting in

domestic tranquility beside the fire, quiet and dejected after returning from theveterinarianwiththeiremaciated,worm-riddencat.

AyoungermathematicianwhoknewHardymuchlater,whenduringthe1920shewas atOxford, says that therewas indeed a “rumor of a youngman” then. Later,whenHardyvisitedAmericainthe1930s,hewouldimpressthemathematicianAlanTuring, himself homosexual, as, in the words of his biographer, Andrew Hodges,“just anotherEnglish intellectualhomosexual atheist.”Andduring thisperiod, too,he would meet an Oxford man many years his junior to whom he would laterdedicateabookandwhomoneaccountsimplyreferstoas“hisbelovedJohnLomas.”

Butallthatcamelater,andifHardywasapracticinghomosexualduringhisearlydays as a Trinity undergraduate and fellow, he was remarkably discreet about it.Severalofthosewhoknewhimwell,whilewillingtoseeinhimhomosexualleanings,notethatheneverdisplayedanyofthestereotypedmannerismsofdressandbehaviorimputed tohomosexuals.Littlewood,whoworkedwithhimforalmost fortyyears,called him “a non-practicing homosexual.” Then, too, no explicit record ofhomosexual activity comes down to us, none of the kind of blunt, gossipyX-was-lovers-with-Ytalesthat,inrecentyears,surroundothervenerableCambridgefigures.Indeed, the only sure knowledge we have of Hardy as sexual being is that oneSaturday night in 1899, when the Apostles debated whether masturbation (theycalledit“self-abuse”)wasbadasanend,asopposedtoameans,Hardyvotedwiththeayes.

Hardywasaproductof theEnglishpublicschools,amonasticenvironmentthatserved as a crucible for homosexual relations among the boys. Everyone knew it,everyoneaccepted it, no one took it seriously, and formost, such adolescent flingsdidn’t carry into adult life. In their pioneering look at homosexuality, SexualInversion, published in 1897, the year afterHardy leftWinchester,HavelockEllisand John Addington Symonds took note of “these school-boy affections andpassions,”notingthatmostpassedwithtime.

IfHardydidhavehomosexualexperiencesduringthistime,hewouldscarcelyhavebeen likely to bruit the news about. A suggestion that homosexuality might beinstinctual,andwasnopropermatterfortheexerciseofmoralcontrol,camethroughEdwardCarpenterinHomogenicLove,publishedin1896,andTheIntermediateSex,published in 1908. But these were nomore than bubbles of tolerance in a sea ofhomophobia. Indeed, the year 1895, while Hardy was in his teens, brought theanguishoftheOscarWildetrials,inwhichthegreatdramatistwassentencedtotwoyears’hard labor forhomosexual acts;hiswife changedhernameand thoseofhertwo sons as well, andWilde’s plays stopped being produced.That was the norm;

CarpenterandHavelockEllis,whoalsowroteofhomosexualityasnatural,werethebarestbreathintheculturalwind.“Thelastfiveyearsofthecenturyweremuchlessopen to discussions of love, sex, andmarriage,” one chronicler of the era, SamuelHynes,haswritteninTheEdwardianTurnofMind.“ItwasasthoughtheVictorianage,initslastyears,haddeterminedtoberelentlesslyVictorianwhileitcould.”

Hardy reached young adulthood in the early 1900s, with its freer Edwardiansensibilities.Buthis attitudeshadbeen formedearlier, in theVictorianera, a timepreoccupiedwithpublicmorality.Itwasatimewhenthefaceyouputonthings,asGertrudeHimmelfarbhasstressedinMannersandMoralsAmongtheVictorians,wastakenwiththeutmostseriousness.Whateverpeopledidonthesly,theywerecarefulto keep it quiet, and led whole lives that way; it wasn’t “hypocrisy,” but simplepropriety. Vastly less play was given “natural” drives. If anything, the idea was tobottle them up, or channel them, but in any case to control them; whatever yourprivatefantasies,youdidn’thavetoactonthem,forgoodnesssake,andshouldn’t.

Hardy’sCambridgeintheearly1900swasacuriousblendoftheoldVictorianismandthenew,freerEdwardianism.IfwithintheApostlesandinCambridgegenerallythe homosexual undercurrent surged toward the surface, it never broke through.Evenamongthese,themostavantoftheavant-garde,Victorianechoessounded.AsHimmelfarb has pointed out, referring to the Bloomsbury movement that largelygrewoutoftheApostles:

It is ironic thatpeoplewhopridedthemselvesontheirhonestyandcandor,especially in regard to theirmuch-vaunted “personal affections”—in contrast, as they thought, toVictorian hypocrisy and duplicity—shouldhavesucceededforsolonginconcealingthetruthaboutthosepersonalaffections.Evensoperceptiveandpsychoanalytic-mindedacriticasLionelTrillingwasabletowriteafull-lengthstudyofForsterin1943withoutrealizingthathewasahomosexual.NordidRoyHarrod,inhisbiographyofKeynespublished in1951(thedefinitivebiography,asitseemedatthetimeandasitremainedformorethanthirtyyears),seefittomentionKeynes’homosexuality—adeliberatesuppression,sinceHarrodwasafriendofKeynesandwasperfectly well aware of his sexual proclivities and activities. Nor did Leonard Woolf, in a five-volumeautobiographythatwasentirelycandidabouthiswife’smentalbreakdowns,giveanyindicationofthefreneticsexualaffairsofeveryonearoundhim.

Given the times out of which they came, then, that Hardy’s long-time friend,Littlewood, called him “a non-practicing homosexual” maymean little. Littlewoodwas himself known to have fathered a child by a married woman but would notpubliclyacknowledgeheruntilnearhisdeathin1977.(WhenhedididlymentionitonedayintheCombinationRoomatTrinity,hewasstunnedthatnooneseemedtocare.)Homosexuality,ofcourse,defiedayetmorerigidtaboo.SoifHardydidleadahomosexuallifeaboutwhichLittlewoodknew,Littlewoodmightwellhavedeniedit.

Thenagain,Littlewoodmayhaveknownnothing,Hardykeepingthevariouspartsofhis life scrupulouslywalled off fromone another.Hardy knew all theApostles,went tomeetings.Andhisname showsup indiaries,memoirs, andbiographies ofStrachey,Forster,Woolf,Russell,andtheothers.Butitdoesn’tshowupmuch;onesenseshimat theedgeof theirworld,not its center.Being amathematician, andapuremathematicianatthat,mayhaveisolatedhim;withintheShakespeareSociety,forexample,hewasgentlyribbedforputting“hisknowledgeofhighermathematics”to use in calculating the tab for a recent dinner at five shillings, one penny—an“alarmingsum.”Intheavant-gardeworldofwhichhewaspart,Hardywastolerated,respected,appreciatedforhisinimitablepersonalcharm;butthecoreofhislife,onegathers,layelsewhere.

Inmathematics?Certainly.Inahomosexualunderworld?Onlyperhaps.ThatHardy’slifewasspentalmostexclusivelyinthecompanyofothermen,that

hescarcelyeversawawoman,was, inthosedays,notuncommon.Afterall,amongHavelockEllis’sthousandorsoBritish“geniuses,”26percentnevermarried.IntheacademicandintellectualcirclesofwhichHardywasapart,suchamonasticsortoflifeactuallyrepresentedonepoleofcommonpractice.

Thus,atCranleighSchool,alltheteachers,exceptfortheHousestaff,weremen,mostofthembachelors;dormitorymastershadtobebachelors.Winchesterwasthesame way. So was Cambridge. “Inmy day we were a society of bachelors,” wroteLeslieStepheninSomeEarlyImpressionsofhis timeatCambridgeduringtheearly1860s. “Idonot rememberduringmy career tohave spoken to a singlewomanatCambridgeexceptmybedmakerandthewivesofoneortwoheadsofhouses.”

NotmuchhadchangedbythetimeHardyreachedCambridgeagenerationlater.Among the twenty or so colleges, two—Girton and Newnham—had beenestablished for women in the previous two decades. But thoughwomen, with thelecturer’sconsentandchaperonedbyawomandon,couldattenduniversitylectures,by1913theystillkeptmostlytothemselvesandplayedlittlepartinundergraduatelife. Until 1882, college fellows couldn’t marry, but even after that most fellowsremainedbachelors.In1887,aproposalwasmadetoofferdegreestowomen;itwassoundly defeated. Ten years later, on a May day in 1897, a straw-hatted mobthronged outside the Senate House, where the matter was again being taken up,demonstratingagainst themeasure.Awomanwashanged ineffigy.Alargebanneradvised (after Act II, Scene I of Much Ado About Nothing) “Get you to Girton,Beatrice.GetyoutoNewnham.Here’snoplaceforyoumaids.”

Itwasanalmostlaughablyartificialenvironment,withdonsleftwoefullyignorantofdomesticlife.OnetimeatSt.John’sCollege,thestorygoes,anelderlybacheloratHighTable congratulated someone on the birth of his son. “How old is the littleman?”heasked.

“Sixweeks,”camethereply.“Ah,” said the bachelor don, “just beginning to string little sentences together, I

suppose.”AbouttheonlytimeHardyandotherfellowsencounteredwomenwasamongthe

bedmakerswhotidiedupcollegerooms—andtheyweresaidtobeselectedfortheirplainness, age, and safely married status, presumably so as to minimize thedistractiontheyrepresentedtostudentsandfellowsofthecolleges.

Withinsoexclusivelymaleasetting,steepedinasternersenseofpublicmorality,and free of today’s pervasive sexual drumbeat, even passionate and devotedfriendships took physical form less frequently. David Newsome, writing of lateVictorianCambridge,alludesto

romanticfriendshipwithinexclusivelymalecommunities,aphenomenonsonormalandrespectedthroughoutthe period . . . that the greatest caremust be taken to avoid slick and dismissive judgements. . . . In thenineteenthcenturythenormalityofbothmenandwomenforminghighlyemotionalrelationshipswiththoseoftheirownsex,of the sameageor sometimesolderor younger . . .wasneitherquestionedasnecessarilyunwholesome, nor felt to inhibit the same relationship with the opposite sex leading to perfectly happymarriage.

Onephysicianinthelate1890swroteEllisandSymondsofseveralsuchcasesofpassionate,yetpresumablynonphysicalrelationships:

In all these, I imagine, the physical impulse of sex is less imperative than in the average man. Theemotional impulse, on the other hand, is very strong. It has given birth to friendships of which I find noadequatedescriptionanywherebutinthedialoguesofPlato;andbeyondacertainfeelingofstrangenessatthegradualdiscoveryofatemperamentapparentlydifferenttothatofmostmen,ithasprovokednokindofself-reproach or shame.On the contrary, the feeling has been rather one of elation in the consciousness of acapacityofaffectionwhichappearstobefinerandmorespiritualthanthatwhichcommonlysubsistsbetweenpersonsofdifferentsexes... .Inallthesecases,aphysicalsexualattractionisrecognizedasthebasisoftherelation,butasamatteroffeeling,andpartlyalsooftheory,theasceticidealisadopted.

Itwas justsuchkindsofrelationships, remotetoAmerican life today, thatC.P.Snow,whoknewHardyaswellasanyone,imputestohim.Hardy,hewrote,didnotnormallyformclose,demonstrativebondsamongeventhosehecalledhisfriends.

But he had, scattered through his life, two or three other relationships, different in kind. These wereintense affections, absorbing, nonphysical but exalted.The one I knew aboutwas for a youngmanwhosenaturewasasspirituallydelicateashisown.Ibelieve,thoughIonlypickedthisupfromchanceremarks,thatthe samewas true of the others.Tomany people ofmy generation, such relationshipswould seem eitherunsatisfactoryorimpossible.Theywereneithertheonenortheother;andunlessonetakesthemforgranted,

onedoesn’tbegin tounderstandthe temperamentofmen likeHardy . . .nor theCambridgesocietyofhistime.

Despite suggestive evidence, then, one cannot conclude that Hardy was apracticing homosexual. And yet, in one sense, it doesn’t matter. Either he led analmost wholly asexual life, scarcely knowing what he was as a sexual being, andsubmerging any sexual desires behind a screen of Victorian propriety, or he led asecret sexual life so elaborately and successfully hidden that even friends knewnothingofitandthosewhodidkeptquiet.Ineithercase,hewouldhaverequiredavastarchitectureofpersonaldefenseworkstopullitoff,heroicactsofwillperformedday in and day out over the years. And thoughmade somewhat easier andmoreordinarybythetimesinwhichhelived,itwould,intheend,havehadtoexactitstoll.

And itdid.Therewasahauntedness toHardythatyoucouldsee inhiseyes. “Isuspect,”rememberedanOxfordeconomist,LionelCharlesRobbins,whoknewhimlater,that“Hardyfoundmanyformsofcontactwithlifeverypainfulandthat,fromavery early stage, he had taken extensive measures to guard himself against them.Certainlyinhisfriendliermoments—andhecouldbeveryfriendlyindeed—onewasconscious of immense reserves.” Always, he kept the world at bay. The obsessionwithcricket,thebrightconversation,thestudiedeccentricity, the fiercedevotiontomathematics—allofthesemadeforabeguilingpublicpersona;butnoneencouragedrealcloseness.HewasafriendofmanyinCambridge,anintimateoffew.

In the years after 1913, Hardy would befriend a poor Indian clerk. Theirfriendship,too,wouldneverripenintointimacy.

6.THEHARDYSCHOOL

In 1900,Hardy became a Fellow ofTrinityCollege. In 1901, hewon one of twoSmith’sPrizes,namedaftera formermasterofTrinityCollege,andsince1769theblueribbonofCambridgemathematics.

In1903,hewasnamedanM.A.,whichatEnglishuniversitieswasnormally thehighest academic degree. (Cambridge didn’t offer the doctorate, a Germaninnovation,untilafterWorldWarI,hopingto lureAmericansotherwisedrawntoGermany.)

In1906,bebecameaTrinitylecturer.Hegavesixhoursaweekoflectures,usuallyintwocourses,elementaryanalysisandthetheoryoffunctions.Heoccasionallygaveinformalclassesduringthisperiod,buthewasneveractuallyacollegetutor.Hewastheretodoresearch.

Hardywouldlatersayheblossomedslowly.Inasense,thatwastrue;mostofhismoreimportantmathematicalcontributionslayinthefuture.Butalreadyinthefirst

decadeofthetwentiethcenturyhewasbattingoutpapersataprodigiousclip,tenoradozenayear,mostofthemonintegralsandseries.Like“ResearchintheTheoryofDivergent Series andDivergent Integrals,” which appeared inQuarterly Journal ofMathematics in 1904; and “On the Zeros of Certain Classes of Integral TaylorSeries,”intheProceedingsoftheLondonMathematicalSocietyin1905.Inmuchofthiswork,hewasrefiningandenhancingideassuggestedbyCamilleJordaninthebookthathadsoinspiredHardyasanundergraduate.

Inlateryears,Hardyhimselfwouldsetlittlestockintheworkhedidduringthisperiod.“Iwroteagreatdealduringthenexttenyears,butverylittleofimportance,”hewouldsayoftheperiodbeforehemetLittlewoodandRamanujan;“therearenotmore than four or five paperswhich I can still rememberwith some satisfaction.”Still,by1907,theyaddeduptoacorpussubstantialenoughthat,onOctober31ofthatyear,hewasputupformembershipintheRoyalSociety.

ManyofthetopnamesintheCambridgemathematicalestablishmentwenttobatforhim.A.E.H.Love,whohadintroducedHardytoCamilleJordanwhenhewasstillanundergraduate,did.SodidE.W.Hobson,oneofthosewhowould,inafewyears, hear from Ramanujan. So did T.J. I’A. Bromwich, whose book on infiniteseriesRamanujanhadbeenurgedtoconsult justbeforehewroteHardy.LikemostwhowouldplacethecovetedF.R.S.aftertheirnames,Hardydidn’tgetitfirsttimeout.Butin1910,hewaselected,attheageofthirty-three.ALondonphotographertook his picture for the occasion and did a little retouching around the eyes andmouth.Butitwasprobablyareflexaction;Hardystilllookedboyish.

Manyyears later,Hardywould insist thatnoneof themathematicshehaddoneduringhiscareerwaseverintheleast“useful.”Butaroundnowcameoneexception.During the previous century an Austrian monk named GregorMendel had doneexperimentsincrossingtallpeaplantswithdwarfs,foundthateachgenerationoftheprogenyborefixed,predictableproportionsofdwarfandtall,andsolaidthebasisforthe science of genetics. At the time, of course, nobody cared, and Mendel’sexperimentslayforgotten.Butthepublicationofhisworkin1900,sixteenyearsafterhisdeath,sparkedaflurryofinterestinhisinsights,andthenextfewyearssawthemthesubjectofmuchactivedebate.

One controversy surrounded the fate of recessive and dominant traits insucceeding generations. A recessive trait was normally “silent”; it needed to berepresented in both parents to show up in the children.A dominant trait, on theother hand, needed only one copy of the gene.An article in theProceedings of theRoyalSocietyofMedicinearguedthatMendeliangeneticspredictedthatadominanttrait, like the stunted finger growth known as brachydactylism, would tend to

proliferateinthepopulation.Thatthisranflatlycountertotheevidence,theauthorasserted,underminedMendel.

Butno,showedHardy,inabrieflettertotheAmericanscholarlyjournalSciencein1908,adominanttraitwouldnotproliferateinthepopulation.Assignsymbolstotheprobabilities of a particular gene type,work through the simple algebra, and you’dfind that the proportion of each gene would tend to stay fixed generation aftergeneration.Inotherwords,ifmatingstookplacerandomly—thatis,notskewedoneway or the other byDarwinian natural selection—dominant traitswould not takeoverandrecessivetraitswouldnotdieout.AGermanphysician,WilhelmWeinberg,showed something similar the same year, and the principle became known as theHardy-Weinberg Law. It exerted amarked impact on population genetics. It wasappliedtothestudyofthegenetictransmissionofbloodgroupsandrarediseases.Itappearstodayinanyscientificdictionaryorgeneticstextbook.

Hardyhimself,ofcourse,setnogreatstoreby it.As forthemathematics, itwastrivial.Besides, itwasnot,well,useless.AndinHardy’smindthatmade itvergeontheexecrable.

HavelockEllisoncewrotethat“byinborntemperament,Iwas,andhaveremained,an English amateur; I have never been able to pursue any aim that no passionateinstinct has drawnme towards.” There was, as in Ellis, a streak of disdain in theEnglishcharacterformerenecessity;theamateur,blesshisheart,didwhathedidforlove,forthesakeofbeautyortruth,notbecausenecessitycompelledit.ThisstreakhadgainedmorereasonedforminthephilosophyofG.E.Moore,Hardy’sApostolic“father.”HisPrincipiaEthica represented, in thewordsofGertrudeHimmelfarb, a“manifestoofliberation”stressinglove,beauty,andtruth.“Andevenlove,beauty,andtruthwerecarefullydelineatedastoremoveanytaintofutilityormorality.Uselessknowledge was deemed preferable to useful, corporeal beauty to mental qualities,presentandimmediatelyrealizablegoodstoremoteorindirectones.”

G. H. Hardy’s mathematics would emerge as the consummate manifestation,withinhisownfield,ofMoore’scredo.

“I have never done anything ‘useful,’ ” is how he would put it years later. “Nodiscoveryofminehasmadeorislikelytomake,directlyorindirectly,forgoodorill,theleastdifferencetotheamenityoftheworld.”Hewouldneversayso,perhapshedid not even see it, but he had taken Moore’s sensibilities and applied them tomathematics. “Hardyism,” someonewould later dignify this doctrine, so hostile topractical applications; andHardy’sMathematician’s Apology, written almost half acenturylater,wouldembodyitoneverypage.

Thatmathematicsmightaidthedesignofbridgesorenhancethematerialcomfortof millions, he wrote, was scarcely to say anything in its defense. For suchmathematics,heboreonlycontempt.

Itisundeniablethatagooddealofelementarymathematics...hasconsiderablepracticalutility.[But]thesepartsofmathematicsare,onthewhole,ratherdull; theyare just thepartswhichhave leastaestheticvalue.The“real”mathematicsofthe “real”mathematicians, themathematicsofFermatandEulerandGaussandAbelandRiemann,isalmostwholly“useless.”

Hardywentontopitythemathematicalphysicistwhomightusemathematicaltoolstounderstandtheworkingsoftheuniverse:wasnothislotinlifealittlepathetic?

Ifhewantstobeuseful,hemustworkinahumdrumway,andhecannotgivefullplaytohisfancyevenwhenhe wishes to rise to the heights. “Imaginary” universes are so much more beautiful than this stupidlyconstructed“real”one;andmostofthefinestproductsofanappliedmathematician’sfancymustberejected,assoonastheyhavebeencreated,forthebrutalbutsufficientreasonthattheydonotfitthefacts.

Notsoinrealmathematics,Hardycontinued,which“mustbejustifiedasartifitcanbe justified at all.” It was in this light that the Hardy-Weinberg Law, hopelesslyusefulasitwas,andbarrenofanythingintheleastmathematicallybeautiful,violatedhisprinciples.

Whiletheseaestheticprinciples,asitwere,maynotyethavebeensofullyformedorsowellarticulatedatthetimeHardyheardfromRamanujaninearly1913,theyhadcertainlybythenjelled—andhadbeguntofindanoutletintheschoolofEnglishmathematicsformingaroundhim.Itwasaschoolwithscantinterestinanythingsodullandpragmaticas,say,genetics;oreven,forthatmatter,withthemathematicalphysicsthathadforyearsbeenthespecialEnglishstrength.Itwasaschool,rather,thatembracedtheContinentalpurityofCamilleJordan.

•••

Aloneintheirislandkingdom,cutofffromtheContinent,theBritishwereaninsularpeople, suspiciousand intolerantofall things foreign.Attheendof thenineteenthcenturyandbeginningofthetwentieth,however,theywereprobablyevenmoreso,baskingintheglowofempire,fatandhappy.“Itmusthaveseemedlikealonggardenpartyonagoldenafternoon—tothosewhowereinsidethegarden,”SamuelHyneshaswritten,inTheEdwardianTurnofMind,ofBritishattitudesduringthisperiod.“Butagreatdeal thatwas importantwasgoingonoutside thegarden: itwas therethatthetwentieth-centuryworldwasbeingmade”—inmathematics,hemighthaveadded,aselsewhere.

Since the seventeenth century, Britain had stood, mathematically, with its backtowardEurope, scarcelydeigning to glanceover its shoulder at it.Back then, IsaacNewtonand theGermanmathematicianGottfriedWilhelmvonLeibnizhadeach,moreorlessindependently,discoveredcalculus.Controversyoverwhodeservedthecrediteruptedevenwhilebothmenlived,thenmushroomedaftertheirdeaths,withmathematicians in England and on the Continent each championing theircompatriots.Newtonwasthepremiergeniusofhisage,themostfertilemind,withthepossibleexceptionofShakespeare’s,ever to issue fromEnglishsoil.Andyethewould later be called “the greatest disaster that ever befell not merely CambridgemathematicsinparticularbutBritishmathematicalscienceasawhole.”Fortodefendhis intellectualhonor, as itwere, generations of Englishmathematicians boycottedEurope—steadfastly clung to Newton’s awkward notational system, ignoredmathematical trails blazed abroad, professed disregard for the Continent’sachievements.“TheGreatSulk,”onechronicleroftheseeventswouldcallit.

In calculus as in mathematics generally, the effects were felt all through theeighteenth and nineteenth centuries and on into the twentieth. Continentalmathematicslaidstressonwhatmathematicianscall“rigor,”thekindtowhichHardyhad first been exposed through Jordan’s Cours d’analyse and which insisted onrefiningmathematical concepts intuitively “obvious” but often litteredwithhiddenintellectual pitfalls. Perhaps reinforced by a strain in their national character thatsniffedatGermanictheorizingandhairsplitting,theEnglishhadlargelyspurnedthisnew rigor. Looking back on his Cambridge preparation, Bertrand Russell, whorankedasSeventhWranglerintheTriposof1893,notedthat“thosewhotaughtmetheinfinitesimalCalculusdidnotknowthevalidproofsofitsfundamentaltheoremsandtriedtopersuademetoaccepttheofficialsophistriesasanactoffaith.IrealizedthattheCalculusworksinpracticebutIwasatalosstounderstandwhyitshoulddoso.”So,itissafetosay,weremostotherCambridgeundergraduates.

Calculus restsona strategyofdividingquantities intosmallerandsmallerpiecesthataresaidto“approach,”yetneverquitereach,zero.Takinga“limit,”theprocessiscalled,andit’sfundamentaltoanunderstandingofcalculus—butalso,typically,alienandslipperyterritorytostudentsraisedonthefirmgroundofalgebraandgeometry.Andyet,itispossibletoblithelysailonpasttheseintellectualperils,concentrateonthemanypracticalapplicationsthatfairlyeruptoutofcalculus,andneverlookback.

In textbooks even today you can see vestiges of the split—whichneatlyparallelsthatbetweenBritainandtheContinentinthenineteenthcentury:theauthorbrieflyintroducesthelimit,assumesahazyintuitiveunderstanding,thenspendssixchapterschargingaheadwithstandarddifferentiationtechniques,maxima-minimaproblems,

and all theothermainstaysofCalc 101 . . . until finally, come chapter7or so, hesteps back and reintroduces the elusive concept, this time covering mine-strewnterrain previously sidestepped, tackling conceptual difficulties—and stretching thestudent’smindbeyondanythinghe’susedto.

Well, the first six chapters of this generic calculus text, it could be said, wereEnglishmathematicswithouttheContinentalinfluence.Chapter7wasthenewrigorsuppliedbyFrench,German,andSwissmathematicians.“Analysis”wasthegenericname for this precise, fine-grained approach. It was a world of Greek letters, ofepsilonsanddeltasrepresentinginfinitesimallysmallquantitiesthatnonethelessthemathematicians found a way to workwith. It was a world in whichmathematics,logic,andTalmudichairsplittingmerged.

FirstGauss,Abel,andCauchyhadrisenabovethelooser,intuitivenostrumsofthepast;laterinthecentury,WeierstrassandDedekindwentfurtheryet.NoneofthemwereEnglish.And theEnglish professednot to care.Why, before the turn of thecentury, Cauchy—the Cauchy, Augustin Louis Cauchy, the Cauchy who hadlaunchedtheFrenchschoolofanalysis,theCauchyoftheCauchyintegralformula—wascommonlyreferredtoaroundCambridgeas“Corky.”

SinceNewton’stime,Britishmathematicshaddivergedoffonadecidedlyappliedroad.Mathematical physics had become the British specialty, dominated by suchnames asKelvin,Maxwell, Rayleigh, and J. J. Thomson. Pure math, though, hadstultified, with the whole nineteenth century leaving England with few figures ofnote. “Rigor inargument,” J.E.Littlewoodwouldrecall, “wasgenerally regarded—therewererareexceptions—withwhatitisnoexaggerationtocallcontempt;nigglingover trifles instead of getting on with the real job.” Newton had said it all; whyresurrect these arcane fine points? Calculus, and the whole architecture ofmathematicalphysicsthatemanatedfromit,worked.

Andso,Englandslept inthedeadcalmof itsTripossystem,whereNewtonwasenshrinedasGod,hisPrincipiaMathematicatheBible.“InmyownTriposin1881,wewereexpectedtoknowanylemma[atheoremneededtoproveanothertheorem]inthatgreatworkby itsnumberalone,”wroteoneprominentmathematician later,“asifitwereoneofthecommandmentsorthe100thPsalm....Cambridgebecameaschool that was self-satisfied, self-supporting, self-content, almostmarooned in itslimitations.”RepliedadistinguishedEuropeanmathematicianwhenaskedwhetherhehadseenrecentworkbyanEnglishman:“Oh,weneverreadanythingtheEnglishmathematiciansdo.”

Thefirstwindsofchangecame inthepersonofAndrewRussellForsyth,whoseTheory of Functions had begun, in 1893, to introduce some of the new thinking—

thoughby this time itwasn’t sonewanymore—fromParis,Göttingen,andBerlin.Written inamagisterial style, itburstonCambridge,asE.H.Nevilleoncewrote,“withthesplendourofarevelation”;somewouldargueithadasgreataninfluenceonBritishmathematicsas anywork sinceNewton’sPrincipia.By the standardsof theContinent, however, it was hopelessly sloppy and was soundly condemned there.“Forsyth was not very good at delta and epsilon,” Littlewood once said of him,referring to the Greek letters normally used for dealing with infinitesimally smallquantities. Still, it helped redirect the gaze of Englishmathematicians toward theContinent.Itchartedacoursetothefuture,butdidnotactuallyfollowit.

ThatwaslefttoHardy.

•••

Asaspokesman for thenewrigor,Hardyexertedhis impactnotalonebywhathehadtosay,butthroughtheforce,grace,andelegancewithwhichhesaidit,bothinprintandinperson.

Inlectures,hisenthusiasmanddelightinthesubjectfairlyspilledover.“Onefelt,”wroteoneofhislaterstudents,E.C.Titchmarsh,“thatnothingelseintheworldbutthe proof of these theorems really mattered.” Norbert Wiener, the Americanmathematical prodigy who would later create the field known as “cybernetics,”attendedHardy’slectures.“Inallmyyearsoflisteningtolecturesinmathematics,”hewouldwrite, “Ihaveneverheard the equalofHardy for clarity, for interest, or forintellectual power.” Around this time, a pupil of E. W. Barnes, director ofmathematicalstudiesatTrinity,soughtBarnes’sadviceaboutwhatlecturestoattend.GotoHardy’s,herecommended.Thepupilhesitated. “Well,” repliedBarnes, “youneednotgotoHardy’slecturesifyoudon’twant,butyouwillregretit—asindeed,”recalledthepupilmanyyearslater,“Ihave.”Otherswhomissedhislecturesmaynot,in retrospect, have felt such regret: so great wasHardy’s personalmagnetism andenthusiasm,itwassaid,thathesometimesdivertedtomathematicsthosewithoutthenecessaryabilityandtemperament.

Butaslucidaswerehislectures,itwashiswritingthatprobablyhadmoreimpact.Later,speculatingaboutwhatcareerhemighthavechosenotherthanmathematics,Hardynotedthat“Journalismistheonlyprofession,outsideacademiclife,inwhichIshouldhavefeltreallyconfidentofmychances.”Indeed,nofielddemandingliterarycraftsmanshipcouldfail tohaveprofited fromhisattention. “Hewrote, inhisownclear andunadorned fashion, some of themost perfect English of his time,”C. P.Snow once said of him. That Hardy’s impressions of Ramanujan would be so

relentlesslyquoted,andwouldgosofartowardfixingRamanujan’splaceinhistory,owesnotalonetohiscloserelationshipwithRamanujanbuttothesheergracewithwhichhewroteabouthim.

Hardy didn’t much like his early style, he decided later, terming it “vulgar.”Ofcourse,hedidn’tmuchlikehisgloriousgoodlooks,either.“EverythingHardydid,”Snow once wrote of him, “was light with grace, order, a sense of style.” And hiswritingexemplifiedthat.Hewrote,fortheCambridgeReview,about thephilosophyof Bertrand Russell. His obituaries of famous mathematicians were rounded,gracious, and wise. He could write about geometry and number theory for layaudiences. And his Mathematician’s Apology, which became a classic, is almostmesmerizinginitslanguage’sholdonthereader.

He applied his gifts even to the most densely mathematical of his work. Incollaborations,itwasalmostalwayshewhowroteupthejointpaperandshepherdeditthroughpublication.“Hesuppliedthegas,”recalledLittlewood,whowascontentifwhat he had to saywas simply correct.Hardywantedmore; the “gas,” as he oncedefinedit,wasthe“rhetoricalflourishes,”theequivalentof“picturesontheboardinthelecture,devicestostimulatetheimaginationofpupils.”Areviewerwouldsayofoneofhismathematical texts thatHardyhad“showninthisbookandelsewhereapowerofbeinginteresting,whichistomymindunequalled.”

Thought,Hardyusedtosay,wasforhimimpossiblewithoutwords.Theveryactofwritingouthislecturenotesandmathematicalpapersgavehimpleasure,mergedhis aesthetic and purely intellectual sides. Why, if you didn’t know math wassupposedtobedryandcold,andhadonlyapagefromoneofhismanuscriptstogoon,youmightthinkyou’dstumbledonaspecimenofsomenewartformbeholdentoChinese calligraphy. Here were inequality symbols that slashed across the page,sweepingintegralsignsaninchandaquarterhigh,sigmasthatresonatedlikethekeysignatures on a musical staff. There was a spaciousness about how he wrote outmathematics, a lightness, as if rejecting the cramped, ungenerous formalities of theprintednotation.Hewas likeaFrench impressionist, intimatingworldswitha fewsplashesofcolor,notamakerofaustereEnglishminiatures.

AllthroughthefirstdecadeofthetwentiethcenturyHardyusedhispentoseduceagenerationof youngEnglishmathematical students into taking seriously thenewContinentalrigor.WhenareviewofBertrandRussell’sPrinciplesofMathematicsranintheTimesLiterarySupplementin1903,itwasHardywhowroteit.WhileEnglishmathematicsoften turnedadeafear to events across theChannel,Hardyused thepages of the Mathematical Gazette to comment on foreign books. In 1903, he

reviewedEinleitungindieFunkionentheorie,byStolzandGmeiner;in1905Leçonssurlesfonctionsdesvariablesréeles,byBorel.

Meanwhile,heheldEnglishandAmericantextstostrictaccountfortheirlapses.InoneMathematicalGazette reviewin1907, forexample,hewroteofanAmericancalculustextbyW.WoolseyJohnson.Oh,itwasn’tbadofitstype,heallowed.ButitstypewasabreedofEnglishbookthat,whileforgivablethirtyyearsbefore,wasnolonger. Perhaps it was all right to pass over theoretical difficulties in laying thefoundationsofcalculus,hewrote.

Buttherearedifferentwaysofpassingoverdifficulties.Wemaysimplyandabsolutelyignorethem:thatisacourseforwhichthereisoftenmuchtobesaid.Wemaypointthemoutandavowedlypassthemby;orwemayexpandalittleaboutthemandendeavourtomakeourconclusionsplausiblewithoutprofessingtomakeourreasoningexact.

But,Hardywenton:

Thereisonlyonecourseforwhichnogooddefencecaneverbefound.Thiscourseistogivewhatprofesstobe proofs and are not proofs, reasoningwhich is ostensibly exact, butwhich reallymisses all the essentialdifficultiesoftheproblem.ThiswasTodhunter’s[theauthorofakindredtext]method,anditisonewhichProf.Johnsontoooftenadopts.

WhereuponHardy launchedintoamathematicalexampletoshowhowtheauthoremployedarguments“entirelydestituteofvalidity.”

Hardyfelthecoulddobetter,andinSeptember1908completedACourseofPureMathematics,thefirstrigorousexpositioninEnglishofmathematicalconceptsothertexts sloughed over in their rush to get to practical applications or cover broadexpansesofmathematicalground.Such rigorwas sorelyneeded, saidHardy inhispreface. “I have [very rarely] encountered a pupil who could face the simplestproblem involving the ideas of infinity, limit, or continuity with a vestige of theconfidencewithwhichhecoulddealwithquestionsofadifferentcharacterandoffargreaterintrinsicdifficulty.”

Like everything elseHardy ever wrote, his textbookwas readable. This was notsimplypageaftergraypageof formula.Hiswererealexplanationsofdifficult ideaspresentedinclear,cogentEnglishprose.Whatinotherhandswouldbeburiedinaseaofabstractions,inHardy’sfairlyjumpedoutatyou,sometimesastheculminationofapassageactuallyvergingonsuspenseful.

Early on, for example, he addressed “rational” numbers, numbers like 6, 2/3,112/3890,or19thatcanbeexpressedasordinary fractionsor integers.Betweenanytwonumbers representing points on a line segment, he showed thatmore rationalpointscanalwaysbesqueezedin.Between1/2and2/3,youcanfita3/5.Between3/5

and2/3, youcan fit5/8.And soon, forever, resulting in an infinityof suchpoints.Thenhegoeson:

Fromtheseconsiderationsthereadermightbetemptedtoinferthattheserationalpointsaccountforallthepointsof the line, i.e. thatevery pointon the line is a rationalpoint.And it is certainly the case that ifweimaginethe lineasbeingmadeupsolelyoftherationalpoints,allotherpoints(ifanysuchtherebe)beingimaginedtobeeliminated,thefigurewhichremainedwouldpossessmostofthepropertieswhichcommonsenseattributestothestraightlineandwould,toputthematterroughly, lookandbehaveverymuchlikealine.

Somethingiscoming,thereaderrightlysuspects,withoutknowingjustwhat.Hardythenshowedthatwithinthesamelinesegmenttherewas,roughlyspeaking,

another infinity of points that could be crowded into the interstices between theserationalnumbers—the“irrational”numbers,whichcannotbeexpressedasfractions,andwhosepropertieshethenproceededtoexplore.

ThislovingattentiontofundamentalswasjustwhatEnglishmathematicsneeded.As one review of the book commented, “When Mr. Hardy sets out to provesomething,then,unlikethewritersoftoomanywidelyreadtextbooks,hereallydoesproveit....Ifthebookiswidelyread,Iforoneshallhopetoavoidinthefuturethemanyweary hours that have usually to be spent in convincingUniversity studentsthat ‘proofs’ which they have laboriously learned at school are little better thannonsense.”

Hardy’s book was widely read. For the next three-quarters of a century, andthroughteneditionsandnumerousreprints, itbecamethe singlegreatest influenceon the teaching of English mathematics at the university level. Through it—andthrough his lectures at Cambridge, through his papers and reviews, through hisrelationshipswithothermathematicians—Hardymaderigornolongerthepreserveof a few Teutonic zealots but something that bordered on the mathematicallyfashionable.

•••

AttherootofBritain’smathematicalbackwardness,Hardywassure,laytheTripossystem.Originallythemeanstoamodestend—determiningthefitnessofcandidatesfor degrees—the Tripos had become an end in itself. As Hardy saw it, Englishmathematicswasbeingsappedbytheverysystemdesignedtoselectitsfutureleaders.

Around1907,hebecamesecretaryofapanelestablishedtoreformit.Butinfact,hechampioned itsreformonlyasa firststeptowarddoingawaywith italtogether,andonlybecausehesawnohope,justthen,formoreradicalchange.AshelatertoldameetingoftheMathematicalAssociation,“Iadheretotheview...thatthesystem

isviciousinprinciple,andthattheviceistooradicalforwhatisusuallycalledreform.IdonotwanttoreformtheTriposbuttodestroyit.”

Hardydidnotopposeexaminationsingeneral;hesawaplaceforthem,asharplylimited one—as a floor, a minimum standard necessary to earn a degree. “Anexamination,” said he, “can do little harm, so long as its standard is low.” But theTripos laid no such meager claims; it meant to appraise, to sift, to grade.Undergraduates,asHardypicturedthem,exhausted“themselvesandtheirtutorsinthestruggle to turnacomfortable second[class] intoamarginal first.”That, inhisview,wastheproblem:theTriposdistortedteachingandlearningalike,andEnglishmathematicswastheloserforit.

With others among the younger dons, Hardy succeeded in forcing changesthrough a reluctant senate, the university’s governing arm.Chief among themwasabolishment of the Order of Merit; a degree candidate still took the Tripos but,beginning in 1910, was ranked only by broad category—as Wrangler, SeniorOptime,orJuniorOptime.TherewouldnolongerbeaSeniorWranglertowhichtoaspire,nolongerthemercilesspressureitcreated,nolongertheambition-drivenneedfor coaches. Overnight, the most notorious abuses of the Tripos system wereeliminated.

ButHardy’smoreambitiousgoalwas futile; theTripos, inmodified form,existsstill—inpart becausewhilemanypointed out its failings, fewdid sowithHardy’sferocity. There was a mild-manneredness in the English personality that Hardy,when it came to the Tripos, trespassed. “It is useless to propose anythingrevolutionary toEnglishmen,” itwould be pointed out to amathematical audiencesome few years later. “Existing institutions always havemerits,which are as deep-seated as their defects are patent. . . . Our English way is to alter the defectiveinstitution a little bit at a time, so that it comes a little nearer towhatwe desire.”Hardy’sfriendLittlewood,whilenofanoftheTripos,wasalsolessheatedaboutit.“Ido not claim to have suffered high-souled frustration,” he wrote of his experiencewithit.“Itookthingsastheycame;thegamewewereplayingcameeasilytome,andIevenfeltasatisfactionofasortinsuccessfulcraftsmanship.”

Hardy’senmity,then,wassomethingdifferent,almostbeyondreason.Plainly,hisownexperience influencedhim.Back in1896, theprospectof twoyearsofTripostediumhadnearlydeflectedhimfrommathematicsaltogether.Butintheend,hehadmeeklysurrendered;hehadacquiescedtoacoach,climbedonboardtheSystem,put“real”mathematicsaside.WhenhedidatlastpithimselfagainsttheTripos,hecouldalmostbesaidtohave“failed”it;forsomeoneascompetitiveashe,that’swhatbeingFourthWranglermeant.TheTripos,inasense,hadbeatenhim.

Hardy’svehemencesuggestsapeculiarriftwithinhispersonality.Herewasaman—afriendwouldonedaylikenhimto“anacrobatperpetuallytestinghimselfforhisnext feat”—who set up rating scales at the least provocation, loved competitivegames,grillednewacquaintancesonwhattheyknew,heldupmathematicalworktothe highest standards—yet swore eternal enmity to theTripos systemwhich, in asense,wastheultimateratingscale,theultimatetest.

In Hardy coexisted a stern, demanding streak with an indulgent liberal-mindedness, a formidable and forbidding exterior with a soft and fragile core.Hewould later claim to have scant interest in his less able students. But this, by allaccounts,wasnine-tenthsbluster;henever failed anyof them. “He simply couldn’tthinkthatway,”MaryCartwright,aformerresearchstudent,toldafriend,“becausehewassokindtotheweakones.”

Hardydisdainedsocialniceties,everkepthisdistance,arrogantlydismissedGod.Yethecouldbekindandendlesslyobliging.Eventheobituarieshewroteshowedalargenessofspiritthat,assomeoneonceputit,“musthavemadeeverymathematicianwishthathecouldhaveseenhisowncareerdescribedinthesamegenerousterms.”

Sohewasdemanding,distant,emotionallyastringent—and large-hearted, caring,andkind.Ittakesnostrainingofthefactstolaythissplittotherespectiveinfluencesofhismother and father.Butwhatever their source, these two contrasting strandswound throughhispersonality always.Andbothwould emerge in his relationshipwithRamanujanoverthenextsevenfatefulyears.

•••

Winter1913.Europestirred,armiesmarshaled.Theworldwasrestlesswithchange.Picasso’sfirstcubistdrawingshadappearedbarelyayearbefore.InParis,Diaghilev,whose Russian Ballet had given its first London performance two years before,prepared for the premiere of Stravinsky’s tempestuous Le Sacre du Printemps, inwhich amaidendancesherself todeath. InEngland,GeorgeVwasKing,Edwardhavingdiedsuddenlyin1910.In1911,theParliamentBillhadstrippedtheHouseofLordsofitsvetopoweronactsofCommons.AllthroughBritain,workersstruckandmilitantsuffragettessmashedwindows.Irelandseethed.

ButinCambridge,thingswereastheyalwayswere.Hardynearedhisthirty-sixthbirthdaywithhis face bearing scarcely amark of it.He’d visitBertrandRussell inNevile’s Court and discuss Bergson and the philosophy of religion; once,NorbertWienerandhisfathermethimthereandtookhimtobeanundergraduate.In1912,Hardy published nine more papers, including his first collaborative one with

Littlewood,“SomeProblemsofDiophantineApproximation.”HisfirstkeypaperonFourier series was coming out later in 1913, the revised edition of his populartextbook the following year. Hardy’s friend from the Apostles, Leonard Woolf,recentlybackfromCeylon,foundCambridgemuchashe’dleftit;onthetrainbacktoLondon, he wrote, “I felt the warmth of a kind of reassurance. I had enjoyedmyweekend. There was Cambridge and Lytton and Bertie Russell and Goldie, theSocietyandtheGreatCourtofTrinity,andHardyandbowls—alltheeternaltruthsandvaluesofmyyouth—goingonjustasIhadleftthemsevenyearsago.”

Hardy held to a regular routine. He read the London Times over breakfast,especiallythecricketscores.Heworkedforfourhoursorsointhemorning,thenhadalightlunchinHall,perhapsplayedalittletennisintheafternoon.Hiscareerwaswellinplace,hislifecomfortable,hisfuturesecure.

ThenthelettercamefromIndia.

CHAPTERFIVE

“IBegtoIntroduceMyself...”

[1913to1914]

1.THELETTER

Theletter,borneinalargeenvelopecoveredwithIndianstamps,wasdated“Madras,16thJanuary1913,”andbegan:

DearSir,IbegtointroducemyselftoyouasaclerkintheAccountsDepartmentofthePortTrustOfficeatMadras

onasalaryofonly£20perannum.Iamnowabout23yearsofage.IhavehadnoUniversityeducationbutIhaveundergonetheordinaryschoolcourse.AfterleavingschoolIhavebeenemployingthesparetimeatmydisposal to work at Mathematics. I have not trodden through the conventional regular course which isfollowed in a University course, but I am striking out a new path for myself. I have made a specialinvestigation of divergent series in general and the results I get are termed by the localmathematicians as“startling.”

Someinsignificantclerk insomebackwaterofanoffice five thousandmilesawayapparentlysoughttoincitebothpityandwonder.Therewasanervinessabouthim:Ihavenot trodden through the conventional regular course.By the secondparagraphhewasinsistinghecouldgivemeaningtonegativevaluesofthegammafunction.BythethirdhewasdisputinganassertioninamathematicalpamphletHardyhadwrittenthreeyearsbefore,partofaseriescalledtheCambridgeTractsinMathematicsandMathematicalPhysics.

ItwascalledOrdersofInfinity:The‘Infinitarcalcul’ofPaulDuBois-Reymond,andinitHardydealtwithhowmathematical functions cangrow toward infinitymoreorless rapidly.Forexample, f(x)=x3 approaches infinity faster thang(x)=3x. Bothfunctions,asxgrowslarger,growwithoutbound;both,itcanbecrudelysaid,“reach”infinity.Butthefirstdoessomorequicklythanthesecond.Bythetimex=100,forexample,thefirstfunctionhasexplodedto1,000,000,whilethesecondisstillmiredat300.Atonepoint,Hardyhadciteda familiarmathematicalexpressionfromthetheory of prime numbers. This expression consisted of, first, a term involvinglogarithmsand,second,anerrorterm,ρ(x),thatsimplyrepresentedhowfarwrong

the first termwas.Onpage36,Hardyhadasserted that “thepreciseorderofρ(x)hasnotbeendetermined.”

Well,RamanujannowwroteHardy,ithadbeendetermined;hehaddeterminedit.“Ihave foundan expression [for thenumberofprimenumbers]which very nearlyapproximates to the real result, the error beingnegligible.”Hewas saying that theprimenumber theorem, as itwasknown in themathematicalworld, and as it hadfirstbeengivenformbyLegendreandthenmorepreciselybyGauss,wasinadequateandincomplete,andthathe,anunknownIndianclerk,hadsomethingbetter.

ThiswasthehookwithwhichRamanujansetouttosnareHardy’sattention.Heconcluded:

I would request you to go through the enclosed papers. Being poor, if you are convinced that there isanythingofvalueIwouldliketohavemytheoremspublished.IhavenotgiventheactualinvestigationsnortheexpressionsthatIgetbutIhaveindicatedthelinesonwhichIproceed.BeinginexperiencedIwouldveryhighlyvalueanyadviceyougiveme.RequestingtobeexcusedforthetroubleIgiveyou.

Iremain,DearSir,

Yourstruly,S.Ramanujan

Thiswasnot,ofcourse, theendof thematter,but thebeginning.The“enclosedpapers” to which Ramanujan referred went on for nine pages (and also probablyincludedacopyofhispublishedpaperonBernoullinumbers).Thefirstpageorsoreadlikeaninventor’spatentclaim,andwiththesamealmostrhythmicringofbrashcertainty:

Ihavefounda functionwhichexactlyrepresentstheno.ofprimenos. lessthanx, “exactly” inthesensethatthedifferencebetweenthefunctionandtheactualno.ofprimesisgenerally0orsomesmallfinitevalueevenwhenxbecomesinfinite.Ihavegotthefunctionintheformofinfiniteseriesandhaveexpresseditintwoways.

Ihavealsogotexpressionstofindtheactualno.ofprimenos.oftheformAn+B,whicharelessthananygivennumberhoweverlarge.

Ihavefoundoutexpressionsforfindingnotonlyirregularlyincreasingfunctionsbutalsoirregularfunctionswithout increase (e.g. the no. of divisors of natural nos.) not merely the order but the exact form. Thefollowingareafewexamplesfrommytheorems....

NowordinaryEnglishvirtuallydisappeared,givingwaytothelanguageofalgebra,trigonometry, and calculus. There were theorems in number theory, theoremsdevoted to evaluating definite integrals, theorems on summing infinite series,theorems on transforming series and integrals, theorems offering intriguingapproximationstoseriesandintegrals—perhapsfiftyoftheminall.

The whole letter, all ten pages of it, was written out in large, legible, roundedschoolboyscriptdistinguishedonlybycrossedt’s thatdidn’tcross.Hishandwritinghadalwaysbeenneat;buthere,ifpossible,itwasneaterstill,asifherealizedthegulfofskepticismthatdividedhimfromHardyanddarednotletanillegiblescrawlwidenit.

Itwasawiseprecaution;thegulfwasindeedgreat.ForHardy,Ramanujan’spagesoftheoremswere likeanalienforestwhosetreeswerefamiliarenoughtocall trees,yetsostrangetheyseemedtohavecomefromanotherplanet;itwasthestrangenessofRamanujan’s theorems that struckhim first,not theirbrilliance.The Indian,hesupposed,was justanothercrank.Hewas forevergettingbizarremanuscripts fromstrangers that, as his friend Snow later put it, “pretended to prove the propheticwisdom of the Great Pyramid, the revelations of the Elders of Zion, or thecryptogramsthatBaconhadinsertedintheplaysoftheso-calledShakespeare.”

And so, after a perfunctory glance, he put the manuscript aside and soon losthimselfintheday’sLondonTimeswhich,inlateJanuary1913,toldofopiumabuseinChina,porthandsinLisbongoneonstrike,theFrenchbattlingArabrebelsnearMogador, theHouse of Lords debating home rule for Ireland. . . .Hemay haveskippedovertheaccountoftheBuxtondivorcetrial,whereMrs.BuxtonwasaccusedofadulterywithHenryArthurMorningtonWellesley,LordCrowley.Buthelikelydidn’t miss the news of England’s one-goal rugby victory over the French beforetwelvethousandspectatorsatTwickenham.

Aroundninethatmorning,hesettoworkonmathematics,keptatituntilaboutone,thenambledovertoHallforlunch.ThenitwasofftotheuniversitycourtsonGrangeRoadforagameof“real”tennis(whichiswhattheEnglishcalledtheindoorvariant that antedated the lawn tennis more popular today). But that day, in thecornerofhismindnormallyleftserenebyvigorousathletics,somethingwaswrong.TheIndianmanuscriptscrapedandtuggedathiscomposurewith,asSnowwrote,its“wildtheorems.Theoremssuchashehadneverseenbefore,norimagined.”

Weretheywildandunimaginablebecausetheyweresilly,or trivial,or justplainwrong,withnothingtosupportthem?Orbecausetheyweretheworkofsomerareflowerofexoticgenius?

Ormaybetheyweremerelywell-knowntheoremstheIndianhadfound insomebookandcleverlydisguisedbyexpressinginslightlydifferentform—makingitjustamatteroftimebeforeHardyfoundthemout?

Orperhaps itwasallapractical joke?Hoaxes,afterall,weremuch invogue justthen. Many Englishmen holding high positions in the Indian Civil Service hadendured the mathematical Tripos or were otherwise versed inmathematics—well

versed enough, perhaps, to pull off such a stunt. And how best to dupe your oldCambridgefriendHardy?Why,you’dgarbfamiliar“theorems” inunfamiliarattire,purposelytwistthemintoweirdshapes.ButwhoinIndiawasadeptenoughtodoit?Maybethehoaxhadoriginated inEurope.Butwouldtheperpetratorhavegone tothetroubleofsecuringagenuineMadraspostmark...?

Vagrant, fragmentary thoughts like these bubbled through Hardy’s head as,returningfromtennis,hewalkedbackacrossoneoftheCambridges,thenovertheexpanseoflawnthatwastheBacks,andthroughthegatewayintoNewCourt.Backin his second-floor suite of rooms, which were built over one of the gateways, heagainsatdownwiththeletterfromIndia.OutsidetheGothicmullionedwindowsofhisroom,thewinterlightbegantofade.

Yearslater,mostoftheformulasinRamanujan’sletterwouldbecomethesubjectsofpapers intheJournalof theLondonMathematicalSociety andothermathematicaljournals.Inthem,theirauthors,includingHardyhimself,wouldtaketwo,orfive,ortenpagestoformallyprovethosenotalreadyknown.Butnow,provingthemwasn’tHardy’saim.Nowhewascontent to see if therewasanything to thematall.Andeven thatwas not apparent—inpart because, asHardywrote later, “somecuriousspecialization of a constant or a parameter made the real meaning of a formuladifficult to grasp.” Roughly speaking, it was as if, instead of stating the aphorism“penny-wise,pound-foolish,”Ramanujanhadforhisownreasonsexpresseditas“twopennies wise, seven-and-a-half pounds foolish”—leaving the listener, distracted bythe particulars, harder pressed to extract its meaning. In any case, it onlycompoundedHardy’sperplexity.

Darkness fell. It was almost time for dinner. The formulas grew no morestraightforward,thequalityofthemanwhohadwrittenthemnoclearer.Geniusorfraud? You couldn’t idly riffle through these pages and tell. Yes, Hardy decided,Littlewoodwouldhavetoseethem,too.

•••

John Edensor Littlewood was just two years older than Ramanujan, but whileRamanujan foundered in India,hehadbeenmathematically schooledbyEngland’sbest.

HecamefromoldEnglishyeomanstock;Littlewoodarchers,itwassaid,foughtatthe Battle of Agincourt in 1415. More recently, his ancestors had been robustlymiddle-class professionals—ministers, schoolmasters, publishers, doctors and thelike. Both his grandfather and father had studied mathematics at Cambridge; his

grandfatherbecameatheologian,hisfatherheadmasterofaschoolinSouthAfrica,whereJohnlivedforeightyears.BackinEnglandwhenhewasfourteen,heattendedSt.Paul’sSchool.Therehecaughtthemathematicsbug.

In hismemoir,AMathematician’sMiscellany, Littlewood would assert, withoutartifice or conceit, that he was a “prodigy.” Prodigy or not, he profited from aneducationthatwaseverythingRamanujan’swasnot.AboutthetimeRamanujanwasstudyingS.L.Loney’sTrigonometry, sowas Littlewood. ButwhereasRamanujan’sformal exposure to mathematical ideas ended there, Littlewood’s had just begun.Over thenext threeyearshemadehisway throughMacaulay’sGeometricalConics,Smith’s Analytical Conics, Edwards’s Differential Calculus, Williamson’s IntegralCalculus, Casey’s Sequel to Euclid, Hobson’s Trigonometry, Routh’s Dynamics of aParticle, Murray’s Differential Equations, Smith’s Solid Geometry, Burnside andPanton’sTheoryofEquations,andmore—allbeforesomuchassittingfortheTrinityCollegeentrancescholarshipexaminDecember1902.Bythistime,RamanujanhadnotevenencounteredCarr.

Two years later, in 1905, Littlewood was Senior Wrangler. But the fellowshipnormallyhisalmostbyrightmysteriouslywenttosomeoneelse.Forthreeyears,heleft to assume a lectureship at theUniversity ofManchester. There, in “exile,” heenduredanoppressiveloadoflecturing,conferences,andpaperwork—thenormallotof facultymembersatprovincialuniversities.After that,beginning in1910,hewasbackatTrinityforgood.

Inthewordsoftwoofhisbiographers,hewas“arough-hewnearthypersonwithacharmofhisown.”Hewasstrong,virile,vigorous.Hehadbeenacrackgymnastatschool, had played cricket, would become an accomplished rock climber and skierand,evenintohiseighties,couldbeseenhikingthroughtheEastAngliancountrysidearoundCambridge.Hewasn’t especially tall, but aphotographofhim lecturing inacademic robes suggests an enormous, hulking masculinity. Another of him andHardytogethershowsLittlewooddominatingthepicture,hatplantedfirmlyatophishead,slope-shouldered,feetspreadasifreadyforafight;thesmallerHardyseemstorecedeintothebackground.

LikeHardy, Littlewood remained a bachelor. But unlikeHardy, he thoroughlyenjoyedthecompanyofwomen.WhileatManchester,hewasthebestdancerinhisgroup.Hewouldwriteofhisgrandmotherthatshewas“aremarkablewomanfroman able family, but unfortunately very saintly.” He was not. His long-termrelationship with amarried woman, with whose family he long shared a house inCornwall, and with whom he had a daughter, would become well known inCambridge.

Asamathematician,Littlewoodwas “themanmost likely to stormand smash areallydeepand formidableproblem: there isnooneelsewhocancommandsuchacombinationofinsight,technique,andpower.”SosaidHardyhimself.Thetwomenfirst met, at least intellectually, in 1906 when an early Littlewood paper to theLondonMathematicalSocietysparkeddisagreementas to itsmeritsand itwent toHardyasreferee.WhenLittlewoodreturnedtoTrinity in1910,heworkedcloselyenough with him for Hardy to acknowledge his help in the preface to Orders ofInfinity,thebookthathadcaughtRamanujan’seyeinIndia.

Their actual collaborationbegan inauspiciously,when a proof they submitted totheLondonMathematicalSociety in June1911 turnedout tobe flawed.But then,thefirstoftheirmorethanonehundredpapersappearedin1912,andafterthattheyweremathematically inseparable. “Nowadays,” somebody said later, “there are onlythree really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.” Because Littlewood disdained bright, sparkling company and stayedaway frommathematicsconferences, some—at least in jest—doubtedhe existed atall.Butexisthedid,andin1913,whenRamanujan’sletterarrived,itwasnaturalthatHardythoughttoshowittohim.

Littlewood had recently moved into rooms on D Staircase of Nevile’s Court.Pausing in the arched doorway at the staircase’s base, he could sight through theportico to the arches across the court framedwithin it. Itwas an arresting view—perhaps one reasonhe remained there for sixty-five years, until his death in 1977.FromHardy’s rooms, itwas just nineteen steps down thewinding stone staircase,thenfortypacesthroughthegateintoNevile’sCourtandaroundtoDStaircase.Yetnormally the twomencommunicatedbymail or collegemessenger anddidnot, inanycase,routinelyrunofftoconferwithoneanotherinperson.

And so, thatwinter evening in1913, to letLittlewoodknowhewished tomeetwithhimafterHall,Hardysentwordbymessenger.

•••

About nine o’clock, as Snow reconstructed the day’s events, theymet, probably inLittlewood’srooms,andsoonthemanuscriptlaystretchedoutbeforethem.Someofthe formulas were familiar while others, Hardy would write, “seemed scarcelypossible to believe.”Twenty years later, in a talk atHarvardUniversity, hewouldinvitehis audience into theday thathad so enrichedhis life. “I should like you tobegin,” he said, “by trying to reconstruct the immediate reactions of an ordinaryprofessionalmathematicianwhoreceivesa letter like this fromanunknownHindu

clerk.” It was a mathematical audience, so Hardy introduced them to some ofRamanujan’stheorems.Likethisone,onthebottomofpagethree:

TheelongatedS-likesymbolappearingontheleft-handsideofthisequation,andin many other equations all through the letter, was an integral sign, a notationoriginatingwithNewton’s competitor Leibniz.An integral—the idea goes back totheGreeks—isessentiallyanaddition,asum,butoneofapeculiar,precise,and,atfirstglance,infuriatingkind.

Imaginecuttingahotdogintodisclikeslices.Youcouldwindupwithtensectionshalfaninchthickorathousandpaper-thinslices.Buthoweverthinyouslicedit,youcould,presumably,reassemblethepiecesbackintoahotdog.Integralcalculus,asthisbranchofmathematicsiscalled,adoptsthestrategyoftakinganinfinitenumberofinfinitesimallythinslicesandgeneratingmathematicalexpressionsforputting themback together again—formaking themwhole,or “integral.”Thispowerful additiveprocesscanbeusedtodeterminethedragforcebuffetingawingasitslicesthroughtheair,orthegravitationaleffectsoftheearthonaman-madesatellite,orindeedtosolve any problemwhere the object is to piece together the contributions ofmanysmallinfluences.

Youdon’tneedintegralcalculustodeterminetheareaofaneatrectangularplotoffarmland;youjustmultiplylengthtimeswidth.Butyoucoulduseit.Andyoucouldusethesameadditivemethodsapplicabletowingsandsatellitestocalculatetheareaof an irregularly shaped plot where length-times-width won’t work. Furnish thefunctionthatmathematicallydefinesitsshape,andinprincipleyoucangetitsareaby“integrating”it—thatis,byperformingtheadditiveprocessinaparticular,preciselydefinedway.

Calculusbooks come litteredwithhundredsofways to integrate functions.Andyet,pick a function at random and chances are it can’t be integrated—at least notstraightforwardly.With“definiteintegrals”likethoseRamanujanofferedinhislettertoHardy,however,you’reofferedaback-doorroutetoasolution.

A definite integral is “definite” in that you seek to integrate the function over adefinitenumericalrange;thelittlenumbersattopandbottomoftheelongatedS—the∞and0inRamanujan’sequation—tellwhatitis.(Inotherwords,youmarkoffapieceofthefarmplotwhoseareayouwantreckoned.)Whenyouevaluateadefinite

integral, you don’t wind up with a general algebraic formula (as you do withindefiniteintegrals)but,inprinciple,anactualnumber.Andsometimes,byapplyingtherightmathematicaltools,youcandeterminethisnumberwithoutintegratingthefunctionfirst—indeed,withoutbeingabletointegrateitatall.

Broadly,thiswaswhatRamanujanwasdoinginthetheoremonpage3ofhislettertoHardyandallthroughthesectionlabeled“IV.TheoremsonIntegrals.”

This particular integral, hewas saying, could be represented in terms ofgammafunctions.(Thegammafunctionis likethemorefamiliar“factorial”—4!,read “fourfactorial,”=4×3×2×1—exceptthat itextendsthe ideatonumbersotherthanintegers.)Hardyfiguredhecouldprovethistheorem.Laterhetried,andsucceeded,thoughitprovedharderthanhethought.NoneofRamanujan’sotherintegralsweretrifling exercises, either, and all would wind up, years later, the object of papersdevoted to them. Still, Hardy judged, these were among the least impressive ofRamanujan’sresults.

Moresoweretheinfiniteseries,twoofwhichwere:

and

Thefirstwasn’tnewtoHardy,whorecognizeditasgoingbacktoamathematiciannamedBauer.Thesecondseemedlittledifferent.Toalayman,infact,itandkindredonesinRamanujan’slettermightseemscarcelyintimidatingatall;saveforpiandthegamma function they were nothing but ordinary numbers. ButHardy and otherswould show how these series were derived from a class of functions calledhypergeometricseriesfirstexploredbyLeonhardEulerandCarlFriedrichGaussandasalgebraicallyformidableasanybodycouldwant.

Sometime before 1910, Hardy learned later, Ramanujan had come up with ageneralformula,latertobeknownastheDougall-RamanujanIdentity,whichunderthe right conditions could be made to fairly spew out infinite series. Just as anordinarybeercan ismade in ahuge factory, theordinarynumbers inRamanujan’sserieswerethedeceptivelysimpleendproductofcomplexmathematicalmachinery.Of course, on thedayhe gotRamanujan’s letter,Hardyknewnothing of this.Heknew only that these series formulasweren’t what they seemed. Compared to the

integrals,theystruckhimas“muchmoreintriguing,anditsoonbecameobviousthatRamanujanmustpossessmuchmoregeneraltheoremsandwaskeepingagreatdealuphissleeve.”

SometheoremsinRamanujan’sletter,ofcourse,didlookcomfortablyfamiliar.Forexample,

Ifαβ=π2,then

Hardyhadprovedtheoremslikeit,hadevenofferedasimilaroneasamathematicalquestion in the Education Times fourteen years before. Some of Ramanujan’sformulasactuallywentback to thedaysofLaplaceand Jacobiacenturybefore.Ofcourse,itwasquitesomethingthatthisIndianhadrediscoveredthem.

Butnow,then,whatwasHardytomakeofthisone,whichhefoundonthe lastpageofRamanujan’sletter?

then

This was a relationship between continued fractions, in which the compressednotationfor,say,thefunctionuactuallymeansthis:

The publication of this result some years hencewould set off a flurry of work byEnglish mathematicians. Rogers would furnish one ten-page proof for it in 1921.Darlingwouldexplore it, too.In1929,Watsonwouldapproachit fromadifferentangle,tryingtosteerclearofthetrickymathematicalterrainofthetafunctions.Butin1913,Hardy couldmake nothing of it, classing it among a group of Ramanujan’stheoremswhich,hewouldwrite,“defeatedmecompletely;Ihadneverseenanythingintheleastlikethembefore.Asinglelookatthemisenoughtoshowthattheycouldonlybewrittendownbyamathematicianofthehighestclass.”Andthen,inaclassic

Hardyflourish,headded:“Theymustbetruebecause,iftheywerenottrue,noonewouldhavetheimaginationtoinventthem.”

AsHardyandLittlewoodprobed the theoremsbefore them, trying tomakeoutwhattheysaid,wheretheyfit intothemathematicalcanon,andhowtheymightbeprovedordisproved,theybegantoreachajudgment.ThattheIndian’smathematicswas strange and individual had been evident from the start. But now they werecoming to see his work as something more. It was not “individual” in the way arebelliousteenagertriestobe,camouflaginghisordinarinessbehindbizarredressorhair.Itwasmuchmore.“ThereisalwaysmoreinoneofRamanujan’sformulaethanmeetstheeye,asanyonewhosetstoworktoverifythosewhichlooktheeasiestwillsoon discover,” Hardy would write later. “In some the interest lies very deep, inotherscomparativelynearthesurface;butthereisnotonewhichisnotcuriousandentertaining.”

The more they looked, the more dazzled they became. “Of the theorems sentwithout demonstration, by this clerk of whom we had never heard,” one of theirTrinitycolleagues,E.H.Neville,wouldlaterwrite,“notonecouldhavebeensetinthe most advanced mathematical examination in the world.” Hardy would rankRamanujan’sletteras“certainlythemostremarkableIhaveeverreceived,”itsauthor“amathematicianof thehighestquality, amanof altogether exceptional originalityandpower.”

Andso,beforemidnight,HardyandLittlewoodbegantoappreciate that for thepast three hours they had been rummaging through the papers of amathematicalgenius.

•••

Itwasn’t the first timea letterhad launchedthecareerofa famousmathematician.Indeed,asthemathematicianLouisJ.Mordellwouldlaterinsist,“Itisreallyaneasymatterforanyonewhohasdonebrilliantmathematicalworktobringhimselftotheattention of themathematicalworld, nomatter howobscure or unknownhe is orhowinsignificantapositionheoccupies.Allheneeddoistosendanaccountofhisresultstoaleadingauthority,”asJacobihadinwritingLegendreonellipticfunctions,orasHermitehadinwritingJacobionnumbertheory.

And yet, ifMordell was right—if “it is really an easymatter”—why hadGaussspurnedAbel?CarlFriedrichGausswasthepremiermathematicianofhistime,and,perhaps,ofalltime.TheNorwegianNielsHenrikAbel,justtwenty-twoatthetimehewroteGauss,hadproved that some equationsof the fifthdegree (likex5+3x4

+...=0)couldneverbesolvedalgebraically.Thatwasarealcoup,especiallysinceleading mathematicians had for years sought a general solution that, Abel nowshowed,didn’texist.YetwhenhesenthisprooftoGauss,themanhistoryrecordsas“thePrinceofMathematics” tossed itasidewithoutreading it. “Here,”oneaccounthashimsaying,dismissingAbel’spaperastheworkofacrank,“isanotherofthosemonstrosities.”

Then,too,if“itisreallyaneasymatter,”whyhadRamanujan’sbrilliancefailedtocastanequalspellonBakerandHobson,theothertwoCambridgemathematicianstowhomhehadwritten?

Certainly Henry Frederick Baker, forty-eight at the time he heard fromRamanujan, qualified as the kind of “leading authority”Mordell had inmind.HeheldaspecialCayleylectureship.HehadbeenelectedaFellowoftheRoyalSociety,attheageofthirty-two,in1898.HehadreceivedtheSylvesterMedalin1910.HehadbeenpresidentoftheLondonMathematicalSocietyuntiltheyearbefore.

But as one biographer noted after his death, Baker “was little affected by therevolutionbrought about amongst theCambridgemathematical analysts byG.H.Hardy in the first decade of the twentieth century. During this period Baker’spositionwasessentiallythatofoneof the leadersof theoldergeneration.”Thathewas immune to Hardy, of course, did not itself explain his indifference toRamanujan; it did, however, suggest some reticence about embracing new ideas.Indeed, Baker was said to so revere the great mathematicians of the past that itchoked his own originality. His upcoming second marriage, which took place in1913,mayalsohave lefthim lessopen to the importuningsofanunknownIndianclerk.

TheotherCambridgemathematician,aSeniorWrangler,wasE.W.Hobson,whowasinhis late fiftieswhenheheard fromRamanujanandmoreeminenteventhanBaker.Hishighforehead,prominentmustache,andstrikingeyeshelpedmakehim,inHardy’swords,“adistinguishedandconspicuousfigure”aroundCambridge.

But he was remembered, too, as a dull lecturer, and after he died his mostimportant book was described in words like “systematic,” “exhaustive,” and“comprehensive,” never in language suggesting great imagination or flair. “An oldstick-in-the-mud,”someoneoncecalledhim.Forsomeyears,hewasaTriposcoach(onestudentwasJohnMaynardKeynes),largelyignoringmathematicalresearch.Hewould take a conventional stand on the coming war, and vehemently opposedgranting degrees to women. These were sensibilities, then, hardly primed forunfamiliartheoremscomingfromanunorthodoxsource.

Of course,Ramanujan’s fate had always hung on a knife edge, and it had nevertakenmorethantheslightestwantof imagination, thebriefesthesitancy, to tip thebalance against him.Only themost stubborn persistence on the part of his friendRajagopalachari had gained him the sympathy of Ramachandra Rao. AndHardyhimselfwasputoffbyRamanujan’sletterbeforehewaswonoverbyit.Thecardsarestacked, against any original mind, and perhaps properly so. After all, many whoclaimthemantleof“newandoriginal”areindeednew,andoriginal—butnotbetter.So, in a sense, it should be neither surprising nor reason for any but themildestrebukethatHobsonandBakersaidno.

NorshoulditbesurprisingthatnooneinIndiahadmademuchofRamanujan’swork.Hardywas perhapsEngland’s premiermathematician, the beneficiary of thefinest education, in touch with the latest mathematical thought and, to boot, anexpert in several fields Ramanujan plowed. . . . And yet a day with Ramanujan’stheorems had left him bewildered. I had never seen anything in the least like thembefore. Like the Indians,Hardydidnot knowwhat tomakeofRamanujan’swork.Likethem,hedoubtedhisownjudgmentofit.Indeed,itisnotjustthathediscernedgeniusinRamanujanthatredoundstohiscredittoday;itisthathebattereddownhisownwallofskepticismtodoso.

That Ramanujan was Indian probably didn’t taint him in Hardy’s eyes. True,Hardy’sknowledgeofIndiamayhavebeenasmiredinimperialstereotypesasthatofmostotherEnglishmen;when hewas ten, back inCranleigh, the schoolmagazinehad featured a day at “An Indian Bazaar,” rife with bejeweled maidens, dagger-bearingGhurkas,andfilthyfakirscursing“Englishdogs”undertheirbreaths.Butby1913, Hardy had already made mathematical contact with several Indians. AprofessorofmathematicsatAllahabad,UmesChandraGhosh,had, in1899,giventhesolutiontooneofhisearliestquestions intheEducationalTimes.And in1908,another of Hardy’s questions, on infinite series, had drawn a response by V.Ramaswami Iyer, who two years before had founded the Indian MathematicalSociety and two years later would befriend Ramanujan. (Then, too, the Indiancricket sensation Ranjitsinjhi had been in his prime when Hardy was anundergraduate,perhapsalsohelpingtooverturnanylingeringprejudices.)

Growing up along Horseshoe Lane and attending Cranleigh School may havemadeHardyreadierthanotherCambridgedonstoseemeritdressedinexoticgarb.All his life, certainly, hewas sympathetic to the underdog.MaryCartwright, whomet him a few years after his Ramanujan period, recalled that, as a womanmathematician,“Iwasadepressedclass”—andsoenjoyedHardy’sfavor.SnowwrotethatHardypreferredthedowntroddenofalltypes“tothepeoplewhomhecalledthe

largebottomed:thedescriptionwasmorepsychologicalthanphysiological....[They]were the confident, booming, imperialist bourgeois English. The designationincluded most bishops, headmasters, judges, and . . . politicians.” When Hardyattendedacricketmatchandknewnoneof thecompetitors,hewould taphisownfavoritesonthespot.These“hadtobetheunder-privileged,youngmenfromobscureschools, Indians, the unlucky and diffident. He wished for their success and,alternatively,forthedownfalloftheiropposites.”

FurthertippingthebalanceinRamanujan’sfavorwasHardy’swillingnesstostrayfromsafe, familiarpaths.HobsonandBakerwereboth fromanearlier generation,more settled, perhaps at a time in their liveswhen theywere less eager to take onsomething new. Hardy, on the other hand, was a generation younger and had apenchantfortheunorthodoxandtheunexpected.HehadleftfamiliarCranleighforWinchester.Hehadallowedasixth-ratenovelist, “AlanSt.Aubyn,” todeflecthimtoward Trinity. He had weighed leaving mathematics; then, he had embracedProfessorLove’ssuggestionthathedipintoCamilleJordan’sCoursd’analyse.Hehadjoined theApostles.Hehad broken precedent to take theTripos after his secondyearinsteadofhisthird.

Each time Hardy had opened himself, he had come away enriched. Now,somethingwildly new and alien had presented itself to him in the form of a long,mathematics-denseletterfromIndia.Onceagain,heopenedhisheartandmindtoit.Onceagain,hewouldbethebetterforit.

2.“IHAVEFOUNDINYOUAFRIEND...”

“Noonewhowas in themathematicalcircles inCambridgeat that timecan forgetthesensation”causedby[Ramanujan’s]letter,wroteE.H.Nevilleyearslater.Hardyshowed it to everyone, sentpartsof it to experts inparticular fields. (Midst all theexcitement,Ramanujan’soriginal cover letter, alongwithonepageof formulas, gotlost.)Meanwhile,Hardyhadsprungintoaction,advisingtheIndiaOfficeinLondonofhisinterestinRamanujanandofhiswishtobringhimtoCambridge.

ItwasnotuntilawindySaturday,Februaryeighth,thedayfollowinghisbirthday,thatHardysatdowntodelivertoRamanujantheverdictonhisgiftsthatCambridgealreadyknew.“TrinityCollege,Cambridge,”hewroteatthetop,andthedate,thenbegan:“DearSir,Iwasexceedinglyinterestedbyyourletterandbythetheorems...”Withanopeninglikethat,Ramanujanwould,atleast,havetoreadon.

Butintheverynextsentence,Hardythrewouthisfirstcaveat:“YouwillhoweverunderstandthatbeforeIcanjudgeproperlyofthevalueofwhatyouhavedone,itisessentialthatIshouldseeproofsofsomeofyourassertions.”

Proof.Itwasn’tthefirsttimethewordhadcomeupinRamanujan’smathematicallife. But it had never before borne such weight and eminence. Carr’s Synopsis,Ramanujan’smodel for presentingmathematical results, had set out no proofs, atleastnonemore involvedthanawordortwoinoutline.ThathadbeenenoughforCarr,andenough forRamanujan.Now,Hardywassaying, itwasnot enough.Themereassertionofaresult,howevertrueitmightseemtobe,didnotsuffice.AndallthroughhislettertoRamanujanhewouldsoundthesameinsistenttheme:

I want particularly to see your proofs of your assertions here. You will understand that, in this theory,everythingdependsonrigorousexactitudeofproof.

Andagain:

assumingyourproofstoberigorous...

And:

Ofcourseinallthesequestionseverythingdependsonabsoluterigour.

On the whole, Hardy’s letter was lavish with encouragement. True, some ofRamanujan’stheoremswerealreadywellknown,orweresimpleextensionsofknowntheorems.Buteven these,Hardyallowed, represented an achievement. “Ineednotsay that ifwhatyousayaboutyour lackof training is tobe taken literally, the factthatyoushouldhaverediscoveredsuchinterestingresultsisalltoyourcredit.”

Then,too,Hardyhazarded,someofRamanujan’sresults,whilethemselvesoflittlenote,wereperhapsexamplesofgeneralmethodsandthusmoreimportantthantheyseemed.“Youalwaysstateyourresultsinsuchparticularformsthatitisdifficulttobesureaboutthis.”

Littlewood was also intrigued by Ramanujan’s work, Hardy mentioned, evenaddingasasortofappendix,“FurtherNotesSuggestedbyMr.Littlewood.”MostofthesedealtwithRamanujan’sworkonprimenumbers,asubjectinwhichLittlewoodhad recentlymade a stunning, if not yet published, advance. So Hardy’s messagefromLittlewoodcarriedan especially ferventplea: “Please send the formula for theno.ofprimes&...”—hereitwasagain—“asmuchproofaspossiblequickly.”

Hardy’s whole letter was like that—shot through with urgency, with a barelycontainedexcitement,thatRamanujanwouldhavebeendullindeednottosense.Atthe bottom of page 6,Hardy wrote, “I hope verymuch that you will sendme asquicklyaspossible”—heunderlineditwithaveritableslashacrossthepage—“afewofyourproofs,andfollowthismoreatyourleisurebyamoredetailedaccountofyourworkonprimesanddivergentseries.”

Andhewenton: “It seems tomequite likely that youhavedonea gooddealofworkworthpublication;andifyoucanproducesatisfactorydemonstration,IshouldbeverygladtodowhatIcantosecureit.”

•••

Hardy’s letter probably arrived late in the third week of February. But hisendorsement of Ramanujan had reached Madras earlier. Almost a week beforewriting Ramanujan,Hardy had contacted the India Office and, by February 3, acertain Mr. Mallet had already written Arthur Davies, secretary to the AdvisoryCommittee for Indian Students inMadras. Later in the month, Davies met withRamanujan and, at Sir Francis Spring’s behest, Narayana Iyer, apprising him ofHardy’swishthatRamanujancometoCambridge.

But asHardy soon learned, Ramanujan wasn’t coming. Religious scruples, or acultural resistance that vergedon it, got in theway;Brahmins andotherobservantHinduswereenjoinednottocrosstheseas.Andthat,itseemedforalongtime,wasthat.

Meanwhile, inMadras, the balance delicately poised sinceRamanujan’smeetingwithRamachandra Rao in late 1910, teetering inconclusively between success andfailure,nowcamedownfirmlyonRamanujan’sside.Allthathadbeenwantingwasfor amathematicianofunimpeachable credentials toweigh inwith a verdict.NowHardyhaddeliveredit.

On February 25, GilbertWalker was shown Ramanujan’s work.Walker was aformerSeniorWrangler,a former fellowandmathematical lectureratTrinity,andwasnow,attheageof forty-five,headoftheIndianMeteorologicalDepartment inSimla. At the time of his appointment, he had no meteorological backgroundwhatever.But so crucialwaspredicting the onset of themonsoon in India, and solively the press furor in the wake of several years of bad predictions, that it wasthoughtaprofessionalmathematicianofWalker’sstanding—hehadjustbeennamedaFellowoftheRoyalSociety—mighthelpdefusethesituation.

Now,asWalkerwaspassingthroughMadras,SirFrancisprevailedonhimtolookthroughRamanujan’s notebooks.The next day,Walker wroteMadrasUniversity,asking it to supportRamanujan as a research student. “The character of theworkthatIsaw,”hewrotetheuniversityregistrar,

impressedme as comparable in originality with that of aMathematics fellow in a Cambridge College; itappearstolack,however,asmightbeexpectedinthecircumstances,thecompletenessandprecisionnecessarybefore theuniversalvalidityof theresultscouldbeaccepted. Ihavenot specialised in thebranchesofpuremathematicsatwhichhehasworked,andcouldnotthereforeformareliableestimateofhisabilities,which

mightbeofanordertobringhimaEuropeanreputation.ButitwasperfectlycleartomethattheUniversitywould be justified in enabling S. Ramanujan for a few years at least to spend the whole of his time onmathematicswithoutanyanxietyastohislivelihood.

Walker, his letter as much as said, was as mystified by Ramanujan’s work aseveryoneelsewas.Heavenknows,hewasnopuremathematician.Asayoungman,he’dshowninterestingyroscopesandelectromagnetics.Earlyprominencehadcomefromhisstudiesofaerodynamicforcesontheboomerang,whichasanundergraduatehe liked to throwon theCambridgeBacks.Now, as India’s chiefweatherman,hismost recentpaperwasentitled “TheColdWeatherStormsofNorthernIndia.” Inotherwords,hewasamathematicianofappliedbentwhosework layas fardistantfromRamanujan’sas itwaspossible tobe.His statement,Ihavenot specialised . . .wasgrossunderstatement,hisadmissionthathecouldnot forma reliable estimateofhisabilitiestheplain,simpletruth.

AndyetnoneofthisdiscouragedhimfromeagerlyrecommendingRamanujanforaspecialscholarship.Becausenow,inanyappraisalofRamanujan,therewasanewfactortoconsider:Hardy.Spring,whohadintroducedWalkertoRamanujan’swork,andeveryoneelseatthePortTrustknewthatRamanujanhadHardy’simprimatur.IfWalkerharboredanydoubtas toRamanujan’smerits,Hardy’s verdicterased it.ThewheelsofRamanujan’scareer,fortenyearsbarelycreakingalong,now,greasedbyHardy’sapproval,begantowhirrandwhinelikeafinelytunedracecarengine.

IfRamanujanhaddoubtsaboutwhatHardy’s endorsementmightmean tohim,theyhadbeendispelledbythelasttwodaysandWalker’sringingendorsement.OnFebruary twenty-seventh, he wrote Hardy a second letter, again packed withtheorems.“Iamverymuchgratifiedonperusingyourletterof8thFebruary1913,”hewrote.“Ihavefoundafriendinyouwhoviewsmylabourssympathetically.”

In fact, it was not justWalker, Spring, and others in theMadrasmathematicalcommunitywhohadbeenfortifiedbyHardy’sletter.ItwasRamanujanhimself.Forallhisconfidenceinhismathematicalprowess,Ramanujanneededoutsideapproval,affirmation.Nowhehadit.Hardy’slettertookhimseriously.AndthepronouncementdeliveredbythisunseenF.R.S.,thismanreputedtobethefinestpuremathematicianinEngland?Itwasnovague,emptyonefilledwithglowingaccoladesthatRamanujanmight, in an anxious moment, dismiss, but rather nine pages of specific, richlydetailedcomment:astatementRamanujanhadwrittenonhissixthpageoftheoremsaboutaseriesexpressibleintermsofpiandtheEulerianconstantcouldbededucedfromatheoreminBromwich’sInfiniteSeries(thebookM.J.M.Hill,inhisletteroftwo months before, had advised that he consult). A theorem on the same page

involving hyperbolic cosines Hardy himself had proved in Quarterly Journal ofMathematics.Hardyknew.

ThatRamanujanhadeverybitthe“invincibleoriginality”withwhichHardywouldlatercredithimdidn’tmeanhedidn’tcarewhatothersthoughtofhim.Hedidcare.In assessing Ramanujan’s work, Hardy had informally broken it down into threebroad categories—those results already known or easily derived from knowntheorems;thosecurious,andperhapsevendifficult,butnotterriblyimportant;andthosepromisingtobe important indeed, if theycouldbeproved. InHardy’smind,plainly,itwasthoseinthisthirdcategorythatweighedmostheavily.

ButnotinRamanujan’s.“WhatIwantatthisstage,”hewroteHardy,“isforeminentprofessorslikeyouto

recognize that there is someworth inme.”And such “worth,” he felt, accruednotalone,orevenprimarily, through the theoremsHardysawasnoveland important,butthosehe’drankedinthefirstcategory,asalreadyknown.Itwasthese,hewrote,“whichencouragemenowtoproceedonward.Formyresultsareverifiedtobetrueeven though I may take my stand upon slender basis.” Before the world whosejudgmentmattered so much to him, he could stand tall and declare, I have beenpronounced by competent authority to be just as I said I was—which was more, forexample,thanhisrepeatedflunkingoutofschoolhadseemedtosay.

In his second long letter to Hardy, Ramanujan seemed buoyant, pumped up,cocky.Hardywantedproof?Well,hewrote

IfIhadgivenyoumymethodsofproofIamsureyouwillfollowtheLondonProfessor[Hill].Butasafact,Ididnotgivehimanyproofbutmadesomeassertionsasthefollowingundermynewtheory.Itoldhimthatthesumofaninfiniteno.oftermsoftheseries:-1+2+3+4+...=−1/12undermytheory.IfItellyouthisyouwillatoncepointouttomethelunaticasylumasmygoal.IdilateonthissimplytoconvinceyouthatyouwillnotbeabletofollowmymethodsofproofifIindicatethelinesonwhichIproceedinasingleletter.Youmay ask how you can accept results based uponwrong premises.What I tell you is this:Verify theresults I give and if they agreewith your results, got by treading on the groove in which the present daymathematiciansmove,youshouldatleastgrantthattheremaybesometruthsinmyfundamentalbasis.

Gotbytreadingonthegroove!Ramanujanwasflyinghigh.Fouryearsofhawkinghismathematicalwareshad lefthimneither shyaboutgoingafterwhatheneedednorabovestoopingtoself-pity:“Iamalreadyahalf-starvingman,”hewroteHardynow.“To preserve my brains I want food and this is now my first consideration. AnysympatheticletterfromyouwillbehelpfultomeheretogetascholarshipeitherfromtheUniversityorfromGovernment.”

InthisveinRamanujancontinuedfortwopages,thenproceededtopileupmoretheorems, expanding on the ideas about prime numbers on which Hardy hadchallengedhim,andgoingontonewwork—inall,ninemoretheorem-stuffedpages.

“Ihavealsogivenmeaningstothefractionalandnegativeno.oftermsinaseriesaswell as in a product,” he wrote, “and I have got theorems to calculate such valuesexactly and approximately. Many wonderful results have been got from suchtheorems....”

Later,manywouldseeinRamanujananappealingandgenuinehumility.Butherewere hints of another Ramanujan, one already dreaming of a place for himself inmathematicalhistory: “Youmay judgemehard that Iamsilenton themethodsofproof,”hewrote.Buthissilencewasdueonlyto lackofspaceinwhichtosetthemdown,notunwillingness todo so.No,hewrote, “Idonotmean that themethodsshouldbeburiedwithme.”

Humble?E.H.Nevillewould later describeRamanujan as “perfect inmanners,simpleinmanner,resignedintroubleandunspoiltbyrenown,gratefultoafaultanddevotedbeyondmeasuretohisfriends.”Nowhere,though,didhecallhimhumble,orsuggestthatRamanujanshrankfromasanguineassessmentofhisowngifts.Nordoes his later remark to Janaki that, in her words, “his name would live for onehundredyears,” suggestunduehumility.Therehadbeena stubborn, self-confidentstreak inRamanujan all along; forhim towork, unrecognized and alone for years,fortifiedonlybyhisdelightintheworkitself,ithadtohavebeenthere.Andhere,inhissecondlettertoHardy,itunabashedlysurfaced.

Hardy, wemay be sure, was not put off by any of this. “Good work,” he oncewrote,“isnotdoneby‘humble’men.”

•••

On March 13, spurred by Walker’s letter, B. Hanumantha Rao, professor ofmathematicsat the engineering college, invitedNarayana Iyer to ameeting of theBoardofStudiesinMathematicstodiscuss“whatwecandoforS.Ramanujan. . . .Youhaveseensomeofhisresults&canhelpustounderstandthembetterthantheauthor himself.” On the nineteenth, the board met and recommended to thesyndicate, the university’s governing body, that Ramanujan receive a researchscholarshipofseventy-fiverupeesamonth—morethantwicehisPortTrustsalary—forthenexttwoyears.

ButwhenthesyndicatemetonApril7,Ramanujan’scaseencounteredasetback.Such scholarships were reserved for those with master’s degrees, and Ramanujanlackedevenabachelor’s;why,he’dflunkedoutofeverycollegehe’deverattended.Byone Indian account, it was the English—led by Richard Littlehailes, an Oxford-educatedprofessorofmathematicsatPresidencyCollegeandlaterportrayedasone

of Ramanujan’s champions—who invoked this technicality and, “with all theirvehementspeeches,”linedupagainsthim.Inanycase,thesyndicate’svice-chancellor,P.R.SundaramIyer,chiefjusticeoftheMadrasHighCourt,thenrose.Didnotthepreamble of the act establishing the university, he asked, specify that one of itsfunctions was to promote research? And, whatever the lapses of Ramanujan’seducation,washenotaprovenquantityasamathematicalresearcher?

Thatargumentwontheday.“TheregulationsoftheUniversitydonotatpresentprovideforsuchaspecial scholarship,” theregistrar laterwrote. “But theSyndicateassumes thatSectionXVof theActof Incorporation andSection3of the IndianUniversitiesAct,1904,allowofthegrantofsuchascholarship,subjecttotheexpressconsentoftheGovernorofFortSt.GeorgeinCouncil.”

It was a measure of how far, six weeks after receipt of Hardy’s letter, MadrasopinionhadswungbehindRamanujan:now,theauthoritieswerestretchingtherulestoaccommodatehim.

ByApril12,Ramanujanhadlearnedthegoodnews.Thescholarshipsethimfreetodomathematics,toattendlecturesattheuniversity,touseitslibrary.Sothatnow,asNevillewouldwrite,he“enteredthePresidencyCollegeinMadrastopracticeasavirtue that singleminded devotion to mathematics which had been a vice inKumbakonamnineyearsearlier.”

3.“DOESRAMANUJANKNOWPOLISH?”

Itwasagreatopenspacethat,asyoucameoutof thealleys leadingto it,offeredabroadbluesweepofsky,alittleasthepiazzaofanItalianhilltowndoes.Butitwasnotapiazza,orasquare,orapark.Itwasa“tank,”asortofreligiousswimmingpool—anexpanseofwater,normallyalmost square,witha sandybottom,granite stepsleadingdowntothewateronseveralsides,andaman-madeislandofrichlydetailedreligioussculptureatitscenter.

This onewas large, perhaps a hundred yards across, and notably handsome. Itslegendarypredecessorgavethewholedistrict itsname—Triplicane,acorruptionofTiru Alii Keni, meaning “sacred lily tank.” Adjacent to it stood the ancientParthasarathyTemple,aVaishnaviteshrinewhoseprincipaldeitywasKrishnaasthe“sarathi,”orcharioteer,inthegreatbattleoftheBhagavadGita,andwhichstillboretheinscriptionofaPallavaking’sgiftoflandinA.D.792.HereMuslim,Dutch,andFrenchsoldiershadencampedoverthecenturies.Herethefaithfulwouldbatheonritualoccasions—sometimesafewatatime,occasionallyhundredsorthousandsofthem,closelypackedtogether,aseaofbare-chestedbrownmen.

Offonestreetborderingthetanktothesouthranalittlelane,HanumantharayanKoilStreet.Andafewdoorsdown,almostwherethestreetturnedabruptlyleft,wasalittlehouse,setbackacourtyard’sdepthfromthestreet.Here,aboutamileandahalf fromPresidencyCollege,Ramanujan andhis familynow lived.With researchscholarshipinhandandonleavefromthePortTrust,theynolongerhadtostayincongestedGeorgetown,andprobablyaroundMayhadmovedbacktoTriplicane.

Ramanujan had nothing to do now but pursue mathematics and, every threemonths,submitaprogressreport.Forthishereceivedseventy-fiverupeesamonth.BackinKumbakonam,thefiveortenrupeesthefamilygotmonthlyfromboardersmadeforasizablechunkoftheirincome.EvenatthePortTrust,he’dstillonlymadetwenty-fiveorthirtyrupees,possiblyasmuchasfiftyatonepoint.Andwhatwithatleast Janaki,Komalatammal, andKomalatammal’sownmother toprovide for, thatdidn’tgofar;sometimeshe’dhadtomoonlight,tutoringcollegestudentsontheside.Now, though, he was almost flush, almost at the hundred rupees a month, forexample, that the IndianMathematical Society set as its threshold for paying fulldues. Ramanujan had friends in high places now, was published inmathematicaljournals,wascorrespondingwithoneoftheWest’stopmathematicians.

Duringtheearlymornings,andthenagainatnight,he’dworkwithNarayanaIyer,no longerbossnow,butcolleague.Sometimeshe’dborrowmathbooks fromK.B.Madhava,a statisticianwho liveddownthestreet.Often,hecouldbe found in thealcoves of the Connemara Library, a wing of which also housed the universitycollection. The Connemara, named for a former governor of Madras, was like asecularchurch,asoaringchamberofarchingstained-glasswindows,filledwithwoodengravings and columns and arches rich with molded plaster ornamentation, anIndo-Saracenictempleoflearning.HereRamanujanwould losehimselfonsomeofthosedeliciouslyfreedaysafterhisscholarshiphadsethimfree.

Itwasalittleliketheperiodbeforehismarriage.Morecompletelythaneverhewasable to throw himself into mathematics, leaving day-to-day cares to others.Sometimes,Janakiwouldrecall, “hewouldaskhismotherorgrandmother towakehimupaftermidnightsothathecouldgoonwithhisworkinthesilentandcoolerhours of the after-night.” Sometimes he “had to be reminded about his food.Onsomeoccasionshisgrandmotherormotherwouldserve, inhishand, foodmadeofcookedricemixedwithsambhar,rasam,andcurdsuccessively.Thistheydidsothathiscurrentofthoughtmightnotbebroken.”Atothertimes,sheorhismotherwouldprepare brinjal in just the way he liked.Normally this teardrop-shaped vegetable,aboutthesizeofapeach,waseatencooked.Buthehadlearnedtoloveitthespecialwayhismothermadeitforhim—quartered,thenquarteredagain,butwithatabof

flesh keeping all eight segments together, the junction steeped in tamarind andmasalaforanhourorso,theneatenraw.

Ramanujannowhadhisownworkroomupstairs.Heand Janakiusually slept atseparatetimes,butitseemedthat“wheneverIopenedmyeyes,”asshewouldrecall,“hewould beworking,” the scratch of stylus on slate sounding through thehouse.Once,on the spurof themoment,he riggedupa little science experiment forher;fromajugofwaterandsomekindoftubing,hemadeasiphonandshowedherhowgravity drew the water to lower points. But otherwise, contact between them wasslight. Janaki was barely fourteen. He worked in the heady realms of puremathematics, whereas she, beyond simple Tamil, had no education at all.Occasionally,whenhetookabreak,hewouldaskherto later joghismemorywithlanguage like, “the one you were working on downstairs,” or “the one you wereworkingonbeforeeating.”But intellectually, theycouldsharenothing.Nordidhetrytoforce-feedher.Shedidn’taskhimto,andhedidn’tvolunteer.

•••

RamanujanandHardy,meanwhile,dancedaroundoneanotheruncertainly.Intheircorrespondence,HardywasdefendingthemathematicalrevolutionhehadwroughtinEngland,tryingtowrestrigorousproofsfromRamanujan.Ramanujanheldback,offeringexcuses.Atleastthatwashowhisstancecouldberead,andbymid-Marchthe situation verged on a real row. “How maddening his letter is in thecircumstances,”LittlewoodwroteHardy.“Irathersuspecthe’safraidthatyou’llstealhiswork.”Takinguptheissueinhisnextletter,Hardytookpainstoreassurehim:

Letmeputthematterquiteplainlytoyou.Youhave inyourpossessionnow3[Ramanujanhadonly two]longlettersofmine,inwhichIspeakquiteplainlyaboutwhatyouhaveprovedorclaimtobeabletoprove.Ihave shown your letters toMr.Littlewood,Dr.Barnes,Mr.Berry, andothermathematicians.Surely it isobviousthat,ifIweretoattempttomakeanyillegitimateuseofyourresults,nothingwouldbeeasierforyouthantoexposeme.Youwill,Iamsure,excusemystatingthecasewithsuchbluntness.IshouldnotdosoifIwerenotgenuinelyanxioustoseewhatcanbedonetogiveyouabetterchanceofmakingthebestofyourobviousmathematicalgifts.

Whether Hardy and Littlewood had misread Ramanujan’s letter or not,Ramanujannowdeniedmisgivings and evenmanaged, rather successfully, to seemhurt. “I am a little pained to see what you have written at the suggestion ofMr.Littlewood,”hewroteinmid-April.

Iamnot inthe leastapprehensiveofmymethodbeingutilisedbyothers.Onthecontrarymymethodhasbeen inmypossessionfor the lasteightyearsandIhavenot foundanyone toappreciate themethod.AsI

wroteinmylastletterIhavefoundasympatheticfriendinyouandIamwillingtoplaceunreservedlyinyourpossessionwhatlittleIhave.

Ramanujanthespecialresearchstudent,RamanujanthefriendofCambridgewasnow a hot topic inMadrasi circles, and people were forever trooping through hishouse,as if to do him homage. InAugust, his name came up at a get-together ofprofessorsandtheirstudents;everyonemarveledhowRamanujanhadgarneredsuchintellectualstanding“without,”asonetherethateveningputit,“thehelpofbooksorteachers.”

In September, Narayana Iyer submitted some theorems on the summation ofseries to the Journal of the IndianMathematical Society, at one point adding: “ThefollowingtheoremisduetoMr.S.Ramanujan,theMathematicsResearchStudentoftheMadrasUniversity.”

OnOctober26,perhapsouttowinoveronewhohadbeforecampaignedagainstRamanujan’sscholarship,NarayanaIyertookRamanujantoseeRichardLittlehailes,professor of mathematics at Presidency College and soon to become Madras’sdirector of public instruction. Ramanujan was never good at explaining his ownmethods,soNarayanaIyerdidthetalking.Littlehailesassured themthat, after thefirstofDecember,helookedforwardtostudyingRamanujan’sresults.

InNovember,themathematicianE.B.RossofMadrasChristianCollege,whomRamanujanhadmeta fewyearsbefore, stormed intoclass,hiseyesglowing. “DoesRamanujanknowPolish?!?”heaskedhisstudents.Ramanujandidn’t,ofcourse.YethismostrecentquarterlyreporthadanticipatedtheworkofaPolishmathematicianwhosepaperhadjustarrivedbytheday’smail.

As a research scholarship holder, Ramanujan’s only obligation was to preparereportseverythreemonthsdetailinghisprogress.Hedeliveredthreeoftheminlate1913andearly1914,alldutifullyontime.Likemuchofhiswork,thetheoremshedescribed had their roots in his notebooks; some went back to notes appearingaroundpage180ofhisfirstnotebook,sometochapters3and4ofthesecond.Mostofthemdealtwithevaluatingdefiniteintegrals.

“At present,” he wrote in his first report, addressed to the Board of Studies inMathematicsanddatedAugust5,1913,“therearemanydefiniteintegralsthevaluesof which we know to be finite but still not possible of evaluation by the presentknownmethods.”The theorem he offered—it would later be called “Ramanujan’sMasterTheorem”—wouldprovidemeansofevaluatingmanyofthem.“Thispaper,”hiscover letternoted,“maybeconsidered the first installmentof the results Ihavegotoutofthetheorem.”More,hepromised,werecomingsoon—asindeedtheywereinthesecondandthirdreports.

As in his letters to Hardy earlier that year, Ramanujan was attacking definiteintegrals that resisted every effort to reduce them to simpler, more useful forms,defeated the whole arsenal of mathematical tools brought to bear on them.Ramanujanwasfashioningnewtools.

Like a screwdriver, saw, or lathe, a mathematical “tool” is supposed to dosomething; those used to evaluate a definite integral perform mathematicaloperationsonitthat,onehopes,getitreadyforthenexttool—thenexttheoremortechnique—and ultimately lead to a solution. But just as a screwdriver tightensscrewsbutcan’tsawwood,amathematicaltoolmayworkforevaluatingoneintegralbutnotothers.Ifyoudon’tknow inadvance,youtry it. If itdoesn’twork,youtrysomethingelse.

Onetool,whichRamanujanhadapparentlyencounteredinan1896textbookonintegralcalculus,wasFrullani’sintegraltheorem.Now,inlate1913,Ramanujanwastellingofapowerfulgeneralizationof itthatcoulddefeatawiderrangeof formerlyunyielding integrals. To apply, Frullani’s theorem demanded that two particularfunctions be equal; in Ramanujan’s generalization, they didn’t have to be, thusexpandingitsapplicabilitytomanymorecases.Back in1902,HardyhadwrittenapaperontheFrullaniintegral.ButhehadneverseeninitwhatRamanujansawnow.

While this time Ramanujan furnished proofs for many of his assertions, moreanalytical mathematicians would later shoot them full of holes. Yet the resultsthemselves—thetheoremsRamanujanofferedastrue—were true.BruceBerndt,anAmerican Ramanujan scholar, would see in that curious split a message formathematicianstoday:“Wemightallowourthoughtstooccasionallyescapefromthechainsofrigor,”headvised,“and,intheirfreedom,todiscovernewpathwaysthroughtheforest.”

4.ADREAMATNAMAKKAL

While life was sweet to Ramanujan during this period, his long-distancecorrespondence with Hardy had soured. In the cover letter to his first quarterlyreport, in August, he cited Hardy much as the author of a book might quotefavorablereviewcomment:“TheintegraltreatedinEx.(v)noteArt.5inthepaper,Mr.G.H.Hardy,M.S.,F.R.S.,ofTrinityCollege,Cambridge,considerstobe‘newandinteresting.’ ”Butbythen, in fact, thetwoof themwerescarcelywritingatall.Perhapsitwasthetartnessintheirearlierexchanges.OrHardy’sdisappointmentatRamanujan’srefusaltocometoEngland,andfrustrationatcommunicatingacrosssoformidable a physical and cultural gap.Or simply the press of otherwork. In any

case,whileRamanujanwrotehimatleastonceduringthisperiod,Hardyformonthsfailedtorespond.

Finally, though, in early January, Ramanujan found a long letter from Hardy,responding to a proof he had earlier supplied, pointing out its flaws, and showinghowtheyhadledhimastray.“Youwillseethat,withallthesegapsintheproof,itisnowonderthattheresult iswrong.”But,Hardywenton,walkingoneggs, “Ihopeyou will not be discouraged by my criticisms. I think your argument a veryremarkable and ingenious one. To have proved what you claimed to have provedwouldhavebeenaboutthemostremarkablemathematicalfeatinthewholehistoryofmathematics.”

Andohyes,therewasonemorething.“Try,”headdedalmostcarelessly,“tomaketheacquaintanceofMr.E.H.Neville,who isnow inMadras lecturing.Hecomesfrommycollegeandyoumightfindhisadviceastoreadingandstudyinvaluable.”

Thiswasnotexactlyalie,butitwasn’tthewholetrutheither.Hardyhadmoreinmind than getting Ramanujan the right books to read.He had madeNeville theinstrumentofhisplan.HehaddeputizedhimtobringRamanujantoEngland.

Of course, that had been his intent from the beginning.As Snowput it, “OnceHardy was determined, no human agency could have stopped Ramanujan” fromcoming.EvenbeforewritingRamanujan,hehadcontacted the IndiaOffice to thateffect.Butthen,theanswerhadcomebackfromMadrasthat,forreligiousreasons,Ramanujanhadnointentionofcoming.

ThewayRamanujantoldthestoryaboutayearlaterinalettertoHardy,hehadgotten a letter in February 1913 from Arthur Davies, secretary to the AdvisoryCommitteeforIndianStudentsinMadras.CouldRamanujanmeethiminhisofficethefollowingnoon?SirFrancisaskedRamanujan’s“superiorofficer”—this,certainly,wasNarayanaIyer—toaccompanyhimandansweranyquestions.

Thenextday,Daviesdroppedthebigquestion:WasRamanujanpreparedtogotoEngland?

Ramanujan’s mind raced. What did the offer mean? Go to England and studymath? Go to England and take examinations, and doubtless fail them? Would itmean...?

Butbeforehecouldspeak,NarayanaIyer—whomRamanujandepictedas“averyorthodoxBrahminhavingscruplestogotoforeignland”—repliedthatno,ofcourseRamanujancouldnotgotoEngland.Then,accordingtoRamanujan,“thematterwasdropped.”

That,atleast,wasRamanujan’slaterversionoftheseevents.Butalmostcertainly,it was a face-saving story concocted to explain his past unwillingness to go.What

blockedhisgoingwas,indeed,Brahminic“scruplestogotoforeignland.”ButitwasnotNarayanaIyer’sscruples,butthoseofRamanujan’s friends,andhis family,andhimself.

•••

“His people shunned him for having lost his caste,” it was recorded of an earlyeighteenth-centurySouthIndianwhohadtraveledtoEuropeindefianceofhiscaste.“Hewaspracticallyadeadmanintheirestimation.”Whileabroad,themanhadbeengrantednumeroushonors,hadbeennamedChevalierof theOrderofSt.Michael.But back in India, none of it counted; he was despised.When, generations later,another Indian, T. Ramakrishna, asked his mother whether he might travel toEngland to study, the story was invoked in warning. Ramakrishna waited for hismothertodiebeforedefyingherwishesandmakingthetrip.

For an orthodox Hindu—and Ramanujan came from a very orthodox Hindufamily—travelingtoEuropeorAmericarepresentedaformofpollution.Itwasinthesame category as publicly discarding the sacred thread, eating beef, or marrying awidow. And, traditionally, it had the same outcome—exclusion from caste. Thatmeantyourfriendsandrelativeswouldnothaveyoutotheirhomes.Youcouldfindno bride or bridegroom for your child. Your married daughter couldn’t visit youwithoutherselfriskingexcommunication.Sometimes,youcouldn’tgo intotemples.You couldn’t even get the help of a fellow casteman for the funeral of a familymember.Herewasthegrim,day-to-daymeaningofthewordoutcaste.

AquartercenturybeforeRamanujan,GandhihadmetsimilarobstaclesingoingtoEnglandforhiseducation.“Willyoudisregardtheordersofthecaste?”itsSheth,orheadman, had asked him. He did—and was pronounced an outcaste. ByRamanujan’stime,scruplesagainstforeigntravelhadrelaxedbutslightly;forallbutthemostadventurous,itwasstilltaboo.

Howevermuch hemay have yearned forEngland,Ramanujan found inMadraslittle reprieve from tradition’s hold. Madras was no oil-fed Houston, or railroad-driven Chicago, where ambitions surged and dreams were meant to be lived. Itspopulationwasstatic,barelychanginginthepastdecade.Itwasflat,builtlow,spreadoutover the countryside; therewerenoHimalayanpeaks todraw the imaginationupward, no great towers to serve as symbols of human aspiration.More than theother great cities of India,Madras clung in spirit to the villages and towns of thesurroundingcountryside.CalcuttaandBombayroiledoverwithunmarriedworkersfrom all over India, bore a sort of rude, masculine, Wild West dynamism. But

Madraswasmoresociallycohesive,withmenfromthevillagesbringingtheirwivesand children with them. Like Ramanujan, the one in sixteen Madrasis who wereBrahminsmostly inhabitedtheareaaroundTriplicane’sParthasarathyTemple.AllaroundRamanujantheylived—orthodoxBrahminswho,likehismother,buttressedtraditionandencouragedconformity.Allaroundhimtheforcesofsocialorderweresecurelyinplace.

AndRamanujanwasnorebel.Ifhewaveredinhisacceptanceofthebaronforeigntravel,hewasnotabouttosayso—andwascertainlynot,onhisown,goingtodefyit.Forhimtogo toEngland, strongoutside forceswouldhave tobebrought tobear.Externalvoicesherespectedwouldhavetosanctionit.

•••

Around New Year’s Day in 1914, Hardy’s man, Eric Harold Neville, arrived inMadras.

Anablemathematicianwhohadneverdone“anythingwhichhasarousedanyrealenthusiasm”:that’showHardyassessedNevilleafewyearslater.“Ishouldexpect[hiswork]tobegood,butheisnotamanlikeLittlewoodofwhomonewouldsayitmustbe good.”That said almost asmuch aboutHardy as it didNeville, of course, forNevillewasalreadyanimportantyoungmathematician.Justtwenty-five,hehadbeenamongthelasttotaketheold-styleTripos—sittingforitayearearlyinordertohaveashotatbecomingthelastSeniorWranglerbeforeHardy’sreformstookhold.(Hecameinsecond.)Twoyearslater,in1911,NevillewontheSmith’sPrize,andayearafterthatwasnamedaFellowofTrinityCollege.Now,inthewinterof1913,he’dcometoMadrastogiveaseriesoflecturesondifferentialgeometryattheuniversity.

He had, of course, one additional charge—to convince Ramanujan to come toEngland.

Itwas intheSenateHouse, theuniversity’s examinationhall andoffices, locatedacrosstheroadfromMarinaBeach,thatinearlyJanuarythetwomenmet.Ontheoutside, the Senate House was an extravagant architectural blend of Italianate,Byzantine,andIndo-Saracenicinfluences,executedinbrickandstone,shotthroughwith rosewindows,minarets,parapets, and chiseled stonepillars.On the inside, itwas littlemore than a vast void—agreathall built to seat sixteenhundredpeople,withonlyacarved,fifty-four-foot-highceilingandstained-glasswindowstorecallitsornateexterior.HereNevillehadcometodeliverthetwenty-onelecturesthat,overthenextmonth,woulddrawmathematiciansfromalloverSouthIndia.

After one of the first lectures,Ramanujanwas introduced tohim.Thiswas not“the uncouth, unshaven, unclean figure of Ramachandra Rao’s picture,” Nevillewouldwritelater,“butamanatoncediffidentandeager.”AsforhisEnglish,oncesopoorithadunderminedhisschoolcareer,“tenyearshadworkedwonders,foramorefluentspeakeroronewithawiderandbetterusedvocabularyIhaveseldommet.”

At least three times they satdown togetherwithRamanujan’snotebook.Nevillewas stunned—somuch so that when, after their thirdmeeting, Ramanujan askedhimwhether he might like to take the notebook away to peruse at his leisure, itstruck him as “the most astounding compliment ever paid to me. The pricelessvolumehadneverbefore[soNevilleassumed]beenoutofhishands:noIndiancouldunderstandit,noEnglishmancouldbetrustedwithit.”

Perhaps,Nevillecametothink,RamanujanmistrustedtheEnglishmenhe’dmetinMadras.PerhapshemistrustedHardy,withwhomhehad correspondedonly at acontinent’s remove. But he,Neville, a fresh wind from afar, bore none of the oldbaggage.Apuremathematician,naturallysympathetictoRamanujan’smathematics,withinayearorsoofhiminage,hehad lavishedthreedaysonthenotebooksthatwereRamanujan’slife’swork.HehadgainedRamanujan’strust.

Andsonowhestruck,actingonHardy’schargetohim:wouldRamanujancometo Cambridge? Anticipating a negative response, Neville silently marshaled hisarguments. Yet now, unaccountably, he didn’t need them. “To my delight andsurprise,”hewrotelater,helearned“thatRamanujanneedednoconvertingandthathisparents’oppositionhadbeenwithdrawn.”

•••

Whatmiraclehadwroughtthistransformation?By one account it was K. Narasimha Iyengar, a family friend with whom

RamanujanhadstayedinMadrasearlyin1911andwithwhomhehadremainedintouch since, who helped get Ramanujan’s mother, certainly a key obstacle, toacquiescetothetrip.SeshuIyeralsoexertedpressureonher,RamaswamiIyerandRamachandraRao onRamanujan. “I lent all theweight ofmy influence to inducehim to go,” recalledRamachandraRao later. So didM.T.Narayana Iyengar, theBangaloremathematicianandeditoroftheJournaloftheIndianMathematicalSocietywho had worked closely with Ramanujan three years before to get his first paperready for publication; scrupulously orthodox himself, his arguments carried addedweight.

Butiftheseinfluencesmaybesaidtohaveultimatelytriumphed,theydidnotbythemselveschangeRamanujan’smind.Somethingmorewasneeded—something,at

least for public consumption, beyond his mother’s merely human will, orRamanujan’s.Neville learned what it was: “In a vivid dream his mother had seen[Ramanujan] surrounded by Europeans and heard the goddess Namagiricommandinghertostandnolongerbetweenhersonandthefulfillmentofhislife’spurpose.”Detailsdiffer,butthisandotherversionsofthestoryagreeinsubstance—thatpermissionforRamanujantogocamepersonallythroughtheinterventionofthegoddessNamagiri,residinginhershrineatNamakkal.

•••

InthevicinityofNamakkal,eightymileswestofKumbakonam,hillsabruptlypushupfromtheotherwiseflatplain.Palmandbananatreescanstillbeseen.Socanricefields.Butoverall, thevegetation ismarkedlythinnerthanalongtheCauvery.Mudandbrickhutshavegivenwaytostone,notmuchseentotheeast,chiseledoutofthesurroundinghillsandfromrockyoutcroppingsjuttingoutfromthefields.

Oneofthoseoutcroppings,more impressivebyfarthantherest,gaveNamakkalitsname.Atownofseventhousandpeople,abouttwentymilesfromthenearesttrainstation(inKarur,nearwhereRamanujan’smotherandgrandparentsoriginallycamefrom),Namakkallayatthefootofagreatwhiterock,twohundredfeetstraightupandhalfamilearound.Throughouttherockweretinysacredgrottos,jonais,formedover the eonswhere rock-rooted greenery and rainfall had conspired towear awayfissures.Atthetop,reachedbynarrowstepshewnintoitssouthwestslope,stoodanoldfortprotectedbybrickbattlementsthreefeetthick.Tosomeone,theverticalgashthat the great rock made in the sky suggested the vertical white streaks of theVaishnavitecastemark,knownasthenamam.Hence,Namakkal.

ItwasforherethatinlateDecemberof1913,Ramanujan,hismother,NarayanaIyer,andNarayanaIyer’s son set out; Janakihad asked to go, too, butRamanujantold her she was too young. They got off the train in Salem, a city of seventythousandsetinapicturesquevalleyrimmedbymountains.TheretheystayedatthehouseofRamaswamiIyer,whowasdeputycollectoraswellasfounderoftheIndianMathematicalSociety.Fromthere,RamanujanwrotebackhometoKumbakonam,then set out alone withNarayana Iyer, probably in a bullock cart, forNamakkal,aboutthirtymilesduesouth.

InNamakkal,theytookalittleroad,flankedwithhousesbuiltuptotheedgeofit,thatgentlyclimbedandcurvedupfromthemiddleoftown.Soon,nearthebaseofthegreatrock,theyreachedthestone-columnedfacadeandgiantwoodendoorsthatguardedthetempleofLordNarasimha,thelion-faced,fourthincarnationofVishnu

—and,inaseparate,smaller,pillaredshrineofftothe left,hisconsort,thegoddessNamagiri.

Ramanujan’s family was not the only one upon whom the goddess exerted apowerfulhold.On certain days of theweek, zealous, frenziedwomenwouldwendtheirwaytotheshrinetobeexorcisedofdevils. “Thehall infrontofthegoddess,”accordingtooneaccount,wouldbe“filledwiththeirshrieksandconvulsions,untilasprinkling of sacred water over their heads by the pujaris [priests] silences them.”Withinthecrampedpassagewhere the sweatingpriestsperformedtheirdevotions,theairwasfullofsmoke,thestonewallsblackwithincense.

For three nights, Narayana Iyer and Ramanujan slept on the temple grounds.Sittingthereon the stoneslab floor, theycould lookupat the sheer rock face thatformed the back wall of the temple, see the scalloped battlements of the old fortsilhouettedagainstthesky.Thefirsttwonights,nothinghappened.Butonthethird,RamanujanrosefromadreamandwokeNarayanaIyerwithwordthat,inaflashofbrilliantlightorsomesuchsimilarrevelation,hehadreceivedtheadesh,orcommand,tobypasstheinjunctionagainstforeigntravel.

NarayanaIyer’s familytodaybelievesthat,withshrewdinsight intoRamanujan’spsycheandmindfulofhisdevotiontothegoddessNamagiri,heconceivedthetriptoNamakkal;thatsostrongwasRamanujan’swishtogotoEnglandandsostronghisdevotiontoNamagiri, that something likewhathappenedhad tohappen;and thatwhenitdid,NarayanaIyercalculated,hewouldbethereto“correctly”interpretanyrevelationRamanujanmighthave.

DidavisionofNamagiriactuallycometoRamanujan,inallearnestness,whilehesleptonthecolumnedtemplegroundsatNamakkal?Or,asNevilleheardit,didhismotherhavethedream,herimportuningstoobeyitsmessagefirmlysettlingmatters?Or did both mother and son dream of Namagiri? Or did Ramanujan now moreardentlythaneverwishtogotoEnglandand,havingdeferredtohisfamilyformostofayear,seekasociallyacceptablewaytodoit?

CertainlyRamanujanalwaysattributed hisdecision todivine inspiration. (SodidJoanofArcattributehers:“IlockedmyselfupintheatticforadayandanightandGodtoldmethatIwastobecomeaGeneralofHeavenandleadthearmiesofFranceand expel theEnglish.”)Known, too, is that inFebruary 1913, so far as the IndiaOfficewasconcerned,RamanujanrefusedtotraveltoEngland;thatinDecember,hevisitedNamakkal;thataweekortwolaterhemetNevilleattheSenateHouseandsurprised himwith hiswillingness to embark for England; that on January 22, hewroteHardy,askinghimandLittlewoodto“begoodenoughtotakethetroubleofgettingmethere—within,”hescribbledbetweenthelines,“averyfewmonths.”

In his letter, Ramanujan sought to distance himself from what, in Cambridge’sview,hadbeenayear’sobduraterefusaltogo.“NowIlearnfromyourletterandMr.NevillethatyouareanxioustogetmetoCambridge,”Ramanujanwrote,asifhearingofitforthefirsttime.“IfyouhadwrittentomepreviouslyIwouldhaveexpressedmythoughts plainly to you.”MeetingwithDavies back inFebruary, he suggested,he’dbeenahelplesspawnundertheswayofhis“superiorofficer.”Why,hehadnotevensuppliedthe languageofhisearlier letters; that, too,hadbeentheworkofhissuperior. As for his own religious scruples, or that of his family, or the trip toNamakkal,Ramanujansaidnothing.

Thewhole schememayhavebeen theworkofNarayanaIyer,orat least carriedoutwithhiswillingcomplicity.Inanycase,itwouldnotbetheonlytimeRamanujanbent the truth to avert embarrassment. Visiting Kumbakonam early in 1914, justbefore leaving for England, he told the family of his friends Anantharaman andSubramanianthathehadcometosaygood-bye.Hewasleaving,heannounced—forCalcutta.Underthecircumstances, the liemayhavebeenwell-advised.Whentheirfatherlearnedthetruth,andfearingthatRamanujan’sexamplemightlurehissonstoEngland,hewenttoMadrastotrytodissuadeRamanujanfromgoing.

•••

Thekeyobstacleremoved,NevillesetaboutaddressingRamanujan’sotherdoubts.Money to get to England and live there? Don’t worry, Neville assured him, thatwouldbe taken care of.HisEnglishwasnot very good? Itwas,Neville said, goodenough.His vegetarianism? That would be respected. And examinations?Havingflunkedvirtuallyeveryonehe’devertakenincollege,hewaspainedattheprospectoftakinganymore—yetknewthatdoingsowastheinevitablelotofIndianstudentsinEurope.No,Nevillereassuredhim,hewouldnothavetotakeany.

InoptingforEngland,Ramanujanstillwentagainsta largebodyofopinion.Hisfather-in-law,forexample,wonderedwhyhecouldnotpursuemathematicsinIndia.Hismother worried that his finicky health might suffer in the English cold; thatremainingvegetarian,withoutgoodIndianfoodavailable,wouldbedifficult;thathemightencounteroutrightprejudicefromthelocals—or,ontheotherhand,thathe’dbebesetbyEnglishgirls.(Beforeheleft,shewasupsetwhensomeEnglishwomen,cometomeetthedarkgeniusboundforCambridge,actuallyshookhishand.)

Some ofRamanujan’s friends,meanwhile, saw the trip, inNeville’swords, as “amean attempt to transfer to the English university the glory that belonged toMadras.”Neville,takingnochances,setaboutwooingthem,too.“Lestheshouldbe

harassedbyattemptstodissuadehim,IaddressedmyselftothetaskofconvincinghisIndian friends that the proposal was in Ramanujan’s own interest, and in factembodiedtheonlychanceofplacinghimonthepinnaclewhere they longed to seehim.”

Next,NevillewroteHardy to say that it was now time to address the financialobstaclestoRamanujan’svisit toEngland.He,Neville,would try to findmoney inMadras. But should he fail, as he later paraphrased his letter, “the money mustsomehow be found in England. . . . Financial difficulties must not be allowed tointerfere.”

HardyapparentlyforwardedNeville’spronouncementtotheIndiaOffice,becausehesoonreceivedfromC.Mallet,secretaryforIndianstudents(whoayearbeforehadrelayed word of Ramanujan’s refusal to go to England), a worried reply. “Mr.Neville’sletterratheralarmedme,becauseitseemedtomethathewasencouragingRamanujantocometoEnglandwithoutanyrealprospectofprovidingforhimwhenhe got here.” Too often, he had found, Indian students arrived without enoughmoney,onlytomeetwith“disappointmentandmisery.”

Inthebluntestterms,MalletadvisedHardythat“nomoneyforthispurposecanbe got from the India Office.” Furthermore, he doubted whether Trinity orCambridgewouldcomeupwithany, andhedidn’t thinkMadras could either.Hewasnotsanguine.AndheinfectedHardywithhispessimism.“I’mwritinginahurrytocatchthemail,”HardywroteNeville,abitfrantically,“andwarnyoutobealittlecareful”; the money had to be there, else Ramanujan couldn’t come. He andLittlewoodmight togethercontribute fiftypoundsayear for thecontemplated twoyears of Ramanujan’s visit—“Don’t tell [Ramanujan] so”—but that came to onlyaboutafifthofRamanujan’sneeds.

Perhaps influencedby correspondencewithHardyor the IndiaOffice thatdoesnot survive,Neville would later castHardy’s concern as a case of intellectual, notfinancial,coldfeet,asjittersaboutRamanujan’srealabilities.“Wehaveheardoftheseunknowngeniusesbefore,”heparaphrased(ormisconstrued,ormisrepresented)theIndiaOffice letterHardy had forwarded. “They dazzle their friends in India, andwhenwebringthemtoEnglandweseethemfortheprecociousschoolboystheyare;in a few weeks they fizzle out, and more harm than good has been done by ourbenevolence.” Neville would laugh at the timidity he imputed to Hardy for,presumably,endorsingsuchdoubts.Ofcourse,headded,“IhadseenthenotebooksandtalkedwithRamanujan,andHardyhadnot.”

In any case, by the time he’d heard from Hardy, Neville already had moneymatters well in hand. Littlehailes had introduced him to people influential in the

universityorgovernment,andeverywherehe talkedupRamanujan. “ThediscoveryofthegeniusofS.RamanujanofMadras,”he’dwrittenFrancisDewsbury,registrarof theuniversity, on January28, “promises tobe themost interesting event of ourtimeinthemathematicalworld.”Itwasathoughtful,rathergrandlystatedletter,allaimedatpreciselyoneend—fundingRamanujan’sstayinEngland.“Iseenoreasontodoubt,”itconcluded,

thatRamanujanhimselfwillrespondfullytothestimuluswhichcontactwithWesternmathematiciansofthehighest class will afford him. In that case, his name will become one of the greatest in the history ofmathematics, and the University and City of Madras will be proud to have assisted in his passage fromobscuritytofame.

Nextday,Littlehaileshimselftookuptheattack,formallyaskingDewsburyfora250-pound-per-year scholarship, coupled with a 100-pound grant to equipRamanujan withWestern clothes and book passage to England. “Ramanujan,” hewrote,“isamanofmostremarkablemathematicalability,amountingImightsaytogenius,whoselightismetaphoricallyhiddenunderabushelinMadras.”

ThefollowingweekLordPentlandhimself,governorofMadras,becamethetargetof this bombardment of Madras officialdom on Ramanujan’s behalf. Sir FranciswrotePentland’sprivatesecretary,C.B.Cotterell:

IamanxioustointeresthiminamatterwhichIpresumewillcomebeforehimwithinthenextfewdays—amatterwhichunderthecircumstancesis,Ibelieve,veryurgent.ItrelatestotheaffairsofaclerkofmyofficenamedS.Ramanujan,who,asIthinkHisExcellencyhasalreadyheardfromme,ispronouncedbyveryhighmathematicalauthoritiestobeaMathematicianofanewandhigh,ifnottranscendental,orderofgenius.

Springhad just learned that theuniversitywasprepared to set aside ten thousandrupees, equivalent to more than six hundred pounds, or enough for two years inEngland. But the decision hinged on higher approval. And here, he said, “HisExcellencymayperhapsbeabletointerferewithadvantage.”

“ThebestgentlemanandbynomeanstheworstbrainweeversenttoIndia,”itwasoncesaidofLordPentland.BornJohnSinclair,hewasaslim,slightmanwithafullmustachewhohadgraduated fromSandhurst, theBritishWestPoint, and servedeight years in theRoyal IrishLancers.Thenhe’d turned to politics,most recentlyserving as secretary forScotlandunderPrimeMinisterDavidLloydGeorge.OnlythepreviousOctoberhe’dcometoMadrasasgovernorofFortSt.George,inwhichcapacity he ruled over forty million people. He believed that the state’s function,according toonewhoknewhim, shouldbe “to secure for all itsmembers the bestprocurableconditions for the fulldevelopmentofpersonality.”Now, in the caseofRamanujan,hehadthechancetoactonit.

He had already gone to bat forRamanujan oncewhen, the year before, he had

consented tohis special research scholarship.Nowhewas ready to “interferewithadvantage” again. “His Excellency cordially sympathizes with your desire that theUniversityshouldprovideRamanujanwiththemeansofcontinuinghisresearchesatCambridge,”hissecretarywrotebacktoSpring,“andwillbegladtodowhathecantoassist.”

Thescholarshipwasapproved.Thelastroadblockwasgone.RamanujanwasgoingtoEngland.

5.ATTHEDOCK

OnFebruary26,Binny&Co.sentRamanujanhissecond-classticket.OnMarch11,SirFranciswrotethesteameragentstomakesurehegotvegetarian

foodenroute.OnMarch14,RamanujanaccompaniedhiswifeandmothertoMadras’sEgmore

Station. There, in its compact waiting room, flanked by two rows of columnedarches, they awaited the train. Ten years before, he had arrived here to begin hisstudies at Pachaiyappa’s College.Now, he wept: he was dispatching his family toKumbakonam,sotheywouldnothavetowitnesshispainful transformation intoaEuropeangentleman.

Janaki,onedaywhilehermother-in-lawwasatthetemple,hadaskedhimtotakeher with him to England. But influenced by Ramachandra Rao, among others,Ramanujan said no, explaining to her that if he had to tend to her in England hewouldnotbeabletoconcentrateonmathematics;that,besides,shewassoyoungandprettyhe’dhaveonlytoturnhisbackandtheEnglishmenwouldbeuponher....

Nobodywho sawRamanujanduring those last busydays recalled seeing inhimany exhilaration, anticipation, or joy. “He was not very jubilant over his futurejourney,”RamachandraRaowouldrecall.“Heseemedto[move]as if . . .obeyingacall.”

HisfriendscoachedhiminWesternways.Stilltakingaproprietaryinterestinhisyoungprotégé,RamachandraRaodecreedthathiskutumi,thelongbunched-upknotofhairatthebackofhishead,hadtogo.Anditwasdone.Further,RamanujanmustwearWesternclothes.SoonRichardLittlehaileswasdrivinghimaroundtownonhismotorcycle,Ramanujaninthesidecar,shoppingforcollarsandtiesandstockingsandshoesandshirts.

For a few days Ramanujan stayed in the country, at the house of a friend ofRamachandraRao’swholivedEuropean-style,learninghowtouseknifeandfork—though “under the strict stipulation,” as his patron observed, “that nothing but

vegetable food should be served.” But even this tentative step into alien ways leftRamanujanunhappy.“Hedidnotrelishfoodbeingservedbystrangeservants.”

RamanujanworriedabouthowhewouldstayvegetarianinEngland.HehatedhisWesternhaircut.Hewasmiserableabouttheclotheshehadtowear.Thedaybeforehewastoleave,hewalkedintothefacultycommonroomatPresidencyCollegewithabigsuitcase,openeditup,andlaidoutonthetabletheWesternclothespurchasedfor him. How, he pleaded, was he to wear them? Making the knot on his tieconfoundedhim,toeveryone’samusement.Ramanujantriedtojokeaboutit,buthisoldKumbakonamfriend,Raghunathan,nowonthestaffof thecollege, thoughthelooked distinctly uncomfortable. Later, when hismother got a photo of him fromEngland,allsqueezedintocollar,tie,andjacket,shewouldn’trecognizehim.

That night,K.Narasimha Iyengar and his cousin hadRamanujan over to theirplaceinTriplicane.Outside,thestreetswerefilledwithcartsdrawnbybullockswithbells jingling frompaintedhorns,withbare-chestedmen indhotis,withwomen insaris, their nose-rings and bangles gleaming against their dark skin. The pungentsmellofburningcowdungfilledtheair.ToRamanujan,itwasallhe’deverknown.ButEngland?Whatwastocome?Allnighthisfriendsstayedupwithhim,triedtocalmhisjitters,preparehimforthegreatadventure.

OnMarch 15, the British India Lines ship S. S.Nevasa had arrived inMadrasthroughthenewnorthernentranceof theharbor,builtduringSirFrancis’stenure.Most ships still tied upwithin the protected breakwaters of the harbor, then hadtheir cargoes transported by lighter to the docks. But a special wharf to bring inpassengers,troops,andhorseshadbeenbuiltatthesouthendoftheharbor,withatimber-decked dock, like a little boardwalk, projecting into the harbor from thebreakwater.ItwasherethattheNevasatiedup.

TheNevasa was brand-new, designed expressly for the India run.Her hull waspaintedblack,exceptforredaccentsandathinribbonofwhiterunningforeandaft.She was a smart-looking vessel, graceful in her way, her single funnel, aboutmidships,setwithamodestrake.

Themorningofitsdeparture,anofficialsend-offwasheldinRamanujan’shonor,organized by Srinivasa Iyengar, the advocate general. On hand were ProfessorMiddlemastand Sir Francis Spring, prominent judges, andKasturirangar Iyengar,publisher of the Hindu. So was Narayana Iyer who had worked so closely withRamanujan,theincessantclickingandscrapingontheirslateskeepingpeopleinhishouse up all night. “My father made a strange request to him,” his son N.Subbanarayananwouldrecordmanyyearslater.“AsamementomyfatherwantedtoexchangehisslatewithRamanujan’sslate,[arequestthat]wasgranted.Perhapsmy

father thought thathemay get an inspiration from the slateduring [Ramanujan’s]absence.”

Ramanujanwas introducedaround.Madras’sdirectorofpublic instruction,J.H.Stone,wishedhimsuccess, toldhimhehadwritten friends inEnglandwhowouldtake care of him. Among the passengers, Ramanujan met a man in the SalvationArmyboundforSouthampton,andaDr.Muthu,atuberculosisspecialist.Hemetthe captain, too,who joked that they’d get along fine so long asRamanujandidn’tbotherhimwithmathematics.

For most everyone, it was a time of good cheer and light banter. But not forRamanujanwho,recalledafriend,“wasintears.”

Finally,therewasnothinglefttodo.Ramanujanwasonboard,hiswell-wishersleftbehind.Ataboutteno’clockonthemorningofMarch17,1914,theNevasaslippedslowlyawayfromthedock.

CHAPTERSIX

Ramanujan’sSpring

[1914to1916]

1.OUTOFINDIA

TheNevasa’spaintstillgleamed.Barelyayearhadelapsedsinceshe’dbeendeliveredtoherownersbyaGlasgowshipyardandsetoutonhermaidenvoyagetotheEast.Atninethousandtons,shewasbyfarthelargestshipintheBritishIndiaLinesfleet.Herfourwidedeckswereairyandcomfortable.Shehadbeendesignedexpresslyforservice in the tropics. Yet Ramanujan could find no comfort aboard her, no relieffromthepitchingseasthatlefthim,onhisfirstoceanvoyage,seasickandunabletoeat.

Temporary relief came with the ship’s first stop in Colombo, capital of Ceylon(today’sSriLanka),thelargeislandjustoffIndia’ssoutheastcoast.LongbeforetheNevasadocked,passengerscouldsmellColombo’scinnamongardens.Bluehillsroseabove the harbor and the houses of the city, with their red roofs and walls ofshimmeringwhite.

OnMarch19,theNevasasteamedoutofport,skirtedsouthofCapeComorin,atthe tipof the subcontinent, andmadedirect forAden, aweek’s passage across theArabian Sea. For Ramanujan, now past his seasickness, the voyage grew pleasant.Now he could enjoy the ship’s roominess. He had his vegetarian food. He’d metseveralamonghistwohundredorsofellowpassengers.Sometimes,thestorygoes,hewithdrew to his second-class cabin and played with numbers suggested by thedimensionsofthecabinorthenumberofpassengers.

Ramanujan was not an introspective man—was not, as Hardy would put it,“particularlyinterestedinhisownhistoryorpsychology.”Butnow, in thecoatandcollarthattorturedhim,midsta limitlessexpanseof sea,asdaybyday theNevasasteamedwestatfourteenknotsandfreshseabreezessweptacrossthedeck,itwouldhavebeenhardforhalf-formedthoughtsnottospilloverintoconsciousness.

Fiveyearsago,hewasaloneonthepialofhishouseinKumbakonam—unknown,unmarried,boymorethanman;now,theMadrasielitehadturnedouttoseehimoff

toEngland.Then,hewas adropout fromGovernmentCollege,Kumbakonam;now,hewas

boundforTrinityCollege,Cambridge.Back then, getting to Madras by train, a matter of about three rupees, was no

triflingexpense;now, to the four-hundred-rupee fare toEngland aboard this greatsteamshipheneedgivenothought.

Then, the thought of violating the bar on overseas travel would scarcely havecrossedhismind;now,influentialmen,orthodoxBrahminsamongthem,hadurgedhimtobreakit.

Whathadwrought suchchanges?Plainly,hehadnotovernightbecomeabettermathematician.Whathad changedwas thathehad thrusthimself onto theworld.Hismotherordaining thathemarry,hehad setout fromdoor tomathematician’sdoorinsearchoflivelihood.Then,withmuchpersistenceandalittleluck,onethinghadledtoanother....

Andthefuture—isitconceivableheneverthoughtofit?Probably,hefeltavaguefearoftheunknown.Possibly,hisdisquiettookmoreconcreteform.Untilnow,eveninthatimpetuousflighttoVizagapatnam,hehadneverleftSouthIndia,wherefair-skinned Englishmen were the one-in-a-thousand exception; soon, in England, hewouldbetheconspicuousone,hisfacewouldstandout,hisaccentwouldseemalien.ThefuturemeantNeville,andHardy,andLittlewood,andCambridge,andTrinity.But onlyNevillewas a face, a person;Hardywas just a few letters.The restwerenames,disembodiedabstractions,mysteries.

Afteraweekofsteadysteaming,morethantwothousandmilesoutofColombo,theNevasaputinatAden,atthesouthernmostendoftheRedSea.NowbeganthepartofthejourneythatmanyamongtheEnglishpassengers,atleast,hadlearnedtodread—thefourteen-hundred-milepassageuptheRedSea,wheredayafterdaytheshipbakedinadesertheat thatneverdeviatedmuchfromahundreddegrees.Thepassagegavethelanguageanewword:if,outwardboundtoIndia,yougotstuckinastarboardcabin,youabsorbedthefullheatofthesunallafternoonandbybedtimeyourcabinsmoldered;portsidecabins,meanwhile,hadthewholeafternoontocooldown. On the homeward passage, it was all reversed. Not surprisingly, VIPsmanaged to land the cooler cabins, which were designated “Port Outward—StarboardHomeward”andgrantedtheacronymPOSH.

Whetherboredby the long voyageor inspiredby theprospectof soonbreakinginto theMediterranean,Ramanujangrewexpansive.FromSuez,at theentrance tothecanal,andPortSaidatitsotherend,exactlytwoweeksoutofMadras,hepostedat least four lettersbacktoIndia.One, toViswanathaSastri,borestampsshowing

thepyramids.OnewenttoR.KrishnaRao,nephewofRamachandraRao.“IdonotknowwhetherIhavetogotoCambridgedirectlyorstayatLondonandthengo,”hewroteaftertellingofthevoyagethusfar.“IshallwritetoyouafterIreachEnglandand everything is settled.My best compliments to your brother and respects andwarmestthankstoyouruncle.”

The next day, the Nevasa sailed from Port Said and steamed into theMediterranean.OnApril 7, after a stop inGenoa, fromwhichRamanujan postedanotherletterhome,sheleftMarseilles.

Then it was through the Strait of Gibraltar and up along the Spanish coastthrough theBayofBiscay toEngland.TheNevasa docked first atPlymouth, thensteameduptheEnglishChannelandarrivedatthemouthof theThamesonApril14.

•••

Itwasabright,lovelyday,alittlewarmerthanusual,withoutsomuchasatraceofovercast—moreoftherunoffineweatherthat,onEasterSunday,twodaysbefore,hadbroughtLondonersout to theparksandstreetsof thecitybythehundredsofthousands. Now, waiting for Ramanujan at the dock was Neville and his olderbrother, who had arrived there by car. They drove to 21 Cromwell Road, in theSouth Kensington district of London, a reception center for Indian students justarrivedinEngland.

Londonwasacityoffivemillionpeople,spillingoveritsancientbordersintothevillagesandhamletsofSurreyandMiddlesex.Inpopulation,itwastentimeslargerthanMadras.MadraswasthecapitalofSouthIndia?ThenLondonwasthecapitaloftheworld,thenervecenterfromwhichtheempirewasdirected.Thecloppingofhorses’ hooves, the jingle of harnesses, and the clatter of hansom cabs overcobblestonedstreetscouldstillbeheardinLondon,butthesehadbeguntogivewaytotheroarandsmokeofStudebakerCabriolets,ofWolsley“TorpedoPhaetons,”ofdouble-decker buses bearing signs advertisingNestlé’s chocolate. Londonwas fast.EvenHardy’s friend LeonardWoolf, who had returned to London in 1911 aftersevenyearsinCeylon,foundLondonaquiver,marchingto“atempoclearlyfasterandnoisierthanwhatIwasaccustomedto.”

BackinMadras,theEnglishmenRamanujanhadknownweremostlyeducatedandupper-class,withaccentstomatch,neverseentostooptomanuallabor.Now,onthestreets of London, he heard the nasal Cockney twang of rag merchants. He sawlamplighterswhopatrolled the streets at duskwith their longpoles.He sawknife

grindersmanning little two-wheeled carts,men sellingmuffinswho heralded theirwaresbyringingabell.Here,therewaseverysortofEnglishman—meninbowlersandflatworkingmen’scaps,womeninfineryandinrags.

Just off the boat and England was already a strange new world for any Indian.CromwellRoad, towhichNevillenow tookRamanujan,was supposed to ease thetransition.Of course it didn’t. The National India Association had offices there.Several rooms in the stately Georgian-styled corner building were available tostudentspassingthrough.Andacrossthestreet,theimposingedificeoftheNaturalHistoryMuseum, fairly glowing in its two-toned stone and adornedwith sculptedgriffins,lentanappropriatelyimperiallustertotheimmigrant’sfirstdaysinEngland.ButCromwellRoad typically failed in itsmission; a study a few years laterwouldchide the reception center for invariably making things worse for newcomers, notbetter.

But unlike most Indians in England, Ramanujan had by his side, in Neville, aCambridgedontohelpsmooththeway.HealsometA.S.Ramalingam,a twenty-three-year-oldengineerfromCuddalore,southofMadras,whohadbeeninEnglandforfouryearsandwhoalsotriedtohelphimfeelathome.Inanycase,Ramanujansurvived whatever rigors Cromwell Road could inflict and onApril 18 went withNevilletoCambridge.SoonhewassettledinNeville’shouseonChestertownRoad,inalittlesuburbofCambridgejustacrosstheRiverCamfromthetownitself.

ChestertownRoadwasa streetof fine townhouses, someturretedandcupolaed,their front yards typically setoff from the streetbywrought-iron fences.Thebay-windowedNevillehousesatoneinfromtheendoftheblockonagentlearcofstreetbesidetheriver,occupyingapeculiarlyshapedplotjustsixteenfeetwideonthestreetsidebutfanningouttomorethanfiftyintheback.Nevilleandhisnewwife,Alice,hadmovedintheyearbeforeandnow,fortwomonthsinthespringof1914,itwasRamanujan’sintroductiontotheEnglishhome.

Ithadbeenamodestlyscaledtwo-storyaffairwhenfirstbuiltaroundthemiddleofthe previous century. Early on, though, it had been enlarged. A long, fingerlikeprojection,builtofthesametanbrick,juttedoutthebackofthehouseandfollowedtheoddlyangledproperty line.Anadded thirdstory,with threemore rooms,gavethehouseaheightthatmadeitlikeanexclamationpointtothelittlegroupofhousesattheendoftheblock.

In any case, the Neville house was by now quite spacious, and wherever theysettled him in the sprawling place, Ramanujan was bound to have a measure ofprivacy he had never enjoyed before. The back of the house faced a large garden,whichhadoncebeenapearorchard.Fromthesecond-floorsittingroom,Ramanujan

couldlookoutovertheRiverCamandVictoriaBridgeandatthebroadexpanseofMidsummerCommonwithitscrisscrossofpedestrianpathsleadingtotheoldstonecourtsandcloistersofthecolleges.

There was, of course, business to attend to—fees to pay, paperwork to workthrough. Hardy and Neville took care of most of it. A printed list of first-yearstudents,preparedafterMichaelmasTerm(pronouncedMih-kel-miss, andstartingin mid-October) of the previous year, listed each student alphabetically. Now,squeezingit inbetweenPugh,F.H.andRawlins,J.D.,someonedippedhispeninblackinkandwroteinRamanujan’snamebyhand.

Thoseearlyweekswererichwithnewpromise,gracedbyawondrousspring.Dayafterdaytheweatherwaslovelyandwarm.MayflowersbloomedinApril.Tractsofopen countrysidewere transformed into great seas of bluebells. At the end of themonth, King George visited Cambridge, where he was greeted by thousands ofschoolchildren, waving tiny Union Jacks, trying to gain a glimpse of the royalDaimler.

Meanwhile, Ramanujan had already set to work with Hardy and Littlewood;Littlewood, for one, saw him about once a week, Hardy much more often.Ramanujan was productive, working hard, happy. “Mr. Hardy, Mr. Neville andothershereareveryunassuming,kindandobliging,”hewrotehomeinJune.

RamanujanhadnotcometoCambridgetogotoschool.Butarriving intimefortheEasterterm,whichbeganinlateApril,hedidattendafewlectures.SomewereHardy’s.Others,onelliptic integrals,weregivenbyArthurBerry,aKing’sCollegemathematicianinhisearlyfifties.Onemorningearlyintheterm,BerrystoodattheblackboardworkingoutsomeformulasandatonepointlookedovertoRamanujan,whose face fairly glowed with excitement. Was he, Berry inquired, following thelecture?Ramanujannodded.Didhecaretoaddanything?Atthat,Ramanujanstood,went to theblackboard, took the chalk, andwrotedown resultsBerryhadnot yetprovedandwhich,Berryconcludedlater,hecouldnothaveknownbefore.

Soon the word was getting out about Ramanujan. W. N. Bailey, then anundergraduate,heard“strangerumorsthathehadbeenunabletopassexaminations,and thathehadrunaway fromsuch terrors.Butapart fromthese rumorsweonlyknew that his name was Ramanujan, and even this was pronounced wrongly,”probably Rah-ma-noo-jn. People didn’t often see him; he was usually busy in hisrooms.Butwhentheydid,theynoticedhim—rememberedhissquat,solitaryfigureas,inthewordsofone,he“waddled”acrossTrinity’sGreatCourt,hisfeetinslippers,unableyettowearWesternshoes.

ItwasHardy’s rooms inNewCourt towhichRamanujanwas apt tobebound.Thissmallerquadranglewas “new”onlybyCambridgestandards,ofcourse,havingbeenbuilt in1823, twocenturiesor soaftermostof the restof thecollege.Hardylived on the second floor of Staircase A, just over the portal leading out to theAvenue,adoublerowoftwo-hundred-year-oldlimetreesparadingacrosstheBacks.ItwasalonghaulfromtheNevilles’toHardy’srooms—acrossthebridgeatthefarendoftowntoMidsummerCommon,alongoneofseveralfootpathscrossingit,andontoParkParade;thenbyoneoranotheroldcobblestonedstreettotheGreatGate,andonlythenintothecollegeitself.Allinall,perhapsatwenty-minutehiketoNewCourtatthefarsouthwesternedgeofthecollege.

That,apparently,wastoofar.InearlyJune,afteraboutsixweeksonChestertownRoad,RamanujanmovedintoroomsonStaircasePinWhewell’sCourt.Itwouldbe“inconvenientfortheprofessorsandmyselfifIstayoutsidethecollege,”hewrotetoafriend.

Probably,hewassadtoleavetheNevilles.NevillewasthefirstEnglishmantowinRamanujan’s confidence and, from themomentRamanujan disembarked from theNevasa, had donemuch to ease his adjustment to English life. Then, too, if lateraccounts are any guide, he and his wife Alice were the consummate hosts, theirhospitalityalegend.Theywereyoung,liberal-minded,andbynowhadanemotionalstakeinRamanujan.Inalllikelihood,theydotedonhim.

In Whewell’s Court, only about five minutes from Hardy’s rooms, Ramanujancould look out his window across to whereHardy had lived as an undergraduatetwenty years before. But Ramanujan had more than twenty years’ worth ofmathematicalcatchinguptodo.Hiseducationhadended,inasense,whenGeorgeShoobridge Carr put the finishing touches on his Synopsis in 1886. And Carr’smathematics was old when it was new, mostly barren of anything developed pastabout1850.

Ramanujan,then,hadmuchtolearn.But,thenagain,sodidHardy.

2.TOGETHER

TogethernowinCambridge,therewasnolongertheneedforthoselong,awkwardletters, across a gulf of culture and geography, with all their chance formisunderstanding.Now,ashewouldforthenextfewyears,RamanujansawHardynearly every day and could showhim themethods he haddeveloped in India thathe’d been loath to describe by international post. Meanwhile, Hardy had thenotebooks themselves before him and, with their author by his side, could studythemasmuchashewished.

Many of the 120 theorems Ramanujan had sent him in those first two letters,Hardycouldseenow,hadbeenpluckedintactfromthenotebooks.Here,inchapter5,section30ofthesecondnotebook,waswhatRamanujanhadwritten inthefirstletter about a class of numbers built up from “an odd number of dissimilar primedivisors.”Fromchapter5alsocamemuchoftheworkgoing intoRamanujan’s firstpublishedpaperonBernoullinumbers.Inchapter6wasthatbizarrestufffromthefirstletteraboutdivergentseriesthat,Ramanujanhadfeared,mightpersuadeHardyhe was destined for the lunatic asylum—the one where 1 + 2 + 3 + 4 + . . .unaccountablyaddedup to−1/12.On its face, thatwas ridiculous;yet it sought togivemeaning to divergent series—which at first glance added up to nothingmorerevealingorprecisethaninfinity.ButnowHardyfoundsomethinglikeRamanujan’sreasoningbehind it,which involved a “constant” that, asRamanujanwrote, “is likethecenterofgravityofabody”—aconceptborrowedfrom,ofallplaces,elementaryphysics.

A few ofRamanujan’s results were,Hardy could see, wrong. Somewere not asprofoundasRamanujanlikedtothink.SomewereindependentrediscoveriesofwhatWesternmathematicians had found fifty years before, or a hundred, or two. Butmany—perhaps a third, Hardy would reckon, perhaps two-thirds, latermathematicians would estimate—were breathtakingly new. Ramanujan’s fat,mathematics-richletters,Hardynowsaw,representedbutthethinnestsampling,thebarest tip of the iceberg, of what had accumulated over the past decade in hisnotebooks. There were thousands of theorems, corollaries, and examples. Maybethree thousand,maybe four.Forpage afterpage, they stretchedon, rarelywatereddown by proof or explanation, almost aphoristic in their compression, all theirmathematicaltruthsboileddowntoalineortwo.

The notebookswould frustratewhole generations ofmathematicians, whowereforeverunderestimatingthesheerdensityofmathematicalrichestheycontained.In1921,afterhavingforsevenyearsbeenexposedtothem,Hardywouldnotethat“amassofunpublishedmaterial”stillawaitedanalysis.Twoyearslater,havingdevoteda paper to Ramanujan’s work in chapters 12 and 13 of his first notebook, onhypergeometric series, he had to report that those were, in fact, “the only twochapterswhich,up to thepresent, Ihavebeen able to subject to a really searchinganalysis.”

Around that time,Hardy was visited by theHungarian mathematician GeorgePolya, who borrowed from him his copy of Ramanujan’s notebooks, not yet thenpublished.Afewdayslater,Polya,insomethinglikeapanic,fairlythrewthembackatHardy.No, he didn’t want them. Because, he said, once caught in the web of

Ramanujan’sbewitchingtheorems,hewouldspendtherestofhislifetryingtoprovethemandneverdiscoveranythingofhisown.

In 1929, G. N. Watson, professor of pure mathematics at the University ofBirmingham and a former Trinity fellow, and B. M. Wilson, who had knownRamanujan in Cambridge and was then at Liverpool University, set out on amathematicalodysseythroughRamanujan’snotebooks.Twoyears later,recountingtheir progress, Watson admitted the task was “not a light one.” A single pair ofmodular equations, for example,had takenhimamonth toprove.YetRamanujanwassorewarding,hewrote,thatheandWilsonthoughtit“worthwhiletospendafairly substantial fraction of our lives in editing the Note Books and makingRamanujan’searlierdiscoveriesaccessible.”Heestimatedthejobmighttakeanotherfive years. In fact, beforehis energywaned in the late1930s,Watsonhaddevotedmostofadecade to the job,producingmore thantwodozenpapersandmassesofnotesneverpublished.(Wilsoncouldgivetheprojectonlyfoursyearsmore;hedied,afterroutinesurgery,in1935.)

In1977, theAmericanmathematicianBruceBerndttookupwhereWatsonandWilsonleftoff.Afterthirteenyearsofwork,havingpublishedthreevolumesdevotedsolelytothenotebooks,heisstillatittoday,thetaskunfinished.

Plainly, then, in the months after Ramanujan arrived in England, Hardy andLittlewood could hardly have more than skimmed the surface of the notebooks,dipping into themat points, lingering over particularly intriguing results, trying toprove this one or simply understand that one. But this first glancewas enough toreinforce the impression left by the letters.After the second letter,LittlewoodhadwrittenHardy,“Icanbelievethathe’satleastaJacobi.”Hardywastoweighinwithatributemore lavishyet.“Itwashis insightintoalgebraicalformulae,transformationofinfiniteseries,andsoforth,thatwasmostamazing,”hewouldwrite.Intheseareas,“Ihavenevermethisequal,andIcancomparehimonlywithEulerorJacobi.”

•••

Euler and Jacobiwere both towering figures on the stage ofmathematical history.Leonhard Euler has been called “the most productive mathematician of theeighteenthcentury,”theauthorofalmosteighthundredbooksandpapers,manyofthemafterhewasblind,ineveryfieldofmathematicsknowninhisday.ItwasEulerwho, in his 1748 book, Introductio in analysin infinitorum, gave the trigonometricfunctionstheformtheyhavetoday.Today’smathematicstextsfairlyspilloverwithEuler’s constant, and Euler’s criterion, and the Euler-Maclaurin formula, and

Eulerian integrals, andEuler numbers.Born twenty years afterEuler’s death, KarlGustav Jacob Jacobiwasnearlyhis equal ingenius.The sonof aBerlinbanker,hepioneered elliptic functions and applied them tonumber theory.His name, too, isenshrinedinmathematics,inJacobi’stheoremandJacobi’spolynomials.

But Euler and Jacobi were not just generic “great mathematicians”; it was notcapriciouslythatHardyandLittlewoodhadcomparedRamanujantothem.Rather,thesetwomenrepresentedaparticularmathematicaltraditionofwhichRamanujan,too,waspart—that of “formalism.”Formal, here, carries no suggestion of “stiff” or“stodgy.” Euler, Jacobi, andRamanujan had (along with deep insight) a knack formanipulating formulas, a delight inmathematical form for its own sake.A “formalresult” suggests one fairly bubbling up from the formulas themselves, almostirrespective of what those formulas mean. Computers today manipulate three-dimensionalcontoursregardlessofwhethertheyrepresenteconomicforecastsorcarbumpers. Somepainters care asmuch for form, line, or texture as they do subjectmatter.Themindofthemathematicalformalistworksalongsimilarlines.

Allmathematicians, of course,manipulate formulas. But formalists were almostmagiciansat it, uncannily selecting just the tricks and techniques needed to obtainintriguingnewresults.Theywould replaceonevariable in an equationby another,thus reducing it to simpler form.Theywould knowwhen to integrate a function,when to differentiate it, when to construct a new function, when to worry aboutrigor,whentoignoreit.

But already by 1914, a faint odor of derision clung to them. For one thing,mathematicians of more finicky tastes clucked at how formalists sometimessteamrolled over certain mathematical niceties. By this light, they were holdoversfromaprerigorouspastwhoseingeniousformulassometimesfailedtostandthetestofclosereasoning.

But more, formalists were seen to inhabit a mathematical backwater. Usefulformulastended tobe foundearly in thedevelopmentofanewmathematical field,pointingthewaytofuturework.Butasafieldmaturedandtheseearlyformulaswereappliedandextended,theyoftengrewtoocomplicatedtobeuseful.ByRamanujan’stime,somethinglikethishadhappenedinbranchafterbranchofmathematics.

Andsothesmartmoney,asitwere,hadlargelyabandonedthesearchforformulas.The formalist came to be seen as one who stopped short of deep thinking andchurnedoutnarrowresults, throughmereconjurer’s tricks, that failed to turnoverimportant new ground.His was a style ofmathematics not somuch profound asclever;thatsmackednotofHighSciencebutLowArt—orBlackMagic.

Ramanujan’smathematics,ifitfitanycategory,fitthisone.Andyet,Hardycouldsee that if Ramanujan possessed conjurer’s tricks, they were ones of almostMephistophelean potency. Ramanujan was a formalist who undermined thestereotypes. “It is possible that the great days of formulae are finished and thatRamanujan ought to have been born 100 years ago,”Hardy would write. But, heacknowledged, “He was by far the greatest formalist of his time,” one whosemathematicalsleightofhandnoonecouldmatch,andwhosetheorems,howeverhegotthem,latergenerationsofmathematicianswouldesteemaselegant,unexpected,anddeep.

It was with some sense, then, of mingled mystery and awe that Hardy andLittlewood came away from their first long look at Ramanujan’s notebooks. “Thebeautyandsingularityofhisresultsisentirelyuncanny,”LittlewoodwouldcommentinareviewofRamanujan’spaperspublishedlater.“Aretheyodderthanonewouldexpectthingsselectedforodditytobe?Themoralseemstobethatweneverexpectenough;thereaderatanyrateexperiencesshocksofdelightedsurprise.”

ForHardy’spart,confrontingthemysteryofRamanujan’smindwouldconstitute,ashisfriendSnowhadit,“themostsingularexperienceofhislife:whatdidmodernmathematicslookliketosomeonewhohadthedeepestinsight,butwhohadliterallyneverheardofmostofit?”

Ramanujan“combinedapowerofgeneralization,afeelingforform,andacapacityfor rapidmodificationofhishypotheses, thatwereoften really startling, andmadehim,inhisownpeculiarfield,withoutarivalinhisday,”Hardywouldconclude.Asforhisultimate influence,Hardy couldn’t, at the timehewrote, say; in a sense, itsvery peculiarity undercut it. “It would be greater,” he suggested, “if it were lessstrange.”

But,headded,“Onegift ithaswhichnoonecandeny—profoundandinvincibleoriginality.”

•••

Having gone so out on a limb to bring Ramanujan to Cambridge, Hardy, afterfamiliarizing himself with the notebooks, probably felt a little relieved, too. Andproud.“Ramanujanwas,”hewouldwrite,“mydiscovery.Ididnotinventhim—likeother greatmen, he invented himself—but Iwas the first really competent personwho had the chance to see some of his work, and I can still remember withsatisfactionthatIcouldrecognizeatoncewhatatreasureIhadfound.”

It didn’t take long to see that much in Ramanujan’s notebooks was well worthpublishing.Editorialworkwas needed, of course; his results needed to be shaped,cast into lucid English, their notation made more familiar. Hardy, ever themathematical journalist, now proceeded to do just that. “All of Ramanujan’smanuscriptspassedthroughmyhands,”hewrote,“andIeditedthemverycarefullyfor publication. The earlier ones I rewrote completely.” (But, he added, hecontributednothing to themathematics itself—an assertionmade entirelycredibleby his readiness to take credit, in their jointly bylined papers, whenever he didcontribute.“Ramanujan,”hewrote,“wasalmostabsurdlyscrupulousinhisdesiretoacknowledge the slightest help.”) Now, in any event, Ramanujan’s bare notebookentriesbegantotakenewform,asmathematicalpapersfittobeseenandreadbytheworld.ByJune,heandHardyhadthebeginningsof twopapers,oneofwhichwasreadyenoughtoshowaround.

The second Thursday of eachmonth normally sawHardy board the 2:15 P.M.train out of Cambridge to attend that evening’s London Mathematical Societymeeting.WhenononeparticularsecondThursday—June11,1914—hegreetedhisfriends at the society’s meeting room near Piccadilly, he bore a manuscript withtheoremsbyRamanujan.Among those there tohear thempresentedwasHobson,oneof those towhomRamanujanhadwritten a year and ahalf before.Bromwichwasthere,too;hisbookoninfiniteserieswastheoneRamanujanhadbeensosternlyadvisedtoread.SowasProfessorLove,Hardy’sadvisorfromhisundergraduatedays.SowasLittlewood.

Ramanujanhimselfwasnot.

•••

The following year, 1915, would see a flood of papers published by Ramanujan,including theoneHardypresented that Juneevening to theLondonMathematicalSociety. But 1914, the year of his arrival in England, saw only one. Comprisingmostly Indianwork, it appeared in theQuarterly Journal ofMathematics under thetitle“ModularEquationsandApproximationstoPi.”

Every schoolchild knows that pi gives the ratio of a circle’s circumference to itsdiameter,about3.14.Whywastetimepursuingnewwaystoapproximateit?Surelynot for the sake of fixing it more precisely; even by the midnineteenth century,mathematicianshaddeterminedpitofivehundreddecimalplaces,farinexcessofanypracticalneed. (TwoCanadianmathematicians, thebrothers JonathanM.Borweinand Peter B. Borwein, have noted that thirty-nine decimal places will fix the

circumference of a circle around the known universe to within the radius of ahydrogenatom.)

Butpi isnotmerelytheubiquitousactor inhighschoolgeometryproblems; it isstitchedacrossthewholetapestryofmathematics,notjustgeometry’slittlecornerofit. Since mathematicians find it more convenient to express angles in pi-based“radians”thanineverydaydegrees,pioccupiesakeyplaceintrigonometry,too.Itisintimately related to that other transcendental number, e, and to “imaginary”numbersthroughEuler’selegantrelationship,

eiπ=−1

which in a single, strange, beautiful statement of mathematical truth tiestrigonometryandgeometry tonatural logarithmsand thence to thewholeworldof“imaginary” numbers.Pi even showsup in themathematics of probability.Drop aneedleontoatablefinelyscoredbyparallellineseachseparatedbythelengthoftheneedleandthechanceofitsintersectingalineis2/pi.Againandagainpipopsup.Sofindingnewways toexpress it can revealhidden linksbetween seeminglydisparatemathematicalrealms.

Ancient societies were usually content to figure pi at, simply, 3. The seventh-centuryIndianmathematicianBrahmaguptaputitasthesquarerootof10,whichisabout3.16. In theWest, early efforts todefinepiwerepursuedgeometrically; youcircumscribecircles,dropperpendiculars,bisect angles,drawparallels andwindupwith pi as the length of some line segment. “Squaring the circle,” as this classicproblem is called, turned out to be impossible. But Archimedes took anothergeometric approach and came up with a value of pi equal to between 310/70 and310/71.

Inthemidseventeenthcentury,thepowerfultoolsofthecalculuswerebroughttobear,leadingtoavarietyofinfiniteseriesthatconvergedtopi.Newtonhimselfcameupwith one that gave pi to fifteen decimal places. “I am ashamed to tell you,” heconfessedtoacolleague,“tohowmanyfiguresIcarriedthesecalculations,havingnootherbusinessatthetime.”

Seriesyieldingpi,orapproximationstoit,canbeofsurpassinggrace,likethisone,attributedvariouslytoLeibniz,ortheScottishmathematicianJamesGregory,ortomathematiciansinKerala:

JohnWalliscameupwiththisinfiniteproductataboutthesametime:

Anotherprettyoneis:

Thuspi,aboutasunrulyanumberasyoucanimagine(nopatterninitsdigits,evenouttomillionsofdecimalplaces,haseverbeenfound),canberepresentedbyseriesofthemostappealingsimplicity.

Ramanujan’s early letters toHardy included several such series approximations.Now,histwenty-three-pagepaperwasfilledwithotherroutestopi.Manyrestedonmodular equations, a subject, goingback to theworkof theFrenchmathematicianLegendre in 1825, that he had exhaustively surveyed in his notebooks.Roughly, amodularequationrelatesafunctionofavariable,x,withthesamefunctionexpressedin termsofx raised to an integralpower (x3orx4, for instance,butnotx3.2).Thetrick,ofcourse,istofindafunction,f(x),tosatisfyit.Notsurprisingly,theyarerare.But when they do show up, it turns out, they often display special propertiesmathematicianscanexploit.Ramanujanfoundthatcertainsuchfunctions,satisfyingcertainmodularequations,gavesolutionsthat,undercertaincircumstances,couldbeusedtocloselyapproximatepi.

Some of his results, it turned out, had been anticipated by Europeanmathematicians, like Kronecker, Hermite, and Weber. Still, Hardy would writelater,Ramanujan’spaperwas“ofthegreatestinterestandcontainsalargenumberofnew results.” If nothing else, it was astounding how rapidly some of his seriesconverged to pi. Leibniz’s series, on page 209, is lovely—but almost worthless forgettingpi;three-decimal-pointaccuracydemandsnofewerthanfivehundredterms.SomeofRamanujan’sseries,ontheotherhand,convergedwithastonishingrapidity.In one, the very first term gave pi to eight decimal places. Many years later,Ramanujan’sworkwouldprovidethebasisforthefastest-knownalgorithm,orstep-by-stepmethod,fordeterminingpibycomputer.

•••

Duringmostofthetenyearssincehe’dencounteredCarr’sSynopsis,Ramanujanhadinhabited an intellectual wilderness. In India, he’d been surrounded by family,friends,familiarfaces.HewasaSouthIndianamongotherSouthIndians,aTamil-speakeramongotherTamil-speakers,aBrahminamongotherBrahmins.Andyet,he

was also amathematical geniusofperhapsonce-in-a-century standing cutoff fromthemathematics ofhis time.He rousedwonder and admiration among those, likeNarayanaIyerandSeshuIyer,whocouldglimpseintohistheorems.Yetnoonehadbeenabletotrulyappreciatehiswork.Hehadbeenalone.Hehadhadnopeers.

Andnow,inLittlewoodandHardy,hedidhavethem.InCambridgehehadatlastfoundanintellectualhome,acommunityofmathematicianswhosawinhisworkallthathesawinit.And,atleastinthebeginning,thatmorethanmadeupforbeingastrangerinanalienland.

TheCambridge intowhichhe stepped thatday inApril 1914was aCambridgewherethePlayhouse,onMillRoad,playedTheFatalLegacy,billedas“agrandandabsorbingdrama,withanexcitingfoxhunt.”Abarbed-wirefencewasgoingupontheOldChestertownRecreationGround toprotect its green for theplayingof bowls.AndthebigstoryinthepaperwasthatCaesar,thelateKingEdward’sfavoritedog,the Irish terrierwhohad accompaniedhisRoyalMaster everywhere, haddied thepreviousSaturday.

To a South Indian, the knives and forks the English used and in whichRamachandra Rao had tried to school Ramanujan back in India seemed like aninvasion—hardmetallic things penetrating the mouth. Feet long unconstrained byanythingmore than sandals felt pinched by shoes; it took months to get used tothem.Englishnamesallrantogetherinablur.Andwhetherovalorsquare,toppedwithbrownhairorblond,theirfacesseemedsoalikeintheiressentialwhiteness;youcouldtalkforhourswithanEnglishmanyetfailtorecognizehimlateronthestreet.Then,ofcourse,aSouthIndianfoundthatwhenhegavethatlittleundulatingjiggleoftheheadthatbackhomemeantsomethingbetweenasimpleacknowledgmentandayes,theEnglishwereapttotakeitasano.

Still, in theglowofRamanujan’sarrivalandhis first fewmonths,any feelingsofhomesickness, loneliness, or frustration must have been fairly swept under theemotionalrug.Yes,wroteNeville,whowitnessedhisadjustmentupclose, “He feltthepettymiseriesoflifeinastrangecivilization,thevegetablesthatwereunpalatablebecause they were unfamiliar, the shoes that tormented feet that had beenunconfined for twenty-six years. But he was a happy man, reveling in themathematicalsocietywhichhewasenteringandidolizedbytheIndianstudents.”Hehadallthemoneyhistastesrequired.Hehadperfectleisuretopursuehiswork.Inthisoldtownofcobbledwalks,grassycourts,andmedievalchapels,wholeuniversesawayfromMadras,Ramanujanhadfoundakindofintellectualnirvana.

Butthen,thefirstcannonsounded.

3.THEFLAMESOFLOUVAIN

TheGreatWarwasbothexpectedandunexpected.Everyone knew itwas coming.All during 1913 and 1914,Europe seethed.The

assassinationoftheAustriancrownprinceFranzFerdinandonJune28,1914setinmotionachainofeventswhichultimatelyprovedirresistible.GermanyandFrance,locked into alliances, declaredwar.WhenGerman armies bound forFrance sweptthroughBelgium,violatingitsneutrality,Englanddeclaredwar,too.Thatcameat11A.M. onAugust 4, 1914.AllEuropewas soon engulfed,Germany and theCentralPowersmarshaledagainstFrance,Britain,andtheotherAlliedPowers.

Tomany,thewarwassomethinghotlysought,achancetoburnofftensionsbuiltupoverfortyyearsofpeace.Europemarchedintowar, flagsflying,tothesoundofmartialmusic.Theuniformswere fresh, the ranksand files still intact.Theenemywouldbetaughtitslessonandthewaritselfwouldbeoverinamonthortwo.Thetroops,somebodysaid,wouldbe“homebeforetheleavesfall.”

Thatwastheunexpectedpart,thewar’sterriblesurprise—thatitwasnobriefbutgloriousorgasmofarmsbutrathergroundalong,monthaftermonth,yearaftercruelyear.DuringAugust,Germanarmies roared throughBelgium, inobedience to theSchlieffenPlanwithwhich they hoped to humble France.The French, with theirPlan Seventeen, took a similarly offensive stance, aiming straight for Berlin. Buthopes for rapid victoryonboth sidesweredashedwithin thewar’s first sixweeks.Thegreatarmiesmet, fought, bled.Planswentwrong.Andafter theBattleof theMarneinSeptember,trenchwarfare replacedgreat sweepingbattlefieldmaneuvers.“RunningfromSwitzerlandtotheChannel likeagangrenouswoundacrossFrenchandBelgianterritory,”BarbaraTuchmanwroteinTheGunsofAugust,“thetrenchesdetermined the war of position and attrition, the brutal, mud-filled, murderousinsanityknownastheWesternFrontthatwastolastforfourmoreyears.”

OnSeptember11,whileGermanyandFrancegrappledattheMarne,Ramanujanwrotehismother,reassuringherthat“thereisnowarinthiscountry.Warisgoingononlyintheneighboringcountry.Thatistosay,wariswagedinacountrythatisasfarasRangoonisawayfromthecity[Madras].”

Thatwasn’ttrue;thewarwasmuchcloser.Infact,Cambridgehadalreadyfeltthewar’simpact.Initsfirstweek,backinearlyAugust,theSixthDivision,fromIreland,convergedonCambridgeandsetuptentsonMidsummerCommon,justacrossfromNeville’s house in Chestertown. Corpus Christi College became temporaryheadquarters for the Officers Training Corps; professors, undergraduates, andfellows all volunteered to help.AtTrinity, the columned area underWren’s greatlibrarywasmadeintoanopen-airhospital;woodenboardslaidontheunevenstone

floor,tokeepthebedslevel,nowmuffledtheechoesthathadbeenafixtureofthosestone precincts for centuries. In the court’s northwest corner, near the windingstaircasethatclimbeduptothelibrary,bathroomswereinstalled.Thesouthcloister,near where Littlewood lived, had lights strung from the ceiling and became anoperatingtheater.Meanwhile,inHardy’sNewCourt,roomsweremadeintooffices.

On August 14, an ambulance with a great red cross on its side bore the firstwounded patient to what was, officially, the First Eastern General Hospital—theopen-airhospitalatTrinity.Laterinthemonth,theGermansburnedLouvain,andtwo British divisions retreated from positions they had defended along theMonsCanal inBelgium. In theninehoursofbattlebefore the retreatbegan, the fightingcostsixteenhundredBritishcasualties.Ofthewounded,manyweresooninNevile’sCourt,indoublerowsofbedsunderthelibrary,whereHenryJackson,thecollege’svice-master,avenerableclassicist,wouldseethemonhiswaytoHall.

“We have a new Cambridge,” wrote Jackson, “with 1700 men in statu pupillariinsteadof3600....MedicalColonelsandMajorsandCaptainsdineinhallinkhaki.”

In early September, the SixthDivision left for France. ButCambridge was stillcrowdedwith men under arms and would remain the final training station for awholesuccessionofarmydivisionsboundforthefront.Whenitrained,horse-drawnguncarriagesandotherarmyvehicleschurnedupinches-deepmudonunasphaltedroads.

OnSeptember 20, theChapelChoirmet on the lawn in themiddle ofNevile’sCourttosinghymnstothewoundedsoldiersinthesurroundingcloisters.

Intheearlydaysof thewar, jingoismhadnotyetbeenburiedbycynicism.“ThedepravationofGermany—itsgospelofiniquityandselfishness—isappalling,”wroteJackson. “For though I never thought the Prussians gentlemen, I had a profoundrespect for their industryandefficiency, and I attributed to themdomestic virtues.Asitis,theirgoodqualitiessubservewhatisevil.”HenryButler,masterofTrinity,hadnotroublebelieving“thattheGermaninfantrycanneithershootnorstandthebayonet.As to the last, they turnandrunandget stuck in theback.Manyof [theNevile’sCourtwounded]assuremethattheyhaveseenwomenandchildrendriveninfrontoftheenemywhentheycharge.”

FeelingagainstGermanyswelled.EvenRamanujanwascaughtupinit,writinghismotherabouttheGermanadvanceacrossBelgium.“Germanssetfiretomanyacity,slaughterandthrowawayallthepeople,thechildren,thewomenandtheold.”

•••

ThepopularEnglishmagazineStrandhadlongcarriedapage,entitled“Perplexities,”devotedtointriguingpuzzles,numberedandcharminglytitled,like“TheFlyandtheHoney,” or “The Tessellated Tiles,” the answers being furnished the followingmonth. Each Christmas, though, “Perplexities” expanded, the author fitting hispuzzlesintoashortstory.Now,inDecember1914,“PuzzlesataVillageInn”tookreaderstotheimaginarytownofLittleWurzelfold,wherethemaintopicofinterestwaswhathadjusthappenedinLouvain.

In late August, pursuing an explicit policy of brutalization against civilianpopulations,GermantroopsbeganburningthemedievalBelgiancityofLouvain,ontheroadbetweenLiegeandBrussels.Housebyhouseand streetby street they setLouvaintothetorch,destroyingitsgreatlibrary,withitsquartermillionbooksandmedievalmanuscripts, andkillingmanycivilians.TheburningofLouvainhorrifiedtheworld, galvanized public opinion againstGermany, and united France, Russia,and Englandmore irrevocably yet. “TheMarch of theHun,” English newspapersdeclared. “Treason toCivilization.” Itwas an early turningpoint of thewar, doingmuchtosetitstone.Louvaincametosymbolizethebreakdownofcivilization.Andnowitreachedeventhe“Perplexities”pageofStrand.

OneSundaymorningsoonaftertheDecemberissueappeared,P.C.MahalanobissatwithitatatableinRamanujan’sroomsinWhewell’sCourt.MahalanobiswastheKing’sCollegestudent,justthenpreparingforthenaturalsciencesTripos,whohadfoundRamanujanshiveringbythefireplaceandschooledhiminthenuancesoftheEnglish blanket.Now, with Ramanujan in the little back room stirring vegetablesoverthegasfire,Mahalanobisgrewintriguedbytheproblemandfiguredhe’dtryitoutonhisfriend.

“Nowhere’saproblemforyou,”heyelledintothenextroom“Whatproblem?Tellme,”saidRamanujan,stillstirring.AndMahalanobisreadit

tohim.

“Iwastalkingtheotherday,”saidWilliamRogerstotheothervillagersgatheredaroundtheinnfire,“toagentlemanabouttheplacecalledLouvain,whattheGermanshaveburntdown.Hesaidheknoweditwell—usedtovisitaBelgianfriendthere.Hesaidthehouseofhisfriendwasinalongstreet,numberedonthissideone,two,three,andsoon,andthatallthenumbersononesideofhimaddedupexactlythesameasallthenumbersontheothersideofhim.Funnythingthat!Hesaidheknewtherewasmorethanfiftyhousesonthatsideofthestreet,butnotsomanyasfivehundred.Imadementionofthemattertoourparson,andhetookapencilandworkedoutthenumberofthehousewheretheBelgianlived.Idon’tknowhowhedoneit.”

Perhapsthereadermayliketodiscoverthenumberofthathouse.

Through trial and error, Mahalanobis (who would go on to found the IndianStatisticalInstituteandbecomeaFellowoftheRoyalSociety)hadfigureditoutinafewminutes.Ramanujanfigureditout,too,butwithatwist.“Pleasetakedownthe

solution,”he said—andproceeded todictateacontinued fraction,a fractionwhosedenominator consists of a number plus a fraction, that fraction’s denominatorconsistingofanumberplusafraction,adinfinitum.Thiswasn’tjustthesolutiontotheproblem,itwasthesolutiontothewholeclassofproblemsimplicitinthepuzzle.As stated, the problem had but one solution—house no. 204 in a street of 288houses;1+2+...203=205+206+...288.Butwithoutthe50-to-500houseconstraint, therewereothersolutions.Forexample,onaneight-house street,no.6would be the answer: 1 + 2 + 3 + 4 + 5 on its left equaled 7 + 8 on its right.Ramanujan’scontinuedfractioncomprisedwithinasingleexpressionallthecorrectanswers.

Mahalonobiswasastounded.How,heaskedRamanujan,hadhedoneit?“ImmediatelyIheardtheproblemitwasclearthatthesolutionshouldobviously

beacontinuedfraction;Ithenthought,Whichcontinuedfraction?Andtheanswercametomymind.”

4.THEZEROESOFTHEZETAFUNCTION

Theanswercametomymind.ThatwasthegloryofRamanujan—thatsomuchcametohimsoreadily,whetherthroughthedivineofficesofthegoddessNamagiri,ashesometimessaid,orthroughwhatWesternersmightascribe,withequalimprecision,to “intuition.” And yet, it was the very power of his intuition that, in one sense,underminedhismathematicaldevelopment.Foritblindedhimtointuition’s limits,gavehimlessreasontolearnmodernmathematicaltools,shieldedhimfromhisownignorance.

“Thelimitationsofhisknowledgewereasstartlingasitsprofundity,”Hardywouldwrite.

Herewasamanwhocouldworkoutmodularequationsandtheoremsofcomplexmultiplication,toordersunheard of,whosemastery of continued fractionswas, on the formal side at any rate, beyond that of anymathematicianintheworld,whohadfoundforhimselfthefunctionalequationoftheZeta-function,andthedominanttermsofmanyofthemostfamousproblemsintheanalytictheoryofnumbers;andhehadneverheardofadoublyperiodicfunctionorofCauchy’stheorem,andhadindeedbutthevaguest ideaofwhatafunctionofacomplexvariablewas.Hisideasastowhatconstitutedamathematicalproofwereofthemostshadowydescription.Allhisresults,neworold,rightorwrong,hadbeenarrivedatbyaprocessofmingledargument,intuition,andinduction,ofwhichhewasentirelyunabletogiveanycoherentaccount.

That mysterious “process” sometimes led him seriously astray. And Hardy’sreferencetothezetafunctionembodiedonenotoriousexampleofit.

In Ramanujan’s first letter, he’d referred to a statement in Hardy’s Orders ofInfinity concerning prime numbers. He had, he wrote, “found a function whichexactly represents the number of prime numbers less than x,” in the form of an

infinite series. Very interesting, Hardy had written back, let’s see some proof. InRamanujan’snextletter,heelaborated.“Thestuffaboutprimesiswrong,”LittlewoodshotbacktoHardywhenhesawit.Andnow,withRamanujan inEngland,Hardycametosee,upclose,justhowhehadgonewrong.

Ramanujanwas not the firstmathematician to be bewitchedbyprimenumbers.Theprimes—numberslike2,3,5,7,and11,butnot6or9(respectively,2×3and3× 3, and so “composite”)—were the building blocks of the number system. Startcountinganditwashardtohelpwondering,whenwouldyouget toaprime?Wasthereanysortofpattern?Ifyoulaidoutallthenumbersonagiantgrid,coloredinthe primes and left the others blank, then stood back and took a picture of theresulting pattern, would there be anything to see, any pattern you could call apattern?

Atfirstglancetherewasn’t,certainlynothingobvious.Andyet,thenagain,therewassomethingthere,rightinfrontofyoureyes:asyoukeptoncounting,youfoundmore primes. There was no last prime. Euclid had proved as much twenty-threehundredyearsago.“Theprimesaretherawmaterialoutofwhichwehavetobuildarithmetic,”Hardywouldwrite,“andEuclid’stheoremassuresusthatwehaveplentyofmaterialforthetask”;justasyouneverrunoutofnumbers,youneverrunoutofprimes.

Well, then, that was something. But could you say anythingmore? Yes: as youkeptcountingitlooked,onaverage,asifyouencounteredalowerdensityofprimes.There were always more of them, but the rate at which you encountered themdroppedoff.Inthefirstonehundrednumbers, forexample,thereweretwenty-fiveprimes; in the second hundred, twenty-one; in the ninth hundred, fifteen.Occasionally,youranintoaspanofhigherdensity;therearesixteenprimesbetween1100and1200,forexample.Butonaveragethefalloffheld.

Anything else?Yes, again.While the density of primes fell off, it fell off slowly.Deliberately, inexorably, the slowing effect could be felt; but it took hold oh-so-gradually. This was not like a campfire roaring away at dusk that is nothing butembersbymidnight,andcoldbymorning;butratheronethatbysunrisestillburnedbrightly,onlynotquitesobrightly.Withprimes,bythetimeyouwereinthebillions,youstillgotfiveoftheminevery100numbers,comparedtofifteenperhundredinthefirstthousand.Sothebrakingeffectwasagentleone.

Could you bemore precise than that?Could you say just how slow the slowingwas?Couldyou,toputitanotherway,makeamathematicalstatementthatgavethenumberofprimesyouencounteredincountingtoanygivennumber?

In time,mathematiciansguessed that the restraining influence, themathematical“force” slowing the increase in the number of primes, was logarithmic; that themathematicalfunctionknownasthelogarithmwassomehowatwork.

To say that something “drops off logarithmically” is the reverse of saying that it“increasesexponentially.”Exponentialgrowthimpliesakindoftakeoff,risingandre-rising,evermorerapidly,onthestrengthofitsowngrowth;compoundinterest,thekind that yields those thick retirement larders thirty years down the road, is anexponential process, rising slowly at first, then accelerating. But with logarithmicgrowth,quitetheoppositeoccurs:yougetlessandlessbangforthebuck.

Thelogarithmicresponseisubiquitousinnature,as,forexample,intherealmofthehumansenses.Doubletheamountoflightinaroomandyouscarcelynotice;theeyecanrespondtoboththeglareofthenoondaysunandtotheflickerofamatchamileawaybecause itsresponseis logarithmic.If light intensityrisesbyafactorofathousand,whichcanbewrittenas103,theresponse,inprinciple,maybeonlyaboutthreetimesasmuchinsteadofathousand.

Well, it was just such a slow logarithmic growth in this braking effect thatmathematicianslongthoughttheysawwithprimenumbers.Or,mathematically

Hereπ(x), read “pi ofx,”means the number of primes encountered in the first xnumbers.Andthis,theequationsays,isequaltosimplyxdividedbythe(“natural”)logarithmofx.Infact,thisisnotquiteright.Rather,

Thisexpressesthesameideaexceptasanapproximation,thoughttogetbetterandbetterasx grows larger. (Gaussactuallywrote it asa logarithmic integral,which issimilarinprinciple.)

In 1896, the French mathematician Hadamard and the Belgian de la Vallée-Poussinprovedthelong-conjecturedprimenumbertheorem,asthisrelationshipwasknown.Andthat’swherethingsstoodin1914:Asxgrewlarge,thetheoremsaid,akey ratio—π(x) ln x/x—approached one. That was pretty good, certainly puttingthingsonafirmmathematicalbasis.Andyet,itwasstillanapproximation,onethatgavenocluetoitsaccuracy;itfailedtospecifyhow“fast”theratioapproachedone,much lessgiveyoua formulathat said,giveme thenumber towhichyoucounted,and I’ll tell youhowmanyprimes you’vepassed along theway. It couldn’t say, forexample,thatamongthefirstmillionintegersthereare,exactly,78,498primes.And

itwasthisthatRamanujaninhisletterstoHardyconfidentlydeclaredhecoulddo:Ihavefoundafunctionwhichexactlyrepresentsthenumberofprimenumberslessthanx.

Well,hehadn’t.Ramanujan’sformula,threeversionsofwhichhegaveinhissecondletter,wasan

infinite series which, for values of x up to 1000 for example, gave virtually exactagreement.Evenlargervalues,Hardyfound later,werestrikinglyclose.Ofthe firstninemillionnumbers,itwasknownthat602,489wereprime.Ramanujan’sformulagaveafigureoffby just53—closerthanthecanonicalversionoftheprimenumbertheorem.Whatwasmore—andthiswasthecrucialpoint—anyerroraccompanyingitsuse,heclaimed,wasbounded,stayedwithingivenlimits.Thiswaswhy,inhisfirstletter, he’d objected toHardy’s assertion that the “precise order” of the error term“hasnotyetbeendetermined”;hisformulas,heinsisted,diddetermineit.

But they didn’t. Back in 1913, shortly afterHardy’s first encouraging letter hadarrived,Ramanujan’sfriendNarayanaIyerhadslippedintotheJournaloftheIndianMathematicalSocietysomeofRamanujan’sresultsonprimes.“Proofs,”headded,“willbe supplied later.” But they never were. Because they didn’t exist—couldn’t exist.BecauseRamanujan’sconclusionswerewrong.

OnceRamanujanwasinEngland,HardystudiedRamanujan’snotebooks,listenedas he worked through his arguments—and came to see where he had stumbled.Ramanujanhadbeenmisledbyunduerelianceonthelowvaluesofxforwhichhe’dtriedhisformula;theerror,forhighervaluesofx,wasmuchlargerthanhethought.These, had he tried them, might have alerted him to the more basic flaws in hisapproach,whichHardywouldfindso illuminatingthatonedayhe’dbuildawholelecturearoundthem.Ramanujan’stheory,Hardywouldwrite,“was(sotosay)whatthetheorymightbeifthezetafunctionhadnocomplexzeroes.”

TheRiemannzetafunctionwasasimpleenoughlookinginfiniteseriesexpressedintermsofacomplexvariable.Here,“complex”meansnotdifficultorcomplicated,but refers to a variable of two distinct components, “real” and “imaginary,” whichtogethercould be thought to range over a two-dimensional plane. In 1860,GeorgFriedrichBernhardRiemannmadesixconjecturesconcerningthezeta function.ByRamanujan’s time, five had been proven. One, enshrined today as the Riemannhypothesis,hadnot.

If you set the zeta function equal to zero, Riemann conjectured, then certainsolutionstotheresultingequation—its“complexzeroes”—would,whengraphed,allliealongaparticular line, oneparallel to the “imaginary” axis andhalf aunit to itsright. And from this hypothesis, if valid, certain important conclusions about the

distribution of primes, going beyond the prime number theorem itself, wouldautomaticallyflow.

ButwastheRiemannhypothesistrue?Tothisday,nooneknows,anditremainsoneofthegreatunprovedconjecturesinmathematics.AroundthetimeRamanujanarrived in England, Hardy proved something just short of it—that an infinitenumberofsolutionslayonthecrucialline.Butthiswasnotthesameassayingtheyalldid.TherenownedGermanmathematicianDavidHilbertoncesaidthat,wereheawakenedafterhavingsleptforathousandyears,hisfirstquestionwouldbe,HastheRiemannhypothesisbeenproved?

RamanujanhadcomeupwithsomethinglikeRiemann’szetafunctiononhisownbuthadmisunderstoodwhathe’dfound.Hehadignoredthecrucialcomplexzeroes,actedasiftheydidn’texist—leadingtoaversionoftheprimenumbertheoremthatwas, simply, wrong. “There are regions of mathematics in which the precepts ofmodernrigourmaybedisregardedwithcomparativesafety,”Hardywouldwrite,“buttheAnalyticTheoryofNumbersisnotoneofthem.”

AsLittlewoodwouldwriteofRamanujan’seffortonprimes,“Theseproblemstaxthelastresourcesofanalysis,tookoverahundredyearstosolve,andwerenotsolvedat all before1890 [sic; heprobablymeant1896,when theprimenumber theoremwasproved];Ramanujancouldnotpossiblyhaveachievedcompletesuccess.Whathedidwas to perceive that an attack on the problems could at least be begun on theformal side, and to reach a point atwhich themain results becomeplausible.Theformulaedonotintheleastlieonthesurface,andhisachievement,takenasawhole,ismostextraordinary.”

Hardy would be similarly indulgent of Ramanujan’s errors. “I am not sure,” hewouldwrite,“that,insomeways,hisfailurewasnotmorewonderfulthananyofhistriumphs.”Helaterregrettedsayingthat,dismissingitasundulysentimental.Buthewas getting at something—that in reachinghis faulty conclusions,Ramanujanhadrediscovered theprimenumber theorem;and that, in trying to reachbeyond it,hehad pursued arguments that, though technically flawed, were, in their own way,brilliant.

Still,Ramanujanwaswrong,whichnow, inEngland,underHardy’s tutelage, hecame to understand. “His instincts misled him,” wrote Hardy. And that was thepoint.Ramanujan’s“instincts,”sureastheywere,insomewaysbetterthanthoseofanyothermathematicianofhisday,werenotgoodenough.

•••

Acarmechanicreliantonmechanicalinstinctmay“know”howanengineworksyetbeunabletosetdownthephysicalandchemicalprinciplesgoverningit.Forawriter,itmaybeenoughto“know”thatonesceneshouldprecedeanotherandnotfollowit,withoutbeingabletoexplainwhy.Butmathematiciansarenotnormallycontenttoguess,orassume,orassertthatsomethingistrue;theymustproveit,orfeeltheyhave—orasHardywouldputit,“exhibittheconclusionastheclimaxofaconventionalpattern of propositions, a sequence of propositions whose truth is admitted andwhicharearrangedinaccordancewithrules.”

Proof isnomere icingonthecake.Takethesequenceof integers31,331,3331,33331,333331,3333331.Each is aprimenumber.So is thenext in the sequence.Havewehituponsomehiddenpattern?No,thepatternself-destructswiththenextinline,theproductof17and19,607,843.Orwhataboutnumbersoftheform22n+1.Forn=0,1,2,3,and4,theresultingnumbersareallprime.Aretheyforalln?PierredeFermatconjecturedasmuch.Butheconjecturedwrong.Because,asEulerfound,eventhenextnumberisnotprime,buttheproductof641and6,700,417.

Manyotherexampleslikethis,whereseemingly“obvious”patternsprovenottobepatternsatall,appearallthroughnumbertheoryandelsewhereinmathematics.OneHardy liked to cite also emerged from the theory of primes. Over the years, incomparingtheapproximationsoftheprimenumbertheoremwiththeactualnumberofprimescalculationrevealed,theapproximationalwaysprovedhigher;theerrorwasalwaysinthesamedirection.Youcouldtryathousand,youcouldtryamillion,youcould try a billion, you could try a trillion, you could try a billion trillion, and italwayscameoutthesame,makingalltheforcesofintuitionarguethatitwasalwaysso—andthatatheoremembodyingitwouldbeagreatonetorunoffandprove.Butnosuchtheoremcouldeverbeproved.Because, intuitivelyobviousthoughitmightseem,itsimplywasn’ttrue.

AyearbeforeheheardfromRamanujan,Littlewoodhadprovedthat,ifyouwentfarenough, theprimenumber theoremwasdestinedtosometimespredict less,notmore, than the actual number of primes. Later, someone found the number belowwhichthisreversalwasguaranteedtotakeplace.Anditwasanumbersobigyouhadto laugh—anumbermore than thenumberofparticles in theuniverse,more thanthepossiblegamesofchess.Itwas,Hardywouldsay,“thelargestnumberwhichhasever served any definite purpose in mathematics.” And it made for the ultimateillustrationofhowintuitioncouldserveyoubadly,andsomustalwaysbesubjecttoproof.

To“prove”something,then,isakindofguarantee—that,formathematicalentitiesA,B,andC,andsubjecttoconstraintsD,E, andF, the theoremholds.A theorem

muststanduptoharduse,willoftenbeappliedtounanticipatednewsituations.Failtopreciselyfixtheconditionsunderwhichitapplies,andyou’reapttogowrong.Amathematicianwith an insufficiently ironclad proof is a little like the brash youngpolicelieutenantinthemoviesconvincedofthebutler’sguiltbutbroughtupshortbyhisboss’scaution,Yeah,butthatwon’tconvinceajury.

Throughouthisnotebooks,allduringhistimeinIndia,andall throughhisearlyletters,Ramanujanhadproclaimeda thousandversionsof thebutlerdid it.Mostofthetimehewasright,andthebutlerhaddoneit:hisresultsweretrue.Andyetbeforecoming toEngland,hewouldhavebeenunable to secure a conviction:hehad—toextend the metaphor—scant knowledge of the rules of evidence, or the relevantcriminallaw,oreventhestandardsofsoundlegalargument.Ramanujan’s“proof”ofhisnewversionoftheprimenumbertheoremwasnoproofatall.

Howcana“proof”bewrong?Howcanyoudutifullymarchthroughitsreasoning,convince yourself that what you’ve said is right, only to have some othermathematician show it is not? What, in other words, can go wrong with amathematicalproofseeminglylaidoutinobeisancetoarelentlessly“logical”sequenceofclear-cutsteps?

Inaword,everything.Perhaps themost familiar, if trivial, exampleof such a failure is the “proof” that

purportstoshowthat2=1.Forstarters,leta=b.Now,multiplybothsidesoftheequationbya,leaving

a2=ab

Adda2−2abtobothsidesoftheequation:

a2+(a2−2ab)=ab+(a2−2ab)

Whichreducesto:

2(a2−ab)=a2−ab

Nowdividebothsidesbya2−ab,leaving

2=1

Voilà.Treatingbothsidesoftheequationwithscrupulousequalitywereacharesultdefyingcommonsense.

What’sgonewrong?Nothingbut that scourgeofmanyanelementaryproof:wehave divided by zero, a mathematically impermissible operation that gives an

“answer”devoidofmeaning.Butnowhere,avoiceinterrupts,doestheproofsayweshoulddividebyzero.Onthecontrary:westartedoffwitha=b,thentowardtheenddividedbya2−ab,

whichiszero.Evenathismostinnocent,Ramanujanwouldnevercommitsuchagaffe.Andyetit

suggestshow,busymanipulatingsymbolsandwithoutappreciatingeverynuanceofwhathewasdoing,hecouldgowrong.

As a mathematician, you can slide into trouble in numerous ways. You candifferentiate a functionwithout realizing the function cannot be differentiated.Oryoucanwriteoff later termsofaserieson theassumption that theyareofa lower“order” than earlier terms,when, in fact, they contribute substantially to the seriessum.Or you can assume that an operation correct for a finite number of terms iscorrectforaninfinitenumber.Oryoucanintegrateafunctionbetweentwopoints,yetfailtonotewheretheintegralmaybeundefined,andsocarrythroughyourproofsuchmeaninglessquantitiesas“infinityminusinfinity.”

Inhisworkwithprimes,Hardywrote,Ramanujan’sproofs

depended upon a wholesale use of divergent series. He disregarded entirely all the difficulties which areinvolvedintheinterchangeofdoublelimitoperation;hedidnotdistinguish,forexample,betweenthesumofaseriesΣanandthevalueoftheAbelianlimit

orthatofanyotherlimitwhichmightbeusedforsimilarpurposesbyamodernanalyst.

Thesewereallquitetechnical,ofcourse—likelegalloopholesofwhichthepolicelieutenantisunaware.Ramanujan’s intuitionsteeredhimclearofmanyobstaclesofwhichhistruncatededucationhadfailedtowarnhim.Butnotall.Theproblemwasnotonlythathewassometimeswrong;itwasthathelackedmathematicalknowledgeenough to tell when hewas right andwhen hewaswrong.He stated correct andincorrect theorems with the same aplomb, the same sweet, naive confidence. Andwhenhedidofferproofs,theyscarcelywarrantedthename.

ForRamanujan,Littlewoodwrotelater,“theclear-cutideaofwhat ismeantbyaproof,nowadayssofamiliarastobetakenforgranted,heperhapsdidnotpossessatall; if a significantpieceof reasoningoccurredsomewhere,and the totalmixtureofevidenceandintuitiongavehimcertainty,helookednofurther.”

ItwasjustRamanujan’sluck,then,tobethrowninwithHardy,whoseinsistenceonrigorhadsenthimoffalmostsingle-handedlytoreformEnglishmathematicsand

towrite his classic text on puremathematics; who had told BertrandRussell twoyearsbeforethathewouldbehappytoprove,reallyprove,anything:“IfIcouldprovebylogicthatyouwoulddieinfiveminutes,Ishouldbesorryyouweregoingtodie,butmysorrowwouldbeverymuchmitigatedbypleasureintheproof.”Ramanujan,IntuitionIncarnate,hadrunsmackintoHardy,theApostleofProof.

Andnow,ashewouldoverthesucceedingmonthsandyears,Hardysettoworkonhim,tryingtoovercomethedeficitthatwasthepriceRamanujanhadpaidforhisintellectualisolation.At twenty-six,Ramanujanhad long-establishedwaysofdoingthings.Nonetheless,heresponded.AsHardywrotelater,

Hismindhadhardenedtosomeextent,andheneverbecameatallan“orthodox”mathematician,buthecouldstill learn to do things, anddo them extremelywell. Itwas impossible to teachhim systematically, buthegraduallyabsorbednewpointsofview.Inparticularhelearntwhatwasmeantbyproof,andhislaterpapers,whileinsomewaysasoddandindividualasever,readliketheworksofawell-informedmathematician.

Byearly1915,aboutthetimeMahalanobisreadhimtheLouvainstreetprobleminStrandmagazine,Ramanujanhadalreadybeguntoshiftgears,toredirecthisworkinwaysHardywasurging.“Ihavechangedmyplanofpublishingmyresults,”hewroteKrishnaRaoinNovember1914.“Iamnotgoingtopublishanyoftheoldresultsinmy note book till thewar is over.After coming here I have learned some of theirmethods. I am trying to get new results by theirmethods.” By early the followingyear,hehad literallysetasidemuchofhisoldwork,wasdrinking in thenew: “Mynotebook is sleeping in a corner for these four or five months,” he wrote hischildhoodfriendS.M.Subramanian,withwhomhehadbrieflyroomedinSummerHousetwoyearsbefore.“IampublishingonlymypresentresearchesasIhavenotyetprovedtheresultsinmynotebookrigorously.”

Proofandrigor:hewasabsorbingtheGospelaccordingtoHardy.

•••

Butwhile taking in the new, asHardywouldwrite,Ramanujan’s “flow of originalideasshowednosymptomofabatement.”InhislettertoSubramanian,forexample,Ramanujanatonepointabruptlybrokeintoaparagraphwith:“Ishallnowtellyou[about]averycuriousfunction,”thenwrotethispatternoffractionsacrossthepage:

1,1/2,2/1,1/3,3/1,2/3,3/2,1/4,4/1,2/4,4/4,3/4,4/3

Guided by it, he ingeniously constructed a function which, bizarrely, wasmathematicallyundefined for the fractions in his series, and existed at all only for“irrational”values,thosenotrepresentableasafraction.“Averycuriousfunction,”he

hadcalled it.Then,again: “There is another curiosityhere . . .”And “Just imagine[how] this function [behaves] . . .”Thiswas vintageRamanujan,hisdelight in thefunction’speculiarbehaviorfairlyspillingacrossthepagesoftheletter.

In what, in some ways, was his greatest achievement, then, Hardy broughtRamanujanmathematicallyuptospeedwithoutmuzzlinghiscreativityordampingthefiresofhisenthusiasm.Itwouldhavebeeneasytosniffathisshortcomingsanddutifullycorrectthem,likeabadeditorwhocrudelyblue-pencilshiswaythroughadelicatemanuscript.Butheknew thatRamanujan’smathematical insightwas rarerbyfarthaneventhemost formidabletechnicalmastery.Itwas finetoknowall themathematical toolsneeded toprovea theorem—butyouhad tohavea theoremtoproveinthefirstplace.

That was easy to forget as you flipped through the Proceedings of the LondonMathematicalSociety.There, as inanymathematics journal, theproofwasmade toseemtheculminationofahundredcloselyreasonedstepsrangingoveradozenpages.There,mathematicscouldseemnomorethananeatlockstepmarchtocertainty,BfollowingdirectlyfromA,CfromB, . . .Z fromY.Butnomathematicianactuallyworked thatway; logic like that reflected the demands of formal proof but hintedlittle at the insights leading to Z. Rather, as Hardy himself would write, “amathematicianusuallydiscoversa theorembyaneffortof intuition; theconclusionstrikeshimasplausible,andhesetstoworktomanufactureaproof.”

Thetheoremitselfwasapttoemergejustasothercreativeproductsdo—inaflashofinsight,orthroughasuccessionofsmall insights,precededbycountlesshoursofslogging through the problem. You might, early on, try a few special cases toinformally“prove”theresulttoyourownsatisfaction.Thenlater,youmightgobackand,with a full arsenal ofmathematicalweapons, supply the kind of fine-texturedproofHardychampioned.Butallthatcamelater—afteryouhadsomethingtoprove.Besides,itwasmostlytechnical, likethe lawsofevidence;youcouldlearnit.Rigor,Littlewoodwould observe, “is not of first-rate importance in analysis beyond theundergraduate state, and can be supplied, given a real idea, by any competentprofessional.”

Givenarealidea—thatwastherarecommodity.“Mathematics has been advancedmost by thosewho are distinguishedmore for

intuition than for rigorous methods of proof,” the German mathematician FelixKleinoncenoted.(AddedLouisJ.Mordell,anAmericanmathematicianwhowouldultimately succeed Hardy in his chair at Cambridge: “To very few othermathematicians areKlein’s remarks . . . so appropriate as toRamanujan.”)A “realidea” wasn’t dished up, like a Tripos problem, by some anonymous mathematical

Intelligence.Ithadtocomefromsomewhere,hadtobeseenbeforeitcouldbeproved.Butwhere did it come from?That was themystery, the source of all the circular,empty, ultimately unsatisfying explanations that have always beset students of thecreativeprocess.Here,“talent”camein,and“genius,”and“art.”Certainlyitcouldn’tbetaught.Andcertainly,wheninhand,ithadtobenurturedandprotected.

Plenty of mathematical technicians, Hardy knew, could follow a step-by-stepdiscursusunflaggingly—yet counted fornothingbesideRamanujan.Years later,hewouldcontriveaninformalscaleofnaturalmathematicalabilityonwhichheassignedhimself a 25 and Littlewood a 30. To David Hilbert, the most eminentmathematicianofhisday,heassignedan80.

ToRamanujanhegave100.“Itwasimpossibletoasksuchamantosubmittosystematicinstruction,totryto

learnmathematicsfromthebeginningoncemore,”Hardywouldwrite.“Iwasafraidtoo that, if I insistedundulyonmatterswhichRamanujan found irksome, Imightdestroyhis confidence or break the spell of his inspiration.” And so he sought anelusive middle ground where, without crimping Ramanujan’s creativity, he couldteachhim,ashewrote,the“thingsofwhichitwasimpossiblethatheshouldremaininignorance....Itwasimpossibletoallowhimtogothroughlifesupposingthatallthe zeroes of the zeta function were real. So I had to try to teach him, and in ameasureIsucceeded,thoughobviouslyIlearntfromhimmuchmorethanhelearntfromme.”TeachingRamanujan,mathematicianLaurenceYounghaswritten, “waslikewritingonablackboardcoveredwithexcerptsfromamoreinterestinglecture.”

Hardyheldinhishandsarareanddelicateflower.Andtheresponsibilityheborefornurturingitwasonlyredoubledbythewar.

5.S.RAMANUJAN,B.A.

“At Cambridge we are in darkness,” wrote Trinity vice-master Jackson in January1915.“Nogasinthestreetsorcourts;fewelectriclightsandthoseshaded;candlesonthehigh tables.The roads intoCambridge areblocked toprevent the approachofmotors such as thosewhich guided theEastCoastZeppelins.Rumor says that anattackonWindsorCastlewasexpectedlastweek.”

InApril, about a year after Ramanujan’s arrival, Jacksonwrote: “In France andFlanderswemakenoprogress. In theDardanellesweareata standstill.Thearmydoesnotgrowasitought.Wehavenotgotammunitionfortheexistingarmy.TheGermanshavebeenpreparingvillainiesforyears.”

WoundedsoldiersfloodedCambridge,almosttwelvethousandbeingadmittedtotheFirstEasternGeneralHospitalbyJuly1915.They’darriveontrainslateinthe

eveningatthestation,wheretheyweremetbywhite-hoodednursesandmotorizedambulances.“Whentheysucceededeachotheratfrequentintervalsforsometime,”aCambridgeschoolgirlrecalledlater,“IknewthattheremusthavebeenheavyfightinginFlandersorFrance.”

Themaimed,hurt,andsickstreamedintoCambridge,thehealthyandstrongout.Of undergraduates, Cambridge was largely deserted; normally some thirty-fivehundred, they now numbered five or six hundred. College fellows served in theForeignOffice,intheWarOffice,intheTreasury,aswellasinthearmyitself.OneTrinitymanwrote,inJackson’swords,that“thefrontwaslikeafirst-rateclub,asyoumet all your friends there.”The university’smedical laboratories,meanwhile,wereputat thedisposalof theFirstEasternGeneralHospital.Thechemical laboratorydid research in gas warfare.The engineering laboratories beganmaking shells andgaugesfortheMinistryofMunitions.

AmongthemanyTrinityfellowstoleavewasLittlewood,nowasecondlieutenantintheRoyalGarrisonArtillery.Hehadlittleuseforthewar,butadapted,justashehadtotheTripossystem,affectinga“cheerfulindifference.”Bylate1915,he’dbeenputtoworkonafreshmathematicalapproachtofashioningantiaircraftrangetables.(“Even Littlewood could notmake ballistics respectable,”Hardywouldwrite, in agibeatappliedmathematics, “and ifhecouldnot,whocan?”)AlmostaloneamongTrinityfellowshewasneverpromoted,andultimatelywasrelievedofroutinechoresandallowedtolivewithfriendsinLondon.“Ballistics,”onebiographerwouldgentlyputit,“didnotfillallofLittlewood’sworkinghoursduringthewaryears”;between1915and1919,hemanagedtocollaboratewithHardyontenjointpapers,inareasofmathematicsasdistantascouldbeimaginedfromthewar.

Still,hewasawayfromCambridge—andawayfromRamanujan.Andhe,andhisformidablemathematical powers, had been a prime reason to bringRamanujan toCambridge in the first place. So that almost before he had caught his breath inEngland,afterbarelyfourmonths,Ramanujanhadbeenthrown,moredependentlythanever,intothearmsofHardy.

Hardywould later register formilitary service under the “DerbyScheme,”LordDerby’s politically shrewd move to forestall unpopular conscription through avoluntary, but socially pressured, “attestation” of readiness to serve. But Hardy,thirty-eightwhentheDerbySchemewaslaunchedinOctober1915,was,accordingtoLittlewood,deemed“unfit”toserve,andspentmostofthewarinCambridge.

In the First World War, unlike the Second, antiwar feeling ran high. Hardy’sactivitieswouldleadatleastoneobituarytoasserthe’dbeenaconscientiousobjector.He was not; indeed, late in the war he would write, “I don’t like conscientious

objectorsasaclass.”Still,asLittlewoodtellsit,Hardy“wrotepassionatelyaboutthenotoriousill-treatment[towhich]objectorsweresubject.”Hebelongedtoanearnest,high-mindedgroupcalled theUnionofDemocraticControl,which focusedon thepeacetofollowwhatwasstillassumedwouldbeabrief,ifbloodyconflict.Hardywassecretary of its Cambridge chapter, his old friend from the Apostles, G. LowesDickinson,itspresident.ItsfirstpublicmeetingwasheldonMarch4,1915.Later,inNovember,when itwasannounced that ameetingwould takeplace in theTrinityrooms of Littlewood—who was a U.D.C. member in absentia and had suppliedwrittenpermission—theauthoritiesmovedtoblockit.

Justoutside thegatesofRamanujan’s castle thewar raged, coming closerdaybyday.ComelyCambridgehadbeentransformedintoatrainingcampandhospital.InMay1915,theLusitaniahadbeensunk,hardeningsentimentagainstGermany.ByJune,foodpriceshadrisen32percentoverthepreviousyear’s.IntellectualcommercewithGermanmathematicianswascutoff.Littlewoodandothermathematiciansweregone.Hardywasdistractedbyextra-mathematicalconcerns.

And yet, so far, the war had not yet reachedRamanujan.Whatever his privaterevulsiontowardit,heyetbaskedinakindofwarmintellectualspring.

Sincehis arrival inEngland,he’dbeenwritinghome regularly—at first, threeorfour times a month, and even now, during 1915, twice a month or so, regularlyassuringhis family thathemaintainedhisvegetarianismandhis religiouspractices.HisletterstofriendsbackinIndiascarcelymentionedthewar,butrathertoldofhisworkandhisprogress,inquiredafterfamilymembers,evenoccasionallygaveadvice.TohistwobrothersbackinIndia,hesentaparcelfullofbooksofEnglishliterature.

Thepersistent bother of finding andpreparing food forhimself underminedhissenseofwell-beingsomewhat.SodidtheEnglishchill,andtheclothesthatcinchedathisfatwaist.Andthepeculiarabsenceof lettersfromJanaki.Butmostly,hestillrodethecrest,keptworking,happilyandhard,atmathematics.

InhisfirstlettertoHardy,hehadsoughthelpinpublishinghisresults.Andnow,during1914and1915,lettershomeshowedhowpreoccupiedhewaswithseeinghisworkinprint.“I...havewrittentwoarticlestillnow,”hewroteinJune1914.“Mr.Hardy is going to London today to read a paper on one ofmy results before theLondonMath.Society.”

“I havewritten three papers till now.Theproof sheets have come. I amwritingthreemorepapers.Allwillbepublishedattheendofthevacation,i.e.inOctober,”hewroteinAugust.

“I am very slowly publishingmy results owing to the present war,” he wrote inNovember1914.

“Itwilltakesomemonthsformetowritethatpapersystematicallyandpublishit,”hewroteNarayanaIyerinNovember1915.

After years as a mathematician known only to himself, then only to Madras,Ramanujan plainly relished the prospect of appearing in prestigious Englishmathematics journals.To appear inprintwas the only tangible signof recognitionyoucouldholdupto familyandfriends, theonlywaytheworldwouldknowwhatyou’ddone.Atonepoint,whenhelearnedthatAnandaRao,amathematicsstudentatKing’swhowas thesonofaMadras judgeandrelativeofR.RamachandraRao,waspreparinganessayforaSmith’sPrize,hewentstraighttoHardytoaskwhetherhe,too,mighttryforit.Doingmathematicssatisfieddeepemotionalandintellectualneeds in Ramanujan. But credit, kudos, appreciation satisfied quite other, just asinsistent, needs. Intellectually, Ramanujan was not, and never could be, “just likeeveryoneelse.”Butinthissocial,publicrealm,hewas:hewantedrecognition,neededit.

Andnowhegotit.In1915,nofewerthannineofhispapersappeared,fiveoftheminEnglish journals, as against six—allbutoneof them in the Journal of the IndianMathematicalSociety—duringtheprevioustwenty-sevenyears.

The advantageshenow enjoyed inEnglandwerehardly lost uponhim.HehadoriginallypromisedtoreturntoIndiaaftertwoyears.AndwhenhewrotehisfriendSubramanian in June1915,he still talkedof returning to India the following year.But, in another letter the followingmonth, he suggested his returnmight be onlytemporary.“Ithinkitmaybenecessary,”hewrote,“tostayhereafewyearsmoreasthereisnohelpnorreferencesinMadrasformywork.”

•••

ForawhileafterhecametoCambridge,apparently,Ramanujan’sshynesswasreadasunfriendliness, and students sometimes taunted him. But by now he was a morepopular, even legendary figure, his room accorded the status of a shrine. “Itwas athrilltometodiscoveronreachingCambridgeinJuly1915thatIwasgoingtobeacontemporary ofRamanujan,” recalledC.D.Deshmukh, until then a student at acollege inBombay. “Ramanujan’s currentachievementswerecommontalkamongstus Indian students.”AnandaRao rememberedRamanujan at tea parties andothersocialgatherings,mixingfreelyamongbothEnglishandIndians.Mahalanobis,withwhom he took Sunday morning strolls, recalled him as reserved in large groups,expansive in small ones. In any case, hewas the toast of the Indian students—themathematical genius, theman whom the English hadmoved heaven and earth to

bring to Cambridge. Two surviving photographs from this period show him in agroup;inbothheoccupiestheverycenterofthecomposition.

Ramanujanmostly stayed inCambridge during the long periods between terms.(The academic calendar came to only about twenty-two weeks a year.) Butsometimeshe’dgettoLondon,wherehevisitedthezooortheBritishMuseum.Onetime, he and his friend G. C. Chatterji went to see Charley’s Aunt, a farce aboutOxbridge undergraduate life. In it, young Lord Fancourt disguises himself asCharley’sauntfromBrazil,therequisitechaperoneforthegirlCharleyhasinvitedtohis rooms for lunch. Since its first performance in 1892, it had become a hardyannual of the London stage, full ofwigs and fluttering eyelashes, of tea poured inhats,ofsillinessandcomicconfusion.Itwas,ofcourse,justRamanujan’sspeed.Helaugheduntilhecried.

Bymid-October of 1915, Ramanujan hadmoved from his rooms inWhewell’sCourttonewonesonStaircaseDofBishop’sHostel,justbehindGreatCourt.Thiswasn’t the original Bishop’sHostel built in 1669, whose sharply sloping roof andgabledwindowsloomedjustoutsidehiseastwindow.Twored-brickstructureshadbeenbuiltadjacenttotheoldbuildingin1878,onthesiteofwhathadoncebeenthecollegestables,andRamanujan’sroomswereonthesecondfloorofoneofthem.Hehadone large sitting room,perhaps fourteen feetwideby twenty long,with a tinybedroom and a small cooking area—actually the “gyp” room, a combination coalcellar,pantry,andkitchen—justoffit.

The layoutwas almost identical tohisold rooms inWhewell’sCourt.Butnow,Hardy was closer yet. Kumbakonam . . . Madras . . . Chestertown Road . . .Whewell’sCourt...itwasasifsomerelentlessforcedrewhimeverclosertoHardy.Now,inBishop’sHostel,allthatdividedthemwasahundredpaces.

•••

Late in 1915, Ramanujan’s big paper on highly composite numbers, his mostimportantbodyofworkduring that firstyearor so inCambridge,appeared in theProceedings of theLondonMathematicalSociety. Back the previous June,HardyhadintroducedhisfriendsattheMathematicalSocietytoit.InNovember,amanuscriptwas ready. But revisionswere required, and it didn’t reach final form untilMarch1915.Now,atlast,itwasinprint.

Solong,andrangingoversovastaterrain,itwasdividedintodiscretenumberedsections and even had its own table of contents page to guide readers through it.Sometimeearlier,RamanujanhadinvitedW.N.Baileyandanothermathematician,

S.Pollard,toseeit.“Hestartedatthebeginning,”Baileywouldrecall,“andquicklyturned over the pages as he explained the ideas and the arguments very briefly.Pollardwrestledmanfullywiththeargumentandwasrewardedbyasevereheadache.Igaveupthestruggleearlier.”

A composite number, recall, is a number that is not prime. The number 21 iscomposite,theproductof3and7.Sois22,whereas23isprime,theproductonlyofitself and one.Now for each composite number, you can list all the numbers thatdivideit.Forexample,21hasthedivisors1,3,7,and21.For22,it’s1,2,11,and22.Twenty-fourcanbedividedby1,or2,or3,or4,or6,or8,or12,or24.

And this last one, 24, was the kind of number with which Ramanujan’s paperdealt. Itseightdivisorsnumberedmore than thoseofanyothercompositenumberlessthan24—andmadeit,inRamanujan’sterminology,“highlycomposite.”Twenty-twohas fourdivisors, 21 has four, 20 six.None less than 24 has even asmany asseven,muchlesseight.Ahighlycompositenumber,then,wasinHardy’sphrase“asunlike a prime as a number can be.”Ramanujan had explored their properties forsometime;intheearliestpagesofhissecondnotebookhe’dlistedaboutahundredhighly composite numbers—the first few are 2, 4, 6, 12, 24, 36, 48, 60, 120—searchingforpatterns.Hefoundthem.

The prime factors from which any composite number, N, was built could bewrittenintheform,

N=2a2×3a35a5...

wherea2,a3,a5, and so on are just the powers towhich the prime numbers, 2, 3,5...,areraised.Thehighlycompositenumber24,forexample,canbeviewedas23

×31.Inthisnotation,then,a2=3,anda3=1.Ramanujanfoundthat,foranyhighlycompositenumber,a2wasalwaysequaltoorlargerthana3,a3wasalwaysequaltoorlarger than a5, and so on. Count as high as you liked, you’d never find a highlycompositenumberlikeN=23×34× . . . .Never.Andheshowedthat,with twoexceptions(4and36),thelastannecessarytoconstructahighlycompositenumberwasalways1.Hewentontoprovetheseandothertruthsthroughfifty-twopagesofreasoningthatHardywouldterm“ofanelementarybuthighlyingeniouscharacter.”

TheproblemRamanujanhadaddressed,Hardyobserved,“isaverypeculiarone,standing somewhat apart from the main channels of mathematical research. Buttherecanbenoquestionastotheextraordinaryinsightandingenuitywhichhehasshownintreating it,noranydoubtthathismemoir isoneof themost remarkablepublishedinEnglandformanyyears.”

•••

AtCambridge, every studenthada tutorwho lookedafterhimandmonitoredhisprogress.Ramanujan’swas E.W.Barnes, whowould call Ramanujan perhaps themost brilliant of all the topTrinity students (which included Littlewood) to havecomebeforehim.ThoughlatertobecomeabishopintheAnglicanchurch,Barneswasnowamathematicianof some standing;hehadbeenoneofAndrewForsyth’searliest disciples, had with Hardy been among the leading advocates of Triposreform, and had made substantial mathematical discoveries of his own. Now, inNovember1915,hewroteFrancisDewsbury,registraroftheUniversityofMadras,of Ramanujan’s progress, which he termed “excellent.He is entirely justifying thehopesentertainedwhenhecamehere.”Histwo-yearscholarship,sooncomingtoanend,oughttobe“extendeduntil,asIconfidentlyexpect,heiselectedtoaFellowshipat theCollege.Such an election I should expect inOctober1917.”Ramanujan, hewassaying,wasinlinetobecomeaFellowofTrinity.

A few days later, Hardy also wrote Dewsbury, calling Ramanujan “beyondquestion the best Indian mathematician of modern times. . . . He will always berathereccentricinhischoiceofsubjectsandmethodsofdealingwiththem....Butofhis extraordinary gifts there can be no question; in some ways he is the mostremarkablemathematicianIhaveeverknown.”

InMadras,SirFrancisSpringjoinedthechorus,specificallyrequestingatwo-yearextension of Ramanujan’s scholarship. Through the late fall and early winter of1915–1916,Madrasauthoritiesdebatedwhetherthescholarshipshouldbeextendedforoneyearortwo.Ifjustone,SpringwroteDewsbury,RamanujanwasinclinednottoreturntoIndiaduringthesummerof1916asplanned;forshouldhisscholarshipendinthespringof1917,he’dbebackinIndiajustninemonthslateranyway.

Theuniversityheld,however,that,iftheTrinityfellowshipwentthrough,itsownscholarshipwouldoverlap.Sooneyear,withthepossibilityoffurtherextensions,itwas.

TheMadrasscholarshipcameto250poundsayear,whichwassupplementedbya60-pound-per-year“exhibition”fromTrinity.In1914,theaverageEnglishindustrialworkermadeabout 75 pounds per year.The threshold for paying income taxes—reachedbylessthan7percentoftheworkingpopulation—was160pounds.Soevenwith the 50 pounds he sent to support his family in India, Ramanujan wascomfortably fixed—especially given what Hardy would call Ramanujan’s “almostludicrouslysimpletastes.”

Ramanujan had no official college duties.He could do as he pleased.He couldimmersehimselfinmathematicswithoutfrettingoverfinancialwant,eitherhisownorhisfamily’s.Yetsomethingstillnaggedathim—hislackofadegree,thetangiblepublicmarkerofacademicachievement.Inhiscase,itwasthemerestformality.Buthewantedit.

Admission as a research student normally meant you already held a universitydiplomaorcertificate.Butinhiscase,therequirementhadbeenwaived.Andnow,inMarch1916,hereceivedaB.A.“byresearch,”onthebasisofhislongpaperonhighlycomposite numbers. He’d put up his five pounds dissertation fee. He’d paid twopoundseachtohisexaminers.Andnow,adozenyearsandtwocollegefailuresafterleavingTownHighSchool,hehadhisdegree.

IntheearlyafternoonofMarch18,Ramanujanposedforaphotographtomarkthe occasion with a group of students in their academic robes outside the SenateHouse.Theshortestandstockiestofthelot,hestoodsquarelyatattention, likeanarmyrecruitinbootcamp,hismortarboardsittingflatatophishead.Histrouserlegswereacoupleofinchestooshort.Hissuitbulged,itsbuttonsstraining.

Whetherbecausethescholarshiphadbeenextendedforonlyasingleyearor forfearoftheU-boatsthenravagingBritishshipping,Ramanujandidn’treturntoIndiainthespring.Overmuchofthenextyear,hecontinuedinsteadtoworkwithHardyon a problem that would indissolubly link their two names in the annals ofmathematics.

ThatJune,HardyfolloweduphisletterofthepreviousyeartoDewsburywithanofficialreport,hisdelightintellingofRamanujan’sprogressmarredonlybythewar:

InonerespectMr.Ramanujanhasbeenmostunfortunate.Thewarhasnaturallyhaddisastrousresultsontheprogressofmathematical research. Ithasdistracted three-quartersof the interest thatwouldotherwisehave been taken in his work, and hasmade it almost impossible to bring his results to the notice of thecontinentalmathematiciansmostcertaintoappreciateit.IthasmoreoverdeprivedhimoftheteachingofMr.Littlewood,oneofthegreatbenefitswhichhisvisittoEnglandwasintendedtosecure.Allthiswillpass;and,in spiteof it, it is already safe to say thatMr.Ramanujanhas justified abundantly all thehopes thatwerebaseduponhisworkinIndia,andhasshownthathepossessespowersasremarkableintheirwayasthoseofanylivingmathematician.

Hardy’saccountofRamanujan’sworkwas“necessarilyfragmentaryandincomplete,”heapologized.But

Ihavesaidenough,Ihope,togivesomeideaofitsastonishingindividualityandpower.Indiahasproducedmanytalentedmathematiciansinrecentyears,anumberofwhomhavecometoCambridgeandattainedhighacademical distinction. They will be the first to recognize that Mr. Ramanujan’s work is of a differentcategory.

The previous December, British, Australian, and New Zealand troops had

sufferedacatastrophicdefeatatGallipoli.U-boatscontinuedtotaketheirbloodytollofAlliedships.ThemachinegunschatteredawayinFrance.Bytheendoftheyear,listsofsometimesfourthousandcasualtiesperdaywouldappearinthenewspaper.

ForRamanujan,inmid-1916,thingscouldhardlyhavebeenbrighter.Buthe,too,wouldultimatelybestruckdownbythewar.

CHAPTERSEVEN

TheEnglishChill

[1916to1918]

1.HIGHTABLE

BeneaththeseeminglyunruffledsurfaceofRamanujan’slifeinwartimeEnglandlaysignsthathisnervesweretautlystrung,hissensitivitiesbalancedonahairtrigger.

Sometime probably in early 1916, he learned that his friendGyanesh ChandraChatterji,atwenty-one-year-oldfromthePunjabregionwhoheldaGovernmentofIndia state scholarship to study in Cambridge, planned soon to marry. To helpcelebratethegoodnews,heinvitedChatterjiandhisfiancéetodinner.

Back inIndia,Ramanujanhadprobablynevercooked inhis life,hadconceivablynevereven stepped into akitchen.Buthere,withneitherwifenormother to servehimandunwillingtotrusttothevegetarianpurityofthecollegekitchen,he’dhadtolearn.Sometimes,onSundays,hehadIndianfriendsoverforrasam,athinpepperysoup, or other South Indian fare. “Delicious,” a friend later recalled.Andonce, S.Kasturirangar Iyengar, owner and editor of South India’s preeminent English-languagenewspaper,theHindu,visitedhiminCambridgeandlavishedpraiseonthepongal,alentilandricedish,thatRamanujanservedhim.

Bynowhetooknolittleprideinhisculinaryskillsand,tohonorChatterjiandhisfiancée,setaboutpreparingthemafeast.

On the appointed day, Chatterji showed up at Ramanujan’s rooms in Bishop’sHostel.WithhimwashisfiancéeIlaRudra,astudentatthelocalteacher’scollege;and,probablyaschaperone,MrilaniChattopadhyaha,awomanfromHyderabadinherearlythirtiesthenstudyingethicsatCambridge’sNewnhamCollege,whowouldgo on to become active in the Indian labor movement and run a school foruntouchables.

Withhis guests seated andhis apartment awash in the cooking smells ofSouthIndia,Ramanujanservedsoup.Allwaswell.

Did his friends wish to have more? asked Ramanujan after a time. They didindeed.

Then:athirdhelping?Chatterjiapparentlyaccepted.Butthistime,hisfiancéeandtheotherwomandeclined.Thedinnerconversationcontinued....

WhereisRamanujan?Abruptly,theylookedaroundandrealizedtheywerealoneinRamanujan’sapartment.Theirhosthadvanished.

Foranhour,Chatterjiandtheotherswaited.Finally,hewalkeddownthesingleflight of stairs, out the door, through the gate into Great Court, and across thecobbledcourtyard to theporter’s lodge,wherehe inquiredafterRamanujan.Why,yes,Chatterjiwastold,Mr.Ramanujanhadcalledforataxi.

Perplexed, he returned to Ramanujan’s rooms and sat there with the others,waiting.Untilteno’clocktheywaited,thetimebywhichguestshadtoleave.StillnoRamanujan.

AndnoRamanujan,either,thenextmorning,whenChatterjicheckedonhim.Forfour days, he heard nothing, grew increasingly fearful. Then, on the fifth day hereceivedatelegramfromOxford,abouteightymilesaway.CouldChatterjiwirehimfivepounds?Theamountwastoday’sequivalentofthreeorfourhundreddollars.

Nextday,RamanujanwasbackinCambridge.“I felthurtandinsultedwhentheladiesdidn’ttakethefoodIserved,”hetoldChatterji.“Ididnotwanttocome[back]inwhiletheywereinthehouse.”Hehadneededtogetasfaraway,asfastashecould;withthemoneyinhispocket,thatwasOxford.

Itwas the same impulse, in the face ofwhat he viewed as humiliation, that haddrivenhimtoVizagapatnamtenyearsearlier.Itwastherash,precipitousgestureofamanstretchedthin.

To all appearances, Ramanujan had made a splendid adjustment to a foreigncountryand an alien life.Mathematically, he had lived up to the fondest hopes ofHardy, Littlewood, Neville, and his other English friends. Socially, too, he hadseemedtoadapt,atleastatfirst.Nevillewouldtellofhisdelightincrackingjokesanddiscussing philosophy and politics, would speak approvingly of his simplicity and“instinctiveperfectionofmanners.”

But inside, Ramanujan was like a checking account from which funds are onlywithdrawn,never deposited.Doingmathematics took vast personal energy.Sodidadjusting to his new life in England, as anyone will attest who has ever tried topenetrate a foreign culture. Together they drained his physical and emotionalreserves.Eventually,theaccountmustrundry.

Sometime in 1916, perhaps around the time of the Chatterji dinner, it did. Aconstellation of forces had conspired to stretch thin his nerves, weaken himphysically, isolate him socially. Indeed, themeal he prepared forChatterji and hisfiancée may have loomed far larger in his mind than it ever did in theirs. For

Ramanujannormallywentwithoutmealtimecompany,unlikeHardyandtheotherTrinityfellows.

•••

The college fellows in their black academic robes solemnly trooped into the greatcandlelitHall.Awebbedunderstructure ofwooden arches graced thehigh ceiling.Dryden and Tennyson, Newton, Thackeray, Francis Bacon, and other Trinitynotablesglowereddownfromtheirportraitsonthewalls.Silveradornedbarewoodtables; the tablecloths of peacetimewere gone now for fear the light they reflectedmightbeckonGermanzeppelins.Onceallthefellowswereseated,thesenioramongthem,sittingatoneendofthelongtable,recitedaLatingrace.Andthemealbegan.

ThiswasHighTable,socalledbecausethetablesatwhichthefellowsateweresetonaplatformbuiltupaboutfourinchesfromthefloor.AtTrinity,aselsewhereinCambridge,itwasthefocalpointofthecollege’ssociallife,andlong-standingnotionsofconversationalgoodformgovernedit.

Foronething,younevergottooserious.“Ifonehasdoneahardday’sthinkingonedoesnotwanttoworkatconversation,”Littlewoodwouldsay.“Dinnerconversationisinfacteasyandrelaxed.Nosubjectisdefinitelybarred,butwedonottalkshopinmixedcompany,and,Heavenbepraised,weabstainfromtheimportantandboringsubjectofpolitics.”Itwasaplacenotofgreatprofunditybutofwit,wine,andrelease—releasefromthehightensionoftranslatingancientGreek,orwritingaboutthefallofConstantinople, or proving a new theorem in the theory of numbers.Here, theImportantrecededintodistantmemory.Here,trifleshadtheirday.

YoucouldseeitinthelittlebookkeptovertheyearsthatrecordedsuggestionsandcriticismsofmealsservedatHighTable.Hardyhadbeenamongseveraltodeclarein1909thatBerlinpuddingandHirsch-hornfrittersoughtneverappearagainonthemenu.Later,heandLittlewooddefendedplumpuddingwithwineandcamperdownsauceagainst thosewhowouldsee itblacklisted;over thedaysandweeks,allianceswere built around the issue, compromises forged. At another point, Hardy wasembroiledinabattleoverwhetherfruitpiesshouldbeservedhot,subscribingtotheview that, as another fellow put it, “amanwhowill eat a hot fruit pie is unfit fordecentsociety.”

It was good fun, evenings of camaraderie that eased days of hard solitarywork.Littlewood got his share of it over the years, before he left for the army. So didNeville. So, of course, did Hardy, for whom High Table, with its light, frothyrepartee,was his natural terrain.During the war, officers attached to the hospital

unit or quartered at the college showed up. So may have a few Indians. But notRamanujan.Forhim,dinner inHallwas a slice ofTrinity life fromwhich hewasexcluded.

“Mutton,sir?Beef,sir?”thewaiterswouldbustlearoundthenoisyhall.Thatwastheproblem.

Ramanujanwas a vegetarianunusually strict in his orthodoxy—if not for SouthIndiansgenerallythenatleastforthoseinEngland.IncrossingtheseashehaddefiedBrahminicalstrictures.Hehadforsakenhistuft.HemostlyworeshoesandWesternclothes. But as he had promised hismother, he clung fiercely to the proscriptionsmostcentraltoBrahminiclife,onfood.

The story is toldof ahungryBrahminwho requests food fromamanhemeetsalong the road.He knows nothing of the man’s caste, or character, but he is toohungrytoask.Bellyfull,hesetsoutagainand,afterafewminutes,reachesthehouseofaBrahmin,whoputshimupforthenight.Thatevening,hespiesagoldstatueinthehouse,achestohaveit,andundercoverofnight,spiritsitaway.Allthenextday,hisguiltmounts.Finally,shakenbyremorse,hereturnstothehousetogiveitback.“Oh,Iknewyouwouldstealsomething,”saysthehost.“Yousee,Isawyoutakefoodfromthatmanyesterday,andIknewhewasathief.”

It was this spirit that was built into Brahminic food prohibitions—that fromwhomyou takeyour foodmatters. Inaccepting food from just anyone,whoknowswhatsinshehascommittedinthisoraformerlife?

Brahmins varied, naturally, in the details of their observance. Some forsworeonionsandgarliconthetheorythatthesefoodsraisedsexualappetites,whileothersextendedtheprohibitioneventocabbageandpotatoes.RemembereddetailsdifferonwhatRamanujanwouldandwouldnoteat.Onefriendhashimeatingeggs,anothernot.Somesaidherefusedonions,oreventomatoes.Butallrecalltheabsoluterigiditywithwhichheclungtohisobservance.

Early in his stay, he had once or twice ordered fried potatoes from the collegekitchen.Friedinlard, jokedanotherIndian,alsoaTamilBrahmin,tryingtogethisgoat. That was all it took.Whether true or not, Ramanujan never again orderedanythingfromthecollegekitchen.

Instead, he cooked in the tiny, small-windowed alcove, equippedwith electricityandalittlegasstove,justoffhissittingroom.Whenhecouldgettheingredients,heatewhathe’deatenbackinIndia—rice,yogurt,fruits,rasam,andsambhar,athick,spicy,potato-lacedvegetablestew.HisfriendMahalanobisrecalledhimstandingoverthat littlestove,stirringvegetablesover the fire.But itwassomethinghedidmuchmoreoftenalone.

Fromhiswindow,Ramanujancouldlookoutovertheroofoftheadjacentbuildingto the spired steeple atop the collegeHall. There, atHighTable during the longwintereveningsof1916,thecandlesflickered,theconversationhummed.Butofallthat,Ramanujannevershared.

2.ANINDIANINENGLAND

Ramanujanwas not the first Indian to come toEngland for an education and feelisolatedfromthealienworldaroundhim.MohandasK.Gandhi,thefutureapostleofnonviolenceandleaderoftheIndianindependencemovement,arrivedinEnglandin1887,theyearofRamanujan’sbirth.Hewrotelater:

Iwouldcontinuallythinkofmyhomeandcountry.Mymother’s lovealwayshauntedme.Atnight thetearswouldstreamdownmycheeks,andhomememoriesofallsortsmadesleepoutofthequestion.Itwasimpossibletosharemymiserywithanyone.AndevenifIcouldhavedonesowherewastheuse?Iknewofnothingthatwouldsootheme.Everythingwasstrange—thepeople, theirways,andeventheirdwellings.IwasacompletenoviceinthematterofEnglishetiquetteandcontinuallyhadtobeonmyguard.

Between 1892 and 1906, CambridgeUniversity admitted about twenty Indiansperyear,duringRamanujan’stimealittlemore.Atanyonemoment,somethinglikeathousandIndianswerescatteredamongEnglishcolleges.Inanycase,theirnumbersandthemagnitudeoftheirproblemsinadjustingtoEnglishlifewereenoughtospurgovernment studies of them in 1907, in 1922, and again in later years. And therecurringthemeofthesestudies,whichisubiquitous,too,instoriestoldbyIndiansovermorethanacentury,wasthemaddeningreserve,theunfathomabledistance,oftheordinaryEnglishman.

Sometimes, to be sure, Indians experienced downright racial prejudice; it wascommonenough,forexample,thatuglyrumorsofitreachedRamanujan’smotherinIndia.Butmore often, it was the peculiar shyness of the English that left Indiansfeelingsoadrift.“Wethinkitmustbeadmitted,”concludedareportbytheLyttonCommitteeonIndianStudentsin1922,“thattheBritishandtheIndianstudenteachhashisracial characteristicswhich imposesan initialbarrier to intimacy.”Whereasthe Indian, for his part, tended to be oversensitive to any hint of patronage, “theBritishhasareservewhichcauseshimtobeslowinmakingfriendsevenwithhisowncountrymen,andheisapttoregardwithsuspicionthefirstattemptsofanystrangertocultivatehisacquaintance.”

Observed a student in amuch later study: “The initial difficulty is to break theextraordinaryreserveof theEnglishpeople, their correctbutcoldbehavior, formal,unemotional,courteousanddecent toadegree,butdetached,both to take sidesorinvolvethemselves.”

Astill later student complainedofhaving to stocka ready supplyofpleases andthank-yous.Tohim, itwas justcrudebarter:yougivemesomething, Igivebackathank-you. Far worse, though, was English indifference. Some Indians learned toaccept,evenembrace,itas“respectforprivacy,”butmostsimplysawitasunfeelingand cold. The conversational task of the Englishman, it could seem, was to bescrupulously correct yet remain untouched and unmoved, taking care always toconveytheclearimpressionthat,atbottom,hejustdidn’tgiveadamn.OneIndianstudentwoulddreamofdying,hisbodygoingundiscovered forweeks, reekingandrottingall thewhile.Nevillewouldreadthe fact “thatnoneofus,as farasIknow,ever pressed [Ramanujan] for the true reason for his initial refusal to come toEngland”in1913astypicalof“thereticenceofhisEnglishfriends.”

Nowherewerethesetraitsmoreprevalent,ofcourse,thaninthepatricianclass.IntheHouseofLordsjustbeforeWorldWarI,oneoftheirnumberperceivedapolite“detachment almost amounting to indifference.” Another complained of the“profound weariness and boredom” its members affected. One later visitor toEngland, the IndianNiradChaudhuri,wrotehow “One evening,whendining at aclub,Itriedinmyinnocencetoopenaconversationacrossthetable,andIadmiredtheskillwithwhichtheintrusionwasfendedoffwithouttheslightestsuggestionofdiscourtesy.”Fendedoff,though,itwas.

Backhome,Indiansrecalled,peoplewouldcomeuptoyou,sitdown,starttalking,and in fiveminutesknowall aboutyou—whetheryouweremarried,had children,whereyouwere from,what kind ofwork youdid.Whereas inEngland, returningIndiansadvisedtheircountrymen,formalintroductionswerederigueur.Onestory,setinIndia,toldofaswimmerwhosecriesforhelpsenteveryonerushingtohisaid.Everyone, that is, save the lone Englishman, who sat where he was, apparentlyunmoved. “Oh,” he replied when asked later why he’d not helped, “were weintroduced?”

Cambridgeboasteditsownbrandofaloofness.AbookaimedatIndianstudentsinEngland told how even college porters went about their duties “without the leastconcern about our new comer and with an air of indifference.” And a student ofRamanujan’stutor,E.W.Barnes,oncesetBarnesapartfrommostotherCambridgedonswho,hesaid,gave“theimpressionthattheywerenotgreatlyinterested.”

Laurence Young, the son of two mathematical contemporaries of Hardy andLittlewoodatCambridge,toldhowwhenLittlewood,asanoldman,visitedhiminWisconsinhewouldrowhimoutontothelaketoviewthesunset.Littlewoodneversaid a word, and Young, in time, surmised that he was bored. But when one dayYoung suggested that the water might be too rough for their excursion, “his face

dropped...andIquicklylookedagainatthelakeandpronounceditsmooth.Thisistypical ofCambridge—what you admire, youmerely do not speak of.” Littlewoodhimself would remark that “When youmake your speech at aTrinity FellowshipElection,donotexpectthemtobreakintoirrepressibleapplause;noonewillblinkaneyelid.”

It was a wall erected around one’s feelings, a great silence of the emotions. InCambridge, the emphasis was on ideas, events, things, work, games—anything, itseemed,butthedeeplypersonal.

AtfirstRamanujanmayhavemarveledatthesepeculiaritiesoftheEnglish,muchasWesterntouristsdothesightofuntendedcattleonthestreetsofDelhiorMadras.Buthewasnotonetoeagerlyembraceforeignways;rather,hewasapttostandapartfrom them. Stubborn, self-driven, self-willed, he was every bit the product of hiscountry and its customs. Unlike some Madrasi intellectuals, he’d never lived aWesternizedlifeinIndia;hewastoomuch,andtoorecently,asonofKumbakonam.Hemaintained an Indian’s deep-felt deference to the wishes of his parents. He’dcriedwhenhiskutumiwas cut off. InEngland, he remained steadfastly vegetarian,kept a poster of an Indian deity in his room, and each morning performed theBrahmin’s fastidious morning ritual. He changed into a new, ritually pure dhoti,appliedthenamamcastemarkonhisforehead,performedhisdevotions,thenwipeditoff.Onlythen,ifheweregoingout,wouldhedonWesternclothes.

Evenifhedidn’talwayslethismindwanderbackhome,thereweretimeswhenthesmall, familiar things of South Indian life insinuated theirway into his awareness.The smells ofhismother’s cookingonSarangapaniSannidhiStreet, or of burningcowdung in the streetsofMadras.Thebrightcolorsof religious festivalsparadingdownthestreetsofKumbakonam,accompaniedbythestrummingandjinglingofthemusicians.Theredsandorangesofsari-cladwomenalongthebanksoftheCauvery,white dhotis setting off the dark brown skin of laborers in the fields.The vibrantgreens of the vegetables, the coconuts and bananas andmangos sold down by themarketneartheriver.Andalways,thebrightblueskyandhighoverheadsun.

AmongtheEnglish,Ramanujancouldnotlongforgethisforeignness.Hismusicalaccent was alien to their ears.His skin was darker bymany shades than theirs—which, the winter chill bringing color to their cheeks, wasmore a rosy pink thananything you could call “white.” Everywherewere churches and chapels, Christiancrosses, and Jesus Christ. Here, in this strange land, families scattered, childrenpaying parents nothing like the respect that was a law of social life in India.Meanwhile, all Cambridge resounded to the sound of tramping feet bound for analienwar.

Even theprevalenceofbodyodors among theEnglishmystifiedhim—until, thestory goes, one day he was enlightened about it at a tea party. A woman wascomplainingthattheproblemwiththeworkingclasseswasthattheyfailedtobatheenough,sometimesnotevenonceaweek.Seeingdisgustwrit largeonRamanujan’sface,shemovedtoreassurehimthattheEnglishmenhemetweresuretobathedaily.“Youmean,”heasked,“youbatheonlyonceaday?”

Ramanujanwasnot thekindofchameleonlike figurewhodoeswellat thetoughjobofreshapinghimselftofitaforeignculture;hewasnotflexibleenough,couldnotsinkdowneffortlesslyintohisnewEnglishlife.Norcouldhelongbeimmunetothatsuccessionofsubtle,slightrebuffstheordinaryEnglishmendispensed,withscarcelyathought,everyday.Hewouldhaveneededtoexperience few incidentsofaloofnessandreservetodamphissunnyopennessandsendhimrunningbacktothecozydenofhismathematicalresearch.

Andthat, it seems, iswhathedid.Hewithdrew. “I rememberhimduring thoseyears,thoughIneverspoketohim,”recalledB.M.Wilsonlater.“Hewasinfactveryrarelyseen.”For longstretches,hescarcely lefthisroom.Back inIndia,he liked towork in the cool of the night, the better to escape the midday heat. Now, evenwithouttheheat,heworkedatnight,alone.

Ramanujanwasnot the first foreigner to retreat intohis shell in anew country;indeed, his was the typical response, not the exceptional one. One later study ofAsianandAfricanstudentsinBritainobservedthatasenseofexclusion“fromthelifeofthecommunity...constitutedoneofthemostseriousproblemswithwhichtheywere confronted . . . [andhad] a seriouspsychological effect” upon them.Anotherstudy, this timeof Indian students inparticular, reported thatwhile83percentofthemsawfriendsmoreor lesseverydayback inIndia, just17percentdidwhile inEngland.

During the EasterTerm of 1916,Hardy was amember of theTrinity SundayEssaySociety.SowasE.H.Neville.SowasBertrandRussell.Ramanujanwasnot.While Hardy played tennis or whatever cricket could still be had in wartimeEngland, Ramanujan remained steadfastly sedentary. When some of his friendsjoined the Majlis, the Indian students’ debating society (where, it has been said,“succeeding generations of scions of Indian aristocracy pickedup their nationalismand radicalism”), Ramanujan did not. When Ramanujan’s paper on a type ofDiophantineequationwasreadtotheCambridgePhilosophicalSocietyonOctober30,1916,itwasHardywhoreadit,notRamanujan.WhenHardyreadajointpaperatameetingof theLondonMathematicalSocietyonJanuary18,1917,Littlewoodwastheretohearit,andsowasNeville,andsowasBromwich—butnotRamanujan.

There,inBishop’sHostel,Ramanujanwasashut-in.Fromthesmallwestwindowofhissleepingalcove,theGothicwindowsandstone

walls of New Court loomed just a few feet away. The east window got whatevermorning sunlightwasn’t blocked by the building across theway.But there usuallywasnotmuchsuntoblock.TheEnglishskieswerenotoriouslyovercast.AndaroundChristmas,Cambridge—which is as farnorth asLabrador—wouldbemostlydarkby four in the afternoon. Even the long summer dayswere, byMadras standards,sunless;atypicalJulybroughtrainadozendaysamonth.

Thewaronlymadethingsworse.CambridgeandsurroundingEastAnglia,whichprojectedoutintotheNorthSeatowardtheContinent,wereparticularlyexposedtozeppelin raids. In part to protectKing’sCollegeChapel, streetswere kept dark atnight.Cambridge,someonewouldremember,“waswrappedinamedievalgloomforsome three years and men had to grope their way about the streets as best theycould.” If the England streetscapewas normally gray and drear, now it was black,fairlypressingyoubackinside.

InSouthIndia,theboundariesbetweeninsideandoutsidewerenotsofixedandimmutableastheywereinEngland,whereyouwereforevertryingtoescapethechill.Walls and windows were more permeable. Insects, smells, and sounds broughtoutside inside. Chipmunks and lizards scampered through window shutters.WhereasinCambridge,amidstthestony,solidpermanenceofitsfive-hundred-year-oldwalls,therewasanever-presentsenseofdemarcationanddivision.

Sothatinthewinter,especially,Ramanujan’sapartmentcouldfeellikeaprison.Aplush-linedprison,perhaps,butaprisonnonetheless.AndwhiledrivenbackintoitbytheEnglishreserve,thewinterchill,andthedarkstreetsandwartimegloom,hewasluredbackintoitbythedelighthegotfromhisworkwithHardy.

3.“ASINGULARLYHAPPYCOLLABORATION”

Mathematician Norbert Wiener would one day note how, in one sense, numbertheoryblurstheborderbetweenpureandappliedmathematics.Insearchofconcreteapplicationsofpuremath,onenormallyturnstophysics,say,orthermodynamics,orchemistry.Butthenumbertheoristhasamultitudeofreal-lifeproblemsbeforehimalways—in the number system itself, a bottomless reservoir of raw data. It is innumber theory, wrote Wiener, where “concrete cases arise with the greatestfrequencyandwhereverypreciseproblemswhichareeasytoformulatemaydemandthemathematician’sgreatestpowerandskilltoresolve.”

In1916,onesuchproblemlayintheareaofnumbertheoryknownas“partitions.”

Onthesurface,theproblemwassosimpleitwentbacktoalmostthefirstdaysofgrammar school: 2+ 2= 4. But that’s just oneway to add up numbers to get 4.Thereareothers.Like1+3.Or1+1+2.Or1+1+1+1.And, lastly, tobescrupulously complete about it, just plain 4 itself. These (aside from mererearrangementsofthesamenumbers)aretheonlywaysofaddingupintegerstoget4.Countthemup,andyougetfivedifferentways,or“partitions.”Mathematicianssaythatthenumberofpartitionsof4is5.Or,

p(4)=5

More generally p(n), read “p of n,” represents the number of partitions of anynumber, n, and is known as the partition function. What, the mathematicianwonders,canbesaidaboutit?Howcouldweevaluateitforanyn?

Inprinciple,it’snotdifficulttogothroughallthepossibilitiesforagivennumberandaddthemup.Inprinciple.Theproblemisthatthenumberofpartitionsrisesveryfast.Thenumberofpartitionsof3,p(3),is just3(3,1+1+1,1+2).Butbythetime you get up to 10, p(10) = 42. And p(50) = 204,226. Two hundred fourthousand, two hundred twenty-six different ways of adding up the integers. Justlistingthemall,oneeveryfiveseconds,wouldtaketwoweeks.Andthat’sjustforn=50. So the question is, can you find a formula for p(n) that sidesteps the awfularithmeticandspitsoutthenumberofpartitionsforanynumberyouplease?

Asusual,itwasEulerwhomadethefirstrealdentintheproblem.Workingfroman area of mathematics which later became known as elliptic modular functions,Euler detoured around the problem to construct what is known as a generatingfunction. In theory,agenerating functionsuppliesnot justaparticularanswer toaparticularproblembutalltheanswerstoallrelatedproblems;itgeneratesanswersasfastasthenumberscanbepluggedin.Withit,youstepoff intonewmathematicalterrain where the object of attention seems no longer to be p(n), but some newfunction,f(x).Exceptthatwithf(x)inhandyoucangobackandgetwhatyoureallywant, p(n). In this case, Euler’s generating function gave rise to a “power series,”whichisjustaseriesoftermseachofsuccessivelyhigherpowersandeachmultipliedbysomething,somecoefficient:

f(x)=1+∑p(n)xn

whichmeans:

f(x)=1+p(1)x1+p(2)x2+p(3)x3+...

It was those coefficients, p(1) and so on, that potentially supplied the answers.Because if you could actually evaluate thepower series, the coefficientswould turnout tobenot anyoldnumbers,but just thep(n)desired. InEuler’s scheme, if youwanted,say,p(50)—whichis204,226—you’djustheadouttothefiftiethtermoftheseries,whereyou’dnotethecoefficientofx50.Soifyouhaddoneitright,thatpartoftheserieswouldread:

...+204,226x50+...

Now,thiswasnotjustblackmagic.Itwasnaturalthatthegeneratingfunctionforpartitionsbeapowerseries,becausepowerscombinebyaddition.Thus4×8=32canbewrittenas:

22×23=25

Theirexponentsaddup.Andaddingthingsupisjustwhatyoudoinpartitions.(“Amoment’s consideration shows that every partition of n contributes just 1 to thecoefficientofxn,”Hardywouldobserveintracingthealgebraiclinksbetweenpowerseries andpartitions.)Squint, then, and evenwithout following the logic in detail,youcanroughlydiscerntheroutebetweenthetwoseeminglyunconnectedrealms.

This, at any rate, was how Euler set up the problem. But Euler didn’t solve it,didn’tsayhowtousehisgeneratingfunctiontoactuallychurnoutthedesiredpowerseries. He offered a strategy for arriving at p(n), but little more. As Hardy andRamanujan would write in their big paper in the Proceedings of the LondonMathematicalSociety,“wehavebeenunabletodiscoverintheliteratureofthesubjectanyallusionwhatevertothequestionoftheorderofmagnitudeofp(n).”Noone,inotherwords,hadaclue.

TheseedsfortheattackRamanujanandHardyweretotakecameinRamanujan’sfirst letter toHardy in January 1913.There, on page 7, he gave a particular thetaseries—apowerseriesofacertainkind,containingonlytermswithsquaredpowers,like x1, x4, and x9—and claimed that to determine its coefficients one had but toevaluatethemathematicalexpressionhefurnishedandselecttheintegerclosesttoit.

This was not quite right. “The function,” Hardy would write, “is a genuineapproximationtothecoefficient,thoughnotatallsocloseasRamanujanimagined.”Andyet,hewenton, “Ramanujan’s false statementwasoneof themost fruitfulheevermade,since itendedby leadingus toallour jointworkonpartitions”;indeed,Ramanujan’sfunctionfrom1913boreastrongfamilyresemblancetotheproblem’sultimatesolution,whichcameonlyin1937.

Inthecourseoftheirwork,RamanujanandHardywereledtowhatwouldcome

tobeknownasthecirclemethod.ThecirclemethodmadeuseofCauchy’stheorem,whichmightseematfirstasinappropriateforthejobasanythingyoucouldimagine.Cauchy’stheoremlayinthedomainof“analysis,”thebroadareaofmathematicsthatincludescalculusanddealswith“continuous”rather than“discrete”quantities;howlongyou’vebeenpregnant isa continuousquantity,howmanychildrenyouhaveadiscreteone.Partitionswereadiscretequantity;youcouldn’thave6.719partitionsjustasyoucouldn’thave6.719children.Ithadtobe6,exactly,or7.

ButCauchy’stheoremhadahistoryofbeingappliedtosuchproblemsbynow,andHardy could be thanked for it; he was foremost among twentieth-centurymathematicians in pioneering the “analytic theory of numbers,” which took thepowerful tools developed over more than two centuries for continuous quantitiesand, through feats of mathematical legerdemain, applied them to the integers ofnumbertheory. InanaccountofhisworkwithRamanujan,Hardynotedthat“theidea[ofusingCauchy’stheoremhadbecome]anextremelyobviousone:itistheideawhich has dominated nine-tenths of modern research in the analytic theory ofnumbers; and it may seem strange that it should never have been applied to thisparticular problem before.” It hadn’t, he went on, in part because of “the extremecomplexity of the behavior of the generating function f(x) near a point of the unitcircle.”

Hardy referred to the fact that the integral at the heart of Cauchy’s theoremcouldn’t be evaluated as it stood, because the “contour” over which it was to beintegratedheld impermissiblepoints,where itsvaluewasn’tdefined.Sotheywouldhave to find an alternative path close to the forbidden “unit circle,” systematicallydissectit,makeapproximationsastheywentalong.Thiswasthenewstrategy.

Basing a result on a series of approximations seemed to guarantee that it woulditself be an approximation. But should they expect anything better?Hadn’t primenumbersresistedalleffortsatexactcalculation,forcingmathematicianstobecontentwith no more than a rough estimation that grew relatively more accurate as nincreased? Hadn’t all Ramanujan’s attempts to make the prime number theoremexact run afoul of mathematical pitfalls? Surely partitions would prove similarlyintractable,makingthemhappywithalmostanydecentapproximation.

Toseehowfarofftheywere—tochecktheworthoftheirapproximationsbyhowclosetheycametothepreciseresult—theyputMajorMacMahontowork.

Thesonofabrigadiergeneral,PercyAlexanderMacMahonwasasixty-one-year-old mathematician who had served in the Royal Artillery—including a stint inMadras back in the 1870s—before seriously taking up mathematics. “With hismoustache,his ‘BritishEmpah’demeanorandworstofallhismilitarybackground,”

itwasoncewrittenofhim,“MacMahonwashardlythetypetobechosenbyCentralCastingfortheroleoftheGreatMathematician.”Butafterleavingthearmy,hehadgoneon tobecomeaprofessoratWoolwich, thearmy school, and since1904hadbeen associated with Cambridge’s St. John’s College. His expertise lay incombinatorics,asortofglorifieddice-throwing,andinithehadmadecontributionsoriginalenoughtobenamedaFellowoftheRoyalSocietyin1890.MacMahonwasawhirlwind of a calculator. Sometimes, in fact, he would take on Ramanujan infriendlyboutsofmentalcalculation—andregularlythrashhim.

But now,MacMahonwas putting his calculating skills to good use and, using asimple formula that led directly from Euler’s earliest work in partitions, hadarduouslyhand-calculated,inimpossible,endlessstreamsofnumbers,thevaluesforthefirsttwohundredp(n).Bytheroughestofanalogies,hewasaddingup29+29+29+. . .1,000,001timestoget29,000,029,whileHardyandRamanujansoughtawaytosimplymultiply29by1,000,001.Buttheydidn’thaveityet,andMacMahon’stediouseffortssuppliedabenchmarkagainstwhichtotestitwhentheydid.

When they tried it early in their work, with their approximation strategy stillrelativelyprimitive, theywerealreadyencouragingly close; forp(50) andp(80), theerror stood at just 5 percent. Further refinements, with correspondingly smallererrors,reinforcedtheirconvictionthattheywereontherighttrack.

Then, inDecember1916, thetwomen,perhapshavinggonetheirseparatewaysuntil the startofLent term in January, cameabig step forward.RamanujanwroteHardy a postcard, squeezing into the tight space one more contribution to theircontinuingmathematicaldialogue:

Itthereforeappearsthatinorderthatp(n)maybethenearestintegertotheapproximatesum,Sneednotbetakenbeyondβ /lognandcannotbetakenbelowα / logn Ihopeyoucaneasilyprove these.Thentheproblemiscompletelysolved.MajorMacMahonwaskindenoughtosendmeatypewrittencopyofthe200numbers.Theapproximationgivestheexactnumber.Ithinkyouknewthisalreadyfromhim.

Ramanujanalludedtoawayofmakingthenumberoftermsoftheseriestheyusedtoapproximatep(n)itselfdependonn.This,asLittlewoodwrote later,was“averygreat step, and involved new and deep function-theory methods that Ramanujanobviouslycouldnothavediscoveredbyhimself.”Whatitseemedtomean,asHardyandRamanujanwouldwrite,wasthat“wemayreasonablyhope,atanyrate,tofindaformulainwhichtheerrorisoforderlessthanthatofanyexponentialofthetypeean;oftheorderofapowerofn,forexample,orevenbounded.”

Abounded error, an error they could fixwithin set limits: theywouldhavebeenhappywiththat.Buttheirresultswerebetterstill:When

weproceededtotestthishypothesisbymeansofthenumericaldatamostkindlyprovidedforusbyMajorMacMahon,we found a correspondence between the real and the approximate values of such astonishingaccuracyastoleadustohopeforevenmore.Takingn=100,wefoundthatthefirstsixtermsofourformulagave

190568944.783 +348.872 −2.598 +.685 −.318 +.064 190569291.996

while p(100)=190569292;

sothattheerroraftersixtermsisonly.004.

Similarprecisionappliedtop(200).Theirmethodwassupplyingananswerwhoseerrorwasnotjust“bounded”—whichcould,afterall,meanboundedbut large—butsmallenoughtoroundoff tothenearest integer. “Theseresults,”wroteHardyandRamanujan, “suggest very forcibly that it is possible to obtain a formula for p(n),which not only exhibits its order ofmagnitude and structure, butmay be used tocalculateitsexactvalueforanyvalueofn.”

Thiswastheshocker.Nothinginthehistoryofthisproblem,nothingintheworkwith primes, had prepared them for anything like it. What they had done withpartitionswasjustwhatRamanujanhadthoughthe’ddonewithprimesbut,Hardyshowedhim,hadn’t.Whatwasmore,theuncannyaccuracyoftheirresultsattestedtothepoweroftheapproximatingtechniquetheyhadusedtogetthem.Thecirclemethod itwouldbecalled, fromhowit letyoudrawoh-so-near,butneveractuallytouch,theforbiddencircularpath.Sosubtleandinspiredweretheapproximationsitpermitted that itwent beyond approximation to promise exactitude.Two decadeslater,sureenough,HansRademachercameupwiththemissingpieceofthepuzzleandmadetheformulaexact.

Did Ramanujan suspect there was an exact solution all along? That’s what theeminentNorwegian number theorist Atle Selberg has suggested. Selberg, in fact,argues that Hardy’s insistence on certain methods of classical analysis actually

impededtheirefforts;andthatlackingfaithinRamanujan’sintuitionhediscouragedasearchforthekindofexactsolutionRademacherproducedtwentyyearslater.

Inanycase,theirpartitionssolutionwasbignews,thecirclemethodthey’dusedtocomeupwithitastunningsuccess.Inlate1916,Hardydashedoffanearlyaccount,under his own name but offered “as the joint work of the distinguished Indianmathematician, Mr. S. Ramanujan, and myself,” to the Quatrième Congrès desMathématiciens Scandinaves in Stockholm. Early the following year, a brief jointpaperappearedinComptesRendus,as“UneFormuleAsymptotiquePourleNombreDesPartitionsDen.”And a one-paragraph reference to theFrench journal articleappearedinProceedingsoftheLondonMathematicalSocietyinMarch.Theforty-pagepapersettingouttheirworkinfulldetaildidn’tappearuntil1918.

•••

Ramanujan andHardy: as a mathematical team, they would remind PennsylvaniaStateUniversitymathematicianGeorgeAndrewsof the storyof the twomen, oneblindandtheotherlackinglegs,whotogethercoulddowhatnonormalmancould.They were a formidable pair. On the strength of their work on partitions alone,whichby itself justifiedRamanujan’s trip toEngland, theirnameswouldbe linkedforeverinthehistoryofmathematics.

CambridgemathematicianBélaBollobáshasobservedthatwhileHardyfurnishedthetechnicalskillsneededtoattacktheproblem,

IbelieveHardywasnottheonlymathematicianwhocouldhavedoneit.ProbablyMordellcouldhavedoneit.Polya could have done it. I’m sure there are quite a few peoplewho could have playedHardy’s role. ButRamanujan’sroleinthatparticularpartnershipIdon’tthinkcouldhavebeenplayedatthetimebyanybodyelse.

Whatever the proper assignment of credit, “We owe the theorem,” Littlewoodwouldwrite,“toasingularlyhappycollaborationoftwomen,ofquiteunlikegifts,inwhicheachcontributed thebest,mostcharacteristic, andmost fortunatework thatwasinhim.Ramanujan’sgeniusdidhavethisoneopportunityworthyofit.”

ForRamanujan, itwasalldeliciouslyaddictive.Adecadebefore,hisdiscoveryofCarrhadlefthimsosingle-mindedlydevotedtomathematicsthathenolongercouldfunctionasanordinarycollegestudent.Now,somethinglikethatwasatworkagain,onlyworse.Fornowitwasnotonlyhisowndelightinmathematicsthatspurredhimon,buttheencouragementhegotfromHardy.Hardy,thisembodimentofall thatwas highest and best in the Western mathematical tradition, with his immensetechnicalprowessandrichknowledgeof thewholemathematicalworldofEngland

and theContinent,was allRamanujan couldwant in a colleague andmentor.Andthat he saw such breathtaking originality in him could do nothing to restrainRamanujan’seagernesstogetonwiththeirwork.

“Acharactersoremarkablyfreefromthepettymeannessesofhumanlife . . . themostgenerousofmen.”That’swhatC.P.SnowoncesaidofHardy.Anothertime,hecalledhim“freer fromtheemotion[ofenvy] thananyman Ihave everknown.”Indeed,thoughHardydeemedRamanujan’snaturalmathematicalabilitysuperiortohisown,nohintsurvivesofsomuchasawispofenvytaintinghisrelationshipwithhim. Despite any private twinges, he seems always to have been Ramanujan’sunalloyed friend and supporter. All his life he championed him, hailed his gifts.RecognizingRamanujan’sgenius,hewantedonlytopushittowarditslimits.

Andthat,ifanything,wasjusttheproblem.HardywasinmanywaysthebestandtruestfriendRamanujaneverhad.Hewas

considerate, loyal,andkindtohim.Andyetinatleastoneway,unintentionally,heprobably did Ramanujan harm. For in the ardent hopes he had for him, in hisunbridledwishtoseethatheliveduptohispotential—inkeepinghimtothemark,drivinghim—HardyonlyfedRamanujan’saddiction.

Hecouldn’tkeepRamanujanfromdeepeningtheholehewasdiggingforhimself.Hemayevenhavehelpedhimdigitdeeper.

4.DEEPENINGTHEHOLE

Hardywasanaristocratoftheintellect,raisedtovaluehighachievementanddismissanythingless.Asamathematician,hewasallRamanujancouldwant.Buthewasalsoaformidableanddistantfigurewhodemandedalways,andonly,thebestfromhim.“His sense of excellence was absolute; anything less was not worthwhile,”rememberedanOxfordeconomist,LionelCharlesRobbins,whoknewHardylater.“WhatDr.JohnsonsaidofBurkeappliedexactlytoHardy:hispresencecalledforthallone’spowers.”

J. C. Burkill, a Cambridge undergraduate beginning in 1919 who later himselfbecameaprominentmathematician,remembersfeelingalwaysintimidatedbyHardy—always “below him,” as he puts it. Whereas Littlewood, say, came across asthoroughly human and accessible, chatting away amiably inHall,Hardywas busybeingbrilliant.“Whenhewasconversing,”saysBurkill,“hefelthehadtobeonahighplane.” The great Hungarian mathematician George Polya told how Hardy onceexpressed disapproval of Polya’s failure to pursue a promising mathematical idea.The two of them, with a thirdmathematician, were visiting a zoo. At one point,Polyarecalled,acagedbear“sniffedatthelock,hititwithhispaw,thenhegrowleda

little, turnedaroundandwalkedaway. ‘Heis likePolya,’ remarkedHardy. ‘Hehasexcellentideas,butdoesnotcarrythemout.’”

Youdidn’tturntoG.H.Hardyforanunjudgingshouldertocryon.Once,whenthemathematician Louis J.Mordell wrote him, complaining that his papers weregetting picked apart by the editors of a mathematical journal because of minorstylisticfaults,Hardyrefusedtoindulgehim.“IknowIhavespentoverthreehoursoverthejournalproofsofanoteofyours,”hewroteback,“andhavemadeoverthirtycorrectionsonapage.All‘trivialities’—sotrivialthatyouhavenevernoticedthem,oratanyratecommentedonthem:butamorning’sworkgonewest.”TheremainderofHardy’s eleven-page letter showed similar irritation.Don’t be so easy on yourself, itsaid.

Hardy,inshort,wasasterntaskmaster.Hiswasapersonalityofexpectations,ofhigh performance. From him, Ramanujan could get encouragement and, in thoseways in which Hardy could express it, friendship—but little in the way of pure,uncriticalnurturing.

Hardy’sreplytoRamanujan’sfirstletterfromIndiashowedthatsameholding-to-account—Prove it—and he never let up once Ramanujan was in England. At onepoint,writingtoRamanujan,whowastheninthehospital,abouttheircurrentwork,his eagerness for the mathematical fray plainly warred with his concern for hisfriend’shealth:“IfIgetoutanymoreIwillwritetoyouagain.Iwishyouwerebetterandbackhere—therewouldbesomesplendidproblemstoworkat.Idon’tknowifyoufeelwellenoughtothinkaboutsuchdifficultthingsyet.”Then,apostscript:“Atpresentyoumustdowhatthedoctorssay.Howeveryoumightbeabletothinkaboutthese things a little: they are very exciting.”Asmaddeningly equivocal as the letterwas,Ramanujanwouldhavebeenobtuseindeedtomissitsmessage:Theworkawaitsyou.

And so, if any part ofRamanujanwanted to relax, pursue his pet philosophicalnotions,investigatethepsychictheoriesoftheEnglishphysicistOliverLodgethatsointriguedhim,takethetrainintoLondontovisitthezoo,ashedidonceortwice,orotherwisestrayfrommathematics,itgotscantsupportfromHardy.

Hardy’surgings,tobesure,foundfertilesoil inRamanujan’sownobsessivebent.Later,hewouldwriteHardythatwhilethehospitalinwhichhewasthenapatientwasuncomfortablycold,

thebathroomsareniceandwarm.Ishallgotothebathroomwithpenandpapereverydayforaboutanhourorsoandsendyoutwoorthreepapersverysoon.Thisthoughtdidnotstrikemebefore.ElseIwouldhavewrittensomethingalready.InaweekorsoImayperhapshaveacomplaintagainstmefromthedoctorthatIamhavingabatheveryday.ButIassureyoubeforehandthatIamnotgoingtobathebuttowritesomething.

Ramanujan,sickandinthehospital,wasapologizingtoHardyforhavingfailedtodomoremathematics!

•••

Hismarriagein1909hadinducedRamanujantostitchhimselfbackintothewidersocialworld.Now,inEngland,thethreadsofconnectionwereonceagainsevered.Byearly1917,hewasamanonamission,propelledtowardhisdestiny,oblivioustoallbutmathematics.AfterthreeyearsinCambridge,hislifewasHardy,thefourwallsofhisroom,andwork.Forthirtyhoursatastretchhe’dsometimeswork,thensleepfortwenty.Regularity,balance,andrestdisappearedfromhislife.

Hewasnotthefirstmantosacrificehishealthonthealtarofmathematics.Jacobigave this retort to a friendwhoworried that excessive devotion tohisworkmightmakehimsick:“CertainlyIhavesometimesendangeredmyhealthbyoverwork,butwhatofit?Onlycabbageshavenonerves,noworries.Andwhatdotheygetoutoftheir perfect wellbeing?” And before Jacobi, there was Newton himself. “Nevercareful of his bodily health,” E. T. Bell wrote of him, “Newton seems to haveforgottenthathehadabodywhichrequiredfoodandsleepwhenhegavehimselfuptothecompositionofhismasterpiece.Mealswereignoredorforgotten.”

In Hardy, Ramanujan had the intellectual companionship he’d long missed. InCambridge,withitsstatelystonechapelsandquietcourtsandgreatlibraries,hehadbeforehimalltherichesofWesterncivilization.Yetnowhewentwithoutmuchelsethathadsustainedhim,perhapswithouthisrealizingit, inIndia.Hewentwithoutfamily.Without dark, familiar faces and open, sunny Indian smiles.Without thesoundoffriendlyTamil.Nowthosehiddenpropstoidentityandself-esteem,soeasytotakeforgrantedordismiss,hadbeenpulledoutfromunderhim.Nowtherewasnooneeventopreparemealsforhim.Noone,asJanakiandhismotherhaddone,toplace food intohishandasheworked.Noone to remindhimtosleep.Noone tocool his fevered brain with a touch, or a sexual embrace. No one to counselmoderation,tourgehim,asitwere,tocomeinoutoftherain.Ramanujanwaslikeabalkythoroughbredwithnoonetogroomorfeedit.Andbyearly1917,therehadbeennoonesincethedayhe’dsteppedaboardtheNevasathreeyearsbefore.

Intellectually,hehadcomehome.ButRamanujanwasmorethanamind;hewasabody,acomplexofmuscleandtissue,hormonesandneurochemicals.Andhisbodyhadneedsthathismindscarcelyknew.

•••

Agenerationbefore, in1890,a smallbookaimedat Indian students, entitledFourYears in an English University, was published in Madras. Its author was S.Satthianadhan, a professor of logic andmoral philosophy atPresidencyCollege inMadras. Apparently based on Satthianadhan’s experiences as a student at CorpusChristiCollege,Cambridge,thebookpurportedtotell Indianstudentswhat lifeatCambridgewasreally like, andwhat Indians could learn from it.Andone point itconveyedwithalltheforceoffreshrevelationwastheEnglishemphasisonsportsandrecreation.

ACantab[fromthemedievalLatinforCambridge]neverfailstotakehistwohours’exerciseperdieminonewayoranother.Seldomdoesonefindastudentinhisroomsintheafternoon,howeverpassionatelyfondofstudyhemaybe.OnewhodoessoandkeepstohisbooksthewholedaylongwillbelookeduponasanabnormalcharacterandbesnubbedbytheotherstudentsoftheCollege.Menssanaincorporesano,—asoundmindinasoundbody,—isamaximofuniversalandpracticalapplication.TheseyoungEnglishmen,whopayasmuchattentiontotheirbodilyastotheirmentaldevelopment,areinnowayworseoffasstudents.Thesemen,whocanwalktwelvemilesadayorrowsixteen,withoutbeingtiredintheleastarejustashard-workingas theGermanstudents;and it is thesestrong,healthy,muscularyoungmenwhoturnoutWranglersandFirstClassClassics.

IfthereisonelessonwhichourstudentsinIndiamustlearnfromEnglishstudents, itisthis—topayasgreatanattentiontotheirbodilyastotheirmentaldevelopment.

Ramanujan, it need hardly be stated, did not. He had no interest in sports; ifHardytriedtointeresthimincricketitdidn’ttake.Hehadalwaysbeenfat, largelyoblivious to his body, almost pathologically sedentary. Still, he was not yetSatthianadhan’svision of the typical Indian student: “A study-worn, consumptive-lookingindividual,withoutanyenergy,appearingtwiceasoldashereallyis,fitrathertobeaninmateofthehospitalthanafrequenterofthelectureroom.”

Itwasashospital“inmates”thatnotafewIndianstudents inEnglandultimatelyfoundthemselves.WhentheLyttonCommitteeonIndianStudentswentaroundthecountryin1921and1922itheardoneOxfordmanobservethatthehealthofIndianstudentstendedtobreakdowntowardtheendof their secondyear,almostas if inobedience to natural law. A Darwinian streak that ran through much committeetestimony suggested why: in their own country, Indians had adapted to theirenvironment,inparticularthehotsunandspicyfood.Now,inEngland,thethinkingwent,theywerefacedwithalienconditions.Thelessfitfailedtoadapt,andsowerestruckdown.

Whateverthescientificvaliditytosuchaview,itwastruethatnationalidentitieswere less blurred in those days, making for more to adapt to; India was moreinviolablyIndia,EnglandmoreEngland.Therewerenot,astherearetoday,IndianrestaurantsandfoodstoreslitteringthestreetsofLondonandCambridge.Asforthe

climate, therewas no air-conditioning or central heating tomoderate its extremes.Youeitherendureditorgotout.Thatwaswhywell-heeledEnglishmenjourneyedtoItalyorSpainduring thewinter;orwhy, in Indiaduring theworstof the summerheat,theyvisitedhillstationslikeSimlaintheNorth,orOotacamundintheSouth.Thosewhocouldnotescape,meanwhile,facedthefullforceoftheheat,easedonlybyfeeblefansor,midstthedampandchillofEnglishwinters,werelefttothemerciesofineffectualcoalfires.

IndianscometostudyinEngland,ofcourse,couldnotescapethealienandhostileclimateandso,itwasbelieved,riskedtheirhealth.ThedirectoroftheUniversityofEdinburgh’s Indian Student Hostel told the committee that the health of Indianstudents there was generally good, “except perhaps,” he said, “in the case of thosecomingfromthesouthernpartsofIndiawhosufferedratherfromtheseverityoftheclimate and were inclined to develop tuberculosis and disease of the chest.” Thepresidentof theRoyalCollegeofPhysicians inEdinburghcommented that amongIndianstudentstuberculosiswascommon,itbeingaggravatedbytheclimateand,inthe words of the report, “the fact that the more religious of the Indian studentsinsisted on keeping to the diet of their cult—a regime unsuited to the conditionsunderwhichtheyfoundthemselves.”

Ramanujandidindeedkeeptothedietofhis“cult,”andfromevenhisfirstdaysinEngland,beforethewar,doingsohadposedproblemsforhim.Inalettertoafriendsoonafterhisarrival,hecomplainedabout“thedifficultyofgettingproperfood.HaditnotbeenforthegoodmilkandfruitshereIwouldhavesufferedmore.NowIhavedeterminedtocookoneortwothingsmyselfandhavewrittentomynativeplacetosendsomenecessarythingsforit.”Inotherletters,heaskedthatcertainprovisionsbesenthimornolongersenthim,gaveaccountsofshipmentsgoneastrayorarrivinginpoorcondition,expressedthanksforthosethatdidarriveintact.Monthly,NarayanaIyer sent him powdered rice in tin-lined boxes.Others sent him spices or pickledfruitsandvegetables.

Back in India, Janaki later recalled, Ramanujan would sometimes abruptly stopeating, or else hurry through it, to pursue a mathematical thought; meals weresomething to be dispensedwith. But now,with no one to cook for him, they hadbecome an awful bother. Preparing from scratch South Indianmeals—assemblingthe ingredients, soakingandgrinding lentils, cuttingvegetables,boiling rice, and soon, all theway through the final cooking—took time.And time,whileworkingonwhat to him were the most seductively challenging problems in the world, heresented having to give up. So, as Neville’s wife, Alice, for one, would report,

Ramanujansometimescookedonlyonceaday,orsometimesonlyonceeveryotherday,andthenatweirdhoursintheearlymorning.

Thesimplest“solution,”ofcourse,wasaspeciesofasceticism.BackinMadras,hehadenjoyedmango,orbanana,orjackfruitwithhisriceandyogurt.Andtherewasbrinjal,preparedthatspecialwayhismotherdidit,and...ButthegreatdaysofgoodSouthIndianmealsweregone.HehadwrittenSubramanianin1915,“Iamnot inneedofanythingasIhavegainedaperfectcontrolovermytasteandcanliveonmerericewithalittlesaltandlemonjuiceforanindefinitetime.”

For a time, certainly, he could. But not indefinitely. Ramanujan’s strictvegetarianismhad cost him, inHighTable, a social outlet thatmight have helpedkeephimonamoreevenkeel.Now,duringwartime,itriskedhishealth,too.

5.“ALLUSBIGSTEAMERS”

Thestreetsweredark.Themoodwasblack.Thewardraggedon.WhenRamanujanbecamea researchstudentatTrinityCollege in1914,hewas

enteredinthesamethickAdmissionsBookasallotherTrinitymen.Hesignedhisname on page 8 of a volume begun just the previous year, then filled in columnsrunningallacross thepagewithhisplaceofbirth, father’sname, school, andothersuchroutineinformation.

But examine that book today, and you find that on the next page afterRamanujan’s, the statelymarch of densely filled columns comes to a chilling halt.Suddenly, for the first time, gaps appear in the record, blank spaces.Thename ofJohn de Vere Loder appears, but not his signature. Islay Makimmon Campbell’sadmission is listed;buthenever signed either.Bothwere youngmenwho, thoughadmitted to Trinity, had taken their places at the front and never reachedCambridge. The next page is worse, like a mouthful of teeth half of which aremissing,orabombedcitywitheveryotherhousereducedtorubble.

Besidemanynamesappears further information:Charles JervoiseDudleySmith,“killed16June1915.”DonaldHolman,“killed8Aug1918.”JohnBrertonHoward,“diedofwounds6Sept1918.”Fortwentypages,itgoesonlikethat,whitespacesintheTrinityAdmissionsBooklikewhitemarbleheadstonesofthemenwhodiedinFlandersandinFrance.

Theuniversity stayedopen,but itwasonly aghostofwhat ithadbeen.Trinitywasdepopulated,itsenrollmentplummetingfromalmostsixhundredbeforethewarto forty-seven in October 1916. By that time, as one young officer training inCambridge wrote, “the pulse of the University had almost stopped beating; therewereveryfewundergraduates in residenceexceptboysundermilitaryage,Asiatics,

andthephysicallyunfit(‘infants,Indiansandinvalids’).”Itwastrue:thosetakingthemathematicalTriposin1916,forexample,includednameslikeTripathi,Mahindra,Prosad,andSaravanamutti.

The war, entered upon with such patriotic fervor, did not end soon, but grewgrimmerinitsceaselesstoll.Theearlyoptimismfaded.Deadlystatictrenchwarfarereplacedsweepingbattlefieldmaneuvers.Togainahundredyardsofmachine-gun-sweptmudmightcostathousandlives.Thesad,stupidrealityoftheconflict,fedbytheawfulcasualtyreports,felllikeapox-riddenblanketacrossBritain.ItmusthaveparticularlygalledHardythatwhentheFirstEasternGeneralHospitalmovedfromNevile’sCourt inMarch1915, itwas to a temporaryhospital built, behindKing’sandClareColleges,onacricketground.Allitselevenacresultimatelyquarteredthesickandwounded.

The war touched everything. “Military Field Boots, Made to Order,” read anadvertisement in theCambridgeMagazine. “BestQualityMarching Boots Kept inStock.” Food prices rose. Shortages set in. By May 1916, the crisp brown stockgracingthecoveroftheGazetteoftheFirstEasternGeneralHospitalhadgivenwayto flimsy blue paper. ByOctober, with the hospital having already treated thirty-threethousandcasualties,thecoverwastheconsistencyoftoiletpaper.

“Neverhavewehadtochronicleso terriblea listof lossesas thisweek’snumber[issue]contains,”wrotetheeditorsoftheCambridgeMagazineinitsissueofOctober14, 1916. “Scarcely a day has passed but names familiar tomany still in residencehave appeared in the official lists.” Then, college by college, it recorded the toll:Hopgood,Hudson,Johnson,Keeling,Knight...

“It is our pride and sad privilege,” said Sir Joseph Larmor two weeks later inaddressing the London Mathematical Society after two years as its president, “torecallthenamesofthecultivatorsofoursciencewho,inresponsetotheircountry’sappealintimeofnationalperil,havealreadylaiddowntheirlivesonherbehalf.InE.K.Wakeford,ScholarofTrinityCollege,Cambridge,notafewofushadrecognizedafutureleaderingeometricalscience.”

Ramanujan’stutor,E.W.Barnes,castacolder,morebitterlightonsuchlosses.Ina sermon towardwar’s end,hedeclared: “Ofmypupils atCambridge at leastone-half,andpracticallyallthebest,havebeenkilledormaimedforlife;theworkthatIdid[teachingmathematicsovertheyears]hasbeenforthemostpartwasted.”

Sometimesitseemedeverybodydied.TheproprietorofagrocerybackinHardy’shometown,RobertCollins,died.

So did artillery battery commanderW.Graham, who five years before back inIndia, had been among those in the Madras administration to contribute to the

debateonRamanujan’smerits.Later, at Winchester College, a War Cloister honoring Wykehamists who fell

duringthewarwouldriseonasitenearwhereHardyhadplayedsoccerinthe1890s.Itwasengravedwiththenamesoffivehundredofthem.

WhatBarbaraWootton,writing inInaWorldINeverMade,best rememberedwas

anendlesssuccessionofmemorialservicesincollegechapelsorCambridgechurches.Inaway,whatdistressedmemost...wastheunashamedpublicgriefofthefathersoftheyoungmen....Accordingtothestandardsofourfamilycircle,tearsweretobeshedonlybychildren.Veryoccasionally,womenperhapsmightweep—butgrownmennever....ThesightofdistinguishedprofessorsandfamousmenwhomIhadbeenbroughtuptoregardwithaweopenlycryinginchurchdisturbedmeprofoundly.

“Thingsherearesadandsorrowful,”Trinityvice-masterHenryJacksonwroteonJanuary25,1917.Hewascheeredalittle,ardentmilitaristthathewas,bythetrampof recruits drilling inNevile’s Court, officers issuing commands,menmarching information.But: “Many friends aredead.Feware lefthere. I amanxious about thehome people. Gout and rheumatism punish me. Deafness incapacitates me. Andthereisalwaysthewar,thewar.”

•••

InFebruary1915,GermanyhadimposedasubmarineblockadeoftheBritishIsles.InOctober1916,itwasintensified;nowshipswouldbesunkwithoutwarning.

BRITISHSHIPSSUNKBYSUBMARINES10,000TONSLOST

readanunexceptionalheadlineinTheTimesofLondononFebruary17,1917.ThesameeditioncarriedwordthatportsinAmerica,whichhadnotyetjoinedthewar,wereundervirtualblockade.FourmillionbushelsofexportwheatwerelockedupinsilosinMinneapolisbecauseEastCoastportswerecloggedwithgrainthatcouldn’tbeloaded.

Ontheeveofthewar,Britaindependedonimportstothetuneoftwo-fifthsofhermeat,four-fifthsofherwheat.RudyardKiplinghadwritten:

Forthebreadthatyoueatandthebiscuitsyounibble,

Thesweetsthatyousuckandthejointsthatyoucarve,

TheyarebroughttoyoudailybyallusBigSteamers—

Andifanyonehindersourcomingyou’llstarve!

Now many of the Big Steamers weren’t getting through. And some Englishmenbegan to starve. There was, it was said, one sure way to tell rich from poor inEngland. You had only to look at them; the poor were shorter, less physicallydeveloped, bore the stigmata of undernourishment. Now poor nutrition becamemore widespread. Bertrand Russell’s second wife Dora, writing in The TamariskTree, recalledhow “peoplebegan to feel the effects of the loss of essential fats andothernecessities.WhenIsawmytwoclergymanunclesinthelatterstagesofthewar,Iwasshockedtoseehowtheserosy,well-coveredmenhadshrunk.”

Foodwas rationed.Prices rose;by1916, the foodbill for aworking-class familyhadclimbed65percentcomparedtobeforethewar.Byearly1917,severeshortageshad set in. “The usual week-end potato and coal scenes took place in Londonyesterday,”theObservernotedonApril8th.

AtWrexhamabigfarm-wagonladenwithpotatoesalreadyweighedintoshillingsworthswasbroughtintothesquarebytheagriculturalistswhoatonceproceededtosellthemtoallcomers.Thewagonwassurroundedbyhundreds of clamouring people, chieflywomen, who scrambled on to the vehicle in the eagerness to buy.Severalwomenfaintedinthestruggle,andthepoliceweresentfortorestoreorder.

EvenHighTablefelttheshortages.“Wearevirtuouslyobedienttothecontrollerof food,” wrote Jackson on March 26, 1917: “fish and potatoes but no meat onTuesdayandFriday;meatbutnopotatoestherestoftheweek:breadrollshalftheirold size: portions strangely dwarfed.” But if for Englishmen it was hard gettingpotatoesandsugar,whatofthespecialtyfoodsRamanujancravedthatwerehardtocomebyeveninpeacetime?InJune1914,hehadwrittenthattheeasyavailabilityof“good milk and fruits” had helped ease his food problems. Now, all fruits andvegetablesweredifficulttoprocure.

ThewarhadreachedRamanujan.The food shortages coupled with his irregular eating habits could hardly have

fortifiedhimagainstanydiseasestowhichisolation,overwork,andclimatemayhavepredisposed him. In his third year in England, he embodied the distinction somephysicians todaymake between being well and merely not being sick. He was anillnesswaitingtohappen.

W. C. Wingfield, medical superintendent of one famous English sanatorium,would observe that the diseasewithwhich hewasmost familiar, tuberculosis,wasbrought on, or aggravated, by faulty modes of life, which he defined as “overwork,overplay,overworry,undernourishment, lackofnecessarysunshineandfreshair,orchronicintemperanceinanyform.”

Except,perhaps,for“overplay,”Ramanujanwasguiltyofallofthem.

6.THEDANISHPHENOMENON

Bothduringhis life and afterward, itwas amatter of somemystery justwhat laidRamanujanlow.Butsomething,inthespringof1917,did.InMay,HardywrotetheUniversity of Madras with news that Ramanujan was sick—afflicted with, it wasthought,someincurabledisease.

Wasthis thetimetosendhimbacktoIndia?The ideawasbroached.ButmanyIndian physicians were on war duty; getting him adequate medical care, it wasthought,mightbe impossible.Then, too,hemightnever reach Indian soil; theU-boats made ocean travel perilous. The future Dora Russell in August 1917accompaniedherfather,ahighAdmiraltyofficial,toNewYork.“Myfatherhadanablepersonalassistant,”shewrote,“butinviewofthesubmarinemenacehefeltthathecouldnotpresshertoaccompanyhim.”SoheaskedDorainstead.“Wetravelledinconvoyandwereneverallowedtobeonemomentwithoutourlifebeltsbesideus.”ThewarhaddeniedRamanujanaccess tomathematicianswithwhomhehadbeenbroughttoEnglandtowork.Ithadunderminedhisnutrition,perhapspriminghimfordisease.Now,sick,ithelpedkeephiminEngland.

Itwas an anxious time.Hewas admitted to a “nursing home”—actually a smallprivatehospitalcateringtoTrinitypatients—onThompson’sLane,overlookingtheCamacrossfromMagdaleneCollege,andwithinastone’sthrowoftheNevillehouse.Hewasveryill.TheprognosiswassopoorthatHardyaskedthemasterofTrinityfor help in getting word to Ramachandra Rao in India by special dispatch. Later,when Ramanujan seemed a little better, he asked Subramanian to contactRamachandraRaoandallayanyfearstheearlierreporthadraised.

Bythattime,Ramanujanwasoutofthehospital,perhapsbackinBishop’sHostel.ThereissomeevidencethatHardyhimselfnursedhimforawhile.Butwhoeverdid,it couldn’thavebeeneasy.BecauseHardy’s letter toSubramanianalreadybore thestampofwhatwastocomplicatealleffortstorestoreRamanujan’shealth:“Itisverydifficulttogethimtotakepropercareofhimself,”Hardywrote.Ramanujan,hewassaying,wasaterriblepatient.

Notthatthiswasanythingnew.Backabout1910,whenRamanujangotsickwhilelivinginMadras,hisfriendRadhakrishnaIyerputhimupforawhile.“Asapatient,”he recalled later, “Ramanujanwas not exemplary; hewas obstinate andwould notdrink hot water and insisted on eating grapes which were sour and bad for him.”Radhakrishna called in a doctor who examined him and, mercifully, ordered thatRamanujanbesentbacktohisparentsinKumbakonam.

Hehadnotchangedinsevenyearsand,all throughhiscare,wasaplagueonhiscaregivers.“Adifficultpatientalwaysinclinedtorevoltagainstmedicaltreatment,”isthewaytwoofhisIndianbiographerswoulddescribehim.“Difficulttomanage,”iswhatHardyhadtosay.“Wheneverhestartedwithadoctorhewasfullofconfidence,faith and hope,” says Béla Bollobás, a Trinity College mathematician who was afriendofLittlewoodandwhohastakenaspecialinterestinRamanujan’slife.“[But]whenherealizedthedoctorcouldn’thelphim,hewenttotheotherextremeandhesawthatthisdoctorwasnogoodatall.‘Ifellintohistrap,how,howcanIgetawayfromhim?’”

Ramanujan was picky about his food, wouldn’t do what he was told, forevercomplainedabouthisachesandpains.Self-willedasever,hehadnofaithinmedicine.“There are very few doctors who would care to have Ramanujan in their nursinghomes,” itwouldbesaidofhim, “and fewerstillwhowouldbotherwithhumoringRamanujan’spalate.”

Ramanujan’spigheadedness,alongwiththeuncertaintysurroundinghisdiagnosis,led him to see at least eight doctors and enter at least five English hospitals andsanatoriumsoverthenexttwoyears.ProbablyaroundOctober,hebecameapatientattheMendipHillsSanatoriumatHillGrove,nearthecityofWells,inSomerset.Therehe cameunder the careofDr.Chowry-Muthu, an Indiandoctorwho, as ithappens,hadaccompaniedRamanujanontheNevasathreeyearsbefore.Dr.Muthuwasatuberculosisspecialist.

•••

TheearliestdiagnosisofRamanujan’sailmentwasagastriculcer,supportforwhichwaxed and waned all through his treatment; at one time, exploratory surgery wasconsidered.

OnedoctorheldthatRamanujan’shydroceleoperationbackinIndiahad,infact,beentoremoveamalignantgrowth,andthatthecancerwasnowspreading;butsinceRamanujan’sconditiondidn’tdeteriorate,mostdoctorsdismissedtheidea.

Blood poisoning was another possibility; an idea heard later was that, perhapsoverly impressed with their presumed sterility, Ramanujan would eat cannedvegetables whose labels gave assurance of their purely vegetarian origin. Bypassingproper pots and pans, the theorywent, Ramanujanwould cook them right in thecans, over the gas flame in his room, perhaps contracting lead poisoning from thesolderedlids.

Itwastuberculosis,though,forwhichRamanujanwastreatedatMendipHillsandforwhichhewouldmostconsistentlybetreatedinthecomingyears,andtuberculosis—consumption, phthisis, the White Plague—which remains the most likelycandidatetoday.

TB accounted for asmany as one in three deaths inEuropean cities during themidnineteenthcenturyand,bytheearlytwentieth,afterhalfacenturyofdecline,stillcausedoneineightdeathsinBritain.In1882,RobertKochshowedthatthediseasestemmedfrominfectionbyaparticularmicroorganism,theslow-reproducingbacillusMycobacterium tuberculosis. The bacillus could infiltrate most any bodily tissue,causinglesions,littlegranulouspockets,inthespine,theeyes,thebones,thekidneys,orthelymphnodes,butmosttypicallyinthelungs.Fever,nightsweatsthatleftthepatient drenched by morning, cough, difficulty breathing, spitting of blood, andweight loss were typical symptoms—but not invariant ones. Diagnosis was alwaystricky.Often, thedisease followed apeculiar course.Sometimes it took an abrupt,violentturn;thiswasthefeared“gallopingconsumption.”Oftenitlaylowforyears,onlytorecur.Sometimesseeminglyadvancedcasesrecoveredspontaneously.

To get tuberculosis you must, in the first place, be infected with the tuberclebacillus.Butmanypeople,probablymost,inIndiaaswellasinEngland,gotinfected,yetnevercamedownsick;theirimmunesystemssuccessfullywardedofftheattack.Heredityclearlyplayedaroleinwhogotsick.Butso,too,almostcertainly,didwhattodaymight be called “lifestyle” factors.While consensus on their influence eludesthe research community, the evidence gives strong credence to Dr. Wingfield’simpression that “overwork, overplay, overworry, undernourishment [and] lack ofnecessary sunshine and fresh air” help transform otherwise failed bacterial attacksintosuccessfulones.

Foronething,aformidablebodyofresearchnowpointstoclosetiesbetweenthenervoussystemandtheimmunesystem,andbetweenstressandillness.Thespousesofbreastcancervictims,achingwithworryanddespair,comedownsickmorethancontrols. Neurotransmitter receptors occupy sites on cells of the immune system,making for a natural communications link between the two systems. Stress—overwork? worry? loneliness?—the evidence powerfully suggests, can weaken the

immune system and offer a ripe field for disease. One study found that FilipinosailorsservingintheAmericannavy—typicallyawayfromhomeforyearsatatime,and formuch longer than other sailors—weremore TB-prone than other sailors.Researchersconcludedthat“emotionalstressassociatedwithseparationfromfamilyandfriends”—simpleloneliness—maywellhavecontributed.

Could Ramanujan’s vegetarianism, made harder to nutritionally maintain bychaoticeatinghabitsandfoodshortages,havesethimupforthedisease?Here,again,evidence is suggestive, with theDanish phenomenon giving the notion a powerfulboost.

In 1908, Dr. H. Timbrell Bulstrode submitted his Report on Sanatoria forConsumptionandCertainOtherAspectsoftheTuberculosisQuestiontoParliament.Init,heincludedachartshowingacomfortinglysteadydeclineinthetuberculosisdeathrate inEnglandandWales, fromclose to300deaths forevery100,000populationaround1850to120by1904.SimilardeclineswerebeingrecordedduringthisperiodinSwitzerlandandGermany,inDenmarkandtheUnitedStates.

ThencametheGreatWar.With it came a wild spasmodic leap in the death rate. In Prussia, tuberculosis

deathsshotupfromabout150perhundredthousandto250.ThesameinBelgium,Italy,andtheotherbelligerentcountries.Englandrecordedalessdramatic,ifequallyunambiguous jump, 17 percent between 1913 and 1917—erasing, in two years,twentyyearsofsteadydecline.

But why did Denmark also show a precipitous 30 percent rise? Denmark wasneutral, untouched by war or civil strife. Moreover, why did the Danish figuresresume their downward course beginning in 1917, two years before the othercountries?

As it happens,Denmark shipped large quantities ofmeat and dairy products toEngland and other belligerents during the first two years of the war. Prices rosedramaticallyathome,butwagesfailedtokeeppace.Danishpercapitaconsumptionplummeted. But with the resumption of unrestricted submarine warfare around1917,DanishfoodpiledupinDenmark.Consumptiononthehomefrontrose—andwithit,apparently,TBresistance.“Thereisampleevidence,acceptedbyvirtuallyallauthorities,”wrotetheEnglishphysicianR.Y.Keers inexplainingthewar’s impacton tuberculosis deaths, “that the main factor in the production of this increasedmortalitywasmalnutrition—aviewsupportedbythefiguresfromDenmark.”

Whatwasthecrucialmissingnutrientthatmeatordairyproductssupplied?Quitepossibly vitamin D—a deficiency of which may have left Ramanujan, amongthousandsofothers,morevulnerabletothetuberclebacillus.

It was not until 1920 that vitaminD’s role in the prevention of rickets, a bonedisease,wasdiscovered.Muchrecentresearch,however,linksitalsototheimmunesystem.Andinwhathasbeencalledapieceof“admirableepidemiologicaldetectivework,”WelshphysicianP.D.O.Daviesin1985showedhowvitaminDdeficiencycouldexplain, through its immunesystemeffects, themuchhigherTBrate amongimmigrantstoBritainfromtheIndiansubcontinent.

Davies noted that the diseasewas thirty times higher in this group than amongnative Britons; that studies showed more than a third of them had vitamin Ddeficiency; that vitamin D clearly played a role in marshaling monocytes andmacrophages—key immune system players. “A Possible Link BetweenVitaminDDeficiencyandImpairedHostDefencetoMycobacteriumTuberculosis,”Daviestitledhis paper in Tubercle, the British tuberculosis journal. “Can this complex[biochemical] pathway really explain the incidence of tuberculosis in dark-skinnedimmigrantstotheUnitedKingdomandthecureofskintuberculosisbyvitaminDtreatment?” asked anothermedical researcher,GrahamA.W.Rook, in evaluatingthelogicofDavies’sargument.Heconcluded:“Itprobablycan.”

Prime sources of vitamin D include egg yolks, organ meats, and fatty fishes.Indeed,codliveroilwasusedasearlyastheeighteenthcenturytotreattuberculosis.Withthepossibleexceptionofegg—theevidencehereiscontradictory—Ramanujanatenoneofthem.

Another source of vitamin D today is milk specifically fortified with it. Butfortifiedmilkwasn’tavailableinRamanujan’sday.

ThereisanotherprimesourceofvitaminD,onethatexplainswhyIndiansbackinIndia,forexample,unlikethoseinBritain,havenormallevelsofthevitamin:thesun.Thesungivesoffnot justvisible lightbutultraviolet radiation,andultraviolet raysactivatecholesterolintheskintomakevitaminD.

Ramanujan got little sun. In Cambridge, high up near the Arctic Circle, therewasn’tmuch tobeginwith.And theEnglish cloud cover blockedmost of the rest.Then, too, Ramanujan didn’t leave his rooms much, often working at night andsleeping by day. Even working by a sunny window would have done no good:ordinarywindowglassabsorbstheultravioletraysthatmakevitaminD.

In the fifth editionof his classicPrinciples andPractice ofMedicine, published in1902, thegreatphysicianWilliamOsler, referring to the sheer capriciousnesswithwhichinfectionbythetuberclebacillusendedindisease,observed:

Thereare tissue-soils inwhichthebacilliare, inallprobability,killedatonce—the seedhas fallenby theway-side.Thereareothersinwhichalodgmentisgainedandmoreorlessdamagedone,butfinallythedayiswiththeconservativeprotectingforce—theseedhasfallenuponstonyground.Thereare tissue-soils inwhich

thebacilligrowluxuriantly;caseationandsoftening,notlimitationandsclerosisprevailandthedayiswiththeinvader—theseedhasfallenupongoodground.

Boththecommonmedicalwisdomofhisowndayandscientificevidencefromourownsuggest that Ramanujan, during his first three years in England, had becomefertilegroundindeedforthegrowthofM.tuberculosis.

•••

Intheyearsbetween1899and1913,MendipHillsSanatoriumclaimedhundredsof“cures.” And five other sanatoriums within a twenty-mile radius apparently alsofound the country just across Bristol Channel and the Severn River from Walesconducive to recovery. Still, geography and cure rate notwithstanding, somethingdidn’t take forRamanujan atMendipHills; hewas there but briefly, and soon, inaboutNovember,hewastransferredtoMatlockHouseSanatoriuminDerbyshire.There,offandonforthebestpartofayear,hewastreatedbyatleastthreedoctorsandranupbillsofatleast£240,theequivalenttodayofperhaps$20,000.

Earlyinhisstay,hewroteHardy:

IhavebeenhereamonthandIhavenotbeenallowedfireevenforasingleday.Ihavebeenshiveringfromcoldmanyatimeandhavenotbeentotakemymealssometimes.InthebeginningIwastoldthatIcouldnotpossiblyhaveanyexceptthewelcomefireIhadforanhourortwowhenIenteredthisplace.Afterafortnightof stay they toldme that they receiveda letter fromyouaboutoneandpromisedme fireon thosedays inwhichIdosomeseriousmathematicalwork.Thatdayhasn’tcomeyetandIamleft inthisdreadfullycoldopenroom.

The Matlock staff wasn’t being flinty eyed or cruel in denying Ramanujan thewarmthofafire; itwas justthatcoldopenroomsliketheoneRamanujanenduredwere,duringthisperiod,themostacceptedtreatmentfortuberculosis.

PhysiciansfacingTBtodaytaketheirpickfromavarietyofdrugregimenstosuitconsiderationsof cost andpossible side effects. Indeveloped countries at least, thediseasehasbeenlargelywipedout.Butitwasn’tuntilthe1950sthatthefirsteffectivedrug, streptomycin, camealong.Until then, tuberculosis remained thegreatWhitePlague,resistanttoallthatmedicalsciencecouldthrowatit.Koch,discovererofthebacillus,hadtrumpetedavaccine—whichdidn’twork.Therewassurgery,involvingtheopening, treatment, anddrainingof tubercle abscesses.Therewas the artificialpneumothorax vogue, in which the tuberculous lung was collapsed, by injecting itwithagas,andallowedto“rest,”presumablyleadingtohealing;oneofRamanujan’sdoctors, the prominent London chest disease specialist Harold Batty Shaw ofBrompton Hospital, would a few years later be among those to remark on thistreatment’srisetopopularity.Goldsalt injectionhaditsproponents.Sodidpatent

medicines like “Tuberculozyne,” takenwithmilk three times a day, and “CrimsonCrossFeverandInfluenzaPowderfortheCureofConsumption.”

But most widely accepted of all during Ramanujan’s time was the open-airtreatmentpioneeredinthelatenineteenthcentury,andcallingforbedrest,typicallyinopenlodgesexposedtothefreshair,lotsoffood,andmeasuredamountsoflightexercise.

“Open-airtreatmentfortuberculosishadprecededthebacteriologicalrevolutioninoriginandwaslittleinfluencedbyit,”LindaBryderhasobservedinBelowtheMagicMountain,herstudyoftuberculosistreatmentinEngland.Tuberculosiswasbredinurbanslums?Thegreatoutdoorswerelinkedtohealth?Thenopenair,faroutinthecountryside,waswhat the lungsof thehaplessconsumptiveneeded.Such,anyway,wasthetreatmentrationale.

Launched inGermany, the open-air strategy heavily influenced British practice,especially in the years around the turn of the century. Indeed, one of the mostinfluentialoftheearlyGermansanatoriums,atNordrach, in theBlackForest, lenttheweightofitsnametooneofthosenearMendipHills.ItwasknownasNordrach-upon-Mendip.TherewasaNordrach-on-Dee,too.

Itwasnolongerpossible,Bulstrodeobservedinhis1908report,tomeaningfullydistinguishsanatoriumsfromordinaryhospitals,“seeingthatpracticallyallhospitalsfor consumptionhavenowmade some attempt towards carrying out the ‘open air’treatment.”Andby1913,therewereatleastfifty-twoBritishsanatoriumsembracingit.MendipHillswasoneofthem.SowasMatlock,thoughasamorerecentconvert;ithadearlierservedasahydropathicestablishmentspecializinginheattherapy.

ByRamanujan’s time, being left to themercies of the coldwas the treatment ofchoice.“Wide-openwindowsandcureonoutsidebalconiesweretheorder,bleakandcoldasthenightordaymightbe,”ReneandJeanDuboswroteinTheWhitePlague.“Theartofwrappingoneselfinblanketsbecameanessentialpartofthecure,almostaritual.”Architectscompetedtofindnewwaysofbringingfresh,coldairtopatients.Somesanatoriumshadopen-airchapels.Moretypically,patientslayonbedslinedupin long, wide open corridors; Liegehallen, the Germans called them. Sometimes,sanatoriumswerewhollyunheated.Evenchildrenwerenot exempt;oneschool forconsumptive children inNorthumberland recordeda temperatureof thirtydegreesFahrenheit in November 1915. At another sanatorium, just after the war, a girldescribedwhatamountedtoahutontopofahill inwhichherconsumptivesisterstayed. “There was no door, no glass in the windows; the window and the rainblowed[sic]through.”

InmiseryatMatlock,Ramanujanwasnotalone.

•••

ButtherewereotherreasonswhyRamanujanwas—andhadbeen,andwouldbe—unhappyduringtheperiodofhiscare.

InSeptember 1917,Hardy hadwrittenSubramanian ofRamanujan’s seeminglyimprovedprospects. Inhis letter,he’dnoted that “itwasonly a fewmonthsago—whenhewasforatimeinaNursingHospitalhere—thatwediscoveredthathewasnotwritingtohispeoplenor,apparently,hearingfromthem.Hewasveryreservedaboutit,anditappearedtousthattheremusthavebeensomequarrel.”

There isa certain shockof revelationhere,a sensealmostofhavingbeencaughtnapping.Wediscovered...apparently...itappeared...somequarrel.Thevaguenessis distinctly un-Hardylike. And in the way Hardy attributes his ignorance toRamanujan’sreserve,thereisevenatraceofdefensiveness.

Alittlelater,anotherhint:writingHardyfromMatlock,Ramanujanseemstobeacquaintinghim for the first timewith some of themost rudimentary facts of hispersonallife:

ItistruethatIpromisedmymotherthatIwasgoinghomeattheendof2years;Iwrotethemseveralletters11/2yearsagothatIwascomingoverthereforthelongvacation;butIhadmanylettersofprotestfrommymothertotheeffectthatIoughtnottocometoIndiatillItookmyM.A.degree.SoIgaveupthe ideaofgoingthere.

ItisnottruethatIamgettinglettersfrommywifeorbrother-in-laworanybody.Ihadonlyafewformallettersfrommywifejustexplainingtomewhyshehadtoleavemyhome....

TheinitialS.inmynamestandsforSrinivasawhichismyfather’sname.Ihaven’tgotasurname,reallyspeaking.

Ramanujanhadnotreturnedhomein1916becausehismotherhadimploredhimnot to? His wife was not writing him? She had left his family’s home? AndRamanujanhadnorealfirstname?

ItwasallnewstoHardy.Therewas,indeed,troublebackhome;“quarrel”wasscarcelythewordforit.And

it had reached across thewaters to upsetRamanujan in England.Makingmattersworse,Hardyhadknownnothingofit,andsocouldhardlyhavedonemuchtoeasehisdistress.

Sometimeinearly1917somethinghadgonebadlywrongwithRamanujan’sbody;hehadcomedownsick.Butbytheendoftheyear,itwasnotonlyhisbodythatwastroubled,buthismind.Bythen,certainly,hewasnotahappyman.Happymendonottrytokillthemselves.

7.TROUBLEBACKHOME

RamanujangotnolettersfromJanakinotbecauseshedidn’twritethembutbecausehismotherinterceptedthem.

One time, with a package destined forRamanujan prepared for pickup and hermother-in-law out of the house, Janaki slipped into it a brief note. ButKomalatammal returned early, spied the note, opened it, read it, dismissed it aschildishorsillyorstupid,andrefusedtosend it. Janakiwasupset.Butwhatcouldshesay?Ordo?Shewasagirlofseventeen,andhermother-in-lawwas,well...hermother-in-law.Soshesaidnothing,consolingherself that thatwas justhowthingswere.

Thatwas justhowthingswereintheIndianextendedfamily.Mother-in-lawandwifetroubleswereagiveninmanyIndianhouseholds,perhapsmost.Theywerethestuff of jokes—and, as usual, the butt of moralistic opprobrium from Westernobservers,whose sensibilitieswereoffendedbysomuchofwhat theysaw in India.The institutionofchild-brides,wroteHerbertCompton in1904,wasabominationenough,butthegirl’scustomaryfateinhernewhomewas,ifanything,evenworse.

Itispitiableforthechild-wife,tornfromahomethatcontainedallsheknewofhappiness,tobeobligedtosubmit herself to the temper, caprice, and often tyranny of her husband, but when to this is added thedespotismandcrueltyofseveralelderlywomen,whooftenavailthemselvesofherhelplessness,andifshefailstofindfavourinherhusband’seyes,almostinvariablytaketheircueofunkindconductfromhim,herlotmaybebetterimaginedthandescribed.Shehasabsolutelynoplacetogotoforcomfortandsympathyifitisnottobefoundinhernewhome.Thereisnoescape,andnomatterwhathersufferings,herparents’homeisclosedtoher.Anappealtothemmeetswitharigidcommandtosubmitherselftoherhusband.

Someyearsbefore,theHinduhadgonetothetroubleofdefendingIndiancultureagainstsuchcriticism.“The tyrannicalmother-in-law,” it argued, “isnot the rule inHindusociety;andevensheisnotsoblackassheispainted.Norisshesopersistentandunchanging inher cruelty to thegirl-wife.” In its verydefensiveness,of course,theeditorialspokesomekerneloftruth.

AnIndianmarriagewasamating—oraclash—offamilies.Thewife,anewcomerto her husband’s family, was apt to be deemed an interloper, a threat to thehouseholdsway longheldbyhermother-in-law.Besides, shewas just a child.Hermother-in-law,whohadundergonethesametrialswhenshewasabride,wastheretoshapeher,justasthehard-bittendrillsergeantdoesnewMarines.Buthere“bootcamp,” as it were, extended over years—until the wife bore her own children andthen,inthecourseoftime,becameamother-in-lawherself.Onlythenmightshetakea dominant place in the household. In themeantime, her husband normally actedtowardher,atleastpublicly,withaspeciesofindifference.Thewife’slotwastodefer,unquestioningly, tohermother-in-lawand theotherolderwomenof thehouse. In

somehomes,shewouldnotevenhandsomethingtoherhusbandwithoutgivingittohermother-in-lawfirst,whointurnwouldpassittohim.

Plainlytherewereseedsofdomestictensioninsuchasituation.AndinthecaseofRamanujan’s family, they grew in more than usually fertile soil. For one thing,Komalatammal’sidentificationwithherson—hisbirthlongsoughtandtheobjectoffervent prayers—was particularly strong.Andwhile young Indianmenwere oftenmore attached to mother than to wife, Ramanujan was more than usually so; helookedlikeher,thoughtlikeher,sharedtheardencyofhertemperament.Then,too,Komalatammal, close to fifty bynow,was a formidable figure of awoman, by oneaccount“insanelyjealous”ofherson,whileJanakiwasameekandcallowteenager.

Sevendecadeslater,Janaki,eighty-eightyearsold,wasastoopedoldwoman,livinginTriplicane,Madras,withherforty-five-year-oldadoptedson,Narayanan,hiswifeandthreechildren.Theirmodesthousestoodbehindalowwallwithanironfence,afewfeetbackfromabusystreetaccentedbycoconutpalmsheavywithfruit.Inside,wrapped inaburgundysari, Janaki satonabarewoodbenchagainstawall,whereshe had only to look up to see a bronze bust of her late husband, garlandedwithflowers, the gift of his admirers from around the world. Her skin was glossy,stretchedoverbonesbarrenoffat.Hunchedandfrail,shegotaroundthehouseonlybypainfullypushingawoodenchair,whichfunctionedasawalker.Nearlydeaf,shecouldhearNarayananonlywhenheshoutedthrougharolled-upmagazineintoherear.As she replied, in loud staccatobursts, to questions askedher, her facewouldsometimesgrowcontortedwiththeeffortofsimplylisteningandspeaking.Atothertimes,itwouldbreakoutintoabroad,captivatingsmile.

According to somewhoknewher, Janakiwasmore confident and assertive nowthan in years past. And yet, such was a daughter-in-law’s place as she had beenbroughtuptoacceptitthatevennow,nearingninetyandknowntobebitterabouthertreatmentatKomalatammal’shands,shetookpainstoshowrespectforherlong-dead mother-in-law. Through Narayanan, she expressed gratitude to her for theopportunity to marry Ramanujan. And she asked that certain difficulties in theirrelationship be couched in properly respectful circumlocutions; that it be said, forexample, that they were simply “not able to see eye to eye”; and that she fled thehouseholdatonepointmerelybecauseshe“wantedtochangetheatmosphere.”

Those close to Janaki, however, suggested some of the basis for her resentment.While Ramanujan remained in India, Komalatammal apparently kept them fromsleepingtogetherashusbandandwife.OncehehadleftforEngland,Janakiwasgivenonlythecoarsestmaterialforsaris.Shegotnomoneyofherown,butdependedonhermother-in-law for themerest trifle. She wasmade to trek across town to the

banksoftheCauveryforwaterwithneverawordofthanks.Shewasthebuttofhermother-in-law’sabusive language.Finally,of course,her letters toRamanujanwereintercepted, as his were to her. At one point, apparently, Ramanujan wrote hismother asking that she have Janaki join him in England. His mother, not tellingJanaki,wrotebackthatitwasoutofthequestion.

Komalatammal’s side of the conflict does not come down to us, except that, bysomeaccounts,sheblamedJanaki,onthebasisofherhoroscope,forRamanujan’sillhealth;hadhemarriedsomeoneelse,shewascertain,hewouldnothavegottensick.It may be, however, that she bore toward Janaki no special animosity at all, butmerelywroteheroffasthechildshestillwas.Almostfortyyearsherseniorandusedtohavingherwayaroundthehouse—andwithRamanujan—she could scarcelybeexpectedtogiveJanakimuchvoiceorautonomyuntilshegrewup.Itdemandslittleimagination, then, to see her dismissing with scarcely a wave of her hand Janaki’spleasandprotestations.

Whatever theprecise familydynamics, itwasa scene rifewithhard feelings andharsh words. Things were so bad that, at one point, even Ramanujan’s half-blindfather,whohadat firstopposedthemarriageof Janaki tohissonandscarcelyeverfiguresinaccountsoffamilylife,stoodupforheragainsthiswife.

Finally,Janaki foundanexcuse togetaway.Heronlybrother,Srinivasa Iyengar,thenworkinginKarachi,inwhatisnowPakistan,wasgettingmarried.TheweddingwouldbebackinherhometownofRajendram,wheresheandRamanujanhadbeenwed.Obviouslyshewouldhavetogo.ThroughmutualfriendsinKumbakonam,herparentshadknownsomethingwasamiss,anditmayhavebeentheirideatousetheweddingaspretexttogetJanakioutofKomalatammal’sclutches.

Alittlelater,Janaki,nowatherbrother’shouseinKarachi,wroteRamanujan,andthistimethelettergotthrough.Couldhesendhersomemoneyforanewsariandforaweddinggift forherbrother?Dutifully,Ramanujan sent themoney.Butbynowbitter at the long silence from hiswife and knowing onlywhat he heard from hismother,heletnowarmthorfeelingslipintohisreply.

Thetroubleathomehadoverfloweditsbanks,distortingRamanujan’srelationshipwithhiswholefamily.Hislettershomefirstdroppedoff,thenstoppedaltogether.In1914, Ramanujan had written home three or four times a month. By 1916,sometimes two or threemonths passed before he wrote. During 1917, the familyheardfromhimnotatall.

For a long time, perhaps shamed at feeling abandoned by his own family,Ramanujankeptsilentaboutit.Butfinally,whileintheCambridgenursinghome,he

couldholditinnolonger.HetoldHardy.AndhetoldhisfriendChatterji.Visitinghimoneday,Chatterjifoundhimlookingunhappy.“What’sthematter?”heasked.

“Oh,myhousehasnotwrittentome,”Ramanujanreplied,usingacommonSouthIndianidiomfor“mywife.”

“Well,”jokedChatterji,thoughfamiliarwiththeidiom,“housesdon’twrite.”

•••

The “quarrel” disturbed Ramanujan’s equanimity, made for a tangled snag in hisemotionallifelinebacktoIndia.Thatwasbadenough.ButthatHardyknewsolittleofhispersonalsideatthislatedate,threeandahalfyearsafterhecametoEngland,testifiedtosomethingmore—and,perhaps,worse.Presumably,Hardywashisbestfriend inEngland.At least beforehe got sick, the twohad seen eachother almosteveryday.Andyet,ofRamanujan’slackoflettersfromhome—and,verylikely,ofthestrainofhis adjustment toEngland, andof thewindsof loneliness that sometimesblewthroughhim—Hardyhad,atleastuntilrecently,knownnothing.

Why, if they were bosom friends, hadn’t Ramanujan been able to tell him longbefore?

Thefactis,theywerenotintimatefriends.RamanujanwascutofffromIndia.Hewas cut off from the English. And, by a chasm of personality, culture, andcircumstances,hewascutofffromHardyaswell.

8.THENELSONMONUMENT

Foronething,askeenaninterestasHardytookinRamanujan,hehadotherthingsonhismind.Hewasaninternationalmathematicalfigure.Hewasinvolvedinmanyareas ofmathematics other than thoseRamanujanpursued; of forty-fivepapershewrote from 1915 to 1918, only four were collaborations with Ramanujan, thoughothers were influenced by their joint work. Hardy was active in the LondonMathematical Society, attended itsmeetings, served as officer, sometimes took onseemingly petty “journalistic” chores for it; in January 1917, for example, heundertook to draft a leaflet advising authors on thewriting of their papers. In theCambridge Philosophical Society, too, he was active. In 1917, he campaigned forsplitting the society’s Proceedings into two separate journals, thereby presumablyupgradingthepuremathematicsoneofthemwouldcarry.

Hardywas deeply involved in theworld outsidemathematics, too, in particularagainst the war.WhenTresilianNicholas, a young Fellow ofTrinity briefly backfromwarservice,showedupinCambridgein1915,hefoundhimselfseatednextto

Hardy inHall. SurprisedNicholas knew nothing of some recent college business,Hardyaskedhim,“Whateverhaveyoubeendoing?”“WhenIsaidIhadbeenonwarservice in the Mediterranean,” Nicholas recalled, “he gave me a look of extremedisapprovalandtalkedtohisotherneighborfortherestofthedinner.”

Fromitsonset,thewarhaddividedCambridge.G.E.Moore,philosophicalguruof the Apostles and Hardy’s “father” from fifteen years before, agonized over hisstance on it. And Hardy, as Paul Levy writes in his biography of Moore, “soonbecameoneofthepeoplewhoseopinionsonthewarmostinterested[him].”Fortwoyears Hardy’s views appeared regularly in his diary. “Hardy just back [fromvacation],”MoorewroteonSeptember25,1914,“thinksweoughttomakepeaceassoonasFranceandBelgiumaresafe.”(Theyneverwere.)

During thisperiod,Trinitywas tornby theBertrandRussell affair.The leadingmathematicalphilosopher of his day,Russell had already become the impassionedantiwar campaigner he would remain on into Vietnam days. A pacifist, he wasunpopularamongconservativeseniorfellowswhonow,withthejuniorfellowsatthefrontorotherwise involved in thewar, ranTrinity. InApril1916, a schoolteachernamedEverett,aconscientiousobjectorgrantedexemptionfromcombatantservice,wascalledupforserviceinthenoncombatantcorps;herefused,wascourtmartialed,and sentenced to two years hard labor. Russell, active in the Non-ConscriptionFellowship,cametohisdefenseinaleaflet,andwentonrecordasitsauthor.Hewasconvictedformakingstatementsprejudicialtorecruitmentanddiscipline,andstifflyfined.

On July 11, 1916, Trinity dismissed Russell from his lectureship. “Trinity inDisgrace,”ranaheadline intheCambridgeMagazine lamenting thecollege’saction.Hardy, who later chronicled the affair inBertrandRussell andTrinity, was amongthose fellows—Littlewood, Barnes, and Neville were others—who protested theaction. (Hardy’s “little book,” one review said of it, “is a reminder of a way of lifewheretheparticipantsdidtheirbesttohurteachotherbydayanddinedtogetherbynight.”)

When,onMay5,1917,CambridgeMagazinecarriedasmalladfortheCambridgeBranch of the Union of Democratic Control—billed as “an Association for theexpressionofindependentopinionconcerningforeignpolicyandthesettlementaftertheWar”—readerswereinformedtheymightseek“furtherinformationfromG.H.Hardy,TrinityCollege.”Earlier,Hardyhadhelpedkeepatleastonemathematicianout of the war by building a case for the importance of his work to the nationalinterest.Andlater,in1918,hewouldprotestthefiring,becauseofhisantiwarviews,of an otherwise competent university librarian. “It is not, so far as I know, that

Dingwall hasdone anything,”Hardywouldwrite. “It is purely and simply that heholdsviewswhichareheldtobeobnoxious...Russell’scase(bitterlyasIresenttheCouncil’saction)wasquitedifferent:hehaddonethings,rightorwrong,butatanyrateperfectlytangibleanddefinite.”

Hardy, in short,wasbusy, busywithmatters that siphoned off time and energyfrom his relationship with Ramanujan. As closely as the two men often worked,Ramanujan was, inevitably, less the beacon of Hardy’s life than Hardy was ofRamanujan’s.

•••

ButevenifHardyweren’tsobusy,animmensepersonalandculturalgapstoodinthewayofrealintimacybetweenthetwomen.

Some years later, the English mathematician Alan Turing would complain ofHardy’s lack of even superficial friendliness. Itwas 1936,Hardywas spending theyearatPrinceton,andTuringfoundhim“verystandoffishorpossiblyshy.ImethiminMauricePryce’sroomsthedayIarrived,andhedidn’tsayawordtome.”Hardylooseneduplater,butasTuring’sbiographer,AndrewHodges,observes,“although‘friendly,’therelationshipwasnotonethatovercameagenerationandmultiplelayersofreserve”—thisthoughHardy“sawtheworldthroughsuchverysimilareyes.”

HardyandRamanujan,whosawtheworldthroughsuchverydissimilareyes,hadfarmoretoovercome.

Whatever itspsychosexualroots,Hardyhad loweredabouthimselfa lovely, lacyveilof personal defenses that was evenmore formidable than that of the ordinaryEnglishman’s.AnIndianadmirerofHardywouldremarkonhis“parentalsolicitude”towardRamanujan.Itwasanaptchoiceofwords;theirrelationshipwasmarkedbydistance,notcomradely intimacy.Tenyearsolderthanhe,Hardyremainedalwaysthe parent, a kind and obliging parent, perhaps, but forbidding, demanding, andremote,too.

If you stand in themiddle of London’sTrafalgar Square,which commemoratesAdmiralNelson’sdefeatof theFrench in1805, at thebaseof theNelsonColumnyou’llsee,upclose,sculpteddepictionsofhisvariousnavalcampaigns.Butasyouliftyourgazetothetopofthegreat167-foot-tallflutedcolumn,Nelsonhimselfisjustacloakedfigureinathree-corneredadmiral’scap,toohightomakeouteventhebareoutline of his features. One day, offered the hypothetical choice between beingcommemorated by such a statue, glorious but distant, and a lower, more

approachable one, Hardy would choose the former. He preferred the safety ofbarriers,privacy,distance.

Theultimatebarrier to their relationship,of course,was, asHardywouldwrite,that“RamanujanwasanIndian,andIsupposethatitisalwaysalittledifficultforanEnglishman and an Indian to understand one another properly.” It was Kipling’sversealloveragain:“EastisEastandWestisWestandneverthetwainshallmeet.”For the English, India was impenetrable, scarcely possible to understand. Butensuring that the cultural gap blocked a closer friendship between them was thatHardyscarcelytriedtobridgeit.

Hardy,C.P.Snowoncesaidofhim,“wouldhavebeenthefirsttodisclaimthathepossessed deep insight into any particular human being.” The untidy contours ofhumanpersonalitywerenothishometurf.Yes,hemightaskRamanujanabouthisknowledge of art or philosophy—the European kinds of things that one mightbrilliantly discuss with advantage at High Table. And he knew something ofRamanujan’stastesinliteratureandpolitics.Buthetalkedtohimscarcelyatallabouthisfamily,orSouthIndia,orthecastesystem,ortheHindugods.Hedidn’tprobe,he didn’t pry. “I rely, for the facts of Ramanujan’s life, on Seshu Iyer andRamachandra Rao,”Hardy began an account of him later—not facts he’d gleanedfromRamanujanhimself.

Even so safelyneutral amatter asRamanujan’smathematical influences in Indianeverprofited fromHardy’squestioning. “Here Imust admit that I am toblame,”Hardywouldwrite,“sincethereisagooddealwhichweshouldliketoknownowandwhichIcouldhavediscoveredquiteeasily. I sawRamanujanalmosteveryday,andcould have cleared upmost of the obscurity by a little cross-examination.” But heneverdid,neversteppedpast themathematicsof themoment, “hardlyaskedhimasinglequestionofthiskind....

Iamsorryaboutthisnow,butitdoesnotreallymatterverymuch,anditwasentirelynatural.[Ramanujan]was a mathematician anxious to get on with the job. And after all I too was a mathematician, and amathematicianmeetingRamanujanhadmore interesting things to think about than historical research. Itseemedridiculoustoworryabouthowhehadfoundthisorthatknowntheorem,whenhewasshowingmehalfadozennewonesalmosteveryday.

Mathematics, then, was the common ground of their relationship—perhaps theonlyoneotherthantheirmutualpleasureinhavingfoundoneanother.LikemanyanEnglishman, Hardy hid behind his reserve, disdaining any too-presumptuous anintrusionintoRamanujan’sprivatelife.HewasnotideallysuitedtodrawoutalonelyIndian, to easehis adjustment to an alien culture, to shelter him from theEnglishchill.

9.RAMANUJAN,MATHEMATICS,ANDGOD

Exemplifying the distance between the two men was Hardy’s refusal to viewRamanujan, in matters of religious belief, as any different from the usual run ofatheists,agnostics,andskepticsheknewamongCambridge intellectuals;or indeed,toseehismindasflavoredbytheEastatall.

•••

In the 1930s, E. T. Bell would remark that Ramanujan had broken the rules bywhichmathematiciansevaluatetheirown.“Whenatrulygreat[algorist,orformalist]liketheHinduRamanujanarrivesunexpectedlyoutofnowhere,evenexpertanalystshailhimasagift fromHeaven,”hewrote, creditinghimwith “all but supernaturalinsight”intohiddenconnectionsbetweenseeminglyunrelatedformulas.

Supernaturalinsight.AgiftfromHeaven.Itisuncannyhowoftenotherwisedoggedrationalistshave,overtheyears,turned

to the languageof the shaman and thepriest to convey somethingofRamanujan’sgifts. Hardy was the first Western mathematician to thoroughly examineRamanujan’snotebooks,butoverthenextseventy-fiveyearsmanyotherswould,too.Andrepeatedlytheyhavebeenreducedtoinchoateexpressionsofwonderandaweinthefaceofhispowers,havestumbledabout,gropingforwords, intryingtoconveythemysteryofRamanujan.

“Wehavenoideahowhedidthemarvelousthingshedid,whatledhimtothem,or anything else,” saysmathematician RichardAskey, a Ramanujan scholar at theUniversity of Wisconsin in Madison. Says Bruce Berndt, after years of workingthrough Ramanujan’s notebooks: “I still don’t understand it all. I may be able toprove it, but I don’t knowwhere it comes from and where it fits into the rest ofmathematics.” He adds at another point, “The enigma of Ramanujan’s creativeprocessisstillcoveredbyacurtainthathasbarelybeendrawn.”

SomethingofthissameenigmaticflavormakesitswayintoLittlewood’saccountofRamanujan’sworkwithpartitions.AttemptingtotracetheprogressofRamanujan’sthinking,heultimatelythrowsuphishands,frustratedandperplexed:

Thereis,indeed,atouchofrealmystery[here].Ifonlyweknew[theresultinadvance],wemightbeforced,byslowstages,tothecorrectformofΨq.ButwhywasRamanujansocertaintherewasone?Theoreticalinsight,to be the explanation, had to be of an order hardly to be credited. Yet it is hard to see what numericalinstances could have been available to suggest so strong a result. And unless the form ofΨq was knownalready,nonumericalevidencecouldsuggestanythingofthekind—thereseemsnoescape,atleast,fromtheconclusionthatthediscoveryofthecorrectformwasasinglestrokeofinsight.

Ramanujan,inthelanguageofthePolishemigrémathematicianMarkKac,wasa“magician,”ratherthanan“ordinarygenius.”

AnordinarygeniusisafellowthatyouandIwouldbejustasgoodas,ifwewereonlymanytimesbetter.Thereisnomysteryastohowhismindworks.Onceweunderstandwhathehasdone,wefeelcertainthatwe,too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in theorthogonal complement of where we are and the working of their minds is for all intents and purposesincomprehensible.Evenafterweunderstandwhattheyhavedone,theprocessbywhichtheyhavedoneitiscompletelydark.

Mystery,magic,anddark,hiddenworkingsinaccessibletoordinarythought;it isthesethatRamanujan’swork invariablyconjuresup,asenseofreasonbuttinghardupagainstitslimits.

Butatreason’slimitsdoessomethingelsetakeover?Dowehereflirtwithspiritualorsupernaturalforcesoutsideourunderstanding?

It’s an unlikely, anachronistic, even heretical notion today, with science andWesternrationalismeverywheretriumphant.ButthereisscantreasontodoubtthatRamanujanhimself thought so. South Indians of otherwise presumably rationalistbent—accountants, lawyers, mathematicians, professors—recall him as wholly ateaseinthespiritualworldtowhichhismotherandgrandmotherintroducedhimasachild,andreadytoseeinitasourceofmathematicalinspiration.

T. K. Rajagopolan, a former accountant general of Madras, would tell ofRamanujan’sinsistencethatafterseeingindreamsthedropsofbloodthat,accordingto tradition, heralded the presence of the godNarasimha, themale consort of thegoddess Namagiri, “scrolls containing the most complicated mathematics used tounfoldbeforehiseyes.”

R.RadhakrishnaIyer,aclassmateatPachaiyappa’sCollege,recalledonedayaskingRamanujanabouthisresearchonlytohavehimreply,inRadhakrishna’swords,“thatLordNarasimhahadappearedtohiminadreamandtoldhimthatthetimehadnotcomeformakingpublicthefruitsofhisresearch.”

AnditwasonadayaboutayearbeforeheleftforEngland,in1912orearly1913,thatRamanujan,showinghisworktomathematicsprofessorR.Srinivasan,madethestatementinwhichhepicturedequationsasproductsofthemindofGod.

Ramanujan’sfriendsinvariablynotedhisinterestinastrologyandhispenchantforinterpretingdreams.HisboyhoodfriendAnantharamanwouldtellhowonce,whenhisbrotherhadadream,Ramanujanreaditasforetellingadeathinthestreetbehindtheirhouse.(Therewas.)

K.Gopalachary, a friendofRamanujan fromMadrasdays, said thatRamanujanevenattributedhisearly interest inmathematics toadream—adreamabout,ofall

things,astreetpeddlerhawkingpills.Ramanujan’sbeliefinhiddenforcesandthepowersofthesupernaturalwasnever,

at leastback inIndia, somethingaboutwhichhe felt theneedtoapologizeorkeepquiet.Itwasnomerematterofprivateconvictiontohim,consignedtotheperipheryofhis life,or somepet theoryaboutwhichhemerely liked to speculate.Timeandagain,heactedonit.

Thus, in the hectic year before he left for England, he found time to prepareastrological projections, fixing auspicious times for religious functions for relativesandfriends.Hewasconvinced, fromstudyingthe linesonhispalm, thathewoulddie before he was thirty-five, and told his friends as much. Anantharaman wouldrecord that he attributed to a temple nearTrichinopoly the power to curementalailmentsandadvisedsuffererstogothere.AndwhilestillatPachaiyappa’sCollege,Ramanujandreamedofafamilywhosechildwasneardeath,wenttoitsparents,andin obedience to the dream, asked them tomove the child to another house. “Thedeathofapersoncanoccuronlyinacertainspace-time-junctionpoint,”hesaid.

One evening before he left for England, Ramanujan returned to his house inTriplicane in an electric streetcar (which had been introduced toMadras in 1895,years before anywhere else in India).The driver, enjoying himself, was alternatingsudden stopswith sharp accelerations, jerking his passengers around unmercifully.SaidRamanujan, sittingwith a friend on the long bench behind the driver: “Thatmanimagineshehasthepowertogosloworfastathispleasure.Heforgetsthathegetsthepowerthroughthecurrent that flows in theoverheadwires. . . .That,”hesaid,invokingtheHindutermfortheillusion,orvanity,thoughttodeflecthumansfromGod,“isthewaymayaworksinthisworld.”

Ramanujan,then,wassteepedinthebeliefsystemofhisculture.Somany stories,fromsomanyquarters,takingsomanyforms,oversomanyyears,adduptonothinglike the skeptical rationalist Hardy imagined, nothing like a man mechanicallyadheringtothequaintorthodoxiesofhisfamilyandhiscaste.

Yet that is indeed what Hardy insisted. “I do not believe in the immemorialwisdomoftheEast,”hewouldonedaydeclareinalecturedevotedtoRamanujan,

and the picture which I want to present to you is that of a man who had his peculiarities like otherdistinguishedmen,butamaninwhosesocietyonecouldtakepleasure,withwhomonecoulddrinkteaanddiscusspoliticsormathematics;thepictureinshort,notofawonderfromtheEast,oraninspiredidiot,orapsychologicalfreak,butofarationalhumanbeingwhohappenedtobeagreatmathematician.

Ramanujan’s religion, Hardy insisted, “was a matter of observance and not ofintellectual conviction, and I rememberwell him tellingme (much tomy surprise)thatallreligionsseemedtohimmoreorlessequallytrue.”

Hewassure,Hardywroteonanotheroccasion, that “Ramanujanwasnomysticandthatreligion,exceptinastrictlymaterialsense,playednoimportantpartinhislife.”

Foryears,C.P.SnowwouldkeepalargephotoofHardyinhisotherwisealmostbare study andotherwise respectedhim in everyway.But fromHardy’s insistencethat Ramanujan “did not believe much in theological doctrine, except for a vaguepantheistic benevolence, any more than Hardy did himself,” Snow took care todistancehimself.“Inthisrespect,”hewrote,“Ishouldnottrusthisinsightfar.”

Still, taking their cue fromHardy,mostWestern observers, and some Indians,have wholly detached Ramanujan’s mystical streak from his mysterious ability toforgenewmathematical linkages.Hailing theone, they’vedismissed theother, andwritten off his credulousness—hisweakness for astrology, his arrant superstitions,his devotion to Namagiri—as an unfortunate eccentricity peripheral to hismathematicalinventivenessbutwhichhassomehowtobestomachedforthesakeofit.

ForRamanujan,though,thesplitwasnotsosharp.ThemanwhomHardymetinEngland inApril 1914was aman still of South India,who had grown up on theIndian gods and the relaxed fluidity ofHindu belief. In him, the natural and thesupernatural,JacobiandNamagiri,NumberandGod,foundacommonhome,stoodinsomethinglikeaneasyintimacy.

DidRamanujan’s religiousbeliefbestowonhimhismathematical gift?Certainlynot, since otherwise all those with kindred beliefs would share it. Nor did henecessarilygainmathematicalinsightsthroughanythinglikethemeanshethoughthedid and towhich he assigned credit.Nor, to state the obvious, does the fact thatRamanujanbelievedwhathebelievedmeanthatwhathebelievedisso.Ontheotherhand,inhowthemysticalstreak inhimsatsidebyside,apparentlyatperfectease,withrawmathematicalabilitymay testify toapeculiar flexibilityofmind,a specialreceptivitytolooseconceptuallinkagesandtenuousassociations.

Despitehisemphasisonrigor,G.H.Hardywasnotblindtothevirtuesofvague,intuitivementalprocesses inmathematics.Bromwich,hewouldwrite, forexample,“wouldhavehad ahappier life, andbeen a greatermathematician, if hismindhadworkedwithlessprecision.Asitwas,eventhebestofhisworkisalittlewantinginimagination. For mastery of technique in a wide variety of subjects, it would bedifficult to find his superior, but he lacked the power of ‘thinking vaguely.’ ” Andsomesuchextraordinarilydevelopedabilityto“thinkvaguely”mayhavebeenamongRamanujan’sspecialgifts.

Ramanujan’s belief in the Hindu gods, it stands repeating, did not explain hismathematical genius.Buthisopenness to supernatural influenceshintedat amindendowedwithslippery, flexible,andelasticnotionsofcauseandeffectthat lefthimreceptive towhat those equippedwithmorepurely logical gifts couldnot see; thatfound union in what others saw as unrelated; that embraced before prematurelydismissing.Hiswasamind,perhaps,whosecriticalfacultywasweakcomparedtoitscreativeandsynthetical.

It is thecritical faculty,ofcourse,thatkeepsmostpeoplesafe—keepsthemfromrashly embracing foolishness and falsehood. InRamanujan, it hadnever developedquite as fully as the creative—thus giving him the credulousness, the appealinginnocence, upon which all who knew him unfailingly remarked. Without thatprotectivescreen,asitwere,heriskedfallingpreytothesillyandthefalse—asmany,over the years,would viewhis belief inpalmistry, astrology, and all the rest of theesotericatowhichhesubscribed.

Andyet,withoutthatscreendidhethusremainmoreopentothemathematicalLight?

•••

Hardywasnot,tosaytheleast,athomeinthismentaluniverse.“Ihavealwaysthoughtofamathematicianas inthe first instanceanobserver”he

said in a Cambridge lecture in 1928, “a man who gazes at a distant range ofmountainsandnotesdownhisobservations.”

Hisobjectissimplytodistinguishclearlyandnotifytoothersasmanydifferentpeaksashecan.Therearesomepeakswhichhecandistinguisheasily,whileothersarelessclear.HeseesAsharply,whileofBhecanobtainonlytransitoryglimpses.AtlasthemakesoutaridgewhichleadsfromA,andfollowingittoitsendhediscoversthatitculminatesinB.Bisnowfixedinhisvision,andfromthispointhecanproceedtofurtherdiscoveries.Inothercasesperhapshecandistinguisharidgewhichvanishesinthedistances,andconjecturesthatitleadstoapeakinthecloudsorbelowthehorizon.

Butabout theveiledprocessbywhichonemightcometodiscern thosepeaks inthe first place,Hardy remained largely silent. Indeed, rarely in a long life of doingmathematicsandwritingaboutitdidhechoosetodiscussthecreativeprocessbehindit,noteven inhisMathematician’sApology, otherwise so richwithhis insights intothemathematician’sworld.Always itwas theproduct of that process, the theoremitself,thatinterestedhim.Hemightwanttoestablish,throughproof,whetherornotitwastrue.Or,perhaps,toevaluateitsplaceinthemathematicalfirmament;intheApology,forexample,hespeaksoftheoremsalmostasanartcriticmighttheworksinagalleryshow,evaluatingthembythisorthatyardstickofmathematicalbeauty.

Almosttheonlytimehedidwriteaboutmathematicalcreativitycamemanyyearslater, near the end of his life, in a review of a book by Jacques Hadamard, ThePsychologyofInventionintheMathematicalField.Whatphilosophersorpoetscouldsayaboutthecreativeprocess inmathematics,Hardyfelt,wasnexttonothing.ButHadamardwasamathematician,andagreatone.Hehadrunupthehighestscoreever recorded on the entrance examination to the Ecole Polytechnique, France’spremierschoolofscience.Hehad,withtheBelgianCharlesJ.delaVallée-Poussin,proved the prime number theorem. What he had to say about “invention in themathematicalfield”wasworthlisteningto.

HardyagreedwithHadamardthat

unconscious activity often plays a decisive part in discovery; that periods of ineffective effort are oftenfollowed, after intervals of rest or distraction, by moments of sudden illumination; that these flashes ofinspirationareexplicableonlyastheresultofactivitiesofwhichtheagenthasbeenunaware—theevidenceforallthisseemsoverwhelming.

Butbeyondthis,Hardyseemedtosayineverywordofhisreview,hethoughtitbestnot to venture. Too soon are we thrust upon the Unknowable; it was better tomeekly sidestep the issue than mire our explanations in foolishness. Indeed, helaudedHadamardforbeing“wiselydiffidentandtentativeinhisconclusions.”

Howunconscious activities “are related to those of amorenormal [sic] kind, tofullyconsciousworkorreflectiononthefringeofconsciousness,howtheyfunctionandwhat is the proper language in which to describe them, are terribly difficultquestions,”Hardywrote.WhenHadamarddidofferatentativeexplanationforwhatHardycalled“themostpuzzlingquestion”ofall—howyouseizeonefromamongthewelter of ideas your unconscious serves up—Hardymade explicit how he felt: “Itmaybeso,”hewrote,“thoughIcannotsaythatIfinditveryconvincing;butIamnopsychologist, and my distaste for all forms of mysticism may be prejudicing meunduly.”

Atonepoint,Hardyadmittedthat,quotingHadamard,“ ‘theunconsciousisnotmerelyautomatic, ithas tactanddelicacy,’ even that ‘itknowsbetterhowtodivinethantheconsciousself,sinceitsucceedswherethathasfailed.’ButIdonotlikethiskindoflanguage,”Hardywenton;tohimitvergedonnonsense.

“Itissomethingofarelief,”hewroteinthemostrevealingadmissionofall,“topasstothelaterchapters,whicharefullofinterestingandlesscontroversialmatter.”

Thissoft,ineffableregionofunconsciousprocesses,ofvague,hazyconnectionsandsuddenly appearing insights, of loose ties and nameless links, was not, Hardy’sremarks suggest, anyplacewherehehimself liked todwell.Hewasuncomfortablediscussing it, uncomfortable thinking about it. That mathematics was a “creative”

activitywasnotthequestion;itwasamongthemostcreative.Butastoitssource—therehedidn’tcaretodelve.Certainly,then,anyresorttoEasternmumbojumbotoexplain Ramanujan’smathematical gifts would not have drawn from him a warmreception.

•••

ButitwasHardy,thededicatedatheist,whorepresentedtheextremeposition,andRamanujanwhowasmoreinlinewiththelargebodyofbeliefandconviction,withinthe Western tradition as well as that of the East, that perceived links betweencreativityandintuitionontheonehandandspiritualforcesontheother.

TheGreeks, for example, invoked themuses—goddesses towhompoets lookedforinspiration.BoththeEnglishandFrenchlanguagesspeakof“divining”thetruth.Hadamardhimselfnoted that the sheer inaccessibilityofunconscious thoughthad,tothinkersoverthecenturies,endoweditwithhigherpowers.

ThatunconsciousnessmaybesomethingnotexclusivelyoriginatinginourselvesandevenparticipatinginDivinity seems already to have been admitted by Aristotle. In Leibniz’s opinion, it sets the man incommunicationwiththewholeuniverse,inwhichnothingcouldoccurwithoutitsrepercussionineachofus;andsomethinganalogousistobefoundinSchelling;again,DivinityisinvokedbyFichte;etc.

Evenmorerecently,awholephilosophicaldoctrinehasbeenbuilton thatprinciple in the firstplacebyMyers,thenbyWilliamJameshimself...,[inwhich]theunconsciouswouldsetmaninconnectionwithaworldotherthantheonewhichisaccessibletooursensesandwithsomekindsofspiritualbeings.

Just as India was not alone in attributing creative insights to divine influence,Ramanujanwasnotaloneamongmathematiciansinholdingstrongreligiousbeliefs.Newton was an unquestioning believer, felt humility in the face of the universe’swonders,studiedtheologyonhisown.Euler,inE.T.Bell’swords,“neverdiscardedaparticle of hisCalvinistic faith,” and grewmore religious as he grewolder.CauchywasforevertryingtoconvertothermathematicianstoRomanCatholicism.Hermitehadastrongmysticalbent.EvenDescartes,thatfatherofEnlightenmentrationality,answeredthecallofthespirit:“Hisreligiousbeliefswereunaffectedlysimpleinspiteof his rational skepticism,” writes Bell. “He compared his religion, indeed, to thenursefromwhomhehadreceivedit,anddeclaredthathefounditascomfortingtoleanupononeasontheother.”(AddsBell,wisely:“Arationalmindissometimesthequeerestmixtureofrationalityandirrationalityonearth.”)

Evenamongmathematiciansnotreligiouslyminded,onefindsevidenceofatleastrespectfulallusiontothedarkterrainbetweenfaithandreason.Gauss,forexample,onceprovedatheorem,ashewrote,“notbydintofpainfuleffortbutsotospeakbythe grace of God.” James Hopwood Jeans, Hardy’s Cambridge classmate and a

famousappliedmathematician,wrote: “From the intrinsic evidenceofhis creation,theGreatArchitectoftheUniversenowbeginstoappearasapuremathematician.”EvenLittlewood,commentingonan incident inwhich“mypencilwrotedown”thesolution to a particularly bedeviling problem, couldwrite, “Ifwemay reject divinebounty, ithappenedexactlyas ifmysubconsciousnessknewthethingall thetime.”Littlewood’s,ofcourse,wastheusualsafe,ironicCambridgeskepticism,perhapsnomore than a stylistic device—just as the other statementsmaybe seen as nomorethanmetaphor.Yeteachcontainedthebarestbreathofambivalenceorhumility inthefaceofthemysteriousoriginsofhumancreativity.

Hardy, though,didnot admit to suchambivalence.Forhim, thewhole spiritualrealm was just so much bunkum. He knew—this was his faith—that whereverRamanujan’s genius came from, therewas something straightforward to explain it.Hewouldwrite:

IhaveoftenbeenaskedwhetherRamanujanhadanyspecialsecret;whetherhismethodsdifferedinkindfromthoseofothermathematicians;whethertherewasanythingreallyabnormal inhismodeofthought.Icannotanswerthesequestionswithanyconfidenceorconviction;butIdonotbelieveit.Mybeliefisthatallmathematiciansthink,atbottom,inthesamekindofway,andthatRamanujanwasnoexception.

Ramanujan’smathematics,hewassaying,wastheproductofthereasonedworkingofareasoningmind,andnothingmoreneededtobesaid.

Someonewould later observe that “Hardy’s deep reverence formathematics andforallthingsofthemindwaspreciselyofthesamekindasimpelsotherpeopletotheworshipofGod;theonlyenigmaaboutHardywasthatthisneverseemedtooccurtohim.”Andatleastforpublicconsumption,itneverdid.HadRamanujanscouredtheBritishIsles,hecouldhavefoundnoonelesssympathetictohisspiritualside,noonewho,inthisonerealm,couldappreciatehimless.

In defending his belief in Ramanujan’s lack of genuine religious feeling, Hardywouldarguethat“ifastrictBrahminlikeRamanujantoldme,ashecertainlydid,thathehadnodefinitebeliefs,thenitis100to1thathemeantwhathesaid.”NomatterthatRamanujan’sstatementwas,inthecontextoftherelaxedtoleranceofHinduism,by no means incompatible with strong religious feeling. Hardy concludes that “ifRamanujan’s friends assumed that he accepted the conventional doctrines of[Hinduism]...andhedidnotdisillusionthem,hewaspracticingaquiteharmless,andprobablynecessary,economyoftruth.”

Aharmlesseconomyof truthwas indeedwhatRamanujanwasperpetratingbut,almostcertainly,itwasHardywhowasitsobject.Ramanujanprobablywasn’tlonginEngland before Hardy let him know, perhaps without even realizing it, thatinvocations ofNamagiri were not apt to be well received. Faced with aman once

described as “an atheist evangelical,” and hardlywishing to provoke his benefactorand friend onwhatwas such touchy ground,Ramanujan simply never revealed tohimtherichnessandextentofhisinnerspirituallife.

Andthatwastheproblem:withHardy,Ramanujancouldnotlethishairdown—hadtodissemble,couldnotbehimself.Thereremainedbetweenthetwomenagreat,unbridgeable gap.Even afterworkingwith him for several years, the conclusion isinescapable,HardyneverreallyknewRamanujan—andthuscouldbenorealbufferagainsttheprofoundlonelinessRamanujanfeltinEngland.

10.SINGULARITIESATX=1

Three-quarters of a century later, in 1989, the black dean of an all-blackwomen’scollegeinGeorgiarecalledherstintasanadministratoratamostlywhitecollegeupnorth.“TheblackstudentsIknewatHaverfordmayhavedonewell,”shesaid,“butInevergotthefeelingtheywerehappy”—becausetheycouldn’tbethemselves,couldn’tbeblack.AndjustsuchasplitappliedtoRamanujaninCambridge.Hehad,underHardy’sguidinghand,donewell;buthewasnothappy.

Hemay not have even realized it himself; Ramanujan was not someone closelyattunedtohisownfeelings.Itwouldhavebeenperfectlypossibleforhimtoexultinmathematics, to derive enormous satisfaction from his intellectual dialogue withHardy,tobetoalloutwardappearancescontent—indeed,tocallhimselfcontent—andyet,forotherneedshecouldscarcelynameandonlydimlysense,feelincomplete.

Seasoned travelers invariably note how adjusting to another culture demandsflexibility,awillingnesstoassumethecolorationofthenewcountry.Theexperiencesof expatriates, immigrants, and visiting students all attest to the same thing. Butflexibility was one trait Ramanujan did not possess in great measure. It tookenormousenergytocastofftheoldandtakeonthenew,nottomentionthewilltodoso.AndRamanujanputallhis energy intomathematics.Hedidnot flexas therawEnglishwindsblewaroundhim.

Sohebroke.Firstinbody,theninmind.

•••

In1916,Ramanujan’stutorBarneshadwrittentheUniversityofMadrasthat,givenRamanujan’sachievements, itseemedlikelyhewouldbeelectedaFellowofTrinityCollege the following October. But October 1917 came and went withoutRamanujan’selection.Atthetime,thecollegewaswrackedwithdissensionovertheBertrandRussellaffair,andRamanujan’schampion,Hardy,wassquarelyintheout-

of-favorcamp.Then,too,itseemscertain,inlightoffutureevents,simpleracismwasafactor;Ramanujan,afterall,wasablackman.

The disappointment left Ramanujan’s mood darker, the whole structure of hispersonalitythatmuchshakier.

ItwasaroundthistimethatheenteredMatlock,whichcouldhardlyhaveliftedhisspirits.Englishsanatoriumsweretypicallypresidedoverbystern,patriarchalfigures,strictdisciplinarianswhoruledwithanironhand.AndMatlockwasinthemold.AfriendwouldlaterrecallthatRamanujanwas“coweddownbyDr.Ram,whoseemstohavetoldhim,‘Aslongasyouareapatientandnotwellyouarenotfreeandthedoctorhascontroloveryourmovements.’”

Matlockwas typical of English sanatoriums in at least one other respect: itwasout-of-the-waygeographically.“Thereismuchevidencetosuggest,”writesBryderinherstudyofBritishtuberculosiscare,“thatinmatesofsanatoriafeltestrangedfromtheoutsideworld.Notonlydidtheirgeographicalisolationmakevisitsdifficultandtherefore infrequent, but social attitudes accentuated that isolation.” At Matlock,located about 150miles northwest ofLondon, in thePeakDistrict ofDerbyshire,Ramanujan enjoyed no steady stream of visitors.Getting there was not easy. Thefollowingyear,afriendwhodidmakethetrip,A.S.Ramalingam,wouldwriteofthe“cold weary journey” on the night train he’d had to endure. Hardy was almostcertainly thinking of the Matlock period when he wrote, a few months afterRamalingam’s visit, of Ramanujan’s “long illness and the spells of comparativesolitude”influencinghismentalstate.

AtMatlock,Ramanujanwas cold andmiserablemuch of the time.TheTrinityfellowshiprejection rankled.Andhewas too sick tobeproductivemathematically,whichalsodistressedhim.Thedoctorsdidn’tsuithim.Hecouldn’tgetfoodtoeatandthefoodhecouldgethedidn’tlike;sometimeshecravedthehotdosai,akindofpancake,thatAnantharaman’smotherwouldcookupforhimbackinKumbakonam.Finally, he was getting little in the way of nurturing and emotional support fromhome.Andhewasn’tgettingitfromHardy,either.

Hegrewprofoundlydepressed.Atonepoint,hehadnightmaresinwhichhewasvisited by images of his own abdomen as a kind of mathematical appendage with“singularities,”pointsinspacemarkedbyindefinablemathematicalsurgeslikethoseheandHardyhadexploredintheirpartitionswork.Intensepainmightshowupatx=1,halfasmuchpainatx=−1,andsoon.Thenightmaresrecurred.

Ramanujan was at a low ebb, balanced precariously on the edge of mentalinstability.

•••

Undeterredby theTrinity rebuffandhoping toboostRamanujan’smorale,Hardysetabouttryingtogethisfriendtherecognitionhefelthedeserved.OnDecember6,1917,RamanujanwaselectedtotheLondonMathematicalSociety.Then,twoweekslater,onDecember18,Hardyandelevenothermathematicians—HobsonandBakerwere among them, as were Bromwich, Littlewood, Forsyth, and Alfred NorthWhitehead,BertrandRussell’scollaboratoronPrincipiaMathematica—togetherputhim up for an honor more esteemed by far than any fellowship of a Cambridgecollege:theysignedtheCertificateofaCandidateforElectionthatnominatedhimtobecomeaFellowoftheRoyalSociety.

TheRoyal Societywas Britain’s preeminent scientific body, going back to 1660whenChristopherWrenandRobertBoyleshelped found it.Therewere, at aboutthetimeHardyputupRamanujan,39foreignmembers,includingtheRussianIvanPavlov, the American Albert Michelson (of Michelson-Morley experiment fame),and6otherNobelPrizewinners.TheRoyalSocietycountedinall464membersinphysics, chemistry, biology, mathematics, and every branch of science. Being anF.R.S.meant that forevermore those three little letterswouldbeappended toyourname,appearonyourownscientificpapers,andonlettersaddressedtoyou.Itwasthe ultimatemark of scientific distinction. Younger scientists lusted after it, olderscientistslamentedtheirlackofit.

C.P.SnowtellsthestoryofH.G.Wells,authorofWaroftheWorlds,TheTimeMachine, many other scientific romances, and serious works of history and socialcommentas well. But while famous and the recipient of numerous honors, “therewas,”asSnowwouldwriteofhim,“preciselyonehonorhelongedfor.Itwentbacktohisyouth,whenheday-dreamedaboutbeingascientist.HewantedtobeanF.R.S.Andthisdesire,insteadofbecomingweakerashegotolder,becamemoreobsessive.”Henever received it, because thoughhe had studied science in school and had, inSnow’swords,“beentheprophetoftwentiethcenturysciencemoreeffectively thananymanalive,”hehadnothimselfactuallymadeoriginalresearchcontributions.

It was this signal honor Hardy sought for his friend and for which he set outRamanujan’s“Qualifications”inhisdistinctivecalligraphichand:

Distinguishedasapuremathematician,particularlyforhisinvestigationsinellipticfunctionsandthetheoryofnumbers.Authorofthefollowingpapersamongstothers:“ModularEquationsandApproximationstoPi,”QuarterlyJournal,vol.45;“NewExpressionsforRiemann’sFunctionsξ(s)andΞ(t),” ibid,vol.46; “HighlyCompositeNumbers,”Proc.LondonMath.Soc, vol. 14 . . . Joint authorwithG.H.Hardy, F.R.S., of thefollowingpapers: “Uneformule asymptotiquepour lenombredespartitionsdeη,”ComptesRendus, 2 Jan.1917...

Thus it continued, listingRamanujan’s papers, and endingwith perhaps themostimportant of all—“Asymptotic Formulae in Combinatory Analysis,” the bigpartitions paper still awaiting publication in the Proceedings of the LondonMathematicalSociety.

On January 24, 1918, the names of Ramanujan and 103 other candidates werereadoutat ameetingof the society. Ifpast experienceapplied,only a fewof themwouldbeelected.

There was no question inHardy’s mind, or Littlewood’s, or anyone else’s, thatRamanujanmeritedthehonor.Still,fewcandidatesmadeitthefirsttimeout,andbynormalpractice his nominationwas premature.Hardy had been thirty-three yearsold when elected in 1910. Littlewood himself had made it only the previousFebruary, also at age thirty-three—more than a decade, and dozens of notablemathematicalpapers,beyondhisearlygloryasaSeniorWrangler.Ramanujan,stilltwenty-nineatthetimeofhisnomination,hadcontributedtoEuropeanmathematicsforjustafewyearsandstillhadamodestpublicationrecord,atleastinnumber.

ButHardy’s concern forRamanujan’s healthmovedhim topresshis claimwithunusualurgency.J.J.Thomson,discovereroftheelectron,winneroftheNobelPrizein 1906, and then president of the Royal Society, had asked him to outline thecircumstancessurroundingRamanujan’scandidature.“IfhehadnotbeenillIwouldhavedeferredputtinghimupayearor so,”Hardyadmitted: “not that there isanyquestion of the strength of his claim, but merely to let things take their ordinarycourse.Asitis,Ifeltnotimemustbelost.”

Ramanujanmightnotmakeittothenextelection,hewassaying,andthesocietywouldhavetoliveforeverwithitsfailuretohonorhim.“Iamnervousabouttryingtorushhim,”Hardycontinued,

andIamawarethatforthetimebeingIamnotanidealsupporter.AndIrealizethattheR.S.hasmanyotherthings to consider.But there is nodoubt that (especially after his disappointment in theFellowships) anystrikingrecognitionnowmightbeatremendousthingforhim.Itwouldmakehimfeelthathewasasuccess,andthatitwasworthwhilegoingontrying.ItisthismuchmorethanthefearoftheR.S.losinghimentirelywhichseemstomeimportant.

Iwriteonthehypothesisthathisclaimsaresuchas,inthelongruninanycase,couldnotbedenied.Thisistomequiteobvious.Thereisanabsolutegulfbetweenhimandallothermathematicalcandidates.

Hardy’slettersuggestedanotherreasonforRamanujan’smentalanguish:itwouldmakehim feel thathewasa success.ForallHardy’sencouragement,Ramanujanhadcometounderstandhowgreatapricehe’dpaidforhis isolationinIndia.Fromthemomenthe’dsteppedoff theboathe’dbeenconfronted, throughHardy,withhowmuchhehadn’tknown, learned,orappreciatedbefore—with function theory,with

Cauchy’s integral theorem,with somuch else thatwas common knowledge in theWestandwhich,byrights,heshouldhavelearnedtenyearsbefore.

“It is perhaps useless to speculate as to his history had he been introduced tomodernideasandmethodsatsixteeninsteadoftwenty-six,”HardywouldwriteafterRamanujan’sdeath.“Itisnotextravaganttosupposethathemighthavebecomethegreatestmathematicianofhistime.”

OfRamanujan’swork in India,Hardywould observe that itwas inevitable thatmuchhadbeenanticipated,sinceRamanujanlaboredunder“animpossiblehandicap,a poor and solitary Hindu pitting his brains against the accumulated wisdom ofEurope.”

And this was what he had to say of the period between when Ramanujan leftGovernmentCollegeandwhenhejoinedtheMadrasPortTrust:“Theyearsbetweeneighteen and twenty-five are the critical years in amathematician’s career, and thedamage had been done. Ramanujan’s genius never had again its chance of fulldevelopment.”

Givenhisrelentlesshonesty,couldHardyhavefailedtoconveysuchsentimentstoRamanujan?AndcouldRamanujanhavefailedtohavebeenwoundedbythem?

Ramanujanmay have known nothing ofHardy’s efforts to have him named anF.R.S.;butifhedidknowofthem,bynowhedoubtlessassumedtheywouldcometonothing, could lead only to the kind of humiliating blow he had suffered whenTrinityturnedhimdowninOctober.

•••

OnFebruary11,endingaperiodofsilencethathadextendedovermorethanayear,RamanujanwrotehisfamilyinIndia.Aboutthissametime,perhapsa littleearlier,duringaperiodwhenhewasbrieflyawayfromMatlock,hetriedtokillhimself.

Today in London, you can buy T-shirts, posters, mugs, and other souvenirsemblazoned with the great labyrinthine grid that represents the LondonUnderground,with itsdozenormoredistinct lines andhundredsof stations.Oneday inJanuaryorFebruaryof1918, itwasatastationsomewhere inthisnetwork,thensmallerandnewer,thatRamanujanthrewhimselfontothetracksinfrontofanapproachingtrain.

Whathappenednextwouldbeeasyenoughtoreadasamiracle.Aguardspottedhimdo it andpulled a switch,bringing the train screeching to a stop a few feet infrontofhim.Ramanujanwasalive,thoughbloodiedenoughtoleavehisshinsdeeplyscarred.

He was arrested and hauled off to Scotland Yard. Called to the scene, Hardy,marshalingallhischarmandacademicstature,madeashowofhowthere,beforethepolice,stoodthegreatMr.SrinivasaRamanujan,aFellowoftheRoyalSociety,andhowaFellowoftheRoyalSocietysimplycouldnotbearrested.

In fact,Ramanujanwasnot anF.R.S.Hewouldhardlyhavebeen immune fromarrestinanycase,andthepolicewerenotfooledforaminute.Buttheyinvestigated,learnedRamanujanwasindeedreputedtobeaneminentmathematician,anddecidedto lethimgo. “We inScotlandYarddidnotwant to spoil [his] life,” theofficer inchargeofthecasesaidlater.

Just what triggered Ramanujan’s desperate bound onto the tracks doesn’t comedownto us.Certainly, though, if themere refusal of his dinner guests to accept athirdhelpingcouldfomentsuchastormofshameinhimthathehadtogetupandleave,deeperhumiliationmightsparkactionmoreprecipitousstill.Andin1917hehadcertainlyexperiencedhisshareofit.RejectedbyTrinity.Seeminglyabandonedbyhiswife.Leftsickanddependentinthesanatorium,helplesseventocommandthefood he wanted.Unable to produce the work he felt his friends expected of him.Confrontedbytheknowledgethatmuchofhispastworkhadbeenrediscoveryand,viewedblacklyenough,awasteoftime.

Andfor the feelingsall thisstirred inhim, therewasnosafetyvalve,at leastnotamonghisEnglishfriends.Littlewoodwasgone.SowasNeville;he’dlosthisTrinityfellowshipin1917,probablyduetohisantiwarviews,andwasoffinLondon.Hardy,meanwhile,wasnoonewithwhomhefeltrelaxedenoughtobarehissoul.

Morethanlikely,hedidnotdeliberatelysetouttokillhimself.Itwasdoubtlessarashact,spurredbysomenewhumiliation,unknowntoustoday,piledontopofallthe others. Once again, Ramanujan had acted impetuously when overcome withshame.

•••

Late in February, back at Matlock, Ramanujan learned of his election, on theeighteenth,totheCambridgePhilosophicalSociety.Thiswasnotsomuchanhonorasjustonesmalllegitimizationofhisstatureinthescientificworld.

About tendays later,hereceiveda telegramfromHardy, sent fromPiccadilly inLondon.

Hereadthetelegramonce.Hereaditasecondtime.

Hereaditagain,thewordsstillcongealing inhismind,makingnosense:Hardywas advising him of what he already knew, that he had been elected to thePhilosophical Society. Which was fine, of course. But he already knew as much.WhatwasHardy’spointinwiringhim?

Oncemorehereadthetelegram,andthistime,finally,anewwordexplodedfromthepage.ItwasnotthePhilosophicalSocietytowhichHardywasadvisinghimhehadbeennamedafellow,buttheRoyalSociety.

Of104candidatesforelectionthatyear,hewas1ofjust15elected.“Mywordsarenotadequatetoexpressmythankstoyou,”hewroteHardy.“Ididnotevendreamofthepossibilityofmyelection.”InMay,hewouldbecomeS.Ramanujan,F.R.S.

India, which soon heard the news, was thrilled. On March 22, the MadrasmembersoftheIndianMathematicalSocietywroteHardyinthanks“fortheaidandguidanceyouhavebeenextendingtoMr.S.Ramanujaninhiswork.”Inapostscript,P.V.SeshuIyeraddedhispersonalthanksforthe“careyouhavebeenbestowingonhimduringthesemonthswhenhishealthhasnotbeengood.”

By May, Ramanujan’s health was still poor. He wrote the Royal Society fromMatlockontheseventeenththathewas, justthen,toosicktotravel toLondonforhisformaladmissionintothesociety.

•••

Aroundthistime,heheardfromA.S.Ramalingam,theSouthIndianengineerhe’dmet,justofftheboatfromIndia,attheCromwellRoadreceptioncenter.Soonafterthedeclarationofwar,Ramalingamhadjoinedthearmyandinearly1916hadbegunworkatashipyardinJarrow,inthenorthofEnglandalmostinScotland.Since1914,thetwomenhadnotbeenintouch.Butnow,perhapslearningoftheRoyalSocietyelection, hewroteRamanujan throughHardy.Hearing nothing in reply, hewroteHardydirectly,learnedofRamanujan’scondition,andwroteRamanujanatMatlock.Evertheengineer,Ramalingamwasnothingifnotpersevering.

So,apparently,washiswholefamily.Inoneofhislettershomehehadmentionedfoodrationing.Thatwasallittook.Fortwomonths,theyhadbeenshoweringhimwithSouth Indian food.Finally, he cabledhome, “Stop sending food.”Bynowhehadparceluponparcelofit,allpiledup.Maybe,hewroteRamanujannow,hemightliketoshareinthebooty?

Ramanujanwrotebackaskingforsomeghee,thespecialbutter,clarifiedbyboilingto resemble oil, and some spicyMadras-style foods. These Ramalingam promptlysent.One thing led to another, and soon, on Sunday, June 16, Ramalingam was

visitinghimatMatlock,wherehestayeduntilafterlunchonTuesday.Forthreedaystheytalked—ofthewar,ofChristianmissions,ofconditionsinIndia,andofmuchelse. Ramalingam had heard Ramanujan’s mental state was impaired, but saw noevidenceforitnow.

His physical state, though,was another story. “Iwas shocked and horrified,” hewroteHardyafterreturningtoJarrowafewdayslater,“tofindhiminthethin,weakand emaciated state I found him in.” His illness, the suicide attempt, his foodproblemsatMatlock,hadtakentheirtoll.

Ramalingam’s letter went on for twelve large lined pages documentingRamanujan’s condition, recording the views of his doctors, detailing his mealschedule, and suggesting improvements to his care. But in all its length, there isnothing like “As you are doubtless aware . . . ,” or “AsRamanujanmay have toldyou...,”or“AsyoumayhavelearnedfromRamanujanwhenyousawhimlast....”Nothing he’d learned from Ramanujan during the three days gave Ramalingamreasontoabridgehis letter intheslightest,or to intimatethatanything in itmightalreadybeknowntoHardy.Rather,itreadslikeamiddlemanager’sreportintendedforahighexecutivepresumedtolackalldetailedknowledgeofthesubject.

Ramalingam’sletter,longasitwas,hadaclearfocus.“ItiswithregardtofoodthatI have to write somewhat harshly and tersely and at a good length,” Ramalingamwrote.Ramanujan,heasmuchassaid,waskillinghimselfbyslowstarvation.

•••

Diet was a big part of any sanatorium “cure.” Tuberculosis patients were called“consumptives” because the disease consumed them. Almost invariably, weight losswentwithit,andoneprevalentnotionheldthatfatteningupthepatientcouldsloworreversethedisease’scourse.

Forbreakfast,MatlocktriedtofeedRamanujanscrambledeggsontoast,withtea.For lunch, rice, chilies, and mustard fried in butter, cucumber and lemon and,sometimes,greenpeas.Whatever itwas, itwasawful, the cooks regularlybotchingthings.RamanujanwroteRamalingamthedayafterhisfriendleft,complainingthatthenewcookhadruinedtheappalappu, rice flour fried inoilorghee;she’dburnedsome,while leavingothers raw. “The curried rice,” he complained, spellingout thewordinTamil,“wasjustlikeakshata”—rawrice,usedonlyonritualoccasions,andnotmeanttobeeaten.Boiledrice:thepoorwomancouldn’tevengetthatright.

Despite its priority claim on rationed foods, the sanatorium couldn’t seem toprocuresomefoodsRamanujanliked.Likebananas.Orcheese,forthemacaroniand

cheese he enjoyed; Ramalingam later sent him some. Butter, meanwhile, was soexpensive in the wake of wartime price hikes that the Matlock administrationgrumbledaboutfryinghispotatoesinit.

Something,Ramalingamcouldsee,hadtogive.Whileatthesanatorium,heaskedifhecouldcooksomethingforhisfriend.“No,youcan’tgotothekitchen,”hewastold.Couldhewriteoutarecipeandhavethecookfollowit?Thissuggestion,too,metresistance.

Sanatoriums,ofcourse,wereusedtofoodcomplaintsandtypicallydismissedthemas manifestations of disease. Tuberculosis patients were supposed to be especiallyfinicky. One whose complaint reached the Ministry of Health, for example, waswrittenoffassufferingfrom“thewarpedtemperamentoccasionallyfoundassociatedwith his malady which makes him impatient and discontented with institutiondiscipline.”

Therealproblem,Ramalingamcouldsee,wasRamanujan’sstubbornness. “He isthinking of his vegetarianism even at the expense of his health and life,” he wroteHardy.“Butonecannotbutthinkofhimascrankyandheadstrongwhenherefusescream and, say, plums on it.”Writing toRamanujan the same day, heminced nowords:

Iwillhavetobeabitharshwithyou.BothfrommytalkwithDr.Ramandaftermysecondthoughts,Iamimpressed with your being so particular about your palate. Well, you will have to choose between . . .controllingyourpalateandkillingyourself.Youmusttryandgetyourselftolikeporridgeoroatmeal,cream,etc.Myfriendshavestronglyadvisedmenottoletyouindulgeinpicklesandchillies....

IamnotgoingtotheextremesandaskingyoutotakebeefteaorBovril[tradenameofaconcentratedbeefextract]thoughconsideringyourlife,myaskingyoutotakesuchathingisquiteexcusable,naydesirable,andevenunavoidable.Bereasonableanddon’tbebigoted.

Iftherewasatimetorelaxhisvegetarianism,thiswasit.OrthodoxJews,whoalsoproscribecertainfoods,mayrelaxthelawsofkashrutwhenotherwisehealthwouldbeimpaired.ButBrahminicalpracticemakesnosuchdispensation,andRamanujanwasnotabouttomakeoneeither.

Sometimebefore,probablyevenbeforeenteringMatlock,RamanujanwasenjoyingoneofhisoccasionalspellsoutofhospitalandstayingatalodginghousecateringtoIndians in London. At breakfast, he drank a commercial beverage, Ovaltine,representedtohimasvegetarian.Laterthatday,idlyexaminingthecan,hesawlistedamong its ingredients some traceof animalproduct.Mortified,hehad to get out ofthere.Abruptly,hepackedhisbagsandwasoutthedoor;itwasVizagapatnamandtheChatterjidinneralloveragain.

Thistime,ashenearedLiverpoolStreetStationtocatchthetrainforCambridge,tonsofbombsfromsilent,high-flyingGermanzeppelinscamerainingdownonthecity.ThiswaslikelytheraidofOctober19,1917,inwhichtwenty-sevenpeoplewerekilled.One bomb fell behind a row of working-class cottages, destroying three ofthem, killing four women and eight children. Yet Ramanujan felt he was beingpersonallypunishedfordrinkingtheOvaltine.Later,writingthelandladytoexplainhisprecipitousdeparture,hepicturedtheraid,inthewordsofafriendwhoheardthestorylater,as“punishmentmetedouttohimbyGodforhavingpartakenofanythingnon-vegetarian.”

Ifanything,Ramanujanhadbecomeevenmorefinicky.Atonepointhe’dbeenonadietlargelyofbreadandmilk;nowhewouldnolongereatthem.NorwouldheeatMatlock’sporridge.HedemandedonlySouthIndianfood.

“Be reasonable,” Ramalingam had written him. But any temptation to be“reasonable”ranhardupagainsttheunsavoryrealityofEnglishcooking.Eventhoseof Ramanujan’s Indian friends less particular than he, or who had forsakenvegetarianism altogether, could have told Ramanujan what all Europe knew—thatEnglish foodwas heavy, ill prepared,maddeningly bland, andmonotonous.WhenGandhi,alsoavegetarian,cametoEnglandagenerationearlier,hewrotethat“eventhe dishes that I could eat were tasteless and insipid.” As for Ramanujan,Nevillewouldrecall:“Ihaveknownhimaskwithunaffectedapologiesifhemightmakehismeal of bread and jam because the vegetables offered to him were novel andunpalatable.”

AfterhisMatlockvisit,somethingDr.KincaidhadsaidgnawedatRamalingam’scomposure.Ramanujan, the doctor assured him, could eat anything he liked, evenspicy pickles and chilies. Yet it was Ramalingam’s understanding that tuberculosispatients shouldnot get foods like that—unless, that is, theyweredying.HewroteHardy:

Whenthepatienthasgonetoofartoberemedied,andwhenitisaquestionofonlyafewweeksormonths,itmatterslittle[whatheeats],aslongasthepatientfeelshappyandcomfortableinthelastdaysofhislife.InpermittingmetogiveanythingtoRamanujantoeat,isthistheobjectofDr.Kincaid?

It is anguishing and breaking one’s heart to feel that Ramanujan, with his wonderful capabilities andvaluablecontributions,shouldbegivenupforsuchashopeless.Warwithallitshorrorsmighthavemadeuscalloustothewhole-saleslaughterandlossoflivesbutsurelyshouldRamanujanbegivenup?

11.SLIPPEDFROMMEMORY

Inthefallof1918,Ramanujan’snamewasonceagainputinforaTrinityfellowship.Hardy,apparentlytoocloselyidentifiedwithRamanujanbynowandlongembroiledin Trinity politics, was this time discouraged from putting up his name. So

Littlewood (who was for a time back in Cambridge, done in by overwork and aconcussion)didinstead.Theracial issueflared.OnefoeofRamanujan’scandidacy,Littlewood wrote later, “went about openly saying that he wasn’t going to have ablackmanasFellow.”

Word of the suicide attempt had gotten around, and Ramanujan’s opponentsseized on it.Didn’t the college bylaws expressly require goodmental health?Andwasn’t a suicide attempt evidence that Ramanujan lacked it? “Grave doubts werebeingfeltabouthismentalstate,”Littlewood’sfriendandformerTriposcoach,R.A.Herman,toldhim.

SoLittlewood,whowashimselftooilltoshowupinperson,wroteareporttakingontherumorsandfurnishingtwomedicalcertificatestotheeffectthatRamanujanwasindeedmentallysound.

“This evidently shocked the moderate members of the Election Committee,”Littlewoodwrote later, “and itwasmoved and passed that they be not read.”TheelectioncouldnowproceedonthemeritsofRamanujan’scase.Ofcourse,Ramanujanhad amost emphatic argument inhis favor—the three letters that now gracedhisname.ForaFellowoftheRoyalSocietytobedeniedaTrinityfellowshipwouldbeascandal. “You can’t reject an F.R.S.,” Littlewood told Herman, who opposedRamanujan.“Yes,”Hermanreplied,“wethoughtthatwasadirtytrick.”Littlewood,wemayimagine,justsmiled.

By now, Ramanujan was out of Matlock. Ramalingam had written Hardy ofRamanujan’swishtoleaveitandgotoLondon,wherehecouldgetIndianfoodmoreeasilyand,perhaps,receivevisitorsmoreconveniently.Ramalingamdidn’tthinkthatwasagoodidea.Whatabouttheairraids?HesuggestedinsteadthatRamanujangotosouthernItaly,orthesouthofFrance,wheretheclimatemightdohimgood,oratleastsomeplaceinEnglandlesspronetoattack.

But Ramanujan got his way and now, learning of the Trinity fellowship, was apatientat a small hospital overlooking a perfectly proportioned little square in theheartofLondon,onwhichGeorgeBernardShawhadlivedinthe1890sandVirginiaWoolf for four years until 1911. FitzroyHousewas a grand five-story townhousewith an oval winding staircase punched up through its center and amassive frontdoorreminiscentofacastlegate.Whilethere,Ramanujansawseveralspecialists,buthisdiagnosisremainedasuncertainasever.Boutsofhighfeverstillcameinirregularbursts.Hehadpainnoonecouldtrace.Hewasontapforatoothextractionaroundthistime,andonedoctorevenattributedallhispaintothat.

“My heartfelt thanks for your kind telegram,” Ramanujan wrote Hardy fromFitzroyHousewhenhegotthegoodnews.“Afteryoursuccessingettingmeelected

bytheRoyalSocietymyelectionatTrinityprobablybecameverymuchlessdifficultthisyear.”

Hewrote that letter on aFriday, probablyOctober 18.The followingMonday,stillelated,hewroteHardy:“PleasetellMr.LittlewoodandMajorMacMahonthatIthankedthemverymuch.Haditnotbeenforyourpainsandtheirencouragement,Iwouldbeneitherthefellowoftheonenorthatoftheother.”Thenheaskedforsomedetailsabouthisfellowship.

Thatwas paragraph one. Paragraph two began: “I have consideredmore or lessexhaustivelyabout the congruency ofp(n) and in general that ofpr(n) . . . by fourdifferent methods”—the first of which bore on a result that, he added, “you arepublishingnow....”

AsHardyhadforeseen,thehonorsaccordedhim,especiallytheF.R.S.,hadliftedRamanujan’sspirits—leadingtowhatNevillewouldterm“abriefperiodofbrilliantinvention” beginning about the spring of 1918.The paperRamanujanmentioned,deliveredtotheCambridgePhilosophicalSocietytwoweeksbefore,represented itsfullestflowering.

•••

“ArecentpaperbyMr.Hardyandmyself,”itbegan,referringtotheirjointworkonpartitionspublishedearlierthatyear

containsatablecalculatedbyMajorMacMahon,ofthevaluesofp(n),thenumberofunrestrictedpartitionsofn,forallvaluesofηfrom1to200.OnstudyingthenumbersinthistableIobservedanumberofcuriouscongruenceproperties,apparentlysatisfiedbyp(n).Thus

(1)p(4),p(9),p(14),p(19),...≡0(mod5),(2)p(5),p(12),p(19),p(26),...≡0(mod7),

andsoon,rightdownthepage.Thepaperwasentitled“SomePropertiesofp(n),theNumberofPartitionsofn,”

andwhatmade it importantwas that until nowmost “properties” of the partitionfunctionhadeludeddiscovery.Thenumberofpartitions,p(n),recall,referstohowmanyways you can add up numbers to getn. But of even such basic facts as, forexample, whether the partition function was odd or even for a particular n,mathematiciansremainedignorant.

The table to which Ramanujan referred was a long, dry list of numbers, theproduct of laborious hand-calculation, that MacMahon had prepared two yearsbefore;thesewerethevalues,all thewayuptop(200), thatRamanujanandHardy

hadusedasbenchmarksagainstwhichtochecktheirgeneral formula forp(n).ButthenRamanujanhadlookeddeepintoMacMahon’ssterilelistandmadeoneofthoseleapsoftheimaginationthatwouldastonishmathematiciansovertheyears.

Tosaywhathesaw,Ramanujanusedthelanguageofcongruences,whichexpressesfactsaboutdivisibility.Twonumbersarecongruentwhentheycanbedividedbythesamenumber, and leave the sameremainder (whichmaybe0).Take, for example,thisstraightforwardcaseofdivision:

14/7=2

Inthespeciallanguageofcongruences,yousay

14≡0(mod7),

whichmeansthatyoucandivide14by7,the“modulus,”withzeroleftover.And

15≡1(mod7)

meansthat15,dividedbythemodulus7,leavesaremainderof1.Thenumber22iscongruentto15becauseit,too,whendividedby7leaves1.

“Congruences are of great practical importance in everyday life,” Hardy withcoauthorE.M.WrightwouldpointoutinAnIntroductiontotheTheoryofNumbers.“Forexample,‘todayisSaturday’isacongruenceproperty(mod7)ofthenumberofdayswhichhavepassedsincesomefixeddate....Lecturelistsorrailwayguidesaretablesofcongruences;inthelecturelisttherelevantmoduliare365,7,and24.”

What Ramanujan had found in MacMahon’s table were certain persistent andintriguingpatternsbestexpressedinthissimplelanguage.Forexample,hefoundthat

p(5m+4)≡0(mod5)

Makemanythingyoulikeand,whateveryoupicked,thenumberofpartitionswould,Ramanujanshowed,alwaysbeexactlydivisibleby5.Forexample, letm=0.Then5m+4isjustequalto4.Howmanypartitionsarethereof4?Theansweris5.Is5exactly divisible by 5? Why, yes. You could make m = 1,000,000 and ask whatp(5,000,004)wasand,withnotaclueastowhatthisastronomicalnumbermightbe,youcouldsaywithabsoluteconfidencethatitcouldbedividedby5.Ramanujanalsocameupwithasimilar identitythatsaidp(7m+5)wasdivisibleby7.Now,inhisCambridge Philosophical Society paper, he proved these results, and conjecturedothers,oneofwhichwouldsuccumbtoprooflaterthatyear.

•••

On the sameday it gotRamanujan’s congruences paper, thePhilosophical Societyreceivedanotherfromhim,onethatbroughtRamanujan’sworkinEnglandfullcircleandtestifiedtoboththegeniusandlostpotentialinhismathematicallife.

Thissecondpaperwentbacktotwostrikingidentitieshehaddiscoveredsometimebefore1913andlatershowntoHardy.(Anidentityisanequationtrueforallvaluesofthevariable.Sothatwhereasx−2=3isanordinaryequation,trueonlyforx=5,(x−2)(x+2)=x2−4istrueforeveryvalueofx.)Oneofthemwas

Theothertookasimilarform.AndsometimeafterRamanujan’sarrivalinEngland,probably in 1915, MacMahon saw in them something Ramanujan had not—somethingthatbore,onceagain,onpartitions.

Studentsoftheadditivetheoryofnumbers,whichistheformalnameforthefield,tookan interestnotonly inpartitionsgenerallybut incertainclassesofthem.Takethe number 10. The number of its partitions—or to invoke a precision that nowbecomesnecessary,thenumberofits“unrestricted”partitions—is42.Thisnumberincludes,forexample,

1+1+1+1+1+1+1+1+1+1=10

and

1+1+1+1+2+1+2+2=10

But, onemight ask,what if you excludedpartitions such as theseby imposing anew requirement, that the smallest difference between numbers making up thepartitionalwaysbeatleast2?Forexample,

8+2=10

and

6+3+1=10

wouldbothqualify,asdofourothers,makingforatotalofsix.Alltheotherthirty-sixpartitionsof10containatleastonepairofnumbersseparatedbylessthan2andarethusineligible.

That’sone classofpartitions.Here’s another, formedbya second,quitedistinctexclusionarytactic:Whatifyouonlyallowedpartitionssatisfyingaspecificalgebraicform?Forexample,whatifyourestrictedthemtothosecomprisingpartstakingtheformofeither5m+1or5m+4(wheremisapositiveinteger)?Ifyoudothat,thepartition

6+3+1=10

fails toqualify.Why?Becausenot all theparts, the individualnumbersmakingupthepartition,satisfythecondition.Thepart6does;itcanbeviewedas5m+1withm=1.Sodoes1,whichcanbeviewedas5m+1withm=0.Butwhatabout3?Makem anythingyouwant andyoucan’t get a3outof either5m+1or5m+ 4(whichtogethercangenerateonlynumberswhosefinaldigitsare1,4,6,or9).

Onepartitionthatwouldqualifyis

6+4=10

Anotheris

4+1+1+1+1+1+1=10

Eachsatisfiesthealgebraicrequirement.Inall,qualifyingpartitionscometosix.Now six also happens to be the number of partitions that fit the first category.

Exceptthatitdoesn’t“happentobe.”Italwaysturnsoutthatway.Pickanynumber.Addupall itspartitionssatisfyingthe“minimaldifferenceof2”requirement.Thenaddupallitspartitionssatisfyingthe“5m+1or5m+4”requirement.Comparethenumbers.They’rethesame,everytime.

This iswhatRamanujan’s identity, suitably interpreted, revealed: two seeminglydistinctmathematical subworlds fused together ina singleunifying relationship. Involume2 of his bookCombinatoryAnalysis, which came out in 1916,MacMahondevoted to it and the other identity awhole chapter, “Ramanujan’s Identities.”Heverifieditbyreamsofhand-calculationasfarasanyreasonablepersoncould,“sothatthereispracticallynoreasontodoubtitstruth;but,”headded,“ithasnotyetbeenestablished”throughformalproof.

Butithadbeen,inapapertwentyyearsbefore.

Onedayin1916or1917Ramanujanwasrummagingthroughthe1894volumeoftheProceedings of theLondonMathematical Society when there, near the bottom ofpage 318, he saw it. Itwas entitled “SecondMemoir on theExpansion ofCertainInfinite Products,” and two of those infinite products were just the identities hethoughthehaddiscovered.BeforeMacMahon’sbook cameout,Hardyhad shownRamanujan’s identities around. Did anyone know proofs for these marveloustheorems?Could anyone furnish any?Noone could.Yethere, inblack andwhite,likeaspecterfromthepast,wasevidencethatsomeonealreadyhad.

“Icanrememberverywellhissurprise,andtheadmirationwhichheexpressed”fortheolderwork,HardywouldsayofRamanujan.Asforanyloss,orevenbittersweetambivalence,thathemayhavealsofelt,Hardysaidnothing.

Themanwho anticipatedRamanujanwas Leonard JamesRogers, a remarkablecharacter if ever therewas one. Born in 1862, inOxford,where his fatherwas aneconomist,he had not only donewell on theOxford equivalent of theCambridgemathematicalTripos,buthadearnedabachelorofmusicdegreein1884.Hewasafine pianist, an exceptional mimic, liked to affect a broad Yorkshire accent. Heknitted.Heskated.Hegardened.Hewasanatural linguist. “Surely,” itwassaidofhim,

nopositionindiplomacywouldhavebeenunattainabletooneendowedwithhiseasymasteryof languages,hisquickintelligence,hissparklingwit,hisfinepresence,hisathleticgrace,hiscourtlycharmthatnowomancouldresist.Yetofwhattheworldcountssuccessheachievedpracticallynothing.

Rogerswas,inspirit,agiftedamateurwho,despitehisabilities,neverpursuedhismathematical careerwith the single-mindeddevotion sonecessary, thenasnow, toestablishabigname:“Hedidthings,anddidthemwell,becausehelikeddoingthem,buthehadnothingoftheprofessionaloutlook,andhisknowledgeofotherpeople’swork in mathematics was vague. He had very little ambition or desire forrecognition.”

Even in the years before Ramanujan rediscovered him, he wasn’t quite themathematical nonentity his obituaries presented him as; for years a mathematicsprofessor at what became the University of Leeds, he had quite a few publishedpapers to his credit. Still, he possessed something of a knack for making notablecontributionsthatwerepromptlyforgotten,atleastintheshortrun.Justthreeyearsearlier, forexample,theProceedingsoftheLondonMathematicalSocietyhadrun thiscorrection:

Prof.Rogers,inhispaper“OntheQuinquisectionalEquation”(Proc.LondonMathSoc.,VolXXXII,pp.199–207),givestheequationinexplicitform.IregretverymuchthatProf.Rogers’swork,whichImusthave

seenatthetimeof[its]publication,hadentirelyslippedfrommymemory,sothatIwasledtostatethattheproblemhadnotbeendealtwithsinceCayley’spaperinVol.XII.

In the case of what later became known as the Rogers-Ramanujan Identities,Hardy suggested later, the Rogers originals appeared “as corollaries of a series ofgeneraltheorems,andpossiblyforthisreason,theyseemtohaveescapednotice, inspite of their obvious interest and beauty.” Then, too, the proofs were tortuous;Rogers’saestheticsensibilitiesdidn’textendtothewrittenword.

Still,theretheywere.Acorrespondenceensued.Inthebooktowhichhe’dputthefinishingtouchesin

April1916,MacMahonhad saidRamanujan’s identitieswere yet unproved.Now,with the falsehood enshrined in black and white, MacMahon wrote Rogers, asRogersrecalledalittlelater,“regrettingthathehadoverlookedmyworkbeforeitwastoolate.”InOctober1917,RogerswroteMacMahonwithanew,simplerproof.InApril1918,probablywhileatMatlock,RamanujanwroteHardywithhisown.

On October 28, 1918, the two proofs, along with Ramanujan’s importantcongruences paper, were read at the annual general meeting of the PhilosophicalSociety.

•••

Twoweekslater,thewarended.TheBolshevikRevolutionhad led to a collapseof theRussian armies facing the

Germans.Germany had withdrawn all but a million men from the eastern front,rushedthemacrosscentralEuropeandsentthemhurtlingwesttowardParis.Onceagain,asin1914,GermanarmiesbreachedtheMarne.OnceagaintheycamewithinsightofParis.Butthistime,therewasadifference.AmillionfreshAmericansoldiershad arrived in France. TheGermans, exhausted by years of war, rolled back, andsoonhadtosueforpeace.ThearmisticewasdeclaredonNovember11,1918.

Very shortly, in thewordsof theCambridgeReview, “therewasa very creditablepre-war bonfire in the Market Place, fed chiefly with packing cases round whichtherewasdancing;butnotall thedancersperformedwithtwosound legs.”Byoneo’clock that afternoon, someGirtonCollegewomen, normally sequestered in theirown campus on the fringes of town, had joined the afternoon revels. At five thatevening,manywent to theThanksgivingService atKing’sCollegeChapel. “Aswestoodinthedimcandlelightofthatwonderfulchapel,”oneofthemwrote,“itseemedas if Earth andHeavenwere no longer divided, and as ifTime andEternitywereone.”

On November 26, Hardy wrote Francis Dewsbury in Madras with word onRamanujan. “I think it isnow time,”hewrote, “that thequestionofhis temporaryreturntoIndiaandofhisfuture,generally,shouldbereconsidered.”

CHAPTEREIGHT

“InSomewhatIndifferentHealth”

[From1918on]

1.“ALLTHEWORLDSEEMEDYOUNGAGAIN”

Ramanujan seemed better and, Hardy wrote Dewsbury, “on the road to a realrecovery.”Hehadgainedalmostfifteenpounds.Histemperaturehadsteadied.Thedoctorsnowfavoredbloodpoisoningasthesourceofhisills,andthis,itseemed,had“driedup.”

Wasitnottime,then,thatRamanujanreturntoIndia?Thereasonsagainstdoingsohaddisappeared.Thesealanesweresafe.HehadachievedallhehadsetouttoinEngland. His Trinity fellowship imposed no residency requirement. He need notstickaroundwhilehisRoyalSocietycandidacywasupintheair;itnolongerwas.So,thethinkingwent,withRamanujanonthemendwhyretardhisrecoverybykeepinghiminEnglandanylonger?

Of course, more was going on behind the scenes. “He has apparently beenapproached(with a view to return)directlyby several friends,”Hardywrote. “It ispossible, I think, that the suggestion has not been made in the most tactful waypossible;atany rate, it seems tohave turnedhimratheragainst the ideaofgoing”;somethingtheysaidhadpushedoneofRamanujan’snumerousbuttons.MindfulofRamanujan’s sensitivities, and of no mind to trespass on them, Hardy advisedDewsbury that “the suggestion would best bemademore or less officially and bylettersimultaneouslytohimandtome.”OfferRamanujanauniversitypositionthatlefthimfreetodoresearchandoccasionallyvisitEnglandandhe,Hardy,wouldfavorhisreturn—andRamanujan,hefeltsure,would,too.

Hardy’s letter bore a reminder that the emotional malaise that had catapultedRamanujanontothesubwaytracksearlierthatyearwasnotwhollycured.“Hewillreturn to India with a scientific standing and reputation such as no Indian hasenjoyedbefore,andIamconfident that Indiawill regardhimas the treasurehe is.Hisnaturalsimplicityandmodestyhasneverbeenaffectedintheleastbysuccess—

indeed,” he added, “all that is wanted is to get him to realize that he really is asuccess.”

Ramanujan still didn’t think he was? InNovember 1918?With close to twentymajorpapersappearinginthepastfouryears?WithanF.R.S.appendedtohisname?And a fellowship at Trinity College? All this did not convince him? In public,RamanujanaffectedaSouthIndianbrandofaw-shucksmodesty.Butinside,hestillwantedsomethingmore.WanteditfromHardy?WasHardy,onwhomthewarhadmade Ramanujan so dependent, in private more niggardly in his encouragement,morealoofanddetached,thanhewasinpublic?

Inanycase,thewheelswerebeingsetinmotionforRamanujan’sreturntoIndiathefollowingyear.Butwashegoingbackbecausehewasreallybetter?Orbecausehewasworse,hischancesforrecoveryinEnglandseenasremote?That,atleast,iswhatoneofhisIndianbiographers implied later. “Mr.Ramanujan’sdiseasehadassumedseriousproportionsbytheChristmasof1918,”wroteP.V.SeshuIyer,referringtoatimeonlyamonthremovedfromHardy’slettertoDewsbury,“andcausedsuchgraveanxietytohisdoctorsinEngland,that,hopingtodohimgood,theyadvisedhimtoreturntohisnativehomeinIndia.”

Whatever was physically wrong with Ramanujan, his progress, or decline, wasglaciallyslow—makingiteasytoreadslightday-to-dayfluctuationsinhisconditionanywayyou liked.So itmayhavebeennogreat change inhishealth towhichhisreturntoIndiawasreallyduebut,moresimply,theendofthewar.

•••

Theweeksandmonthsjustafterthearmisticewereatimeofendingsandbeginnings,ofvastreliefallacrosstheblood-drenchedsoilofEurope.Streetlightsblazedagain.OnDecember7,Britonswererestoredtherighttomakecakesandpastriesandtosmearthemwithchocolatetotheirhearts’content.Rationingcontinued,asitwouldforsomeproductsinto1920,butadoublemeatrationwasannouncedforChristmas1918.OnDecember9,demobilizationbegan.

Duringthewar,2162Cambridgemenhadbeenkilled,almost3000wounded—altogether about a third of all who served. Some 80,000 wounded had come inthroughtherailstationoverthefouryears.Butnow,withinmonthsofwar’send,theuniversitywasalreadyreturningtosomethinglikeitsprewarproportions.

TheNevasa,theBritishIndiaLinesshipthathadcarriedRamanujantoEnglandin1914,wentbacktocarrying travelersandtourists.Shehadbecomea troopshipearlyinthewar.Then,herfunnelpaintedyellowandabroadbandpaintedaround

herwhitehullingreen,shewasmadeoverintoahospitalship,with660beds,whichfortwoyearsborethesickandwoundedbetweenSuez,Basra,andBombay.Later,atroopship again, she carried American soldiers to France, at least twice having tooutrunU-boats.Now,in1919,shewasapassengershiponcemore.

Hardy’sfriend,thehistorianG.M.Trevelyan,returnedtoEngland“togetonwithwritinghistorybooksagain.Ihadnootherambitioninlife.Thedelightfuldelusionthatwehaddonewithtotalwaratleastforageneration,perhapsforever,gaveazesttodomesticandpersonalhappiness.IshallneverforgettheexhilarationofaCornishholidaywithmywife and girl andboy atEaster 1919,when all theworld seemedyoungagain,andthesandsandrockyheadlandsrejoiced.”

•••

About the time of the armistice, Ramanujan had left Fitzroy Square for anothernursinghome,thisoneinthesuburbofPutney,afewmilessoutheastofLondononthesouthbankoftheThames.

ItwascalledColinetteHouse.Fromtheoutsideitwasabig,boxy,undistinguishedstructure, in plan almost square, part of an early suburban development offreestandingbrickhousesbuiltinthe1880s;LeonardWoolfhadlivedinoneofthemonhis return fromCeylona fewyears earlier, finding it somethingof a comedownfrom theAnglo-Indian luxury towhich he had become accustomed.The interiorsweremore impressive, though, laced as theywerewith elaboratemoldings, stainedglass, grandly scaled rooms andhandsomewinding staircases.DuringRamanujan’stime, the eight-bedroom house at 2 Colinette Road had been made into a smallnursinghomepresidedoverbyoneSamuelMandevillePhillips.

Compared toMatlock, Ramanujan was much more accessible here, and Hardy(whosemotherhaddiedinCranleighafewweeksafterthearmistice)couldvisithimmoreeasily;beyondtheusualtwohoursonthetrainintoLondon,Putneywasjustacabrideaway.Once,inthetaxifromLondon,Hardynoticeditsnumber,1729.HemusthavethoughtaboutitalittlebecauseheenteredtheroomwhereRamanujanlayinbedand,withscarcelyahello,blurtedouthisdisappointmentwith it. Itwas,hedeclared,“ratheradullnumber,”addingthathehopedthatwasn’tabadomen.

“No,Hardy,” saidRamanujan. “It is a very interestingnumber. It is the smallestnumberexpressibleasthesumoftwocubesintwodifferentways.”

Findingnumbersthatwerethesumofonepairofcubeswaseasy.Forexample,23

+33=35.Butcouldyougetto35byaddingsomeotherpairofcubes?Youcouldn’t.Andasyoutriedtheintegersonebyone,itwasthesamestory.Onepairsometimes,

twopairnever—never,thatis,untilyoureached1729,whichwasequalto123+13,butalso103+93.

How did Ramanujan know? It was no sudden insight. Years before, he hadobservedthislittlearithmeticmorsel,recordeditinhisnotebookand,withthateasyintimacywithnumbersthatwashistrademark,rememberedit.

WhileRamanujanwasatColinetteHouse,hegot somegoodnews.All throughthewar,theauthoritiesinMadrashadkeptupwithhimthroughHardy,periodicallyextending his leave of absence from the Port Trust and his fellowships from theuniversity.In1917,Ramanujan’smotherhadcometoMadrasandlearnedthathersonwouldbenamedUniversityProfessor,earningatleast400rupeespermonth—sixtimeswhathemadeasaresearchscholarbeforeheleft.Now,inthewakeofhisF.R.S.,theMadrasauthoritieswerefairlyfallingalloverthemselvestoseewhattheycoulddoforhim.AndinlateDecember1918orearlyinthenewyear,helearnedtheuniversityhadgrantedhima250-pound-per-year fellowship.Thiswason topof alikeamountawardedhimbyTrinity.Anditwasgoodforsixyears.AnditpermittedhimperiodictripsbacktoEngland.

Ramanujan too well remembered the days in Kumbakonam when he’d lost hisscholarship and had to drop out of school; and the period in Madras when he’dscraped by on Ramachandra Rao’s patronage and the few rupees he made fromtutoring.Besides, hewas embarrassedbyhis comparative lackofproductivity overthepastyearandahalf.Andso,onJanuary11,1919,hewrotetoDewsbury:

Sir,I beg to acknowledge the receipt of your letter of 9th December 1918, and gratefully accept the very

generoushelpwhichtheUniversityoffersme.I feel,however, thataftermyreturn toIndia,whichIexpect tohappenas soonasarrangementscanbe

made,thetotalamountofmoneytowhichIshallbeentitledwillbemuchmorethanIshallrequire.Ishouldhopethat,aftermyexpensesinEnglandhavebeenpaid,£50ayearwillbepaidtomyparentsandthatthesurplus,aftermynecessaryexpensesaremet,shouldbeusedforsomeeducationalpurpose,suchinparticularasthereductionofschool-feesforpoorboysandorphansandprovisionofbooksinschools.Nodoubtitwillbepossibletomakeanarrangementaboutthisaftermyreturn.

Ifeelverysorrythat,asIhavenotbeenwell,Ihavenotbeenabletodosomuchmathematicsduringthelasttwoyearsasbefore.IhopethatIshallsoonbeabletodomoreandwillcertainlydomybesttodeservethehelpthathasbeengivenme.

Ibegtoremain,Sir,Yourmostobedientservant,

S.Ramanujan

About amonth later, onMonday, February 24,Ramanujanwaswell enough totend to the business of getting his passport. “Age: 30,” the clerk recorded theinformation.“Profession:ResearchStudent.”Sittingforhispassportphoto,heheldhishead,withitsfullshockofstraightblackhair,cockedalittletooneside,and,with

luminouseyes,peeredslightlyupatthecamera.TheresultingHollywood-handsomeimagewas not theRamanujanhis friends knewback in India, nor theRamanujanHardyknewin1914,butthatofanill,muchthinnerman.Hisshirt,buttonedtothetop, was loose around the neck. Layers of fat no longer pressed against his jacketwhich—nowillfittingintheotherdirection—wastwosizestoobigaroundhisneckandshoulders.

About two weeks later, on March 13, 1919, two short notes by RamanujanappearedintheProceedingsoftheLondonMathematicalSociety; inthem,herevealednewcongruencepropertiesofthepartitionfunctionandanewlinkbetweenthefirstand second Rogers-Ramanujan identities. That same day, lightened by notebooksandotherpapershehadleftwithHardy,butencumberedwith,amongotherthings,at least adozenbooks, a box of raisins for his younger brothers, and a big leathertrunkfilledwithpapers,heboardedthePacificandOrientLinesshipNagoya.

TheNagoya, of the same recent vintage as the Nevasa only a bit smaller, wassteamingforBombay.

•••

TheIndiatowhichRamanujanwasreturninghadnotgoneunscathedbytheconflictthathadbledEurope.Indeed,Madrasitselfhadbecomeabattleground.Immediatelyuponthedeclarationofwar,theGermanlightcruiserEmdenhadstrucksealanesinthe Indian Ocean, taking prizes and destroying merchant ships. One night sheappeared outside Madras. Wartime or not, the harbor lights burned brightly,illuminating the red-striped white tanks of the Burma Oil Company. Mistakenlyadvised of theEmden’s sinking, British officials were at a dinner party celebratingwhen theGermanmarauderbegan to shell theharbor, setting fire to theoil tanks.Later,fromninetymilesoutatsea,thenightskywasstillaglowfromthefiresashore.

The attack had little strategic impact, but it struck terror in the local citizenry,whichmayhaveincludedRamanujan’sparentsandwife.Manynatives,fearfuloftheEmden’sreturn,fledthecity.

ThewarpurgedIndiaofallbutaboutfifteenthousandBritishtroops.AvolunteerIndianarmyofmorethanamillionmenhadbeenraised,manyofwhom,atleastinMadras, were untouchables; local recruiters promised them, “When you wear theKing-Emperor’suniform,youwillbeabletowalkthroughtheBrahminquarterandspitwhereyoulike.”

Politically,itwasatimeofgreatnewstridestowardindependence.In1915,AnnieBesant, English social reformer and president of Madras’s Theosophical Society,

began agitating forHomeRule with the publication of her daily newspaper,NewIndia, and the formation of the Home Rule League. In the same year, Gandhi,alreadyfamousforthenonviolentmethodsofcivildisobediencehehadforgedforthedefense of Indian immigrants in South Africa, returned to India, where he beganusinghisnew socialweapons to formamassmovement; five years later, hewouldbecomeheadoftheIndianNationalCongress.

Change—achafingat thebondsof thecastesystem,anawakeningof theIndianmasses,a resurgent Indian nationalism, a rediscovery of indigenous Indianways—marked theyearsRamanujanwas away.All across the subcontinent, the giantwasfeelingitsstrength.

Intellectual India, too, felt it. In late December 1916, the IndianMathematicalSociety held its first conference, in Madras. Lord Pentland himself opened theproceedings,beingmetonthestepsofPresidencyCollegebyR.RamachandraRao,the society’s fourth president. “At the present time,” Lord Pentland told theconference,“ayoungIndianstudent,Mr.S.Ramanujan”—cheerserupted fromtheaudience—“is studying at Cambridge whose career we in Southern India arewatching with keen interest and high anticipation. You know the story of thediscovery of his unusual talent and all herewill be glad to hear how entirely he isjustifyingtheeffortswhichweremadetogiveitfullscope.”

Everyoneknewthestorybynow.BackinMay1914,theNevasahavingbornehimaway barely two months before, Ramanujan was already being heralded in theMadraspapers.

Mr.S.RamanujanofMadras,whoseworkinHigherMathematicshasexcitedthewonderofCambridge,isnowinresidenceatTrinity.HewillreadmainlywiththetwoFellowsoftheCollege—Mr.HardyandMr.Littlewood.They are going throughmasses ofworkhehas alreadydone, and aremaking some surprisingdiscoveriesinit!

Whenhereceivedhisdegree,theyheardaboutitbackhome.WhenhewasnamedanF.R.S.,Madrasrolledouttheredcarpetforhiminabsentia,intheformofameetingtohonorhimatPresidencyCollege.OneclassmateofhisfromPachaiyappa’sCollegedays,K.Chengalvarayan, later recalledhowwhenhemetold school friendsduringtheseyears,“theemergenceofRamanujanintofamewasusuallyoneofthetopicsoftalks.”

Ten weeks before Ramanujan’s arrival in Bombay, the Indian MathematicalSocietymetthereforitssecondconference.Ramanujan’s“brilliantcareer,”his“veryhumbleorigin,”andhiselevation to theRoyalSocietywereonevery speaker’s lips.Hewasreturningatabadtime,though,attheheightofafluepidemicthatwouldkilltenmillionpeople;amongthemathematicalsociety’ssmallmembershipalone,ithad

claimedfivelivesalready.OtherswhohaddiedduringRamanujan’stimeabroadwereE.W. Middlemast, the Presidency College mathematics professor who had givenhimoneof his earliest recommendations; and, at the age of forty-two, SingaraveluMudaliar,hismathematicsprofessoratPachaiyappa’sCollege.

Nowtheleadpageofthesociety’sJournal,datedApril1,1919,borethenewsofRamanujan’s return.But hewas coming back, it advised, “in somewhat indifferenthealth.”

2.RETURNTOTHECAUVERY

“Where is she?”askedRamanujanofhismotherashe steppedof the ship into themawofBombayonMarch27,1919.ShewasJanaki.Hismotherwasthere,andhisbrother Lakshmi Narasimhan; the two of them had set out for Bombay on thetwenty-first.Butnothiswife.WhyfretoverJanaki?snipedKomalatammal.Scarcelyoff the boat,Ramanujanhaddropped into the family snake pit.Domestic conflicthad sabotaged his last three years in England. Now they dampened his arrival inIndia.

Infact,thetwosidesofthefamilyhadbeenoutoftouchformorethanayear.NooneknewjustwhereJanakiwas.PerhapsbackinRajendram.Or,ifshehadlearnedofRamanujan’sarrival,maybewithhersister inMadras.So,LakshmiNarasimhanwas dispatched to write two identical letters to Janaki asking that she comemeetRamanujaninMadras.

She was back in Rajendram, it turns out, where she’d gone for her brother’sweddingmore than a year before. Shedid knowofRamanujan’s return—but onlythankstotheMadraspapers,notbecausehermother-in-lawhadbotheredtoinformher.Herbrother,R.Srinivasa Iyengar,had advised her not to rejoin the family inMadras. Komalatammal hated her; why place herself within the tiger’s jaws onceagain? But then the letter came from Lakshmi Narasimhan. Ramanujan, it said,wantedher.Thatwasallittook.Sheandherbrothersetoff.

“WhenIgobackIshallneverbeaskedtoafuneral,”RamanujanhadtoldNevillebeforeheleft.Itwasoneformofthetaintheexpectedtobear,intheeyesofstrictlyorthodoxBrahmins, for going toEngland.To help remove it,Komalatammal hadplannedtotakehersondirectlytoRameswaram,forpurificationceremoniesinthegreattempletowhichthewholefamilyhadgoneforapilgrimagein1901.Butnow,inBombay,onelookathimdecidedheragainst it;Rameswaram,all thewaysouthalmosttoCeylon,wasanotherfivehundredmilesandanotherharddayonthetrainbeyond Madras. He was just too sick. And so, after a few nights in Bombay,

Ramanujan,hismother,andbrotherboardedtheBombayMailfortheovernighttriptoMadras.

•••

“WhenImethimalightingfromtherailwaytrain,”RamachandraRaowouldrecallofRamanujan’s arrival inMadras on April 2, “I foresaw the end.”He looked awful.Makingthingsworse,therewasstillnoJanaki.Why?heaskedhismother.ThistimeKomalatammal said something about Janaki being off to tendher father,whowasunwell.

Ramanujanwasbundledontoajutka,atwo-wheeled,horse-drawnvehicle,whichsoonpulledawayfromthebustlingareaaroundCentralStation.HisoldfriendK.S.ViswanathaSastri, oneof thosewho’d seenhimoff fromMadras five years beforeand whom Ramanujan had once tutored in math, followed on behind by bicycle.About three miles south of the station, they reached a large, lovely bungalow onEdwardElliotsRoadownedbyawealthytriallawyer.WhenViswanathaSastrigottherealittlelater,Ramanujanwasalreadyeatingyogurtandsambhar.“IfIhadthisinEngland,”saidRamanujan,“Iwouldnothavegottensick.”

Theweekendbeforehisarrival inMadras, the localpaperscarriedanaccountofhis life prepared by Dewsbury’s office. “This,” Sir Francis Spring scrawled on aninterofficememo, “does not seem to go far enough in satisfying legitimate publicinterest inamanwhosemathematicalgeniusmayyetdogreatthingsfortheworldandhasalready.”HedirectedNarayanaIyer torummagethroughthePortTrust’sRamanujan files and assemble amuch fuller biography.This appeared onApril 6.“Mr.S.Ramanujan is anativeofKumbakonam in theTanjoreDistrictofMadrasPresidency.Hewasbornin1888[sic]ofpoorand,sofarasEnglishgoes, illiterateparentsoftheVaishnavasectofBrahmins....”

Ramanujanwasofferedauniversityprofessorship,whichhesaidhewouldacceptwhen his health improved. Madrasi notables trooped by to visit the convalescinggenius,SouthIndia’sconqueringherooftheintellect,whohadshowntheBritishersthe stuff of which South Indians were made. Now and over the next year, theyrushed to pick up hismedical and other expenses.They offered him their homes.Toppeople fromtheHinducameby.SodidRamachandraRao,ofcourse,andSirFrancisSpring,andNarayanaIyer.

Toshieldhimfromvisitorsalittle,Ramanujan’sdoctor,M.C.NanjundaRao,hadhimmovedahalfmileor so south toaplace calledVenkataVilasonLuzChurchRoad, named for a nearby Portuguese church (known locally as Kattu Kovil, or

JungleTemple).Now,farfromtheteemingTriplicaneandGeorgetownofhisyouth,he was in the very heart of cultured Madras, peopled by high-born Brahminintellectuals, lawyers, and scholars, who lived in large compounds, luxuriant withbananatreesandbetelgardens.Itwashere, finally,onApril6, that Janakiandherbrother caught upwith him, followed, about a week later, byRamanujan’s father,grandmother,andyoungerbrother,fromKumbakonam.

Laterthatmonth,onthetwenty-fourth,RamanujanwroteDewsburyaskingforanaccountofhisexpensesinEnglandandthetripbackandrequestingpaymentofhisfellowshipinmonthlyinstallments.Hewassettlinginforthelonghaul.

ForthreemonthsRamanujanstayedatthebungalowonLuzChurchRoad.Andhere,heandJanakibegantoforgesomethinglikearealrelationship.Janakihadbeenjust thirteenwhenhe left.Now shewas eighteen.As theynever really hadbefore,they began to talk, perhaps now each coming to discover howKomalatammal hadinterceptedtheirletters,perhapsfindingtimeforphysicalintimacyaswell.

ButthemanJanaki,hisfamily,andfriendssawwasmuchchangedfromthemantheyhadknown.Therewerethesmallsuperficialchanges,ofcourse.Hedrankcoffeenow,whichhehadn’tmuchbeforedespite itspopularity (more than tea) inSouthIndia.Histuftwasgone.Hewasabitlighterincomplexion.Hewasmuchthinner;Janaki,seeinghimsoemaciatedandcoughingupphlegm,onlynowrealizedhowsickhewas.

Butitwasthechangeinhispersonalitythatupsethisfriendsmost.K.NarasimhaIyengar,withwhomRamanujanhad lived for a time inMadras around1912,hadbeentheretogreethimatthetrainplatform.“Ifoundhimnotacheerful,chummyandaffectionateRamanujan,”he said later, “buta thoroughlydepressed, sullenandcold Ramanujan, even after seeing me, a close and affectionate friend.” RecalledAnantharaman,whovisitedhimatLuz,bearingabunchofKumbakonambananas:“Hewasnot theoriginalRamanujan.Hecould [scarcely] speakandhis illnesshadmade him peevish.” He was bitter, impatient, sour. His faith was tarnished, too.Once, Anantharaman’s older brother, Ganapathy, said something about gods andtemples,andRamanujansnappedbackthat“itwas foolishtalk,andtheywereonlydevils.”

Janakiwasnotimmunefromhiswrathandfoundhimquicktojumponherwhendisturbedwhile working. In his passport photo, she thought, he seemed troubled,andthesamelookcrossedhisfeaturesoftenoverthenextyear.Otheraccountshavehimyellingathisbrotherforwastingmoney,orimpetuouslyscatteringthingsaroundthehouse,orevenchewingonthermometersplacedinhismouth.

The family’s own record of events in Ramanujan’s life, reconstructed after hisdeath (probably by his brother LakshmiNarasimhan) is over the next year oftenambiguous,itscrypticscrawlbearingfragmentaryreferenceslike“Tamilbooks,”and“paradoxabout points,” and “rumor that hewasmad.” It records the comings andgoingsoffamilymembers,visitstothehouse,thereceiptofletters,andmuchotherdaily minutiae, most of it not fleshed out enough to reconstruct today. But onenotation, recorded over and over during the year, leaves little ambiguity: quarrel.Sometimes the quarrels were between Janaki and Komalatammal, sometimesbetween Janaki and Ramanujan’s grandmother. Sometimes the antagonists wentunrecorded.TwiceduringAprilthefamilyscribementionedquarrels,twiceagaininJuly, another time inAugust, again in September, again inOctober, and at othertimeslater.

Formuchof theyear,duringwhich the familymoved repeatedly, thehouseholdwasa hellhole of simmering resentment.Once, a battle apparently broke out overwhetherRamanujanshouldgotoasanatorium,anothertimeoversomedonationoranother. Ramanujan’s letter from Colinette House, suggesting that some of hisfellowship money go toward scholarships, was another irritant; Komalatammalpreferredthathersonshowermoreofhisnewaffluenceonthefamily,notstrangers.Janakiwasprobablynomoreimmunetomoney’sallurethanKomalatammal;yearslatersherecalledhowRamanujan,inhisfinaldays,“saidhehadfivethousandrupeesinhissavingstobuymediamondeardropsandagoldbelt.”

•••

With the approach of the Madras summer and its daytime temperatures over ahundreddegrees,Ramanujan’sdoctorsadvisedhimtoheadinlandtoescapetheheatandhumidity.OnepossibilitywasCoimbatore,acityaboutthesizeofKumbakonamlocatedinhillcountrymostofthewaywestacrossthesouthernspineoftheIndiansubcontinent.Coimbatorewouldbedrier,andtendegreescooler.ButRamanujan’smotheropted instead forKodumudi, a sleepy little townof a few thousandpeopleknownforitsMagudeswaratemple,whichwasunusualinofferingcommonworshiptoBrahma,Vishnu,andSiva.Kodumudi formedonevertexof a twenty-mile-widetriangleofwhichNamakkalandErode,Ramanujan’sbirthplace,weretheothertwo.ForKomalatammal’ssideofthefamily,especially,Kodumudiwaslikegoinghome.

The district comprising Kodumudi, with rainfall only half as heavy as that ofMadras, wasmostly brown and bare. But the town itself, occupying a site on thesouthernbankoftheCauvery,wasgreen.Theriver,besidewhichafertileleveehad

builtupovertheyears,wasbroadhere,alineofpalmtreesonthenorthshoredistantenoughtosavorofaforeigncountry.

Througharrangementsmadebytheuniversity,RamanujanwasstayinginahouselocatedonEastAgraharamStreet,nearthetownregistrar’soffice;agraharammeanttheBrahminquarter.CrossingtheTofAgraharamStreetwasastreetthatdroppeddown to theCauvery at a spot wheremen andwomen came to wash clothes andbathe.Andthere,onAugust11,1919,at the timeof theSravanamceremony thatmarkedtheannualchangingofthesacredthread,Ramanujanopenlyrebelledagainsthismother.

Troublehadbeenbrewingforweeksbetweenthetwoofthem,fromevenbeforethey’darrived in town;Ramanujanhadwanted to travel first class, buthismotherhad insisted on second or third class.Now, Ramanujan was heading down to theriver to bathe, as part of the Sravanam rites. Janaki wanted to go with him.Ramanujansaidyes.Komalatammalsaidno.

AndRamanujaninsisted,yes.Itwasaturningpoint.Allthehostilitypentupoverthepastfewyears,thediverted

letters,theendlessquarrels—andnowthis.Hespokerespectfully,butfirmly:Janakiwasgoingwithhim.

After that, Janaki began to occupy a larger place in her husband’s life.He grewfreerinhercompany.Hetoldher,morethanonce,“IfonlyyouhadcomewithmetoEngland perhaps I would not have fallen ill.” On Friday afternoons, he wouldsometimeswatchheroil andwashherhair, shaking itoutand letting itdry in thesun.Thesightgavehimgreatpleasure,sometimesenoughtodrawhimawayfromhiswork. Incomingmonths, itwouldbe Janakiwho, increasingly,wouldcare forhimduringtheday,givehimhismedication,nursehimatnight.When,atonepoint,hismotherurgedhimtosendJanakipacking,backtoherparents,Ramanujanrefused.

FortwomonthsRamanujanstayedinKodumudi.EverySundaythedoctorvisited—aC.F.Fearnside,DivisionalMedicalOfficer.Anditmayonlyhavebeennow,inquestioninghim,thatKomalatammalgraspedthefullextentofherson’scondition.At one point Fearnside suggested that Ramanujan go to Coimbatore. At another,Thanjavur, near Kumbakonam, was the recommendation. (Ramanujan would nothear of it. “They wantme to go toTan-savu-ur,” he said, punning on the word’sTamil components, “the place-of-my-death.”) Butwith theworst of summer over,thepressurewasstrongtogosomewherelessout-of-the-way.

KomalatammaldecreedthatitbeKumbakonam.In lateAugust, perhaps grateful to get away from a domestic situation that had

turned against her, she went on ahead to look for a house, the family home on

Sarangapani Sannidhi Street apparently being deemed inappropriate for her son’scare.On September 3, Ramanujan and the rest of the family left Kodumudi and,aroundduskofthefollowingday,arrivedinKumbakonam.

The surroundings of his childhood and adolescence may have mellowedRamanujan a little because stories from this period bear a softer quality, even anostalgic tinge. One of Ramanujan’s former classmates told how “he insisted onseeingmyaunt,whoisanoldwidowonthewrongsideof70,whomheknewwhileatschool some20years ago.”Old friendsdroppedby.Balakrishna Iyer, oneof thoseRamanujan had approached for a job around 1910, sent him dried brinjal slices.RadhakrishnaIyer, thePachaiyappa’sCollegeclassmatewhohadnursedhimwhilehewassickin1909,recalledRamanujanasjust“abundleofbones,”lyingonhiscot.“Ramanujanhadpowerfulandpenetratingeyes.Foramomenthiseyesflashedalookofrecognition,”andhemumbledRadhakrishna’sname.

Ramanujan had a new doctor now, P. S. Chandrasekar, brought down fromMadrasespeciallytocareforhim.Hardy,throughtheRoyalSociety,hadapproachedtheauthoritiesinMadras.ThesurgeongeneralhadturnedtoChandrasekar,fifty,aprofessor of hygiene and physiology atMadrasMedical College and a prominenttuberculosisspecialist.Atonepoint,ChandrasekaraccompaniedoneofRamanujan’soldestfriends,Sarangapani(theboywhohadoncebestedhiminanarithmeticexam)tothehouseonBhaktapuriStreetwhereRamanujanlaysick.FormorethananhourChandrasekar examined him. It was tuberculosis, he was sure, notions of bloodpoisoningnotwithstanding.“Ihaveafriendwholovesmemorethanallofyou,whodoes not want to leave me at all,” Ramanujan told Sarangapani later. “It’s thistuberculosisfever.”

WhenRamanujanboardedtheNagoyatoreturntoIndiaearlythatspring,hopesfor recovery, at least in Hardy’s estimation, had been high. But the kind ofsanatoriumstreatinghimforthepasttwoyearshadoneessentialfailing:theydidn’tcure tuberculosis.Of thosedischarged fromEnglishsanatoriums, one study found,between one-and two-thirdswere deadwithin five years,most of themwithin thefirst two. InRamanujan,Chandrasekarnowconcluded, thediseasehad reachedanadvancedstage.HisfatelaymorewithGodthaninanythingmedicinecoulddo.

Sincecomingdownsicktwoandahalfyearsbefore,Ramanujanhadbeenthroughthe wringer. In England, he’d seen many doctors, heard countless conflictingdiagnoses,beenshuffledaroundtonumeroushospitalsandnursinghomes.NowinIndia, itwas the same thing.Twoplaces inMadras.Kodumudi.Perhapsbriefly inCoimbatore.Kumbakonam.All inninemonths.Moredoctors.Moreexaminations.

More diagnoses. Through it all, his condition grew gradually worse. Now, inKumbakonam,thedownwardslideaccelerated.

JustbeforeChristmasof1919,whilewritingHardywithdetailsofRamanujan’saccounts,Dewsburyadded:“Heisstillinverybadhealthanddifficulttodealwith,livingup countrywithhis family.Mr.RamachandraRao is doingwhat he can forhim,butMr.Ramanujanhimselfwillnotconsent to live ina suitableenvironmentunderpropertreatment.Itisagreatpity.”Byoneaccount,herefusedtobefurthertreated,byanotherhehadsimplygivenupthewill to live. Itwas likepulling teetheventogethimtogotoMadras,whereChandrasekarwantedhim.Itwasthecoolest,mostpleasanttimeoftheyear;theheatthathadoriginallydrivenhimtoKodumudihadgivenwaytodaytimetemperaturesintheeighties.Finally,Ramanujanrelented.

Sometimeprobablyalittleafterthefirstoftheyear,Ramanujan,hismother,wife,andbrother-in-law,Srinivasa,setoffforMadras—but,forthemoment,gotonlyasfarastherailwaystation.Theyhadmissedthetrain,thenextonenotdueforanothersixorsevenhours.RatherthansubjectRamanujantoanotherbumpy,two-mileridetoBhaktapuriStreet,onlytoturnaroundafewhourslater,theycampedoutatthestationontheedgeoftown.Periodically,JanakiandKomalatammalreturnedtothehouseforfoodorotherprovisions.

The train ultimately came, of course, and they reached Madras. On Januaryfifteenth, when Dewsbury wrote Hardy again, he gave Ramanujan’s address as“Kudsia,”HarringtonRoad,Chetput,Madras.

3.THEFINALPROBLEM

Ramanujanwas living in one of a group of substantial stucco houses built for theBritisharoundtheturnofthecenturyinawesternsuburbofthecity.Givennameslike Sydenham, Ravenscroft, and Lismoyle, and sometimes supplied with tenniscourts, they were popular with middle-rank Indian Civil Service officers andcompanyofficials.SeveraldozenofthemhadgoneupalongalaneperpendiculartoHarrington Road, now called Fifteenth Avenue, but during Ramanujan’s timeunnamed.“Kudsia”wasoneofthem,thoughapparentlyRamanujandidn’tstaytherelong.Foratime,hewasinstalledinaconsiderablylargerone,“Crynant,”inaculde-sacattheendofthedustylane.Ultimately,hewouldbemovedtoanother,knownas“Gometra.”

It was from one of these houses that on January 12, 1920, Ramanujan wroteHardyforthefirsttimeinalmostayear.

I am extremely sorry for not writing you a single letter up to now. . . . I discovered very interestingfunctionsrecentlywhichIcall“Mock”thetafunctions.Unlikethe“False”thetafunctions(studiedpartiallyby

Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as the ordinary thetafunctions.Iamsendingyouwiththislettersomeexamples.

Inhowitwouldexcitetheactive interestofmathematiciansupuntilthepresent, itwas a letter much like the one Ramanujan had writtenHardy seven years earlieralmosttotheday.Byoneestimate,itrepresented“oneofthemostoriginalpiecesofmathematics,andinsomewaystheverybest,whichRamanujandid.”

AswithsomanyothersubjectsthatinterestedRamanujan,thetafunctionscouldberepresentedas infinite series.Strict rulesgoverned the formationof these series,andsolongasyouobservedthem,theyalwayshadparticularlyintriguingproperties.For example, they were “quasi doubly periodic” and they were “entire functions.”OvertheyearssinceJacobihadfirststudiedthem,thetafunctionshadcometoexertaprofound impact on fields ranging frommathematical physics tonumber theory.Booksdevotedtothemtypicallywoundupbridgingvastterrain,yieldingconnectionstofieldstowhichatfirsttheymightseemwhollyunrelated.

But theta functionswere fragile.Theproperties thatmade themso intriguing tomathematiciansvanished if, in trying to construct them,youdepartedeven slightlyfromtheircanonicalform—forexample,bymerelychangingthesignsofsomeoftheterms; at a glance, they might seem indistinguishable from theta functionsthemselves, yet they lost most of their interesting properties. True, certain “falsethetafunctions”exploredbyRamanujan’sintellectualcousin,L.J.Rogers,inapaperafewyearsbefore,heldsomeinterest;laterstudywouldapplythem,forexample,topartitions. But compared to the real, or “classical,” theta functions thatmathematiciansfoundsorichwithmeaning,theywerepalesubstitutes.

Now,inhis lettertoHardy,Ramanujanwastellinghowanewclassoffunctionscouldbe constructed that were not theta functions but bore crucial similarities tothemandwere just as interesting.Theywerenot “false” theta functions, like thoseRogershadinvestigated,butwhatRamanujantermed“mock”thetafunctions.

Classical theta functions could be put into a variant “Eulerian” form, in whichtermslike(1−q)or(1−q3)appearedinthedenominator.Thatmeantthatwhenq= 1, say, or q3 = 1, the functions were mathematically undefined and had to beexplored in alternative ways (in a way loosely analogous to how Ramanujan andHardyhadexploredthepartitionfunctionthroughtheir “circlemethod”).Offeringtwo theta functions as examples,Ramanujan noted that at such “singularities”—atthepointsdemandingspecialstudy—“weknowhowbeautifullytheasymptoticformofthefunctioncanbeexpressedinaveryneatandclosedexponentialform.”

All this was spadework, with which Hardy was familiar. But now Ramanujanrangedbeyondit:

Now a very interesting question arises. Is the converse of the statements concerning the [two thetafunctions]true?Thatistosay:SupposethereisafunctionintheEulerianformandsupposethatalloraninfinityofpointsareexponentialsingularities,andalsosuppose...

andonhewent likethis, fashioninghis functionbeforeHardy’seyes.Whenithadsatisfied a host of such special conditions itwas, he said, a “mock theta function.”Then he went on to offer a whole slew of them—four third-order mock thetafunctions,tenfifth-orderfunctions,andsoon.

“The first threepages inwhichRamanujanexplainedwhathemeantbya ‘mocktheta function’ are veryobscure,”G.N.Watson, oneof the first of those to studythem,would allow. But hazy and poorly expressed as theywere, they held withinthemprofoundmathematicaltruthsthatbecamethesubjectofWatson’spresidentialaddress to the London Mathematical Society sixteen years later and would keepothermathematiciansbusyforyears.WroteWatson:

Ramanujan’sdiscoveryofthemock-thetafunctionsmakes itobviousthathisskilland ingenuitydidnotdeserthimattheoncomingofhisuntimelyend.Asmuchasanyofhisearlierwork,themock-thetafunctionsare an achievement sufficient to cause his name to be held in lasting remembrance.To his students suchdiscoverieswill be a source of delight andwonder until the time shall comewhenwe too shallmake ourjourneytothatGardenofProserpinewhere

Pale,beyondporchandportalCrownedwithcalmleaves,shestandsWhogathersallthingsmortalWithcold,immortalhands.

In1893,ArthurConanDoyle,creatorofSherlockHolmes,wroteastoryinwhichthefamousdetective,grapplingwithhisarchfoeProfessorMoriartyatReichenbachFallsinAustria,plungestohisdeathontherocksbelow—orsoHolmes’sfriend,Dr.Watson,is ledtobelieve.Now,fourdecades later,discussingRamanujan’ssurgeofmathematical creativity in the months before his death, this other Watson, themathematician, resurrected the titleConanDoyle hadused forSherlockHolmes’slastcase.Mockthetafunctions,hesaid,wereRamanujan’s“FinalProblem.”

AllthatyearinIndiaRamanujanworkedonmockthetafunctions,“q-series,”andrelatedareas,fillingpageafterpagewiththeoremsandcomputationalfragmentsandinfiniteseriesgoingoffineverydirection,byonereckoningabout650formulasinall.WhentheAmericanmathematicianGeorgeAndrews began poring through themhalfacenturylater,hewasstunnedbytheirrichness,thesurprisestheyoffered,wasleftmystified over how anyone could think them up. “It is very difficult even forsomebody trained inmathematics,butnotanexpert, to tell themapart,”hewouldsay.Amongone groupof five superficially similar formulas, “the first one tookme

fifteen minutes to prove, the second an hour. The fourth one followed from thesecond.Thethirdandfifthtookmethreemonths.”

Ramanujan’sworkduringtheyearbeforehediedcouldbeseentosupportanoldnostrum of the tuberculosis literature—that the tuberculous patient, as hesuccumbed, was driven to an ever-higher creative pitch; that approaching deathinspiredafinalflurryofcreativityimpossibleduringnormaltimes.Theideahasbeensoundlytrouncedbymodernscholarsquicktociteartistsandothercreativeworkerswhosegreatestworklongprecededtheirillnessandwhosedeathbedproductivitywasnil.Still,duringRamanujan’stime,theideahadafollowing.AndatleastoneaccountofRamanujan’s lastdays,byP.V.SeshuIyer,boreitsflavor:“Therearenopapersandresearchesofhismorevaluednormoreintuitive,”hewrote,“thanthosewhichhethoughtoutduringthesefatefuldays.Hisphysicalbodywasfailingnodoubt,buthisintellectualvisiongrewproportionatelykeenerandbrighter.”

ButGeorgeAndrews,forone,couldexplainRamanujan’sremarkableproductivityin more prosaic, if doubtless sounder, terms. “He had the full power of hiscollaboration withHardy behind him and the sort of wild eccentric genius of hisyouthbehindhim,”too.Ramanujan,inotherwords,shouldhavebeenattheheightofhispowers. “Andthisnotebook,” saidheof theproductof that finalyear, “bears itout.”

•••

Sometime during late winter or early spring of 1920, while staying in “Crynant,”Ramanujan complained to his mother that the “cry” of that name seemedinauspicious.KomalatammalwenttoNamberumalChetty,ownerofthebungalows(as well as of a small railroad) and a friend of Sir Francis Spring. OmittingRamanujan’s real reasons,Komalatammal toldhimher sonneededaquieterplace.Namberumal obliged, and Ramanujan was moved a little down the road to“Gometra,”whichmeanssomethinglike“FriendofCows,”areferencetoKrishna.

Itwasahousefullofdoors.Therewerebanksofdoorseverywhere,doorsleadingbackoutsidetoanarchedportico,doorstoaninteriorcourtwherethecookingwasdone in a separate outbuilding, literally dozens of doors ranging over the house.Whentheywereallopen,theinsideofthehouse,withitsflooroflargegraniteslabs,was like a cool, shadedglenopen toany straypassingbreeze.You’d enter throughwhat from the street seemed a side entrance butwhich, leading you under one ofseveral arches, actually brought you into the center of the house. There stood astairway,itsbanistersfashionedfromBurmateak,thatleduptothesecondfloor.To

yourright, towardthe frontof thehouse,wasa large livingroom.Ramanujanmaybrieflyhave stayed there,but in timehewas shifted toa small corner roomon theothersideofthestairsatthebackofthehouse.

Enhancingitssenseofopennesswasthebungalow’sscantfurniture.Ramanujan’sbedwas amattress and a pillow laid out upon the bare granite floor.He scarcelybudged from it.Lying there,he could lookupandseeamassivewoodbeam, fromwhich,likeaspine,thesecondfloorwassupported,spanningthelengthoftheroom.It was quiet back here away from the street. Visitors moved silently so as not todisturbhim.Tirunarayanan,who fondly recalled his older brother horsing aroundwithhim,tellinghimstoriesandcarryinghimonhisshoulders,nowfoundaccesstohimjealouslyguarded,withnooneallowedtobotherhim.

Still,Ramanujancouldscarcelyhavegoneunawareofthetensionsthatgrippedthehousehold.Thewholefamilywasinastateofjanglednerves.Andthathewassick,miserable,angry,andpronetosuddenoutburstsdidnothingtoeasematters.Againandagain, Janaki andKomalatammal clashed, absorbing energies thatmightbetterhavegonetowardRamanujan’scare.Now,moreover,thecontestwasnotsounequalasbefore.Janakiwastwenty,enjoyedanewandstrongerpositioninthehousehold,andhadherbrotherSrinivasabyherside,too.

AroundMarch,Komalatammal appeared at the door ofG.V.NarayanaswamyIyer, a former student ofmathematician P.V. Seshu Iyer, Ramanujan’s long-timefriendandchampion,whosenoteofintroductionshenowbore.NarayanaswamyIyerwasahighschoolteacherinTriplicane,butitwasnotforhispedagogicabilitiesthatKomalatammalwantedtoseehim.Hewasalsoanotedastrologer.

Narayanaswamy Iyer asked for a copy of the horoscope she wanted read.Komalatammaldictateditfrommemory.This,hesaidafterstudyingitforawhile,wasthecharteitherofamanofworldwidereputationapttodieattheheightofhisfame,oronewho,ifhedidlivelong,wouldremainobscure.“Whoisthisgentleman?Whatishisname?”heasked.

Itwas, of course,Ramanujan, andKomalatammal, in tears, replied that shehaddrawnthesameinferencefromhishoroscope.

Narayanaswamy backpedaled.Ramanujan! “I am sorry I was so hasty,” he said.“PleasedonotcarrywhatIsaidtoanyofhisrelatives.”

“IamthemotherofRamanujan,”shereplied.Gropingforawayoutoftheuncomfortablescene,Narayanaswamysuggestedthat

maybe his wife’s horoscope might mitigate the bad news. “Can you bring herhoroscope next time?” he asked. A separate visit was hardly necessary:Komalatammalrattleditoffforhim.

The astrologer looked for something—anything—bywhich to inspire hope. Butafterhalfanhour’sstudyofthetwohoroscopes,laidsidebyside,hewasreducedtonothing beyond suggesting that perhapsRamanujan and Janakimightwish to liveapartforawhile.

That, of course,was justwhatKomalatammalwanted tohear. “I have been . . .pleadingwithmysontosendherawaytoherparents’house,”shesaid.“But,alas,heutterlyrefuses.Hehasbeenalwaysanobedientson.Butinthismatter,hisobstinacyisunbreakable.”

•••

DidRamanujannowknowhewasdying?According to another of those enduringmythsthatsurroundedtuberculosis,thevictimwasthelasttorealizehisapproachingdeath. “Buoyed up by spes phthisca, a delusive hope of recovery,” one student oftuberculosis, Nan Marie McMurry, recounts, “the consumption patient wassupposedtoanticipateareturntohealthonthedeathbed.”

ButRamanujansufferednosuchdelusionsandoftenduringhisfinalmonthstoldhisdoctorhehadlostthewilltolive.Earlier,itistrue,hehadsaidhewouldacceptauniversitypositionofferedhimoncehehadrecoveredhishealth.AndasrecentlyasJanuary,aroundthetimehewroteHardyaboutmockthetafunctions,heexpressedinterestinsubscribingtonewmathjournals.Butthedarksuperstitioussideofhimsawthingsmoreblackly—and, as ithappens,more accurately.Hehad longbeforereadhisownhoroscopeaspredictinghisdeathbeforeagethirty-five.AndbroughttoHarringtonRoad,hepunnedonthenameoftheMadrassuburb,Chetput,inwhichitwaslocated;“chat-pat,”inTamil,meant“itwillhappensoon.”

Duringhis lastmonths,Ramanujandrew closer to Janaki,withwhom,whenhewasn’texplodingatherinafit,henowhadawarmer,morerelaxedrelationship.“Hewasuniformlykindtome,”Janakirecalled.“Inhisconversationhewasfullofwitandhumor,”was forever cracking jokes.As if trying to cheerherup,hepliedherwithtales of England—of his visits to the BritishMuseum, and the animals he’d seenthere,ofthetimeinCambridgewhenanEnglishguesteatingaSouthIndiandishhehadpreparedchompeddownonapieceofhotpepper...

Hislifecamedownalittlefromtheheightsofmathematicstosmallthings,humanthings.HewouldsummonJanakiwithalittlebell,orwouldtapwithastick.HetoldC. S. Rama Rao Sahib, Ramachandra Rao’s son-in-law, howmuch he craved therasam he had enjoyed almost a decade before in those destitute days of 1910 atVictoriaHostel—andloveditwhensomebodybroughthimsome.

Still, it was not an easy death. Whatever cheerfulness or equanimity he couldmuster—and some of his friends preferred later to remember him that way—paperedoveragrimdespair.Hewassullenandangrymuchofthetime.Hismoodwasvolatile,restingonahairtrigger.Janakifeltlater—sohadHardyinEngland,andsodidmanyofhis friends inIndia—thathis illnesshadaffectedhismind.Hewasforever raving at one thing or another. There was a tray kept nearby with lentils,spices, and rice, and once, in a fit of anger and pain, Ramanujan pounded it alltogetherwitha stick.NarasimhaIyengar,hurtat findingRamanujanso sullenandcold when he greeted him at the Central Station, visited him now again onHarringtonRoad. “I found tomy great grief that though physically living, he wasmentallydeadtotheworld,eventohisoncedearfriends.”

Toward the end, “he was only skin and bones,” Janaki remembered later. Hecomplainedterriblyof thepain. Itwas inhis stomach, inhis leg.When itgotbad,Janaki would heat water in brass vessels and apply hot wet towels to his legs andchest;“fomentation,”itwascalled,standardtherapyatthetime.Butthroughallthepainandfever,throughtheendlesshouseholdsquabbles,throughhisowndisturbedequanimity,Ramanujan,lyinginbed,hisheadproppeduponpillows,keptworking.When he requested it, Janaki would give him his slate; later, she’d gather up theaccumulated sheets ofmathematics-covered paper towhich he had transferred hisresults andplace them in thebig leatherboxwhichhehadbrought fromEngland.“Hewouldn’ttalktoanyonewhocametothehouse,”saidJanakilater.“Itwasalwaysmaths....Fourdaysbeforehediedhewasscribbling.”

EarlyonApril26,1920,helapsedintounconsciousness.Fortwohours,Janakisatwith him, feeding him sips of dilutemilk.Aroundmidmorning or perhaps a littleearlier,he died.With himwere his wife, his parents, his two brothers, and a fewfriends.Hewasthirty-twoyearsold.

Atthefunerallaterthatday,mostofhisorthodoxBrahminrelativesstayedaway;Ramanujanhad crossed thewaters and, too sick onhis return tomake the trip toRameswaram for the purification ceremonies his mother had planned, was stilltaintedintheireyes.RamachandraRaoarrangedthecremation,throughhisson-in-law and Ramanujan’s boyhood friend, Rajagopalachari. At about one in theafternoon,hisemaciatedbodywasputto the flameson thecremationgroundnearChetput.

The next day, assigning it Registration No. 228, a government clerk officiallyrecordedhisdeath.

4.ASONOFINDIA

TheastrophysicistSubrahmanyanChandrasekhar grewup inLahore, inwhatwasthen northern India, moved to Madras when he was eight, attended PresidencyCollegethere,thenCambridge’sTrinityCollege(wherehemetHardy),wentontopropose the theoreticalunderpinnings forblackholes, andwasawarded theNobelPrizein1983.In1920,hewasjustnineyearsold,buthewellrecallshowonedayinlateAprilofthatyearhismother,readingaboutitinthenewspaper,toldhimofthedeath of Ramanujan. “Though I had no idea at that time of what kind of amathematician Ramanujan was or indeed what scientific achievement meant,”ChandrasekharwouldtellanAmericanaudiencealmostsevendecadeslater,

IcanstillrecallthegladnessIfeltattheassurancethatonebroughtupundercircumstancessimilartomyowncouldhaveachievedwhatIcouldnotgrasp.

ThefactthatRamanujan’searlyyearswerespentinascientificallysterileatmosphere,thathislifeinIndiawas not without hardships, that under circumstances that appeared to most Indians as nothing short ofmiraculous, he had gone toCambridge, supported by eminentmathematicians, andhad returned to Indiawitheveryassurancethathewouldbeconsidered,intime,asoneofthemostoriginalmathematiciansofthecentury—thesefactswereenough,morethanenough,foraspiringyoungIndianstudentstobreaktheirbondsofintellectualconfinementandperhapssoarthewaythatRamanujanhad.

In the India of the 1920s, he would say at another time, “We were proud ofMahatma Gandhi, of Nehru, of [the Nobel Prize-winning poet Rabindranath]Tagore,ofRamanujan.Wewereproudofthefactthatanythingwecoulddowouldequate to anything else in the world.” Within Indian mathematics, of course,Ramanujan’s influence extended correspondingly deeper. “I think it is fair to say,”Chandrasekhar would observe, “that almost all the mathematicians who reacheddistinctionduring the three or four decades followingRamanujanwere directly orindirectly inspired by his example.” (One brilliant young mathematics student atMadras’s Presidency College, T. Vijayaraghavan, deliberately neglected his studiesandfailedhisexaminations,themoreperfectlytofollowinRamanujan’sfootsteps.)

SuchwasRamanujan’simpactonIndiaduringtheyearsafterhisdeath.Butbackinthefamily,preoccupiedbythedailycaresoflifeandfilledwithpainfulmemoriesofthelastdifficultyears,hisdeathleftawakemuchsmaller,yetjustaskeenlyfelt.

Inthe1930s,Chandrasekar,thetuberculosisexpertwhohadtreatedRamanujanin his final months, was asked by an old Madras friend of Ramanujan whetheranything could have been done to save him. Yes, he replied, flying into a rage,Ramanujancouldhavebeensaved,shouldhavebeen,andhismotherandwifeweretoblame,atleastinpart.ThedayafterRamanujandied,hewroteinhisdiary:

Ifhehadbeenallowedtofollowmyinstructions,thisdoubletragedyneednothavetakenplace.TheneglectofRamanujanduringhisearlyphase—perhapspartlyduetotheignoranceofhiscontemporaries,aswellas

his relatives’ (mother’s and wife’s) contributory (I almost feel like using the stronger word “criminal”)negligencehavecontributedtothisdoubletragedy—atragedywhichistoodeepfortears.

Chandrasekar(whohadlosthismostillustriouspatient,andsomayhavehadhisown axe to grind) pictured both wife and mother wallowing in the mire ofmaterialism.ButbothbeforeRamanujan’sdeathandnow,afterit,theywerescarcelyin a position to ignore the economic facts of their lives. Whatever else he was,Ramanujanwas, throughhis ample fellowships, a breadwinner.Nowhewas gone,andyes:therewasagrapplingforwhatseemedmightbethespoils.

OnApril29,withRamanujan’sashesscarcelycold,hiseighteen-year-oldbrother,LakshmiNarasimhan,wroteHardywith thenewsofhisdeath (whichHardyhaddoubtlessalreadyheard).Using languageappropriate toabattledispatch,LakshmiNarasimhancomplainedthatRamanujan’sbooksandpaperswere“entirelyunderthecontrolofhiswifehere.”HehadheardthroughRamachandraRao’sson-in-law,hesaid,thatthegovernmentwouldbeprovidingJanakiasmallmonthlystipend.“Iamvery sorry that such an arrangement was notmade formy family.” Janaki, plainlyenough,wasnotpartofthefamily.

It was a pathetic plea, and one that hinted at the pressures Ramanujan hadwithstoodasayoungman inorder topursuemathematics rather thanestablishinghimself in a good job.His father had gone blind about when Ramanujan left forEngland,saidLakshmiNarasimhan.Hisgrandmotherwaslame.“Besidesthese,”saidthe young man in whose hands the family’s fortunes now rested, “I have got acorpulent mother who resembles my brother in all his physical features.” As forhimself,hehadsuspendedeffortsonhisownbehalfover thepastyear, inorder tohelphisbrother.

Ihavenounclesorcousinstoprotectme.Wehavenoproperty,asyoumighthaveknownverywell.IhaveagreatdesiretostudyandIwish“toneglectworldlyends,alldedicatedtoclosenessandthebetteringofmymind” (Tempest). Ihaveno taste, sorry to say, forMathematics. I like to readShakespeare,Wordsworth,Tennyson,andwish to travel in the fairy land “half flyinghalfon foot.” Idonotknowhow to feed them.Therefore,IhumblyrequestyoutowritetotheMadrasUniversitytogiveamonthlyallowancetous.IhavebeentoldthatmybrotherisentitledtogetasumfromtheCambridgeUniversity.

In short, I ama youngman.Onedaypasseswithgreatdifficulty. Idonotknowhow toprotect them.Unlessanyarrangementismadetosupportus,wehavetogoa-beggingfromdoortodoor.Ientrustthewholecaseintoyourhands.Itisyourboundendutytoprotectus.

ThereisnoreasontothinkHardydidanythingofthekind,andthefamilydriftedbackintosomethingliketheobscuritythatwas,savefortheaccidentofRamanujan’sbirth,itsnaturallot.

AfewmonthsafterRamanujan’sdeath,hissixty-five-year-oldfathergotsickandwasbrought to thePycroft’sRoadhouseofNarayana Iyer.There,Komalatammal

andNarayanaIyer’swife lookedafterhimuntil, inNovember, in thehousewhereRamanujanandNarayanaIyerhadstayeduplatedoingmathematics,hedied.

Komalatammal,meanwhile, never recovered from her son’s death. “She lost heroptimism and became very often sullen,” said one old friend of the family, K.SrinivasaRaghavan,whohadbecomeacollegeprofessor.Aftermakingnoefforttocontacthimforthepastsevenyears,inAugust1927shewroteHardy,invokingthehelp of a scribe or family friend to cast her Tamil into lucid English. Her sonLakshmiNarasimhan, she reported, had passed his Intermediate examination (thenewnamefortheF.A.examRamanujanhadalwaysfailed)andnowworkedforthepostofficeinTriplicane.TirunarayananattendedPresidencyCollegeandhadearnedhisB.A.degreeearlythatyear.

Most of the rest of the three-page letter outlined the family’s fragile economicfooting,detailedthemeagerpensionsitreceived,andrecountedhertroubles.“IsentmysontoEnglandin1914thinkingthatmyfamily,everweddedtopoverty,wouldbecome rich, and that my son would become famous. Like Achilles, he woneverlastingfame;likeAchilles,hediedyoung.”Now,sheaskedHardytointercedeonher sons’ behalf with the government and with the India Office in London. Shewantedforthemhighpositions inthepostofficedepartment—thebetter-educatedTirunarayanan as probationary superintendent of post offices, and LakshmiNarasimhan as inspector of post offices in Madras. Whether or not Hardyinterceded,Tirunarayanan did become assistant postmaster.Themore flamboyantLakshmiNarasimhandiedwhilestillyoung.

Long-timefamilyfriendAnantharaman,atwhosehouseRamanujanateoftenasaboy, had been in bed recovering from a leg operation when his uncle told him ofRamanujan’s death. It came as a great shock. Later, during the late 1930s,Komalatammalwould visit him and his family inTriplicane and, hewould recall,“consoleherselfbyseeingus,andsayshewasfeelingasthoughshewereseeinghersonChinnaswamihimself.”

Janaki, meanwhile, had gone her own way. Ramanujan had worried about howshe’dbetreatedbythefamilyafterhisdeath.Shewasatwenty-year-oldwidow,andwidowswerevirtuallyanoppressedclassinIndia,apttobeill-treatedanddespised.She had no education, no skills. Her brother and mother, who had arrived inChetputtwodaysbeforeRamanujan’sdeath,figuredshecouldexpectnohelpfromKomalatammalandtherestofRamanujan’sfamily.

SoafterthecremationshereturnedwithhermothertoRajendram.Then,forthenext six years she lived with her brother, who became an income tax officer, inBombay.Whenshelearnedofthetwenty-rupee-per-monthpensionawardedherby

theuniversityinreturnforherrightstoRamanujan’spapers,shereturnedtoMadras,brieflystayingwithhersisterinTriplicane.Soonshefoundherownplace—ahouseonHanumantharayanStreet two doors fromwhere she andRamanujan had livedbeforehe’dleftforEngland.Thereshestayedformostofthenexthalfcentury.

ShehadlearnedtoembroiderandworkasewingmachineinBombay,andsonowshescratchedoutalivingmakingclothesandteachingtailoringtogirls.In1937,S.Chandrasekhar,theastrophysicist,askedbyHardytotrytofindagoodphotographofRamanujannext timehewas inIndia, trackedherdowninTriplicane.Shewas,Chandrasekhar reported back, “having a rather difficult life, some of herunscrupulous relatives having swindled her out of such financial resources asR[amanujan]had lefther,” andunable even to getholdof a copyofherhusband’sCollectedPapers,publishedafewyearsbefore.

Around1948,Janakibegantakingcareofasmallboy,Narayanan,whosemotherwas in the hospital andwho himself suffered from typhus. She visited him in thehospital, nursed him there, brought him books for school. Later, both his parentsdied,andNarayananwenttolivewithher.Whenshecouldn’tbothtakecareofhimandsupportherselfasaseamstress,hebrieflylefttoattendaresidentialschool.Butwhenhewasaboutfifteen,hecamebacktolivewithherforgood,andsheadoptedhimasherson.

•••

At the time of Ramanujan’s death in April 1920, the editor of the Journal of theIndianMathematicalSocietyhadfallensofarbehindhispublicationschedulethattheissue bearing the newswas datedDecember 1919. Into copies of that issue, smallolivegreenslipsofpaper,borderedinblack,wereinserted:

THELATEMR.S.RAMANUJAN

Wedeeply regret to announce theuntimelydeathofMr.S.Ramanujan,B.A.,F.R.S., onMonday, the26thofApril1920, athis residence atChetput,Madras.Anaccountofhis life andworkwill appear in asubsequentissueofthisjournal.

Sevenmonthslater,thejournalcarriedtwoobituarynotices.One,byP.V.SeshuIyer, furnished the facts and dates; it may have been at his request that LakshmiNarasimhan had assembled the family record of Ramanujan’s life. The other, byRamachandraRao,reprintedfromanothermagazine,wasmorelyrical:“Andheisnomore,”itbegan—

Hewhosenamesheda lustreonall India,whosecareer isunderstoodas theseverestcondemnationof thepresentexotic[sic]systemofeducation:whosenamewasalwaysappealedto, ifanyoneforgetfulofIndia’spastventuredtodoubtherintellectualcapabilities.

There was something endlessly appealing about Ramanujan’s life to Indiansensibilities.Somemight see his early school failures, asHardy did, as “the worstinstancethatIknowof thedamagethatcanbedonebyan inefficientand inelasticeducational system.”Toothers among theEnglish,his lifemight seem the stuff ofcinema,arags-to-intellectual-richesstory.ButtomanyIndians,itwastheideaoftheluminous light of his intellect lurking behind the rags that was so compelling, hisseemingdisdain for the shallowplauditsof theworld. “Evenwhen twoContinentswerepublishinghisresults,”wroteRamachandraRao,

heremainedthesamechildishman,withnostyleindressoraffectationofmanner,withthesamekindface,withthesamesimplicity.PilgrimscametoRamanujan’schambersandwonderedifthiswashe....IfIamtosumupRamanujaninoneword,Iwouldsay,Indianality.

Evenbeforehisdeath,hewasbeingseenasaboontoemergingIndianidentity:“Byhisuniquemathematical talentsandbytheamountofusefulandoriginalwork[hehasperformed],”theJournalof the IndianMathematicalSociety saidofhim, “hehasraised India in the estimation of the outside world.” S. Chandrasekhar wouldobserve, accurately enough, that “Ramanujan represents so extreme a fluctuationfromthenormthathisbeingbornanIndianmustbeconsideredtoalargeextentasaccidental.”AndIndiacouldtakepride inmanyothersamong itscountrymenwhohadmadeitintheWest,oruponthegreatstageoftheworldgenerally.Still,atleastamongitsscientists,fewweremore“Indian”thanRamanujan.

India was a shapeless mass of poverty, of ceaseless struggle for the materialnecessitiesoflife?Ramanujan,ifonlyreluctantlyandonlyinordertowork,sharedinthatstruggle.

IndiawastheEssentialEast,standingapartfrom,andindependentof,theWest?Ramanujan had spent his formative years, and done much of his most individualwork, within a day’s rail journey of his hometown. He had lived a life almostuntouchedbytheseductivecharmsoftheWest,hadneverhadanyparticulardesiretoleaveIndia.Andwhenthetimecamethathedid,hebristledathistransformationintoanEnglishgentleman.

Indiawasaplaceofspiritualvaluestriumphant?Ramanujan,atleastupuntiltheveryendofhislifeandperhapseventhen,neverforsooktheIndiangods.Heinvokedthe name of the goddessNamagiri.Hewas steeped in, and accepted,most of thevalues,beliefs,andwaysoflifeofSouthIndia.

The Indian psychologist Ashis Nandy has made a similar point in his bookAlternative Sciences, contrasting Ramanujan with the Indian physicist and plantphysiologist J. C. Bose. Bose, he wrote, was ever troubled by the split within himbetweenEast andWest,whileRamanujan, themore autonomous of the two,wasnot. “Inhisnationalism, asmuch as in hisWesternizedmodernity, [Bose]was farmore deeply bound up with the West, both in admiration and in hatred.”Ramanujan,ontheotherhand,waswhohewas—aSouthIndianBrahminthroughand through. Less thanmany other Indians was Ramanujan’s head turned by theWestanditsways.

Not through anything he professed but in the very model of his life and hisachievements,RamanujanlivedoninthesoulofIndia.“TheBritishthoughtIndianswere inferior, andRamanujan showedotherwise,” saysP.K.Srinivasan, aMadrasimathematics teacherwhohas compiled abookof reminiscences aboutRamanujan.“Heboostedourmorale.”InE.H.Neville’sviewitwastheparticularplayingfieldonwhich Ramanujan triumphed—mathematics, in all its austere purity—thatmagnifiedhisimpact.In1941,NevillewroteoutaradiolectureonRamanujan.Histalk was broadcast but, whether due to constraints of time or politics, he did notdeliverallhehadprepared.Amongcommentsleftunairedwerethese:

Ramanujan’s career, just because he was amathematician, is of unique importance in the development ofrelationsbetweenIndiaandEngland.Indiahasproducedgreatscientists,butBoseandRamanwereeducatedoutsideIndia,andnoonecansayhowmuchoftheir inspirationwasderived fromthegreat laboratories inwhichtheirformativeyearswerespentandfromthefamousmenwhotaughtthem.Indiahasproducedgreatpoetsandphilosophers,butthereisasubtletingeofpatronageinallcommendationofalienliterature.Onlyinmathematicsarethestandardsunassailable,andthereforeofallIndians,RamanujanwasthefirstwhomtheEnglishknewtobeinnatelytheequaloftheirgreatestmen.Themortalblowtotheassumption,soprevalentinthewesternworld,thatwhiteisintrinsicallysuperiortoblack,theoffensiveassumptionthathassurvivedcountlesshumanitarianargumentsandpoliticalappealsandpoisonedcountlessapproaches tocollaborationbetweenEnglandandIndia,wasstruckbythehandofSrinivasaRamanujan.

InLondon,atameetingoftheMathematicalSocietyonJune10,1920,accordingtoitsminutes, “ThePresidentreferredtothe loss that theSocietyhassuffered inthedeath of Mr. S. Ramanujan and Major MacMahon spoke on the subject of Mr.Ramanujan’smathematicalwork.”

Hardy, after so recently getting the seemingly upbeat letter on mock thetafunctions,wasshockedtolearnthatRamanujanhaddied.HewroteabriefobituaryforNature, to which Neville added more information some months later, then amuchlongeronefortheProceedingsoftheLondonMathematicalSociety.

In America, the American Mathematical Monthly, expanding the section itnormally reserved for dry accounts of papers appearing in foreign journals, gave

ampleplaytotheromanticstoryofRamanujan’sdiscovery,evendowntohisphysicalappearance: “In conversationhebecameanimated, andgesticulatedvividlywithhisslenderfingers.”

Ramanujan’s papers had somehow wound up with the University of Madras;JanakilaterchargedSeshuIyerwithspiritingthemawayduringthefuneralitself.Afewyearslater,onAugust30,1923,DewsburysentallbuttheoriginalnotebookstoHardy.

In1921,HardybroughtoutthelastofRamanujan’spapers,andthesameyear,hebegantotakeupRamanujan’sworkinpapersofhisown.Theearly1920ssawothersdosoaswell.In1922,Mordellcameoutwith“NoteonCertainModularRelationsConsideredbyMessrs.Ramanujan,DarlingandRogers.”Thefollowingyear,B.M.Wilson’s“ProofsofSomeFormulaeEnunciatedbyRamanujan”waspublishedintheProceedingsoftheLondonMathematicalSociety.

Soon after learning of Ramanujan’s death,Hardy had writtenDewsbury: “Is itpossible that Madras would consider the question of publishing the papers in acollected form? There should be some permanent memorial of so remarkable agenius;andthismemorialwouldcertainlybethemostappropriateform.”Finally,in1927, after protracted correspondence,CambridgeUniversityPress cameoutwithRamanujan’s Collected Papers, 355 pages with almost everything he had everpublished. The early Indian work was there. So was the partition function, andhighlycompositenumbers,evenquestionshehadposedreadersoftheJournaloftheIndianMathematicalSocietyandthemathematicalpartsofhisletterstoHardy.

And with its publication, as the wider mathematical world took notice ofRamanujan’swork, the floodgatesopened.Over thenext fewyears,paper followedpaper, dozens and dozens of them, with titles like “Two Assertions Made byRamanujan,” and “Noteon a Problem ofRamanujan,” and “Note onRamanujan’sArithmetical Function τ(n).” In 1928, Hardy handed over to G. N. WatsonRamanujan’snotebooksalongwithothermanuscripts,letters,andpapers,andheandB.M.Wilsonsetoutonthemathematicaladventure—editingthenotebooks—thatwas to consumeWatsonuntil the eve ofWorldWar II andwould result inmorethantwodozenmajorpapers.

Sofarasthemathematicalcommunitywasconcerned,Ramanujanlived.“During our generation no more romantic personality than that of Srinivasa

Ramanujan has moved across the field of mathematical interest,” wrote AmericannumbertheoristRobertCarmichaelin1932.“Indeeditistruethattherehavebeenfewindividualsinhumanhistoryandinall fieldsof intellectualendeavorwhodrawour interestmore surely thanRamanujanorwhohave excitedmore fully a certain

peculiar admiration for their genius and their achievements under adverseconditions.”

Writing in a kindred vein a few years later, Mordell wrote that “few othermathematiciansforsomegenerationspasthavebeenso fullofhuman interest.Thestory of his life is that of the rise of an obscure Indian in the face of the greatestdifficulties to the position of the most famous mathematician that India has everproducedandofhisearlydeathjustafterhehadwonthemostcoveteddistinctions.”

ButifRamanujan’slifeexertedapeculiarholdonmathematicians,muchmoresodid his work. In Hungary, in 1931, an eighteen-year-old University of Budapestprodigy,PaulErdos,hadwrittenapaperonprimenumbers.Histeachersuggestedhereada similarproof inRamanujan’sCollectedPapers, “which I immediately readwith great interest.” Then, the following year, he saw a Hardy-Ramanujan paperconcernedwiththenumberofprimefactorsinaninteger.

Somenumbers,recall,aremorecompositethanothers,asubjectRamanujanhadexploredinhislongerpaperonhighlycompositenumbers.Anumberlike12=2×2×3hasmoreprime factors, three, thananumber like14=7×2,whichhasonlytwo.Thenumber15hastwo,while16hasfour.Asyoutesteachinteger,thenumberof its divisors varies considerably. Well, Hardy and Ramanujan had said, we willexaminenothowthisnumbervariesbutseekitsaveragevalue—inthesamewaythatyoucannotpredictthenextthrowofthediceyetcanpredict,onaverage,howoftenparticular dice combinations will appear. Their result, loosely speaking, was thatmostintegershaveaboutloglognprimefactors:noten,takeitslogarithm,thentakeits logarithm, andyouwindupwith a crude estimate that improves asn increases.Butasroughastheresultwas,itbeatanythinganyonehadbefore,andtooktwenty-threepagesofclose-grainedmathematicalreasoningtoproveit.

Foralmosttwentyyears,HardylatertoldErdos,theirtheoremseemeddeadinthewater, no progress being made in improving it. Then, in 1934, the problem wasresurrected,and in 1939Erdos andMarkKacwere led to a theorem that took itmuchfurther.Itwasonlythenthatmathematicianscouldlookbackandpronouncethe Hardy-Ramanujan paper of 1917 the founding document of the field thatbecameknownasprobabilisticnumbertheory.

•••

InNorway,in1934,aschoolboynamedAtleSelberg,whowastobecomeoneoftheworld’smost famousnumbertheorists,cameuponanarticleaboutRamanujaninaNorwegian mathematical journal. The article termed Ramanujan “a remarkable

mathematical genius,” but as Selberg told an audience in Madras years later, theNorwegian word most aptly translated as “remarkable” also had connotations of“unusualandsomewhatstrange.”

The article, with some of the results it included, “made a very deep and lastingimpressiononme and . . . fascinatedme verymuch.”ThenSelberg’s brother, alsoflirtingwithmathematics,broughthomeacopyofRamanujan’sCollectedPapers.ToSelberg,thiswas“arevelation—acompletelynewworldtome,quitedifferentfromanymathematicsbookIhadeverseen—withmuchmoreappealtotheimagination.”Overtheyears,itretaineditsexcitementandairofmystery.“Itwasreallywhatgavetheimpetuswhichstartedmyownmathematicalwork.”

Later,hisfathergavehimasapresenthisowncopyoftheCollectedPaperswhich,ashetoldtheBombayaudience,hecarriedwithhimstill.

•••

InEngland,in1942,FreemanDyson,asecond-yearstudentatHardy’salmamater,Winchester,won a schoolmathematicsprize and selected abookon the theoryofnumbersbyHardyandE.M.Wright. “Thechapter inHardyandWrightwhichIloved the most,” he would recall many years later, “was chapter 19 with the title‘Partitions,’ ” and featuring the congruence properties of the partition functiondiscoveredbyRamanujan.Dysonwasintrigued,speculatedonwhatthecongruencepropertiesmightimply,andconceivedthenotionof“rank.”Therankofapartition,as he defined it, was its greatest part minus the number of its parts. Thus, onepartitionof9is

6+2+1=9.

Its “rank” is 3—the largest part, 6, less the total number of parts, 3. WhatRamanujan’s congruence properties implied, speculated Dyson, was that certainpartitionsbrokedownneatlyintoequal-sizedcategoriesbasedontheirrank.

“That was the wonderful thing about Ramanujan,” he would say later. “Hediscoveredsomuch,andyetheleftsomuchmoreinhisgardenforotherpeopletodiscover. In the forty-four years since that happy day, I have intermittently beencoming back to Ramanujan’s garden. Every time when I come back, I find freshflowersblooming.”

Twoyearslater,DysonwasworkingfortheRoyalAirForceBomberCommandasastatistician,wherehesawupclose,asalltheAlliedpropagandacouldnotdeny,thestaggeringlossesoverGermany.“Itwasalong,hard,grimwinter,”hewrotelater.In

theevenings,hecorrespondedwithanothermathematicianonideasfirstadvancedbyRamanujan. “In the cold dark evenings, while I was scribbling these beautifulidentitiesamidthedeathanddestructionof1944,IfeltclosetoRamanujan.Hehadbeen scribbling even more beautiful identities amid the death and destruction of1917.”

•••

Over the years, then, Ramanujan was never forgotten. A 1940 listing noted 105papersdevotedtohisworksincehisdeath.Inthelate1950s,whenMorrisNewmandelivered a paper at an Institute in the Theory of Numbers conference at theUniversityofColoradoatBoulder,hebegan:“Aswithsomuchinanalyticnumbertheory,thestudyofcongruencepropertiesof thepartitionfunctionoriginatedwithRamanujan,” and used Ramanujan’s early papers as a jumping-off point. Andcertainlybythetimeofthecentennialofhisbirthin1987,Ramanujan’sreputationwassecure.

InhisIntroductiontotheHistoryofMathematics,HowardEvesfashionedanoutlinechroniclingtheseminalmoments inmathematics throughtheages.Theyear1906,forexample,markedtheworkofFréchet,theyear1907thatofBrouwer.Then,byEves’sreckoning,camealongdryspell,withnothing,fornineyears,thatinthesweepofmathematicalhistoryrankedsufficientlyhightoinclude.In1916,thedryspellwasover with Einstein’s general theory of relativity. Then, for 1917, came Eves’s nextentry:

HardyandRamanujan(analyticalnumbertheory)

Still, through most of the middle years of the century, a sense of tragedy andunfulfilledpromiseclungtoRamanujan’sname,ofregretthathehadnotbeengreaterstill. Hardy himself set the tone, pointing to how Ramanujan had inhabited amathematical desert for so many years. “He would probably have been a greatermathematicianifhehadbeencaughtandtameda little inhisyouth,”hewrote;“hewould have discovered more that was new, and that, no doubt, of greaterimportance.”Littlewood echoed that sentiment in reviewing theCollectedPapers in1927: “How great amathematicianmightRamanujan have been 100 or 150 yearsago?WhatwouldhavehappenedifhehadcomeintotouchwithEulerat therightmoment?”

Buthehadn’t,ofcourse;itwasashame,andthatwasthat.

For someyears,hisworkwent intoeclipse, asnewareasofmathematics,whollydistant from those Ramanujan had pursued, became fashionable. The CollectedPapers,whileithaditsdisciples,wasnobest-seller,evenbyscholarlystandards.Just42 books were sold the first year, 209 the second, and an officer of CambridgeUniversityPresspredictedinalettertoHardylatein1929thatitwouldbeanothertenyearsbeforethefirstprinting,of750,wassoldout.“WhenIcametotheUnitedStates[afterWorldWarII],”FreemanDysonwouldrecall,“Iwasallbymyselfasadevotee.” To the mathematical avant-garde of the day, Ramanujan’s work was nomorethana“backwater,”avestigeofthenineteenthcentury.

Butthat,intime,wouldchange.

5.RAMANUJANREBORN

One milestone came in the late 1950s. S. Ramaseshan, whose father had knownRamanujan,wasvisitingfriendsinBombay,whenhewastakentoaninnerroomoftheirprintingplant.There,hewas

shownastackofbrownedoldpaperwithmagicsquaresandbeautifulmathematicalformulaesystematicallywritteninaneleganthand.IcouldnotbelievethatIwasfacetofacewithRamanujan’snotebook,theoneIhadheardaboutfirstfrommyfatherin1937,thefamous“frayednotebook”ofRamachandraRao.Iranthetipsofmy fingersgentlyover theoldpaper—to feel the sheetswhichRamanujanhimselfhad filledwithasmileonhisfacewhenhewaswithoutajobandeverythingelseinhislifeseemedsobleak.

His friends at the Commercial Printing Press had been given “one of themostexciting jobs they had ever undertaken.” It was 1957 and the Tata Institute ofFundamental Research, in a publishing venture financed by the Sir Dorabji TataTrust, was bringing out Ramanujan’s notebooks in a facsimile edition of twophysicallydauntingvolumesthattogetherweighedinatmorethantenpounds.

Until then, onlyRamanujan’s publishedpapers, alongwithhis letters toHardy,hadbeenpublished.But those ledLittlewood to think,whenhe reviewed them in1929,that “thenotebookswouldgiveanevenmoredefinitepictureof theessentialRamanujan.”ThepublishedpapersmostlyshowedRamanujanburnishedbyHardy’sediting,hiswork all gussiedup and tied in a ribbon.Thenotebooks, on the otherhand,wereRamanujanintheraw.

•••

On October 8, 1962, a group of men met at the three-hundred-year-oldMallikeswarar Temple, at the northern end of Linghi Chetti Street, in Madras’sGeorgetown district, whose streets Ramanujan had walked half a century before.

Here,intheshadowofthetemple’sornategopuram,P.K.Srinivasan,amathematicsteacheratMuthalpietHighSchool,broughthisfriendstogethertolaunchaproject.HehadfirstreadaboutRamanujantwentyyearsbefore.Eversince,hehadtriedtoinspirestudentswithhisexample.Then,eightyearsbefore,afriendhadtakenhimtomeet Janaki and Tirunarayanan, Ramanujan’s surviving brother. Now, as theseventy-fifth anniversary of Ramanujan’s birth approached, he was determined tobringoutamemorialbook,filledwithlettersandreminiscences,tohonorhim.

Recruiting thehigh school’s alumni,or “oldboys,” tohelphim,heplacedads inlocalpapers,interviewedpeoplewhohadknownRamanujan,gatheredletters.Whenhe’dget some flickerof interest fromanadorcontact,he’d immediately followup.Often,hefoundhimselfjustpatientlysittingthere,whilesomeonerummagedaroundin an old trunk for some half-remembered letter. Sometimes he’d bring in astenographer,skilledinbothEnglishandTamil,torecordtheconversation.

Ramanujan’sseventy-fifthbirthdaywasobservedacrossSouthIndia.TownHighSchool,inKumbakonam,namedoneofitsbuildingsafterhim.Astampwasissuedinhishonor; twoandahalfmillion copiesofhispassportphoto, reduced to inch-highform,coloredsienna,andvaluedatfifteennewpaise,soldoutthedaytheywereissued. InMadras, around the time of the anniversary, a birthday celebration washeldinhishonor,andmanyofthosewhohadbeenclosetohimorhisfamilywereintown.Srinivasanexploitedtheopportunity,stationingoldboysattheentranceofthehalltosolicitcomments,correspondence,andreminiscences.

At another point, he visited Ramanujan’s old house in Kumbakonam;Tirunarayanan had given him permission to look through the almirah, a sort ofwardrobe,keptinaseparatelockedroomofthehouse.Inthepresenceofthetenant,Srinivasanunlockedtheroom.Whenheopenedthealmirah,whichwascoveredwithdust and cobwebs, cockroaches swarmed out. But in it, despite Tirunarayanan’sassurancethatanysuchfindwasunlikely,hefoundaletterRamanujanhadwrittenhisfatherfromEngland.

P.K.Srinivasan’scompilationoflettersandreminiscencescameoutin1968.BriefbiographiesofRamanujanappeared,inEnglish,in1967,1972,and1988;inTamilin1980and1986;andinHindi,Kannada,Malayalam,amongotherIndianlanguages.

•••

Then,in1974,Deligneprovedthetauconjecture.Almost sixty years before, in 1916, Ramanujan had published a paper with the

unprepossessingtitle,“OnCertainArithmeticalFunctions.”Anarithmeticalfunction

is one that originates in trying to learn certain properties of numbers; pi (n), thenumber of prime numbers, and p(n), the number of partitions, both problems onwhichRamanujanworked,weretwoofthem.Thetaufunction,τ(n),wasanother.

“Letσs(n)denotethesumofthesthpowersofthedivisorsofn,”Ramanujanhadbegun.Ifn=6,forexample,itsdivisorsare6,3,2,and1.Sothatif,say,s=3,“thesumofthesthpowersofthedivisors,”σ3(6),isjust63+33+23+l3=252.Buthowcalculateσs(n)generally?ThatquestionledRamanujan,afterfifteenpages,tothetaufunction, whose properties, Hardy would write twenty years later, “are veryremarkable and still very imperfectly understood.” As in so much of analyticalnumbertheory,steppingthroughtheopendoorofasimple-seemingproblemhadledintoamathematicallabyrinthofformidablecomplexity.

Ramanujan’s hypothesis, as Hardy called it, or the tau conjecture as it becamemoregenerallyknown,didnotofferanexplicitformulaforτ(n).Rather, itmerelystatedthatτ(n)was“ontheorderof...”something.Inotherwords,liketheprimenumber theorem, itwas a kind of approximation. “There is reason for supposing,”Ramanujanwrote,“thatτ(n)isoftheform0(n11/2+∈).”Thenotationwasawayofsayingthatitsvalue,whateveritactuallywas,wasalwayslessthansomething.Whatwas “something”? Leaving out constants, it was, at most, n to some power onlyslightlygreaterthan11/2(whichmeansthesquarerootofntotheeleventhpower).

Ramanujan’s conjecture, in the words of S. Raghavan of the Tata Institute ofFundamental Research in Bombay, “kept at bay a whole galaxy of distinguishedmathematiciansfornearlysixdecades,”remainingoneofthemajoropenproblemsinnumbertheory.AnaccountbyHardyin1940hintedatboththeinterestitgeneratedanditsresistancetosolution.Ramanujanhimself,hereported,hadprovedthatτ(n)=0(n7);but7,thepowerofnhere,wasmuchmorethan11/2,andsostillfarshortofprovingthetauconjecture.Twoyearslater,Hardyhimselfcutalittlecloser,to0(n6).Kloosterman got closer in 1927, Davenport and Salie closer still in 1933. TheScottishmathematicianRobertRankin,astudentofHardy’s,provedin1939thatτ(n)=0(n29/5).

Allthiswasalittlelikeshowing,first,thatthemurdererlivedatahouseonUnionStreetnumbered less than2170; then, that thehousenumberwas less than2160;then,less than2158 . . .Andyet still youcouldn’tprovehe livedat2155,which iswhereyourprimesuspectlived.

Itwasonlyin1974thattheBelgianmathematicianPierreDeligne,inwhatwouldbedescribedas “oneof thecelebratedeventsof20thcenturymathematics,”provedthe conjecture using powerful new tools supplied by the field known as algebraic

geometry.Deligne was subsequently awarded the Fields Medal, the mathematicalcommunity’scounterparttotheNobelPrize,forhistriumph.

Andwhenhedidit,ithelpedsolidifyRamanujan’sreputationallthemore.

•••

In 1988, looking back upon the fall and rise of Ramanujan’s reputation over theyears,BruceBerndtcomparedhimtoJohannSebastianBach,whoremainedlargelyunknown for years afterhisdeath in1750.ForBach, thebig turnaround cameonMarch11,1829withFelixMendelssohn’sperformanceof theSt.MatthewPassion.For Ramanujan, suggested Berndt, the roughly analogous event was GeorgeAndrews’sdiscoveryoftheLostNotebookin1976.

InlateAprilofthatyear,AndrewswasayoungUniversityofWisconsinvisitingprofessorboundforaone-weekconferenceinFrance.Theairlinefarestructuremadeitcheapertogoforthreeweeksthanone,sohelookedforsomethingtokeephim,hiswife, and his two daughters occupied during the extra time. A side trip toCambridge?Acolleague therehad suggestedhemight be interested in rummagingthrough some papers left behind by G. N. Watson at his death in 1965. SoCambridgeitwas.

Watson,whohadworkedonRamanujan’s papers formany years beforeWorldWar II,was amajor classical analyst from the prewar period and a Fellow of theRoyal Society. When he died, the society asked J. M. Whittaker, son of one ofWatson’s collaborators, to write his obituary. Whittaker wrote Watson’s widow:couldhecomeseehispapers?Sheinvitedhimtolunch,thentookhimupstairstothestudy.There,Whittakerrecalled,papers

covered the floor of a fair sized room to a depth of about a foot, all jumbled together, and were to beincineratedinafewdays.Onecouldonlymakeluckydips[intotherubble]and,asWatsonneverthrewawayanything, the resultmight be a sheet ofmathematics butmore probably a receipted bill or a draft of hisincometax return for1923.Byanextraordinarystrokeof luckoneofmydipsbroughtup theRamanujanmaterial.

This“material,”all140pagesofit,waspartofabatchofpapersDewsburyhadsentHardy in 1923 that had later gone to Watson. After his “lucky dip,” WhittakerpassedthemontoRobertRankin,whoin1968handedthemtoTrinityCollege.

Whittaker and Rankin, both professional mathematicians of high standing, butwhosebackgroundsill-suitedthemtodistinguishthismaterialfromwhathadalreadyappearedinthepublishednotebooks,hadfailedtoseeinitwhatAndrewssawnow.Withinafewminutes,herealizedthatsomeofitboreonmockthetafunctions,the

subjectofhisownPh.D.thesis,andrelatedsubjects.WhichmeantthesewerepapersRamanujancouldhavegeneratedonlyduringthelastyearofhislifeinIndia.

Hewasthrilled,“extremelyexcitedthatIhadmyhandsonsomethingspectacular.”Butwhatnow?Getthepapersphotocopied,ofcourse.

TheinterioroftheTrinityCollegelibraryisavisuallyarrestingplacethatowesitsbeauty to the great English architect Christopher Wren. After its completion in1695, someone said of theWrenLibrary that it “touches the very soul of any onewho first sees it.”Everywherearedelicatewoodcarving, stained-glasswindows, andstatuesfillingapsesonthewall.BustsofCambridgegreatsarearrayedalongitlength,andThorwaldsen’s statueofLordByron standsat the far end. It is at a long tablebehind the Byron statue that library staff seat scholars studying medievalmanuscripts,orthepapersofNewton,orthepoemsofMilton.Andin1976,GeorgeAndrewswasoneofthem.Tohim,theWrenwas“ashrine,withbustsofNewtonglowering down on you. The idea of going up to the desk and having somethingxeroxedterrifiedme.”

Hemustereduphiscourageandtoldthemwhathewanted.“Thatwillbeaboutsevenpounds”forairmailpostagetotheUnitedStates,theywarnedhim.“Willthatbeallright?”

Itwouldbefine,heassuredthem.“Iwasreadytotakeasecondmortgageonmyhousetogetit.”

Inapaperappearingsoonafterhehadunearthedit,Andrewsstyledhisfind“theLostNotebook.”Itsdiscovery,mathematicianEmmaLehmerwasmovedtosay,was“comparable to the discovery of a complete sketch of the tenth symphony ofBeethoven.”

But the appellation ruffled feathers in Britain. Robert Rankin pointed out, forexample, that itwas not a “notebook” but loose papers; and that, pristinely securewithin the walls of the Wren Library, it had never been lost. Still, as Andrewsobserved later, “the manuscript and its marvelous results disappeared from anymentionoraccountbythemathematicalcommunityformorethan55years.”Inthatsense, they had indeed been “lost.”His contribution lay “not in saving them fromoblivion as [Whittaker and Rankin] did,” but in recognizing them for what theywere.

Sometime before, in January 1974, Bruce Berndt had been on sabbatical at theInstituteforAdvancedStudyandcameuponpapersbearinguponhisownworkoftwoyearsbeforethatprovedsomeformulasinRamanujan’snotebooks.Theinstitutelibraryhadnocopyofthefacsimileedition,butneighboringPrinceton’sdid.Init,hefoundaslewofrelatedformulasthatseemedsimilartotheothersbutthat,tryashe

might,hecouldn’tprove.“Andthat,”herecalled,“botheredme.”Hesetaninformaltaskforhimself—toprovealltheresultsinchapter14.Thattookhimayear.Later,Andrews visited Illinois andmentioned thatWatson andWilson had spent yearstrying toprove the theorems in thenotebooks, and that their ownnoteswere stillaround. Ever since, beginning in May 1977, Berndt has worked on editingRamanujan’snotebooks,“andIhaven’tdoneanythingelsesince.”

Forthedecadeendingin1988,acomputersearchoftheliteraturerevealed,somethree hundred papers referred toRamanujan in their titles or their abstracts.Tiesand cross-links to other areas of mathematics, what Freeman Dyson calls“connectionstodeepstructureandmoregeneralabstractnotions,”wereshowingupeverywhere.Overthepasttwentyyearsorso,Dysonsays,“IthasbecomerespectableagaintotakeRamanujanseriously.Somuchthatheconjecturedwasnotjustprettyformulasbuthadsubstanceanddepth.”Thegreat tree thatwasRamanujan’sworksentitsrootsdowndeepandfar.

Hehadplanted it forhimself,not tobetter thematerialconditionofIndiaoroftheworld.Andyet,itsundergroundtendrilsrangedintofieldsfardistantfrompuremathematics—andintoapplicationswhich,Hardymighthavecringedtolearn,werebynomeans“useless.”

6.BETTERBLASTFURNACES?

“IfIamaskedtoexplainhow,andwhy,thesolutionoftheproblemswhichoccupythebestenergiesofmy life isof importance to thegeneral lifeof thecommunity, Imustdeclinetheunequalcontest,”wroteHardy,justafterthegunsofWorldWarIhadstilled.

I have not the effrontery to develop a thesis so palpably untrue. Imust leave it to the engineers and thechemiststoexpound,withjustlypropheticfervour,thebenefitsconferredoncivilizationbygas-engines,oil,andexplosives.IfIcouldattaineveryscientificambitionofmylife,thefrontiersoftheEmpirewouldnotbeadvanced,notevenablackmanwouldbeblowntopieces,noone’sfortuneswouldbemade,andleastofallmyown.Apuremathematicianmustleavetohappiercolleaguesthegreattaskofalleviatingthesufferingsofhumanity.

Whether and how the engineers and chemistsmight indeed apply Ramanujan’sworktothecommonpurposesoflifestrikesasensitivechordinIndia,besetasitisbypractical problems of great urgency and less naturally inclined to trust in researchwhose rewards may accrue only decades or centuries later. “Several theorems ofRamanujan are now being widely used in subjects like particle physics, statisticalmechanics, computer science, cryptology and space travel in the United States—subjectsunheardofduringRamanujan’stime,”TheHindu assured its readers in its

December 19, 1987 issue. But efforts to justify Ramanujan’s work on utilitariangroundsgobackalmosttohislifetime.

AttheThirdConferenceoftheIndianMathematicalSociety,heldinLahoresoonafterhisdeath,Ramanujan’slifeandworkwereoneveryspeaker’slips.“MentionofRamanujan’s name,” declared the society’s president, Balak Ram, “suggests thequestion of the organization and endowment of scientific research in India,” inparticularthebalancebetweenappliedandpure.Theengineermightberewardedforhisinventiveness,becomerichthewayEdisondid.Butwhatoflearningforlearning’ssake?Oughtsocietyreserveaplaceforit?Yes,heinsisted.

Everysettledcommunity issubconsciouslyorconsciouslyconvincedthatattemptsatprogressareapttobewastefulunlessguidedbythinkersandteacherswhohaveleisuretothinkandteachclearly;therefore,wefindthesecommunitiesgivingspecialprotectionandadvicetolearnedmenandthusprovidingthemwithleisureandthepsychologicalincentiveswhichreplacethemissingpecuniaryreward.

If India was to be assured of “progress,” he was saying, it had to allow for itsRamanujans.Inthelongrun,itwouldpayoff.

When the Ramanujan stamp came out in 1962, the Indian postal service tookpainstopointoutthepotentialapplicationsofhiswork:“HisworkandtheworkofothermathematiciansonRiemann’szetafunction,doneinanothercontext,hasnowbeengearedtothetechnologicalmill.Ithasbeenappliedtothetheoryofpyrometry,theinvestigationof furnacesaimedatbuildingbetterblast furnaces.”Andhisworkonmockthetafunctions,modularequations,andinotherrealmswasbeingstudiedforitspossibleapplicationtoatomicresearch.

Down through the years,Ramanujan’smathematics has indeed been brought tobearonpracticalproblems, ifsometimestangentially.Forexample,crystallographerS. Ramaseshan has shown how Ramanujan’s work on partitions sheds light onplastics.Plastics,ofcourse,arepolymers,repeatingmolecularunitsthatcombineinvariousways;conceivably,youmighthaveonethat’samillionunitslong,anotherof8251, another of 201,090, and so on. Ramanujan’s work in partitions—on howsmallernumbers combine to form larger ones—plainly bears on the process.As itdoes, for example, in splicing telephone cables, where shorter subunits, of varyinglengths,againadduptomakeawhole.

Blastfurnaces?Plastics?Telephonecable?Cancer?At a meeting of the southeastern section of the American Physical Society in

Raleigh,NorthCarolina, inNovember1988, threeUniversityofDelhi researcherspresented a paper entitled “A Study of Soliton Switching in Malignancy andProliferationofOncogenesUsingRamanujan’sMock-ThetaFunctions.”Theywere

using Ramanujan’s mathematics to help understand cancer, if only as a small,tangential contribution to a vast and complex subject.When theHindu noted thepaper,however,itassignedtheheadline“Ramanujan’sMathsHelpFightCancer.”

Inother fields,Ramanujan’smathematicshasplayedamoredecisiverole—as in,for example, string theory, which imagines the universe as populated byinfinitesimally short stringlike packets whose movement produces particles.Groundedintherealworldornot—thejuryisstillout—themathematicsrequiredtodescribethesestringsdemandstwenty-sixdimensions,twenty-threemorethanthethreeonwhichwemanageineverydaylife.PartitiontheoryandRamanujan’sworkintheareaknownasmodularformshaveprovedessentialintheanalysis.

An important problem in statistical mechanics has also proved vulnerable toRamanujan’s mathematics—a theoretical model that explains, for example, howliquidheliumdisperses through a crystal lattice of carbon.As it happens, the sitesheliummoleculesmayoccupyinasheetofgraphite,say,canneverlieadjacenttooneanother.Sinceeachpotentialsiteissurroundedbysixneighborsinahexagonalarray,onceitisfilled,thesixarounditdefineanunbreachablehexagonalwall.Reviewingin1987 thework forwhichhe received theprestigiousBoltzmannMedal sevenyearsearlier,R. J. Baxter of theAustralianNationalUniversity inCanberra set out thethinkingbehindhis“hardhexagonmodel.”Mathematically,heshowed, itwasbuilton a particular set of infinite series. And “these series,” he observed, “are preciselythose that occur in the famous Rogers-Ramanujan identities” (though he hadn’trealizeditbackin1979whenhe’dfoundthemonhisown).Basedonthem,hefounda way to determine the probability that any particular site harbored a heliummolecule; predictions borne of the model agreed closely with experiment. “TheRogers-Ramanujan identities,”Baxter concluded, “are perhaps not so remote from‘ordinaryhumanactivity’asHardywouldhaveliked!”

Computers, scarcely the dream ofwhich existed in 1920, have also drawn fromRamanujan’s work. “The rise of computer algebra makes it interesting to studysomebodywhoseems likehehada computer algebrapackage inhishead,”GeorgeAndrewsoncetoldaninterviewer,referringtosoftwarethatpermitsreadyalgebraicmanipulation.SometimesinstudyingRamanujan’swork,hesaidatanothertime,“IhavewonderedhowmuchRamanujancouldhavedoneifhehadhadMACSYMAorSCRATCHPAD or some other symbolic algebra package. More often I get thefeeling that he was such a brilliant, clever and intuitive computer himself that hereallydidn’tneedthem.”Then, too,amodularequation inRamanujan’snotebooksledtocomputeralgorithmsforevaluatingpithatarethefastestinusetoday.

•••

“Hype,” FreemanDyson calls some of what he deems undue fanfare for practicalapplications of Ramanujan’s work. To apply his mathematics to string theory is“stretching the point,” he says. “You don’t have to read Ramanujan to do stringtheory.”Itwasalltrueenough,andcompletelyvalidasfarasitwent—butperipheral,ifnotirrelevant,tomostmathematicians.

WhatmakesRamanujan’sworkso seductive isnot theprospectof itsuse in thesolution of real-world problems, but its richness, beauty, and mystery—its sheermathematicalloveliness.Hardywasenrapturedbyit.AndDysonwas,andSelberg,andErdos,andmany,manyothers.“ThebestseemtoappreciateRamanujanearly,”says Richard Askey, a University of Wisconsin mathematician deeply involved inRamanujan’s work. “The rest of us have to need some of his work before reallyappreciatingit.”GeorgeAndrewsoncetoldaninterviewerhow,asayoungman,hewas beguiled by theRamanujan-Hardy partition formula. “Iwas stunned the firsttime I saw this formula. I could not believe it. And the experience of seeing itexplained,andunderstandinghowittookshape. . .convincedmethatthiswastheareaofmathematicsIwantedtopursue.”

Becauseitliesonacool,etherealplanebeyondtheeverydaypassionsofhumanlife,andbecauseitcanbefullygraspedonlythroughalanguageinwhichmostpeopleareunschooled,Ramanujan’sworkgrantsdirectpleasuretoonlyafew—afewhundredmathematiciansandphysicistsaroundtheworld,perhapsafewthousand.Therestofusmusteithersitonthesidelinesand,ontheauthorityofthecognoscenti,cheer,orelserelyonvague,metaphoric,andnecessarilyimpreciseglimpsesofhiswork.

SomeRamanujanexpertshaveresortedtothe languageofother,moreaccessiblerealmsofknowledgetosuggesthisholdonthemathematicalimagination.

EmmaLehmer,recall,likenedTheLostNotebooktoaBeethovensymphony.WatsonconcludedhispresidentialaddresstotheLondonMathematicalSocietyin

1937 by saying that one Ramanujan formula gave him “a thrill which isindistinguishablefromthethrillwhichIfeelwhenIentertheSagrestiaNuovaoftheCapelle Medicee and see before me the austere beauty of the four statuesrepresenting, ‘Day,’ ‘Night,’ ‘Evening,’ and ‘Dawn’whichMichelangelohas set overthetombs”oftheMedicis.

Berndt comparedRamanujan toBach and, alluding toRamanujan’s devotion tomathematics,quotedfromShelley’s“HymntoIntellectualBeauty”:

IvowedthatIwoulddedicatemypowersTotheeandthine;haveInotkeptthevow?

For the layman, to be sure, this is an ultimately unsatisfying way to confrontRamanujan’smathematics,foritkeepsusatseveralremovesfromwhathedid,leavesus having to take others’ word for it, looking at his mathematical achievementsthroughablurryfilmofmetaphor,poetry,and,yes,ignorance.True,thecompositionofasonatamaybeequallymysterious;buttheresultmoreintimatelyinvolvesthefivesenses.

WhatRamanujandidwill liveforever.Itwillnot,tobesure,liveintheheartsofthemassesofmen,liketheworkofGandhi,Shakespeare,orBach.Still,hisideasanddiscoveries,percolatingthroughthosefewmindstunedtothem,willminglewiththeintellectual energy of the cosmos, and thence into the deep, broad pool of humanknowledge.“Whatwedomaybesmall,butithasacertaincharacterofpermanence,”wroteHardyoftheworkofpuremathematicians,“andtohaveproducedanythingofthe slightest permanent interest, whether it be a copy of verses or a geometricaltheorem,istohavedonesomethingutterlybeyondthepowersofthevastmajorityofmen.”

•••

Bythetimeofthecentennialofhisbirthin1987,Ramanujan’sreputationshonewithanewluster.InIndia,hewascomparedtoNehruandNobelPrize-winningphysicistC.V.Raman, both of whose centennials were being celebrated at about the sametime. Three Indian films were made about his life. A Ramanujan MathematicalSociety,startedin1986,publishedthefirstvolumeofitsjournal.

Celebrationswereheld all acrossSouth India.Andrews,Askey, andBerndt, thethree American mathematicians who had most contributed to the restoration ofRamanujan’sname,werekeptbusyshuttlingallover thecountry,giving lecturesatAnnamalainagar,andBombay,andPune,andGorakhpurandMadras.

InMadras,whentheNarosaPublishingHouseissuedTheLostNotebook,PrimeMinisterRajivGandhiwastheretosignthefirstcopyandpresentittoJanaki.

InKumbakonam, the framed and garlandedposter-sizedportrait ofRamanujan,fat again and wearing the mortarboard he wore when he received his Cambridgedegree, was borne through the streets atop a gaily decorated elephant, to theaccompanimentoftraditionalstreetmusicians.TheNationalCadetCorpswasthere,andgirlscoutsinpinkblousesandrose-coloredskirts,andtraditionaldancers,allofthem showered with flower blossoms as they paraded down the street in front ofRamanujan’shouse.LoudspeakersshoutedpraiseofKumbakonam’s favorite son totheassembledthrongsinthestreet.

AtAnnaUniversity,whichnameditscomputercenterforRamanujan,Andrews,his voice choked with emotion, presented Janaki with a shawl. It was she whodeserved the credit for theLostNotebooks, he said, since shehadkepthis paperstogetherwhilehelaydying.

AfterWorldWarII,theuniversityhadnoticedthatJanaki’s20-rupee-per-monthpension no longer went very far and raised it to 125. Janaki financed her sonNarayanan’sway through college.He later got a jobwith theStateBankof India,married,andhadthreechildren.Recently,heretiredfromthebankjobheheldfortwenty-fiveyearsandbegantospendmoretimecaringforhismother.

Afterahardlifeandyearsofanonymity,Janakiherselfbegantogarnerattentionasher husband was rediscovered. Along the way, she grewmore outspoken. At onepoint,presentedapensionbytheUniversityofMadras,sheremarkedthatwhilethecashwasfine,itwouldhavedonehermoregoodsixtyyearsearlier.Alittleearlier,in1981,shehadtoldanIndiannewspaperreporter,“Theysaidyearsagoastatuewouldbeerectedinhonorofmyhusband.Whereisthestatue?”

DickAskey learned of Janaki’s lament. “If shewants a bust of her husband,” hethought, “we owe that much to Ramanujan, and to her.” We meant themathematicians of theworld.Askey knew theMinnesota sculptorPaulGranlund,fromwhomhehadboughtsomeworks,andcontactedhim.Granlundagreedtodoit,providedatleastthreebustswouldmaterializefromtheproject.(Ultimately, tendid.)Askeyandhiswifewouldbuyone.SowouldChandrasekhar,theastrophysicist,towhomAskeyhadwritten about the idea.Butwherewould the$3000 come forJanaki’s?Askeygotsomeofitfrominstitutions,includingTrinityCollege.Butmostcameinindividual$25contributionsfrommathematiciansaroundtheworld.Today,Granlund’sbronzebustofRamanujan,basedonthepassportphotograph,standsonapedestalinJanaki’shouseinMadras.

With the approach of the centennial, the house became a pilgrimage site.MathematicianspassingthroughMadraspaidherhomage.SheappearedinaBritishtelevision special about Ramanujan, Letters from an Indian Clerk, later shown inAmericaaswell.InAugustofthecentennialyear,afoundationpresentedherwithapurseoftwentythousandrupeesandamonthlypensionofonethousandrupees;sheasked that aSrinivasaRamanujanTrustbe created for awards and scholarships tobrightyoungmathematicsstudents.Early the followingyear,TrinityCollegemadeits own gesture, its council inFebruary1988 agreeing to “a grant of £2,000 a yearuntilfurthernotice.”

Asthecentennialapproached,T.V.Rangaswami,aTamil language journalist inMadras, set outonwhatbecamea thirty-one-part seriesof articlesonRamanujan.

Hespentmonths collecting letters, documents, and photographs, and interviewingJanaki,wholivednearhiminTriplicane.Eachafternoontheywouldmeetandtalk.Oneafternoon,heshowedheraphotographof“Gometra,”thehouseoffHarringtonRoadwhereshehadnursedRamanujanandinwhichhehaddied.Thewrinkledoldwoman, removed bymore than sixty years from the events that took place there,brokedownandcried.

7.SVAYAMBHU

A bittersweet tang sometimes slipped into the encomiums India lavished onRamanujan over the years, sad reminders of the poverty, bureaucracy, andinstitutionalrigiditythatalmostcrushedhimin1905.RamanujanwasaninspirationtoIndia,yes—butalsoarebuke.HowcouldIndialethimcomesoclosetobeinglostto the world? Why hadn’t he gotten more encouragement? Why was it left toforeignerstomakehimfamous?

J.B.S.Haldane,thedistinguishedEnglishbiologistwholivedinIndiatowardtheendofhislife,intheearly1960scomplainedthat

todayinIndiaRamanujancouldnotgetevenalectureshipinaruralcollegebecausehehadnodegree.MuchlesscouldhegetapostthroughtheUnionPublicServiceCommission.ThisfactisadisgracetoIndia.IamawarethathewasofferedachairinIndiaafterbecomingaFellowoftheRoyalSociety.ButitisscandalousthatIndia’sgreatmenshouldhavetowaitforforeignrecognition.IfRamanujan’sworkhadbeenrecognizedinIndiaasearlyasitwasinEngland,hemightneverhaveemigratedandmightbealivetoday.Wecancasttheblame for Ramanujan’s non-recognition on the British Raj. We cannot do so when similar cases occurtoday....

Onhisbirthdayin1974,aProfessorSrinivasaRamanujanInternationalMemorialCommittee published a commemoration volume littered with advertisementsexpressing “respectful homage” and like sentiments by every little South Indiancompany thatwanted to briefly bask inRamanujan’s reflected glory, fromMadrasAluminumCo.inCoimbatoretoSmartDressesonRanganathanStreetinMadras.PrimeMinisterIndiraGandhiwrotefromNewDelhithatRamanujan’s“untutoredgenius”dazzledscienceandthat“hisachievementwill inspiresuccessivegenerationsofIndianyouth.”ButeditorS.Ramakrishnanadded:“LetnotFreeIndialosesightofherlivingRamanujans,languishinginobscurity.”

Languishinginobscurity.TherewasanothersidetoRamanujan’sserviceassymbolofIndiaand,in1946,NehruhimselfhadreferredtoitinhisDiscoveryofIndia:

Ramanujan’s brief life and death are symbolic of conditions in India.Of ourmillions how few get anyeducationatall;howmanyliveonthevergeofstarvation...Iflifeopeneditsgatestothemandofferedthemfood andhealthy conditions of living and education and opportunities of growth, howmany among these

millionswouldbeeminentscientists,educationists,technicians,industrialists,writers,andartists,helpingtobuildanewIndiaandanewworld?

R. Viswanathan, a later headmaster of Ramanujan’s alma mater, Town HighSchool inKumbakonam,would insist that, given the resources, he could turn outmanyRamanujans.Takenliterally,hewasquitewrong;allthewealthoftheBritishEmpire,alltherichintellectualtraditionofEurope,allthefreedomandopportunityofAmerica,havemadeforbutahandfulofRamanujansthroughthecenturies.Still,Viswanathanhadexpressedalargertruth—thatIndiaheldvaststoresoftalentandability denied the means to develop fully. Ramanujan represented his country’sintellectualandspiritualstrengths—butalsoitsuntappedpotential.

•••

But didn’t Ramanujan’s story prove, quite to the contrary, that genius in the endovercomes?IfRamanujan,withallhisdisadvantages,couldcommandtheattentionof the world and leave so indelible a mark on it, couldn’t anyone endowed withspecialgiftsdoit,too?

Notso.SolongdidRamanujanlanguish,somanytimesdidhisfuturehangonaknife edge, so close did he come to dying unknown—and so plainly was his fullpromiseneverrealized—thathislife’slessonbearsasmuchonthestumblingblockshefacedasonhissuccess,suchasitwas,inovercomingthem.

RamanujanhadmuchmoregoingforhimthanmillionsofothersinIndiadid.Hisfamily was poor but hardly destitute. He was a Brahmin, part of a culture thatencouragedlearning.Hismotherwastolerantofhiswhimsandforcefulinadvancinghisinterests.His innocentcharmwonover thosehiseccentricitiesmightotherwisehaveputoff.Andheenjoyedapeculiarlystubbornfaithinhimselfandhispowers.

What,onemustask,ifhehadhadnoneofthese?Whatifhehadbeeneveryinchthegeniushewas,with just asmuch togive theworld,buthismotherhadbeenalittle less supportive?Orhehadbeen thebarestbit less likeableor less sureofhisabilities? Doubtless he would have wound up like his brothers, an anonymousgovernment bureaucrat, or otherwise consigned to obscurity. For those who havebiographieswrittenaboutthem,theSystembydefinitionworks;themeasureof itsfailureliesinthosewhoneverbaskinthewarmglowoftheworld’sacclaim.Thoseyouneverhearabout.

Ramanujan, then, was an embarrassment to India as well as an inspiration, areminder of the gauntlet India’s other Ramanujans must run in order to achieve

anything.AssymbolofSouthIndiangenius,hewasadelighttocontemplate.Inthesometimesharshrealityofhislife,muchlessso.

When inNovember 1968, on receiving the Srinivasa RamanujanMedal of theIndianNationalScienceAcademy,S.ChandrasekharrevealedthatRamanujanhadtried to commit suicide, his comments sparked a furor. “I was shocked andsurprised,”hewrotelater,“thatIwasaccusedbyseveral[includinghisuncle,C.V.Raman] of defaming Ramanujan’s name.” Someone charged him with trying toenhancehisownreputationatRamanujan’sexpense.Chandrasekharhadsinned:hehadtorndownRamanujanasicon,replacedhimwithRamanujantheman.

Today,asinRamanujan’sdayandinalltheyearsinbetween,manyofIndia’sbestmindsleavetogooverseas,wheretheyarenourishedbyWesternideasandsidestepthe sort of “inefficient and inelastic” educational system that stifled Ramanujan.India,apoorcountry,canlavishpraiseonthelong-deadRamanujanmoreeasilythanitcanlavishresourcesonfindingandnurturingnewRamanujans.

In1951,awealthymerchantandpatronofhighereducation,AlagappaChettiar,foundedtheRamanujanInstitute.Afterhisdeath,theinstitutefacedseverefinancialproblems and almost shut down. In 1957, it was absorbed by the University ofMadras,whereitbecametheCentreforAdvancedStudiesinMathematics.In1972,it moved into a low-slung modern building just across a little concrete bridgespanning the Buckingham Canal from the main university campus. The institute,withnineteenteachers, lecturers,andresearchstudents,remainstheretoday.But itdoesnot specialize inareasofmathematicsRamanujanpursued,nordoes itsnamebearanythinglikethelusterofitsnamesake.

Even the centennial festivities were marred by reminders of the limitedopportunitiesIndiaoffered itsRamanujansandtheobstacles itplacedintheirway.Why, somewondered,wereRamanujan’s research reports from1914 lostwhile inIndian care? Why was it left to American, more than Indian, mathematicians torestoreRamanujan’sreputation?“Iwantyoualltositbackandthinkaboutthis,”onespeaker,S.Ramaseshan,askedhislistenersatacentennialeventinKumbakonam:

Howmanyregistrarsinthiscountrytoday,orforthatmatterhowmanyvicechancellorsoftoday,100yearsafterRamanujanwasborn,wouldgivea failedpre-universitystudentaresearchscholarshipofwhat isnowequivalentofRs.2000/-orRs.2500/-today?Thisisafter40yearsofindependence,whenwecannolongerblameacolonialpowerfornotencouragingIndiantalent.

AwriterfortheIllustratedWeeklyofIndia,referringtoRamanujan,suggestedthat“perhaps the luminescence of hismind is tooharsh for Indians in this age of theirintellectualbleakness....”

•••

Ramanujan as inspiration.Ramanujan as rebuke.Ramanujan as . . . ?Ramanujan’slifecanbemadetoserveasparableforalmostanylessonyouwanttodrawfromit.AtanearlyIndianMathematicalSocietyconference,hisnamewasinvokedtopointup the society’s meager resources: “When the famous mathematician, Mr.Ramanujan, F.R.S., sought some modest help from our Society to enable him todevotehisattentiontomathematicalstudiesandresearch,ourbankruptcywasmademanifest. . . .” In the 1950s, a library director recounted Ramanujan’s story forreadersofWilsonLibraryBulletin,emphasizingtheimpactonhimofCarr’sSynopsis.Ramanujan’s life, hewas saying,was a testimonial to the books that had nurturedhim.Allacrosstheyearsanduptothepresent,itwaslikethat—Ramanujaninvokedas model, inspiration, warning, or instructive case history. Ramachandra Rao haddoneit.Hardyhad.Nehruhad.

Cut cruelly short, Ramanujan’s life bore something of the frustration that acheckedswingdoesinbaseball;itlackedfollow-through,roundedness,completion.Itneverhadasecondhalf togive itshape.Sowecontinuetogive itshapenow,yearsafterhisdeath.

Hislifewastruncated,likeaconeslicedoffshortofitsvertex.Orlikeaneconomicgraph that stopswith thepresent, leaving forecasters to fill in a vast anduncertainfuture.Ramanujan’slife,litteredwithwhat-might-have-beens,waslikethat:itwassoeasy to see in it what you wanted to see. Its bare facts fairly cried out forinterpretation.

WashisfailureinschooltestimonytoIndia’sfailuretonurtureitsown?Orwashisrescue by Ramachandra Rao and Narayana Iyer proof that, in the end, Indiarecognizedandappreciatedhim?

Washeanexampleoftheoppressivenessoftheraj,acaseofnativegeniusnearlyquashed—then making its contribution, when it did emerge, to “Englishmathematics”?OrwashisdiscoveryatestamenttoBritishbeneficence?

Would he have achievedmore had he foundmentors early on?Would he havebecomethenextGaussorNewton?Ordidworkingonhisown,underlessthanidealconditions, make him more mathematically resourceful, even contribute to hisstunningoriginality?

Was his genius the product of sheer intellectual power, different only in degreefromotherbrilliantmathematicians?Orwasitsteepedinsomethingofthemysticalorthesupernatural?

WasRamanujan’s life a tragedy of unfulfilled promise?Or did his five years inCambridgeredeemit?

In each case, the evidence left ample room to see it either way. In this sense,Ramanujan’slifewas liketheBible,orShakespeare—arich fundofdata, lushwithambiguity,thatholdsupamirrortoourselvesorourage.

•••

There were no mathematicians in Ramanujan’s family, no strain of unusualmathematicalaptitude.Sowheredidhecomefrom?

LegionarethosewhohavetakencreditfordiscoveringRamanujanorforotherwisefiguring largely in his life. Ramaswami Iyer would later say he was proud of twothings—starting the Indian Mathematical Society and discovering Ramanujan.Narayana Iyer’s family today points with pride to its patriarch’s role in rescuingRamanujanfromoblivion,andnumberlesspeoplelaterclaimedcreditforconvincinghimtogotoEngland.Then,too,manysawRamanujanasthecreationofhismother;thetwoofthem,plainly,weremadefromthesamecloth,andherheartwasboundupinhissuccess.

AmongtheEnglish,HardywasnotaloneinsavoringhisroleinRamanujan’slife;Neville, too, would sweetly recall his days with him in Madras a quarter centurybefore,andpointoutthathadhe“failedtowintheconfidenceofRamanujanandhisfriends,” Ramanujan might never have reached England. But Hardy remains theprime candidate. Paul Erdos has recorded that whenHardy was asked about hisgreatest contribution to mathematics, he unhesitatingly replied, “The discovery ofRamanujan.” At another time, indeed, Hardy went on record as calling him “mydiscovery.”

In a sense, of course, hewas.But amore satisfyingway to explainRamanujan’semergence as a figure on the world mathematical stage was given in an articleappearing in India during the centennial, by a writer forThe IllustratedWeekly ofIndiaknownonlyas“RGK.”Ramanujan,hesuggested,wassvayambhu—“self-born.”He had sprouted up out of the soil of India of his own accord. He had createdhimself.

YoucannotsaymuchaboutRamanujanwithoutresortingtothewordselfHewasself-willed, self-directed, self-made. Indeed, some might conceivably label him“selfish”forhispreoccupationwithdoingthemathematicshelovedwithoutanygreatconcernforthebettermentofhisfamilyorhiscountry.

Ramanujandidwhathewishedtodo,wenthisownway.Itwasonlylater,afterhehadindulgedinanorgyofmathematicalcreation,thathemightwakeupandrealizehow far he had strayed from the common run of human intercourse. Only thenmighthebegintocare,sometimespainfullymuch,howothersthoughtofhim.

Whenhewasateenager,helentsupporttothelocalcrazyman,thoughthewholerestofKumbakonamdismissedhimasacrank.Hegavehimselfovertomathematics,throwingaside everything else, even the degree-as-meal-ticket hismother somuchwantedforhim.HeknockedondoorsalloverSouthIndia, introducinghimself toonemathematician after another.Then,when he had exhausted themathematicalresourcesof India,he turned toEngland.HewroteBaker.HewroteHobson.HewroteHardy.

HardydiscoveredRamanujan?Notatall:aglanceat thefactsof1912and1913showsthatRamanujandiscoveredHardy.

And what of the dream ofNamakkal, when the goddessNamagiri presumablygaveherblessingforhistriptoEngland?Whatfiercedrivetoliveouthisother,truer“dream”didRamanujanneedtocontrive,subconsciously,thatitturnoutasitdid?

Ramanujanwasamanforwhom,asLittlewoodputit,“theclear-cutideaofwhatismeantbyproof...heperhapsdidnotpossessatall”;oncehehadbecomesatisfiedofatheorem’struth,hehadscantinterestinprovingittoothers.Thewordproof,here,appliesinitsmathematicalsense.Andyet,construedmoreloosely,Ramanujantrulyhadnothingtoprove.

Hewashisownman.Hemadehimself.“Ididnot inventhim,”HardyoncesaidofRamanujan.“Likeothergreatmenhe

inventedhimself.”Hewassvayambhu.JustwhatdidRamanujanwant?Hewantednothing—andeverything.Hesoughtnowealth,certainlynonebeyondwhatheneededtocarryouthiswork,

andtogivetohisfamilywhathefeltwasexpectedofhim.He did crave respect, understanding, perhaps even a favorable judgment from

history.ButwhatRamanujanwantedmore,morethananything,wassimplythefreedom

todoashewished,tobeleftalonetothink,todream,tocreate,tolosehimselfinaworldofhisownmaking.

That,ofcourse,isnomodestwishatall.Hewanted“leisure.”Andhegotit.InSouth India today, everyonehasheardofRamanujan.Collegeprofessors and

bicycle rickshaw drivers alike know his story, at least in sketchy outline, just aseveryoneintheWestknowsofEinstein.Fewcansaymuchabouthiswork,andyet

somethinginthestoryofhisstruggleforthechancetopursuehisworkonhisownterms compels the imagination, leaving Ramanujan a symbol for genius, for theobstacles it faces, for the burdens it bears, for the pleasure it takes in its ownexistence.

Epilogue

BythetimehelearnedofRamanujan’sdeath,HardyhadalreadyleftTrinity.InDecember1919,aboutwhenRamanujan,athisdoctor’sinsistence,preparedto

leaveKumbakonamforMadras,HardywaswritingJ.J.Thomson,masterofTrinityCollege,with the news that he had accepted theSavilianProfessorship atOxford.“ThepostcarrieswithitaFellowshipatNewCollege,theacceptanceofwhichwillvacatemyFellowshiphereautomatically,”hewrote.Onereasonforthemovewastheincreasing loadof administrative responsibilityhebore atCambridge. “If Iwish topreserve full opportunities for the researches which are the principal permanenthappinessofmylife,”hehaddecided,hewouldneedapositionoffering“moreleisureandlessresponsibility.”AtOxford,he’dbeenassured,hewouldgetthat.

Unmentionedintheletterbutprobablyweighingmoreonhimthanadministrativechoreswerethehardfeelingsleftoverfromthewar,theinfightingthatsurroundedthe Russell affair, and the departure of Ramanujan. “If it had not been for theRamanujancollaboration, the1914–1918warwouldhavebeendarker forHardy,”wroteSnow. “Itwas theworkofRamanujanwhichwasHardy’s solaceduring thebittercollegequarrels.”NowRamanujanwasgone.Trinity,hishomeforthirtyyears,hadgrownuglytohim.Hescarcelyspokewithsomeofhiscolleagues.Earlier,hehadurgedW.H.Young,anoldermathematicianwhohadspentmuchofhisprofessionallife abroad, to apply for theSavilian chair, only to askhim towithdrawhisname,which Young did. “Hardy,” recalled Young’s son, Laurence, also amathematician,“felthemustgetaway.”

OxfordwastheothergreatEnglishuniversity,lessthanahundredmilesaway.InCamfordObserved, Jasper Rose and John Ziman tried to bring it and Cambridgewithin the compass of a single account: “Oxford is a city of wide and noblethoroughfares,theHigh,theBroad,St.Giles; inCambridgeallthestreetsstraggle.ThegreatbuildingsofOxfordfacethestreets,tallandimposing,andformaseriesofbreathtaking vistas. The great buildings of Cambridge are more isolated, lessemphatic,more secretive, giving on to college courts and gardens.Oxford ismorecoherent; Cambridge more diffuse. Oxford overwhelms—Cambridge beguiles.”Academically,Cambridge tipped slightlymore toward the sciences,Oxford towardtheclassics.

NewCollegewasoneofOxford’stwodozenorsodistinctcolleges.Theplacewaslikeawalled city,withmedieval battlements, pierced by tall, narrow slots throughwhicharchers could fire theirbows, still enclosing two cornersof it and forming abackdroptotheshrubs,trees,andbushesoftheCollegeGarden.InmovingtoNewCollege, Hardy was coming full circle. The college was founded by William ofWykehamin1379,eightyearsbeforehefoundedWinchesterasafeedertoit.Ithadbeen the destination of many of Hardy’s abler Winchester classmates twenty-fiveyearsbeforeandwouldprobablyhavebeenhisaswellhadhenot,hisheadturnedbythatSt.Aubynbook,optedforTrinityinstead.

Hardy had been inOxford just a fewmonths when he received the news fromMadras:

Bydirectionofthe[University]Syndicate,Iwritetocommunicatetoyou,withfeelingsofdeepregret,thesadnewsofthedeathofMr.S.Ramanujan,F.R.S.,whichtookplaceonthemorningofthe26thApril.

“ItwasagreatshockandsurprisetometohearofMr.Ramanujan’sdeath,”Hardyreplied in a letter toDewsbury. But was there, in what he wrote next, the barestbreathofdefensiveness?

Whenhe leftEngland the general opinionwas that,while still very ill, he had turned the corner towardsrecovery;hehadevengainedoverastone inweight(atoneperiodhehadwastedawayalmosttonothing).AndthelastletterIhadfromhim(abouttwomonthsago)wasquitecheerfulandfullofmathematics.

Ramanujan had, after all, been entrusted to Hardy’s care. Was Hardy—in a notuncommonsortofresponsetoalovedone’sdeath—nowtryingtoassurehimselfthatRamanujan’sfinaldeclinehadcomeonlyafterhehadbeenplacedsafelyaboardtheshiptoIndia?

AbouttheimpactofRamanujan’sdeathonHardytherecanbenodoubt:

Formypart, it isdifficult forme to saywhat Iowe toRamanujan—hisoriginalityhasbeena constantsourceofsuggestiontomeeversinceIknewhim,andhisdeathisoneoftheworstblowsIhaveeverhad.

AtOxford,thespecteroftheGreatWarstillhungheavilyoverHardy,asitdidallacrossEurope.FeelingagainstGermanyrandeep.“Letustrust,”oneEnglishscientisthadwrittenNatureintheclosingmonthsofthewar,“thatforthenexttwentyyearsatleastallGermanswillberelegatedto thecategoryofpersonswithwhomhonestmen will decline to have any dealings.” Mathematicians were not immune to thebitterness;manyinEnglandandFrancefelt thatCentralEuropeanmathematiciansshouldbebannedfrominternationalmathematicalcongresses.

Hardyhadbeenrevoltedbythewar’sstupidsavagery,hatedthewholeideaofoldmensendingboysofftodie,andfeltcruellycutofffromhismathematicalfriendson

the Continent. Now, the war over, he tried to heal the wounds. He wrote theLondonTimesprotestingsomeofthevengeful imbecilitiesbeingbruitedabout.Hecooperated in the peacemaking efforts ofGöstaMittag-Leffler, long-time editor ofActaMathematica, a Swedishmathematical journal founded in 1882midst similartensionsamongmathematiciansfollowingtheFranco-PrussianWar.HewrotewithhisviewsofthewartothegreatGermanmathematicianEdmundLandau;hisownviews,Landauwrotebackwithamathematician’stouch,hadbeenthesame—except“withtrivialchangesofsign.”

WhilevisitingGermanyin1921,HardywroteMittag-Leffler:“Formypart,Ihaveinnorespectmodifiedmyformerviews,andaminnocircumstancespreparedtotakepartin,subscribeto,orassistinanymannerdirectlyorindirectly,anyCongressfromwhich, for good reasons or for bad, mathematicians of particular countries areexcluded.”Hehadboycottedone such congress inStrasbourg in1920 fromwhichGermans, Austrians, and Hungarians had been kept out, and he would boycottanotherinTorontofouryearslater.

The armistice, the departure of Ramanujan, his own move to Oxford, andRamanujan’sdeathhadallcomewithineighteenmonths.Butbyallaccounts,HardyfellineasilyatNewCollege,feltathomethereinawayheneverhadinCambridge.It was, Snow tells us, “the happiest time of his life.” He was accepted. His newOxfordfriendsmadeafussoverhim.Hisconversationalflamboyancefoundnewandappreciative ears. Sometimes, it seemed, everyone in the Common Room—theOxford term forwhatback atCambridgewas theCombinationRoom—waited tohearwhatHardywasgoingtotalkabout.

Meanwhile, his collaboration with Littlewood, conducted largely by mail,continued.Hewasattheheightofhismathematicalpowers,thezenithofhisfame.MaryCartwrightwouldrecallhowherbarementionof“ProfessorHardy’sclass”toacollege porter drew a response revealing “a far greater respect forHardy than thecustomarydeferenceofthosedaysofanycollegeportertoanydon.”

For the academic year 1928–1929, Hardy exchanged places with Princeton’sOswald Veblen and spent the year in the United States, mostly at PrincetonUniversity.WhileinAmerica,hekeptupabusylectureschedule;hespokeatLehighUniversity,forexample,onJanuaryeleventh,atOhioStateontheeighteenth,attheUniversity ofChicagoon the twenty-first.DuringFebruary andMarch, hewas inresidenceatCalifornia InstituteofTechnology.At theendof theyear,Princeton’spresidentaskedhimtostayabitlonger.Hardywrotebackthatthoughhe’dhad“adelightfultime,”hisdutiesatOxforddemandedhisreturn.

OnthisoroneofhisothertripstoAmerica,hedevelopedatasteforbaseball.BabeRuth, it was said of him, “became a name as familiar in hismouth as that of thecricketerHobbs.”Onetime,hewassentabook, inscribedby“IronMan”Coombs,stuffedwithproblemsinbaseballtactics.“Itisawonderfulbook,”hewroteapostcardsaying.“Itrytosolveoneproblemaday(e.g.1out,runnersat1stand2nd,batsman[thecricketterm]hitsamoderatepacedgrounderratherwideto2ndbaseman’slefthand—hebeingrighthanded.Shouldhetryforadoubleplay1sttosecondor2ndtofirst?Ithinktheformer.)”

Hisloveofcricketandtennis,ofcourse,continuedunabated.Intennis,hesteadilyimprovedhisgame.InonesnapshottakenwhileHardywasatOxford,itisabright,sunnyday in latespringorearlyfallandHardy, lookingabsolutelysmashing inhiswhite tennis gear, standsmidst a group of about a dozen other players. They areBeautifulPeople,ca.1925orso,andHardy,clutchinghisracket,wearinglong-leggedtennistogsandaheavyshawlsweaterunderajacket,isoneofthem.

ThoughhappyatOxford,by1931Hardywasback inCambridge, asSadleirianProfessor, following thedeathofE.W.Hobson;Cambridgewas,afterall, still thecenter,farmorethanOxford,ofEnglishmathematics,andhewasnowbeingoffereditsseniormathematicalchair.Anotherreason,accordingtoSnow,wasthatthetwouniversitieshaddifferentrulesaboutretirement;whereasOxfordwouldturnhimoutofhisroomsatsixty-five,atTrinityhecouldoccupythemuntilhedied.

Foratime,HardywouldperiodicallyreturntoOxfordforafewweeksatatimetocaptaintheNewCollegeSeniorCommonRoomcricketteam.And,ofcourse,hewasalwaysthereatLord’sfortheannualmatchbetweenCambridgeandOxford.“Therehewasathismostsparkling,yearafteryear,”wroteSnow.“Surroundedbyfriends,men andwomen, hewas quite released from shyness; hewas the centre of all ourattention, which he by no means disliked; and one could often hear the party’slaughterfromaquarterofthewayroundtheground.”

Hardy’sformerlyunpopularantiwarviewswerenow,whennotforgotten,actuallyapplauded.TheyoungCambridgemathematicians,Snowrecords,“weredelightedtohave him back: he was a real mathematician, they said, not like those Diracs andBohrsthephysicistswerealwaystalkingabout:hewasthepurestofthepure.”Itwas,as Laurence Young portrayed it later, a golden age of Cambridge mathematics.“Spiritually and intellectually,Cambridgewas suddenly at least the equal of Paris,Copenhagen,Princeton,Harvard,andofWarsaw,Leningrad,Moscow.”Asprinkleof foreign visitors to Cambridge had now, as Jews and others sought escape fromHitler’sGermany,becomeatorrent.

Beginningaround1933,Hardy,incooperationwiththeSocietyfortheProtectionofScienceandLearning,usedhisinfluencetogetJewsandothersdrivenfromtheirjobstoEnglandandothersafehavens.MathematiciansofthestatureofRiesz,Bohr,and Landau were among those who got out. “Hardy, in many ways, was other-worldly,”A.V.Hillwrote,“butinhisdeepsolicitudeforthedangersanddifficultiesofhiscolleaguesheshowednotonlyabroadhumanitybutafineandresoluteloyaltytotheuniversalintegrityandbrotherhoodoflearning.”

Hardy resigned from at least oneGerman organization ofwhich he had been amember—not because it was German, but because of what it did. “My attitudetowards German connections of this kind,” he wrote Mordell in the early Naziperiod, “is that I do nothing unless I ampositively forced to; but if anti-Semitismbecomesanostensiblepartoftheprogrammeofanyperiodicalorinstitution,thenIcannotremaininit.”

In 1934,Hardywrote toNature responding to aUniversity of Berlin professorwho purported to show the influence of blood and race upon creative style inmathematics.Therewere,itseems,“J-type”and“S-type”mathematicians,theformerofgoodAryanstock,thelatterFrenchmenandJews.HardyicilysurveyedProfessorBieberbach’s assertions,made a showof seeking ground onwhich to excuse them,finally found himself “driven to the more uncharitable conclusion that he reallybelievesthemtrue.”

Hardy’ssympathieslayinvariablywiththeunderdog,andhispoliticalviewsweredecidedly left-wing. Until about 1927, he was active in the National Union ofScientificWorkers, evenmade recruiting speeches on its behalf. In one, as J.B.S.Haldaneparaphraseditlater,hesaidtohisaudienceofscientists“thatalthoughourjobswereverydifferentfromacoalminer’s,weweremuchclosertocoalminersthancapitalists.At leastwe and theminerswere both skilledworkers, not exploiters ofotherpeople’swork,andiftherewasgoingtobealine-uphewaswiththeminers.”VisitorstoHardy’sroomsoftennotedthatonhismantelpiecestoodphotographsofEinstein,thecricketerJackHobbs—andLenin.

Butwithinthemathematicalcommunity,he,Littlewood,andthoseintheircampstoodsquarelyintheEstablishment.Englishmathematicians,Hardywrotein1934,nolongerlaboredunder“thesuperstitionthatitisimpossibletobe‘rigorous’withoutbeingdull,andthatthereissomemysteriousterrortoexactthought.”Therevolutionhe had helped usher in a quarter century before had won the day. Indeed, somewouldgrumble later that ithadactually impededprogress insuchfieldsasalgebra,topology, functional analysis, and other topics within pure mathematics. By the1930s,inanycase,Hardywasseenaspartoftheoldergeneration.

Duringtheseyears,thehonors,largeandsmall,rolledin.OnMarch6,1929,theone-hundredthanniversaryofthedeathofthegreatNorwegianmathematicianAbel,Hardy,inthepresenceofthekingofNorway,receivedanhonorarydegreefromtheUniversityofOslo.

OnDecember27,1932,hegottheChauvenetPrize,awardedeverythreeyearsforamathematicalpaperpublishedinEnglish,forhis“AnIntroductiontotheTheoryofNumbers.”

OnFebruary29,1934,he receiveda letter,on thehammer-and-sickle embossedstationeryof the SovietUnion, from J.Maisky, the Soviet ambassador to Britain,congratulating him on his election as an honorary member of the Academy ofSciencesinLeningrad.

The universities of Athens, Harvard, Manchester, Sofia, Birmingham, andEdinburghawardedhimdegrees.HereceivedtheRoyalMedaloftheRoyalSocietyin1920,itsSylvesterMedalin1940.Hewasmadeanhonorarymemberofmanyofthe leading foreign scientific academies. Without a doubt, he was the mostdistinguishedmathematicianinBritain.

To this period, his prime, much Hardy lore is owed. One year, Hardy’s NewYear’sresolutionswereto:

1.ProvetheRiemannhypothesis.2.Make211noout in the fourth innings of the last testmatch at theOval [whichwas something like

hittingagrandslamhomerunwhilebehindbythreerunsintheninthinningoftheWorldSeries’finalgame].

3.FindanargumentforthenonexistenceofGodwhichshallconvincethegeneralpublic.4.BethefirstmanatthetopofMt.Everest.5.BeproclaimedthefirstpresidentoftheU.S.S.R.ofGreatBritainandGermany.

6.MurderMussolini.

Another story neatly combined his love of cricket, his pleasure in the sun, hiswarfarewithGod,andhismadcapbent.Oneofhiscollaborators,MarcelRiesz,wasstayingattheplaceHardysharedwithhissister inLondon.Hardyorderedhimtostepoutside,openumbrellaclearlyinview,andyelluptoGod,“IamHardy,andIam going to the BritishMuseum.”This, of course, would draw a lovely day fromGod,whohadnothingbettertodothanthwartHardy.Hardywouldthenscurryoffforanafternoon’scricket,fineweatherpresumablyassured.

InlongtalkswithHardybeginningin1931andextendingoverthenextfifteenorsoyears,C.P.SnowcameawaysteepedinHardy’s“oldbrandy”sensibilities.Byoldbrandy Hardy meant any “taste that was eccentric, esoteric, but just within theconfinesof reason.”For example,heoncewroteSnow that “the half-mile fromSt.George’sSquaretotheOval[inLondon]ismyoldbrandynominationforthemost

distinguished walk in the world.”Old brandy was a sort of studied eccentricity—youthful foolishness transformed intoa “mature” form,madea little self-conscious,ossified...

Andthat, indeed, iswhathadhappenedtoHardy.Somehow,hehadbecomeanoldman.Evenbythefallof1931,whenhewasfifty-four,youcouldseesignsof it.BackinCambridgefortheyear,NorbertWienernoticedthat“bynow,HardyhadbecomeanagedandshriveledreplicaoftheyoungmanwhomIhadmetinRussell’srooms”twentyyearsbefore.Hardyknewit,too.OnhisreturntoCambridge,hewasdistressedbyallthenew,youngfaceshesawamongthemathematicians.“There is,”hewrote, “somethingvery intimidating toanolderman insuchyouthfulquicknessandpower.”

Onedayin1939,whiledustinghisbookcase,Hardyhadhisfirstheartattack.Hewassixty-twoatthetime.Initswake,hecouldnolongerplaytennis,orsquash,orcricket.Hiscreativitywaned.One listingofhismost importantpapers (it includedeveryoneofthoseonwhichhehadworkedwithRamanujan)includednonebeyond1935.Now,hisoutputdeclinedbyevenacrudequantitativeyardstick—fromhalfadozenorsoperyearinthelate1930stooneortwoperyear.

His waning mathematical powers depressed him. So did the new war withGermany.But around 1941, when young FreemanDyson came up to CambridgefromHardy’soldschool,Winchester,andfortwoyearsattendedhislectures,Dysoncouldn’tseeit.TohimHardywasstillagod.HeandthreeotheradvancedstudentsallsataroundatableinasmallroomintheoldArtsSchool, listened,andwatchedHardyfromafewfeetaway:

He lectured likeWanda Landowska playing Bach, precise and totally lucid, but displaying his passionatepleasuretoallwhocouldseebeneaththesurface....Eachlecturewascarefullyprepared,likeaworkofart,withtheintellectualdenouementappearingas ifspontaneously inthelastfiveminutesofthehour.Formetheselectureswereanintoxicatingjoy,andIusedtofeelsometimesanimpulsetohugthatlittleoldmaninthewhitecricket-sweatertwofeetawayfromme,toshowhimsomehowhowdesperatelygratefulwewereforhiswillingnesstogoontalking.

HardyretiredfromtheSadleirianChairin1942.Theyearbefore,aphotographerfortheBritishmagazinePicturePostsnappedhis

pictureatarugbymatch.Therehewasonachillywinterday,cigaretteinhand,allrolled up in flannels,watchingCambridge defeatOxford, 9 to 6.The photographlaterappearedinoneofthevolumesofhiscollectedpapers.Hissisterdidn’tlikeit.“Itmakeshimlooksoold,”shesaid.

Buthewasold.

By1946,hewasvirtuallyaninvalid.Snowpicturedhimas“physicallyfailing,shortofbreathafterafewyards’walk.”Hissistercametonursehim(thoughtheTrinityrulesweresostrictthatshehadtoleavehisroomsatnight).

Inearly1947,hetriedtokillhimselfbyswallowingbarbiturates.Buthe tooksomany that he vomited themup, hit his head on the lavatory basin, andwoundupwithanuglyblackeyeforhistrouble.

Later thatyear, theRoyalSocietynotifiedhim thathewas to receive itshighesthonor,theCopleyMedal.“NowIknowthatImustbeprettyneartheend,”hetoldSnow. “When people hurry up to give you honorific things there is exactly oneconclusiontobedrawn.”

OnNovember24,SnowwrotehisbrotherPhilip:“Hardyisnowdying(howlongitwilltakenooneknows,buthehopesitwillbesoon)andIhavetospendmostofmysparetimeathisbedside.”

Itwassoon.HardydiedonDecember1,1947,thedayhewastobepresentedtheCopleyMedal.He left his substantial savings and the royalties of his books, oncehavingprovidedforhissister,totheLondonMathematicalSociety.“Hisloss,”wroteNorbertWiener,“broughtusthesenseofthepassingofagreatage.”

•••

The 1939heart attack began the long physical and emotional slide that led to hissuicideattempt.Anditwasinitswake,aboutamonthafterFrancefelltotheNazis,that he put the finishing touches to A Mathematician’s Apology, his paean tomathematics.SnowsawtheApology “asabookofhauntingsadness,”theworkofamanlongpasthiscreativeprime—andknowingit.“Itisamelancholyexperiencefora professional mathematician to find himself writing about mathematics,” wroteHardy.Paintersdespisedartcritics?Well,thesamewentforanycreativeworker,amathematicianincluded.Butwritingaboutmathematics,ratherthandoingit,wasallthatwaslefthim.

And yet, the sadness is at the prospect of a rich, full life nearing its end, notbitternessata life ill-spent.Pride runs through theApology, too,andpleasure,anddeepsatisfaction.

IstillsaytomyselfwhenIamdepressed,andfindmyselfforcedtolistentopompousandtiresomepeople,“Well,Ihavedoneonethingyoucouldneverhavedone,andthatistohavecollaboratedwithbothLittlewoodandRamanujanonsomethinglikeequalterms.”

Ramanujan.Allthesetwentyyearslater,Ramanujanremainedpartofhim,abrightbeacon,luminousinhismemory.

“Hardy,” saidMaryCartwright, his studentduring the1920s andwhomHardywould describe as the best woman mathematician in England, “practically neverspokeofthingsaboutwhichhefeltstrongly.”Yetatoneremovefromhislistener,ontheprintedpage,hebecamealittlefreer.AndthereherevealedRamanujan’sholdonhim: “I owemore to him,” hewrote, “than to any one else in theworldwith oneexception[Littlewood?]andmyassociationwithhimistheoneromanticincidentinmylife.”

Intheyearsafterhisdeath,HardybeganrummagingthroughRamanujan’spapersandnotebooks.That, asmanyothermathematicianswere to learn, couldbe toughgoing.AfterarrivinginOxford,HardywroteMittag-LefflerthathehadpreparedashortpaperfromRamanujan’smanuscripts,“butitwashardlysubstantialenoughforthe Acta [the journal Mittag-Leffler edited]. I am now trying to make a moreimportantone.Butitisnotpossibletodoitveryrapidly,asallofRamanujan’sworkrequiresmostcarefulediting.”

By1921, hehad culled fromRamanujan’s papers enough to prepare a sequel toRamanujan’s work on congruence properties of partitions. The manuscript fromwhichhewasworking,Hardywroteinanoteappendedtothepaper,whichappearedinMathematischeZeitschrift,“isveryincomplete,andwillrequireverycarefuleditingbefore it canbepublished in full. Ihave taken from it the three simplestandmoststrikingresults,asashortbutcharacteristicexampleoftheworkofamanwhowasbeyondquestiononeofthemostremarkablemathematiciansofhistime.”

Hardy’s own papers over the years were fairly littered with Ramanujan’s name:“Note on Ramanujan’s Trigonometrical Function cq(n) and Certain Series ofArithmetical Functions” appeared in 1921; “A Chapter From Ramanujan’sNotebook” in 1923; “Some Formulae of Ramanujan” in 1924. Then more in themid-1930s: “AFormulaofRamanujan in theTheoryofPrimes”; “AFurtherNoteonRamanujan’sArithmetical functionτ(n).”Hardy appreciated the debt he owedRamanujanandLittlewood:“Allmybestwork,”hewrote,“hasbeenboundupwiththeirs,anditisobviousthatmyassociationwiththemwasthedecisiveeventofmylife.”

Hardywasthirty-sevenwhenhemetRamanujan,livingouthisboyhooddreamasaFellowofTrinity,alreadyanF.R.S.ButhiscollaborationwithLittlewoodhadonlyjustbegun,andhewouldcometoviewhisearlycontributiontomathematics,thoughformidablebystandardsotherthanhisown,asunspectacular.

Then,abruptly,Ramanujanenteredhislife.Ramanujanwas,ifnothingelse,aliving,breathingreproachtotheTripossystem

Hardy despised. Sheer intuitive brilliance coupled to long, hard hours on his slate

madeup formostofhis educational lacks.Andhewas sodevoted tomathematicsthathecouldn’tbothertostudytheothersubjectsheneededtoearnacollegedegree.This“poorandsolitaryHindupittinghisbrainsagainsttheaccumulatedwisdomofEurope,”asHardycalledhim,hadrediscoveredacenturyofmathematicsandmadenewdiscoveriesthatwouldcaptivatemathematicians for thenextcentury. (AndallwithoutaTriposcoach.)

IsitanywonderHardywasbeguiled?Fromthenon,overthenextthirty-fiveyears,Hardydidallhecouldtochampion

Ramanujan and advance his mathematical legacy. He encouraged Ramanujan.Heacknowledgedhis genius.He brought him toEngland.He trainedhim inmodernanalysis. And, during Ramanujan’s life and afterward, he placed his formidableliteraryskillsathisservice.

“HardywroteexquisiteEnglish,”theManchesterGuardianwouldsayofhim,citingespecially his obituarynotice ofRamanujan as “among themost remarkable in theliterature aboutmathematics.”TomathematicianW.N.Bailey, itwas “one of themostfascinatingobituarynoticesthatIhaveeverread.”AnditwasHardy’sbookonRamanujan,morethananythingheknewabouthimotherwise,thatconvincedAshisNandytomakeRamanujanaprimesubjectofhisownbook.Hardy’spenfiredtheimagination,shapingRamanujan’sreceptionbythemathematicalworld.

Itbeganin1916,whenHardyreportedtotheuniversityauthoritiesinMadrasonRamanujan’sworkinEngland;onelookatitandtheyaskedthatitbepreparedforpublication.Init,Hardywroteofthe“curiousandinterestingformulae”Ramanujanhad inhispossession,ofhowRamanujanpossessed “powers as remarkable in theirway as thoseof any livingmathematician,” thathis giftswere “sounlike thoseof aEuropeanmathematician trained in the orthodox school,” that his work displayed“astonishing individuality and power,” that in Ramanujan “India now possesses apuremathematicianofthefirstorder.”Thiswasscarcelythesortoflanguageapttopassunnoticedamongmathematicians,IndianorBritish,accustomedtothesortofflat, gray prose normally appearing in their journals. Someone once said ofHardythat “conceivably he could have been an advertising genius or a public relationsofficer.”Herewastheevidenceforit.

Hardy’slongobituaryofRamanujanappearedfirstintheProceedingsoftheLondonMathematicalSocietyin1921,alittlelaterintheProceedingsoftheRoyalSociety,thenagaininRamanujan’sCollectedPapers in1927. In it,he toldRamanujan’s story.Heinvested itwith feeling.His language lingered inmemory. “One gift [Ramanujan’swork] has which no one can deny,” he concluded—“profound and invincibleoriginality.”

Hewouldprobablyhavebeenagreatermathematicianifhehadbeencaughtandtamedalittleinhisyouth;hewouldhavediscoveredmorethatwasnew,andthat,nodoubt,ofgreaterimportance.Ontheotherhandhewouldhavebeen lessofaRamanujan,andmoreofaEuropeanprofessor,and the lossmighthavebeengreaterthanthegain.

Snow,whofirstmetHardyin1931,revealedthatforallHardy’sshyness,“abouthisdiscoveryofRamanujan,heshowednosecrecyatall.”MaryCartwrightrecalledthat “Hardy was terribly proud, and rightly, of having discovered Ramanujan.”Ramanujanhadenrichedhislife.Hedidn’twanttoforgetRamanujan,andhedidn’t.

OnFebruary 19, 1936,Hardywrote S.Chandrasekhar fromCambridge: “I amgoing to give some lectures (here and at Harvard) on Ramanujan during thesummer.”TheywouldbecomethebasisforRamanujan:TwelveLecturesonSubjectsSuggestedByHisLifeandWork.“Alaboroflove,”onereviewercalledit.

TheHarvard lecture was part of the great university’s celebration of the three-hundredth anniversary of its founding. The bash culminated in three grandTercentenaryDays,fromSeptember16to18.HarvardYard,nowagreatoutdoortheaterwith seventeen thousand seats, was awashwith silk hats, crimson bunting,andcolorfulacademiccostumes.Onthesecondevening,at9:00P.M.,upwardofhalfamillionpeoplelinedbothbanksoftheCharlesRiverfortwohoursoffireworks.

The following morning, under a steady drizzle and dark, brooding clouds (theleadingedgeofahurricanemovinguptheAtlanticCoast), sixty-twoof theworld’smostdistinguishedbiologists,chemists,anthropologists,andotherscholarsreceivedhonorary degrees. Theymarched in a procession fromWidener Library and tooktheirplacesonstandserectedinfrontofthepillarsofMemorialChurchattheyard’snorthend.PsychoanalystCarlJungwasamongthem.SowasJeanPiaget,thepioneerstudentofchilddevelopment.SowasEnglishastrophysicistSirArthurEddington.So was Hardy. The citation honoring him, slipped within the red leatherpresentation book stamped with Harvard’s Veritas seal, called him “a Britishmathematicianwhohas led theadvance toheightsdeemed inaccessiblebypreviousgenerations.”

DuringhisstayatHarvard,Hardywasputupatthehouseofaprominentlawyer,who later became aUnitedStates senator.Bothhe andhis host, according to oneaccount,wereworried:whateverwouldtheytalkabout? “The lawyerwasnobetterprepared to discussZeta functions than themathematician to comment upon therule inShelley’s case.”So theyseizedon their commonenthusiasm—baseball.TheRedSoxwereintown,andHardywasatFenwayoftentowatchthem.

In the weeks prior to the grand finale, a Tercentenary Conference of Arts andSciences brought toHarvardmore than twenty-five hundred scholars for lectures

underbroadrubricsofknowledgelike“ThePlaceandFunctionsofAuthority,”and“TheApplicationofPhysicalChemistrytoBiology.”Einstein’swifewasill,sohesentword that he couldn’t come. Nor could Werner Heisenberg, author of theuncertaintyprinciple,whowasadvisedatthelastminutethathewasneededforeightweeks’ service in Hitler’s army. Their absence notwithstanding, it was an augustgroup,includingnofewerthanelevenNobelPrizewinners.“HighbrowsatHarvard,”Timeheadeditsaccount.TheNewYorkTimescoveredsomeofthepubliclectures,includingHardy’s.

Ataboutnineintheeveningoftheconference’sfirstday,Hardy—wizened,gray,andalmost sixty now—rose before his audience inNew LectureHall. “I have setmyself a task in these lectures which is genuinely difficult,” he told them, in themeasuredcadencesthatwerethemarkofhisspeechandofhisprose.

and which, if I were determined to begin bymaking every excuse for failure Imight represent as almostimpossible.Ihavetoformmyself,asIhaveneverreallyformedbefore,andtotrytohelpyoutoform,somesort of reasoned estimate of themost romantic figure in the recent history ofmathematics; amanwhosecareer seems full of paradoxes and contradictions, who defies almost all the canons by which we areaccustomedtojudgeoneanother,andaboutwhomallofuswillprobablyagreeinonejudgmentonly,thathewasinsomesenseaverygreatmathematician.

AndthenHardy,thememorystillfreshofthedayaquartercenturybeforewhenanenvelopestuffedwithformulasarrivedinthemailfromIndia,begantotellabouthisfriend,Ramanujan.

Thepassportphoto.Ramanujan in1919, onhiswayback to India. “He looks rather ill,”G.H.Hardywrotewhenhe first saw the photo in 1937, “but he looks all over the genius hewas.”Master and Fellows ofTrinityCollege,Cambridge

Komalatammal,Ramanujan’smother,andthedecisiveinfluenceonhiminhisyouth.NophotoofK.SrinivasaIyengar,Ramanujan’sfather,isknowntoexist.Ragami’sCollections,Madras,SouthIndia

Ramanujan’shouse,onSarangapaniSannidhiStreet,Kumbakonam,SouthIndia.Once,whileinhighschool,hefoundthataformulahehadthoughtoriginalwithhimactuallywentback150years.Mortified,hehidthepaperonwhichhehadwrittenitintheroofofthehouse.Ragami’sCollections,Madras,SouthIndia

Arecentphotoofthepial,orfrontporch,ofRamanujan’shouseinKumbakonam.Herehewouldsitforhoursandworkonmathematicswhilehisfriendsplayedinthestreet.

Ramanujan scored high on this examination, which he tookwhen hewas nine. But later, once he discoveredmathematicsand lost interest inall else,he regularly failedhisexams. (Here, theEnglishspellingofhisTamilnamewasrenderedasRamanujam.)Ragami’sCollections,Madras,SouthIndia

Arecentphotoof theSarangapaniTemple, justupthestreet fromRamanujan’shouse,whichisvisibleontheright.

AtTownHighSchool, inhishometownofKumbakonam,Ramanujanwasstillaconventionallygoodstudent,earningprizesandwinningpraisefromhiselders.Here,theschool’scampusinarecentphotograph.

TheycallitRamanujanHalltoday,inhonorofTownHighSchool’smostdistinguishedalumnus.The“m”attheendreflectshowRamanujan’sTamilnameoftengetstransliteratedintoEnglish.ItwasunderthisspellingthatitfirstappearedinIndianmathematicaljournals.

S.Tirunarayanan,bornin1905whenRamanujanwas17.Hisolderbrotherwouldteasehim,carryhimonhisshoulders,tellhimstories.Ragami’sCollections,Madras,SouthIndia

Janaki, Ramanujan’s wife, in a picture taken after Ramanujan’s death. As a widow, and mostly cut off fromRamanujan’sfamily,shesupportedherselfasaseamstress.Ragami’sCollections,Madras,SouthIndia

NarayanaIyer,Ramanujan’s immediatebossattheMadrasPortTrust,oneofhismostardentchampions,andhimselfafinemathematician.Whenthetwoworkedonmathematics,oftenuntillateintothenight,Ramanujan’spenchantforcollapsingmanystepsintoonelefthimdazed.“Youmustdescendtomylevelofunderstanding,”hewouldcomplain.Ragami’sCollections,Madras,SouthIndia

P.V.SeshuIyer.OneofRamanujan’smathematicsteachersatGovernmentCollegeinKumbakonamandlateroneofhissupporters.ButonefriendrecalledRamanujancomplainingthatSeshuIyer,likeeveryoneelse,hadatfirstbeen“indifferent”tohimandhiswork.Ragami’sCollections,Madras,SouthIndia

AfterRamanujanmarried,hescouredSouthIndia fora joborapatron,his friendsoftenputtinghimupforawhile.Onenight,while stayingwith a friend atVictoriaHostel inMadras, shownhere in a recent photo, hecomparedhis impoverished lot to thatofGalileo,persecutedby the Inquisitionandmisunderstood inhisowntime.

Soon afterRamanujan receivedhis first encouraging letter fromHardy inCambridge, hewas named researchscholarattheUniversityofMadras.Freedfrommoneyworriesforthefirsttimeinhislife,hewouldcomehere,totheConnemaraLibrary,seenhereinarecentphoto,andlosehimselfinmathematics.

Shrine to thegoddessNamagiri,Ramanujan’s familydeity, in a recentphoto. Itwashere in theSouth IndiantownofNamakkalthatRamanujancameinlate1913withNarayanaIyer.HestayedonthetemplegroundsforthreenightsandbytheendhadresolvedtogotoEngland,indefianceofHindutradition.

E. W. Hobson (previous) and H. F. Baker. Both were eminent Cambridge mathematicians. Both receivedRamanujan’sappealsforhelp.Bothdismissedthem.MasterandFellowsofTrinityCollege,Cambridge

The tank,a large ritualpool,opposite theParthasarathyTemple, thecentral religious shrineoftheTriplicanedistrictofMadras.ItwasinTriplicane,downthestreetfromthetank,thatRamanujanlivedintheperiodbeforeheleftforEngland.

The first of nine pages ofmathematical results Ramanujan sentG.H.Hardy from India in 1913.Syndics ofCambridgeUniversityLibrary

AmoretypicalpageofRamanujan’sfirstlettertoHardy.SyndicsofCambridgeUniversityLibrary

Sophia(previous)andIsaacHardy,G.H.Hardy’sparents.Shewasupright,pious,andstern.Hewasa“WhiteKnight,”who“neverutteredanunkindword.”MasterandFellowsofTrinityCollege,Cambridge

G.H.Hardy,alittlebeforeheheardfromRamanujan,atthetimehebecameaFellowoftheRoyalSociety.“Hisfacewas beautiful,”C. P. Snow oncewrote of him, “with high cheek bones, thin nose, spiritual and austere.”MasterandFellowsofTrinityCollege,Cambridge

Hardy’ssister,Gertrude.Shecalledherbrother“Harold,”hismiddlename.Neitherofthemevermarried.Bothspenttheirlivesinacademicsettings.Bothwereenchantedbyintellectandcontemptuousofreligion.MasterandFellowsofTrinityCollege,Cambridge

The“Mt.Pleasant”cottages,inoneofwhichHardylivedasachild,acrosstheroadfromCranleighSchool.Theseparateentrancenook,attheextremerightofthisrecentphoto,isnew.

CranleighSchool,conceivedasa“middleclassschool,”inarecentphoto.Hardy’sfatherwasdrawingteacher.Hismotherdirectedthepreparatoryschoolacrosstheroad.Hardyhimselfattendedforafewyears,butbythetimehewasthirteenhadleftforamoreacademicallychallengingschool.

FlintandStone.AcourtatWinchesterSchool,atraditionalEnglishpublicschoolgoingbacktothefourteenthcentury.Hardyhatedtheplaceandapparentlyneverreturnedforavisit.

J. E. Littlewood, “Senior Wrangler,” 1905. That honor went to the man scoring highest on the notoriousMathematicalTriposexam,andinEnglishacademiccircleswasabout likebeingnamedAll-American,RhodesScholar, and Bachelor of the Year all at once. It was Littlewood to whomHardy turned when Ramanujan’smystifyingletterarrivedfromIndia.MasterandFellowsofTrinityCollege,Cambridge

E.H.Neville,Hardy’s emissary toRamanujan.Hearrived inMadras in thewinterof1913 togive a seriesoflectures.ButHardyhadgivenhimanadditional task—tobringRamanujan toEngland.MasterandFellows ofTrinityCollege,Cambridge

Long the first destination in England ofmost Indian students: 21Cromwell Road, in the South KensingtondistrictofLondon,shownhereinarecentphoto.Ramanujanwasthereinmid-April1914.

Ramanujan’sintroductiontotheEnglishhome:113ChestertownRoad,inarecentphoto.E.H.Nevilleandhiswifeboughtitin1913,andRamanujanlivedherefortwomonthsin1914.

NewCourt, Trinity College, whereHardy lived during the time he knew Ramanujan.Master and Fellows ofTrinityCollege,Cambridge

TheWrenLibrary,TrinityCollege,inarecentphototakenfromtheBacks.Hardylivedonthesecondfloorofthebuildingtotheright,roomsextendingoverthegateintoNewCourt.

Bishop’sHostel. From about 1915 to 1917,Ramanujan lived in the building closest to the camera. From thewindowwiththestrongreflectiononthesecondfloor(directlyovertheghostlyfigurebelow)Ramanujancouldseethespiredsteepleatopthecollegedininghall.ButhisrigidvegetarianismmeantthatheneveratetherewithotherTrinityCollegescholars.MasterandFellowsofTrinityCollege,Cambridge

The last page of a letterHardywrote toRamanujan inFebruary1918,whileRamanujanwas sick and in thehospital,abouttheircurrentwork.MasterandfellowsofTrinityCollege,Cambridge

A.S.Ramalingam,anengineerwhomRamanujanmetinthedaysafterhearrivedinEngland.Threeyearslater,RamalingamvisitedhimatMatlock,atuberculosissanatorium,wherehisfriend’semaciatedconditionandfinickyeatinghabitslefthimworried.Ragami’sCollections,Madras,SouthIndia

TheIndianstampissuedin1962tohonorRamanujan.

“Gometra,”thehouseoffHuntingtonRoad,intheChetputsectionofMadras,whereRamanujandied.Ragami’sCollections,Madras,SouthIndia

P.S.Chandrasekar,Ramanujan’sphysicianbackinIndia.Ramanujan’sdeath,hewroteinhisdiarythefollowingday,was“atragedytoodeepfortears.”Ragami’sCollections,Madras,SouthIndia

Hardy inhisprime. “Tosit thatway,” someoneoncesaidofhim, “youhave tohavebeeneducated inapublicschool.”(Hewas.)MasterandFellowsofTrinityCollege,Cambridge

Aphoto taken in1941, snappedby someone fromThePicturePost, asHardywatched a rugby gamebetweenCambridgeandOxford.Hewas64yearsold—andafteraseeminglyendlessyouth,lookedit.MasterandFellowsofTrinityCollege,Cambridge

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Notes

I have supplied notes formost quotations and formost statements of fact that the inquiring readermight bemovedtoquestion.

I have not usually supplied citations for amply documented historical events, such as World War I.InformationaboutSouthIndiantownsandregions,population,geography,climate,temples,agriculture,literacystatistics,andthelikeislargelydrawnfromstandardgazetteers,publishedbothinRamanujan’stimeandintheyearssince,includingtheImperialGazetteer,MadrasPresidency,1908,andvariousdistrictgazetteers.

For many of the book’s supporting characters, I have not usually cited biographical information from theDictionary of National Biography, from the series of published biographical memoirs of Fellows of the RoyalSociety,orfromsimilarstandardbiographicalreferences.

Inthecaseofunpublishedletters,Ihavetriedtonotethecollegeorotherinstitutioninwhosefilestheymaybefound,orthepersonwhosuppliedthemtome.

ReferencestoHardy’sCollectedPapersaretovolume7unlessindicatedotherwise;thosetoP.K.Srinivasanaretovolume1ofRamanujanMemorialNumber,unlessindicatedotherwise.ThosetoRagami,thepennameofT.V. Rangaswami, carry no page number but are based on informal translations from the Tamil of his bookRamanujan, theMathematical Genius, by V. Anantharaman and his wifeMalini, of Baltimore. The “ReadingManuscript” refers to the original version of E.H.Neville’s broadcast talk onRamanujan, now stored at theUniversityofReading,inEngland.

MuchoftheinformationcontainedintheFamilyRecordisambiguous,andIhavereferredtoitonlywhen,infullcontext,theevidenceseemsclear.

Information drawn from written materials and personal interviews has been supplemented by personalobservationandinformalconversationinEnglandandIndia.

PROLOGUE

“yourwonderfulcountrymanRamanujan.”S.R.Ranganathan,80–81.“wasneartoreligion.”OxfordMagazine,2January1948.“athoughtofGod.”S.R.Ranganathan,88.

CHAPTERONE

September1887.FamilyRecord.Andsowidelyobserved.Slater,SouthernIndia,121.“wetskull.”CoimbatoreDistrictGazetteer,1966.TeppukulamStreet.Interview,T.V.Rangaswami.Smallpoxincident.FamilyRecord;Ragami;P.K.Srinivasan,Margosaisalsoknownasneem.casestudyinthedamningstatistics.BirthsanddeathsinFamilyRecord.caseofitchingandboils.FamilyRecord.hescarcelyspoke.Ragami.AtsharaAbishekam.Ragami.bristledatattending.FamilyRecord.

Ramanujanwasfondofasking.SeshuIyer,81.crushhimtopieces.FamilyRecord.shuffledbetweenschools.FamilyRecord.armsfoldedinfrontofhim.Ragami.loandispute.Interview,T.V.Rangaswami.bouncedbacktohismaternalgrandparents.FamilyRecord.thefamilyenlistedalocalconstable.FamilyRecord.BackinKumbakonambymid-1895.FamilyRecord.asmallcourtyard.Padfield,14;Hemingway,TanjoreDistrictGazetteer.Mahammakham festival. T. R. Rajagopalan, 34; Hemingway, Tanjore District Gazetteer, supplies 1897 as a

festivalyear.three-quartersofamillionpilgrims.Slater,SouthernIndia,117.itswaterlevelwassaidtoriseseveralinches.Urwick,68.seventy-two-bedhospital.ImperialGazetteer,MadrasPresidency,1908.Ricegrowing.Interview,J.M.Victor.SeealsoHemingway,TanjoreDistrictGazetteer.littleroominwhichtograze.Hemingway,TanjoreDistrictGazetteer.thatsamenumberofvillages.ImperialGazetteer,MadrasPresidency.Kumbakonamsaris and silk. Interviews inKumbakonam, especiallywithR.Viswanathan,headmaster,Town

School,andwithclothmerchants.twothousandsmalllooms.ImperialGazetteer,Madras,1908.couldcostasmuchasahundredrupees.Hemingway.ayear’s incometomanypoor families.Forexample,Compton,164–165,pictures fiveannasperdayasatypical

wageforunskilled labor(16annas=1rupee).Fuller,63,saysfield laborerscouldbehiredfortwoorthreepenceaday,theEnglishpencebeingequaltoananna.AmorerecentSouthIndiandistrictgazetteer,lookingback,givestheaverageagriculturalwagebetween1901and1912asthreerupees,eightannaspermonth,oragainsomethinglikethreeannasperday.ImperialGazetteer,1908,alsogivesthegoingrateforunskilledlaboratthreeannasperday(withskilledlaborworthsevenoreightannasperday).Thus,ifwefigurethreehundredworking days a year at, say, four annas per day,we get about seventy-five rupees as a representative year’sincome.

normallythehusbands.Slater,SouthernIndia,119.Clerk’slife.BasedoninterviewsinKumbakonam.Srinivasawasgoodatappraisingfabrics.K.R.Rajagopalan,4.Indianfather’srole.SeeCarstairs,67–69.“Veryquiet.”Bharathi,51.“weightless.”Nandy,102.reminderstokeepupthehouse.P.K.Srinivasan,170.Ramanujanwrotehismother.Ibid.,168.GoatsandTigers. Interview,M.Vinnanasan,Kumbakonam. Janaki, inP.K.Srinivasan, 171, saysRamanujan

andhismother played the “15 points game,” another name forGoats andTigers.A recent SalemDistrictgazetteerreferstoagame,calledPulikatlam,whichmaybethesamething.

“a shrewd and cultured lady.” Seshu Iyer, 81.Other information aboutRamanujan’smother and her family isderived from biographical material in Port Trust File and interviews with S. Sankara Narayanan, K.Bhanumurthi,Janaki,andothers.

managedachoultry.Interview,T.V.Rangaswami.

Komalatammalfedhimhisyogurt.ThispicturedrawnlargelyfrominterviewwithK.Bhanumurthi.intotheprincipal’soffice.P.K.Srinivasan,85.“Anexceptionallygiftedlady.”Ibid.,114.Castesystem.SeeThurston;Mayo;Bhattacharaya;Fuller;Padfield;Compton;M.N.Srinivas.tosecuredivinegrace.See,forexample,Chopraetal.Brahminfamilies...wouldpullovertothesideoftheroad.Fuller,147.Interview,A.Saranathan.Hindueatingpractices.Fuller,147.Numerousinterviews.thatgavefoodareddishcastreminiscentofblood.Interview,A.Saranathan,Kumbakonam.“IfIseemeat,Ibeginto

vomit.”“Asthechildlearned...”Carstairs,67.afastidiousnessaboutHindulife.SeeCarstairs,80.“Asceticismandmysticism...”InSinger,8.“Simplelivingandhighthinking.”Interview,T.S.Bhanumurthy.of650graduatesoftheUniversityofMadras.CitedinM.N.Srinivas,102.“alanguagemadebylawyersandgrammarians.”QuotedinSlater,SouthernIndia,132.distinctfrom...Hindi.G.Ramakrishnaetal.,459.boastedaverseformreminiscentofancientGreek.Slater,SouthernIndia,136.almosttwentymillionpeople.Thurston,122.11percentofTamilBrahminsliterate.CitedbyM.N.Srinivas,179.TownHighSchool. Interview,R.Viswanathan, thecurrentheadmaster.Seealso “HistoryofOurSchool,” in

CentenaryCelebrationSouvenir.partial to impromptu strolls between classes. “My Reminiscences,” R. Kandaswamy Moopanar, in Centenary

CelebrationSouvenir.comingtohimforhelp.S.R.Ranganathan,61.“Butiszerodividedbyzeroalsoone?”S.R.Ranganathan,105.Boarders.P.K.Srinivasan,84.Loney’sTrigonometry.S.R.Ranganathan,105.Mostlikely,accordingtoRichardAskey,thiswasonlypart1of

thetwo-volumetext.learnedfromanolderboy.MadrasPortTrust.understandtrigonometricfunctions.SeshuIyer,82.toanynumberofdecimalplaces.Ibid.finishinhalftheallottedtime.Ibid.solvethemataglance.P.K.Srinivasan,84.RamanujandidGanapathiSubbier’sjob.P.K.Srinivasan,104.Thecurrentheadmaster,R.Viswanathan,gives

the number of students in the school at about one thousand. N. Govindaraja Iyengar, quoted in P. K.Srinivasan,putsthefigureatfifteenhundred.

deservedhigherthanthemaximumpossiblemarks.P.K.Srinivasan,121.SarangapaniTemple.SeeT.R.Rajagopalan,36;G.Ramakrishnaetal.,258;Das,135;Balasubrahmanyan,196.templebuiltbyNayakkings.Das,137.fallasleep in themiddleof theday.Ragami, “Ramanujan, ‘AGiftFromHeaven,’ ”IndianExpress,19December

1987.Once a year during the years hewas growingup.Seshu Iyer, 82. Interviews, at the temple,withS.Govindaraja

BattachariarandP.Vasunathan.FormoreonUppiliapanKoilseeDas,143.

Sacredthreadceremony.Padfield,63.MoonlightwalktoNachiarkovil.S.R.Ranganathan,66.God,zero,andinfinity.Ibid.,84.“Immenselydevout.”Ibid.,73.“Atruemystic.”Ibid.,88.deityoftheanthill.Whitehead,15.“BrahminicalHinduismisherealivingreality.”Hemingway.“asWestminsterAbbeyandSt.Paul’saretotheotherchurchesofLondon.”Urwick,72.Rameswaram.SeeG.Ramakrishnaetal.,383.Hindu deities. SeeDas; FriedhelmHardy, “Ideology andCultural Contexts of the SrivaishnavaTemple,” in

Stein,119;Fuller.OnecontemporaryEnglishaccount.Fuller,160.“fusionofvillagedeities...”Chopraetal.directthemtowardsomethinghigherandfiner.V.Subramanyam,amemberofNarayanaIyer’sfamilyinMadras,

offersadelightfulmetaphor:Consideranarchitect’sdrawing,whosearcanevisuallanguagemightrepresentamasterpiece.One ignorantof that languagemay,asheexamines thedrawing,beunable to see the genius itembodies.Andyet,hecanappreciatethecareandreverencewithwhichthedrawingisunrolled,handled,andpreserved,perhapsbeinspiredtosearchoutitshiddenmeaning.Likewise,astonedeity.

toenteratrance.S.R.Ranganathan,13.abizarremurderplot.P.K.Srinivasan,98.speakthroughherdaughter’sson.S.R.Ranganathan,13.

CHAPTERTWO

It first came into his hands. Seshu Iyer andRamachandraRao, xii. But S.R.Ranganathan, 19, says itwas an“elderlyfriend.”

“notinanysenseagreatone.”Hardy,Ramanujan,2.Tripos.Seechapter4.G.S.Carr.Hardy,Ramanujan,3.Carr,iv.CambridgeUniversityrecords.fromhisdeskinHadley.Carr,x.firststatementonthefirstpage.Ibid.,33.“Ihave,inmanycases...”Ibid.,iv.“hismethods.”Littlewood,Miscellany,87.“Throughthenewworldthusopenedtohim...”SeshuIyerandRamachandraRao,xii.Government College. Hemingway, Tanjore District Gazetteer. Imperial Gazetteer, Madras Presidency, 1908.

GovernmentCollegeCalendar for1975–76,partB.bridgetodayspanningtheriver.T.V.Rangaswami, inaninterview, claims that during Ramanujan’s time there was a feeble footbridge across the river, perhapsconstructedfrompalmyrapalms.ButGovernmentCollegeCalendar,5,recordsthebuildingofthebridgeonlyin1944,replacingaferryservice.

ahostelforseventy-twostudents.Hemingway,TanjoreDistrictGazetteer.“Collegeregulationscouldsecurehisbodilypresence.”Neville,“Ramanujan”(Nature149),292.“Hewasquiteunmindful.”P.K.Srinivasan,122.SeealsoS.R.Ranganathan,20.foreign-languagemathtexts.P.K.Srinivasan,106.lefthimtodoashepleased.Ibid.,122.“ingeniousandoriginal.”SeshuIyer,82.

Professordemandedbook’sreturn.P.K.Srinivasan,122.SeshuIyer“indifferent.”Ibid.,99.“Tothecollegeauthorities...”Neville,“Ramanujan”(Nature149),292.wenttoseetheprincipal.P.K.Srinivasan,85.Tuitionwasthirty-tworupeesperterm.Hemingway,TanjoreDistrictGazetteer.InearlyAugust1905.FamilyRecord.two-thirdstheload,attwo-thirdsthespeed.Hemingway,TanjoreDistrictGazetteer.couldstilltakethreeweeks.KameshwarC.Wali,Chandra(UniversityofChicagoPress,1990),42.“Whenyougettothethird-classrailwaycarriage...”Compton,27.Vizagapatnam.Thurston,40;Urwick,83;ImperialGazetteer,MadrasPresidency,1908.Fragmentaryaccounts.ThescantyinformationavailableaboutRamanujan’sflighttoVizagapatnamisdrawnfrom

FamilyRecord; SureshRam, 10; Seshu Iyer, 83; S.R.Ranganathan, 46, 94;K.R.Rajagopalan, 11.Also,interviewswithP.K.SrinivasanandT.V.Rangaswami.

probablybySeptember.FamilyRecord.Primaryexamhumiliation.P.K.Srinivasan,106.Ragami.hesecretedthepapers...intheroofofhishouse.SeshuIyerandRamachandraRao,xii.“toosorryforhisfailure.”P.K.Srinivasan,27.“Anobligatoryaspectofshame...”Wurmser,52.“winceatourselvesinthemirror...”Ibid.,17.“Hidingisintrinsic...”Ibid.,29.a“mentalaberration.”QuotedinNandy,108“atemporaryunsoundnessofmind.”MadrasPortTrust.SeealsoSeshuIyer,whosaysthat“beingtoosensitiveto

askhisparentsforhelp”afterhisfailureatGovernmentCollege,heleftforVizagapatnam.hebreathlesslyinquired.S.R.Ranganathan,77.PachaiyappaMudaliar.Muthiah,186.modeledontheTempleofTheseus.Ibid.,185.arrivedatEgmoreStation.Interview,T.V.Rangaswami,whosayshehearditfromJanaki.“Toappearandsucceed...”Fuller,175.halffailed.Fuller,176.livedafewblocks...offthefruitbazaar.S.R.Ranganathan,73.boutofdysentery.FamilyRecord.atPachaiyappa’sCollege.Ibid.,66,73.showhowtosolveit.S.R.Ranganathan,65–67.SingaraveluMudaliar.Ibid.,65.Ramanujan’sgifts. “Of theproblemsgiven inour textbooks ingeometry, algebra, and trigonometry,” recalled a

Pachaiyappa’sclassmate,RamayanaRatnakaraT.SrinivasaRaghavacharya,“heusedtoremark,‘Theseareallmentalsums.’”S.R.Ranganathan,75.

“Attheupperleft-handpartofthestomach...”FosterandShore,17.“Procurearabbit.”Ibid.,13.abig,anesthetizedfrog.S.R.Ranganathan,68.“...theDigestionchapter.”Ibid.,69.He’dtakethethree-hourmathexam.P.K.Srinivasan,119.

December1906.Thereissomedisagreementhere.Forexample,accordingtoK.R.Rajagopalan,RamanujandidnottaketheF.A.examinationinDecember1906,butonlythefollowingyear.AndtheFamilyRecordhashimtakingan“IntermediateExam”in1908.Noneofwhichmuchaffectsthestory:Ramanujantookexams,alwaysfailedthem,andwoundupcredentialless.

“fallfarshortofthemultitude.”Compton,171.Ramanujan’scap.P.K.Srinivasan,117.Agriculturalworkersinsurroundingvillages.Seenoteforchapter1,page17.anoldwoman...wouldinvitehiminforamiddaymeal.S.R.Ranganathan,22.Ramanujanfeddosai.P.K.Srinivasan,96.RamanujantutorsViswanathaSastri.Ibid.,89.TutortoGovindarajaIyengar.S.R.Ranganathan,62.“Inprovingoneformula,hediscoveredmanyothers...”Neville,“Ramanujan”(Nature149)(1942):292.a successionofhandwrittenaccounts.This iswhat Ihavecalled theFamilyRecord.“apeculiar green ink.”G.N.

Watson,“Ramanujan’sNoteBooks,”JournaloftheLondonMathematicalSociety6(1931):139.“Twomonkeyshavingrobbedanorchard...”QuotedbyA.C.L.Wilkinson,“PresidentialAddress,”Journalofthe

IndianMathematicalSociety,February1919,24.“Everyrationalinteger.”Hardy,inRamanujan,CollectedPapers,xxxv.“fromwhichmostofuswouldshrink.”Lecturenotes,B.M.Wilson.TrinityCollege.WilliamThackeray.MadrasCivilServants(London:Longman,Orme,Brown,andCo.,1839).“veryproperthatinEngland.”QuotedinChopraetal.,vol.3.“intotheicywater.”Bell,330.towell-worn paths. For a kindred instance, considerWeierstrass: “The creative ideas with which he fertilized

mathematics were for themost part thought out while he was an obscure schoolteacher in dismal villageswhere advancedbookswereunobtainable. . . .Beingunable to afford postage,Weierstrasswas barred fromscientific correspondence. Perhaps it is well that he was; his originality developed unhampered by thefashionable ideas of the time. The independence of outlook thus acquired characterized his work in lateryears.”Bell,416.

“thecarefreedays.”Neville,“Ramanujan”(Nature149):293.Ramanujanwarnedtheparentsofasickchild.S.R.Ranganathan,85.“theprimordialGodandseveraldivinities.”P.K.Srinivasan,92.SatyapriyaRao.S.R.Ranganathan,85;P.K.Srinivasan,90. “Hewasa sturdy strongmanandhe taught the

studentsnativeexercises—‘Dandal,’‘Baski,’etc.”CentenaryCelebrationSouvenir.TheoryofZeroandInfinity.S.R.Ranganathan,82.“spokewithsuchenthusiasm.”Ibid.,83.he’ddropbythecollege.Ibid.,61.legspulledintohisbody.Ibid.,62.“theIndiancharacterhasseldombeenwanting.”Hindu,11February1889.“ablankandvacantlook.”Bharathi,51.“thattime-testedIndianpsychotherapy.”Nandy,109.

CHAPTERTHREE

Ramanujan’s marriage. Ragami; P. K. Srinivasan, K. R. Rajagopalan, 1718; Nandy; Suresh Ram; S. R.Ranganathan.InterviewswithJanaki,Mr.andMrs.T.U.Bhanumurthy.

Thefamilyhadoncebeenbetteroff.Interview,Janaki.

wouldnotsomuchasglimpsehisface.Inaninterview,Janakiconfirmsthisstandardpractice.Father’sresponsetowordofson’smarriage.Interviews,Janaki,Mr.andMrs.T.S.Bhanumurthy.afamilyinKanchipuram.FamilyRecord.eversensitivetolocalcustom.Compton,117.Mysoreban.Ibid.,119.sixmonths’income.Fuller,143.localusurers.Marvin,102.adoublewedding.Interview,Janaki.FamilyRecord.forthreeyears.FamilyRecord.Hydrocele.Interview,JacekMostwin.Dr. Kuppuswami volunteered to do the surgery for free. P. K. Srinivasan, 100; K. R. Rajagopalan, 19; Family

Record.Aschloroformadministered.SeshuIyer,82.thewoundbegantobleed.P.K.Srinivasan,100.hewouldoftenhavetodepend.Interview,T.V.Rangaswami.morethanaweek’spay.Compton,27,putsthethird-classfare(in1904)ataboutafarthingpermile.Afarthing

wasone-quarterofapenny,andapennyequaledananna.Thus,atriptoMadras,aboutafour-hundred-mileround-trip fromKumbakonam,wouldhavebeenaboutahundred annas, or six rupees.Ramanujan’s fathermadetwentyrupeespermonth.Plainly,gettingaroundSouthIndiaduringthisperiodmusthaverepresentedarealfinancialburdentoRamanujanandhisfamily.

atthehouseofafriend.S.R.Ranganathan,87.hestayedwithViswanathaSastri.P.K.Srinivasan,90.onthesufferanceoftwooldKumbakonamfriends.S.R.Ranganathan,69.Narasimhasqueakedby.P.K.Srinivasan,108.Onedayprobablysoonafterthis.S.R.Ranganathan,74.“incorrigibleidler.”Neville,“Ramanujan”(Nature149):292.hadmadehimmarry.S.R.Ranganathan,24.“Likeregimentswehavetocarryourdrums.”Chaudhuri,101.neveramounttoanything.P.K.Srinivasan,123.“sofriendlyandgregarious.”Ramaseshan,4.“Hewouldopenhisnotebooks.”P.K.Srinivasan,123.“suchasimplesoul.”S.R.Ranganathan,61.puppetshows.P.K.Srinivasan,97.“knewnothingofmathematics.”S.R.Ranganathan,22.Sometimelatein1910...S.R.Ranganathan,23;P.K.Srinivasan,99;FamilyRecord.Everyonecalledhim“Professor.”K.G.Ramanathan,20.“Iwasstruckbytheextraordinarymathematicalresults.”P.K.Srinivasan,129.Notseenoneanothersince1906.SeshuIyer,83.“Iwasnotbigenough.”P.K.Srinivasan,102.RamachandraRao.Bhargava,Who’sWhoinIndia;Who’sWhoinMadras,1934.RamachandraRaomeeting.P.

K.Srinivasan,86;RamachandraRao;S.R.Ranganathan,24,74;FamilyRecord;SeshuIyer,83.Neville,“TheLate”(Nature106):66.

heprobablydidn’t“workitout.”RichardAskeysuppliedthisinsight.Fermat’sleisure.Bell,59.

fifth-largestcityintheBritishEmpire.AfterLondon,Calcutta,Bombay,andLiverpool.Lewandowski,51.Originsofname.Thurston,2;KrishnaswamiNayadu,1.HistoryofMadras.SeeSlater,SouthernIndia;Urwick;Steevens;Lewandowski;Srinivasachari;Lanchester.morethanfiftyfeetabovesealevel.Lanchester,90.“hutments.”Srinivasachari,292.thecityhadexpandedhorizontally.Lewandowski,46.largeruraltracts.Singer,142.“Madrasismorelostingreen.”Steevens,299.aBrahmin-runrestaurant.Bharathi,50.spinoccultstories.S.R.Ranganathan,16.arefreshingGangasnanam.Ibid.,67.PublicworksprojectsinMadras.Srinivasachari,298.aletterfromRamaswamiIyer.JournaloftheIndianMathematicalSociety11(April1919):42–44.Indian mathematics. See, for example, Eves, 161–183; M. S. Rangachari, “The Indian Tradition in

Mathematics,”JournalofIndianInstituteofScience(RamanujanSpecialIssue1987):3–9.“amixtureofpearlshellsandsourdates.”Eves,171.“Ramanujan’sfirstlove.”Letter,BruceBerndttoauthor.RamanujanhadstumbledonBernoullinumbers.MadrasPortTrustArchives,inAskey.“SomePropertiesofBernoulli’sNumbers.”Ramanujan,CollectedPapers,1–14.Byonereckoning.K.SrinivasaRao,“SrinivasaRamanujan:HisLifeandWork,”inNagarajanandSoundararajan,

4.“Mr.Ramanujan’smethodsweresoterseandnovel.”SeshuIyer,83.“bynomeanslight.”P.K.Srinivasan,131.“numericalevidenceassufficient.”Mordell,Nature(1941);645.“theycallyouagenius.”S.R.Ranganathan,25.“apieceofthinstring.”Bharathi,50.onpaperalreadywrittenupon.S.R.Ranganathan,76.prevailed onhim to copy it over. S.R.Ranganathan, 58.Here, “during thisperiod”meansbetween early1911,

whenRamanujanfirstcameunderRamachandraRao’sinfluence,andearly1913,whenheleftforEngland.“Paper,TheGreatImmortalizer.”Bharathi,v.Foraboutayear.Neville,“Ramanujan”(Nature149),292.grewtobotherhim.S.R.Ranganathan,26.“ItwasRamanujan,unwillingtobelongeraburden,whobroughtto

anendhisdependence”bytakingthePortTrustjob:Neville,ReadingManuscript.atemporaryjobintheMadrasAccountantGeneral’sOffice.FamilyRecord.Srinivasan,176,putsthedatesofthe

job as January 12 to February 21, 1912, but there is some reason to doubt this. Conceivably, it was thepreviousyear.

“aclerkshipvacant.”P.K.Srinivasan,31.“quiteexceptionalcapacityinMathematics.”Ibid.,49.TheportofMadras.SeeThurston;Urwick;Srinivasachari;Lanchester;ThePortofMadras:Past,Presentand

Future;gazetteers.Interview,V.Meenakshisundaram.hisownmotorcar.Muthiah,183.SirFrancisSpring.SeeThePortofMadras:Past,PresentandFuture,8.NarayanaIyer.InterviewswithhisfamilyinMadras.

SirFrancisreliedonhimheavily.“WhateverNarayanaIyersaid,”saysS.SankaraNarayanan,“heagreed.”“themostconvenientdress.”Hindu,24April1896.latein1912.P.K.Srinivasan,172.FamilyRecord.athirdofthecity’spopulation.Lewandowski,46.threerupeespermonth.Interview,Janaki.“sinecurepost.”RamachandraRao,87.Bharathi,47.“runningtohisoffice.”Bharathi,50.Ramanujan’s Port Trust routine. Interview, Janaki. “Since a year [after taking his job at the Port Trust]

Ramanujan still found his clerkship a bondage fromwhich he craved release, hemust have been applyinghimselfconscientiouslytodutieswhichhispatrondidnotforamomentintendhimtotakeseriously”:Neville,ReadingManuscript.

establishingcashbalances.Interview,T.V.Rangaswami.“pilotagefundclerk.”Internalmemo,11February1913,MadrasPortTrust.Ramanujan’sregularjobwasasa“billclerk.”prowlingforpackingpaper.S.R.Ranganathan,76.sternlyregardinghisaide.Ibid.,27.slatesproppedontheirknees.Bharathi,48.“Youmustdescendtomylevel.”Ibid.Ramanujanasdiamond.P.K.Srinivasan,172.TheBritishRajandtheIndianCivilService.SeeWorswickandEmbree;Fuller;Compton.“treasontoourtrust.”QuotedinWorswickandEmbree,140.two-thirdsthoseoftheBritish.Fuller,276.“Plato’sidealrulers.”WorswickandEmbree,142.“awakenedbyresponsibility.”Fuller,276.“regardedwithawe,notaffection.”Urwick,55.“theEnglishofficialisanincomprehensiblebeing.”Ibid.AnEnglishmantypicallyhadhisownwasherman.Compton,190.brushandfoldhisclothes.Ibid.,246.stewardstoopedtoservehimtea.Ibid.,207.“noassimilationbetweenblackandwhite.”Ibid.,247.NarasimhahadintroducedhimtoE.B.Ross.FamilyRecord.E.W.Middlemastrecommendation.P.K.Srinivasan,49.“cunningandcontentiousinargument.”Compton,40.“youcanpolishtheHinduintellect.”Ibid.,171.“constantlydisconcerted.”Fuller,179.“hardworking,docileandenduring.”Urwick,50.atRamachandraRao’sbehest.FamilyRecordreferstoaletterfromRamachandraRaotoGriffithon3November

1912.SeealsoNeville,ReadingManuscript.“Youhaveinyouroffice.”P.K.Srinivasan,50.“Ifhisgeniusissoelusive.”Ibid.,51.“Hegivesmetheimpressionofhavingbrains.”Ibid.,52.“IthinkIwasrightinwritingtoProf.Hill.”Letter,GriffithtoSpring,28November1912.MadrasPortTrust.Ramanujanhadfallenintosomepitfalls.P.K.Srinivasan,53.

“evidentlyamanwithatasteformathematics.”Letter,GriffithtoHill,7December1912.MadrasPortTrust.ManyhadadvisedRamanujantowritetoEngland.S.R.Ranganathan,70;P.K.Srinivasan,91.aslongastwoweeks.Compton,178.lettersdraftedwiththehelp.Hardy(“Obituary,”494)reportsthatRamanujantoldhimthepersonalintroduction

hadbeenwrittenby“afriend.”AndS.R.Ranganathan,32,arguesthattheletterwas“largelywordedforhimbytheseniorswhohadhelpedhim,”Ramanujanpresumablybeingtooshyandhumbletowriteanythingsoboastful.Plainly,Ramanujangothelp.ButasIarguelater,Idon’tthinkhewasashumbleassomethink.

InRamanujan’slettertoHardyofJanuary22,1914,heclaimsthat“all letterswrittentoyou,exceptthisone [and one other], did not containmy language. Those were written by the superior officer mentionedbefore, though themathematical results andhandwritingweremyown.”But there is reason tobelieve (seechapter5)thatRamanujanwasdissembling.Besides,Ramanujanwasalwaysscrupulousaboutacknowledgingthehelpofothersandmayhavedeemedanyhelpfromhisfriendstomeanthatthelanguagewasnothis“own.”Finally,eventhisletter,theletterheacknowledgesashisown,wasnotsodramaticallydifferentfromtherest.Inshort,IbelievetherewasinvariablyalargechunkofRamanujaninanyletterthatborehissignature.

BakerandHobsonsaidno.Snow,Apologyforeword,33–34:“Imentionedthatthereweretwopersonswhodonotcomeoutofthestorywithcredit.Outofchivalry,HardyconcealedthisinallthathesaidorwroteaboutRamanujan.Thetwopeopleconcernedhavenowbeendead,however,formanyyears,anditistimetotellthetruth. It is simple.Hardywasnot the first eminentmathematician to be sent theRamanujanmanuscripts.Therehadbeentwobeforehim,bothEnglish,bothofthehighestprofessionalstandard.”ButSnowwentnofurther. In Alternative Sciences, 146–147, Nandy reports that Littlewood identified them as “Baker andPopson.” In “Ramanujan—AGlimpse,” 78,Bollobás corrects the error. In a letter toBruceBerndt, Indianmathematician K. Venkatachaliengar dismisses the assertion that Baker was among those who ignoredRamanujan. But he seems to do so solely on the basis that anyonewhowould, as Baker at one point did,recommendnotanEnglishmanbutanIndian,C.HanumanthaRao,toapositionatanIndianuniversitycouldscarcely have spurned Ramanujan. Baker, writes Venkatachaliengar, was “a very sincere man of uprightconduct. . . . [He] would have acknowledged his error if it were so to [Ramanujan] direct or throughHanumanthaRao.”Thisassertion,however,isnotenoughtoundercutourconfidenceinwhatcomesdowntousdirectlythroughLittlewoodwho,accordingtoBollobás(“Ramanujan—AGlimpse,”78), “often chuckledovertheembarrassmentofhiscolleagueswhofailedtorecognizeagenius.”

CHAPTERFOUR

“He . . . looks a babe of three.” Michael Holroyd, Strachey: A Critical Biography, vol. 1, The Unknown Years(London: Heinemann, 1967), 516, footnote. Hardy was sometimes refused beer. Bollobás, Littlewood’sMiscellany,120.Hiscollegeroomshadnomirrors.Snow,Apologyforeword,16.SeealsoAlexanderson,65.

“RedIndianbronze.”Snow,Apologyforeword,9.Hardytooksixpagestoattacktheplan.Letter,HardytoJackson,9November1910.TrinityCollege.enthusiasms,peeves,andidiosyncrasies.Titchmarsh,452.“There’samatchduetobegin.”Collins,27.leavingthebenighted.M.H.A.Newman,inBBCobituary.TrinityCollege.Sisterreadtohimaboutcricketwhen

hedied.Snow,Apologyforeword,58.Chapelincident.Ibid.,20.“happiesthoursofmylife.”Snow,Apologyforeword,21.wouldnotshakehands.Interview,MaryCartwright.“strangeandcharmingofmen.”Woolf,Sowing,123.“themostextravagantandfanaticalkind.”Hardy,CollectedPapers,532.“theonegreatpermanenthappinessofmylife.”Letter,HardytoThomson,December1919.CambridgeUniversity.Headmittedthepro-Godpositionwasstronger.Interview,MaryCartwright.

“whathisrealopinionswere.”Titchmarsh,450.“moredelicate,lesspadded,finer-nerved.”Snow,Apologyforeword,16.“aslightlystartledfawn.”Woolf,Sowing,124.“Tositthatway.”Alexanderson,64.“personalpowersandcapabilities.”A.Ostrowski,inExperentia5(1949):131—132.Middle-classschools.CranleighSchoolMagazine,July1887,133,198.SeealsoMegahey.“thelabourer’sson.”Megahey,12.55werethesonsoftradesmen.Minutes,CranleighSchool,29January1880.IsaacHardy.SurreyAdvertiserandCountyTimes,7September1901;TheCranleighan, 1901,267–271;Surrey

CountySchoolRegister,1872,1;otherCranleighSchoolrecords.anextrafiftypoundsperyear.Minutes,SurreyCountySchool,23October1874.richindustrialists.Brandon,97.GenealogyofHardy’sparents.InformationfurnishedbyRobertA.Rankin,Glasgow.“withmorethanatouchoftheWhiteKnight.”Snow,VarietyofMen,204.singinglessons.SurreyCountySchoolRegister,Christmas1872.earnestlyandsincerelymourned.SurreyAdvertiserandCountyTimes,7September1901.“rareandprecioussouls.”TheCranleighan,1901,270.SophiaHallHardy.InformationfurnishedbyRobertA.Rankin,Glasgow,throughD.H.J.Zebedee,Lincoln

DiocesanTrainingCollege,1862–1962.tochurchtwoorthreetimes.Interview,MaryCartwright.taughtpiano.Minutes,CranleighSchool,29January1890.Handelconcert.CranleighSchoolMagazine,July1878.GertrudeHardy.ReferencestoherarescatteredthroughSt.Catherine’sSchoolMagazineovermorethanthirty

years.“There is a girl I can’t abide.” Gertrude Hardy, “Lines Written Under Provocation,” St. Catherine’s School

Magazine(October1933),575.“alittleobsessive.”Snow,Apologyforeword,14.Only“good”books.Titchmarsh,447.writingdownnumbersintothemillions.Ibid.EustaceThomasClarke.Cranleighan,June1905,18.Cranleighan,1937,120.givewronganswers.Snow,Apologyforeword,14.“Heseemstohavebeenbornwiththreeskinstoofew.”Ibid.,15.Gertrude“shyanddiffident.”St.Catherine’sSchoolMagazine,obituary,August1963,6.“Ateacherofoutstanding

ability,shecombinedkeeninterestinherbrighterpupilsand,asIhavegoodreasontoremember,boundlesspatiencewiththeslow.Herreproofswereastringent,butneverundeserved.Herscrupulousfairnessgavethatfeelingofsecuritywhichissovaluabletotheyoung.”

“Iwasveryshy.”St.Catherine’sSchoolMagazine,October1938,229.Theclergymanandthekite.StanislawUlam,AdventuresofaMathematician(NewYork:Scribners,1976),60.Gertrude the “Mohammedan.” Letter, Marjorie Dibden to Robert A. Rankin, 24 November 1983. Robert

Rankin.losthereyeasachild.Interview,MaryCartwright.Running itwouldbeMr.andMrs.Hardy.SchoolMinutes,29January1880.At leastaround1881.Thecensus

records,obtainedattheGuildfordLibrary,areRG79297bschedule153.Hardyislistedas“Godfred.”TherecordsclearlyindicatetheHardyhouseas“Mt.Pleasant,”cottagesthatstillstand.However,schoolrecordsrefertothetwenty-fourboysas“atMr.Hardy’shouse,”whichcouldonlymeanthemuchlargerHouse.Itis

conceivable, then, thatMr. andMrs.Hardy actually lived, along with their children, at the House, whilemaintainingofficialresidenceintheMt.Pleasantcottage.

ubiquitousinSurreysincetheseventeenthcentury.NairnandPevsner,14.SeealsoJekyll,4.low-ceilingedsittingroom.Tourofthehouse.Mythankstoitscurrentresident,whosenameIfailedtorecord.whohelpedMrs.HardyattheHouse.SketchoftheHistoryoftheSurreyCountySchool,17.torentoutsleepingspace.

ItisunlikelythattheservantsmentionedinthecensusrecordsservedtheHardyfamily.Megahey,16,reportsthatanupper-middle-class incomeoffivehundredpoundsperyearwasenoughtohire threeservants. IsaacHardymadelessthanone-fifthofthat.

athousandpoundsayear.Megahey,16.secondmastermadeonlyahundred.Ibid.,17.“atypicalVictoriannursery.”Titchmarsh,447.littlemorethanamudtrack.Megahey,20.“Cranleighboysmaywanderwheretheyplease.”Ibid.,27.“enlightened,cultivated,highlyliterate.”Snow,Apologyforeword,14.asixth-formerwhowastwenty.Megahey,31.reachedthesixthformattwelve.Hewasplacedinthethirdformatten.Megahey,52.Ward’s assessment of youngHardy.Cranleigh SchoolRegister,Report to theCouncil of the SurreyCounty

School,26July1889.137boyscompeted.Sabben-Clare,54.Hardyplacedfirst.Megahey,52.beginningtoshed.Ibid.,44.“Connel.”RegisterofWinchesterSchool,courtesyRobertRankin.originallyendowed.DunningandSheard,48.“gentlemanlyrebelsandintellectualreformers.”Bishop,18.architecturaldetails.WinchesterCollege:AGuide,schoolpublication,rev.andrepr.May1984.“notions.” See, for example, Sabben-Clare, 144; Bishop, 26;WinchesterCollegeNotions, “by Three Beetleites”

(Winchester:P.&G.Wells,1904).thirtystrokes,withaground-ashstick.Sabben-Clare,44.“MorningHills.”Ibid.,151.circledthroughthestone-archedwalkways.Ibid.,152.“Mrs.Dick.”Ibid.,45.Cricket. John andEmmaLeigh, ofCranleigh, assureme that a foreigner possesses the essential knowledge of

cricketwhenhefullyunderstandsthefollowing:Youhavetwosides.Oneoutinthefieldandonein.Eachmanthat’sinthesidethat’singoesoutandwhenhe’southecomesinandthenextmangoesin

untilhe’sout.Whentheyarealloutthesidethat’soutcomesinandthesidethat’sbeeningoesoutandtriestoget

thosecominginout.Sometimesyougetmenstillinandnotout.WhenbothsideshavebeeninandoutincludingthenotoutsThat’stheendofthegame.

(withacknowledgmentstotheMaryleboneCricketClub)HardysawRichardsonandAbel.Snow,“TheMathematician,”67.practicesweresacrosanct.Megahey,27.“thevillagedidmuchbetter.”CranleighSchoolMagazine,October1888,251.

“athingofpersonalartandskill.”NevilleCardus,PlayResumedwithCardus(LondonSouvenirPress,1979),14.“Canvassing,Clarendontype,andprofessionalcricket.”RecordsofMagpie&Stump,966thmeeting,3June1910.

CambridgeUniversity.withwalkingstickandtennisball.Woolf,Sowing,124.“onelonggameofcricket.”Wykehamist,20June1893.“small, taut, and wiry.” C.J. Hamson, text of a talk given in the Senior Combination Room at Trinity, 24

November1985,andpublishedinTrinityReview,1986,23.AccountsofHardyasathlete.Wykehamist,December1895.frustrating...abrilliantcricketcareer.Snow,Apologyforeword,48.crushedanyartisticability.Titchmarsh,447.sosickhealmostdied.Snow,Apologyforeword,17.Hardyandmutton.Titchmarsh,452.Bollobás,120.neverreturnedtovisit.Wykehamist,18February1948.Oftwenty-sixclasshours.Bishop,32.Thefiguresarefrom1900.Richardson.Sabben-Clare,62.DuncanPrize.WinchesterSchoolrecords.physicsonhisown.TheWykehamist,18February1948.aweaknessfordetectivestories.Interview,MaryCartwright.Hardytoldherhehadoncereadthirty-sixofthemin

aweek.(Hissistersharedtheweakness:St.Catherine’sSchoolMagazine,August1963,6.)Onedaywhenhewasaboutfifteen.Hardy,Apology,145.“HerbertwasbackagainatTrinity.”AFellowofTrinity,AlanSt.Aubyn(London:ChattsandWindus,1892),

261.“adecentenoughfellow.”Hardy,Apology,146.WykehamiststoCambridge,Oxford.Wykehamist,December1896.“Congratulationsaredue.”Wykehamist,December1895.theschool’sannualSpeechDay.SurreyAdvertiser,1August1896.74percenttoOxfordorCambridge.Ellis,144.assignedaroominWhewell’sCourt.TrinityCollegerecords.Tripos.See“OldTriposDaysatCambridge,”byA.R.Forsyth,MathematicalGazette19(1935):162–179;“Old

CambridgeDays,”byLeonardRoth,AmericanMathematicalMonthly78(1971):223–226(bothofthesearereprinted inCampbellandHiggins,vol.1,81–103);RouseBall;Littlewood,Miscellany, and “MathematicalLifeandTeaching”;Hardy,CollectedPapers,527–553.

“themostdifficultmathematicaltest.”Roth,inCampbellandHiggins,97.inearliestform,to1730.RouseBall,11.“wishthemJoyoftheirHonour.”QuotedinOxfordEnglishDictionary,entryfor“Wrangler.”WoodenSpoon.CambridgeFolkMuseumdisplay.“All-American,aRhodesscholar,andBacheloroftheYear.”CampbellandHiggins,vol.1,81.“You,sir.”Roth,inCampbellandHiggins,98.1881 Tripos results. Littlewood, “Mathematical Life and Teaching,” 21. Typical Tripos problems. See, for

example,Forsyth,inCampbellandHiggins,86–89;Roth,inCampbellandHiggins,97–98;Thomson,56–60;Littlewood,Miscellany,72–75.

Mustbeatrick.ThefutureSeniorWranglerwasLittlewood.HetellsthestoryinhisMiscellany,74.excellenttraining—forthebar.Thomson,58.likearacehorse.Indeed,practicallyallwhohavewrittenaboutthecoachingsystememploythismetaphor.

twodozenofthem.Forsyth,inCampbellandHiggins,86.“beyondthepaleofaccessiblecriticism.”Forsyth,inCampbellandHiggins,84.handedovertoR.R.Webb.Titchmarsh,448.“altogethertoomuchlikeacrosswordpuzzle.”QuotedinR.Clark,LifeofBertrandRussell,43.“withoutseriousprejudicetohiscareer.”Hardy,CollectedPapers,537.“theverdictofmyelders.”Hardy,Apology,144.Hardy’sboyhoodhistoryofEngland.Titchmarsh,447.Hardy’sessayimpressedhisexaminers.Littlewood,“Reminiscences,”12.HeadmasterFearon’streatment.Titchmarsh,448.Love’ssuggestion.Hardy,Apology,147.“atlastin[the]presenceoftherealthing.”Hardy,CollectedPapers,722.intotworegions.SeeIbid.,723.“thatremarkablework.”Hardy,Apology,147.“eventherebelHardy.”Littlewood,quotedinYoung,270.“totakesomuchtroubleandtolearnnomore.”Hardy,CollectedPapers,530.“Itisagreattriumph.”Letter,G.M.TrevelyantoHardy,14June1898.TrinityCollege.“heoughttohavewonit.”Snow,Apologyforeword,24.ShakespeareSocietymeeting.Minutes,23May1901.HardyinCranleigh.SurreyAdvertiserandCountyTimes,7September1901.“finalandirrevocable.”TrinityCollege,recordsofclubsandsocieties.TheApostles.SeeDeacon;Levy;Woolf,Sowing.“averynicescientist.”Deacon,33.Hardyinducted.Levy,195.“tenacityofabulldog.”Woolf,Sowing,148.“hispeculiarpassionfortruth.”Woolf,BeginningAgain,24.“DoesYouthApproveofAge?”Levy,196.“absolutefreedomofspeculation.”QuotedinDeacon,70.“breathethemagicair.”QuotedinDeacon,59.G.L.Dickinson.SeeFurbank,59.whoonceadvisedHardy.Snow,Apologyforeword,31.“WaltWhitmanesquefeelingsofcomradeship.”Levy,227.“TheHigherSodomy.”Deacon,55.“thewomaniserspretendtobesods.”Deacon,62.R.K.Gaye.Woolf,Sowing,124;CambridgeUniversityrecords.“theywereneverseenapart.”Ibid.,124.“rumorofayoungman.”Interview,MaryCartwright.“justanotherEnglishintellectualhomosexualatheist.”Hodges,117.“his beloved JohnLomas.” Snow, “TheMathematician,” 72.Deacon, 73, reports that on oneprevious occasion

Hardytriedtocommitsuicide,“thefirsttimefollowingthedeathofaclosemalefriend.”“anon-practicinghomosexual.”Interview,BélaBollobás.Voteon“self-abuse.”Levy,207.“school-boyaffections.”EllisandSymonds,39.“relentlesslyVictorianwhileitcould.”Hynes,185.

“freneticsexualaffairs.”Himmelfarb,42.fatheredachild.Seechapter5.“hisknowledgeofhighermathematics.”Minutes,ShakespeareSociety,16February1903.26percentnevermarried.Ellis,152.WritesEllis:“Apassionatedevotiontointellectualpursuitsseemsoftentobe

associatedwithalackofpassionintheordinaryrelationshipsoflife,whileexcessiveshynessreallybetraysalsoafeeblenessoftheemotionalimpulse.”

Cranleighteachersweremen.Megahey,38–39.Winchesterwasthesameway.WritesBishop,28:“TheEnglishboardingschoolwasnottheonlysysteminhistory

totreatwomenandthefamilyaspotentiallysubversiveelements:ithasevenbeenarguedthatvirtuallyeverytraining establishment which has exalted direct service to the community has guarded itself against thefemininethreat.LiketheancientSpartans,theTurkishjanissarycorpsandtheChinesecommune,theBritishpublic school has tacitly recognized thatwomen ‘are incapable of putting the interests of any outside bodyabove the interestsof those they love.’ ”Thequote is by JohnPringle, “TheBritishCommune,”Encounter,London,February1961.

“societyofbachelors.”QuotedinFowlerandFowler,233.“GetyoutoGirton.”Photo,CambridgeFolkMuseum.“stringlittlesentencestogether.”Barham,17.selectedfortheirplainness.Ibid.,1.“perfectlyhappymarriage.”QuotedinDeacon,58.“theasceticidealisadopted.”EllisandSymonds,40.“scatteredthroughhislife.”Snow,Apologyforeword,26.“manyformsofcontactwithlifeverypainful.”BuxtonandWilliams,117.hopingtolureAmericans.BiographicalMemoirsofFellowsoftheRoyalSociety19(December1983):fn,495.sixhoursaweekoflectures.Titchmarsh,448.“butverylittleofimportance.”Hardy,Apology,147.HardyF.R.S.RoyalSocietyrecords.ThearticleobjectingtoMendel.ProceedingsoftheRoyalSocietyofMedicine.1:165.Hardy’sreply.Hardy,CollectedPapers,477.Hardy-WeinbergLaw.SeeentryforHardy,byVictorCassidy, in

ThinkersoftheTwentiethCentury.a“manifestoofliberation.”Himmelfarb,35.“neverdoneanything‘useful.’”Hardy,Apology,150.“Hardyism.”DavisandHersh,87.“The‘real’mathematics.”Hardy,Apology,119.“theydonotfitthefacts.”Ibid.,135.“alonggardenpartyonagoldenafternoon.”Hynes,4.“thegreatestdisaster.”CampbellandHiggins,vol.1,96.“TheGreatSulk.”Ibid.“whyitshoulddoso.”R.Clark,43.“Corky.”Hardy,CollectedPapers,829.Editor’sNote.“insteadofgettingonwiththerealjob.”Littlewood,“MathematicalLifeandTeaching,”15.“maroonedinitslimitations.”CampbellandHiggins,vol.1,85.“Oh,weneverreadanything.”Bell,433.“with the splendour of a revelation.” E.H.Neville, obituary ofAndrewRussell Forsyth. Journal of the London

MathematicalSociety17(1942),245.

“notverygoodatdeltaandepsilon.”QuotedinCampbellandHiggins,vol.1,100.“nothingelseintheworld.”Titchmarsh,451.“neverheardtheequal.”Wiener,Ex-Prodigy,190.“butyouwillregretit.”Barnes,35.Hardy sometimesdiverted tomathematics those ill-equipped for it.See, for example,ManchesterGuardian, 2

December1947.“theonlyprofession.”Hardy,Apology,150.“someofthemostperfectEnglishofhistime.”Snow,“TheMathematician,”68.Hardy’searlystyle“vulgar.”Bollobás,Littlewood’sMiscellany,118.“lightwithgrace,order,asenseofstyle.”Snow,Apologyforeword,49.HardywroteaboutRussell.“TheNewNewRealism,”CambridgeMagazine,11and18May1912.Hardy’sobituaries.SeetheCollectedPapers.wroteupthejointpaper.CitedinBatemanandDiamond,31.Butinaninterview,MaryCartwrightreportsthat

Littlewoodwroteuptheirverylastone.“devicestostimulatetheimagination.”Hardy,CollectedPapers,598.“whichistomymindunequalled.”QuotedinH.T.H.Piaggio,“ThreeSadleirianProfessors,”TheMathematical

Gazette15(1931):464.Thoughtimpossiblewithoutwords.Hardy,CollectedPapers,837.Actofwritinggavehimpleasure.Bollobás,Littlewood’sMiscellany,118.HardyonJohnson’sbook.Hardy,CollectedPapers,819.“offargreaterintrinsicdifficulty.”Hardy,PureMathematics,vi.“lookandbehaveverymuchlikealine.”Ibid.,4.“littlebetterthannonsense.”ArthurBerry,TheMathematicalGazette5(July1910):304.“toreformtheTripos.”Hardy,CollectedPapers,537.“solongasitsstandardislow.”Ibid.,528.“intoamarginalfirst.”Ibid.,537.“OurEnglishway.”H. B.Heywood, “TheReform ofUniversityMathematics,”TheMathematicalGazette 12

(1925):323.“high-souledfrustration.”Littlewood,Miscellany,71.“anacrobat.”C.J.Hamson,[untitled],TrinityReview(1986):23.“hewassokindtotheweakones.”AvariantofthestoryappearsinM.L.Cartwright,“SomeHardy-Littlewood

Manuscripts,”BulletinoftheLondonMathematicalSociety13(1981),294.“inthesamegenerousterms.”Titchmarsh,454.HardywoulddiscussBergson.R.Clark,169.anundergraduate.Wiener,Ex-Prodigy,183.“truthsandvalues.”Woolf,BeginningAgain,20.Hardy’sroutine.Snow,Apologyforeword,31.

CHAPTERFIVE

coveredwithIndianstamps.Snow,Apologyforeword,30.Unlessotherwisenoted,mostoftheaccountdescribingthereceiptofRamanujan’sletterisdrawnfromSnow,30–38.

“Ibegtointroducemyself.”Ramanujan,CollectedPapers,xxiii.“enclosed papers.” Ramanujan’s letters are in the Cambridge University Library. Most of them appear in

Ramanujan,CollectedPapers,xxiii–xxxand349–355.

late January1913. Snow,Apology foreword, 31, gives the time as simply “January.”Ramanujanwrote him onJanuary16.TimeformailtoreachEnglandwasusuallylessthantwoweeks.AdaybetweenJanuary26andJanuary31seemsmostlikely.

a practical joke. Snow, 30–31, speaks of Hardy getting manuscripts from “cranks,” viewing Ramanujan’s aspossiblya“fraud.”Neville,intheReadingManuscript,expandsonthisthemeconsiderably.

“somecuriousspecializationofaconstant.”Hardy,“Obituary,”494.“prodigy.”Littlewood,Miscellany,66.ThebooksLittlewoodstudied.Ibid.“arough-hewnearthyperson.”BatemanandDiamond,33.“unfortunatelyverysaintly.”Littlewood,“Reminiscences.”long-termrelationshipwithamarriedwoman.The relationshipwasoneofmanydecades’ standing.Thewoman

was the wife of Dr. Streatfeild, Bertrand Russell’s physician. Littlewood’s daughter is Ann Johannsen.Bollobás,Littlewood’sMiscellany,18.InterviewswithBélaBollobás,CharlesBurkill.

“stormandsmashareallydeepandformidableproblem.”Hardy,quotedinBurkill,61.itwenttoHardyasreferee.Littlewood,Miscellany,78.turnedouttobeflawed.BatemanandDiamond,29.“onlythreereallygreatEnglishmathematicians.”Thishasbecomeacommonplaceinmathematicalcircles.recentlymovedintoroomsonDStaircase.TrinityCollegerecords.probablyinLittlewood’srooms.Snow,33,saysitwasHardy’srooms.ButBollobás,in“Ramanujan—AGlimpse,”

76, says it was Littlewood’s. Mary Cartwright, who was Littlewood’s student and collaborator, picturesLittlewoodasinvariablyexpectingpeopletocometoseehim.ShedeemsitunlikelythattheymetinHardy’srooms.

“seemedscarcelypossibletobelieve.”Hardy,“Obituary,”494.“Ishouldlikeyoutobegin.”Hardy’sresponsetoRamanujan’sletteroverthenextfewpagesislargelydrawnfrom

Hardy,Ramanujan,7–10.“curiousandentertaining.”Hardy,inRamanujan,CollectedPapers,xxv.“theoremssentwithoutdemonstration.”Neville,“Ramanujan”(Nature149),293.“amanofaltogetherexceptionaloriginalityandpower.”Hardy,Nature(1920):494–495.beforemidnight.Thisisthecanonicalversion,accordingtoSnow.Butinaninterview,BélaBollobásemphasizes

thatRamanujan’s formulaswere complicated, his results astounding; that evenHardy andLittlewoodmayhaveneededmore than an evening to evaluate them; that,while true to the spirit of the events,Snowmayconceivablyhavecollapsedseveralevenings’workintoone.

“reallyaneasymatter.”Mordell,“Ramanujan,”643.“anotherofthosemonstrosities.”Bell,314.“adistinguishedandconspicuousfigure.”Hardy,CollectedPapers,751.suggestinggreatimaginationorflair.H.T.H.Piaggio,“ThreeSadleirianProfessors,”TheMathematicalGazette15

(1931):463.“Anoldstick-in-the-mud.”Interview,MaryCartwright.newandoriginal—butnotbetter.IowetoinventorJacobRabinowthisinsightintothenaturalandnormalhuman

resistancetothenew.“IndianBazaar.”CranleighSchoolMagazine,May1887,112.“Iwasadepressedclass.”Interview,MaryCartwright.“thelargebottomed.”Snow,Apologyforeword,42.“forthedownfalloftheiropposites.”Snow,“TheMathematician,”70.

“forgetthesensation.”Neville,“Ramanujan”(Nature149),293.Hardyhad sprung intoaction.Snow,Apology foreword,34, says itwas thenextday. Inanycase,Hardygot in

touchwiththeIndiaOfficebeforeheevenwroteRamanujan.“Iwasexceedinglyinterestedbyyourletter.”Letter,HardytoRamanujan,8February1913.Laterinthemonth,DaviesmetwithRamanujan.Letter,RamanujantoHardy,22January1914.OnFebruary25.Memo,SirFrancisSpring.S.R.Ranganathan,29.“anxietyastohislivelihood.”P.K.Srinivasan,55.Hardy’s imprimatur. ThatWalker’s assessmentwas formed largely byHardy’s letter is not theway the story

usuallycomesdowntous.Hardy(CollectedPapers,703)saysWalker“wasfartoogoodamathematiciannottorecognize [the quality of Ramanujan’s work], little as it had in common with his own. He broughtRamanujan’s case to the notice of the Government.” But the contrary conclusion, that Walker’srecommendationcameonlyinthewakeofHardy’sletter,seemsunavoidable.Foronething,inSpring’smemo(SureshRam,27–28),henotesthatWalker“disclaimedabilitytojudgeMr.Ramanujan’sworkandsaidthatMr. Hardy of Trinity College, Cambridge, was in his opinion the most competent person to arrive at ajudgementofthetruevalueofthework.Mr.RamanujanhadalreadybeenincorrespondencewithMr.Hardy,aletterofwhosedated8February1913—just17daysbeforeDr.Walker’svisithere—isinthisfile.”LiningupWalkerbehindRamanujansmacksofanorchestrationofMadrasopinionbySpringandNarayanaIyer.

“whoviewsmy labours sympathetically.”Letter,Ramanujan toHardy,27February1913.CambridgeUniversityLibrary.

fortifiedbyHardy’sletter.Hardy’slettertotheSecretaryforIndianStudentshadalsoreachedMadras.SeeS.R.Ranganathan,34.

“invincibleoriginality.”Hardy,CollectedPapers,720.Ramanujan’ssecondletter.27February1913.“perfectinmanners,simpleinmanner.”Neville,“TheLate”(Nature106),662.“hisnamewouldliveforonehundredyears.”Transcript,Nova1508,18.“notdoneby‘humble’men.”Hardy,Apology,66.“whatwecandoforS.Ramanujan.”SureshRam,29.syndicate’saction.SeeSureshRam,28–31;S.R.Ranganathan,30–31; letter,K.Venkatachaliengar toBruce

Berndt,suppliedbyBerndt.“withalltheirvehementspeeches.”Letter,K.VenkatachaliengartoBruceBerndt.ByApril 12,Ramanujan had learned the good news. Letter,Ramanujan toRegistrar,University ofMadras, 12

April1913.MadrasPortTrust.“aviceinKumbakonam.”Neville,“Ramanujan”(Nature149),293.acorruptionofTiruAlliKeni.KrishnaswamiNayadu,30.Thedescriptionof theTriplicanetankandadjacent

templeisdrawnfrompersonalobservationandDas,245.Pallavaking’sgiftofland.KrishnaswamiNayadu,30.movedbacktoTriplicane.FamilyRecord.thresholdforpayingfulldues.JournaloftheIndianMathematicalSociety(February1917),19.workwithNarayanaIyer.S.R.Ranganathan,31.mathbooksfromK.B.Madhava.P.K.Srinivasan,132.RamanujanattheConnemara.S.R.Ranganathan,31.Servefoodinhishand.Ibid.,90.hismothermadeitforhim.Interview,Janaki.alittlescienceexperiment.Interview,Janaki.AshisNandy,131,speaksofRamanujan“teachinghertheelementsof

science,”butplacesthisafterRamanujan’sreturnfromEngland,notbefore.

“the one you were working on before eating.” Interview, Janaki. Nandy pictures Ramanujan as “using her as asecretary,”againreferringtothepost-Englandperiod.

Shedidn’taskhimto,andhedidn’tvolunteer.Interview,Janaki.“Howmaddeninghisletteris...”Letter,LittlewoodtoHardy,inRamanujan,TheLostNotebook,383.“yourobviousmathematicalgifts.”P.K.Srinivasan,56.“whatlittleIhave.”Letter,RamanujantoHardy,17April1913.CambridgeUniversityLibrary.Augustget-together.S.R.Ranganathan,12.submittedsometheorems.JournaloftheIndianMathematicalSociety5,no.5:185.perhapsouttowinover...Littlehailes.Letter,K.VenkatachaliengartoBruceBerndt.helookedforwardtostudyingRamanujan’sresults.Memo,NarayanaIyer,31October1913.MadrasPortTrust.“DoesRamanujanknowPolish?!?”S.R.Ranganathan,12.QuarterlyReports.SeeBerndt,Ramanujan’sNotebooks,partI,295;andBerndt,“QuarterlyReports.”“todiscovernewpathwaysthroughtheforest.”Berndt,“QuarterlyReports,”516.“‘newandinteresting.’”Berndt,Ramanujan’sNotebooks,partI,297.“theresultiswrong.”P.K.Srinivasan,57.“nohumanagency.”Snow,Apologyforeword,34.Ramanujan’sversion.Letter,RamanujantoHardy,22January1914.CambridgeUniversityLibrary.“adeadmanintheirestimation.”T.Ramakrishna.“disregardtheordersofthecaste?”Gandhi,37.withmenfromthevillagesbringingtheirwivesandchildren. J.ChartresMolony,ABookofSouthIndia(London:

Methuen&Co.,1926).CitedinLewandowski,53.the area around Triplicane’s Parthasarathy Temple. A detailed map in Lanchester, unpaged, shows residential

patternsinMadras.“notamanlikeLittlewood.”Letter,HardytoMittag-Leffler,about1920.RobertRankin,Glasgow.SenateHouse.Muthiah,106;personalobservation.“amanatoncediffidentandeager.”Neville,ReadingManuscript.“opposition...withdrawn.”Neville,“Ramanujan”(Nature149),293.InfluencesonRamanujantogotoEngland.S.R.Ranganathan,36.“weightofmyinfluence.”RamachandraRao,88.“thefulfillmentofhislife’spurpose.”Neville,“Ramanujan”(Nature149),293.inlateDecemberof1913.FamilyRecord.Ramanujanwrotebackhome.FamilyRecord..threenights.P.K.Srinivasan,114;Bharathi,48.“Ilockedmyselfupintheattic.”QuotedbyC.T.Rajagopal,MathematicsTeacher(India)11A(1975),119.“withinaveryfewmonths.”Letter,RamanujantoHardy,22January1914.hehadcometosaygood-bye.P.K.Srinivasan,100.Ramanujan’s other doubts. Letter, Ramanujan toHardy, 22 January 1914; andNeville, “Ramanujan” (Nature

149),293.Father-in-law’sconcerns.K.R.Rajagopalan,33.Mother’sconcerns.Ibid.,32–33.actuallyshookhishand.K.R.Rajagopalan,34.“theglorythatbelongedtoMadras.”Neville,ReadingManuscript.“wheretheylongedtoseehim.”Neville,ReadingManuscript.

NevillewroteHardy.Neville,“Ramanujan”(Nature149),294.aworriedreply.Letter,MallettoHardy,11February1914.TheLibrary,UniversityofReading.“I’mwritinginahurry.”Letter,HardytoNeville.TheLibrary,UniversityofReading.“unknowngeniuses.”Neville,“Ramanujan”(Nature149),294.“themostinterestingevent.”P.K.Srinivasan,59.“hiddenunderabushelinMadras.”Ibid.,61.IfVenkatachaliengar’saccountofthesyndicatemeetingisaccurate,

Littlehaileshadbynowchangedhistune.“ifnottranscendental,orderofgenius.”Ibid.,64.“bynomeanstheworstbrain.”QuotedinSlater,SouthernIndia,262.“forthefulldevelopmentofpersonality.”Ibid.,263.“HisExcellencycordiallysympathizes.”S.R.Ranganathan,35.OnMarch11.S.R.Ranganathan,37.OnMarch14.Interview,Janaki.“Threedaysbeforeheleft...”Now,hewept.Interview,Janaki.while hermother-in-lawwasat the temple.KalyanalakshmiBhanumurthy, “TheManBehind theMathematics,”

TheHindu,20December1987.shewassoyoungandpretty.K.R.Rajagopalan,35.“asif...obeyingacall.”RamachandraRao,88.drivinghimaroundtownonhismotorcycle.S.R.Ranganathan,36.“nothingbutvegetablefood.”RamachandraRao,88.How,hepleaded,washetowearthem?P.K.Srinivasan,134;S.R.Ranganathan,62.shewouldn’trecognizehim.K.R.Rajagopalan,35.hisfriendsstayedupwithhim.S.R.Ranganathan,71.a special wharf. The Port of Madras: Past, Present and Future, 8. Ramanujan’s friend K. Narasimha Iyengar

reports,inS.R.Ranganathan,70,thatSpringprevailedonRamanujanandNevilletoleavefromMadras,notthemoreusualBombay.

TheNevasa.Detailsdrawnfrom(British)NationalMaritimeMuseumInformationFile;BICentenary, 1856–1956,byGeorgeBlake(London:Collins,1956);“TheNotable‘Nevasa,’”byJ.H.Isherwood,inSeaBreezes,newseries21(1956):28–31;“Endofthe‘Nevasa,’”SeaBreezes,newseries5(1948):148–149.Ramanujan’ssend-off.P.K.Srinivasan,91–92.

“hemaygetaninspiration.”Bharathi,49.

CHAPTERSIX

seasickandunabletoeat.P.K.Srinivasan,3.OnMarch19.ThepictureofRamanujan’svoyage toEngland isdrawn fromFamilyRecord;detailsaboutthe

Nevasa(seechapter5);Ramanujan’slettersinP.K.Srinivasan;S.R.Ranganathan,38;K.R.Rajagopalan,36;P&OPocketBook(London:AdamandCharlesBlack,1908);LloydsWeeklyShippingIndexfortheperiodofthevoyage.

withdrewtohis...cabin.K.R.Rajagopalan,36.“hisownhistoryorpsychology.”Hardy,Ramanujan,11.four-hundred-rupeefare.The second-class farebetweenLondonandBombaywas32pounds in1914,or about

480rupees.BritishPassengerLinersoftheFiveOceans,byC.R.VernonGibbs(London:Putnam,1963),63.orthodoxBrahminsamongthem.S.R.Ranganathan,36.

POSH.AHundredYearHistoryoftheP&O,1837–1937,byErnestAndrewEwart(London:I.NicholsonandWatson,1937),112—113.

hepostedatleastfourlettersbacktoIndia.FamilyRecord.stampsshowingthepyramids.P.K.Srinivasan,92.“warmestthankstoyouruncle.”Ibid.,3.ReceptionbyNeville.Letter,RaymondNevilletoRobertA.Rankin,15May1982.London.See,forexample,GeoffreyMarcus,47.“atempoclearlyfaster.”Woolf,BeginningAgain,16.CromwellRoad.Reportofthe[Lytton]CommitteeonIndianStudents.metA.S.Ramalingam.Rankin,“RamanujanasaPatient,”84.ChestertownRoad.Informationonthestreet,andspecificallythehouseat113,isdrawnfromaninterviewwith

ConstanceWillis,thecurrentowner,andtourofherhouse;photosfromtheCambridgeshireCollection;andrecordsfurnishedthroughJ.D.Webb,BuildingControlOfficer,CambridgeCityCouncil.

for two months. In his dedication to Ramanujan of The Farey Series of Order 1025 (Cambridge: CambridgeUniversityPress,1950),xxvii,Nevilleremembersitasthreemonths.ButthisconflictswithRamanujan’slettertoKrishnaRaoof11 June1914, inP.K.Srinivasan,7.There,he sayshe’s alreadymoved into the college(afterlessthantwomonthsinCambridge).

HardyandNevilletookcareofmostofit.P.K.Srinivasan,7.dippedhispeninblackink.TermsBook,1899–1915.TrinityCollege.gracedbyawondrousspring.Marcus,112.KingGeorgevisitedCambridge.CambridgeIndependentPress,1May1914.Littlewood...sawhimaboutonceaweek.P.K.Srinivasan,154.“veryunassuming,kindandobliging.”Ibid.,7.Berrystoodattheblackboard.P.K.Srinivasan,145.“even this was pronounced wrongly.” Quoted in K. G. Ramanathan, “Ramanujan: A Life Sketch,” ScienceAge,

December1987,22.he“waddled”acrossTrinity’sGreatCourt.Lecturenotes,B.M.Wilson.TrinityCollege.Wilsonlatercrossedout

“waddled,”perhapsbecausehethoughtthewordundignified,butitisclearlyvisibleintheoriginalmanuscript.“inconvenientfortheprofessors.”P.K.Srinivasan,7.sadtoleave.Interview,MaryCartwright.consummate hosts. T. A. A. Broadbent, “EricHaroldNeville,” Journal of the London Mathematical Society 37

(1962):482.nearlyeveryday.Hardy,Ramanujan,11.perhapsathird.Ibid.,10.“amassofunpublishedmaterial.”Hardy,CollectedPapers,718.“a really searching analysis.” Hardy, “A Chapter from Ramanujan’s Notebook,” Proceedings of the Cambridge

PhilosophicalSociety21(1923):503.Polya.Berndt,Ramanujan’sNotebooks,partI,14.“notalightone.”G.N.Watson,“Ramanujan’sNotebooks,”JournaloftheLondonMathematicalSociety6(1931):

140.amonthtoprove.Ibid.,150.“atleastaJacobi.”Letter,LittlewoodtoHardy,probablyearlyMarch,1913.TrinityCollege.“EulerorJacobi.”Hardy,CollectedPapers,720.“greatestformalistofhistime.”Hardy,Ramanujan,14.

“delightedsurprise.”Littlewood,Miscellany,87.“neverheardofmostofit.”Snow,Apologyforeword,36.“withoutarival.”Hardy,CollectedPapers,720.“whatatreasureIhadfound.”Hardy,Ramanujan,1.“Ieditedthemverycarefully.”Hardy,CollectedPapers,717.the2:15P.M.trainoutofCambridge.Hardy,CollectedPapers,751.LondonMathematicalSocietymeeting.Minutes.withintheradiusofahydrogenatom. JonathanM.BorweinandPeterB.Borwein,ScientificAmerican, February

1988,112.“havingnootherbusinessatthetime.”Ibid.,114.“newresults.”Hardy,CollectedPapers,494.“anexcitingfoxhunt.”CambridgeDailyNews,18April1914.Caesardied.CambridgeDailyNews,19April1914.“Buthewasahappyman.”Neville,“Ramanujan”(Nature149),294.“nowarinthiscountry.”P.K.Srinivasan,168.muffledtheechoes.FirstEasternGeneralHospitalGazette,6July1915,135.strungfromtheceiling.Ibid.,22June1915,100.ambulancewithagreatredcross.Ibid.,97.wouldseethemonhiswaytoHall.Parry,200.“WehaveanewCambridge.”Ibid.,95.inches-deepmudonunasphaltedroads.Keynes,182.OnSeptember20.Butler,200.“ThedepravationofGermany.”Parry,96.“driveninfrontoftheenemy.”Butler,200.“Germanssetfire.”P.K.Srinivasan,168.“Nowhere’saproblemforyou.”S.R.Ranganathan,81.Mahalanobisdidn’tcitetheDecember1914issue,butthe

contentoftheproblem,andtheissueinwhichitappears,squareperfectlywithhisrecollection.“Thelimitationsofhisknowledge.”Hardy,CollectedPapers,714.“foundafunctionwhichexactlyrepresents.”Ramanujan,CollectedPapers,349.“Thestuffaboutprimesiswrong.”Letter,LittlewoodtoHardy,probablyearlyMarch1913, inRamanujan,The

LostNotebook,380.“Euclid’stheoremassuresus.”Hardy,Apology,99.DelaVallée-Poussin.AprofessorattheUniversityofLouvain;manyofhisworkswerelostwhenthelibrarywas

settothetorchinthefirstmonthofthewar.JournaloftheIndianMathematicalSociety10:27.“Proofswillbesuppliedlater.”JournaloftheIndianMathematicalSociety5(1913):50–51.cametoseewherehehadstumbled.SeeHardy,Ramanujan,chapters1and2.“nocomplexzeroes.”Hardy,CollectedPapers,706.Riemannhypothesis.Foraniceintroduction,accessibletothelayperson,seeCampbellandHiggins,vol.2,149–

153.HastheRiemannhypothesisbeenproved?ThisisoneofmanybitsofRiemannhypothesislore.“theAnalyticTheoryofNumbersisnotoneofthem.”Hardy,CollectedPapers,706.“hisachievement...ismostextraordinary.”Littlewood,Miscellany,87.“morewonderfulthananyofhistriumphs.”Hardy,CollectedPapers,706.

“Hisinstinctsmisledhim.”Hardy,Ramanujan,38.“astheclimaxofaconventionalpatternofpropositions.”Ibid.,16.Take the sequenceof integers.These illustrative examples aredrawn fromIvarsPeterson, “AShortageofSmall

Numbers,”ScienceNews,9January1988.OneHardylikedtocite.Hardy,Ramanujan,16.“thelargestnumberwhichhaseverservedanydefinitepurposeinmathematics.”Hardy,Ramanujan,17.The2=1“proof.”Ahighschoolclassic.slideintotroubleinnumerousways.ExamplesduetomathematiciansClareFriedman,RichardAskey,andBruce

Berndt.“Hedisregardedentirelyallthedifficulties.”Hardy,CollectedPapers,706.“ofwhatismeantbyaproof.”Littlewood,Miscellany,88.“verymuchmitigated.”R.Clark,176.“hardenedtosomeextent.”Hardy,Ramanujan,10.“Ihavechangedmyplan.”P.K.Srinivasan,15.“Mynotebookissleepinginacorner.”Ibid.,21.“nosymptomofabatement.”Hardy,CollectedPapers,715.“averycuriousfunction.”P.K.Srinivasan,23.“hesetstoworktomanufactureaproof.”Hardy,Ramanujan,16.“first-rateimportance.”Littlewood,Miscellany,88.“Mathematicshasbeenadvanced.”Klein,inMordell,“Ramanujan,”647.aninformalscale.Berndt,Ramanujan’sNotebooksI,14.“breakthespellofhisinspiration.”Hardy,CollectedPapers,715.“excerptsfromamoreinterestinglecture.”Young,282.“AtCambridgeweareindarkness.”Parry,98.“InFranceandFlanderswemakenoprogress.”Ibid.Cambridgeschoolgirl.Keynes,62.“thefrontwaslikeafirst-rateclub.”Parry,96.“cheerfulindifference.”Williams,15.“EvenLittlewood.”Hardy,Apology,140.“didnotfillallofLittlewood’sworkinghours.”Burkill,63.Hardydeemed“unfit”toserve.OnwhatbasisIhavenotlearned.atleastoneobituary.NorbertWiener,“GodfreyHaroldHardy,”BulletinoftheAmericanMathematicalSociety55:

72–77.TherecordwassetstraightinasubsequentlettertotheBulletinfromLittlewoodandothers.“Idon’tlikeconscientiousobjectorsasaclass.”Letter,HardytoJenkinson,15June1918.CambridgeUniversity.Hardy“wrotepassionately.”Littlewood,“Reminiscences,”13.Itsfirstpublicmeeting.Hardy,Russell,12.theauthoritiesmovedtoblockit.Ibid.,19.risen32percent.Marwick,125.threeorfourtimesamonth.FamilyRecord,55.aparcelfullofbooks.P.K.Srinivasan,162.preoccupied...withseeinghisworkinprint.ThelettersareallinP.K.Srinivasan,7,11,13,and32,respectively.hewentstraighttoHardy.S.R.Ranganathan,77.thefollowingyear.P.K.Srinivasan,28.

“nohelpnorreferencesinMadras.”Ibid.,29.studentssometimestauntedhim.S.R.Ranganathan,71.“athrilltometodiscover.”Ibid.,78.Ramanujanatteaparties.Ibid.,76.reservedinlargegroups.Ibid.,83.Ramanujanduringvacations.Ibid.,76.Charley’sAunt.SureshRam,43.Bymid-October.TrinityCollegerecords.Highlycompositenumberspaper.SeeRamanujan,CollectedPapers,78–128.“Igaveupthestruggleearlier.”W.N.Bailey,quotedinGeorgeE.Andrews,Q-Series.“ofanelementarybuthighlyingeniouscharacter.”Hardy,CollectedPapers,497.“extraordinaryinsightandingenuity.”Ibid.,499.perhapsthemostbrilliant.Barnes,33.“entirelyjustifyingthehopes.”P.K.Srinivasan,66.“themostremarkablemathematicianIhaveeverknown.”SeshuIyerandRamachandraRao,xvii.SpringwroteDewsbury.This and other correspondence concerningRamanujan’s possible return to India is in

MadrasPortTrust.about75poundsperyear.Marwick,23.thresholdforpayingincometaxes.Ibid.,21.“almostludicrouslysimpletastes.”Hardy,CollectedPapers,714.Ramanujan,B.A.Rankin,“Ramanujan’sManuscriptsandNotebooks,II,”364.Ramanujandidn’treturntoIndia.Butthereisathirdreason,beyondthetwogiven.Seechapter7.“mostunfortunate.”Hardy,CollectedPapers,491.fourthousandcasualtiesperday.Marwick,133.

CHAPTERSEVEN

Chatterjidinner.SureshRam,40–42.Details aboutChatterji and theotherguests come fromWho’sWho inIndia, 1937–38. Chatterji was apparently still in India, at Punjab University, until late 1915, and hewasmarriedin1916,leadingtomyguessonthedateofthedinner.

Kasturirangar’sappraisal.S.R.Ranganathan,90.“instinctiveperfectionofmanners.”Neville,“Ramanujan”(Nature149),294.thetableclothsofpeacetimeweregonenow.ToldtomeatHighTable.“onedoesnotwanttoworkatconversation.”Littlewood,“Reminiscences.”littlebookkeptovertheyears.TrinityCollegeKitchens:SuggestionsandComplaints,1909–1955.TrinityCollege.unusually strict in his orthodoxy. S. Chandrasekhar has estimated that even back in India, Ramanujan’s family

would have been in the top quarter of some imaginaryOrthodoxy Scale.Those traveling toEnglandweretypicallyfarlessorthodox.

Traveler’sstory.Interview,T.M.Srinivasan.Friedinlard.SureshRam,42.SeealsoS.R.Ranganathan,78.alittlegasstove.Ibid.,76.stirringvegetablesoverthefire.SeealsoSnow,Apologyforeword,35:“Hardyusedtofindhimrituallychangedin

hispyjamas,cookingvegetablesrathermiserablyinafryingpaninhisownroom.”“thinkofmyhomeandcountry”Gandhi,40.

abouttwentyIndiansperyear.Thesefiguresarederivedfromthe“governmentstudies”cited.Fuller,186,givesafigureofaboutseventeenhundredforthewholeUnitedKingdom.

“Theinitialdifficulty.”AmarKumarSingh,IndianStudentsinEngland.Astilllaterstudent.Interview,RajivKrishnan,Christ’sCollege,Cambridge.dreamofdying.Ibid.“thereticenceofhisEnglishfriends.”Neville,ReadingManuscript.“detachment almost amounting to indifference.” Quoted in Barbara Tuchman, The Proud Tower (New York:

Macmillan,1966),66.“withouttheslightestsuggestionofdiscourtesy.”Chaudhuri,90.“Oh,werewe introduced?” Interview,S.SankaraNarayanan.Asimilarstory is toldbySudinHowtoBecomea

BarristerandTakeaDegreeatOxfordorCambridge.“anairofindifference.”Satthianadhan,28.“thattheywerenotgreatlyinterested.”Barnes,35.“hisfacedropped.”Young,271.“noonewillblinkaneyelid.”Bollobás,Littlewood’sMiscellany,149.rituallypuredhoti.Interview,T.V.Rangaswami.couldnotlongforgethisforeignness.SeealsoS.R.Ranganathan,47.“youbatheonlyonceaday?”DavidE.Fisher,Fire&Ice(NewYork:Harper&Row,1990),46.ToldtoFisher,

hereports,byaPolishmathematician,atCornell,inthe1960s.“veryrarelyseen.”B.M.Wilson.Unpublishednotes.TrinityCollege.inthecoolofthenight.SureshRam,40.“aseriouspsychologicaleffect.”ColonialStudentsinBritain,94.just17percentdidwhileinEngland.AmarKumarSingh,IndianStudentsinEngland.TrinitySundayEssaySociety.TrinityCollege.RamanujannotinMajlis.SureshRam,44.“theirnationalismandradicalism.”Nandy,125.“wrapped in amedieval gloom.”W. E.Heitland, After Many Years, quoted in CambridgeCommemorated, ed.

LaurenceandHelenFowler(CambridgeUniversityPress,1984),283–284.“greatestpowerandskilltoresolve.”NorbertWiener,BulletinofAmericanMathematicalSociety55(1949):77.“Amoment’sconsideration.”Hardy,LecturesbyGodfreyH.HardyontheMathematicalWorkofRamanujan[Notes

byMarshallHall](AnnArbor,Michigan,1937),25.“unabletodiscover.”Ramanujan,CollectedPapers,277.“asRamanujanimagined.”Hardy,Ramanujan,9.“anextremelyobviousone.”Hardy,QuatrièmeCongrèsdesMathématiciensScandinaves,46.“hardly the type to be chosen by Central Casting.” Gian-Carlo Rota, Introduction to Collected Papers of P. A.

MacMahon,ed.GeorgeAndrews,xiii.andregularlythrashhim.Ibid.,xiv.“Thentheproblemiscompletelysolved.”Bollobás,Littlewood’sMiscellany,98.“averygreatstep.”Littlewood,Miscellany,89.“orevenbounded.”Ramanujan,CollectedPapers,283.“weproceededtotestthishypothesis.”Ibid.Selberg’sargument.AtleSelberg,“RamanujanandI,”ScienceAge,February1988,39.twomen.Hindu,22December1987.“nottheonlymathematician.”Transcript,Nova1508,12.“asingularlyhappycollaboration.”Littlewood,Miscellany,90.SeealsoA.Ostrowski,inExperentia5(1949):131–

132:“Theywereverydifferentscientificpersonalities . . .Ontheoneside,amanwhohadcompletelygiven

himselfuptocalculation,whoguessedintuitivelyandworkedoutthemostcomplicatedandhiddenformulae,who overflowed with such discoveries, only to become entangled again and again in the net of analyticaldifficulties.Ontheotherside,amanwhowasasuprememasterofallthefinessesandmostsubtleargumentsofmodernanalysis,andwhowascontinuallyenrichingthembynewinspirations,devices,andsubtleties.”

“anymanIhaveeverknown.”Snow,VarietyofMen,194.“Hissenseofexcellencewasabsolute.”LionelCharlesRobbins,inBuxtonandWilliams,117.SeealsoPhilipSnow,

C.P.Snow’sbrother:“Hardywasasterncriticoftheirrelevant,theignorant,thebrash.”PhilipSnow,44.“onahighplane.”Interview,CharlesBurkill.Polya’szoostory.GeorgePolya, “SomeMathematicians IHaveKnown,”AmericanMathematicalMonthly 76

(1969):746–753.“amorning’sworkgonewest.”Letter,HardytoMordell,undated.St.John’sCollege.apersonalityofexpectations.Theexpectationsappliedfullytohimself:“InHardy’sletters[toLittlewood],‘Please

check,’‘Pleasecheckverycarefully,’seemslikeathemesong,andwhenhewaswrongheswore,whichconfirmsmy impression that he really minded very much when he himself made bad mistakes.” Mary Cartwright,“SomeHardy-LittlewoodManuscripts,”BulletinoftheLondonMathematicalSociety13(1981):273–300.

“somesplendidproblemstoworkat.”Ramanujan,TheLostNotebook,389.OliverLodge.P.K.Srinivasan,143.“thebathroomsareniceandwarm.”Bollobás,“Ramanujan—AGlimpse,”79.Forthirtyhoursatastretch.Unpublishednotes,B.M.Wilson.TrinityCollege.“Onlycabbageshavenonerves.”Bell,329–330.“Mealswereignoredorforgotten.”Bell,109.“theirmentaldevelopment.”Satthianadhan,38–39.“study-worn,consumptive-looking.”Ibid.towardtheendoftheirsecondyear.LyttonCommittee,testimonyofS.M.Burrows.“inclinedtodeveloptuberculosis.”LyttonCommittee,testimonyofS.S.Singara.“aregimeunsuitedto...conditions.”LyttonCommittee,testimonyofSirR.W.Philip.“tocookoneortwothingsmyself.”P.K.Srinivasan,5.powderedriceintin-linedboxes.Bharathi,48.Ramanujan’seatinghabits.Interview,Janaki.weirdhoursintheearlymorning.S.R.Ranganathan,39.“alittlesaltandlemonjuice.”P.K.Srinivasan,27.“ ‘infants, Indians and invalids.’ ” BasilWilley,Spots of Time: A Retrospect of theYears 1897–1920 (London:

ChattoandWindus,1965),118.on a cricket ground. Historical Register of the University of Cambridge, Supplement, 1911–1920 (Cambridge:

CambridgeUniversityPress,1922),203.Marchingboots.CambridgeMagazine,5May1917.“ourprideandsadprivilege.”Larmoraddress,2November1916.ProceedingsoftheLondonMathematicalSociety,

2dseries,16(1917):6.“forthemostpartwasted.”Barnes,p.35.RobertCollins.Seymour&Warrington,caption,fig.41.ThedeathofGrahamisrecordedinMadrasTimes, 6

April1919.“memorialservices.”Wootton,InaWorldINeverMade,quotedinFowlerandFowler,284.“Thingsherearesadandsorrowful.”Parry,101.two-fifthsofhermeat.Marwick,18.

Kiplingpoem.CitedinMarcus,41.onesurewaytotellrichfrompoor.Marwick,24:“Poordietinggavemanymembersofthelowerordersyetanother

characteristicwhichdistinguishedthemfromtherestofsociety:theirsmallstature.”“rosy,well-coveredmen.”Russell,51.hadclimbed65percent.Marwick,125.“AtWrexham.”QuotedinMarwick,192.“portionsstrangelydwarfed.”Parry,107.“goodmilkandfruits.”P.K.Srinivasan,5.“overwork,overplay,overworry.”R.C.Wingfield,ModernMethods in theDiagnosisandTreatment ofPulmonary

Tuberculosis,1924.QuotedinBryder,166.“Wetravelledinconvoy.”Russell,54–55.NursinghomeonThompson’sLane.Rankin,“Ramanujan’sManuscriptsandNotebooks,”95.byspecialdispatch.P.K.Srinivasan,73.Hardy...nursedhim.K.R.Rajagopalan,41.“totakepropercareofhimself.”P.K.Srinivasan,69.“Ramanujanwasnotexemplary.”S.R.Ranganathan,74.“Adifficultpatient.”SeshuIyerandRamachandraRao,xviii.“Wheneverhestartedwithadoctor.”BélaBollobás,inNova1508,14.Thetranscriptattributesthisstatementto

S.Chandrasekhar,butisinerror.FewdoctorscouldabideRamanujan.Letter,A.S.RamalingamtoHardy,inRankin,“RamanujanasaPatient,”

87.ProbablyaroundOctober.Ibid.,81.Chowry-Muthu.IamguidedintheseassertionsbyRankin,Ibid.SeealsoP.K.Srinivasan,110.exploratorysurgery[forgastriculcer]considered.Rankin,“RamanujanasaPatient,”87,92.Nothydrocele,butcancer.Ibid.,91.Bloodpoisoning.P.K.Srinivasan,76.Tuberculosis—disease,treatment,andhistory.See,forexample,Bryder,Smith,Dubos,Keers.oneineightdeathsinBritain.Bryder,1.Nervoussystem-immunesystemties.See,forexample,RobertKanigel,“WhereMindandBodyMeet,”Mosaic,

Summer1986,52–60.“emotional stress associatedwith separation.” GeorgeW. Comstock et al., “TuberculosisMorbidity in theU.S.

Navy:ItsDistributionandDecline,”AmericanReviewofRespiratoryDisease110(1974):572–580;interview,GeorgeW.Comstock.

wildspasmodicleap.Bryder,109–110.TheDanishphenomenon.SeeBryder,109–112.“supportedbythefiguresfromDenmark.”Keers,148.“admirable epidemiological detective work.” Graham A. W. Rook, “The Role of Vitamin D in Tuberculosis,”

AnnualReviewofRespiratoryDisease138(1988):768.Daviespaper.Tubercle66(1985):301–306.“Itprobablycan.”Rook,769.workingatnightandsleepingbyday.SureshRam,40.Osler.QuotedinBulstrode,3.“thisdreadfullycoldopenroom.”Bollobás,“Ramanujan—AGlimpse,”79.

treatment’srisetopopularity.Smith,141.HaroldBattyShawismentionedinRankin,“RamanujanasaPatient,”91.

“waslittleinfluencedbyit.”Bryder,23.“towardscarryingoutthe‘openair’treatment.”Bulstrode,131.atleastfifty-twoBritishsanatoriums.Keers,93.ButBryderreportsninety-six“open-air”sanatoriumsinEngland

andWalesby1907.MendipHillswasoneofthem.Bryder,25.Matlock,ashydropathicestablishment.Rankin,“RamanujanManuscriptsandNotebooks,”95.“bleakandcold.”Dubos,181.Architectscompeted.Bryder,50.“therainblowedthrough.”Bryder,53.“somequarrel.”P.K.Srinivasan,71.“Ihaven’tgotasurname.”Bollobás,“AGlimpse,”79.slippedintoitabriefnote.Interview,Janaki.Mother-in-lawtroubles.Interview,S.SankaraNarayanan.“Itispitiableforthechild-wife.”Compton,119.“Thetyrannicalmother-in-law.”TheHindu,12May1899.whointurnwouldpassittohim.Carstairs,45,66.moreconfidentandassertive.Interview,S.Chandrasekhar.ThoseclosetoJanaki.Janaki’slitanyofabusebasedoninterviewswithT.V.Rangaswami,“Hari”ofMadras,Mr.

andMrs.T.S.Bhanumurthy.outofthequestion.K.R.Rajagopalan,41.Ramanujan’sfatherstuckupforJanaki.Interview,Janaki.Janakifoundanexcusetogetaway.Interview,Janaki.moneyforanewsari.K.R.Rajagopalan,44.Interview,Janaki.Ramanujan’slettershome.FamilyRecord.hetoldhisfriendChatterji.SureshRam,44.leafletadvisingauthors.Minutes,LondonMathematicalSociety,14March1918.“alookofextremedisapproval.”TresilianNicholasRemembers,”TrinityReview,Easter1969,13.“whoseopinionsonthewar.”Levy,279.RussellandTrinity.See,forexample,Hardy’sownRussellandTrinity.“littlebook.”DailyTelegraph,30April1970.Hardykeptonemathematicianoutofthewar.Letter,HardytoLarmor,28March1916.RoyalSociety.ThemathematicianwasChapman.“perfectlytangibleanddefinite.”Letter,HardytoJenkinson,12June1918.CambridgeUniversity.“verystandoffishorpossiblyshy.”Hodges,117.“multiplelayersofreserve.”Ibid.,118.“parentalsolicitude.”SeshuIyer,85.Hardywouldchoosetheformer.Hardy,Apology,153.“alwaysalittledifficult.”Hardy,Ramanujan,1.“thefirsttodisclaim.”Snow,“TheMathematician,”69.Ramanujan’stastesinliteratureandpolitics.Hardy,CollectedPapers,715.“Irelyforthefacts.”Hardy,Ramanujan,2.

“Iamtoblame.”Hardy,Ramanujan,11.“allbutsupernaturalinsight.”Bell,140.“Wehavenoidea.”IvarsPeterson,“TheFormulaMan,”ScienceNews,25April1987,266.“thathasbarelybeendrawn.”BruceBerndt,“Ramanujan—100YearsOld,”29.“atouchofrealmystery.”Littlewood,Miscellany,90.“Anordinarygenius.”MarkKac,EnigmasofChange(NewYork:Harper&Row,1985),xxv.“unfoldbeforehiseyes.”S.R.Ranganathan,87.“appearedtohiminadream.”Ibid.,73.penchantforinterpretingdreams.P.K.Srinivasan,97.astreetpeddlerhawkingpills.S.R.Ranganathan,84.timetoprepareastrologicalprojections.Ibid.,91.Predicteddeath.P.K.Srinivasan,123.atemplenearTrichinopoly.Ibid.,101.“space-time-junctionpoint.”S.R.Ranganathan,85.“thewaymayaworksinthisworld.”Ibid.,89.“Idonotbelieve.”Hardy,Ramanujan,5.“equallytrue.”Hardy,CollectedPapers,715.“Ramanujanwasnomystic.”Hardy,Ramanujan,5.“Ishouldnottrusthisinsightfar.”Snow,Apologyforeword,35.“thinkingvaguely.”Hardy,CollectedPapers,733.“whogazesatadistantrangeofmountains.”Ibid.,598.“theevidenceforallthisseemsoverwhelming.”Forthisandwhatfollows,seeIbid.,834–838.“withsomekindsofspiritualbeings.”Hadamard,40–41.“neverdiscardedaparticle.”Bell,144.strongmysticalbent.Bell,457.“thequeerestmixture.”Bell,43.“bythegraceofGod.”“TrueGenius,”byRGK,IllustratedWeeklyofIndia,20December1987,31.“theGreatArchitectoftheUniverse.”JamesJeans,TheMysteriousUniverse(NewYork:Macmillan),1930.“Ifwemayrejectdivinebounty.”Bollobás,Littlewood’sMiscellany,146.“noexception.”Hardy,CollectedPapers,719.“Hardy’s deep reverence for mathematics.” E. A. Milne, obituary of Hardy in Monthly Notices of the Royal

AstronomicalSociety108(1948):45.“ifastrictBrahmin.”Hardy,Ramanujan,4.“aquiteharmless...economyoftruth.”Ibid.,5.“atheistevangelical.”Hodges,118.“Inevergotthefeelingtheywerehappy.”NewYorkTimes,January20,1988.racismwasafactor.Seepage299.“coweddownbyDr.Ram.”Rankin,“RamanujanasaPatient,”86.“make visits difficult.” Bryder, 200. War-borne slashes in rail services around January 1917 must have left

Ramanujanallthemoreisolated(Marwick,195).“coldwearyjourney.”Rankin,“RamanujanasaPatient,”85.“comparativesolitude.”P.K.Srinivasan,76.hecravedthehotdosai.Ibid.,96.

“singularities.”Transcript,Nova1508,15.OnDecember6,1917.Minutes,LondonMathematicalSociety.RoyalSociety.Year-bookoftheRoyalSocietyofLondon,1919(London:HarrisonandSons,1919).“onehonorhelongedfor.”Snow,VarietyofMen,84–85.OnJanuary24,1918.Year-bookoftheRoyalSociety1919,212.“notimemustbelost.”Rankin,“RamanujanasaPatient,”91.“uselesstospeculate.”Hardy,“Obituary,”495.“apoorandsolitaryHindu.”Hardy,Ramanujan,10.“thecriticalyears.”Ibid.,6.OnFebruary11.FamilyRecord.Oneday in January orFebruary of1918.The account ofRamanujan’s suicide attempt is derived almost solely

fromS.ChandrasekharinnotesdepositedintheRoyalSocietyandappearinginNotesandRecordsoftheRoyalSociety 30 (1976).Chandrasekhargives the timeof the incident asFebruary1918.Nandy,127, says itwas“some time in the secondhalfof1917.”But this seemscontradictedby thedates surroundingRamanujan’selectiontotheRoyalSociety,whichareintimatelyboundupwithChandrasekhar’sstory.Ontheotherhand,lateJanuary,ratherthanFebruary1918,cannotberuledout.Bythen,Ramanujanhadalreadybeenputupformembership.OnJanuary25,HardywroteDewsburyandinhisreplyofMarch5DewsburythankedHardyfor “theveryclearandexplicit statementon thepositionofMr.Ramanujan.”Further,hepromised to treatHardy’s letter “as personal and confidential.” Could Hardy have been advising Dewsbury of the suicideattempt?(Dewsbury’sletterisatTrinityCollege.)

“WeinScotlandYard.”Chandrasekharaccount.Hereadthetelegramonce.Bollobás,Littlewood’sMiscellany,137.“theaidandguidance.”LettertoHardyatTrinityCollege.toosicktotravel.Letter,RamanujantoRoyalSociety,15May1918.DuplicatedinIndianExpress,19December

1987.A. S. Ramalingam. Information about him, and most of the account that follows, is drawn from the

correspondenceappearinginRankin,“RamanujanasaPatient,”79–100.fatteningupthepatient.Bryder,53.“warpedtemperament.”Bryder,208.Bovril.OxfordEnglishDictionary.nosuchdispensation.OrsomySouthIndianinformantsadviseme.Abruptly,hepackedhisbags.S.R.Ranganathan,79.likelytheraidofOctober19,1917.ThereareinconsistenciesintheDesmukhaccount,whichrecordsabombing

raid“inearly1918,whentheZeppelinraidswereattheirworst.”ButtheTimesofLondonaccountsrecordnozeppelinraidsonthecapitalthen,norindeedraidsofanykind.ThemostrecentsubstantialraidapttoimpressRamanujanasDesmukhrecordsisthatonthedategiven.

“tastelessandinsipid.”Gandhi,40.“novelandunpalatable.”Neville,“TheLate”(Nature106),662.“shouldRamanujanbegivenup?”Rankin,“RamanujanasaPatient,”88.Ramanujan’s Trinity fellowship. Bollobás, Littlewood’s Miscellany, 136–138; Bollobás, “Ramanujan—A

Glimpse,”79.painnoonecouldtrace.Bollobás,Littlewood’sMiscellany,137.“Myheartfeltthanks.”Ibid.“yourpainsandtheirencouragement.”Letter,RamanujantoHardy,undated.TrinityCollege.“abriefperiodofbrilliantinvention.”Neville,“Ramanujan”(Nature149),294.

“Arecentpaper.”Ramanujan,CollectedPapers,210–213.“greatpracticalimportance.”HardyandWright,49.“practicallynoreasontodoubt its truth.”PercyAlexanderMacMahon,CombinatoryAnalysis, vol. 2 (Cambridge:

CambridgeUniversityPress,1916),33.Ramanujan was rummaging. Hardy, Ramanujan, 91, puts the year at 1917. But Richard Askey thinks the

correspondencesurroundingthediscoverypointsmoreforciblyto1916.someonealreadyhad.“Itisunlikely,”GeorgeAndrewshaswrittenofRogers’swork,“thatanyimportantseriesof

paperseverdroppedfurtherfromsightthanthesedidforthenext20years.”“rememberverywellhissurprise.”Hardy,Ramanujan,91.“nopositionindiplomacy.”Nature132(1933),701.“verylittleambition.”ObituaryNoticesofFellowsoftheRoyalSociety.1(1932–1935),299.Thecorrection.ProceedingsoftheLondonMathematicalSociety14,Series2,(1915),endpaperofboundvolume.“theyseemtohaveescapednotice.”Hardy,inRamanujan,CollectedPapers,344.yetunproved.MacMahon,CombinatoryAnalysis,vol.2,33.“regrettingthathehadoverlookedmywork.”Letter,RogerstoF.H.Jackson,13February[1918],quotedinGeorge

E.Andrews, “L. J.Rogers and theRogers-Ramanujan Identities,”MathematicalChronicle 11 (1982), 1–15.LetterfurnishedAndrewsbyLucySlater,CambridgeUniversity.

“averycreditablepre-warbonfire.”CambridgeReview,quotedinHowarth,17–18.“asifTimeandEternitywereone.”Howarth,18.“Ithinkitisnowtime.”P.K.Srinivasan,76.

CHAPTEREIGHT

“ontheroadtoarealrecovery.”P.K.Srinivasan,76.“andcausedsuchgraveanxiety.”SeshuIyer,85.restoredtherighttomakecakesandpastries.Marwick,271.Cambridgecasualties.Howarth,16.“rocky headlands rejoiced.”Trevelyan,AnAutobiography&Other Essays (London: Longmans, Green and Co.,

1949),37.ColinetteHouse. Interview with its current owners, Deborah and Bryan J. B. Gauld; tour of the house and

grounds.SamuelMandevillePhillips.Rankin,“RamanujanasaPatient,”82.afewweeksafterthearmistice.“ShewaslaidtorestonDecember6thinthegravewhereherhusbandliesinthe

churchyard.”TheCranleighan13(1919),76.1729.Hardy,CollectedWorks,719–720;Snow,Apologyforeword,37.wouldbenamedUniversityProfessor.Memo,NarayanaIyertoSirFrancisSpring,12March1918.MadrasPort

Trust.“thehelpthathasbeengivenme.”P.K.Srinivasan,46.Passportphoto.Photo,P.K.Srinivasan,opposite136.RamanujanbroughtbacktoIndia.TheHindu,19December1987;P.K.Srinivasan,162;Berndt,“Ramanujan—

100YearsOld,”26.thenightskywasstillaglow.Hoyt,101.ShellfragmentsfromtheattackcanbeseentodayintheFortSt.George

Museum,Madras.included Ramanujan’s parents and wife. Jagjit Singh, “Srinivasa Ramanujan: A Short Biography—Part I,”

MathematicalEducation,July-September1987,15.

“WhenyouweartheKing-Emperor’suniform.”Slater,SouthernIndia,291.“keeninterestandhighanticipation.”JournaloftheIndianMathematicalSociety9,no.1(February1917):13.“somesurprisingdiscoveries.”S.R.Ranganathan,37.“theemergenceofRamanujanintofame.”Ibid.,64.claimedfivelivesalready.JournaloftheIndianMathematicalSociety10,no.6(December1918):1.RamanujaninBombayandmovementsafterthat.Interview,Janaki.FamilyRecord.K.R.Rajagopalan,46–48.WhyfretoverJanaki?Interview,Janaki.twoidenticalletters.Ibid.Brotheradvisedher.Ibid.“askedtoafuneral.”Neville,“Ramanujan”(Nature149),294.boardedtheBombayMail.P.K.Srinivasan,92.“Iforesawtheend.”RamachandraRao,88.Komalatammal’sexcuse.Interview,Janaki.bundledontoajutka.P.K.Srinivasan,92.“doesnot...gofarenough.”Springmemo,March27,1919.MadrasPortTrust.whenhishealthimproved.Berndt,“Ramanujan—100YearsOld,”26.Madrasinotables.K.R.Rajagopalan,47–48.LuzChurch.Muthiah,137.onApril6.FamilyRecord.inmonthlyinstallments.P.K.Srinivasan,182.arealrelationship.Interview,Janaki.drankcoffeenow.Ibid.“notacheerful...Ramanujan.”S.R.Ranganathan,71.“nottheoriginalRamanujan.”P.K.Srinivasan,101.“onlydevils.”Ibid.he seemed troubled. Letter, S. Chandrasekhar toHardy, 4 August 1937. Trinity College. Other accounts. See

Nandy,131.anotherirritant.Letter,A.Ranganathan,12January1981.RoyalSociety.“diamondeardrops.”Transcript,Nova1508,16.Coimbatorevs.Kodumudi.InterviewwithJanaki.FamilyRecord.Butinaninterview,T.V.Rangaswamiclaims

Ramanujanwas, at least briefly, inCoimbatore, perhaps later in the year.AndK.R.Rajagopalan, 48, alsomentionsastayinCoimbatore.Butifso,itwasbrief.

Ramanujan’sstayinKodumudi.Interviewswithtownspeople,includingR.ChandrasekharandK.Elangovan.registrar’soffice.Interview,Janaki.Ramanujanopenlyrebelled.Thestory, its timing,andthe trouble leadingupto it,arebasedon interviewswith

JanakisupplementedbytheFamilyRecord.“Ifonlyyouhadcome.”Janakihasrepeatedthismanytimesovertheyears,asintheHindu,19December1987,

andplainlyclingstothememoryofit.oilandwashherhair.Interview,“Hari”ofMadras,whoreportsbeingtoldthestorybyJanaki.sendJanakipacking.Nandy,130.EverySunday.FamilyRecordFearnside.P.K.Srinivasan,182.Thanjavurpun.Thisappearsinmanyforms.See,forexample,S.R.Ranganathan,93.

Komalatammaldecreed.Interview,Janaki.wentonaheadtolookforahouse.FamilyRecord.“insistedonseeingmyaunt.”SeshuIyer,82.driedbrinjalslices.P.K.Srinivasan,103.“powerfulandpenetratingeyes.”S.R.Ranganathan,75.Dr.Chandrasekar.K.R.Rajagopalan, 49;P.K. Srinivasan, 108; letters,A.Ranganathan, 12 January and 26

March1981.RoyalSociety.“thistuberculosisfever.”K.R.Rajagopalan,48.deadwithinfiveyears.Keers,144.“stillinverybadhealth.”Letter,DewsburytoHardy,22December1919.TrinityCollege.refusedtobefarthertreated.SeshuIyerandRamachandraRao,xviii.givenupthewilltolive.Letter,A.Ranganathan,12January1981.RoyalSociety.ResistedgoingtoMadras.SeshuIyerandRamachandraRao,xviii.probablyalittleafterthefirstoftheyear.Letter,DewsburytoHardy,15January1920.TrinityCollege.gotonlyasfarastherailwaystation.Interview,Janaki.knownas“Gometra.”Interview,Mr.andMrs.T.C.Krishna,currentownersofGometra.“whichIcall‘Mock’thetafunctions.”Ramanujan,CollectedPapers,xxxi.“insomewaystheverybest.”S.Chandrasekhar,Transcript,Nova1508,16.“weknowhowbeautifully.”G.N.Watson, “TheFinalProblem,” Journalof theLondonMathematicalSociety 11

(1936):57.“cold,immortalhands.”Ibid.,80.“Itisverydifficult.”Interview,GeorgeE.Andrews,in“SpectacularGenius,”TheHindu,21January1987.finalflurryofcreativity.SeeMcMurryforaperceptivestudyofthemythssurroundingtuberculosis.“proportionatelykeenerandbrighter.”SeshuIyer,85.“wildeccentricgeniusofhisyouth.”Interview,GeorgeE.Andrews,in“SpectacularGenius,”TheHindu,21January

1987.Crynantinauspicious.Interview,T.V.Rangaswami.NamberumalChetty.Who’sWhoinMadras,1934.“Gometra?”TourofthehouseconductedbyMr.andMrs.T.C.Krishna.maybrieflyhavestayedthere.Janaki,in

aninterview,recallsalarge“hall”inwhichRamanujanstayed,whichcouldhavebeenthelargelivingroom.scantfurniture.P.K.Srinivasan,163.Visitorsmovedsilently.“‘HisPapersDisappearedMysteriously,’”TheHindu,21June1981.Accessjealouslyguarded.P.K.Srinivasan,162.Janaki’sbrotherwithher.Ibid.,163.NarayanaswamyIyer.S.R.Ranganathan,14.“Buoyedupbyspesphthisca.”McMurry,140.subscribingtonewmathjournals.Letter,DewsburytoHardy,15January1920.TrinityCollege.Chetputpun.S.R.Ranganathan,93.“uniformlykindtome.”S.R.Ranganathan,91.JanakiconfirmsthatRamanujanwassurehewasgoingtodie.pieceofhotpepper...K.R.Rajagopalan,50.Bellandstick.Interview,Janaki.cravedtherasam.P.K.Srinivasan,136.poundeditalltogether.Interview,Janaki.

“mentallydeadtotheworld.”P.K.Srinivasan,111.“skinandbones.”S.R.Ranganathan,91.TheaccountofRamanujan’sfinaldaysisdrawnlargelyfrominterviews

withJanakiandfromvariousaccountsshehasgivenovertheyears.relativesstayedaway.K.R.Rajagopalan,51.RamachandraRaoarrangedthecremation.P.K.Srinivasan,88.officiallyrecordedhisdeath.TheHindu,3January1988.SubrahmanyanChandrasekhar.SeeKameshwarC.Wali,Chandra(Chicago:UniversityofChicagoPress,1990).“Icanstillrecallthegladness.”Andrewsetal.,3.“WewereproudofMahatmaGandhi.”Muthiah,6.“inspiredbyhisexample.”Andrewsetal.,5.neglectedhisstudies.S.R.Ranganathan,20.“toodeepfortears.”Letter,A.Ranganathan,12January1981.RoyalSociety.“entirelyunderthecontrol.”Letter,LakshmiNarasimhantoHardy,29April1920.TrinityCollege.DeathofRamanujan’sfather.Interview,V.Viswanathan.IhavenotlearnedwhenRamanujan’smotherdied,but

asisterofV.Viswanathan (andgranddaughterofNarayana Iyer) reports thatwhenMadraswasevacuatedduringWorldWar II, probably in 1942, she sawKomalatammal inKumbakonam. She would have beenseventy-four.

“becameveryoftensullen.”Bharathi,51.Komalatammal’sletter.Letter,KomalatammaltoHardy,25August1927.TrinityCollege.“consoleherselfbyseeingus.”P.K.Srinivasan,101.worriedabouthowshe’dbetreated.Interview,Janaki.twodaysbeforeRamanujan’sdeath.Ibid.nexthalfcentury.In1931,accordingtoT.V.Rangaswami,sheoncemetA.S.RamalingaminMadras.workasewingmachine.Interview,Janaki.SeealsoK.R.Rajagopalan,53.“relativeshavingswindledher.”Letter,S.ChandrasekhartoHardy,4August1937.TrinityCollege.JanakiandNarayanan.Interview,Janaki.SeealsoK.R.Rajagopalan,53.“Andheisnomore.”RamachandraRao,87.“inefficientandinelasticeducationalsystem.”Hardy,Ramanujan,7.“Indianality.”RamachandraRao,89.“raisedIndiaintheestimationofthe...world.”JournaloftheIndianMathematicalSociety11(April1919),1.“afluctuationfromthenorm.”S.Chandrasekhar,“OnRamanujan,”inG.E.Andrewsetal.,4.“moredeeplyboundupwiththeWest.”Nandy,139.“boostedourmorale.”InterviewwithP.K.Srinivasan.“bythehandofSrinivasaRamanujan.”Neville,ReadingManuscript.“somepermanentmemorial.”P.K.Srinivasan,79.WilsonandWatson.“MyrespectforR[amanujan]hasincreasedconsiderablyinthelastthreemonths,”Watson

wroteWilsonon28June1929.“IhaveretiredfromtheJ.M.B.togaintimeforR.”TrinityCollege.“no more romantic personality.” R. D. Carmichael, “Some Recent Researches in the Theory of Numbers,”

AmericanMathematicalMonthly39,no.3(1932):140.“sofullofhumaninterest.”Mordell,“Ramanujan,”642.“whichIimmediatelyreadwithgreatinterest.”PaulErdos,“RamanujanandI,” inNumberTheory,Madras1987,

ed.K.Alladi, no. 1395 in the series,LectureNotes inMathematics, ed.A.Dold andB.Eckmann (Berlin:Springer-Verlag,1989),1.TheHardy-Ramanujanpaperisno.35inRamanujan,CollectedPapers.

“unusualandsomewhatstrange.”AtleSelberg,“RamanujanandI,”ScienceAge,February1988,37.

“Thechapter...Ilovedthemost.”FreemanDyson,“AWalkThroughRamanujan’sGarden,”inG.E.Andrewsetal.,9.

“destructionof1917.”Ibid.,15.“originatedwithRamanujan.”MorrisNewman,“CongruencePropertiesofthePartitionFunction,”Reportofthe

InstituteintheTheoryofNumbers(UniversityofColorado,1959).“tamedalittle.”Hardy,CollectedPapers,720.“intotouchwithEuler.”Littlewood,Miscellany,86.anothertenyears.Letter,R.J.L.KingsfordtoHardy,23August1929.TrinityCollege.“allbymyselfasadevotee.”Interview,FreemanDyson.“astackofbrownedoldpaper.”S.Ramaseshan, “SrinivasaRamanujan,”Proceedingsof theRamanujanCentennial

InternationalConference,ed.R.Balakrishnanetal.(Madras,1988),6.“evenmoredefinitepicture.”Littlewood,Miscellany,86.Srinivasan’sproject.Interview,P.K.Srinivasan.SeealsoP.K.Srinivasan,vi–x.“OnCertainArithmeticalFunctions.”Ramanujan,CollectedPapers,136–162.“veryimperfectlyunderstood.”Hardy,Ramanujan,161.“kept at bay.” S. Raghavan, “Impact of Ramanujan’s Work on Modern Mathematics,” SrinivasanRamanujan

Centenary1987[specialissueoftheJournaloftheIndianInstituteofScience],46.AnaccountbyHardy.Hardy,Ramanujan,170.Bachcomparison.Berndt,“Ramanujan—100YearsOld,”24–29.DiscoveryoftheLostNotebook.Interview,GeorgeAndrews.Transcript,Nova1508,17.Andrews,Q-Series.“Andthatbotheredme.”Interview,BruceBerndt.“substanceanddepth.”Interview,FreemanDyson.“askedtoexplain.”Hardy,SomeFamousProblems,4.“missingpecuniaryreward.”JournaloftheIndianMathematicalSociety13,no.3(June1921):100.“betterblastfurnaces.”Bharathi,93.Plastics.S.Ramaseshan,“SrinivasaRamanujan,”11.Cancer.MeetingheldNovember10–12,1988.TheHinduarticlewasdated23December1988.“hardhexagonmodel.”R.J.Baxter,“Ramanujan’sIdentities inStatisticalMechanics,” inAndrewsetal.,69–84.

SeealsoGurneyWilliamsIII,“TheMasterofMath,”Omni,December1987,58–64.“a computer algebra package in his head.” Saraswathi Menon, “Beautiful Important Work,” The Hindu, 22

December1987.“hereallydidn’tneedthem.”Andrews,Q-Series.“Hype.” Interview, FreemanDyson. Bruce Berndt has writtenme with “another ‘application’ ofRamanujan’s

mathematics thatDyson calls ‘hype.’ ”WilliamBeyer, a physicist at LosAlamosNational Laboratory, hasshownhowoneofRamanujan’sformulascanbe“veryusefulinpredictingnuclearwar.”Theformuladoesnot,of course, truly “predict”nuclearwar,butrathersupplies thebasis fora theoreticalmodel forestimating itslikelihood.

“appreciateRamanujanearly.”Askey,72.“Iwasstunned.”Transcript,Nova1508,12.“athrill...indistinguishable.”G.N.Watson,“TheFinalProblem,”JournaloftheLondonMathematicalSociety11

(1936):80.“acertaincharacterofpermanence.”Hardy,SomeFamousProblems,4–5.framedandgarlanded...portrait.Notedinavideotapemadeofthecelebration.

theuniversityhadnoticed.S.R.Ranganathan,55.S.ChandrasekharalsoplayedaroleinseeingtoitthatJanakiwasbettercaredfor.

JanakifinancedInterview,Janaki.“Whereisthestatue?”Hindu,21June1981.Granlundsculpture.Interview,RichardAskey.Eachafternoon.Interview,T.V.Rangaswami.Inthemornings,hereports,Janakiwasoccupiedwithdevotionsor

cooking.brokedownandcried.Interview,T.V.Rangaswami.lefttoforeigners.Ramanujan,saidP.K.Srinivasan inan interview,“wasborn inIndia,reborn intheU.K.,and

nowbornagainintheU.S.”“couldnotgetevenalectureship.”J.B.S.Haldanequotedinsidebarto“TrueGenius,”by“RGK,”30.“languishinginobscurity.”Bharathi,36.“anewIndiaandanewworld?”Nehru,DiscoveryofIndia(London:MerideanBooks),1960.manyRamanujans.Interview,R.Viswanathan,Kumbakonam.“of defaming Ramanujan’s name.” S. Chandrasekhar, “An Incident in the Life of S. Ramanujan . . . ,” notes

depositedinRoyalSociety.RamanujanInstitute.Interview,K.S.Padmanabhan.“Howmanyregistrars.”S.Ramaseshan,3.“luminescence.”“TrueGenius,”by“RGK,”31.“ourbankruptcywasmademanifest.”ReportofJointSecretaryD.D.Kapadia,SecondConferenceoftheIndian

MathematicalSociety,11–13January1919,reportedintheSociety’sJournal.“towintheconfidence.”Neville,TheFareySeriesofOrder1025(Cambridge:UniversityPress,1950),xxvii.Hardy’sgreatestcontribution.QuotedinTheHindu,19December1987.svayambhu.“TrueGenius,”by“RGK,”26.“Ididnotinventhim.”Hardy,Ramanujan,1.

EPILOGUE

“principalpermanenthappiness.”Letter,HardytoThomson,13December1919.CambridgeUniversity.hard feelings left over from the war. “Life in College was through all these years [of the war] . . . definitely

unpleasant, and the recollection of them was an important factor in my own decision to try to move toOxford.”Hardy,RussellandTrinity,10.

“darkerforHardy.”Snow,Apologyforeword,38.“felthemustgetaway.”Young,287.“Cambridgebeguiles.”RoseandZiman,44–45.“BydirectionoftheSyndicate.”LettertoHardy,29April1920.TrinityCollege.“agreatshockandsurprise.”P.K.Srinivasan,78.“all Germans will be relegated.” Quoted in Joseph W. Dauben, “Mathematicians of World War I: The

International Diplomacy of G. H. Hardy and Gösta Mittag-Leffler as Reflected in Their PersonalCorrespondence,”HistoriaMathematica7(1980):261–288.

“trivialchangesofsign.”Thestoryhasbeentoldinmanyplaces.Young,280,isoneofthem.“modifiedmyformerviews.”Letter,HardytoMittag-Leffler,quotedinDauben,“MathematiciansofWorldWar

I.”“happiest timeofhis life.”Snow,Apology foreword, 40. “AfterHardy’s social discomforts at theHighTable of

Trinity,Hardyexpandedandmellowedinawonderfulway,inwhatwastohimthemorebenignatmosphere

of New College. His popularity there was immediate and assured, and he gave back in the intellectualexhilarationofhisconversationwhathereceivedinsympatheticfriendship.”E.A.Milne,obituaryofHardyinMonthlyNoticesoftheRoyalAstronomicalSociety108(1948):46.

“customarydeferenceofthosedays.”MaryCartwright,“MomentsinaGirl’sLife,”unpublishedmanuscript,5.“adelightfultime.”Letter,HardytopresidentofPrincetonUniversity,24April1929.(Theletterheadisthatof

theMayflowerHotel,Washington,D.C.)RuthasfamiliarasHobbs.“Hehadasimilarpassionforbaseball[asforcricket],whichhewatchedwheneverhe

couldwhenhewasintheStates.”Fromajournalreferee’sreportcommentingonanarticleaboutRamanujanandHardy.RobertRankin,Glasgow.

“awonderfulbook.”Postcard,HardytoR.P.Boas,10January1940.Oxfordtennisphoto.TrinityCollege.retirement.Snow,Apologyforeword,43.“onecouldoftenheartheparty’slaughter.”Snow,“TheMathematician,”72.“arealmathematician.”Snow,Apologyforeword,9.“suddenlyatleasttheequal.”Young,293.“deepsolicitude.”A.V.Hill,obituarynoticeinTheMathematicalGazette22(May1948),51.“thenIcannotremaininit.”Letter,HardytoMordell.St.John’sCollege.“moreuncharitableconclusion.”Hardy,CollectedPapers,611.“hewaswiththeminers.” J.B.S.Haldane, “A ‘Pure’Scientist—andaGreatOne,”DailyWorker (London),29

December1947.“somemysteriousterrortoexactthought.”Hardy,CollectedPapers,260.somewouldgrumble later. J.C.Burkill, reviewofCollectedPapers ofG.H.Hardy, vol. 7,Bulletin of theLondon

MathematicalSociety12(1980):226.CongratulationsfromSoviets.Letter,SovietAmbassadortoHardy,20February1934.TrinityCollege.Honorarydegrees.AftertheceremonyinwhichhereceivedhisdegreefromEdinburgh,Hardywasstrollingwith

a group ofmathematicianswhen he spotted amouse.His colleagues, reportedThe Scotsman (3December1947),“weregrantedtheprivilegeofwatchingoneoftheworld’sgreatestmathematiciansstalkingit,onhandsandknees,aroundatree.”

Hardy’sNewYear’sresolutions.J.C.Burkill,entryonHardyinDictionaryofScientificBiography.hismadcapbent.Young,276.“oldbrandy.”Snow,Apologyforeword,48.“agedandshriveledreplica.”Wiener,IAmaMathematician,152.“youthfulquicknessandpower.”Hardy,CollectedPapers,745.whiledustinghisbookcase.Interview,MaryCartwright.hismostimportantpapers.Titchmarsh,458–461.“likeWandaLandowskaplayingBach.”Letter,FreemanDysontoC.P.Snow,22May1967.FreemanDyson.“makeshimlooksoold.” Interview,MaryCartwright.Written inblue ink,almostcertainlybyhissister,onthe

photointheTrinityarchives:“Heneverlookedasoldasthis!”Hardy’ssuicideattempt.Snow,Apologyforeword,54.“oneconclusiontobedrawn.”Ibid.,57.“Hardyisnowdying.”PhilipSnow,95.“thepassingofagreatage.”NorbertWiener,BulletinoftheAmericanMathematicalSociety55(1949):77.“bookofhauntingsadness.”Snow,Apologyforeword,50.“Itisamelancholyexperience.”Hardy,Apology,61.“onsomethinglikeequalterms.”Ibid.,148.

“aboutwhichhefeltstrongly.”MaryCartwright,“MomentsinaGirl’sLife,”unpublishedmanuscript,7.“theoneromanticincident.”Hardy,Ramanujan,2.“mostcarefulediting.”Letter,HardytoMittag-Leffler,about1920.“ashortbutcharacteristicexample.”Hardy,inRamanujan,CollectedPapers,232.“thedecisiveeventofmylife.”Hardy,Apology,148.“poorandsolitaryHindu.”Hardy,Ramanujan,10.“amongthemostremarkable.”ManchesterGuardian,2December1947.“oneofthemostfascinatingobituarynotices.”QuotedinAndrews,Q-Series.aprimesubjectofhisownbook.Nandy,

3.Hardy’sreportonRamanujan.Hardy,CollectedPapers,491–503.“greaterthanthegain.”Hardy,CollectedPapers,720.“nosecrecyatall.”Snow,Apologyforeword,30.“Hardywasterriblyproud.”Interview,MaryCartwright.“Iamgoingtogivesomelectures.”Letter,HardytoChandrasekhar,19February1936.TrinityCollege.“laboroflove.”Mordell,“Ramanujan,”642.Harvard Tercentenary celebration. See, for example, Harvard Alumni Bulletin, 30 September 1936. The

TercentenaryofHarvardCollege(Cambridge,Mass.:HarvardUniversityPress,1937),compiledbyJeromeD.Greene;HarvardMagazine,May-June1986.

“whohasledtheadvance.”TrinityCollege.“theruleinShelley’scase.”Newman,TheWorldofMathematics,vol.4(NewYork:SimonandSchuster,1956),

2036.AccordingtoNewman,thestoryistoldbySupremeCourtJusticeFelixFrankfurter.inHitler’sarmy.“HighbrowsatHarvard,”Time,14September1936.aboutnineintheevening.Conferenceschedule,inTheTercentenaryofHarvardCollege,AppendixN,465.“averygreatmathematician.”Hardy,Ramanujan,1.

SelectedBibliography

Alexanderson,G.L.ThepolyaPictureAlbum:EncountersofaMathematician.Boston:Birkhäuser,1987.Andrews, George E. “An Introduction to Ramanujan’s ‘Lost’Notebook.”American Mathematical Monthly 86

(1979):89–108.———.NumberTheory.Philadelphia:W.B.SaundersCompany,1971.———.Q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and

Computer Algebra. Providence, R.I.: American Mathematical Society, Regional Conference Series inMathematics,No.66,1986.

———,RichardA.Askey,BruceC.Berndt,K.G.Ramanathan,andRobertA.Rankin.RamanujanRevisited.SanDiego:AcademicPress,1988.

Askey, Richard A. “Ramanujan and Hypergeometric and Basic Hypergeometric Series.” In RamanujanInternationalSymposiumonAnalysis,editedbyN.K.Thakare.MacmillanIndia,1989.

AssociationofMathematicsTeachersofIndia.RamanujanCentenaryYearSouvenir.Proceedingsof22dannualconference,3–6December1987.

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Author’sNoteandAcknowledgements

InwritingthelifeofRamanujan,Ifacedthebarriersoftwoforeigncultures,achallengingdiscipline,andadistanttime. As I am expert in none of these, I owe a debt of gratitude to the many persons who have helped mesurmount those barriers—who have consented to interviews, spent hours explaining recondite areas ofmathematicsor Indiancultural life,guidedme tooutof thewaydocuments in librariesandarchives, read andcriticizedearlydrafts,befriendedmeinEnglandandIndia—and,backinBaltimore,offeredasupportivehandorwordofadvice.IamabashedathowmuchspaceIrequiretosimplysaythankyou,butitisanaptmeasureofmydebt.

First,thanksaredueJudy,whohasbornemorehusbandlychangesofmoodthananypersonshouldhavetobear,andtowhomIowemuchinthewayofnurturing,encouragement,andsupport.Thoseintangiblesareeasytotakeforgrantedwhenyouhavethem,butalmostimpossibletogetalongwithwhenyoudon’t.

ToJudyandmysisterRachele,andtheirrefusaltotellmewhatIwishedtohearatacrucialpoint,IoweanapproachtowritingaboutthemathematicstowhichotherwiseImightnothaveturned.

ThanksgotoDavy,whoseDaddywasgoneinEnglandandIndiaforthreemonthsandthen,fortheyearandahalfittooktowritethisbook,wastoomuchintheofficeandnotenoughwithhim.“Sohow’stheRamanujanbook?”heaskedmeonedaywhenhewasfive.Ihopethatonedayhe’llreaditandbeabletoanswerforhimself.

Tomyparents,BeaandCharlesKanigel, for leavingme fascinatedequallybywords,numbers, people, andideas.

ToMichael,Kevin,andJonathan,toHarryandRachele,andtoEliseandLiz—allofthem,eachintheirownway,irreplaceablepartsofmylife.

ToBillStumpforencouragingayoungwritertwentyyearsago.ToV.Viswanathan, ofMadras,whomade room for a confusedAmerican in an already overstuffed auto-

rickshaw.Tohim,hisbrotherV.MeenakshisundaramoftheMadrasPortTrust,toS.SankaraNarayananandV.Subramanyam,andtothemanyothermembersofhisfamilywhobefriendedmeinMadras,IoweadebtofkindnessIcannotpossiblyrepay.

ToSambandam,Vijaya,Mahalingam,andalltheirfamilyinKumbakonam,fortheirboundlesshospitality.Tothe “GangofThree”—threeAmericanmathematicians,admirersandstudentsofRamanujan,whohave

helpedmetounderstandhiswork,andwhohavereadthemanuscriptalongtheway,invariablypepperingtheirexcellentadvicewithdollopsofneededencouragement:GeorgeAndrews,PennsylvaniaStateUniversity;RichardAskey,UniversityofWisconsin;andBruceBerndt,UniversityofIllinois.Withoutthehelpofthesethreementhisbookcouldnothavebeenwritten.

To Robert Rankin of Glasgow, Scotland, whose research into the lives of Ramanujan and Hardy hasmaterially contributed to this book, andwho has been unfailingly patient in responding tomy trans-Atlanticqueries.

ToFreemanDyson,fortakingsuchinterestinthebook,roundingupoldletters,readingthemanuscript,andmakingimportantsuggestions.

ToBarbaraGrossman,whoseideathisbookwas,andwhohasplacedthefullforceofherpersonalitybehindit. To Joy Smith for her diligence and unfailing good nature.To ErichHobbing,David Frost, and others atScribner’sfortheirhelpandtalents.ToZoëEnglishKharpertianforherfinejobofcopyediting.

ToVickyBijur,my agent, for the important role she played in bringing this book into being, and for thealmostfrighteningefficiencywithwhichshehasactedinmybehalfalongtheway.

ToJaneAlexander,whosentmetoIndiathefirsttimeandhasbeenafriendsince.

Somanyhavehelped, in smallwaysand large, tomake thisbookpossible that it seemsscarcelypossible torememberthemall.IapologizetoanyImayhaveinadvertentlyomitted.

InAmerica:SudarshanBhatiaandAshaRijhsinghani.V.AnantharamanandMalini.AshvinRajan.RanjanRoy,BeloitCollege.HenryS.Tropp,HumboldtStateUniversity.S.Chandrasekhar,UniversityofChicago.ArthurMagida. Gary Leventhal. Ann Finkbeiner and CalWalker. Ken Gershman. Lee and Phyllis Jaslow. MildredFoster.Carolyne,Kathy,andeveryoneattheRedBalloon.WilliamDyalandThomasSlakey,St.John’sCollege,Annapolis.WarrenKornberg,Mosaic magazine. JacekMostwin, JohnsHopkinsHospital. Steve Fisher. AlanSea. Adrianne Pierce, Johns Hopkins University. John Halperin, Vanderbilt University. Maurice St. Pierre,Morgan State University. Suzanne Holland, Harvard magazine. George W. Comstock, Johns HopkinsUniversitySchoolofHygieneandPublicHealth.S.Bhargava,Universityof Illinois,Urbana.WayneMarkert,UniversityofBaltimore.

InBritain: Emma and Jonathan Leigh,Cranleigh School.KevinGray,TrinityCollege. Béla Bollobás,TrinityCollege. Charles Burkill, Cambridge. Mary Cartwright, Cambridge. Rajiv Krishnan, Christ’s College. RogerDavidHughCustance,WinchesterCollege.GuyNewcombe,TrinityCollege.VinceDarley,TrinityCollege.JohnVickers,St.John’sCollege.R.Robson,TrinityCollege.ConstanceWillis,Cambridge.DeborahandBryanJ. B.Gauld, Putney.Theodor Schuchat and LouiseHarper, London. Paul andClare Friedman, London. S.J.Mann,CranleighSchool.SusanM.Oakes,LondonMathematicalSociety.J.D.Webb,CambridgeCityCouncil.TomDoig,CambridgeFolkMuseum.PatKattenhorn,IndiaOfficeLibrary.

In India: Janaki Ammal, the widow of Ramanujan, and her son, W. Narayanan, and his family, Triplicane,Madras. T. V. Rangaswami (“Ragami”), Triplicane, Madras. A. P. Victor, Kuala Lumpur, Malaysia. P. K.Srinivasan,AssociationofMathematicsTeachersofIndia,Madras.T.M.SrinivasanandT.M.Kasturirangan,Madras.JohnHerbetAnand,Mylapore,Madras.R. Janarthanan,Kumbakonam.R.Viswanathan,headmaster,andV.Vaidynathan,assistantheadmaster,TownHigherSecondarySchool,Kumbakonam.S.Subbarathinan,Erode.A.NazimuddinandH.Sharmila,Madras.K.Elangovan,Kodumudi.R.Chandrasekhar,Kodumudi.A.Sanguttuvan,Erode.L.Rajagopolan,S.ElangoandD.G.Ramamurthy,Kumbakonam.A.Saranathanandhisfamily,Kumbakonam.S.GovindarajaBattachariarandP.Vasunathan,UppiliapanKoil,Thirunageswaram.T.U.BhanumurthyandKalyanalakshmi,Madras.M.Vinnanasan,Kumbakonam.K.S.Padmanabhan,RamanujanInstitute,Madras.K.NarayananandA.V.Chandrasekhar,TheHindu.A.Ranganathan,Madras.K.Rajamani,Kumbakonam.T.C.Krishnaandhis family,Madras.BhamaSrinivasan,Universityof Illinois,Chicago.P.P.Kulkarni,Nagpur.And,ofcourse,theunforgettableHari,ofMadras.

•••

IwishalsotoexpressmygratitudetothestaffsofthemanylibrariesintheUnitedStates,England,andIndiawhohavehelpedmeinthequiet,faceless,butunfailinglycompetentwayweallexpectofthem.That,ofcourse,istheproblem:it’ssoeasytotakethelibraryforgrantedwhenithasthebookordocumentyouneed,andtogrumblewhen itdoesn’t.But theveryconceptof the library,asaplace tostore,preserve,andgiveaccess tobooks,wasthrownintosharpfocusduringmyfiveweeksinIndia.There, librariescannotalwaystreattheirtreasureswiththeexpensivecareWesternlibrariescanlavishontheirs.Inone,Ifoundbooksandjournalssetoutonthefloor,piledthiswayandthat,thepagesofevenrecentbookscrumbling,dustyandmildewed.AndyetneverhaveIseenlibrariessointensivelyused,bookssohungrilydevoured.InonesmalllibraryinErode,Isaweveryseatateverytabletaken,andmanypeoplestandingintheaislestoread.

Myappreciation,then,goestolibrarians,archivists,andotherstaffatthefollowinginstitutions:Baltimore: The Milton S. Eisenhower Library of Johns Hopkins University. William H. Welch Medical

Library.EnochPrattFreeLibrary—centrallibrary,St.PaulStreetbranch,andtelephonereferenceservice.Washington,D.C.:LibraryofCongress.

Cambridge,Mass.:HarvardUniversityArchives.Princeton,N.J.:SeelyG.MuddManuscriptLibrary.Cambridge, England: University Library of Cambridge University, and the libraries of these Cambridge

colleges: Trinity, St. John’s, Gonville & Caius. Scientific Periodicals Library. Cambridgeshire Collection.CambridgeFolkMuseum.

London: Royal Society Library. India Office Library. University of London.NationalMaritime Museum,Greenwich.

Elsewhere in England: Guildford Public Library (Local Studies Collection). Cranleigh School, library andarchives.CranleighLibrary.St.Catherine’sSchoolLibrary,Bramley.NewCollegeLibrary,Oxford.UniversityofReading.

Erode:PublicLibrary.Madras:ConnemaraLibrary,UniversityofMadras.ArchivesofTheHindu.BritishEmbassyLibrary.FortSt.

GeorgeMuseum.MadrasMuseum.

Index

Abel,NielsHenrik,147,150,169–70,366ActaMathematica(journal),363,370Additivetheoryofnumbers,303AdvisoryCommitteeforIndianStudents,174,184Agincourt,Battleof,163Algebra,41–44,46,53,60,86,119,149,161,205,366

computer,349geneticsand,146

Algebraicgeometry,344Allen,G.C.,126AlliedPowers,212AlternativeSciences(Nandy),335AmericanMathematicalMonthly,336AmericanPhysicalSociety,348AnalyticalClub,85Analyticalnumbertheory,249,340AnandaRao,230Anantharaman,55,72,190,282,290,318,332–33Andrews,George,6,252,325,326,344–46,349,350–52AnnaUniversity,351–52Apostles.SeeCambridgeApostlesAppliedmathematics,347–49Archimedes,209AreConjectandi(Bernoulli),90Aristotle,287Arithmeticalfunctions,343Arnold,Thomas,112,121Arybhata,86Askey,Richard,280,350–52Athens,Universityof,366Atomicresearch,348AtsharaAbishekam(ritualpractice),13AustralianNationalUniversity,349

Bach,JohannSebastian,344,350,351Bacon,Francis,239Bailey,W.N.,202,231,371Baker,H.F.,106,170–72,291,358BalakrishnaIyer,S.,78,321Ballistics,228Barnes,E.W.,151,233,243,261,277,290

Bauer(mathematician),167Baxter,R.J.,349Bayeuxtapestries,133Beethoven,Ludwigvan,346,350Belgium

inGreatWar,211–14,227,260,277tuberculosisin,267

Bell,E.T.,65,81,256,280,287BelowtheMagicMountain(Bryder),270Benedictineorder,126Bergson,Henri,157Berlin,Universityof,365Berndt,Bruce,183,204–5,280,344,346,350,351Bernoulli,Jacob,89Bernoullinumbers,89–92,105,161,203Berry,Arthur,201–2BertrandRussellandTrinity(Hardy),277Besant,Annie,314Bhagavad-Gita,30,179Bhaskara,86BhavaniswamiRao,106Bieberbach,Professor,366Birmingham,Universityof,204,366BirthofBritain,The(Churchill),133Bloomsburyliterarymovement,111,137,141Bohr,Niels,365Bollobás,Béla,253BolshevikRevolution,306Borel(mathematician),153Borwein,JonathanM.,208Borwein,PeterB.,208Bose,J.C.,335,336Boundederrors,251–52Bourne,A.G.,103Boyles,Robert,291Brahmagupta,86,209Brahmins,20–26,29–33,36,46,51,54,77,81–82,112,174,210,354

dietaryregulationsof,240–41,298foreigntravelforbiddenfor,174,185–86,198,240,316,329morningritualof,244

BrightonRailway,113BritishEastIndiaCompany,82,99BritishIndiaLines,195,311Bromwich,T.J.I’A.,104,105,145,176,245,284,291Brouwer,Luitzen,340Bryder,Linda,270,290Budapest,Universityof,338Buddhism,32Bulstrode,H.Timbrell,266–67,270Burkill,J.C.,254

BurmaOilCompany,314Butler,Henry,213Byron,GeorgeGordon,Lord,2,345

Calculus,46,89,148–50,153,161,249differential,56integral,166,183piin,209

CaliforniaInstituteofTechnology,364CambridgeApostles,137–41,147,157,172,228,276CambridgeMagazine,260,277CambridgePhilosophicalSociety,130,245,276,295,301,303,306CambridgeReview,152,307CambridgeShakespeareSociety,136,137,142CambridgeTractsinMathematicsandMathematicalPhysics,160CambridgeUniversity,1–4,6,40,66,100,105–7, 109–11, 124, 126–33, 145, 154, 157, 163, 170, 226, 253,

285,332,344,345,361–65,367,368,372arrangementstobringRamanujanto,172,174,190–94debatingsocietiesat,137andGreatWar,212–13,227–29,244,246,259–61,311homosexualsat,140–44Newton’sinfluenceat,148–50Ramanujanat,198,200–207,210–15,219–20,228–31,233–35,237–46,252–59,289–91,315,330,351,

357Tripossystemat,seeTripossystemSeealsospecificcolleges

CambridgeUniversityPress,337,341CamfordObserved(RoseandZiman),361Campbell,IslayMakimmon,259Cancerresearch,348Cardus,Neville,123Carmichael,Robert,337Carnot’sCycle,131Carpenter,Edward,140Carr,GeorgeShoobridge,39–46,57,58,72,90,92,105,164,173,203,253,356Carstairs,G.Morris,23Cartwright,Mary,156,171,364,369,372CaseofthePhilosopher’sRing,The(Collins),109Castesystem,20–23,112,314Cauchy,AugustineLouis,150,287Cauchy’stheorem,249CentralCollege(Bangalore),91CentralPowers,212Chandler,Laura,118Chandrasekhar,Subrahmanyan,329–30,333,335,352,354,372Chandrasekar,P.S.,321,322,330Charley’sAunt(play),231Chatterji,G.C.,230–31,237–39,275Chattopadhyaha,Mrilani,237–38Chaudhuri,NiradC.,75,243

Chengalvarayan,K.,315Chettiar,Alagappa,355Chicago,Universityof,364CholaEmpire,15Churchill,Winston,133Circlemethod,249,252,324Clarke,EustaceThomas,116Classsystem,British,112,113ColinetteHouse,311–12,319Collins,Randall,109Collins,Robert,261Colorado,Universityof,atBoulder,340Combinatorics,250CombinatoryAnalysis(MacMahon),305CommercialPrintingPress,341Compositenumbers,231–32,234,337ComptesRendus,252Compton,Herbert,55,101,102,272Computers,349Congruences,302–3,306,313,339,370ConnemaraLibrary(Madras),180Continuedfractions,57,215Continuousfunctions,134Continuousquantities,249Convergentseries,88ConversazioneSociety,137Coombs,“IronMan,”364CorpusChristiCollege,Cambridge,212,256Cotterell,C.B.,193CourseinPureMathematics,A(Hardy),153Coursd’analysedeI’EcolePolytechnique(Jordan),134,149,172CranleighSchool,113–20,123,126,136,142,171,172Crystallography,348Cubicequations,27Cubism,157Curzon,Lord,99

Darling(mathematician),168Davenport(mathematician),344Davies,Arthur,174,184,190Davies,P.D.O.,267–68Decemviridebatingsociety,137Dedekind(mathematician),150Definiteintegrals,90,182–83Delhi,Universityof,348Deligne,Pierre,343,344Denmark,tuberculosisin,267Denton,Eliza,118Derby,Lord,228Descartes,René,287

Deshmukh,C.D.,230Dewsbury,Francis,192–93,233,234,307,309–12,317–18,322,337,345,362Diaghilev,Serge,157Dickinson,GoldsworthyLowes,138,228DiscoveryofIndia(Nehru),354Discretequantities,249Divergentseries,80,174,203Dougall-RamanujanIdentity,167Doyle,ArthurConan,109,325Dravidianpeoples,32,33Dryden,John,239Dubos,Jean,270Dubos,René,270DudleySmith,CharlesJervoise,260Dyson,Freeman,339–41,346,349,368

EcolePolytechnique,285Eddington,SirArthur,372Edinburgh,Universityof,366

IndianStudentHostel,258Edison,ThomasAlva,347EducationTimes,77,168,171EdwardVII,KingofEngland,101,136,157,211EdwardianTurnofMind,The(Hynes),140,148EinleitungindieFunkionentheorie(StoltzandGmeiner),153Einstein,Albert,111,340,359,366,373Ellipticfunctions,61,169,205

modular,247Ellipticintegrals,80,201Ellis,Havelock,126,140,142,143,146–47Emden(ship),314England

afterarmistice,310–11,363conditionsforIndianstudentsin,256–58cookingin,299inGreatWar,212–14,235,262tuberculosisin,265–68,270

Enlightenment,the,287Erdos,Paul,338,350,358Eton,112,113,120Euclid,86,131,216–17Euler,Leonhard,50,147,205,208,221,247–48,250,287,340Euler-Maclaurinsummationformula,90Eves,Howard,340Exponentialgrowth,217

Fearnside,C.F.,320Fearon,W.A.,121,122,124,133FellowofTrinity,A(St.Aubyn),124–25Fermat,Pierrede,81,147,221

Fichte,JohannGottlieb,287FirstEasternGeneralHospital,213,227,260FirstIndianWarofIndependence,99Fitzgerald,Edward,2FitzroyHouse,300,311Formalism,205–6Forster,E.M.,138,141Forsythe,AndrewRussell,132,150–51,233,291Foster,Michael,54Fourierseries,157FourYearsinanEnglishUniversity(Satthianadhan),256–57Fractions,continued,57,215France

afterarmistice,364inGreatWar,211–14,227,235,260,277,307,311Nazioccupationof,369

Franco-PrussianWar,363FranzFerdinand,Archduke,211Frechet(mathematician),340Frullani’sintegraltheorem,183Fuller,SirBampfylde,102Functionalanalysis,366Functionalnotation,61–62

Gallipoli,Battleof,235Gammafunctions,167

negativevaluesof,159Ganapathy,M.M.,318Gandhi,Indira,353–54Gandhi,MohandasK.,99,185,241,299,314,330,351Gandhi,Rajiv,351Gauss,KarlFriedrich,147,150,160,167,169–70,218,287Gaye,R.K.,136,139Generatingfunctions,247Genetics,146Geometry,86,119,149,152,208–9

algebraic,344GeorgeV,KingofEngland,96,99,157,200Germany

afterarmistice,363inGreatWar,211–14,227,229,262,298,306–7,314Nazi,365,368,369tuberculosisin,267,270inWorldWarII,340

GirtonCollege,Cambridge,142,307Gmeiner(mathematician),153Gonville&CaiusCollege,Cambridge,126Gopalachary,K.,282Gosh,UmesChandra,171GovernmentCollege(Kumbakonam),27,28,45–47,49,51,53,54,77,90,101,198,293

GovindarajaIyengar,N.,56Graham,W.,103–4,261Gramadevata(villagegods),33,35Granlund,Paul,352Grant,Duncan,139Grant-Duff,MountstuartGeorge,84GreatWar,18,211–14,227–29,234–35,244,246,259–64,267,276–77,298,306–7,310–11,314–15,347,

361,363,365Greeks,ancient,86,88,165,287Gregory,James,209Griffith,C.L.T.,103–5Groupingliketerms,42GunsofAugust,The(Tuchman),212

Hadamard,Jacques,218,285–87Haldane,J.B.S.,353,366HanumanthaRao,B.,178Hardy,G.H.,4,5,7,109–57,197,198,213,216–17,221,222,232,238,242,302,303,305,326, 329, 333,

334,337–41,343–47,349–51,356–73inApostles,137–41arrangesforRamanujantocometoCambridge,191–92atheismof,109,117,286birthof,113childhoodof,114–20collaborativepaperswithLittlewood,157,165,228,264collaborativepaperswithRamanujan,248–52,276,301,338,340,368correspondencewithRamanujan,107,159–79,181,183–84,187, 190, 198, 209, 216, 218–19, 229, 248,

255,323–24,328,337,341andcultofuselessness,146–48deathof,369electedtoRoyalSociety,145–46andGreatWar,228,229,234,260,276–77,361,363,365homosexualityof,139–44honorsreceivedby,366–67,369,373–74importanceofRamanujan’smemoryto,369–73inLondonMathematicalSociety,208,231,245,276,291andmealsatTrinity,239,240natureofrelationshipwithRamanujan,271–72,275–80,289Nevilleand,186,191andRamanujan’snotebooks,204,280,313atOxford,361–65,370physicalappearanceof,109politicalviewsof,366andRamanujanatCambridge,201–7,210,219–20,230,231,233–35,245,246,252–57,289–91,315andRamanujan’sdeath,331,332,336,337,361–63andRamanujan’sillness,263,264,271,290,295–300,312,321,322,328onRamanujan’sintuition,215–16,220,223–27andRamanujan’sreturntoIndia,307,309–10andRamanujan’sspirituality,280,283,284,286,288–89rigorof,151–54

andRoyalSociety,291–96,300,301suicideattemptby,368–69andTripos,128,129,132–36,155–56onunconsciousactivity,285–86inUnitedStates,364atWinchester,120–26winsTrinityCollegescholarship,125–26writingstyleof,151–52,371–72

Hardy,GertrudeEdith(sister),113–18,136,139,368,369Hardy,Isaac(father),113–18,120,126,136Hardy,SophiaHall(mother),113–18,136,139Hardy-WeinbergLaw,146,148HariRao,N.,46,76Harold,EarlofWessex,133Harrod,Roy,141Harrow,112,113,120Hartley,L.P.,5HarvardUniversity,165,366,372–73Hastings,Battleof,133HaverfordCollege,289Heisenberg,Werner,373HenryVIII,KingofEngland,128Herman,R.A.,300Hermite(mathematician),169,210,287Highlycompositenumbers,231–32,234,337Hilbert,David,220,226Hill,A.V.,365Hill,M.J.M.,103–6,176Himmelfarb,Gertrude,140,141,147Hindu(newspaper),67–68,95,195,237,273,317,347Hinduism,7,11,17,19–21,23,24,29–31,33–35,54,71–72,96,282,283,288

banonforeigntravelin,185mathematicsin,86

HistoryoftheEnglish-SpeakingPeoples(Churchill),133Hitler,Adolf,365,373Hobbes,Thomas,109Hobbs,Jack,109,364,366Hobson,E.W.,106–7,145,170–72,291,358,364Hodges,Andrew,139,278Holman,Donald,260HomeRuleLeague,314HouseofCommons,157HouseofLords,157,242–43Howard,JohnBrerton,260Huxley,Julian,124Hynes,Samuel,140,148Hyperboliccosines,176Hypergeometricseries,57,80,167,204

Identities,303,305–6,313,340

Illinois,Universityof,346IllustratedWeeklyofIndia,356,358Imaginarynumbers,59,208InaWorldINeverMade(Wootton),261IndiaOffice,172,174,184,190–92,332IndianCivilService(ICS),99–100,102,162,322IndianMathematicalSociety,77–79,85,86,96,98,171,180,189,295,315,347,356,357

Journalof,85–87,90–93,182,188,219,230,315,334–35,337IndianMeteorologicalDepartment,174IndianNationalCongress,314IndianNationalScienceAcademy,355IndianStatisticalInstitute,215IndianUniversitiesAct(1904),178Infiniteseries,47,87–89,145,161,167,171,177,216

applicationsof,349Bernoullinumbersand,90piand,209thetafunctionsas,323

Infinity,66,160Ingalls,Daniel,23InstituteforAdvancedStudy,346InstitutesofManu,21Integers,59,61Integrals,145

definite,90,182–83elliptic,80,201theoremson,161,165–67

IntermediateSex,The(Carpenter),140Introductioinanalysininfinitorum(Euler),205IntroductiontotheHistoryofMathematics(Eves),340IntroductiontotheTheoryofNumbers,An(HardyandWright),302Irrationalnumbers,59,154

Jacobi,KarlGustavJacob,65,168,169,205,256,283,323Jackson,Henry,213,227,261,263James,William,287Janaki(Ramanujan’swife),69–71,96–98,178,180–81,189, 194, 229, 256, 258, 316–20, 322, 327–31, 333,

336,342,351–53Komalatammaland,96,97,271–75,316,318–20,327

Jeans,JamesHopwood,136,287–88JesusCollege,Cambridge,126Jews,365–66JoanofArc,190Johnson,W.Woolsey,153Jordan,Camille,134–35,145,148,149,172Jung,Carl,372

Kac,Mark,281,338KapalaswaraTemple,82KasturirangarIyengar,S.,195,237

Keers,R.Y.,267Kelvin,Lord,131,150Keynes,JohnMaynard,138,139,141,170Kincaid,Dr.,299King’sCollege,Cambridge,1,126,138,214,230,307Kipling,Rudyard,101,262,278–79Klein,Felix,226Kloosterman(mathematician),344Koch,Robert,265,269Komalatammal(Ramanujan’smother),11–14,18–20,34,36–37,47,54,65,68–73,180, 256, 259, 312, 322,

354,357Janakiand,96,97,271–75,316,318–20,327orthodoxyof,186andRamanujan’sdeath,326–27,329–33andRamanujan’sreturntoIndia,316andtriptoEngland,188–91,194,195,271

KrishnaRao,R.,78,199,224KrishnaswamiIyer,C.R.,84KrishnaswamiIyer,S.,26,27Kronecker(mathematician),210Kshatriyas,20,21Kuladevata(familydeity),35–36Kuppuswami,Dr.,72Kupuswamy(Ramanujan’sgrandfather),17

LakshmiNarasimhan(Ramanujan’sbrother),12,316,318,331,334Landau,Edmund,363,365Laplace,MarquisPierreSimonde,168LaPoussin,CharlesJ.de,285Larmor,SirJoseph,260–61LaVallée-Poussin(mathematician),218Leçonssurlesfonctionsdevariablesréeles(Borel),153Lee,Alice,118Legendre,AdrienMarie,160,169,210LehighUniversity,364Lehmer,Emma,346,350Leibniz,Gottfriedvon,89,148,165,209,210,287Lenin,V.I.,366LettersfromanIndianClerk(televisionspecial),4–5,352Levy,Paul,276LincolnDiocesanTrainingCollege,113,114Littlehailes,Richard,178,182,192–94Littlewood,J.E.,135,139,141,145,151,152,163–65,186,192,198,213,220,221,224,226, 233, 238–40,

242,251,254,280–81,288,295,340,341,358,366,370collaborativepaperswithHardy,157,165,228,264inGreatWar,228,229,234,277andLondonMathematicalSociety,208andRamanujanatCambridge,201,205–7,210,315onRamanujan-Hardycollaboration,253andRamanujan’sletters,163,165,168–69,173,181,216

onrigor,150andRoyalSociety,291,292andTrinityFellowshipforRamanujan,299–301andTripossystem,132,133,155–56

LiverpoolUniversity,204LloydGeorge,David,193Loder,JohndeVere,259Lodge,Oliver,255Logarithms,217–18Lomas,John,139London,Universityof,115LondonMathematicalSociety,104–6,111,164,170,208,229,245,261,276,291,324,336,350,369

Proceedingsof,145,162,225,231,248,252,291,305,306,313,336,337,371LondonObserver,263Loney,S.L.,27,44,163Louvain(Belgium),213–14Love,A.E.H.,134,145,172Lusitania(ship),229LyttonCommitteeonIndianStudents,242,257

Macaulay,Lord,2,101,163Mach,Ernst,90MacMahon,MajorPercyAlexander,250,251,301–3,305,306,336McMurry,NanMarie,328Madhava,K.B.,180Madras,Universityof,24,52,233,263,290,311,331,336,352

CentreforAdvancedStudiesinMathematics,355SeealsoPresidencyCollegeMadrasChristianCollege,74,75,92,101,182

MadrasEngineeringCollege,103MadrasHighCourt,178MadrasMedicalCollege,321MadrasPortTrust,94–98,102,159,175,178–80,293,312,317MadrasPresidency,15–17,95,99MagdaleneCollege,Cambridge,126,264Magpie&Stumpdebatingsociety,137Mahabharata,20,29Mahalanobis,PrasanthaChandra,1–4,66,214–15,224,230,241Mahammakhamfestival,15Maisky,J.,366Majlis,245Mallet,C.,191–92MallikeswararTemple,342Manchester,Universityof,164,366ManchesterGuardian,371MannersandMoralsAmongtheVictorians(Himmelfarb),140Marlborough,112Marne,Battleofthe,212Marshall,Francis,124MaskofShame,The(Wurmser),50MathematicalAssociation,155

MathematicalGazette,47,152–53Mathematician’sApology(Hardy),147,152,285,369Mathematician’sMiscellany,A(Littlewood),163MathematischeZeitschrifi(journal),370MatlockHouseSanatorium,269–71,290,294,296–300,306,311Maxwell,JamesClerk,130,137,150Maynard,Catherine,118Mechanics,statistical,348–49Mendel,Gregor,146Mendelssohn,Felix,344MendipHillsSanatorium,265,269,270MenofMathematics(Bell),65Michelangelo,350Michelson,Albert,291Middlemast,E.W.,94,101,102,195,315Milton,John,345Mittag-Leffler,Gösta,363,370Modularequations,204,210,348Modularfunctions,elliptic,247Moghulempire,32Mordell,LouisJ.,169,170,226,253,254,337,338,365Moore,G.E.,137,147,276–77MuchAdoAboutNothing(Shakespeare),142Muslims,32,33,86MuthalpietHighSchool,342Muthu,Dr.,196,265Myers(philosopher),287

Nagoya(ship),314,321NamberumalChetty,326Nandy,Ashis,68,335,371NanjundaRao,M.C.,317Narasimha,101,106NarasimhaIyengar,K.,188,195,318,328–29NarayanaIyengar,K.,19,74NarayanaIyengar,M.T.,91,188NarayanaIyer,S.,95–98,102–3,106,174,178,180,182,184–85,189–90,195–96,210,219,229,258,317,

332,357Narayanan,W.,333,352Narayanaswami,Dr.,74NarayanaswamyIyer,G.V.,327NarosaPublishingHouse,351NationalCadetCorpsofIndia,351NationalIndiaAssociation,200NationalUnionofScientificWorkers,366Nature(magazine),336,363,365Nayakkings,29Negativenumbers,59Nehru,Jawaharlal,99,330,351,354,356Nelson,Lord,278

Nestingofsquareroots,86–87Nevasa(ship),195–99,202,256,265,311,314,315Neville,Alice,200,202,259Neville,E.H.,46,47,57,65,75,78,150,169,171,177–79,184,186–87,190–92,198–202,211,212,238–40,

242,245,264,277,295,299,316,335–36,357–58NewCollege,Oxford,125,361–63NewIndia(newspaper),314Newman,Morris,340NewnhamCollege,Cambridge,142,238Newsome,David,143Newton,Isaac,2,89,131,148–51,165,209,239,256,287,345NewYorkTimes,The,373Nicholas,Tressilian,276Notationsystems,60–63

Bernoulli’s,90Newton’s,149

Non-ConscriptionFellowship,277Numbertheory,6,63,152,161,169,246,249,339,340,343

additive,303analytical,249,340probabilistic,338

Numericalcomputation,63

OhioStateUniversity,364OrdersofInfinity(Hardy),160,164,216Osler,William,268Oslo,Universityof,366OxfordUniversity,100,126,131,139,144,257,305,368

Hardyat,361–65,370

Pachaiyappa’sCollege,52–55,57,65,70,74,93,106,194,281,282,315,321ParliamentBill,157ParthasarathyTemple,179,186Partitions,246–52,280,290,291,301–6,311,323,337,339,340,343,348,350,370PassagetoIndia,A(Forster),138Pavlov,Ivan,291Peasants’Revolt,121PembrokeCollege,Cambridge,126PennsylvaniaStateUniversity,252Pentland,Lord,193,315Pepys,Samuel,126PeterhouseCollege,Cambridge,126Phillips,SamuelMandeville,311PhysiologyforBeginners(FosterandShore),54Pi,208–10,349Piaget,Jean,372Picasso,Pablo,157PicturePost(magazine),368PlanSeventeen,212Plastics,348

Plato,143PlatonicConceptionofImmortalityandItsConnexionwiththeTheoryofIdeas(Gaye),139Pollard,S.,231Polya,George,204,253,254Powerseries,248PresidencyCollege(Madras),52,53,73,74,77,78,84,94,101,102,175,178–87,195,256,315,329,330,332Primenumbers,46,61,173–74,250,252,338,343

theoryof,160–61,177,216–23,285PrincetonUniversity,278,346,364PrincipiaEthica(Moore),137,147PrincipiaMathematica(Newton),150,151PrinciplesofMathematics(Russell),152,291PrinciplesandPracticeofMedicine(Osler),268Probabilisticnumbertheory,338Probability,209ProfessorSrinivasaRamanujanInternationalMemorialCommittee,353Proofs,92,221–22

ofCarr’sequations,41–44,92andIndianmathematics,86ofRamanujan’s theorems,163, 167, 168, 173–74, 177, 181, 183, 184, 204, 219, 221–26, 255, 303, 305,

343–44,346,358–59Pryce,Maurice,278PsychologyofInventionintheMathematicalField,The(Hadamard),285Puremathematics,60,146–48,346–51,365,366

numbertheoryand,246Pyrometry,348

Quadraticequations,60QuarterlyJournalofMathematics,145,176,208QuatrièmeCongrèsdesMathématiciensScandinaves,252

Rademacher,Hans,252RadhakrishnaIyer,R.,32,74,78,264,281,321Radicals,nested,87Raghavan,S.,343Raghunathan,N.,76,195Railroads,Indian,48–49Rajagopalachari,C.V.,78–79,171,329Rajagopolan,T.K.,281Ram,Balak,347Ram,Dr.,290,298RamachandraRao,R.,78–82,84,92–94,96,97,102,171,174,187,188,194,199, 211, 230, 264, 279, 311,

315–17,322,328,329,331,334,356,357Ramakrishna,T.,185Ramakrishnan,S.,354Ramalingam,A.S.,200,290,296–300Raman,C.V.,336,351,355Ramanujachariar,N.,53Ramanujan:TwelveLecturesonSubjectsSuggestedByHisLifeandWork(Hardy),372Ramanujan,Srinivasa

arrangementsfortriptoEngland,187–94arrivalinLondon,199–200B.A.grantedto,234onBernoullinumbers,89–92,105birthof,11,17,36andBritishIndia,101–6bustof,352atCambridge,1–4,6,66,198,200–207,210–15,219–20,228–31,233–35,237–46,252–59,289–91,315,

330,351,357Carr’sinfluenceon,39–45casteof,21–23,112andCauveryRiver,9–10centennialofbirthof,340,351–53,356,358childhoodof,11–20,24collaborativepaperswithHardy,248–52,276,301,338,340,368CollectedPapersof,333,337–41,371andconflictsbetweenJanakiandKomalatammal,272–76,318–20,327correspondencewithHardy,159–79,181,183–84,187,191,198,209,216, 218–19, 222, 229, 248, 255,

323–24,328,337,341andcultofuselessness,148deathof,327–34,336,361–63disappearancesof,48–49,238,298documentaryfilmabout,4–5earlyeducationof,13–14,19,25formalismof,205–6andGreatWar,212,213,229,234atGovernmentCollege(Kumbakonam),45–47,49,51,53,54,90,101Hardy’swritingsabout,151,372inhighschool,25–28,101illnessesof,72,74,254,263–72,290,295–300,310–12,318,320–22,325–27importancetoHardyof,369–73inIndianMathematicalSocietyJournal,86–87,90–93influenceonmathematicsof,330,334,337–51jobhuntingby,73–76intuitionof,215–16,220,223–27,252letterstoBritishmathematicians,106–7,145,170inMadras,81–85atMadrasPortTrust,94,96–98marriageof,19,68–73,75,96–98,198,255atNamakkal,188–90natureofrelationshipwithHardy,271–72,275–80,289notebooksof,56–64,75–78,80,87,88,90,92,93,182,187,192,203–7,219,222,224–25,232,280,312,

313,326,337,341–42,344–46,350–52,370atPachaiyappa’sCollege,52–55,57,65,70papersby,207–10,229–32,245,291,301–6,343personalityof,76–77physicaldescriptionof,67atPresidencyCollege,178–87psychologyof,50–52RamachandraRaoand,78–82,92–94

refusaltogotoEngland,174,185–86,242returntoIndia,307,309–18,321asRoyalSocietyFellow,291–96,300,301,309,310,315spiritualityof,7,19–20,28–34,36–37,65–67,189–90,280–89,335suicideattemptby,272,294–96,300,310,355assvayambhu,357–59assymbolofIndia,353–57astutor,55–56,73,74vegetarianismof,22–23.54,191,194,229,240–41,244,258–59,263,266,268,298–99onvoyagetoEngland,195–99Westernizationof,194–95,244

RamanujanInstitute,355RamanujanMathematicalSociety,351Ramanujan’sMasterTheorem,182RamaRaoSahib,C.S.,328Ramaseshan,S.,341,348RamaswamiIyer,N.,93RamaswamiIyer,V.,77–78,81,85,86,171,188,189,357Ramayana,12,20,29Rangammal(Ramanujan’sgrandmother),19,36,97Rangaswami,T.V.,353Rangaswamy,(Janaki’sfather),70–71Ranjitsinjhi,171Rankin,Robert,344–46Rationalnumbers,59,154Rayleigh(mathematician),150Richardson,George(“Dick”),124Richardson,Sarah,122Riemann,GeorgFriedrichBernhard,147Riemannzetafunction,219–20,348Riesz,Marcel,365,367Rigor,151–54,224–26,366Robbins,LionelCharles,144,254Rogers,LeonardJames,168,305–6,323Rogers-Ramanujanidentities,306,313,349Rook,Graham,A.W.,268Rose,Jasper,361Ross,EdwardB.,74–75,101,182Routh,E.J.,132RoyalAirForceBomberCommand,339RoyalArtillery,250RoyalCollegeofPhysicians,258RoyalGarrisonArtillery,228RoyalGeographicSociety,114RoyalIrishLancers,193RoyalSociety,106,111,134,145,170,174,215,250,321,344,353

Hardyelectedto,145Hardyhonoredby,366,369Proceedingsof,271Ramanujanelectedto,291–96,300,309,315

RoyalSocietyofMedicine,Proceedingsof,146Rudra,Ila,237–38RugbySchool,112,120Ruskin,John,124Russell,Bertrand,2,132,137,138,141,149,152,157,224,245,262,277,290,291,361,367Russell,Dora,262,264Russia

AcademyofSciencesof,366inGreatWar,214,306

RussianBallet,157Ruth,Babe,364Rutherford,Ernest,2

SacreduPrintemps,Le(Stravinsky),157Sadagopan(Ramanujan’sbrother),12St.Aubyn,Alan,124–25,172St.Catherine’sSchool,115,117St.John’sCollege,Cambridge,116,119,126,142,250St.Joseph’sCollege(Trichinopoly),95St.MatthewPassion(Bach),344St.Paul’sSchool,163Saldhana,Professor,79,80Salie(mathematician),344Sandhurst,193SarangapaniIyengar,K.,50,74,321Sarangapanitemple,28–29Satthianadhan,S.,256–57SatyapriyaRao,66Schelling,FreidrichWilhelmJosephvon,287SchlieffenPlan,212Science,146ScotlandYard,294Selberg,Atle,252,338–39,350SepoyMutiny,99Series,145

convergent,88divergent,80,174,203hypergeometric,57,80,167,204power,248summationof,182theoremson,161SeealsoInfiniteseries

Seshan(Ramanujan’sbrother),12SeshuIyer,P.V.,47,77–78,81,82,106,188,210,279,295,310,325,327,334,336SexualInversion(EllisandSymonds),140Shaivism,34Shakespeare,William,148,351Shamesyndrome,50–51Shastras,31Shaw,GeorgeBernard,300

Shaw,HaroldBatty,270Shelley,PercyBysshe,350Shore,LewisE.,54SidneySussexCollege,Cambridge,128SingaraveluMudaliar,P.,53,74,106,315Singularmodule,57Snow,C.P.,109,111–12,114,116,119,123,135,144,151,152,161,162,165,171–72,184,207,253,279,

283,291–92,361,363–65,367–69,372Snow,Philip,369SocietyfortheProtectionofScienceandLearning,365Sofia,Universityof,366SomeEarlyImpressions(Stephen),142SouthAfrica,341SouthIndianRailways,49,95Spring,SirFrancis,95–98,101–4,106,174–76,193–95,233,317,326Squareroots,nestingof,86–87Squaringthecircle,209Sravanamceremony,320Srinivasa(Ramanujan’sfather),11,17–18,36,47,55,65,68,70,275,329,331,332Srinivasacharya,T.,93SrinivasaIyengar,195SrinivasaIyengar,R.,275,316,322,327Srinivasan,K.S.,92Srinivasan,P.K.,335,342–43Srinivasan,R.,282SrinivasaRaghavan,S.,332SrinivasaRamanujanTrust,352Statisticalmechanics,348–49Stephen,J.T.,136Stephen,Leslie,142Stone,J.H.,196Strachey,Lytton,109,111,136–39,141,157Strandmagazine,214,224Stravinsky,Igor,157Stringtheory,348,349Stolz(mathematician),153StudyofBritishGenius,A(Ellis),126Symonds,JohnAddington,140,143SynopsisofElementaryResultsinPureandAppliedMathematics,A(Carr),39–46,57,58,90,173,203,356Subbanarayanan,N.,195–96Subbier,Ganapathi,27Subramanian,S.M.,55,190,224–25,230,259,264,271Sudras,20,21SundramIyer,P.R.,178Sykes,Christopher,5

Tagore,Rabindranath,330TamariskTree,The(Russell),262Tamillanguage,13,14,16,24,25,33TataInstituteofFundamentalResearch,341,343

Tauconjecture,343–44Teachers’College,78Telugulanguage,14,25Tennyson,Alfred,Lord,2,137,239Thackeray,WilliamMakepeace,2,64,81,239Thambuswami,Mudaliar,25TheoryofFunctions(Forsyth),150TheoryofInfiniteSeries(Brownwich),104TheosophicalSociety,314Thetafunctions,168,248,323–25,336,345,348ThomastheApostle,St.,82–83Thomson,J.J.,130,131,150,292,361Thomson,William,131Thorwaldsen(sculptor),345TimeMachine,The(Wells),291Timemagazine,373TimesLiterarySupplement,152TimesofLondon,130,157,161,262,363Tirunarayanan(Ramanujan’sbrother),12,332,342Titchmarsh,E.C.,151Todhunter(mathematician),153Topology,366TreatiseontheMathematicalTheoryofElasticity(Love),134Trenchwarfare,212,260Trevelyan,G.M.,135,311Trigonometry,27,44,46,50,53,86,119,161,205

infiniteseriesin,88–89piin,208

Trilling,Lionel,141TrinityCollege,Cambridge,2,6,109,111,112,123,127–28,136,139,141,145,151,164,169,174,183–84,

192,198,243,264,295,329,345,352,361,362,368Apostlesfrom,138duringGreatWar,212–13,227,259–61,263,276–77mealsat,239–41Nevilleat,186novelabout,124–25,172Ramanujanat,198,233,239,290,294,299–301,309–11,315retirementpolicyof,365scholarshipsto,125–26SundayEssaySociety,245

Tripossystem,39–40,124,128–36,149,150,155–56,162,170,172,186,214,228,233,260,305,370Tuberculosis,258,265–71,290,297,321,325,327–28Tuchman,Barbara,212Turing,Alan,139,278TwelfthNight(Shakespeare),136,137,139Tyndall,John,124

UnionofDemocraticControl,228,277UnitedStates

inGreatWar,307,311

tuberculosisin,267UniversityCollege(London),104Untouchables,20,21Upanishads,31UppiliapanKoiltemple,31Uselessness,cultof,146–48,346

Vaishnavism,21,34,179,189,317Vaisyas,20Veblen,Oswald,364Vedas,31Vedicceremonies,85–86Victoria,QueenofEngland,5,101,113,136VietnamWar,277Vijayalakshmi,70Vijayathasami(openingdayofschool),13Vikramadityalegends,20ViswanathaSastri,K.S.,56,73–74,199,317Vijayaraghavan,T.,330Viswanathan,R.,354

Wakeford,E.K.,261Walker,Gilbert,174–76,178Wallis,John,209Ward,J.T.,119–20WaroftheWorlds(Wells),291Watson,G.N.,168,204,324–25,337,344,346,350Webb,R.R.,132Weber(mathematician),210Weierstrass(mathematician),150Weinberg,Wilhelm,146Wells,H.G.,291–92Whitehead,AlfredNorth,137,291WhitePlague,The(Dubos),270Whittaker,J.M.,344–46Wiener,Norbert,151,157,246,367Wilde,Oscar,140William,PrinceofNormandy,133Wilson,B.M.,4,63,204,245,337,346WilsonLibraryBulletin,356WinchesterSchool,112,120–26,128,133,140,142,172,261,339,362,368Wingfield,Dr.,266Wisconsin,Universityof,280,344,350Woolf,Leonard,111,112,136,137,139,141,157,199,311Woolf,Virginia,111,136,141,300Woolwich,250Wootton,Barbara,261Wordsworth,William,101,126WorldWarI,seeGreatWarWorldWarII,339–40,368,369

Wren,Christopher,2,212,291,345Wright,E.M.,302,339Wurmser,Leon,50–51Wykeham,Williamof,126,362

Young,Laurence,227,243,361,365Young,W.H.,361

Zero,66,85Zetafunction,216–20,227,348Ziman,John,361

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Themanwhonewinfinity:alifeofthegeniusRamanujan/RobertKanigel.p.cm.Originallypublished:NewYork:CharlesScribner’sSons,1991.Includesbibliographicalreferencesandindex.ISBN:0-671-75061-51. Ramanujan, Aiyangar Srinivasa, 1887–1920. 2. Hardy, G. H. (Godfrey Harold), 1877–

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ISBN:978-1-43912186-3(eBook)Theauthorgratefullyacknowledgespermissionfromthefollowingsourcestoreprintmaterialintheircontrol:TheMasterandFellowsofTrinityCollege,Cambridge,forS.RamanujanandG.H.Hardydocuments residing inTrinityLibrary;St.Catherine’sSchoolmagazine, 1933, forG.E.Hardypoem;ReadingUniversityArchivesforE.H.Nevillemanuscript;FreemanDysonforhis lettertoC.P.Snow;CambridgeUniversityPressformaterialfromAMathemetician’sApologybyG.H.Hardy,copyright1940;TheLondonMathematicalSocietyforG.H.HardydocumentsresidinginTrinityLibrary;theSyndicsofCambridgeUniversityLibrary forS.Ramanujandocuments residing inCambridgeUniversityLibrary.Author’s failure toobtainanecessarypermissionfortheuseofanyothercopyrightedmaterialincludedinthisworkisinadvertentand will be corrected in future printings following notification in writing to the Publisher of such omissionaccompaniedbyappropriatedocumentation.