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Contents

Cover

TitlePage

Copyright

Dedication

Introduction

1DoNotDisturbMyCircles•Archimedes

2MasteroftheWay•LiuHui

3DixitAlgorismi•Muhammadal-Khwarizmi

4InnovatoroftheInfinite•MadhavaofSangamagrama

5TheGamblingAstrologer•GirolamoCardano

6TheLastTheorem•PierredeFermat

7SystemoftheWorld•IsaacNewton

8MasterofUsAll•LeonhardEuler

9TheHeatOperator•JosephFourier

10InvisibleScaffolding•CarlFriedrichGauss

11BendingtheRules•NikolaiIvanovichLobachevsky

12RadicalsandRevolutionaries•ÉvaristeGalois

13EnchantressofNumber•AugustaAdaKing

14TheLawsofThought•GeorgeBoole

15MusicianofthePrimes•BernhardRiemann

16CardinaloftheContinuum•GeorgCantor

17TheFirstGreatLady•SofiaKovalevskaia

18IdeasRoseinCrowds•HenriPoincaré

19WeMustKnow,WeShallKnow•DavidHilbert

20OverthrowingAcademicOrder•EmmyNoether

21TheFormulaMan•SrinivasaRamanujan

22IncompleteandUndecidable•KurtGödel

23TheMachineStops•AlanTuring

24FatherofFractals•BenoitMandelbrot

25OutsideIn•WilliamThurston

Abouttheauthor

Bythesameauthor

MoreadvancepraiseforSignificantFiguresMathematicalPeople

FurtherReading

Notes

Index

ToJohnDavey,editorandfriend(19April1945–21April2017)

Introduction

ALL BRANCHES OF SCIENCE can trace their origins far back into the mists ofhistory,butinmostsubjectsthehistoryisqualifiedby‘wenowknowthiswaswrong’ or ‘this was along the right lines, but today’s view is different’. Forexample,theGreekphilosopherAristotlethoughtthatatrottinghorsecanneverbeentirelyofftheground,whichEadweardMuybridgedisprovedin1878usinga line of cameras linked to tripwires. Aristotle’s theories of motion werecompletelyoverturnedbyGalileoGalileiandIsaacNewton,andhistheoriesofthemindbearnousefulrelationtomodernneuroscienceandpsychology.

Mathematics isdifferent. Itendures.When theancientBabyloniansworkedouthow tosolvequadraticequations–probablyaround2000BC, although theearliest tangible evidence dates from 1500 BC – their result never becameobsolete.Itwascorrect,andtheyknewwhy.It’sstillcorrecttoday.Weexpresstheresultsymbolically,butthereasoningisidentical.There’sanunbrokenlineofmathematicalthoughtthatgoesallthewaybackfromtomorrowtoBabylon.WhenArchimedesworkedout thevolumeofa sphere,hedidn’tusealgebraicsymbols,andhedidn’tthinkofaspecificnumberπaswenowdo.Heexpressedthe result geometrically, in terms of proportions, as was Greek practice then.Nevertheless,hisanswerisinstantlyrecognisableasbeingequivalenttotoday’sπr3.To be sure, a few ancient discoveries outside mathematics have been

similarly long-lived. Archimedes’s Principle that an object displaces its ownweightofliquidisone,andhislawoftheleverisanother.SomepartsofGreekphysics and engineering live on too. But in those subjects, longevity is theexception, whereas in mathematics it’s closer to the rule. Euclid’s Elements,laying out a logical basis for geometry, still repays close examination. Itstheoremsremaintrue,andmanyremainuseful.Inmathematics,wemoveon,butwedon’tdiscardourhistory.

Beforeyouallstarttothinkthatmathematicsisburyingitsheadinthepast,Ineedtopointouttwothings.Oneisthattheperceivedimportanceofamethodoratheoremcanchange.Entireareasofmathematicshavegoneoutoffashion,orbecomeobsoleteasthefrontiersshiftedornewtechniquestookover.Butthey’restilltrue,andfromtimetotimeanobsoleteareahasundergonearevival,usuallybecauseofanewlydiscoveredconnectionwithanotherarea,anewapplication,or a breakthrough inmethodology.The second is that asmathematicianshavedeveloped their subject, they’ve not only moved on; they’ve also devised agiganticamountofnew,importantbeautiful,andusefulmathematics.

That said, the basic point remains unchallenged: once a mathematicaltheoremhasbeencorrectlyproved,itbecomessomethingthatwecanbuildon–forever.Even thoughourconceptofproofhas tightenedupconsiderablysinceEuclid’sday,togetridofunstatedassumptions,wecanfillinwhatwenowseeasgaps,andtheresultsstillstand.

Significant Figures investigates the almost mystical process that brings newmathematicsintobeing.Mathematicsdoesn’tariseinavacuum:it’screatedbypeople.Amongthemaresomewithastonishingoriginalityandclarityofmind,thepeopleweassociatewithgreatbreakthroughs–thepioneers,thetrailblazers,the significant figures. Historians rightly explain that the work of the greatsdepended on a vast supporting cast, contributing tiny bits and pieces to theoverall puzzle. Important or fruitful questions can be stated by relativeunknowns;majorideascanbedimlyperceivedbypeoplewholackthetechnicalability to turn them into powerful new methods and viewpoints. Newtonremarkedthathe‘stoodontheshouldersofgiants’.Hewastosomeextentbeingsarcastic;severalofthosegiants(notablyRobertHooke)werecomplainingthatNewtonwasnotsomuchstandingontheirshouldersastreadingontheirtoes,bynotgivingthemfaircredit,orbytakingthecredit inpublicdespitecitingtheircontributionsinhiswritings.However,Newtonspoketruly:hisgreatsynthesesofmotion, gravity, and light dependedon a hugenumber of insights fromhisintellectual predecessors. Nor were they exclusively giants. Ordinary peopleplayedasignificantparttoo.

Nevertheless, the giants stand out, leading the way while the rest of usfollow.Throughthelivesandworksofaselectionofsignificantfigures,wecangaininsightintohownewmathematicsiscreated,whocreatedit,andhowtheylived.Ithinkofthemnotjustaspioneerswhoshowedtherestofustheway,but

astrailblazerswhohackedtraversablepathsthroughthetangledundergrowthofthe sprawling jungle ofmathematical thought. They spentmuch of their timestruggling through thornbushesandswamps,but fromtime to time theycameacrossaLostCityoftheElephantsoranElDorado,uncoveringpreciousjewelshiddenamong theundergrowth.Theypenetrated regionsof thoughtpreviouslyunknowntohumankind.

Indeed, they created those regions. The mathematical jungle isn’t like theAmazonRainforest or theAfricanCongo.Themathematical trailblazer isn’t aDavidLivingstone,hackingaroutealongtheZambeziorhuntingforthesourceof the Nile. Livingstone was ‘discovering’ things that were already there.Indeed,thelocalinhabitantsknewtheywerethere.Butinthosedays,Europeansinterpreted ‘discovery’ as ‘Europeans bringing things to the attention of otherEuropeans.’ Mathematical trailblazers don’t merely explore a pre-existingjungle.There’sasenseinwhichtheycreatethejungleastheyproceed;asifnewplants are springing to life in their footsteps, rapidly becoming saplings, thenmighty trees.However, it feels as if there’s apre-existing jungle, becauseyoudon’tgettochoosewhichplantsspringtolife.Youchoosewheretotread,butyoucan’tdecideto‘discover’aclumpofmahoganytreesifwhatactuallyturnsupthereisamangroveswamp.

This,Ithink,isthesourceofthestillpopularPlatonistviewofmathematicalideas:thatmathematicaltruths‘really’exist,buttheydosoinanidealforminsomesortofparallel reality,whichhasalwaysexistedandalwayswill. In thisview,whenweprove a new theoremwe just findoutwhat has been there allalong.Idon’tthinkthatPlatonismmakesliteralsense,butitaccuratelydescribestheprocessofmathematicalresearch.Youdon’tgettochoose:allyoucandoisshakethebushesandseeifanythingdropsout.InWhatisMathematics,Really?ReubenHershoffersamorerealisticviewofmathematics:it’sasharedhumanmental construct. In this respect it’s much like money. Money isn’t ‘really’lumpsofmetalorpiecesofpaperornumbersinacomputer;it’sasharedsetofconventions about how we exchange lumps of metal, pieces of paper, andnumbersinacomputer,foreachotherorforgoods.

Hershoutragedsomemathematicians,whozoomedinon‘humanconstruct’and complained that mathematics is by no means arbitrary. Social relativismdoesn’t hack it. This is true, but Hersh explained perfectly clearly thatmathematics isn’t any human construct. We choose to tackle Fermat’s LastTheorem, but we don’t get to choose whether it’s true or false. The humanconstruct that is mathematics is subject to a stringent system of logical

constraints, and somethinggets added to theconstructonly if it respects thoseconstraints.Potentially,theconstraintsallowustodistinguishtruthfromfalsity,butwedon’tfindoutwhichofthoseappliesbydeclaimingloudlythatonlyoneof them is possible. The big question is: which one? I’ve lost count of thenumberoftimessomeonehasattackedsomecontroversialpieceofmathematicsthattheydislikebypointingoutthatmathematicsisatautology:everythingnewis a logical consequence of things we already know. Yes, it is. The new isimplicitintheold.Butthehardworkcomeswhenyouwanttomakeitexplicit.Ask Andrew Wiles; it’s no use telling him that the status of Fermat’s LastTheoremwasalwayspredeterminedbythelogicalstructureofmathematics.Hespentsevenyearsfindingoutwhatitspredeterminedstatusis.Untilyoudothat,being predetermined is as useful as asking someone the way to the BritishLibraryandbeingtoldthatit’sinBritain.

SignificantFigures isn’tanorganisedhistoryof thewholeofmathematics,butI’vetriedtopresentthemathematicaltopicsthatariseinacoherentmanner,sothat the concepts build up systematically as the bookproceeds.On thewhole,this requires presenting everything in roughly chronological order.Chronologicalorderbytopicwouldbeunreadable,becausewe’dbeperpetuallyhopping from one mathematician to another, so I’ve ordered the chapters bybirthdateandprovidedoccasionalcross-references.1

My significant figures are 25 in number, ancient and modern, male andfemale, eastern andwestern. Their personal histories begin in ancientGreece,with thegreatgeometerandengineerArchimedes,whoseachievementsrangedfromapproximating π and calculating the area and volumeof a sphere, to theArchimedean screw for raisingwater and a crane-likemachine for destroyingenemy ships.Next come three representatives of the far east, where themainmathematicalactionoftheMiddleAgestookplace:theChinesescholarLiuHui,thePersianmathematicianMuhammad ibnMusaAl-Khwarizmi,whoseworksgave us the words ‘algorithm’ and ‘algebra’, and the Indian Madhava ofSangamagrama, who pioneered infinite series for trigonometric functions,rediscoveredinthewestbyNewtonamillenniumlater.

The main action in mathematics returned to Europe during the ItalianRenaissance,whereweencounterGirolamoCardano,oneofthebiggestroguesevertogracethemathematicalpantheon.Agamblerandbrawler,Cardanoalsowroteoneof themost importantalgebratextseverprinted,practisedmedicine,

and led a life straight out of the tabloid press. He cast horoscopes, too. Incontrast, Pierre deFermat, famous for hisLastTheorem,was a lawyerwith apassionformathematicsthatoftenledhimtoneglecthislegalwork.Heturnednumbertheoryintoarecognisedbranchofmathematics,butalsocontributedtoopticsanddevelopedsomeprecursors tocalculus.Thatsubjectwasbrought tofruitionbyNewton,whosemasterwork ishisPhilosophiaeNaturalisPrincipiaMathematica (Mathematical Principles of Natural Philosophy), usuallyabbreviated toPrincipia. In it, he stated his laws of motion and gravity, andappliedthemtothemotionofthesolarsystem.Newtonmarksatippingpointinmathematicalphysics,turningitintoanorganisedmathematicalstudyofwhathecalledthe‘SystemoftheWorld’.

ForacenturyafterNewton, the focusofmathematicsshifted tocontinentalEuropeandRussia.LeonhardEuler,themostprolificmathematicianinhistory,turnedoutimportantpapersatajournalisticrate,whilesystematisingmanyareasof mathematics in a series of elegant, clearly written textbooks. No field ofmathematics evaded his scrutiny. Euler even anticipated some of the ideas ofJosephFourier,whoseinvestigationofthetransmissionofheatledtooneofthemostimportanttechniquesinthemodernengineer’shandbook:Fourieranalysis,which represents a periodic waveform in terms of the basic trigonometricfunctions ‘sine’ and ‘cosine’. Fourierwas also the first to understand that theatmosphereplaysanimportantroleintheEarth’sheatbalance.

Mathematics enters the modern era with the peerless researches of CarlFriedrichGauss, a strongcontender for thegreatestmathematicianof all time.Gaussbegan innumber theory, sealedhis reputation in celestialmechanicsbypredicting the reappearance of the newly discovered asteroidCeres, andmademajoradvancesregardingcomplexnumbers,least-squaresdatafitting,andnon-Euclideangeometry,thoughhepublishednothingonthelatterbecausehefearedit was too far ahead of its time and would attract ridicule. Nikolai IvanovichLobachevsky was less diffident, and published extensively on an alternativegeometry to that of Euclid, now called hyperbolic geometry. He and JanósBolyaiarenowrecognisedastherightfulfoundersofnon-Euclideangeometry,which can be interpreted as the natural geometry of a surface with constantcurvature. Gauss was basically right to believe that the idea was ahead of itstime,however,andneitherLobachevskynorBolyaiwasappreciatedduringhislifetime.WeroundoffthiserawiththetragicstoryoftherevolutionaryÉvaristeGalois, killed at the age of twenty in a duel over a young woman. Hemademajor advances in algebra, leading to today’s characterisation of the vital

conceptofsymmetryintermsoftransformationgroups.A new theme now enters the story, a trail blazed by the first female

mathematicianweencounter.Namely,themathematicsofcomputation.AugustaAdaKing,CountessofLovelace,actedasassistanttoCharlesBabbage,asingle-mindedindividualwhounderstoodthepotentialpowerofcalculatingmachines.HeenvisagedtheAnalyticalEngine,aprogrammablecomputermadeofratchetsandcogwheels, now the central gimmickof steampunk science fiction.Ada iswidelycreditedwithbeingthefirstcomputerprogrammer,althoughthatclaimiscontroversial.ThecomputerthemecontinueswithGeorgeBoole,whoseLawsofThought laiddowna fundamentalmathematical formalismfor thedigital logicoftoday’scomputers.

Asmathematicsbecomesmorediverse,sodoesourtale,hackingitswayintonew regions of the ever-growing jungle. Bernhard Riemann was brilliant atuncovering simple, general ideas behind apparently complex concepts. Hiscontributions include the foundations of geometry, especially the curved‘manifolds’ upon which Albert Einstein’s revolutionary theory of gravitation,GeneralRelativity,depends.Buthealsomadehugestepsinthetheoryofprimenumbers by relating number theory to complex analysis through his ‘zetafunction’.TheRiemannHypothesis, about thezerosof this function, isoneofthegreatestandmostimportantunsolvedproblemsinthewholeofmathematics,withamillion-dollarprizeforitssolution.

NextcomesGeorgCantor,whochangedthewaymathematiciansthinkaboutthe foundations of their subject by introducing set theory, and defined infiniteanaloguesofthecountingnumbers1,2,3,…,leadingtothediscoverythatsomeinfinities are bigger than others – in a rigorous,meaningful, and useful sense.Like many innovators, Cantor was misunderstood and ridiculed during hislifetime.

Our second woman mathematician now appears on the scene, theprodigiously talentedSofiaKovalevskaia.Her lifewasrathercomplicated, tiedup with Russian revolutionary politics and the obstacles that male-dominatedsocietyplacedinthepathofbrilliantfemaleintellectuals.It’samazingthatsheaccomplished anything in mathematics at all. In fact, she made remarkablediscoveriesinthesolutionofpartialdifferentialequations,themotionofarigidbody,thestructureoftheringsofSaturn,andtherefractionoflightbyacrystal.

Thestorynowgatherspace.Aroundtheturnofthenineteenthcentury,oneofthe world’s leading mathematicians was the Frenchman Henri Poincaré. Anapparent eccentric, he was actually extremely shrewd. He recognised the

importanceof thenascentareaof topology–‘rubber-sheetgeometry’ inwhichshapescanbedistortedcontinuously–andextendeditfromtwodimensionstothreeandbeyond.Heappliedittodifferentialequations,studyingthethree-bodyproblem forNewtoniangravitation.This ledhim todiscover thepossibilityofdeterministicchaos,apparently randombehaviour inanon-randomsystem.HealsocameclosetodiscoveringSpecialRelativitybeforeEinsteindid.

AsaGermancounterpart toPoincaréwehaveDavidHilbert,whosecareerdivides into five distinct periods. First, he took up a line of thought thatoriginatedwithBoole,about‘invariants’–algebraicexpressionsthatremainthesamedespitechangesincoordinates.Hethendevelopedasystematictreatmentof core areas of number theory. After that, he revisited Euclid’s axioms forgeometry, found themwanting, andaddedextraones toplug the logicalgaps.Next,hemovedintomathematicallogicandfoundations,initiatingaprogrammetoprovethatmathematicscanbeplacedonanaxiomaticbasis,andthat this isbothconsistent(nologicaldeductioncanleadtoacontradiction)andcomplete(every statement can either be proved or disproved). Finally, he turned tomathematical physics, coming close to beating Einstein to General Relativity,andintroducingthenotionofaHilbertspace,centraltoquantummechanics.

EmmyNoether isour thirdand final femalemathematician,who livedat atime when the participation of women in academic matters was still frownedupon by most of the incumbent males. She began, like Hilbert, in invarianttheory, and later worked with him as a colleague. Hilbert made strenuousattempts to smash the glass ceiling and secure her a permanent academicposition, with partial success. Noether blazed the trail of abstract algebra,pioneering today’s axiomatic structures such as groups, rings, and fields. Shealso proved a vital theorem relating the symmetries of laws of physics toconservedquantities,suchasenergy.

Bynowthestoryhasmoved into the twentiethcentury.Toshowthatgreatmathematicalabilityisnotconfinedtotheeducatedclassesofthewesternworld,we follow the life and career of the self-taught Indian genius SrinivasaRamanujan,whogrewup in poverty.His uncanny ability to intuit strangebuttrueformulaswasrivalled,ifatall,onlybygiantssuchasEulerandCarlJacobi.Ramanujan’sconceptofproofwashazy,buthecouldfindformulasthatnooneelsewouldeverhavedreamedof.Hispapersandnotebooksarestillbeingminedtodayforfreshwaysofthinking.

Twomathematicianswithaphilosophicalbentreturnustothefoundationsofthesubjectanditsrelationtocomputation.OneisKurtGödel,whoseproofthat

any axiom system for arithmetic must be incomplete and undecidabledemolished Hilbert’s programme to prove the opposite. The other is AlanTuring,whoseinvestigationsintotheabilitiesofaprogrammablecomputerledtoasimplerandmorenaturalproofoftheseresults.Heis,ofcourse,famousforhiscodebreakingworkatBletchleyParkduringWorldWarII.HealsoproposedtheTuringtestforartificialintelligence,andafterthewarheworkedonpatternsin animal markings. He was gay, and died in tragic and mysteriouscircumstances.

I decided not to include any living mathematicians, but to end with tworecentlydeceasedmodernmathematicians:onepureand theotherapplied (butalsounorthodox).ThelatterisBenoitMandelbrot,widelyknownforhisworkonfractals, geometric shapes that have detailed structure on all scales ofmagnification. Fractals often model nature far better than traditional smoothsurfaces suchas spheresandcylinders.Althoughseveralothermathematiciansworkedonstructuresthatwenowseeasfractal,Mandelbrotmadeagreat leapforwardbyrecognisingtheirpotentialasmodelsofthenaturalworld.Hewasn’tatheorem-prooftypeofmathematician;instead,hehadanintuitivevisualgraspofgeometry,which ledhim to see relationships and state conjectures.Hewasalso a bit of a showman, and an energetic promoter of his ideas. That didn’tendear him to some in the mathematical community, but you can’t pleaseeveryone.

Finally, I’ve chosen a (pure) mathematician’s mathematician, WilliamThurston.Thurston,too,hadadeepintuitivegraspofgeometry,inabroaderanddeepersensethanMandelbrot.Hecoulddotheorem-proofmathematicswiththebestofthem,thoughashiscareeradvancedhetendedtofocusonthetheoremsandsketchtheproofs.Inparticularheworkedintopology,wherehenoticedanunexpectedconnectionwithnon-Euclideangeometry.Eventually, thiscircleofideasmotivated Grigori Perelman to prove an elusive conjecture in topology,duetoPoincaré.HismethodsalsoprovedamoregeneralconjectureofThurstonthatprovidesunexpectedinsightsintoallthree-dimensionalmanifolds.

Inthefinalchapter,I’llpickupsomeofthethreadsthatweavetheirwaythroughthe25 storiesof these astonishing individuals, andexplorewhat they teachusaboutpioneeringmathematicians–whotheyare,howtheywork,wheretheygettheircrazyideas,whatdrivesthemtobemathematiciansinthefirstplace.

For now, however, I’d just like to add twowarnings. The first is that I’ve

necessarilybeen selective.There isn’t enoughspace toprovidecomprehensivebiographies,tosurveyeverythingthatmytrailblazersworkedon,ortoenterintofine details of how their ideas evolved and how they interacted with theircolleagues. Instead, I’ve tried to offer a representative selection of their mostimportant – or interesting – discoveries and concepts, with enough historicaldetail topaintapictureofthemaspeopleandlocatethemintheirsociety.Forsomemathematiciansofantiquity,eventhathastobeverysketchy,becausefewrecords about their lives (and often no original documents about their works)havesurvived.

The second is that the25mathematicians I’ve chosenarebynomeans theonlysignificantfiguresinthedevelopmentofmathematics.Imademychoicesformany reasons– the importanceof themathematics, the intrinsic interestofthearea,theappealofthehumanstory,thehistoricalperiod,diversity,andthatelusivequality, ‘balance’. Ifyour favouritemathematician isomitted, themostlikelyreasonislimitedspace,coupledwithawishtochooserepresentativesthatarewidelydistributedin thethree-dimensionalmanifoldwhosecoordinatesaregeography, historical period, and gender. I believe that everyone in the bookfully deserves inclusion, although one or twomay be controversial. I have nodoubt at all that many others could have been selected with comparablejustification.

1DoNotDisturbMyCircles

Archimedes

ArchimedesofSyracuse

Born:Syracuse,Sicily,c.287BC

Died:Syracuse,c.212BC

THE YEAR: 1973. The place: Skaramagas naval base, nearAthens.All eyes arefocusedonaplywoodmock-upofaRomanship.Alsofocusedontheship:theraysoftheSun,reflectedfromseventycopper-coatedmirrorsfiftymetresaway,eachametrewideandhalfashighagain.

Withinafewseconds,theshipcatchesfire.IoannisSakkas, amodernGreek scientist, is recreating a possiblymythical

piece of ancient Greek science. In the second century AD the Roman authorLucianwrotethatattheSiegeofSyracuse,around214–212BC,theengineerandmathematician Archimedes invented a device to destroy enemy ships by fire.Whether this device existed, and if so, how it worked, is highly obscure.Lucian’s story could just be a reference to the common use of fire arrows orburningragsshotfromacatapult,butit’shardtoseewhythiswouldhavebeenpresented as a new invention. In the sixth century, Anthemius of Trallessuggested,inhisBurningGlasses,thatArchimedeshadusedahugelens.Butinthemostprevalentlegend,Archimedesusedagiantmirror,orpossiblyanarrayofmirrorsarrangedinanarctoformaroughparabolicreflector.

The parabola is a U-shaped curve, well known to Greek geometers.Archimedescertainlyknewaboutitsfocalproperty:alllinesparalleltotheaxis,when reflected in the parabola, pass through the same point, called the focus.Whether anyone realised that a parabolic mirror would focus light (and heat)from theSun in the sameway is less certain, becauseGreekunderstandingoflight was rudimentary. But, as Sakkas’s experiment shows, Archimedeswouldn’tactuallyhaveneededaparabolicarrangement.Alotofsoldiers,eacharmedwithareflectingshield, independentlyaimingit todirect theSun’sraystowardsthesamepartoftheship,wouldhavebeenjustaseffective.

The practicality of what is often called ‘Archimedes’s heat ray’ has beenhotly debated. The philosopher René Descartes, a pioneer in optics, didn’tbelieveitcouldhaveworked.Sakkas’sexperimentsuggestsitmighthavedone,but his fake plywood ship was flimsy, and coated in a tar-based paint, so itwouldburneasily.Ontheotherhand, inArchimedes’stimeitwascommontocoat ships with tar to protect their hulls. In 2005 a bunch of MIT studentsrepeatedSakkas’sexperiment,eventuallysettingawoodenmock-upofashiponfire – but only after focusing the Sun’s rays on it for ten minutes while itremained totally stationary. They tried it again for the TV showMythbustersusing a fishing boat in San Francisco, and managed to char the wood andproduceafewflames,butitdidn’tignite.Mythbustersconcludedthatthemythwasbust.

Archimedes was a polymath: astronomer, engineer, inventor, mathematician,physicist.Hewasprobablythegreatestscientist(tousethemodernterm)ofhis

age.Aswellasimportantmathematicaldiscoveries,heproducedinventionsthatarebreathtakingintheirscope–theArchimedeanscrewforraisingwater,block-and-tackle pulleys to lift heavy weights – and he discovered Archimedes’sprinciple on floating bodies and the law (though not the apparatus, whichappeared much earlier) of the lever. He’s also credited with a second war-machine, the claw. Allegedly he used this crane-like device at the Battle ofSyracusetoliftenemyshipsfromthewaterandsinkthem.The2005televisiondocumentarySuperweapons of the AncientWorld built its own version of thedevice,anditworked.AncienttextscontainmanyothertantalisingreferencestotheoremsandinventionsattributedtoArchimedes.Amongthemisamechanicalplanetarycalculator,muchlikethefamedAntikytheramechanismofaround100BC,discoveredinashipwreckin1900–1901andonlyrecentlyunderstood.

WeknowverylittleaboutArchimedes.HewasborninSyracuse(Siracusa),ahistoricSiciliancitylocatedtowardsthesouthernendoftheisland’seastcoast.Itwasfoundedineither734or733BCbyGreekcolonists,supposedlyunderthesemi-mythical Archias when he exiled himself from Corinth. According toPlutarch,ArchiashadbecomeinfatuatedwithActaeon,ahandsomeboy.Whenhis advances were rejected he tried to kidnap the lad, and in the struggle,Actaeon was torn to pieces. His father Melissus’s pleas for justice wentunanswered,soheclimbedto the topofa templeofPoseidon,calleduponthegodtoavengehisson,andflunghimselfontotherocksbelow.Aseveredroughtand famine followed these dramatic events, and the local oracle declared thatonly vengeance would propitiate Poseidon. Archias got the message, exiledhimself voluntarily to avoid being sacrificed, headed for Sicily, and foundedSyracuse.LaterhispastcaughtupwithhimanywaywhenTelephus,whoasaboyhadalsobeenanobjectofArchias’sdesires,killedhim.

Thelandwasfertile,thenativesfriendly,andSyracusesoonbecamethemostprosperous and powerfulGreek city in the entireMediterranean. InThe SandReckoner, Archimedes says that his father was Phidias, an astronomer.According to Plutarch’sParallel Lives, he was a distant relative of Hiero II,tyrantofSyracuse.Asayoungman,Archimedes is thought tohavestudied inthe Egyptian city of Alexandria on the coast of the Nile delta, where heencounteredCononofSamosandEratosthenesofCyrene.AmongtheevidenceishisstatementthatCononwasafriend;alsotheintroductionstohisbooksTheMethod of Mechanical Theorems and the Cattle Problem are addressed toEratosthenes.

There are some tales about his death, too, andwe’ll come to those in due

course.

Archimedes’smathematicalreputationrestsonthoseworksthathavesurvived–allaslatercopies.QuadratureoftheParabola,whichtakestheformofaletterto his friend Dositheus, contains 24 theorems about parabolas, the final onegiving the area of a parabolic segment in terms of a related triangle. Theparabolafiguresprominentlyinhiswork.It’satypeofconicsection,afamilyofcurvesthatplayedamajorroleinGreekgeometry.Tocreateaconicsection,useaplanetocutadoublecone,formedbyjoiningtwoidenticalconesattheirtips.Therearethreemaintypes:theellipse,aclosedoval;theparabola,aU-shapedcurve;andthehyperbola,twoU-shapedcurvesbacktoback.

Thethreemaintypesofconicsection.

On Plane Equilibria consists of two separate books. It establishes somefundamentalresultsaboutwhatwenowcallstatics:thebranchofmechanicsthatanalysesconditionsunderwhichabodyremainsatrest.Thefurtherdevelopmentof this topic underpins the whole of civil engineering, making it possible tocalculatetheforcesthatactonthestructuralelementsofbuildingsandbridges,toensurethattheyreallydoremainatrest,ratherthanbucklingorcollapsing.

Thefirstbookconcentratesonthelawofthelever,whichArchimedesstatesas:‘Magnitudesareinequilibriumatdistancesreciprocallyproportionaltotheirweights.’Oneconsequenceisthatalongleveramplifiesasmallforce.PlutarchtellsusthatArchimedesdramatisedthis inaletter toKingHeiron:‘Givemeaplace to stand, and I will move the Earth.’ He’d have needed a very long,perfectly rigid, lever, but the main downside of levers is that although theapplied force is amplified, the far end of the lever travels a much smallerdistance than the forcedoes.Archimedescouldhavemoved theEarth throughthe same (very tiny) distance just by jumping. Nonetheless, a lever is very

effective,andsoisavariantthatArchimedesalsounderstood,thepulley.WhenascepticalHeironaskedhimtodemonstrate,Archimedes

fixedaccordinglyuponashipofburdenoutoftheking’sarsenal,whichcouldnotbedrawnoutofthedockwithoutgreat labourandmanymen;and, loadingherwithmanypassengersanda fullfreight,sittinghimselfthewhilefaroff,withnogreatendeavour,butonlyholdingtheheadofthepulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, assmoothlyandevenlyasifshehadbeeninthesea.

Thesecondbookismainlyaboutfindingthecentreofgravityofvariousshapes–triangle,parallelogram,trapezium,andsegmentofaparabola.On the Sphere and Cylinder contains results of whichArchimedeswas so

proud that hehad them inscribedonhis tomb.Heproved, rigorously, that thesurfaceareaofasphereisfourtimesthatofanygreatcircle(suchastheequatorofasphericalEarth);thatitsvolumeistwothirdsthatofacylinderfittingtightlyround the sphere; and that the area of any segment of the sphere cut off by aplane is the same as the corresponding segment of such a cylinder.His proofused a convoluted method known as exhaustion, which was introduced byEudoxus todealwithproportions involving irrationalnumbers,whichcan’tberepresented exactly as a fraction. Inmodern terms, he proved that the surfaceareaofasphereofradiusris4πr2anditsvolumeis πr3.

Mathematicians have a habit of presenting their final polished results inbeautifully organised fashion, while concealing the often messy and muddledprocess that led to them.We’re fortunate to have some extra insight into howArchimedesmadehisdiscoveriesabout thesphere, recorded inTheMethodofMechanicalTheorems.Thisworkwas long thought tobe lost,but in1906 theDanishhistorianJohanHeibergdiscoveredanincompletecopy,theArchimedespalimpsest.Apalimpsestisatextthatwasrubbedoutorwashedoffinantiquityto allow theparchmentor paper tobe reused.TheworksofArchimedeswerecollectedtogetherbyIsidorusofMiletusaround530inConstantinople(modernIstanbul), capital of theByzantineEmpire. Theywere copied in 950 by someByzantinescribe,atatimewhenLeotheGeometerwasrunningamathematicalschool studying Archimedes’s works. The manuscript made its way toJerusalem, where in 1229 it was disassembled, washed (not very effectively),foldedinhalf,andreboundtomakea177-pageChristianliturgy.

Inthe1840sthebiblicalscholarConstantinvonTischendorfcameacrossthistext, by now back in aGreek orthodox library inConstantinople, and noticedfaint tracesofGreekmathematics.He tookonepageawayanddeposited it in

Cambridge University Library. In 1899 Athanasios Papadopoulos-Kerameus,cataloguing theLibrary’smanuscripts, translated part of it.Heiberg realised itwasbyArchimedes,andtrackedthepagebacktoConstantinople,wherehewasallowed to photograph the entire document.He then transcribed it, publishingthe resultsbetween1910and1915, andThomasHeath translated the text intoEnglish.Afteracomplicatedseriesofevents,includinganauctioncontestedbyalawsuitoverownership,itwassoldtoananonymousAmericanfortwomilliondollars.The newownermade it available for study, and itwas subjected to avarietyofdigitalimagingtechniquestobringouttheunderlyingtext.

Thetechniqueofexhaustionrequiresadvanceknowledgeoftheanswer,andscholarshadlongwonderedhowArchimedesguessedtherulesfortheareaandvolumeofasphere.TheMethodprovidesanexplanation:

Certainthingsfirstbecamecleartomebyamechanicalmethod,althoughtheyhadtobeprovedbygeometryafterwardsbecausetheirinvestigationbythesaidmethoddidnotfurnishanactualproof.Butitisofcourseeasier,whenwehavepreviouslyacquired,bythemethod,someknowledgeofthequestions,tosupplytheproofthanitistofinditwithoutanypreviousknowledge.

Archimedesimagineshangingasphere,acylinder,andaconeonabalance,andthencuttingthemintoinfinitelythinslices,whichareredistributedinawaythatkeeps the balance level. He then uses the law of the lever to relate the threevolumes(thoseofthecylinderandconewereknown)anddeducestherequiredquantities. It has been suggested that Archimedes was pioneering the use ofactualinfinitiesinmathematics.Thismaybereadingtoomuchintoanobscuredocument,butit’sclearthatTheMethodanticipatessomeideasofcalculus.

Archimedes’sotherworks illustratehowdiversehis interestswere.OnSpiralsproves some fundamental results about lengths and areas related to theArchimedean spiral, the curve described by a point moving at uniform speedalong a line rotating at uniform speed.OnConoids and Spheroids studies thevolumesofsegmentsofsolidsformedbyrotatingconicsectionsaboutanaxis.OnFloatingBodiesistheearliestworkinhydrostatics,equilibriumpositions

of floating objects. It includes Archimedes’s principle: a body immersed in aliquid is subjected to a buoyancy force equal to theweight of fluid displaced.ThisprincipleisthesubjectofafamousanecdoteinwhichArchimedesisaskedtodeviseamethodtodeterminewhetheravotivecrownmadeforKingHieroII

istrulymadeofgold.Sittinginhisbath,heissuddenlystruckwithinspiration,becoming so excited that he rushes off down the street crying ‘Eureka!’ (I’vefound it!) – omitting to get dressed first. Public nudity would not have beenparticularlyscandalousinancientGreece,mindyou.Thetechnicalhighpointofthe book is a condition for a floating paraboloid to be stable, a forerunner ofbasicideasinnavalarchitectureonthestabilityandcapsizingofships.MeasurementofaCircleappliesthemethodofexhaustiontoprovethatthe

areaofacircleishalftheradiustimesthecircumference–πr2inmodernterms.Toprovethis,Archimedesinscribesandcircumscribesregularpolygonswith6,12, 24, 48, and 96 sides. By considering the 96-gons, he proves a resultequivalenttoanestimateforthevalueofπ:itliesbetween3 and3 .

TheSandReckoner is addressed toGelo II, tyrant of Syracuse, the son ofHeiroII.ThisaddsevidencethatArchimedeshadroyalconnections.Heexplainsitsobjective:

Therearesome,kingGelo,whothinkthatthenumberofthesandisinfiniteinmultitude…ButIwill trytoshowyou…that,ofthenumbersnamedbymeandgivenintheworkwhichIsenttoZeuxippus,someexceednotonlythenumberofthemassofsandequalinmagnitudetotheEarthfilledup,butalsothatofthemassequalinmagnitudetotheuniverse.

Here Archimedes is promoting his new system for naming large numbers bycombating the commonmisuse of the term ‘infinite’ tomean ‘very large’.Hehasaclearsenseofthedistinction.Histextcombinestwomainideas.Thefirstis an extension of the standard Greek number words to allow much largernumbersthanamyriadmyriad(100million,108).Thesecondisanestimateforthesizeoftheuniverse,whichhebasesontheheliocentric(Sun-centred)theoryofAristarchus.Hisfinalresultisthat,intoday’snotation,itwouldtakeatmost1063sandgrainstofilltheuniverse.

There’s a long recreational tradition in mathematics, featuring games andpuzzles. Sometimes these are just fun, and sometimes they’re light-heartedproblems that illuminate more serious concepts. The Cattle Problem raisesquestions that are still studied today. In 1773 Gotthold Lessing, a Germanlibrarian,cameacrossaGreekmanuscript:a44-linepoeminvitingthereadertocalculatehowmanycattlethereareintheSungod’sherd.ThetitleofthepoempresentsitasaletterfromArchimedestoEratosthenes.Itbegins:

Compute, O friend, the number of the cattle of the Sunwhich once grazed upon the plains ofSicily,dividedaccordingtocolourintofourherds,onemilk-white,oneblack,onedappledandoneyellow.Thenumberofbulls isgreater thanthenumberofcows,andtherelationsbetweenthemareasfollows.

Itthenlistssevenequationsalongthelinesof

whitebulls blackbulls+yellowbulls

andcontinues:

If thou canst give,O friend, the number of each kind of bulls and cows, thou art no novice innumbers, yet cannot be regarded as of high skill. Consider, however, the following additionalrelationsbetweenthebullsoftheSun:Whitebulls+blackbulls=asquarenumber,Dappledbulls+yellowbulls=atriangularnumber.Ifthouhastcomputedthesealso,Ofriend,andfoundthetotalnumberofcattle,thenexultasa

conqueror,forthouhastprovedthyselfmostskilledinnumbers.

Squarenumbersare1,4,9,16,andsoon,foundbymultiplyingawholenumberby itself. Triangular numbers are 1, 3, 6, 10 and so on, formed by addingconsecutivewholenumbers–forinstance,10=1+2+3+4.Theseconditionsconstitute what we now call a system of Diophantine equations, named afterDiophantusofAlexandria,whowroteaboutthemaroundAD250inArithmetica.The solution must be given in whole numbers, since the Sun god would beunlikelytohavehalfacowinhisherd.

Thefirstsetofconditions leads toan infinitenumberofpossiblesolutions,thesmallestgiving7,460,514blackbullsandcomparablenumbersoftheotheranimals.Thesupplementaryconditionsselectamongthosesolutions,andleadtoatypeofDiophantineequationknownasthePellequation(Chapter6).Thisasksfor integers x and y such that nx2 + 1 = y2 where n is a given integer. Forexample,whenn=2theequationis2x2+1=y2,withsolutionssuchasx=2,y= 3 and x = 12, y = 17. In 1965 HughWilliams, R.A. German, and CharlesZarnke found the smallest solution consistent with the two extra conditions,usingtwoIBMcomputers.It’sapproximately7·76×10206544.

There’s no way Archimedes could have found this number by hand, andthere’s no evidence that he had anything to dowith the problem, beyond thepoem’s title.The cattleproblemstill attracts the attentionofnumber theorists,andhasinspirednewresultsonthePellequation.

Thehistorical recordofArchimedes’s life is flimsy,butweknowa littlemoreabouthisdeath,assuminganyofthetalesisaccurate.Theyprobablycontainatleastagrainoftruth.

In the Second Punic War, around 212 BC, the Roman general MarcusClaudius Marcellus besieged Syracuse, capturing it after two years. Plutarchrelates that the elderlyArchimedeswas looking at a geometric diagram in thesand. The general sent a soldier to tell Archimedes to meet him, but themathematician protested that he hadn’t finishedworking on his problem. Thesoldier lost his temper and killed Archimedes with his sword; the sage’s lastwordswere allegedly ‘Donotdisturbmycircles!’Knowingmathematicians, Ifind this entirely plausible, but Plutarch gives another version in whichArchimedes tries to surrender to a soldier, who thinks the mathematicalinstrumentsheiscarryingarevaluableandslaughtershimtostealthem.Inbothversions, Marcellus was somewhat peeved at the death of this reveredmechanicalgenius.

Archimedes’s tombwas decorated with a sculpture depicting his favouritetheorem,fromOntheSphereandCylinder:asphereinscribedinacylinderhastwo thirds its volume and the same surface area. More than a century afterArchimedes’s death, theRoman oratorCicerowas a quaestor (state-appointedauditor) inSicily.Hearingof the tomb,heeventually found it in adilapidatedstateneartheAgrigentinegateinSyracuse.Heordereditsrestoration,whichlethimreadsomeofitsinscriptions,includingadiagramofthesphereandcylinder.

Today, the location of the tomb is unknown, and nothing appears to havesurvived. But Archimedes lives on through his mathematics, much of it stillimportantmorethantwothousandyearslater.

2MasteroftheWay

LiuHui

LiuHui

Flourished:CaoWei,China,thirdcenturyAD

THE ZHOU BI SUAN JING (ArithmeticalClassic of theGnomon and theCircularPathsofHeaven) is themostancientknownChinesemathematical text,datingfromtheperiodoftheWarringStates,400–200BC.Itopenswithaneatpieceofeducationalpropaganda:

Longago,RongFangaskedChenZi‘Master,IhaverecentlyheardsomethingaboutyourWay.Isit really true that your way is able to comprehend the height and size of the Sun, the areailluminated by its radiance, the amount of its dailymotion, the figures for its greatest and leastdistances,theextentofhumanvision,thelimitsofthefourpoles,theconstellationsintowhichthestarsareordered,andthelengthandbreadthofheavenandEarth?’‘Itistrue,’saidChenZi.RongFangasked,‘AlthoughIamnotintelligent,Master,Iwouldlikeyoutofavourmewithan

explanation.CansomeonelikemebetaughtthisWay?’Chen Zi replied, ‘Yes. All things can be attained to you by mathematics. Your ability in

mathematics is sufficient to understand such matters if you sincerely give repeated thought tothem.’

ThebookgoesontoderiveafigureforthedistancefromtheEarthtotheSun,using geometry. Its cosmologicalmodel was primitive: a flat Earth beneath aplane circular sky. But its mathematics was quite sophisticated. Essentially itusedthegeometryofsimilartriangles,appliedtoshadowscastbytheSun.

TheZhouBi shows the advanced state ofChinesemathematics around thetimeof theGreekHellenisticperiod, fromthedeathofAlexander theGreat in323BCto146BCwhentheRepublicofRomeaddedGreecetoitsempire.ThisperiodwasthepeakofancientGreekintellectualdominance;thetimeofmostofthe great geometers, philosophers, logicians, and astronomers of the classicalworld. Even under Roman dominion, Greece continued to make cultural andscientific advances until about AD 600, but the centres of mathematicalinnovationmovedtoChina,Arabia,andIndia.Thecuttingedgeofmathematicalprogressdidn’treturntoEuropeuntiltheRenaissance,althoughthe‘darkages’weren’tasdarkasthey’resometimespainted,andlesseradvancesweremadeinEuropetoo.

The Chinese advances were stunning. Until recently most histories ofmathematics adopted a Eurocentric viewpoint and ignored them, until GeorgeGheverghese Josephwrote about the earlymathematicsof theFarEast inTheCrestofthePeacock.AmongthegreatestoftheancientChinesemathematicianswasLiuHui.Adescendant of theMarquis ofZixiangof theHandynasty, helived in the state of CaoWei during the Three Kingdoms period. In 263, heeditedandpublishedabookwithsolutionstomathematicalproblemspresentedin thefamousChinesemathematicsbookJiuzhangSuanshu (NineChaptersontheMathematicalArt).

His works include a proof of Pythagoras’s Theorem, theorems in solid

geometry, an improvement on Archimedes’s approximation to π, and asystematic method for solving linear equations in several unknowns. He alsowrote about surveying, with especial application to astronomy. He probablyvisited Luoyang, one of the four ancient capitals of China, andmeasured theSun’sshadow.

EvidencefortheearliesthistoryofChinacomesfromafewlatertexts,suchasthe Han dynasty scribe Sima Qian’s vast Records of the Grand Historian(around110BC)andtheBambooAnnals,ahistoricalchroniclewrittenonslipsofbamboo,buriedinthegraveofKingXiangofWeiin296BCanddugupagaininAD 281.According to these sources,Chinese civilisation began in the thirdmillennium BC with the Xia Kingdom. Written records start with the Shangdynasty, which ruled from 1600–1046 BC and left the earliest evidence ofChinesecountingintheformoforaclebones–markedbonesusedforfortune-telling.AsuccessfulinvasionbytheZhouledtoamorestablestatewithafeudalstructure,whichbegantofallapartthreecenturieslaterasothergroupstriedtomusclein.

By476BCvirtualanarchyruled,aperiodknownastheWarringStates thatlastedovertwocenturies.TheZhouBiwaswrittenduringtheseturbulenttimes.Its main mathematical contents are what we now call Pythagoras’s Theorem,fractions, and arithmetic; it also includes a lot of astronomy. Pythagoras’sTheoremispresentedinaconversationbetweenDukeChouKungandthenobleShangKao.Theirdiscussionofrighttrianglesleadstoastatementofthefamoustheoremandageometricproof.ForatimehistoriansthoughtthatthisdiscoverybeatPythagorasbyhalfamillennium.Thegeneralviewtodayisthatitwasanindependentdiscovery,predatingPythagoras,butnotbymuch.

AnimportantsuccessorfromthesamegeneralperiodistheaforementionedJiuzhang,which contains awealth ofmaterial such as the extraction of roots,solutionofsimultaneousequations,areasandvolumes,andagainrighttriangles.AcommentarybyChangHeng inAD130gives theapproximationπ .ChaoChunChin’scommentaryonZhouBisometimeinthethirdcenturyADaddedamethodforsolvingquadraticequations.ThemostinfluentialdevelopmentfromJiuzhangwasmadebyChina’sgreatestmathematicianofantiquity,LiuHui inAD263.Heintroducedthebookwithanexplanation:

In the past, the tyrant Qin burnt written documents, which led to the destruction of classical

knowledge.Later,ZhangCang,MarquisofPeiping,andGengShouchang,Vice-Presidentof theMinistry of Agriculture, both became famous through their talent for calculation. Because theancient texts had deteriorated, ZhangCang and his team produced a new version removing thepoorpartsandfillinginthemissingparts.Thus,theyrevisedsomeparts,withtheresultthattheseweredifferentfromtheoldparts.

In particular, Liu Hui provided proofs that the book’s methods work, usingtechniques that today we wouldn’t consider rigorous, akin to those ofArchimedes inTheMethod.Andhesuppliedadditionalmaterialonsurveying,also published separately as Haidao Suanjing (Sea Island MathematicalManual).

ThefirstchapteroftheJiuzhangexplainshowtocalculatetheareasoffieldsofvariousshapes,suchasrectangles,triangles,trapeziums,andcircles.Itsrulesarecorrect,exceptforthecircle.Evenheretherecipeisright:multiplytheradiusbyhalfthecircumference.However,thecircumferenceiscalculatedas3timesthediameter,ineffecttakingπ=3.Asapracticalmatter,theruleunderestimatestheareabylessthan5percent.

Late in the firstcenturyBC the rulerWangMang instructed theastronomerandcalendar-makerLiuHsingtocomeupwithastandardmeasureforvolume.LiuHsingmadeaveryaccuratecylindricalbronzevessel, toact asa standardreferencemeasure.ThousandsofcopieswereusedalloverChina.Theoriginalvessel is now in a museum in Beijing, and its dimensions have led some tosuggest thatLiuHsing ineffectusedavalue forπ somewherearound3·1547.(Quitehowthefigurecanbeobtainedtothisdegreeofaccuracybymeasuringabronze pot is unclear, tome at least.) TheSui Shu (official history of the Suidynasty)containsastatementequivalenttoLiuHsinghavingfoundanewvalueforπ.LiuHuiremarkedthataroundthesametime, thecourtastrologerChangHeng proposed taking π to be the square root of 10,which is 3·1622.Clearlyimprovedvaluesforπwereintheair.

InhiscommentaryontheJiuzhang,LiuHuipointsoutthatthetraditional‘π= 3’ rule is wrong: instead of the circumference of the circle, it gives theperimeterofaninscribedhexagon,whichisvisiblysmaller.Hethencalculatedamore accurate value for the circumference (and, implicitly, for π). In fact, hewent further, describing a computational method to estimate π to arbitraryaccuracy. His approach was similar to that of Archimedes: approximate the

circlebyregularpolygonswith6,12,24,48,96,…sides.Inordertoapplythemethod of exhaustion, Archimedes used one sequence of approximatingpolygonsinscribedinthecircle,andasecondsequencefittingoutsideit.LiuHuiused only inscribed polygons, but at the end of the calculation he gave ageometricargumenttoplacebothlowerandupperboundsonthetruevalueofπ.Thismethodgivesarbitrarilyaccurateapproximations toπusingnothingmoredifficultthansquareroots.Thesecanbecalculatedsystematically;themethodislaboriousbutnomorecomplexthanlongmultiplication.Askilledarithmeticiancouldobtaintendecimalplacesofπinaday.

Later,aroundAD469,TsuCh’ungChihextendedthecalculationtoshowthat

3·1415926<π<3·1415927

Theresultwasrecorded,buthismethod,whichmayhavebeenexplainedinhislostworkSuShu(MethodofInterpolation),wasnot.Itcouldhavebeendoneby continuing Liu Hui’s calculation, but the book’s title suggests it involvedestimatingamoreaccuratevalue fromapairofapproximations,one toosmallandtheothertoobig.Methodslikethatcanbefoundinmathematicsrightuptothepresentday.Notsolongagotheyweretaughtinschools,forusewithtablesof logarithms. Tsu came up with two simple fractions approximating π: theArchimedean22/7,accuratetotwodecimalplaces,and355/113,accuratetosixdecimalplaces.Thefirstiswidelyusedtoday,andthesecondiswellknowntomathematicians.

One reconstruction ofLiuHui’s proof ofPythagoras’sTheorem, basedon theinstructions in his book, is an ingenious and unusual dissection. The righttriangleisshowninblack.Thesquareononesideissplitintwobyadiagonal(light grey). The other square is cut into five pieces: one small square (darkgrey),apairofsymmetricallyarrangedtrianglesthesameshapeandsizeastheoriginal right triangle (medium grey) and a pair of symmetrically arrangedtrianglesfillingtheremainingspace(white).Thenallsevenpiecesareassembledtomakethesquareonthehypotenuse.

Other,simplerdissectionscanalsobeusedtoprovethistheorem.

Possiblereconstructionof

LiuHui’sproofof

Pythagoras’sTheorem.

TheancientChinesemathematicianswereeverybitascapableastheirGreekcontemporaries, and the courseofChinesemathematics afterLiuHui’speriodincludes many discoveries that predate their appearance in Europeanmathematics.Forexample,theestimatesforπfoundbyLiuHuiandTsuCh’ungChihwerenotbetteredforathousandyears.

JosephexamineswhethersomeoftheirideasmighthavebeentransmittedtoIndiaandArabiaalongsidetradegoods,andthencepossiblyeventoEurope.Ifso, the laterEuropean rediscoveriesmightnothavebeenentirely independent.There were Chinese diplomats in India in the sixth century, and Chinesetranslations of Indian mathematics and astronomy books were made in theseventhcentury.AsregardsArabia,theProphetMuhammadissuedahadith–apronouncementwithreligioussignificance–saying‘Seeklearning,thoughitbeas far away asChina.’ In the fourteenth centuryArab travellers report formaltradelinkswithChina,andtheMoroccantravellerandscholarMuhammadibnBattutawroteaboutChinesescienceandtechnology,aswellasculture,inRihla(Journey).

We know that ideas from India and Arabia made their way to medievalEurope,asthenexttwochaptersillustrate.Soit’sbynomeansimpossiblethatChineseknowledgedidlikewise.TheJesuitpresenceinChinaintheseventeenthand eighteenth centuries inspired some of Leibniz’s philosophy, by way ofConfucius. There may well have been a complex network, transmittingmathematics,science,andmuchmore,betweenGreece,theMiddleEast,India,andChina. If so, theconventionalhistoryofwesternmathematicsmayrequiresomerevision.

3DixitAlgorismi

Muhammadal-Khwarizmi

MuhammadibnMusaal-Khwarizmi

Born:Khwarizm(modern-dayKhiva),Persia,c.780

Died:c.850

AFTERTHEDEATHoftheProphetMuhammadin632,controloftheIslamicworldpassedtoaseriesofcaliphs.Inprinciple,caliphswerechosenonmerit,sothecaliphate’spowersystemwasn’texactlyamonarchy.However, thecaliphwasverymuchincharge.By654underUthman,thethirdcaliph,thecaliphatehadbecomethelargestempiretheworldhadeverseen.Itsterritory(inpresent-daygeography) included theArabian Peninsula,NorthAfrica fromEgypt throughLibya to easternTunisia, theLevant, theCaucasus, andmuchofCentralAsiafromIranintoPakistan,Afghanistan,andTurkmenistan.

The first four caliphs constituted theRashidun caliphate, succeeded by theUmayyaddynasty,whichwasinturnsucceededbytheAbbasiddynasty,whichoverthrew the Umayyads with Persian assistance. The centre of government,originallyinDamascus,movedtoBaghdad,acityfoundedbyCaliphal-Mansurin762.Itslocation,closetoPersia,wasinpartdictatedbytheneedtorelyontheservicesofPersian administrators,whounderstoodhow thevarious regionsofthe IslamicEmpire interacted.Thepositionofvizierwascreated,allowing thecaliphtodelegateadministrativeresponsibility;thevizierinturndelegatedlocalmatters to regional emirs. The caliph’s position slowly became that offigurehead, with the real power residing in the vizier, but the early Abbasidcaliphsexertedconsiderablecontrol.

Around 800 Harun al-Rashid founded the Bayt al-Hikma, or House ofWisdom, a library in which writings from other cultures were translated intoArabic. His son al-Ma’mun pushed the project through to completion,assembling a huge collection of Greek manuscripts and a large number ofscholars.Baghdadbecameacentreforscienceandtrade,attractingscholarsandmerchants from places as distant as China and India. Among them wasMuhammadibnMusaal-Khwarizmi,akeyfigureinthehistoryofmathematics.

Al-KhwarizmiwasborninornearKhwarizminCentralAsia,nowKhivainUzbekistan.Hedidhismainworkunderal-Ma’mun,helping tokeepalive theknowledge that Europewas fast losing.He translated keyGreek and Sanskritmanuscripts,made his own advances in science,mathematics, astronomy, andgeography,andwroteaseriesofbooksthatwewouldnowdescribeasscientificbestsellers. On Calculation with Hindu Numerals, written around 825, wastranslated into Latin as Algoritmi de Numero Indorum, and it almostsinglehandedly spread the news of this amazing newway to do arithmetic tomedievalEurope.Alongtheway,AlgoritmibecameAlgorismi,andmethodsforcalculatingwiththesenumeralswerecalledalgorisms.Intheeighteenthcentury,thewordchangedtoalgorithm.

Hisal-Kitabal-mukhtasarfihisabal-jabrwa-l-muqabala(TheCompendiousBook onCalculation byCompletion andBalancing), written around 830,wastranslated into Latin in the twelfth century by Robert of Chester as LiberAlgebrae etAlmucabola.As a result,al-jabr,Latinised to ‘algebra’, became aword in itsown right. It now refers to theuseof symbols suchasx andy forunknown quantities, together with methods for finding those unknowns bysolvingequations,butthebookdoesn’tusesymbols.

TheAlgebracameaboutwhenCaliphal-Ma’munencouragedal-Khwarizmi towrite a popular book about calculation. Its author describes its purpose asexplaining

what is easiest and most useful in arithmetic, such as men constantly require in cases ofinheritance, legacies,partition, lawsuits, and trade, and inall theirdealingswithoneanother,orwherethemeasuringoflands,thediggingofcanals,geometricalcomputations,andotherobjectsofvarioussortsandkindsareconcerned.

Thatdoesn’tsoundmuchlikeanalgebrabook.Infact,algebraoccupiesonlyasmallpart.Al-Khwarizmibeginsbyexplainingnumbersinverysimpleterms–units, tens, hundreds – on the grounds that ‘when I consider what peoplegenerally want in calculating, I find that it always is a number’. It wasn’t alearnedtreatiseaimedatscholars;itwasapopularmathematicsbook,oneoftheeducational kind that tries to teach the general reader aswell as inform them.That’s what the caliph wanted, and that’s what he got. Al-Khwarizmi didn’tconsiderhisbooktobeatthefrontiersofresearchmathematics.Butthat’showwe now see the part on al-jabr. This is the deepest section of the book: asystematic development of methods for solving equations in some unknownquantity.Al-jabr,usuallytranslatedas‘completion’,referstotheadditionofthesame

termtoeachsideof theequation,with theaimofsimplifying it.Al-muqabala,‘balancing’, refers to the removalof a termonone sideof theequation to theother side (butwith theopposite sign)and to thecancellationof like termsonbothsides.

Forexample,iftheequation,expressedinmodernsymbolicnotation,is

x−3=7

thenal-jabrallowsustoadd3toeachside,obtaining

x=10

whichinthiscasesolvestheequation.Ifit’s

2x2+x+6=x2+18

then al-muqabala lets us move 6 on the left side to the right, as long as wesubtractit,whichyields

2x2+x=x2+12

Asecondal-muqabalaletsusmovex2ontherightovertotheleftandsubtractthat,getting

x2+x=12

whichissimpler,thoughnotyettheanswer.Irepeatthatal-Khwarizmidoesn’tusesymbols.Thefatherofalgebradidn’t

actuallydowhatmostofus thinkofasalgebra.Hestatedeverythingverbally.Specificnumberswereunits,theunknownquantitythatwecallxwasroot,andourx2wassquare.Thepreviousequationwouldread:squareplusrootequalstwelveunits

withoutanysymbols.So thenext job is toexplainhowtogofromthissortofequation to the answer. Al-Khwarizmi classifies equations into six types, atypicalcasebeing‘squaresandrootsequaltonumbers’,suchasx2+x=12.

Geometricsolutionof‘squaresandrootsequalto

numbers’.

Hethenproceeds toanalyseeach type in turn,solving theequationusingamixtureofalgebraicandgeometricmethods.Thustosolvetheequationx2+x=12,al-Khwarizmidrawsasquaretorepresentx2(left-handpicture).Toaddtherootx, headjoins four rectangles, eachof sidesx and¼ (middlepicture).The

resulting shape leads to the idea of ‘completing the square’ by adjoining foursmallsquares,eachofside¼andarea .Soheadds4× =¼to the left-handsideoftheequationaswell(right-handpicture).Bytheruleofal-jabr,hemust also add¼ to the right-hand side of the equation, which becomes 12¼.Now

(x+ )2=12¼=49/4=(7/2)2

Takethesquareroottoget

x+ =7/2

sox=3.Todaywewouldalsotakethenegativesquareroot,−7/2,andobtainasecondsolutionx=−4.Negativenumberswerestartingtobeunderstoodinal-Khwarizmi’sday,buthedoesn’tmentionthem.

BoththeBabyloniansandtheGreekswouldhaveunderstoodthisapproach,because they’dalreadydonemuch the same. In fact, there’s somecontroversyaboutwhether al-Khwarizmiwas aware ofEuclid’sElements.He shouldhavebeen,becauseal-Hajjaj,anotherscholarattheHouseofWisdom,hadtranslatedEuclidintoArabicwhenal-Khwarizmiwasayoungman.Butontheotherhand,themainjoboftheHouseofWisdomwastranslation,anditsworkerswerenotobligedtoreadtheworkstranslatedbytheirfellows.Somehistoriansarguethatal-Khwarizmi’sgeometryisnotpresentedinEuclid’sstyle,suggestingalackoffamiliarity. But, I repeat, the Algebra is a popular mathematics book, so itwouldn’t follow Euclid’s axiomatic style, even if al-Khwarizmi knew Euclidbackwards. In any case, completing the square goes right back to theBabylonians,andwasavailablefrommanysources.

Why,then,domanyhistoriansconsideral-Khwarizmithefatherofalgebra?Especially when he uses no symbols? There’s a strong competitor, the GreekDiophantus.HisArithmetica, a seriesofbookson the solutionof equations inwholeorrationalnumberswrittenaround250,didusesymbols.OneansweristhatDiophantus’smain interestwas number theory, and his symbols are littlemorethanabbreviations.Adeeperpoint,whichIfindmoreconvincing, is thatal-Khwarizmi often, though not always, provides general recipes. The typicalpresentational style of his predecessors was to use an example with specificnumbers,tellyouhowtosolvethat,andleaveyoutoinferthegeneralrule.Sotheupshotofthegeometricargumentabovemighthavebeenpresentedas‘take1,divideitby2toget½,squarethattoget¼,thenadd¼tobothsides’,leaving

the reader to infer that the general rule is to replace the initial 1 by half thecoefficient of x, square that, and add the result to both sides, and so on. Thislevel of generality would of course have been made clear by the tutor, andreinforcedbymakingthestudentworkoutlotsofotherexamples.

Sometimes al-Khwarizmi appears to do the same, but he tends to bemoreexplicit about the rule that’s being applied. So the deeper reason for creditinghimwiththeinventionofalgebraisthathefocusesmoreonthegeneralitiesofmanipulating algebraic expressions than on the numbers they represent. Forexample,hestatesaruleforexpandingaproduct

(a+bx)(c+dx)

in terms of the square x2, the root x, and numbers.We would write his rulesymbolicallyas

ac+(ad+bc)x+(bd)x2

andthisiswhathestates,verbally,withoutusingspecificnumbersfora,b,c,ord.He is tellinghis readershowtomanipulategeneralexpressions innumbers,roots, and squares. They’re not thought of as coded versions of an unknownnumber,butasanewkindofmathematicalobject,whichyoucancalculatewithevenwhenyoudon’tknowtheactualnumbers.It’sthissteptowardsabstraction–ifweacceptitassuch–thatunderpinstheclaimthatal-Khwarizmiinventedalgebra.There’snothinglikeitinArithmetica.

Other topics in the book aremore prosaic: rules for areas and volumes offigures such as rectangles, circles, cylinders, cones, and spheres. Here al-Khwarizmifollows the treatment inHinduandHebrewtexts,andnothing in itlooksmuchlikeArchimedesorEuclid.Thebookendswithmoredown-to-earthmatters:anextensive treatmentofIslamicrulesfor the inheritanceofproperty,requiring division in various proportions, but nothing more complicatedmathematicallythanlinearequationsandbasicarithmetic.

Al-Khwarizmi’smost influentialwork, at the time hewrote it and for severalhundredyearsafterwards,wasOnCalculationwithHinduNumerals,which,asalreadyremarked,gaveustheword‘algorithm’.Thephrase‘dixitAlgorismi’–‘thusspokeal-Khwarizmi’–wasapotentargumentinanymathematicaldispute.Themasterhasspoken:heedhiswords.

Hindu numerals, of course, are the early versions of decimal notation, inwhichanynumbercanbewrittenasasequenceoftensymbols,0123456789. As the book’s title indicates, al-Khwarizmi gave credit to the Hindumathematicians,butsogreatwashisinfluenceinmedievalEuropethattheideahas becomeknown asArabic numerals (or sometimesHindu-Arabic,which isstillunfair to theHindus.)TheArabicworld’smaincontributionwasto inventits own number symbols, related to but distinct from the Indian ones, and todisseminate thenotation and encourage its use.The symbols for the tendigitshavechanged repeatedlywith thepassageof time,anddifferent regionsof themodernworldstillusedifferentsymbols.

Todayanalgorithmisastep-by-stepprocedurethatcomputessomespecificquantity,orproducessomespecificoutput,withaguaranteethatitgetstherightanswer and stops. ‘Keep trying numbers at random until one works’ isn’t analgorithm: if it gets an answer, it’s correct, but it might keep trying foreverwithout finding anything. For an early example of an algorithm, recall that aprimenumberhasnofactorsotherthanitselfand1.Thefirstfewprimesare2,3, 5, 7, 11, 13.Any other positivewhole number greater than 1 is said to becomposite. For example, 6 is composite because 6 = 2 × 3. The number 1 isdeemed to be special, and is called a unit in this context. The sieve ofEratosthenes, from around 250BC, is an algorithm forwriting down all primenumbersuptoagivenlimit,asfollows.Startbylistingpositivewholenumbersuptothatlimit.Removeallmultiplesof2except2,thenremoveallmultiplesofthenextsurvivingnumber3,asidefrom3itself, thendothesamefor thenextsurviving number 5, and so on. After a number of steps that’s less than thechosenlimit,theprocessendsbylistingpreciselytheprimenumbersuptothatlimit.

Algorithms have become central to modern life, because computers aremachines that run algorithms.Algorithms post cute cat videos to the internet,calculate your credit rating, decidewhich books to try to sell you, implementbillionsofcurrencyandstockmarkettradeseverysecond,andtrytostealyouronline banking password. Ironically, the place where algorithms are the mostsignificantfeatureofal-Khwarizmi’sworkisnotinOnCalculationwithHinduNumerals, though every method for arithmetical calculation is of course analgorithm. It’s his algebra book, whose claim to fame is the specification ofgeneral procedures to solve equations. Those procedures are algorithms, andthat’swhatmakesthemimportant.

Al-Khwarizmiwroteongeographyandastronomyaswellasmathematics.HisKitab surat al-ard (Book of the Description of the Earth) of 833 updates thepreviousstandardworkonthistopic,Ptolemy’sGeographyofaround150.Thisis a kind of do-it-yourself atlas of the then known world: outlines of thecontinents on three alternative types of coordinate grid, with instructions forwhere to putmajor cities and other prominent features. It also discusses basicprinciplesofmap-making.Hisrevisionexpandedthelist to2402locationsandcorrectedsomeofPtolemy’sdata,inparticularreducinghisoverestimateforthelength of the Mediterranean. While Ptolemy showed the Atlantic and Indianoceansasseassurroundedbyland,al-Khwarizmileftthemunbounded.Zij al-Sindhind (Astronomical Tables of the Sindhind), which dates from

around820,containsoverahundredastronomicaltables,mainlytakenfromtheworks of Indian astronomers. They include tables of the motion of the Sun,Moon,andthefiveplanets,togetherwithtablesoftrigonometricfunctions.Itisthought that he also wrote on spherical trigonometry, which is important innavigation.Risalafi istikhrajta’rikhal-yahud(ExtractionoftheJewishEra)isabouttheJewishcalendar,anddiscussestheMetoniccycle,a19-yearperiodthatis very close to a commonmultiple of the solar year and the lunarmonth. Inconsequence, solar and lunar calendars,which tend to diverge as time passes,almost come back into alignment every 19 years. It’s named after Meton ofAthens,whointroduceditin432BC.

Alongside the mathematicians of ancient China (Chapter 2) and India(Chapter 4), al-Khwarizmi’s achievements add to the weight of evidence thatduringtheMiddleAges,whenEurope’ssciencemostlystagnated,thecentreofscientific and mathematical advances moved to the Far and Middle East.Eventually, during the Renaissance, Europe woke up again, as we’ll see inChapter5.Al-Khwarizmihadblazedanewtrail,andmathematicswouldneverlookback.

4InnovatoroftheInfinite

MadhavaofSangamagrama

Irinnarappilly(or

Irinninavalli)Madhava

Born:Sangamagrama,Kerala,India1350

Died:India,1425

‘THE WATER IN HURRICANE RITA weighed as much as 100 million elephants.’Todaythemediaoftenuseelephantsasaunitofweight,nottomentionBelgiumand Wales as measures of area, Olympic swimming pools as measures ofvolume,andLondonbusesforlengthorheight.Sowhatdoyoumakeofthis?

Gods (33), eyes (2), elephants (8), snakes (8), fires (3) qualities (3), vedas (4), naksatras (27),elephants(8),andarms(2)–thewisesaythatthisisthemeasureofthecircumferencewhenthediameterofacircleis900,000,000,000.

Anythingspringtomind?Actually,it’satranslationofapoemaboutπ,writtenaround 1400 by Madhava of Sangamagrama, probably the greatest of the

medieval Indianmathematician-astronomers. The gods, elephants, snakes, andso on, are number symbols, whichwould have been drawn as small pictures.Collectively(workbackwardsthroughthelist)theyrepresentthenumber

282,743,388,233

which,dividedby900billion,gives

3·141592653592222…

Thisshouldlookmorefamiliar.Theratioconcernedisthegeometricdefinitionofπ,whichis

3·141592653589793…

Thetwofiguresagreeto11decimalplaces(roundingupthe589toget59forthetenth and eleventh decimal places). At the time this was one of the bestapproximations known.By 1430, the Persianmathematician Jamshid al-Kashihad broken the recordwith 16 decimal places inMiftah al-hisab (TheKey toArithmetic).

SomeofMadhava’sastronomical textshavesurvived,buthismathematicalwork is known only through later commentaries. The perennial problem ofgiving the great founder andmaster credit for results foundbyhis intellectualdescendants (so that, for example, anything discovered by a member of thePythagoreancult isattributedbydefault toPythagoras)meansthatwecan’tbesure exactlywhich resultswere discovered byMadhava. Inwhat follows, I’lltakehissuccessorsattheirword.

Hisgreatestachievementwastointroduceinfiniteseries,therebytakingearlystepstowardsanalysis.HefoundwhatinthewestisknownasGregory’sseriesfor the inverse tangent function, leading to expressions forπ as infinite series.Hismostimpressivediscoveriesareinfiniteseriesforthetrigonometricsineandcosinefunctions,whichwerefoundinthewestovertwohundredyearslater,byNewton.

LittleisknownofMadhava’slife.HelivedinthevillageofSangamagrama,andthis is conventionally appended to his name to distinguish him from otherMadhavas, such as the astrologer Vidya Madhava. The village had a templedevotedtoagodofthesamename.Itisbelievedtohavebeenlocatednearthe

modernBrahminvillageofIrinjalakuda.This isclose toCochin in thestateofKerala,alongthinregionnearthesoutherntipofIndia,sandwichedbetweentheArabianSeaalongitswesterncoastandtheWesternGhatmountainstotheeast.In late medieval times Kerala was a hotbed of mathematical research. Mostearlier Indian mathematics originated further north, but for some unknownreasonKeralaunderwentanintellectualrevival.Mathematicswasgenerallyseenas a branch of astronomy in ancient India, andMadhava founded the Keralaschoolofastronomyandmathematics.

This included a number of unusually proficient mathematicians.Parameshvara was a Hindu astronomer who used observations of eclipses tochecktheaccuracyof thecomputationalmethodsof theday.Hewroteat least25 manuscripts. Kelallur Nilakantha Somayaji wrote a major astronomy textTantrasamgraha in 1501, consisting of 432Sanskrit verses organised as eightchapters. In particular, it includes his modifications of the great IndianmathematicianAryabhata’stheoryofthemotionofMercuryandVenus.Healsowrote an extensive commentary Aryabhatiya Bhasya on other work ofAryabhata, inwhichhediscussedalgebra, trigonometry, and infinite series fortrigonometric functions. Jyesthadeva wrote Yuktibhasa, a commentary onTantrasamgrahathataddedproofsofitsmainresults.Someconsideritthefirstcalculustext.MelpathurNarayanaBhattathir,amathematicallinguist,extendedPanini’s axiomatic system of 3959 rules for Sanskrit grammar in Prkriya-sarvawom.HeiscelebratedforNarayaneeyam,asongofpraisetoKrishnastillinusetoday.

Trigonometry, the use of triangles formeasurement, goes back to the ancientGreeks, especially Hipparchus, Menelaus, and Ptolemy. There are two maintypesofapplication:surveyingandastronomy.(Later,navigationwasaddedtothelist.)Theessentialpointis thatdistancesareoftenhardtomeasuredirectly(in the case of astronomical bodies, impossible), but angles can be measuredwheneverthere’saclearlineofsight.Trigonometrymakesitpossibletodeducethelengthsofthesidesofatrianglefromitsangles,providedatleastonelengthisknown.Insurveying,onecarefullymeasuredaccessiblebaselineandalotofanglesleadtoanaccuratemap,andthesamegoesforastronomy,withtacticaldifferences.

LetABbeanarcofa

circleofradius1,

centreO.Thechordof

angleAOB(whosesize

is2θ)isthelengthofAB.Thesineofangle

AOC(whosesizeisθ)isthelengthofAC.

ThecosineofθisthelengthofOC,andthe

tangentisAC/OC.

TheGreeksworkedwith the chord of an angle, see the illustration above.Hipparchus produced the first table of chords in 140BC, and used it for bothplaneandsphericaltrigonometry.Thelatterisabouttrianglesformedbyarcsofgreat circles on a sphere, and it’s essential for astronomy because stars andplanetsappeartolieonthecelestialsphere,animaginaryspherecentredontheEarth.Moreprecisely,thedirectionstothesebodiescorrespondtopointsonanysuch sphere. In the second century Ptolemy included tables of chords in hisAlmagest,andhisresultswerewidelyusedforthenext1200years.

ThemathematiciansofancientIndiabuiltontheGreekworktomakemajoradvancesintrigonometry.Theyfounditmoreconvenienttousenotchords,butthecloselyrelatedsine(sin)andcosine(cos)functions,whichwestilldotoday.Sines first appeared in theSuryaSiddhanta, a seriesofHinduastronomy textsfrom about the year 400, and were developed by Aryabhata in Aryabhatiyaaround500.SimilarideasevolvedindependentlyinChina.TheIndiantraditionwas continued by Varahamihira, Brahmagupta, and Bhaskaracharya, whoseworksincludeusefulapproximationstothesinefunctionandsomeofthebasicformulas,suchasVarahamihira’s

whichisthetrigonometricinterpretationofPythagoras’sTheorem.Until recently scholars thought that Indian mathematics stagnated after

Bhaskaracharya,followedonlybycommentariesrehashingtheclassicalresults.OnlywhenBritain added India to its burgeoningempiredidnewmathematicsappearthere.ThismayhavebeentrueforlargepartsofIndia,butnotforKerala.Joseph2remarksthat‘thequalityofthemathematicsavailablefromthe[Keralaschool] texts… is of such a high level compared towhatwasproduced in theclassical period that it seems impossible for the one to have sprung from theother.’However, theonlycomparable ideasare thosedevelopedcenturies laterin Europe, so no plausible ‘missing link’ is evident. The Kerala school’sadvancesseemtohavebeensuigeneris.

Jyesthadeva’s commentary Yuktibhasa describes a series attributed toMadhava:

Thefirsttermistheproductofthegivensineandradiusofthedesiredarcdividedbythecosineofthe arc. The succeeding terms are obtained by a process of iteration when the first term isrepeatedlymultipliedby the square of the sine anddividedby the square of the cosine.All thetermsarethendividedbytheoddnumbers1,3,5,…Thearcisobtainedbyaddingandsubtractingrespectivelythetermsofoddrankandthoseofevenrank.

Translatedintomodernnotation,andrememberingthat the tangent tanθ is thesinedividedbythecosine,thisbecomes

Which (rewritten in terms of the inverse tangent) iswhatwe in thewest callGregory’sseries–discoveredinourcivilisationbyJamesGregoryin1671,orperhapsalittleearlier.AccordingtotheMahajyanayanaPrakara(MethodsfortheGreatSines),Madhavausedthisseriestocalculateπ.Aspecialcase(θ=π/4=45°)ofthepreviousseriesgivesaninfiniteseriesforπ,thefirstexampleofitstype:

Thisisn’tapracticalwaytocalculateπ,becausethetermsdecreaseveryslowlyandhugenumbersoftermsareneededforevenafewdecimalplaces.Takingθ=π/6=30°instead,Madhavaderivedavariantthatconvergesfaster:

Hecalculatedthefirst21termstoobtainπto11decimalplaces.Thisserieswasthe first new method for computing π after Archimedes’s use of ever finerpolygons.

One aspect ofMadhava’sworks is surprisingly sophisticated.He estimatedtheerrorwhentheseriesistruncatedatsomefinitestage.Infact,hestatedthreeexpressionsfortheerror,whichcanbeaddedinasacorrectiontermtoimprovetheaccuracy.Hisexpressionsfortheerrorafteraddingntermsoftheseriesare:

He used the third one to obtain an improved value for the sum, getting 13decimal places of π. Nothing similar occurs anywhere in the mathematicalliteratureuntilmoderntimes.

In1676Newtonwrotea letter toHenryOldenburg,Secretaryof theRoyalSociety,informinghimoftwoinfiniteseriesforthesineandcosine:

whichhehadderivedbya roundaboutmethod,usingcalculus.Wenowknowthat these expressions, long assumed to have originated with Newton, wereobtainedbyMadhava,nearlyfourcenturiesearlier.Detailsof thederivationofthese series are given in Yuktibhasa. The method is complicated, but can beviewed as an early anticipation of the calculusmethod of integrating a seriestermbyterm.

Indeed,itisarguedthatMadhavadevelopedsomebasicnotionsofcalculus,long before Newton. Namely differentiation, the integral as the area under acurve, and term-by-term integration. He found methods for expandingpolynomials in algebra, devised a numerical method for solving equations byiteration,andworkedoninfinitecontinuedfractions.

Joseph asks whetherMadhava’s ideasmight have percolated into Europe. Hepointsout thatEuropeanexplorers suchasVascodaGamaknewKeralawell,

becauseitisausefulstoppingpointforshipscrossingtheArabianSeaenroutetoChinaandotherplacesintheFarEast.ItsroleasatradecentregoesbacktoBabylonian times. Itsgeographical isolation,hemmed inby theWesternGhatsand the Arabian Sea, protected it from the turbulent politics of the rest ofmedievalIndia,anaddedbonusforforeigntravellers.ItdoesseemthatsomeofKerala’stechnology,anditsgoods,madetheirwaytoEuropeatthattime,butsofar no evidence of direct transfer ofmathematical ideas has been found.Untiland unless new evidence comes to light, it appears that Kerala and Europediscoveredmanyimportantmathematicalideasindependently.

Theworkofgreat Indianfigures likeAryabhataandBrahmaguptahas longbeen recognised in Europe. That of the Kerala school was first brought toEuropeanscholarlyattentionasrecentlyas1835,whenCharlesWhishwroteanarticle about four major texts: Nilakantha’s Tantrasamgraha, Jyesthadeva’sYuktibhasa, Putumana Somayaji’s Karana Paddhati, and Sankara Varman’sSadratnamala. Whish put the cat among the pigeons with the claim thatTantrasamgraha contains the basis of fluxions, Newton’s term for calculus(Chapter7): that it ‘aboundswith fluxional formsandseries tobefound innoworkofforeigncountries’.InthedayswhentheEastIndiaCompanycontrolledtradewithIndia,andthecountryitselfwasseenasripeforconquest,thisclaimwent down like a lead balloon. Keralamathematics was essentially forgotten.Over a century later, in the1940s, its advancednature finally re-emerged in aseries of articles by Cadambur Rajagopal and collaborators, analysing Keralamathematics and demonstrating that Hindu mathematicians discovered manyimportant resultsmuchearlier than theEuropeans towhomtheyhadgenerallybeencredited.

5TheGamblingAstrologer

GirolamoCardano

Girolamo(Gerolamo,

Geronimo)Cardano/

HieronymusCardanus

Born:Pavia,DuchyofMilan,24September1501

Died:Rome,21September1576

AT A VERY EARLY PERIOD in my life, I began to apply myself seriously to the practice of

swordsmanship of every class, until, by persistent training, I had acquired some standing evenamong themostdaring.Bynight,evencontrary to thedecreesof theDuke, Iarmedmyselfandwent prowling about the cities in which I dwelt. I wore a black woollen hood to conceal myfeatures,andputonshoesofsheep-pelt.OftenIwanderedabroadthroughoutthenightuntildaybroke,drippingwithperspirationfromtheexertionofserenadingonmymusicalinstruments.

SuchwaslifeinRenaissanceItalyaround1520–atleast,forGirolamoCardano,whorevealedtheseactivitiesandmuchmoreinafrankautobiographyTheBookofMyLife.Cardano,apolymathespeciallygiftedinmathematicsandmedicine,

enjoyed(if that is theword)acareerstraightoutof thesoapsandtabloids.Hefrittered away the family fortune, became addicted to gambling, and enduredruinand thepoorhouse.Suspectinganothermanwascheating,he slashed theplayer’s facewith a knife.Hewas accused of heresy and imprisoned; his sonwas executed forwife-poisoning.ButCardano also restored the speech of theBishopofStAndrews,whohadbecomemute, earninga rewardof1400goldcrowns. Returning to Italy in triumph, he was admitted into the College ofPhysicians,whichhadspentdecadestryingdesperatelytokeephimout.

Mostimportantlyofall,hewasamastermathematicianwhowroteoneoftheall-timegreat textbooks,ArsMagna (TheGreatArt). Its subtitle:TheRulesofAlgebra. In Ars Magna, algebra came of age, acquiring both symbolicexpressionandsystematicdevelopment.Cardanocanbeviewedasyetanothercandidateforthetitle‘fatherofalgebra’.But,truetoform,hedidnotattainthisstatuswithoutcontroversyanddouble-dealing.

Cardano was an illegitimate child. His father Fazio, a lawyer with strongmathematicaltalentsandahair-triggertemper,livedinPaviaandwasafriendofLeonardo da Vinci. He habitually wore an unusual purple cloak and a smallblack skullcap, andhad lost all his teethby the ageof 55.Girolamo’smotherChiara (néeMicheria), a youngwidowwith three children,married his fathermanyyearslater.Shewasfat,withatempertorivalFazio’s,andquicktotakeoffence.Shewasalsodeeplyreligiousandhighlyintelligent.Whenshebecamepregnant,plagueappearedinMilan,soshemovedtothecountryside,whileherthree elder children remained in the city and died of the plague. Cardano’simminent arrivaldidnotprovoke joy: ‘Althoughvariousabortivemedicines…weretriedinvain,Iwasnormallybornonthe24thdayofSeptemberintheyear1500.’

Fazio, though a lawyer by trade, was sufficiently adept at mathematics tohaveadviseddaVinciaboutgeometry,andhetaughtgeometryattheUniversityof Pavia and at the Piatti Foundation in Milan. He passed on his skill inmathematics and astrology to his bastard son: ‘My father, in my earliestchildhood,taughtmetherudimentsofarithmetic,andaboutthattimemademeacquaintedwiththearcane…HeinstructedmeintheelementsoftheastrologyofArabia…After I was twelve years old he taughtme the first six books ofEuclid.’

Girolamo was a sickly child, and his father’s plans to bring him into the

family’slegalbusinessfailed.EnrollingasamedicalstudentatPaviaUniversity,he performed brilliantly, and although many found his outspoken natureoffensive,hewaselectedrectoroftheuniversitybythemarginofasinglevote.Successwent tohishead.Thiswas theperiodwhenheroamedthecitystreetsarmed with his sword and musical instruments, and turned to gambling. Hismathematical understanding of chance gave him a distinct advantage, andaround1564hewroteoneoftheearliestbooksonprobability:BookonGamesofChance,finallypublishedin1663.Hisabilitytoplaychess–formoney–alsohelped. But as he became more dissolute, he lost both his luck and hisinheritance.

Stillhepressedon.Nowinpossessionofamedicaldegree,hetriedtogainentry toMilan’sCollege of Physicians, the gateway to a profitable professionandacomfortablelifestyle.Thistimehistendencytospeakhismindfranklylethim down, and he was refused entry, so he took a position as a doctor in anearbyvillage.Itbroughtinjustaboutenoughtoliveon,andhemarriedLuciaBandarini, daughter of a militia captain. Rejected again by the college, herevertedtohisearlieroccupation,andlostafortune.Afterhe’dpawnedalltheirpossessions, including Lucia’s jewellery, they ended up in the poor house. ‘Iruined myself! I perished!’ Cardano wrote. He and Lucia had a child, whosuffered from various minor birth defects but did not, at that time, count asdeformed.BynowFaziohaddied,andGirolamowasappointedhissuccessor;thingswerefinallylookingup.In1539eventheCollegeofPhysiciansstoppedtrying to keep him out. He was developing a new string to his bow as well,publishing several mathematics books. One of them placed him firmly in theranksofthemathematicaltrailblazers.

Mostareasofmathematicsemerge throughacomplexandconfusinghistoricalprocess that lacks any clear direction, precisely because the direction itself isbeingcreatedas fragmentary ideasbegin to link together.The junglegrowsasyou explore it. A few features of algebra can be traced back to the ancientGreeks,wholackedaneffectivenotationevenforwholenumbers.Byinventingabbreviated notation for unknown quantities, Diophantus gave proto-algebra abigboost,buthisfocuswasonsolvingequationsinwholenumbers,whichledmoredirectly tonumber theory.GreekandPersiangeometerssolvedproblemswe now consider to be algebraic by purely geometric means. Al-Khwarizmiformalisedalgebraicprocesses,butfailedtointroducesymbolicrepresentations.

Long before any of this had happened, the Babylonians had alreadydiscovered the first genuinely important technique in algebra: how to solvequadraticequations.Thiskindofquestion,wenowappreciate,opensthedoortoalgebraintheformithadacquiredbythenineteenthcentury,whichisthebulkofwhatgoesbythatnameinschoolmathematics.Namely,deducingthevalue(orashortlistofpossiblevalues)ofanunknownquantityfromsomenumericalrelationshipbetweenthatquantityandits‘powers’–itssquare,itscube,andsoon.Thatis,solvingapolynomialequation.

Ifthehighestpoweroftheunknownthatappearsisitssquare,theequationisquadratic. The scribe-mathematicians of ancient Babylon knew how to solvesuchthings,andtheytaughtthemtoschoolboys.Wehavetheclaytablets,withtheirarcanecuneiformcharacters, toprove it.Themostdifficult step is takingthesquarerootofanappropriatequantity.

Withhindsight,thenextstepisclear:cubicequations,involvingthecubeoftheunknownaswellasitssquareandtheunknownitself.OneBabyloniantablethintsataspecialmethodforsolvingcubics(thenicknamethatmathematiciansuse in place of the cumbersome ‘cubic equations’), but that’s allwe know oftheirdiscoveriesinthisarea.TheGreekandPersiangeometricmethodsdidthetrick;themostdetailedtreatmentisthatofOmarKhayyam,morefamousforhispoetry, especiallyRubaiyat (and evenmore especially inEdward FitzGerald’stranslation).Apurelyalgebraicsolutionseemedbeyondreach.

AllthatchangedintheheadydaysoftheItalianRenaissance.Around1515ScipionedelFerro,aprofessorinBologna,discoveredhowto

solve some types of cubic. The distinction into types arose because negativenumbers were not then recognised, so equations were arranged with positivetermsonbothsides.DelFerropassedonsomenotestohisson-in-lawAnnibaledelNave,whichshowthathecouldsolve thecase‘cubeplusunknownequalsnumber’. In all likelihoodhecouldalso solve twoother types,whichbetweenthemeffectivelycoverallpossibilitiesaftersomepreparatorymanipulation.Hismethodinvolvedbothsquarerootsandcuberoots.

AlongwithdelNave,themethodfortheaforementionedcasewasknowntodel Ferro’s student Antonio Fior. Independently, Niccolò Fontana (generallyknown by the now politically incorrect nickname of Tartaglia, ‘stammerer’)rediscovered the solution for the samecase.Fior,whowas intending to setupshop as amathematics teacher, got a bright idea: engageTartaglia in a publicbattle,where eachwould challenge the other to solvemathematical problems.Thissortofintellectualcombatwascommonatthetime.Butthescamblewup

inhisfacewhenTartaglia,spurredonbyrumoursthatallthreecaseshadbeensolved,anddeeplyworried thatFiorknewhow,madeahugeeffortandfoundthesolutions just in time for thecontest.Belatedlydiscovering thatFiorcouldsolveonlyonetype,Tartagliathensethimonlycasesthathecouldn’tsolve,andwipedthefloorwithhim.

Thenewswas juicyand spread rapidly, reaching theearsofCardano,whowasassiduouslycollectingmaterial forArsMagna.He,keepingaweathereyeopenforanyinterestingnewmathematicsthatmightimprovehisintendedbook,was quick to spot a golden opportunity.Del Ferro’s earlierwork had by thenbeenall but forgotten, soCardanovisitedTartaglia, begginghim to reveal thesecret of the cubic. Eventually Tartaglia gave in. Legend has it that he sworeCardanotosecrecy,but thisseemsa littleunlikely in thecontextofCardano’sintention towrite an algebra book.At any rate,whenArsMagna appeared, itcontainedTartaglia’ssolutionofthecubic.Creditedtohim,butthatcameapoorsecond to being scooped. The irate Tartaglia hit backwithDiverseQuestionsand Inventions, which included all of the correspondence between him andCardano. The book claimed that in 1539 Cardano had sworn a solemn oath‘nevertopublishyourdiscoveries’.Now,theoathwasbroken.

As might be expected, the full story is probably more complicated. Sometimeafterwards,LodovicoFerrari,wholaterbecameCardano’sstudent,claimedhe’dbeenpresentatthemeeting,andCardanohadnotagreedtokeepTartaglia’smethodsecret.On theotherhand,Ferrariwashardlyadispassionateobserver.His response to Tartaglia’s claim of a broken oath was to issue a cartello –challengingTartagliatodebatewithhimonanytopichewished.InAugust1548a large crowd gathered in a church to watch the contest. I doubt many weredrawn there by the mathematics, or even understood it: what most of themwantedwasagoodoldset-to.Althoughnorecordsof theoutcomeareknown,Ferrari was soon offered the post of tutor to the Emperor’s son; in contrast,Tartaglianeverclaimedawin,losthisjobinBrescia,andkeptwhingeingabouttheresult.Sowecanmakeaneducatedguess.

Theironyisthatnoneofithadbeennecessary.DuringthepreparationofArsMagna, Cardano and Ferrari had seen del Ferro’s Bologna papers, whichcontained his prior solution of the cubic. This, theymaintained, had been theactual source of the method. The reason Cardano had mentioned Tartaglia’sworkhadbeentoexplainhowhehadheardofdelFerro’s.Thatwasall.

Perhaps.ButwhydidCardanobegTartagliatorevealthesecret,ifhealreadyknewitfromanearliersource?Maybehedidn’tbeg.WehaveonlyTartaglia’s

wordforit.Ontheotherhand,somethingheldCardanoback,foratime,becausehe didn’t just need the solution of the cubic in its own right. Ferrari, underCardano’sguidance,hadmanagedtotakeeverythingonestepfurtherbysolvingthe quartic equation (fourth power of the unknown, aswell as lower powers).But,crucially,hissolutionoperatedbyreducingeverythingtoarelatedcubic.SoCardano couldn’t reveal to theworld the solution of the quartic, without alsotellingthemhowtosolvecubics.

PerhapsitwasallasCardanoandFerrariclaimed.Tartaglia’sdefeatofFiormadeCardanoawarethatcubicscouldbesolved.ThenabitofdiggingledhimtodelFerro’smanuscript,whichgavehimthemethodheneededforhisbook.Stimulatedbythisdiscovery,Ferrarithenconqueredthequartic.Cardanoputthelot inhisbook.Ferrari, ashis student,couldhardlycomplainabouthis resultsbeingincluded,andseemstohavetakenpridethattheywere.OutofdeferencetoTartaglia,Cardanogavehimcreditforrediscoveringthemethodanddrawingittohisattention.ArsMagnaissignificantforoneotherreason.Cardanoappliedhisalgebraic

methods to find twonumberswhose sum is10andproduct is40, andgot theanswer5+ and5− .Sincenegativenumbershavenosquareroots,hedeclaredthisresulttobe‘assubtleasitisuseless’.Theformulaforcubicsalsoleads to such quantities when all three solutions are real, and in 1572 RafaelBombelliobservedthatifyouignorewhatsuchexpressionsmightmeanandjustdothesums,youget thecorrectrealsolutions.Eventually this lineof thinkingledtothecreationofthesystemofcomplexnumbers,inwhich−1hasasquareroot. This extension of the real number system is essential to today’smathematics,physics,andengineering.

Inthe1540sCardanowentbacktopractisingmedicine.Then(likeIsaid,soapsand tabloids) tragedy struck. His eldest son Giambatista had secretly marriedBrandonia di Seroni,who inCardano’s opinionwasworthless and shameless.Her parents were gold-diggers, andGiambatista’s wife taunted him in public,claiming thathewasn’t the fatherof their threechildren.Hepoisonedherandpromptly confessed. The judge insisted that the only way to avoid the deathpenaltywasifCardanoagreedcompensationwiththediSeronis.Thesumtheydemandedwassohugethathecouldn’tpay,sohissonwastortured,hadhislefthandcutoff,andwasthenbeheaded.

Cardano, a tough cookiewho’d seen it all,was forced tomove, becoming

professor of medicine at Bologna. There, his arrogance made enemies of hismedical colleagues, and they tried to get him fired. His younger son Aldoamassedhugegamblingdebts,andbrokeintohisfather’shousetostealmoneyandjewels.Cardanofelthehadnochoicebuttoreportthetheft,andAldowasbanished from Bologna. Even so, Cardano remained an optimist, writing thatdespite these tragic events, ‘I still have so many blessings, that if they wereanother’shewouldcounthimselflucky.’Butmoredisasterswereinthepipelineoffate,andtheircausewashisinvolvementinastrology.In1570hecastJesus’shoroscope. He also wrote a book praising Nero, who had martyred earlyChristians.Thecombinationledtoachargeofheresy.Hewasimprisoned,andthenreleased,butbarredfromanyacademicpost.

He went to Rome, where to his surprise he was greeted warmly. PopeGregoryXIIIhadapparentlyforgivenhim,andgrantedhimapension.HewasadmittedtotheCollegeofPhysiciansthere,andwrote,butdidnotpublish,hisautobiography. It finally appeared more than sixty years after his death.Accordingtolegend,hediedbyhisownhandbecausehehadpredictedhisowndateofdeath,andprofessionalpriderequiredtheforecasttobecorrect.

6TheLastTheorem

PierredeFermat

PierredeFermat

Born:Beaumont-de-Lomagne,France,17August1601(or31October–6December

1607)

Died:Castres,France,12January1665

FEWMATHEMATICIANSMANAGE to pose a problem that remains unanswered for

centuries,especiallyonethatturnsouttobeofcentralimportancetoareasofthesubject thatdidn’tevenexistwhen thequestionwas first asked.PierreFermat(the‘de’wasaddedlaterwhenhebecameagovernmentofficial)isperhapsthebest known among this exalted company. But he wasn’t exactly amathematician;hehadalawdegreeandbecameacouncillorattheparliamentinToulouse. On the other hand, it would be stretching a point to call him anamateur.Perhapsheisbestthoughtofasanunpaidprofessionalwhosedayjobwaspractisinglaw.

Fermat published very little, possibly because his non-mathematical dutieslefthardlyanytimetowriteuphisdiscoveries.Whatweknowofthemcomesmainly from his letters tomathematicians and philosophers, such as Pierre deCarcavi,RenéDescartes,MarinMersenne,andBlaisePascal.Fermatknewwhataproofwas,andinparticularthesoleincorrectstatementinhissurvivingpapers(about a formula that he thought always yielded a prime number) wasaccompanied by the assertion that he lacked a proof. Very few of his proofssurvive, themain one being a proof that two squares can’t add up to a fourthpower,accomplishedbyanovelmethodthathecalled‘infinitedescent’.

Fermathasmanyclaimstomathematicalfame.Hemademajoradvancesingeometry,developedprecursors tocalculus,andworkedonprobabilityandthemathematical physics of light. His foremost contribution, however, is hisseminalworkinnumbertheory.There,hestatedtheconjecturethatalsoensuredhis fameamong thegeneralpublic, thanks inpart to a televisiondocumentaryand a bestselling book.Namely, hisLast Theorem.This simple but enigmaticstatement acquired its namenotbecausehegasped it out onhisdeathbed, butbecause Fermat’s successors, over the next hundred years or so, managed toprove (or in one case disprove) every theorem he had stated, with this soleexception. Itwas the last to hold out against the onslaught, and it baffled thefinestminds.

Among themwasGauss,oneof the finestof themall.Nearly twohundredyears after Fermat’s marginal note, Gauss dismissed Fermat’s Last Theorem,declaringittypicalofahugerangeofstatementsaboutnumbersthatareeasytoguess but virtually impossible to prove or disprove. Gauss normally hadimpeccabletastewhenitcametomathematics,butthisassessmentturnedouttobearareunderestimateofmathematicalsignificance.InGauss’sdefence,mostmathematiciansfeltthesamewayforthefirstthreeandaquartercenturiesafterFermatstatedtheproblem.Onlywhensubtlelinkstoother,morecentralareasofmathematics,werespotted,diditstrueimportanceemerge.

Today,Beaumont-de-LomagneisaFrenchcommune(anadministrativedistrict)in theMidi-Pyrénées region of southern France. It was founded in 1276 as abastide–oneof a seriesof fortifiedmedieval towns in that area, and it had aturbulenthistory.ThetownwascapturedtemporarilybytheEnglishduringtheHundred Years War and then lost 500 of its citizens to the plague. It wasCatholic,hemmedinbythreeneighbouringProtestanttowns.HenriIIIsoldittothe future Henri IV, who attacked it in 1580 andmassacred a hundred of itspeople. Louis XIII put it under siege in the early 1600s; it took part in therebellion against the king andwas subject tomilitary occupation in 1651 andheavilyfined;thenplaguestruckagain.

Neatlybookendedby theseeventswas thebirthof the town’smost famousinhabitant:PierreFermat,sonofarichleathermerchantDominiqueandhiswifeClaire (née de Long) who hailed from a family of lawyers. There’s someuncertaintyabout theyearofhisbirth (either1601or1607)becausehemighthavehadanelderbrother, alsonamedPierre,whodiedyoung.His fatherwasalso the second consul of Beaumont-de-Lomagne, so Fermat grew up in apolitical family. His father’s position makes it a racing certainty that Fermatgrewup in the townofhisbirth, and if so,hemusthavebeeneducated in itsFranciscanmonastery. After a spell at the University of Toulouse he went toBordeaux,wherehismathematicalinterestsbegantoflourish.Firstheproduceda tentative restoration ofOn Plane Loci, a lost work of the Greek geometerApollonius; then he wrote on maxima and minima, anticipating some earlydevelopmentsincalculus.

His legal career also blossomedwith a law degree from the University ofOrléans.In1631heboughthimselfapositionasacouncillorintheparliamentatToulouse,whichentitledhimtoadd‘de’tohisname.Heactedinthiscapacity,andasalawyer,fortherestofhislife,livinginToulousebutworkingfromtimeto time in Beaumont-de-Lomagne and Castres. Initially he was in the lowerchamberofparliament,butherosetoahigherchamberin1638andthence, in1652,tothetoplevelofthecriminalcourt.Helpedinpartbytheplague,whichkilledmanyseniorofficials in the1650s,Fermatcontinued tobepromoted. In1653 it was reported that Fermat had died of the plague, but (as with MarkTwain)thereportwasgreatlyexaggerated.ItlooksasthoughFermatwasbitingoffmore than he could chew; his interest inmathematicswas distracting himfrom legalduties.Onedocument reads: ‘He is ratherpreoccupied,hedoesnotreportcaseswellandisconfused.’

His 1629 Introduction to Plane and Solid Loci pioneered the use ofcoordinates to link geometry and algebra. This idea is generally credited toDescartes in his 1637 essay La Geometrie, an appendix to Discours de laMéthode,but itwashintedat inmuchearlierwritings,all thewayback to theGreeks.Itusesapairofcoordinateaxestorepresenteachpointoftheplanebyauniquepair of numbers (x,y), amethodnow so commonplace that it scarcelyrequiresdiscussion.

Inhis1679OnTangentstoCurvedLines,Fermatfoundtangentstocurves,ageometricversionofdifferential calculus.Hismethod for findingmaximaandminimawasanother forerunnerofcalculus. Inopticshestated theprincipleofleast time: a light ray followswhichever pathminimises the total travel time.Thiswasanearly step towards thecalculusofvariations, abranchofanalysisthatseekscurvesorsurfaces thatminimiseormaximisesomerelatedquantity.For example, which closed surface of fixed volume has the smallest surfacearea?Theanswerisasphere,andthisexplainswhysoapbubblesarespherical,becausetheenergyofsurfacetensionisproportionaltothesurfacearea,andthebubbletakeswhichevershapemakesthatenergysmallest.

Inasimilarvein,FermatarguedwithDescartesoverthelatter’sdeductionofthelawofrefractionforlightrays.Descartes,probablyannoyedthatFermatwasgetting credit for coordinates, which Descartes considered his own invention,respondedbycriticisingFermat’sworkonmaxima,minima,andtangents.Thedispute became so heated that the engineer and pioneering geometer GirardDesargueswas roped in as referee.Whenhe saidFermatwas right,Descartesgrudgingly conceded: ‘If you had explained it in this manner at the outset, Iwouldhavenotcontradicteditatall.’

Fermat’s greatest legacy is in number theory. His letters contain manychallengestoothermathematicians.Theseincludedprovingthatthesumoftwoperfectcubescan’tbeaperfectcube,andsolvingthemisnamed‘Pellequation’nx2+1=y2,wherenisagivenwholenumberandwholenumbersxandymustbefound.LeonhardEulererroneouslyattributedasolutionbyLordBrounckertoJohn Pell. In fact, Brahmagupta’s Brahmasphutasiddhanta (CorrectlyEstablishedDoctrineofBrahma)of628includesamethodforsolvingit.

One of Fermat’s most important and beautiful results characterises thosenumbersthatcanbeexpressedasthesumoftwoperfectsquares.AlbertGirardstatedtheanswerinworkpublishedposthumouslyin1634.Fermatwasthefirst

toclaimaproof,announcingitinalettertoMersennein1640.Themainpointisto solve the problem for prime numbers. The answer depends on the type ofprime,inthefollowingsense.Theonlyevenprimeis2.Oddnumbersareeitheramultipleof4with1added,oramultipleof4with3added;thatis,theyareofthe form 4k + 1 or 4k + 3. The same goes for odd primes, of course. Fermatprovedthat2,andeveryprimeoftheform4k+1,aresumsoftwosquares;ontheotherhand,thoseoftheform4k+3arenot.

Ifyouexperiment,it’seasytoguessthis.Forexample13=4+9=22+32,and13=4×3+1.Ontheotherhand7=4×1+3,andasumoftwosquarescan’tequal7.ProvingFermat’stwo-squarestheorem,though,isdistinctlyhard.Theeasiestbitistoshowthatthe4k+3primesarenotsumsoftwosquares;I’llshowyouhowinChapter10usingatrickthatGaussdevelopedtosystematiseabasicmethodofnumbertheory.Showingthatthe4k+1primesaresumsoftwosquares is considerably harder. Fermat’s proof hasn’t survived, but proofs areknownthatusemethodsthatwouldhavebeenaccessibletohim.Eulergavethefirstknownproof,announcingitin1747andpublishingitintwoarticlesof1752and1755.

Theupshot is thatawholenumber is thesumof twosquares ifandonly ifeveryprimefactoroftheform4k+3appearstoanevenpowerwhenthenumberisresolvedintoprimefactors.Forinstance,245=5×72.Thefactor7isoftheform4k + 3, andoccurs to an evenpower, so245 is the sumof two squares.Indeed,245=142+72.Incontrast,35=5×7,andthefactor7occurstoanoddpower, so35 is not the sumof two squares.This resultmay seeman isolatedcuriosity, but it sparked several lines of research, flowering into Gauss’s far-reaching theory of quadratic forms (Chapter 10). Inmodern times it has beentaken much further. A related theorem, proved by Lagrange, states that anywholenumber is thesumof foursquares (where0=02 isallowed).This, too,hasextensiveramifications.

ThestoryofFermat’sLastTheoremhasbeen toldmany times,but Imakenoapologyfortellingitagain.It’sagreatstory.

It’s perhaps ironic that Fermat’s greatest fame rests on a theorem that healmost certainlydidn’t prove.He apparentlyclaimed a proof, and the result isnowknowntobetrue,buttheverdictofhistoryisthatthemethodsavailabletohimweren’tuptothetask.Hisclaimtopossessaproofexistsonlyasamarginalnoteinabook,whichdoesn’tevensurviveasanoriginaldocument,soitcould

havebeenmadeprematurely.Inmathematicalresearchit’snotunusualtowakeupinthemorningconvincedyou’veprovedsomethingimportant,onlytoseetheproofevaporatebynoonwhenyoufindamistake.

The book concerned was a French translation of the Arithmetica ofDiophantus, the first greatwork on number theory, unless you count Euclid’sElements,whichdevelopsmanybasic properties of primenumbers and solvessomeimportantequations.CertainlyArithmeticaisthefirstspecialisttextonthetopic.Recall that this book gavemathematics the technical term ‘Diophantineequation’ for a polynomial equation that must be solved in whole or rationalnumbers.Diophantusmadeasystematiccatalogueofsuchequations,andoneofthecentralexhibitsistheequationx2+y2=z2forso-calledPythagoreantriples,because a triangle with sides x, y, and z has a right angle by Pythagoras’sTheorem.Thesimplestsolutioninnonzerowholenumbersis32+42=52, thecelebrated 3–4–5 triangle. There are infinitely many solutions: Euclid gave aproceduretogenerateallofthem;Diophantusincludedit.

FermatownedacopyofClaudeBachetdeMéziriac’s1621LatintranslationofArithmetica,andhejottedobservationsinthemargins.AccordingtoFermat’sson Samuel, the Last Theorem is stated as a note attached to Diophantus’sQuestionVIIIofBookII.WeknowthisbecauseSamuelissuedhisowneditionofArithmetica,whichincludedhisfather’snotes.Thedatesofthenotesarenotknown,butFermatstartedstudyingArithmeticaaround1630.Thedateisoftengiven as 1637, but this is a guesstimate. Presumably musing on potentialgeneralisations of Pythagorean triangles, Fermat was led to his epic marginalannotation:

It is impossible todivideacubeinto twocubes,orafourthpower into twofourthpowers,or ingeneral, any power higher than the second, into two like powers. I have discovered a trulymarvellousproofofthis,whichthismarginistoonarrowtocontain.

Thatis,theDiophantineequationxn+yn=znhasnowholenumbersolutionsifnisanintegergreaterthanorequaltothree.

There’scircumstantialevidence thatFermatsubsequentlychangedhismindabouthavingaproof.Incorrespondence,heoftensethistheoremsaspuzzlesforothermathematicians to solve (andat leastoneof themcomplained theyweretoohard).However,noneofhisextantlettersmentionsthistheorem.Evenmoretellingly,hedidpose twospecial cases, cubesand fourthpowers, asproblemsforhiscorrespondents.Whydothisifhecouldproveamoregeneralresult?Itseemscertainthathecouldprovethecubiccase,andweknowhowheproved

thetheoremforfourthpowers.Infact, thatproofis theonlyproofinallof theworks andpapershe left.His actual statement is: ‘Theareaof a right trianglecannotbeasquare.’HeclearlyintendedthistorefertoPythagoreantriples.TheEuclid–Diophantus solution easily implies that this problem is equivalent tofindingtwosquaresthatsumtoafourthpower.Ifasolutionofx4+y4=z4withexponent 4 existed, then both x4 and y4 would be squares (of x2 and y2respectively);Fermat’sstatementthenimpliesthatnosuchsolutionexists.

Hisproofisingenious,andatthetimewasaradicalinnovation.Hecalleditthe method of infinite descent. Suppose a solution exists, apply the Euclid–Diophantussolution,messaround forabit,andyoucandeduce thatasmallersolution exists. Therefore, said Fermat, you can construct an infinite chain ofever smaller solutions. Since any descending chain of this kind, formed fromwhole numbers, must eventually stop, this is a logical contradiction. So thehypotheticalsolutionthatwestartedfromcan’tactuallyexist.

Fermat may have concealed his proofs deliberately. He seems to have beenrather mischievous, and liked to torment other mathematicians by setting hisresultsaspuzzles.Hismarginalnoteisnottheonlyonetoannouncesomethingimportant followed immediately by an excuse for not proving it. DescartesconsideredFermattobeabraggart,andWallisreferredtohimas‘thatdamnedFrenchman’. Be that as it may, the tactic – if such it was – worked. AfterFermat’sdeath– indeed,duringhis life too–greatmathematiciansmade theirmark by polishing off one or other of the puzzles he had posed to posterity.Euler,forinstance,claimedaproofthattwocubescan’taddtoacubeinaletterof1753 tohis friendChristianGoldbach.Wenow realise thathisproofhadagap,butthiscanbefixedupfairlyeasily,soEulergenerallygetscreditforthefirstpublishedproof.Adrien-MarieLegendreprovedtheLastTheoremforfifthpowersin1825,andPeterDirichletproveditfor14thpowersin1832,clearlyafailedattempt toprove it forseventhpowers thatcouldbesalvagedbyaimingforsomethingweaker.GabrielLamédealtwithseventhpowersin1839,andin1847heexplainedthemainideasoftheprooftotheParisAcademyofSciences.It involved an analogue of prime factorisation for a special type of complexnumber.

Immediately after his talk, Joseph Liouville stood up and pointed out apossible flaw in Lamé’s method. For the usual kind of number, primefactorisation is unique: aside from the order in which the factors are written,

there’sonlyonewaytodoit.Forexample,theprimefactorisationof60is22×3×5andnothingessentiallydifferentworks.Liouvillewasworried thatuniquefactorisation might not be valid for Lamé’s class of complex numbers.Eventuallyhisdoubtswerejustified:thepropertyfirstfailsfor23rdpowers.

ErnstKummermanagedtofixupthisideabythrowingnewingredientsintothemix,whichhecalled‘idealnumbers’.Thesebehavelikenumbers,butaren’t.He used ideal numbers to prove Fermat’s Last Theorem for many powers,includingallprimesup to100except37,59, and67.By1993Fermat’ sLastTheoremwasknowntobe trueforallpowersup to4million,but thiskindofincreasinglydesperatescramblewasn’t sheddingany lighton thegeneralcase.Newideasstartedtoshowupin1955whenYutakaTaniyamawasworkingonadifferent,apparentlyunrelatedareaofnumbertheorycalledellipticcurves.(Thename is misleading and the ellipse isn’t one of them. An elliptic curve is aspecial kind of Diophantine equation.) He conjectured a remarkable linkbetweenthesecurvesandcomplexanalysis,thetheoryofmodularfunctions.Foryears hardly anyone believed he was right, but evidence slowly piled up thatwhatisusuallycalledtheShimura–Taniyama–Weilconjecturemightactuallybetrue.

In 1975 Yves Hellegouarch noticed a relationship between Fermat’s LastTheorem and elliptic curves, suggesting that any counterexample to Fermat’sLastTheoremwouldleadtoanellipticcurvewithverystrangeproperties.Intwopaperspublishedin1982and1986,GerhardFreyshowedthatthiscurvemustbeso strange that it can’t exist. Fermat’s Last Theorem would then follow bycontradiction, except that Freymade essential use of the Shimura–Taniyama–Weil conjecture, which was still up for grabs. However, these developmentsconvincedmanynumbertheoriststhatHellegouarchandFreywereontherighttrack. Jean-Pierre Serre predicted that someone would prove Fermat’s LastTheorembythisroute,aboutadecadebeforeithappened.

AndrewWilestookthefinalstepin1993,announcingtheproofofaspecialcaseof theShimura–Taniyama–Weil conjecture, powerful enough to completetheproofofFermat’sLastTheorem.Unfortunatelya logicalgap thencame tolight,oftenaprelude to total collapse.Wileswas lucky.Helpedbyhis formerstudentRichardTaylor, hemanaged to repair thegap in1995.Now theproofwascomplete.

PeoplestilldebatewhetherFermathadaproof.AsI’vesaid,circumstantialevidencestronglysuggestshedidn’t,becausehewouldsurelyhaveposeditasachallengetoothers.Morelikely,hethoughthehadaproofwhenhescribbledhis

marginalnote,butlaterchangedhismind.Intheunlikelyeventthathedidhaveaproof, it can’thavebeenanything likeWiles’s.Thenecessaryconcepts, andtheabstractviewpoint,simplydidnotexistinFermat’sday.It’slikeexpectingNewton to have invented nuclear weapons. Still, it’s conceivable that Fermatspottedsomemethodofattackthateveryoneelsehadmissed.Suchthingshavehappened.However,noone isgoing to findsuchaproofunless theyhave themathematicaltalentsofPierredeFermat,andthat’satallorder.

7SystemoftheWorld

IsaacNewton

SirIsaacNewton

Born:Woolsthorpe,England,

4January1643

Died:London,31March1727

IN 1696 THE ROYAL MINT, responsible for the production of England’s money,acquiredanewwarden,IsaacNewton.HehadbeengrantedthepositionbytheEarl of Halifax, Charles Montagu, who at that time was Chancellor of theExchequer–thegovernment’sheadoffinance.Newtonwasputinchargeoftherecoining of the realm.At that time, Britain’s coinagewas in a shoddy state.Newtonestimatedthatabout20percentof thecoinsincirculationwereeithercounterfeitorclipped;thatis,sliversofgoldorsilverhadbeenshavedofftheiredges,tobemelteddownandsold.Inprinciple,counterfeitingandclippingwereactsof treason,punishableby the judicial tortureofbeinghanged,drawn, andquartered.Inpractice,hardlyanyonewaseverconvicted,let’alonepunished.

As Lucasian Professor of Mathematics at Cambridge University, the newwardenwas an ivory-tower academicwho had devotedmost of his life to theesoteric subjects of mathematics, physics, and alchemy. He had also writtenreligioustractsaboutinterpretationsoftheBible,andhaddatedtheCreationto4000 BC. His track record of public service was patchy. He had served asMemberofParliament forCambridgeUniversity in1689–90andwoulddo soagainin1701–2,butit’sclaimedthathissolecontributiontodebateswasthatheonce observed that the chamber was cold and asked for the windows to beclosed. So it was easy to assume that, having obtained the post as a sinecurethroughpoliticalpatronage,Newtonwouldbeapushover.

Withinafewyears,28convictedcoinerswouldrealisethatthiswasnotthecase.Newtonwent in search of evidence in amanner thatwould do credit toSherlockHolmes.Hedisguisedhimself as a frequenterofdisreputable tavernsand alehouses, spying on the customers, observing criminal activity.RealisingthattheobfuscatorynatureofEnglishlawwasoneofthebiggestobstaclestoasuccessfulprosecution,Newtonfellbackon thecountry’sancientcustomsandlegal precedents. The office of justice of the peace had considerable legalauthority,able toopenprosecutions,cross-examinewitnesses,andprettymuchactasjudgeandjury.SoNewtongothimselfappointedasajusticeofthepeacein all of the counties adjacent to London. In eighteen months, starting in thesummer of 1698, he cross-examinedmore than a hundredwitnesses, suspects,andinformers,securingtheaforementioned28convictions.

Weknowaboutthis,incidentally,becauseNewtonstuckadraftletteraboutitinhisowncopyofhismasterworkPrincipia, inwhichhe effectively foundedmathematicalphysicsbystatingthelawsofmotionandthelawofgravity,andshowinghowtheyexplainavastrangeofnaturalphenomena.

The tale illustrates that when Newton turned his mind to something, heusuallyachievedgreatthings,thoughnotinalchemyandprobablynotinbiblicalscholarship.Hewenton tobecomeMasterof theMint,Presidentof theRoyalSociety,andwasknightedbyQueenAnnein1705.Hisgreatestcontributionstohumanity,however,wereinmathematicsandphysics.Heinventedcalculusandusedit toexpressfundamental lawsofnature,fromwhichhededuced–asthesubtitleofBook3of thePrincipia states– theSystemof theWorld.Howtheuniverseworks.

Hisbeginnings,however,werefarhumbler.

NewtonwasbornonChristmasDay1642.At least, thatwas thedatewhenhewas born. But it was determined by the Julian calendar, and when that wasreplacedbytheGregoriancalendar,notoriousforits‘lostdays’,theofficialdatebecame4January1643.Asachild,helivedonafarm;WoolsthorpeManorinthe tiny village ofWoolsthorpe-by-Colsterworth in Lincolnshire, not far fromGrantham.

Newton’sfather,alsonamedIsaac,diedtwomonthsbeforehissonwasborn.TheNewtonswere an established farming family, and IsaacNewton the elderwascomfortablywealthy,owningalargefarm,ahouse,andmanyfarmanimals.HismotherHannah(néeAyscough)managedthefarm.WhenIsaacwastwosheremarried to Barnabas Smith, minister of the church in the nearby village ofNorth Witham. The boy was cared for at Woolsthorpe by his grandmotherMargeryAyscough.Itwasn’tahappychildhood,andIsaacdidn’tgetonterriblywell with his grandfather James Ayscough. He got on even worse with hismotherandstepfather:whenheconfessedhissinsattheageof19,hementioned‘threateningmyfatherandmotherSmithtoburnthemandthehouseoverthem’.

Hisstepfatherdiedin1653,andIsaacstartedattheFreeGrammarSchoolinGrantham, where he lodged with the Clarke family. William Clarke was anapothecary, and his house was on the High Street next to the George Inn.Newtonbecamenotorious among the townsfolk for his strange inventions andmechanicaldevices.Hespenthispocketmoneyontools,andinsteadofplayinggames, he made things from wood – dolls’ houses for the girls, but also aworkingmodelofawindmill.Headdedatreadmill,runbyamouse,todrivethemill.Hemadeasmallcartwhichhecouldsit inandmovebyturningacrank.And he fixed a paper lantern to a kite to startle the neighbours at night.According to his biographerWilliamStukeley, this ‘wonderfully affrighted alltheneighboring inhabitants for some time, andcaus’dnot a littlediscourseonmarketdays,amongthecountrypeople,whenovertheirmugsofale.’

Historians have since discovered Newton’s source for most of theseinventions, TheMysteries of Nature and Art by John Bate. One of Newton’snotebooks contains numerous extracts from this book. But the inventionsillustratehisearlyfocusonscientificmatters,eveniftheywerenotoriginal.Hewasalsofascinatedbysundials–ColsterworthChurchhasoneattributedtohim,supposedly constructedwhen hewas only nine – and young Isaac distributedsundialsliberallythroughoutClarke’shouse.Hehammeredpegsinthewallstomarkthehours,half-hours,andquarter-hours.Helearnedhowtospotsignificantdates,suchassolsticesandequinoxes,withenoughsuccessthatthefamilyand

theirneighboursoftencametolookatwhattheycalled‘Isaac’sdials’.Hecouldtell the time by observing the shadows in a room.He also took advantage oflivinginanapothecary’sshopbyinvestigatingthecompositionofmedicines,anearlyintroductiontochemistrywhichpavedthewaytohisextensivealchemicalinterests in later life. He drew impressively convincing birds, animals, ships,evenportraitsonthewallsofhisroomincharcoal.

He was clearly a clever young man, but he showed no special signs ofmathematicaltalent,andhisschoolreportsdescribehimasidleandinattentive.Atthispointhismothertookhimoutofschooltohavehimtrainedtomanageherestate,astandardtaskforaneldestson,butheshowedevenlessinterestinthat.AnunclepersuadedherthatIsaacshouldgotouniversityinCambridge,soshesenthimbacktoGranthamtocompletehiseducation.

He entered Trinity College at Cambridge University in 1661, intending toobtainadegreeinlaw.ThecoursewasbasedonAristotelianphilosophy,butinthe third year he was allowed to read works by Descartes, the philosopher-scientist Pierre Gassendi, the philosopher Thomas Hobbes, and the physicistRobertBoyle.HestudiedGalileo,learningaboutastronomyandtheCopernicantheorythattheEarthgoesroundtheSun.HereadKepler’sOptics.HowNewtonwas introduced toadvancedmathematics ismoreopaque.AbrahamdeMoivrewrotethatitallstartedwhenNewtonboughtanastrologybookatafairground,and couldn’t understand the mathematics. Trying to master trigonometry, hediscoveredhedidn’t knowenoughgeometry, so hepickedup a copyof IsaacBarrow’seditionofEuclid.Thatstruckhimastrivial,untilhegottoatheoremabouttheareasofparallelograms,whichimpressedhim.Hethenracedthroughaseries of major mathematical works – William Oughtred’s The Key ofMathematics,Descartes’sLaGéométrie,theworksofFrançoisViète,FransvanSchooten’sGeometryofRenéDescartes,andJohnWallis’sAlgebra.Wallisusedindivisibles–infinitesimals–tocalculatetheareacontainedbyaparabolaandahyperbola.Newton thought about this, andwrote: ‘ThusWallis doth it, but itmaybedonethus…’Alreadyhewascomingupwithhisownproofsandideas,inspired by the great mathematicians, but not subservient to them. Wallis’smethodswereinteresting,butbynomeanssacred.Newtoncoulddobetter.

In1663Barrow tookup theLucasian chair, becomingaFellowofTrinity,whereNewtonwas based, but there’s no evidence that he noticed any specialtalent in the young student. It came to flower in 1665, when the university’sstudentsweresenthometoavoidthegreatplague.InthepeaceandquietoftheLincolnshire countryside, no longer distracted by the hustle and bustle of the

city,Newtonturnedhisattentiontoscienceandmathematics.Between1665and1666 he developed his law of gravity,which explained themovements of theMoon and planets, devised laws of mechanics to explain moving bodies,inventedthedifferentialandintegralcalculus,andmadesignificantdiscoveriesinoptics.Hepublishednoneofthiswork,butreturnedtoCambridgetotakehismaster’s degree, andwas elected aFellowofTrinityCollege. In1669hewasappointed Lucasian Professor of Mathematics when Barrow resigned, and hebecameaFellowoftheRoyalSocietyin1672.

From1690NewtonwrotemanytractsontheinterpretationoftheBibleandcarried out alchemical experiments. He held important administrative posts,eventuallybecomingMasteroftheRoyalMint.HewaselectedPresidentoftheRoyalSocietyin1703,andknightedin1705whenQueenAnnepaidavisit toTrinity College, Cambridge. The only scientist to have been knighted beforethenwas FrancisBacon.He lost a fortunewith the collapse of the South SeaBubble,andwent to livewithhisnieceandherhusbandnearWinchesteruntildying in his sleep inLondon in 1727.Mercury poisoninghas been suspected:traces of the metal were found in his hair. It fits with his experiments inalchemy,andcouldexplainwhyhebecameeccentricinhisoldage.

One of Newton’s early discoveries shows him to be a master of coordinategeometry. By then, conic sections were known to be defined by quadraticequations.Newtonstudied thecurvesdefinedbycubicequations.Hefound72species (we now recognise 78), and grouped them into four distinct types. In1771JamesStirlingprovedthateverycubiccurvebelongstooneofthesetypes.Newtonclaimedthatallfourtypesareequivalentunderprojection,andaproofwas found in 1731. In all of these discoveries,Newtonwaswell ahead of histime, and the broad context into which they fit – algebraic and projectivegeometry–becameapparentonlycenturieslater.

According to a possibly apocryphal tale, one of Newton’s practicalinventionscameintobeingduringhisearlyworkonoptics,around1670.Everyschoolchildistoldthataglassprismsplitswhitesunlightintoallthecoloursofthe rainbow. This discovery goes back to Newton, who performed theexperiment in his attic. However, there was a snag. He had a cat, whichapparently was rather rotund because its master, absorbed in his scientificresearch, failed tocontrolhowmuch itate.Thecathadahabitofpushing theatticdooropen tofindoutwhat Isaacwasdoing,which let light inandruined

theexperiment.SoNewtoncutaholeinthedoorandhungapieceoffeltoverit,inventingthecatflap.Whenthecathadkittens,headdedasmallerholenexttothebigone.(Thismaynothavebeenasabsurdasitappears;perhapsthekittensfounditdifficulttopushpastaheavypieceoffelt.)Thesourceforthisanecdotehasbeen identifiedonlyasa ‘countryparson’,and itmayjustbeashaggycatstory.Butin1827JohnWright,wholivedinNewton’sformerroomsatTrinityCollege,wrotethat thedoorhadoncehadtwoholes,of therightsizeforacatandakitten.

Newton’s biggest contributions to mathematics, however, are calculus andthePrincipia. Hiswork in opticsmademajor strides in physics, butwas lessinfluential in mathematics, so I won’t discuss it further. Logically, calculuscomes before the Principia, but historically both are intertwined in complexways,madeallthemoreobscurebyNewton’sreluctancetopublish.Hehadaninstinctive dislike of criticism, and the easy way to avoid it was to keep hisdiscoveries to himself. The end result, as it happened, was a much greaterbarrage of criticism and a huge public controversy, because the GermanmathematicianandphilosopherGottfriedLeibnizhadverysimilarideasaroundthesametime,eventuallytriggeringaprioritydispute.

TheoriginsofcalculuscanbetracedbacktoArchimedes’sMethod,Wallis’s1656Arithmetic of the Infinite, andworks of Fermat (Chapter 6) The subjectdividesintotwodistinctbutrelatedareas.

Differential calculus is a method for finding the rate of change of somequantity that varies with time. For example, velocity is the rate of change ofposition (how many kilometres your position changes as an hour passes).Acceleration is the rateofchangeofvelocity (areyouspeedingupor slowingdown?).Thebasic issue indifferentialcalculus is to find therateofchangeofsomefunctionoftime.Theresultisalsoafunctionoftime,becausetherateofchangecanbedifferentatdifferenttimes.

Integralcalculusisaboutareas,volumes,andsimilarconcepts.Itproceedsbycutting the object into very thin slices, estimating the area or volume of eachslicebyignoringanyerrorthatismuchsmallerthanthethicknessoftheslices,adding everything together, and letting the slices become arbitrarily thin. Asboth Newton and Leibniz discovered, integration is essentially the reverseprocesstodifferentiation.

Both processes involve a philosophically tricky idea: quantities that can bemade arbitrarily small. These were known as infinitesimals, and they requireverycarefulhandling.Nospecificnumbercanbe‘arbitrarilysmall’, since that

wouldmakeitsmallerthanitself.However,anumberthatvariescanbecomeassmallaswewish.Butifsomethingvaries,howcanitbeanumber?

Supposeweknowexactlywhereacaris locatedatanyinstantoftime,andwewant toworkout itsvelocity.If,overaperiodofonehour, ithastravelledsixty kilometres, the average velocity during that time is sixty kilometres perhour.Butthevelocitycouldhavebeenfasteratsometimesandsloweratothers.Reducing the time interval to one second gives a more precise estimate, theaverage velocity over one second. But the velocity might still have changedslightly during that time.We can approximate the instantaneous velocity, at agivenmoment,byfindingouthowfarthecarmovesinaveryshortintervaloftime,anddividingthatdistancebythetimeittook.Howeversmallwemakethatinterval,though,theresultisonlyanapproximation.Butifyoutrythisusingaformula for the position of the car, it turns out that if you make the intervalcloserandclosertozero,theaveragevelocityoverthatintervalgetscloserandcloser to some specific value. We define that value to be the instantaneousvelocity.

Theusualwaytodothesumsrequiresustodividethedistancetravelledbythetimeelapsed.CriticssuchasBishopGeorgeBerkeleywerequicktopointoutthat when the elapsed time becomes zero, this fraction is 0/0, which ismeaningless. Berkeley published his criticisms in 1734 in a pamphlet TheAnalyst, a Discourse Addressed to an Infidel Mathematician, referringsarcastically to Newton’s fluxions (the instantaneous velocities) as ‘ghosts ofdepartedquantities’.

Newton and Leibniz had answers to such objections. Newton employed aphysicalimageoftheintervalflowingtowards0,butnotactuallygettingthere.Thedistance travelled flows towards0aswell,and theaveragevelocity flowstoo. What matters, Newton said, is what it flows towards. Getting there isirrelevant. So he called his method ‘fluxions’ – things that flow. Leibnizpreferredtotreatthetimeintervalasaninfinitesimal,bywhichhemeantnotafixed nonzero quantity that can be arbitrarily small (which makes no logicalsense)butavariablenonzeroquantitythatcanbecomearbitrarilysmall.ThisisprettymuchthesameviewpointasNewton’s.Itis,giveortakesomeprecisionin terminology, the viewpointwe use today, known as ‘taking a limit’. But ittook several centuries to sort all this out. It’s subtle.Even today,mathematicsundergraduatestakeawhiletogetusedtoit.

BishopBerkeleymayhavebeenunhappyaboutthefoundationsofcalculus,butmathematicians are alwayswilling to ignorephilosophers, especiallywhen thephilosopherstellthemtostopusingamethodthatworksperfectlywell.No,thebigargumentaboutcalculuswasaprioritydisputeaboutwhocreatedit.

NewtonhadwrittenhisMethodofFluxionsandInfiniteSeries in1671,buthad not published it. It finally saw the light of day in 1736 in an Englishtranslation of the Latin original by John Colson. Leibniz published ondifferentialcalculusin1684,andonintegralcalculusin1686.Newtonpublishedhis Principia in 1687. Moreover, although many of its results depended oncalculus, Newton chose to present them in a more classical geometric form,using a principle he called ‘prime andultimate ratios’.Here’s howhedefinedequalityoffluxions:

Quantities,andtheratiosofquantities,whichinanyfinitetimeconvergecontinuallytoequality,andbeforetheendofthattimeapproachnearertoeachotherthanbyanygivendifference,becomeultimatelyequal.

Today’sformulationofthelimitconceptinanalysisisequivalenttothis,butthemeaning is made more explicit. Newton’s critics never understood thisdefinition.

Newtonusedgeometry insteadofcalculus in thePrincipia toavoidgettingtangledup in issues about infinitesimals, but bydoing so, hemissed a goldenopportunity to reveal calculus to the world. Those ideas circulated informallyamongBritishmathematicians, butwent largelyunnoticed in thewiderworld.SowhenLeibnizbecamethefirsttopublishoncalculus,hecausedanoutcryinBritain. A Scottish mathematician named John Keill set the ball rolling bypublishing an article inTransactions of theRoyal Society accusingLeibniz ofplagiarism.WhenLeibniz read it in 1711 he demanded a retraction, butKeillupped the antebyarguing thatLeibnizhad seen two letters fromNewton thatcontained the main ideas of differential calculus. Leibniz asked the RoyalSociety to mediate, and it set up a committee. This came down in Newton’sfavour–butthereportwaswrittenbyNewton,andnoonehadaskedLeibniztopresenthisowncase.Big-namemathematiciansincontinentalEuropejoinedin,complainingthatLeibnizwasn’tgettingfairtreatment.LeibnizstoppedarguingwithKeill,onthegroundsthatherefusedtoarguewithanidiot.Itallgotoutofhand.

Laterhistoriansconsiderthegametohavebeenadraw.NewtonandLeibnizdevised their methods pretty much independently. They were to some extent

aware of each other’s work, but nobody stole anyone else’s ideas. Variousmathematicians,includingFermatandWallis,hadbeencirclingaroundthemfora century or more. Unfortunately, this senseless controversy caused Britishmathematicianstoignorewhattheircontinentalcousinsweredoingforthenexthundredyearsorso,whichwasapitybecauseitincludedmostofmathematicalphysics.

ThePrincipia built on thework of earlier scientists, especiallyKepler,whosethreebasiclawsofplanetarymotionledNewtontoformulatehislawofgravity,andGalileo,whoinvestigatedthemotionofafallingbodyexperimentally,andspottedelegantpatternsinthenumbers.Hepublishedhisdiscoveriesin1590inOnMotion.ThisinspiredNewtontostatethreegenerallawsofmotion.Thefirsteditionof thePrincipiawaspublished in1687; furthereditions,withadditionsandcorrections,followed.In1747AlexisClairautwrotethatthebook‘markedtheepochofagreatrevolutioninphysics’.Inthepreface,Newtonexplaineditsbigtheme:

RationalMechanicswillbe thescienceofmotions resulting fromanyforceswhatsoever,andofthe forces required to produce anymotions…and thereforeweoffer thiswork asmathematicalprinciplesofphilosophy.Forall thedifficultyofphilosophy seems toconsist in this– from thephenomena of motions to investigate the forces of Nature, and then from these forces todemonstratetheotherphenomena.

Itwasaboldclaim,but inhindsight itsoptimismwas fully justified.Withinacentury Newton’s early insights had grown into a massive area of science:mathematical physics. Many of the equations developed during this periodremain in use today, with applications to heat, light, sound, magnetism,electricity,gravity,vibrations,geophysics, andsoon.We’vegonebeyond that‘classical’styleofphysicswithrelativityandquantumtheory,butit’samazinghowimportantNewtonianphysicsremains.Andhisideaofdescribingnaturebydifferential equations is used throughout the sciences, from astronomy tozoology.

Book 1 of the Principia tackles motion in the absence of any resistingmedium–nofriction,noairresistance,nofluiddrag.Thisisthesimplesttypeofmotionwiththemostelegantmathematics.Itstartsbyexplainingthemethodoffirst and last ratios,uponwhichall else rests.Asexplained, this is calculus in

geometricdisguise.Itestablishesearlyonthataninversesquarelawofattractionis equivalent to Kepler’s laws of planetary motion. At first sight, the logicalequivalenceofNewton’slawwithKepler’sthreelawssuggeststhatallNewtonachievedwastoreformulateKepler’slawsinthelanguageofforces.Butthere’sone further feature, a prediction rather than a theorem. Newton, like Hookebefore him, claims that these forces areuniversal. Every body in the universeattracts every other body. This lets him develop principles applicable to theentiresolarsystem,andhemakesastartontheproblemofthreebodiesmovingundergravitationalattraction.

Book2tacklesmovementinaresistingmedium,includingairresistance.Itdevelops hydrostatics – the equilibria of floating bodies – and compressiblefluids.Astudyofwavesleadstoanestimateforthespeedofsoundinair,1088feetpersecond(331metrespersecond),andhowitvarieswithhumidity.Themodern figure, at sea level, is 340 metres per second. Book 2 ends bydemolishing Descartes’s theory of the formation of the solar system throughvortices.

Book 3, subtitled On the System of the World, applies the principlesdeveloped in the first two books to the solar system and astronomy. Theapplicationsareastonishinglydetailed:irregularitiesinthemotionoftheMoon;themovement of Jupiter’s satellites, ofwhich fourwere then known; comets;tides;precessionoftheequinoxes;andespeciallytheheliocentrictheory,whichNewtonformulatedinaverythoughtfulmanner:‘Thecommoncentreofgravityof the Earth, the Sun and all the Planets is to be esteem’d the Centre of theWorld…[andthiscentre]eitherisatrest,ormovesuniformlyforwardinarightline.’ByestimatingtheratiosofthemassesoftheSun,Jupiter,andSaturn,hecalculated that thiscommoncentreofgravity isveryclose to thecentreof theSun,withanerroratmostthediameteroftheSun.Hewasright.

TheinversesquarelawofattractionwasnotactuallyoriginaltoNewton.Kepleralludedtothistypeofmathematicaldependenceinthecontextoflightin1604,arguing that as a bunchof light rays spreadsout, it has to illuminate a spherewhoseareagrowsasthesquareofitsradius.Iftheamountoflightisconserved,brightnessmustbeinverselyproportionaltothesquareofthedistance.Healsosuggestedasimilarlawfor‘gravity’,butwhathemeantwasahypotheticalforceexertedby theSun thatpropelledplanets along theirorbits, andhebelieved itwasinverselyproportionaltodistance.IsmaëlBullialdusdisagreed,arguingthat

thisforcemustbeinverselyproportionaltothesquareofthedistance.Gravitationalattraction, itsuniversality,andtheinversesquarelawwereall

very much in the air around 1670. It is also a very natural relationship, byanalogywiththegeometryoflightrays.InalecturetotheRoyalSocietyin1666RobertHookesaid:

Iwill explain a system of theworld very different from any yet received. It is founded on thefollowingpositions. 1.That all theheavenlybodieshavenotonly agravitationof their parts totheir own proper centre, but that they also mutually attract each other within their spheres ofaction.2.Thatallbodieshavingasimplemotion,willcontinuetomoveinastraightline,unlesscontinually deflected from it by some extraneous force, causing them to describe a circle, anellipse,orsomeothercurve.3.Thatthisattractionissomuchthegreaterasthebodiesarenearer.Astotheproportioninwhichthoseforcesdiminishbyanincreaseofdistance,IownIhavenotdiscoveredit.

In1679hewroteaprivate letter toNewton,3 proposingan inverse square lawdependenceforgravityinthissense.HewasdistinctlymiffedwhenexactlythatlawappearedinthePrincipia,eventhoughNewtongavehimcreditforit,alongwith Halley and ChristopherWren.We can sympathise with Hooke because,despite that, Newton got the lion’s share of the credit. In part this happenedbecausethePrincipiabecamesoinfluential,butthere’sanotherreason.Newtondidn’tmerely suggest such a law.He deduced it fromKepler’s laws, therebyputting it on a sound scientific footing. Hooke agreed that only Newton hadgiven ‘theDemonstrationof theCurvesgenerated thereby’, that is, thatclosedorbits are elliptical. (The inverse square law also permits parabolic andhyperbolicorbits,butthesearenotclosedcurvesandthemotiondoesnotrepeatperiodically.)

Nowadays we tend to see Newton as the first great rational thinker. WedisregardhisstrongbeliefinGodandhisbiblicalscholarship,andwesteadfastlyignore his extensive researches into alchemy, the rather mystical attempts toconvert matter from one form into another.Most of his writings on alchemywereprobablylostwhenhislaboratorycaughtfire,andtwodecadesofresearchwent up in smoke.Apparently his dogwas the cause:Newton is said to havescolded the animal, saying ‘Oh Diamond, Diamond, thou little knowest themischiefthouhastdone.’

Bethatasitmay,enoughbooksandpaperssurvivetosuggesthewasseekingthephilosopher’sstone,whichwouldturnleadintogold.Andpossiblytheelixiroflife,thekeytoimmortality.Onetitleis:NicholasFlammel,HisExpositionof

theHieroglyphicallFigureswhichhecaused tobepainteduponanArch inStInnocentsChurch-yard inParis. Togetherwith the SecretBooke ofArtephius,And the Epistle of Iohn Pontanus: Containing both the Theoricke and thePractickeofthePhilosophersStone.Here’sanexcerpt:

ThespiritofthisearthisyefireinwchPontanusdigestshisfeculentmatter,thebloodofinfantsin

wchye & baththemselves,theuncleangreenLionwch,saithRipley,isyemeansofjoyning

yetincturesof and ,thebrothwchMedeapouredonyetwoserpents,theVenusbymeditation

ofwch vulgarandthe of7eaglessaithPhilalethesmustbedecocted.

Herethesymbolshavethefollowingmeanings: =Sun, =Moon, =Mercury.Tothemoderneye,thislookslikemysticalnonsense.ButNewtonwasblazingtrails,andknewnotwheretheymightlead.Asithappens,thisonewasadeadend. In notes for a lecture that he never actually gave,4 the economist JohnMaynard Keynes called Newton ‘the last of the magicians… the lastwonderchild to whom the Magi could do sincere and appropriate homage’.Todaywemostly ignoreNewton’smystical aspect, and rememberhim for hisscientific and mathematical achievements. But by doing so, we lose sight ofmuchthatdrovehisremarkablemind.BeforeNewton,humanunderstandingofnature was deeply entwined with the supernatural. After Newton, we cameconsciously to recognise that the universe runs on deep patterns, expressiblethrough themediumofmathematics.Newtonhimselfwas a transitional figurewith a foot in each world, leading humanity away from mysticism towardsrationality.

8MasterofUsAll

LeonhardEuler

LeonhardEuler

Born:Basel,Switzerland,15April1707

Died:StPetersburg,Russia,18September1783

TODAY,LEONHARDEULERPROBABLYranksasthemostimportantmathematiciantobevirtuallyunknownto thegeneralpublic.Butduringhis lifetime,sogreatwas his reputation that in 1760, when Russian troops wrecked his farm inCharlottenburgduring theSevenYears’War,General IvanSaltykovpromptlypaidforthedamage.EmpressElizabethofRussiaaddedanother4000roubles,ahugeamount at that time.Thatwasn’t the endof thematter, either.EulerhadbeenamemberoftheStPetersburgAcademyfrom1726until,concernedaboutthedeterioratingpoliticalstateofRussia,heleftforBerlinin1741.In1766hereturned,havingnegotiateda3000roubleperyearsalary,agenerouspensionforhis wife, and promises that his sons would be looked after with lucrativepositions.

Lifewas by nomeans rosy, however. Euler had been suffering from poor

visionafterhelostthesightofhisrighteyein1738;nowhislefteyedevelopeda cataract and he went almost totally blind. However, he was the fortunatepossessorofanastonishingmemory;hecouldrecite thewholeofVirgil’sepicpoemtheAeneid,andgivenapagenumbercouldtellyouthefirstandlastlineonthatpage.Once,unabletogettosleep,Eulerfoundthetraditionalmethodofcountingsheeptootrivial,andpassedthetimebycalculatingthesixthpowersofallnumbersup toahundred.Severaldays later,hecould still remember themall.His sons JohannandChristophoftenactedas scribes, andsodidacademymembersWolfgangKrafftandAndersLexell.Euler’sgrandson-in-lawNikolaiFusshelpedtoo,becominganofficialassistantin1776.Allofthesepeoplehadsolid mathematical backgrounds and Euler discussed his ideas with them. Sosuccessfulwerethesearrangementsthathisalreadyprodigiousoutputincreasedsignificantlyafterhelosthissight.

Virtually nothing stopped Euler fromworking. In the 1740s, at the BerlinAcademy, he undertook a huge amount of administration, supervised thebotanicalgardens andobservatory,hired employees, handled the finances, anddealtwith publications such asmaps and calendars.He acted as consultant toKingFredericktheGreatofPrussiaontheimprovementoftheFinlowCanalandthe hydraulic system at the royal summer home of Sanssouci. The king wasunimpressed. ‘Iwanted tohave awater jet inmygarden:Euler calculated theforce of thewheels necessary to raise thewater to a reservoir, fromwhere itshouldfallbackthroughchannels,finallyspurtingoutinSanssouci.Mymillwascarried out geometrically and could not raise amouthful of water closer thanfiftypacestothereservoir.Vanityofvanities!Vanityofgeometry!’5

HistoricalrecordsshowthatFrederickwasblamingthewrongpersonandthewrongsubject.Theking’sarchitectforSanssouciwrotethathewantedalotoffountains,includingahugeonespurting30metresintotheair.TheonlysourceofwaterwastheriverHavel,1500metresaway.Euler’splanwastodigacanalfrom the river toapump,poweredbyawindmill.Thiswould raisewater toareservoir that created a difference in height of about 50 metres, providingenough pressure to drive the fountain. Construction began in 1748 andproceeded without any problems until the pipeline from the pump to thereservoirwasinstalled.Thiswasmadefromstripsofwoodheldbyironbands,likebarrels.Assoonas thebuildersstartedpumpingwaterup to thereservoir,thepipesburst.Drilledtree-trunksalsofailed,sometaltubeshadtobeused,buttheseweretoonarrowtoprovideanadequaterateofflow.Attemptstosortitallout continued until 1756, paused during the Seven Years War, and briefly

resumed. Then the King lost patience and the project was abandoned. ThearchitectblamedFrederick,whohadahabitofconceivingmagnificentstructuresbutfailingtoprovidethenecessarymoney.Hisreportlistseveryoneresponsibleforthefailure.Eulerisnotamongthem.

In fact, Euler’s work on the design initiated the theory of hydraulic flowthroughpipes,analysinghowthemotionofthewateraffectsthepressureinthepipe.Inparticular,heshowedthatmotioncausesthepressuretoincrease,evenwhen there isnodifference inheight.Traditionalhydrostaticsdoesnotpredictthis. Euler calculated the pressure increase, made recommendations about thepump and the pipeline, and gave explicit warnings that the builders werebunglersandtheprojectwouldinevitablyfail.Hewrote:

Ihavemadecalculationsaboutthefirsttrialsatwhichthewoodentubesburst,assoonasthewaterreachedaheightof[20metres].Ifindthatthetubesactuallysustainedapressurecorrespondingtoa[100metre]highwatercolumn.Thisisacertainindicationthatthemachineisstillfarfromitsperfection…atallcostsonehastouselargertubes.

He insisted that lead tubes should be used, not wooden ones, and that thethickness of the lead should be deduced from experiments. His advice wasignored.

Frederick never had any great respect for scientists, preferring artisticgeniuses such as Voltaire. Hemade fun of Euler’s blind eye, calling him the‘mathematical Cyclops’. When Frederick wrote about the Sanssouci fiasco,thirtyyearshadpassed,andthelong-departedEulerwasaconvenientscapegoat.Thelingeringbeliefthathewasanivory-towermathematicianwithnopracticalabilitiesiscompletenonsense.Headvisedthegovernmentoninsurance,finance,artillery,andthelottery.Hewasthemathematicalfix-itmanofhisday.Andallthetimehekeptupasteadyflowofpenetratingoriginalresearchandtextbooksthatbecameinstantclassics.

Hewasstillworkingthedayhedied.Inthemorning,muchasusual,hegaveone of his grandchildren amathematics lesson,made some calculations aboutballoons on two small chalkboards, and discussed the recent discovery of theplanet Uranus with Lexell and Fuss. Late that afternoon he suffered a brainhaemorrhage,said‘Iamdying’,andexpiredsixhourslater.InhisEulogyforMrEuler,NicolasdeCondorcetwrote‘Eulerceasedtoliveandcalculate.’Forhim,mathematicswasasnaturalasbreathing.

Euler’sfatherPaulstudiedtheologyatBaselUniversityandbecameaProtestantminister.HismotherMargaret (née Brucker)was the daughter of a Protestantminister.ButPaulalsotooklecturesfromthemathematicianJacobBernoulli,inwhosehousehelivedasanundergraduate,alongwithJacob’sbrotherJohann,afellowstudent.TheBernoullis are thearchetypal exampleof amathematicallytalentedfamily,andforfourgenerationsnearlyallofthemstartedoutpursuingmoreconventionalcareers,butendedupdoingmathematics.

Euler became a student at BaselUniversity at the age of 13, in 1720.Hisfatherwantedhimtobeapastor.By1723hehadcompletedamaster’sdegreecontrasting the philosophies of Newton andDescartes, but although hewas adevoutChristian, theologyfailedtoappeal,andneitherdidHebrewnorGreek.Mathematicswasanothermatteraltogether:Eulerlovedit.Andheknewhowtogo about making a career in it, too. His unpublished autobiographical papersincludethispassage:

IsoonfoundanopportunitytobeintroducedtoafamousprofessorJohannBernoulli…True,hewasverybusyandsorefusedflatlytogivemeprivatelessons;buthegavememuchmorevaluableadvice to start reading more difficult mathematical books on my own and to study them asdiligentlyasIcould;ifIcameacrosssomeobstacleordifficulty,Iwasgivenpermissiontovisithim freely every Sunday afternoon and he kindly explained to me everything I could notunderstand.

Johannquicklyspotted theyoungman’sastonishing talent,andPaulagreed tolethissonchangesubjecttostudymathematics.NodoubtPaul’slong-standingfriendshipwithJohannhelpedgreasethewheels.

Eulerpublishedhisfirstpaperin1726,andin1727heenteredapaperfortheParis Academy’s annual grand prize, which on that occasion was to find theoptimalarrangementofmastsonasailingship.PierreBouguer,anexpertinthearea,won,butEulercamesecond.ThisachievementcametotheattentionofStPetersburg,andwhenNicolausBernoullidiedandhispositionbecamevacant,itwasofferedtoEuler.Aged19,hesetoffforRussia,aseven-weektrip:alongtheRhinebyboat,thenbywagon,andbacktoaboatforthefinalleg.

Between1727and1730healsoservedasamedicallieutenantintheRussiannavy,butwhenhewasmadeafullprofessorheleftthenavy,andsoonbecameapermanentmemberoftheacademy.In1733DanielBernoulliresignedfromhischairatStPetersburgtoreturntoBasel,andEuler tookoverasamathematicsprofessor. His finances had improved enough for him to marry, and he dulyspliced the knot with Katarina Gsell, the daughter of an artist at the local

Gymnasium(highschool).Eventuallythecoupleproducedthirteenchildren,ofwhicheightdied in infancy,andEuler remarked thathehaddonesomeofhisbestworkwhileholdingababyandsurroundedbychildrenplaying.

Hehadpersistenteyesightproblems,exacerbatedbya fever in1735whichnearlykilledhim.Asalreadyremarked,hebecamenearlyblindinoneeye.Thishadvery little effect onhis productivity–nothing everdid.Hewon theParisAcademy’sgrandprizein1738andagainin1740;eventuallyhewonittwelvetimes. In 1741, as Russian politics became increasingly turbulent, he left forBerlin,becomingtutortoanieceofFredericktheGreat.His25yearsatBerlinproduced 380 papers. He wrote books on analysis, artillery and ballistics,calculus of variations, differential calculus, themotion of themoon, planetaryorbits, shipbuilding and navigation, and even popular science, in Letters to aGermanPrincess.

When Pierre Louis Moreau de Maupertuis died in 1759, Euler becamepresidentoftheBerlinAcademyinallbutthetitle,whichherefused.FouryearslaterKingFrederickofferedthepresidencytoJeanleRondd’Alembert,whichwasn’tgreatlytoEuler’sliking.D’Alembertdecidedhedidn’twanttomovetoBerlin, but the damage was done, and Euler decided it was time to head forpasturesnew.Or,inthiscase,pasturesold,forhewentbacktoStPetersburgatthe invitation of Catherine the Great. And there he ended his days, havingenrichedmathematicsbeyondmeasure.

It’s almost impossible to convey either Euler’s brilliance, or the variety andoriginality of his discoveries, in anything shorter than a book. Even then, itwouldbeachallenge.Butwecangraspa littleofwhatheachieved,andgainsome insight intohis remarkable abilities. I’ll startwithpuremathematics andmoveontoapplied, ignoringchronologytomaintainsomekindofflowof theideas.

Firstandforemost,Eulerhadanamazingintuitionforformulas.Inhis1748Introduction toAnalysisof the Infinitehe investigated therelationbetween theexponential and trigonometric functions for complex numbers, leading to theformula

eiθ=cosθ+isinθ

From this, setting θ = π radians = 180°, it’s possible to derive the famous

equation

eiπ+1=0

relatingthetwoenigmaticconstantseandπ,andtheimaginarynumberi.Heree= 2·178… is the base of natural logarithms and i is the symbol that Eulerintroduced for the square root of minus one, still standard today. Now thatcomplexanalysisisbetterunderstood,thisrelationshipdoesn’tcomeasmuchofa surprise, but in Euler’s day it was mind-blowing. Trigonometric functionscame from the geometry of circles and the measurements of triangles; theexponential functioncamefromthemathematicsofcompound interestand thecalculating tool of logarithms. Why should these things be so intimatelyconnected?

Euler’suncannyknackwithformulasledtoatriumphthatbroughthimgreatfameat theageof28,whenhesolved theBaselproblem.Mathematicianshadbeen finding interesting formulas for the sums of infinite series, perhaps thesimplestbeing

TheBaselproblemwastofindthesumofthereciprocalsofthesquares

Many famousnameshad sought theanswerwithout success:Leibniz,Stirling,de Moivre, and three of the most proficient Bernoullis: Jacob, Johann, andDaniel. Euler trumped them all by proving (or, at least, doing a calculationindicating–rigourwasnothisstrongpoint)thatthesumisexactlyπ2/6.

Asimplerinfinitesum,the‘harmonicseries’ofreciprocalsoftheintegers,is:

and thisdiverges– its sum is infinite.Euler, unfazed, foundahighly accurateapproximateformula:

whereγ,nowcalledEuler’sconstant,is,to16decimalplaces:

0·5772156649015328…

0·5772156649015328…

Eulerhimselfcalculateditsvaluetothatmanydecimalplaces.Byhand.Number theory naturally attracted Euler’s attention. He took much of his

inspiration from Fermat, and correspondence with his friend Goldbach, anamateurmathematician, provided furthermotivation.His solution of theBaselproblem led him to a remarkable relation between primes and infinite series(Chapter 15). He obtained proofs of several basic theorems stated by Fermat.Onewastheso-called‘LittleTheorem’,asdistinctfromtheLastTheorem.Thisstatesthatifnisaprimeandaisnotamultipleofn,thenan–aisdivisiblebyn.Innocuous as this statement may appear, it’s now the starting point for someallegedlyunbreakablecodes,widelyusedontheinternet.Healsogeneralisedtheresulttocompositen,introducingthetotient(orEuler)functionφ(n).Thisisthenumberofintegersbetween1andnthathavenoprimefactorincommonwithn.Heconjecturedthelawofquadraticreciprocity,laterprovedbyGauss(Chapter10);characterisedallprimesthatarethesumoftwosquares(2,alloftheform4k+1,noneoftheform4k+3),andimprovedLagrange’stheoremthateverypositiveintegeristhesumoffoursquares.

His textbooks on algebra, calculus, complex analysis, and other topicsstandardised mathematical notation and terminology, much of it still in usetoday,suchasπforpi,eforthebaseofnaturallogarithms,iforthesquarerootofminusone,theΣnotationforasum,andf(x)forageneralfunctionofx.HeevenbroughttogetherNewton’sandLeibniz’snotationsindifferentialcalculus.

Iliketodefineamathematiciannotas‘someonewhodoesmathematics’,butas‘someone who spots an opportunity for doingmathematics when no one elsewould’.Eulerseldommissedanopportunity.Twoexampleskick-startedtheareanowknownascombinatoricsordiscretemathematics,which isaboutcountingandarrangingfiniteobjects.

Thefirst,in1735,wasapuzzleaboutthecityofKönigsberginPrussia(nowKaliningrad in Russia). Situated on the river Pregel, the city had two islandsconnectedtoeachotherandthebanksoftheriverbysevenbridges.Thepuzzlewastofindapaththroughthecitythatcrossedeverybridgeonceandonceonly.Thestartandfinishcouldbeindifferentplaces.Eulerprovedthatnosuchpathexists,bytacklingthemoregeneralquestionforanyarrangementofislandsandbridges.Heprovedthatapathexistsifandonlyifatmosttwoislandsareattheendsofanoddnumberofbridges.Todayweinterpretthistheoremasoneofthe

firstingraphtheory,thestudyofnetworksofpointsconnectedbylines.Euler’sproofwasalgebraic,involvingasymbolicrepresentationofthepathusinglettersforislandsandbridges.It’seasytoprovehisconditionisnecessaryforapathtoexist;theharderpartistoproveitsufficient.

MapofthesevenbridgesofKönigsberg,fromEuler’s

‘Solutioproblematisadgeometriamsituspertinentis’.

The second combinatorial problem, which he posed in 1782, was the 36officerspuzzle.Givensixregiments,eachcomprisingsixofficersofsixdifferentranks,cantheybearrangedina6×6squaresothatnoroworcolumncontainstwoofficers in the same regiment or of the same rank?Euler conjectured thatthis is impossible, a resultwhoseproof had towait forGastonTarry in 1901.TheunderlyingstructurehereisaLatinsquare,inwhichncopiesofnsymbolsmustbearrangedinann×nsquaresothateachsymboloccursexactlyonceineach row and column. The 36 officers are required to form two ‘orthogonal’Latin squares, one for the regiment and another for the rank,with all possiblepairs included. Latin squares have applications to experimental design forstatistical tests, andwidespreadgeneralisationsknownasblockdesignsappearinseveralbranchesofmathematics.Sudokuisavariationonthetheme.

TheresultsI’vediscussedbarelyscratchthesurfaceofEuler’sprodigiousoutputofpuremathematics,buthewasatleastasprolificinappliedmathematicsandmathematicalphysics.

In mechanics, he systematised and advanced the state of the art for themotionofaparticleinhisMechanicsof1736.Amajorinnovationwastheuseofanalysisinplaceofgeometry,whichunifiedthetreatmentofpreviouslydiverseproblems. He followed this with a book about ship design, beginning withhydrostatics, which also introduced differential equations for the motion of a

rigidbody.Thisthemewasdevelopedin1765inTheoryoftheMotionofSolidBodies, inwhich he defined a coordinate system now known asEuler angles,relatingthemtothebody’sthreeaxesofinertiaanditsmomentsofinertiaaboutthose axes. The axes of inertia are distinguished lines representing specialcomponents of the body’s spin; the corresponding moment determines theamountofspinrelativetothechosenaxis.InparticularhesolvedhisequationsfortheEulertop,abodywithtwoequalmomentsofinertia.

InfluidmechanicshesetupbasicequationsnowcalledtheEulerequations,whichremainofinteresteventhoughtheyignoreviscosity.Hestudiedpotentialtheory, with applications to gravity, electricity,magnetism, and elasticity. Hiswork on light was instrumental in the success of the wave theory, whichprevailed until the appearance of quantum mechanics in 1900. Some of hisresults in celestial mechanics were used by the astronomer Tobias Mayer tocalculatetablesofthemotionofthemoon.In1740hewroteMethodforFindingCurvedLines–thefulltitleismuchlonger–initiatingthecalculusofvariations.This seeks curves and surfaces that minimise (or maximise) some relatedquantity, such as length or area. All of his books are clear, elegant, andorganised.

Otherworkscovertopicssuchasmusic,map-making,andlogic–therearevery few areas of mathematics that didn’t attract Euler’s attention. LaplacesummedupEuler’sroleperfectly:‘ReadEuler,readEuler,heisthemasterofusall.’

9TheHeatOperator

JosephFourier

JeanBaptisteJosephFourier

Born:Auxerre,France,21March1768Died:Paris,France,16May1830

IT WAS 1804, and mathematical physics was in the air. Johann Bernoulli hadapplied Newton’s laws of motion, combined with Hooke’s law for the forceexertedbyastretchedspring, to thevibrationsofaviolinstring.His ideas ledJean le Rond d’Alembert to formulate the wave equation. This is a partialdifferential equation, relating the rates of change of the shape of the string

relativetobothspaceandtime.Itgovernsthebehaviourofwavesofallkinds–waterwaves,soundwaves,vibrations.Similarequationshadalsobeenproposedformagnetism,electricity,andgravity.NowJosephFourierdecidedtoapplythesame methods to another area of physics, the flow of heat in a conductingmedium.Afterthreeyearsofresearch,heproducedalengthymemoiraboutthepropagationofheat. Itwas read to theParis Institute, tomixed reactions, soacommitteewassetup toexamine it.When the reportwaswritten, itwasclearthatthecommitteewasn’thappy.Theyhadtworeasonsforthis,onegood,onebad.

Jean-BaptisteBiothadalerted themtowhatheclaimedwasan issue in thederivation of the equation for the flow of heat. In particular, Fourier hadn’tmentionedan1804paperofhis.Thiswasthebadreason,becauseBiot’spaperwas wrong. The good reason was that a key step in Fourier’s argument,transforming a periodic function into an infinite series of sines and cosines ofmultiples of a given angle, had not been establishedwith due rigour. Indeed,Euler and Bernoulli had been arguing about the same idea for years in thecontextof thewaveequation.Fourierhastenedtoclarifyhisreasoning,but thecommitteeremainedunsatisfied.

Nevertheless, the problemwas considered important andFourier hadmadesignificant inroads into it, so the institute announced that itsprizeproblem for1811wouldbetheflowofheatinasolid.Fourieraddedsomefurtherresultstohismemoir,oncoolingandradiationofheat,andsubmittedit.Anewcommitteeawarded him the prize, but stated the same reservation about trigonometricseries:

Themanner inwhich the author arrives at these equations is not exempt of difficulties and hisanalysistointegratethemstillleavessomethingtobedesiredonthescoreofgeneralityandevenrigour.

Itwasnormalfor theprizewinningmemoir tobepublished,but thecommitteedeclinedtodoso,becauseofthiscriticism.

In 1817, Fourierwas elected amember of theParisAcademyof Sciences.Fiveyearslaterthesecretaryforthemathematicssection,JeanDelambre,died.FrançoisArago,Biot, andFourier applied for theposition, butAragodroppedoutandFourierwonbyalandslide.Soonafter,theacademypublishedFourier’sAnalytic Theory of Heat, the memoir that had won the prize. This looks likeFourierhavingaslydigatthecommittee,butithadbeenDelambrewhosentitforpublication.Still,itmusthavegivenFourieragreatdealofsatisfaction.

Fourier’sfatherwasatailor,whosefirstmarriageproducedthreechildren.Whenhis wife died, he remarried, and the second marriage produced no less thantwelvechildren,ofwhomJosephwas theninth.When theboywasnineyearsold his mother died, and his father died a year afterwards. He started hiseducationataschoolrunbyAuxerrecathedral’smusicmaster,studyingFrenchandLatin, atwhich he excelled. In 1780, aged 12, hemoved on to the city’sÉcoleRoyaleMilitaire.Hedidwell in literature,butby theageof13his realtalentwasemerging:mathematics.Hereadadvancedtexts,andwithinayearhehad worked his way through all six volumes of Étienne Bézout’s Course ofMathematics.

In1787,intendingtobecomeapriest,hewenttotheBenedictineabbeyofStBenoit-sur-Loire,butremainedabsorbedinmathematics.Hedecidednottotakereligious vows, left the abbey in 1789, and presented a paper on algebraicequations to the academy.A year after that heworked as a teacher at his oldschool.Tocomplicatematters,hebecameamemberofthecity’srevolutionarycommitteein1793,writingthatitwaspossible‘toconceivethesublimehopeofestablishing among us a free government exempt from kings and priests’ anddedicating himself to the revolutionary cause. However, the violence of theTerrorduringtheearlydaysoftheFrenchRevolutionrepelledhim,andhetriedtoresign.Thisprovedpoliticallyimpossible,andhewasirrevocablytiedupintherevolution.Factionalinfightingwascommonamongtherevolutionaries,allofwhomhaddifferentideasaboutthecoursethattherevolutionshouldfollow,and Fourier became involved in the public support of one faction inOrléans.This led to his arrest and the prospect of Madame Guillotine. At that pointMaximilien Robespierre, one of the most influential revolutionaries, wasguillotined,thepoliticalatmosphereshifted,andFourierwassetfree.

His mathematical career flourished under the watchful eyes of the greatFrench mathematicians of the period. He attended the École Normale, beingamongitsfirststudentswhenitopenedin1795.HetookcoursesfromLagrange,who he considered to be the top scientist in Europe; Legendre, who didn’tgreatly impress him; and Gaspard Monge. He obtained a post at the ÉcoleCentraledesTravauxPublics, later renamed theÉcolePolytechnique.Hispastcaught upwith him, and hewas arrested oncemore, and jailed.Hewas soonreleased, however, for reasons that remain obscure, but probably involved aflurryofbehind-the-scenesactivitybyhisstudentsandcolleagues,plusanotherchange in the political scenario. By 1797 he had come up smelling of roses,

inheritingLagrange’schairinanalysisandmechanics.NapoleonnowinvadedEgypt.Fourierjoinedhisarmyasascientificadviser,

alongsideMongeandÉtienne-LouisMalus.AfterNapoleonhadenjoyed someearlysuccesses,HoratioNelsondestroyed theFrenchnavy in theBattleof theNileandNapoleonwasstuck inEgypt.Fourierbecameanadministrator there,set up an educational system, and did some archaeology. He was a foundermemberofthemathematicsdivisionoftheCairoInstitute,organisingreportsonthe expedition’s scientific discoveries. He introduced Jean-FrançoisChampolliontotheRosettaStone,akeystepinChampollion’sdeciphermentofhieroglyphs.

In1799NapoleonlefthisarmybehindinEgyptandreturnedtoParis.Fourierfollowed him in 1801, and resumed his professorship. But Napoleon decidedFourierwas such an able administrator that he should bemade prefect of thedepartment of Isère. It was an offer that the reluctant Fourier felt unable torefuse,sohemovedtoGrenoble.ThereheoversawthedrainingoftheBorgoinswamps, supervised the construction of the Grenoble–Turin highway, andworkedonNapoleon’smassiveDescriptionofEgypt,publishedin1810.Fouriermoved toEngland in 1816, but soon returned to France, becoming permanentsecretaryof theacademy.While inEgypt,hehadexperiencedheartproblems,whichcontinuedafterhisreturntoFrance,withfrequentboutsofbreathlessness.InMay1830hefellonthestairs,makingtheconditionmuchworse,andhediedshortlyafterwards.Hisnameisoneofthe72inscribedontheEiffelTower.Butasfarasmathematicsisconcerned,it’sFourier’stimeinGrenoblethathadthemost importantconsequences,because itwas there thathecarriedouthisepicresearchonheat.

Fourier’sheatequationdescribes,symbolically,theflowofheatinaconductingrod–say,onemadeofmetal.Ifpartof therodishotter thanitssurroundings,theheatspreadsintonearbyregions;ifthatpartiscolderthanitssurroundings,itgets hotter at the expense of nearby regions. The greater the temperaturedifference, the faster the heat spreads. The rate at which heat flows alsodetermines how quickly the entire rod cools down. Fourier’s heat equationdescribeshowtheseprocessesinteract.

Initially, different parts of the rod can be heated or cooled to differenttemperatures,creatingatemperatureprofileorheatdistribution.Solutionsoftheequationdescribehow this initialdistributionofheatalong the rodchangesas

timepasses.ThepreciseformoftheequationledFouriertoasimplesolution,ina special case. If the initial temperature distribution is a sine curve, with amaximumtemperatureinthemiddlewhichtailsawaytowardstheends,thenastimepasses the temperaturehas thesameprofile,but thisdecaysexponentiallytowardszero.WhatFourierreallywantedtoknow,however,waswhathappensforanyinitialtemperatureprofile.Suppose,forexample,thatinitiallytherodisheatedalonghalfitslength,andkeptmuchcooleralongtheotherhalf.Thentheinitialprofileisasquarewave.That’snotsinusoidal.

Howtogetasquarewavefromsinesandcosines.Left:Thecomponentsinusoidalwaves.Right:Theirsumandasquarewave.ThefirstfewtermsoftheFourierseriesare

shown:additionaltermsmaketheapproximationtoa

squarewaveascloseaswewish.

To obtain solutions despite this obstacle, Fourier exploited an importantfeature of his equation: it’s linear. That is, any two solutions can be addedtogether to give another. If he could represent the initial profile as a linearcombination of sine curves, then the solution would be the correspondingcombinationofexponentiallydecayingsinecurves.Hediscoveredthatasquarewave can be represented in this form, provided you take infinitelymany sinecurvesandcombineprofilesoftheformsinx,sin2x,sin3x,sin4x,andsoon.Togetanexactsquarewave,youneed infinitelymany terms like this. In fact,forarodoflength2π,theformulais

whichisreallyratherpretty.Fourier’s calculations convinced him that if you use cosine terms as well,

infinite trigonometric series can represent any initial temperature profile,howevercomplicated,evenifithasdiscontinuitiesliketheoneoccurringinthesquarewave.Sohecouldwritedownasolutionofhisheatequationinthesameform.Eachtermdecaysatadifferentrate;themorewigglesinthesineorcosinecurve,thefasteritscontributiondecays.Sotheprofilechangesitsshapeaswellas its size.Healsoderivedageneral formula for the terms in theseries,using

integration.The committee was sufficiently impressed to award him the prize, but its

members were worried about Fourier’s claim that his method applies to anyinitial profile, even one with many jumps and other discontinuities, like thesquarewaveonlyworse.Fourierappealed tophysical intuitionas justification,but mathematicians always worry that intuition involves hidden assumptions.Indeed,neitherthemethodnortheproblemitraisedwerereallynew.Thesameissue had already arisen in connectionwith thewave equation, causing a rowbetween Euler and Bernoulli, and Euler had published the same integralformulas as Fourier for the series expansion,with a simpler andmore elegantproof.ThebigdifferencewasFourier’sassertionthathismethodwasvalidforall profiles, continuous or discontinuous, a claim that Euler had shied awayfrom.Thisquestionwasalessseriousissueforwaves,becauseadiscontinuousprofilewouldmodelabrokenviolinstring,whichwouldn’tvibrateatall.Butforheat, profiles such as the square wave had sensible physical interpretations,subject to idealised modelling assumptions. That said, the underlyingmathematical issue was the same in both cases, and at that time it remainedunresolved.

Withhindsight,bothsidesinthedisputewerepartlyright.Thebasicproblemis that of convergence of the series: does the infinite sum have a sensiblemeaning?For trigonometric series, this is a delicate issue, complicated by theneedtoconsidermorethanoneinterpretationof‘converge’.Acompleteanswerrequired three ingredients: a new theory of integration developed by HenriLebesgue, the language and rigourof set theory as inventedbyGeorgCantor,andaradicallynewviewpoint foundbyBernhardRiemann.Theupshot is thatFourier’smethod isvalid forabroadbutnotuniversalclassof initialprofiles.Physicalintuitionprovidesagoodguideforthese,andthey’readequateforanysensiblephysicalsystem.Butmathematically,youshouldn’tclaimtoomuch,forthere are exceptions. So Fourier was right in spirit, but his critics had validpointstoo.

Inthe1820s,Fourierwasoneof thepioneersofresearchintoglobalwarming.Notchangesinclimatecausedbyman-madeglobalwarming,however;hejustwanted tounderstandwhy theEarthwaswarmenough to sustain life.To findout, he applied his understanding of heat flow to our home planet. The onlyobvioussourceofheatwas the radiation that theEarth received from theSun.

Theplanetradiatessomeofthisheatbackintospace,andthedifferenceshouldaccount for theobservedaveragesurface temperature.But itdidn’t.Accordingto his calculations, the Earth ought to be noticeably colder than it actually is.Fourier deduced that other factors must be involved, and published papers in1824 and 1827, investigating what theymight be. Eventually he decided thatextra radiation from interstellar space was themost likely explanation, whichturned out to be hopelesslywrong. But he also suggested (and discarded) thecorrect explanation: that the atmosphere can act as a kind of blanket, keepingmoreheatinandallowinglesstoradiateaway.

HisinspirationwasanexperimentcarriedoutbythegeologistandphysicistHorace-Bénédict de Saussure. Investigating the possibility of using the Sun’sraystocookfood,deSaussurediscoveredthataninsulatedboxwiththreelayersofglass,widelyseparatedbylayersofair,wasthemostefficientofhisdesigns,and that it could reach 110°C, both on the warm plains and high in the coldmountains.Thereforethewarmingmechanismlargelydependedontheairinsidetheboxandtheeffectoftheglass.FourierguessedthattheEarth’satmospheremight act in the same manner as de’ Saussure’s solar oven. The phrase‘greenhouseeffect’mayderivefromthissuggestion,butitwasfirstusedbyNilsEkholmin1901.

Ultimately, Fourier was unconvinced that this effect was the answer hesought,inpartbecausetheboxprecludesconvection,whichtransportsheatoverlargedistancesintheatmosphere.Hedidn’tappreciatethespecialroleofcarbondioxideandother‘greenhousegases’,whichabsorbandemitinfraredradiationinamannerthattrapsmoreheat.Theprecisemechanismiscomplicated,andtheanalogy with a greenhouse is misleading, because a greenhouse works byconfiningwarmairinanenclosedspace.

Fourieralsodevelopedaversionofhisequationforheatflowinregionsoftheplane,orofspace,intermsofwhatwenowcalltheheatoperator.Thiscombineschangesintemperatureatagivenlocationwithdiffusionofheatintooroutofitsvicinity.EventuallymathematicianssortedoutthesenseinwhichFourierseriessolve the heat equation, for spaces of any dimension. By then it had alreadybecomeapparentthatthemethodhasfarbroaderapplications–nottoheatatall,buttoelectronicengineering.

This is a typical example of the unity and generality ofmathematics. Thesametechniqueappliestoanyfunction,notjustaheatprofile.Itrepresentsthat

function as a linear combinationof simpler components,making it possible toprocess the data and extract information from some rangeof components. Forexample,aversionofFourieranalysisisusedforimagecompressionindigitalcameras – encoding an image as a combination of simple patterns based oncosinefunctions,whichreducesthememoryrequiredtostoreit.

Nearly two hundred years on, Fourier’s initial insight has become anindispensable tool for mathematicians, physicists, and engineers. Periodicbehaviour is widespread, and whenever it happens, you can work out thecorrespondingFourierseriesandseewhereitleads.Ageneralisation,theFouriertransform, applies to non-periodic functions. A discrete analogue, the fastFourier transform, is one of the most widely used algorithms in appliedmathematics, with applications to signal processing and high-precisionarithmetic incomputeralgebra.Fourier serieshelpseismologists tounderstandearthquakesandcivilengineerstodesignearthquake-proofbuildings.Theyhelpoceanographers tomap thedeepoceans andoil companies to prospect for oil.Biochemistsusethemtoworkoutthestructureofproteins.TheBlack–Scholesequation,whichtradersuse topricestockmarketoptions, isacloserelativeoftheheatequation.Thelegacyoftheheatoperatorisalmostunbounded.

10InvisibleScaffolding

CarlFriedrichGauss

JohannCarlFriedrichGauss

Born:Braunschweig,DuchyofBraunschweig-Wolfenbüttel,30April1777

Died:Göttingen,KingdomofHanover,23February1855

THEYEARIS1796,thedate30March.TheyoungCarlFriedrichGausshasbeentryingtodecidewhethertostudylanguagesormathematics.Nowhehasmadea

very significantbreakthrough,usingalgebraicmethods touncoverageometricconstruction that had gone unnoticed since the time of Euclid,more than twothousandyearsago.Usingonlythetraditionalgeometricinstrumentsofrulerandcompasses,hecanconstructaregularheptadecagon.Thatis,a17-sidedpolygonwithallsidesequalandallinterioranglesequal.Notjustapproximately–that’seasy–butexactly.Fewpeoplearegiventheopportunitytodiscoversomethingthat no one else had even suspected for two millennia; even fewer take it.Moreover,despiteitsesotericnature,themathematicsisveryoriginalandofthehighestbeauty,thoughofitselfithasnopracticalimportance.

Euclid’s Elements sets the scene. It gives constructions for an equilateraltriangle,asquare,aregularpentagon,aregularhexagon:regularpolygonswiththree,four,five,andsixsides.Whataboutsevensides?No,nothing.Ofcourseeightiseasy–drawasquaresurroundedbyacircleandcutitssideinhalf;thenextendradiithroughthosemidpointstocreatefournewcornersonthecircle.Ifyoucanconstructanyregularpolygon,thesametrickconstructsonewithtwiceasmanysides.Nine?No,Euclid remainsmute.Ten is easyagain: justdoublefive. Nothing about eleven. Twelve is twice six, straightforward. Thirteen,fourteen–no.Fifteencanbedonebycombiningtheconstructionsforthree-andfive-sidedpolygons.Sixteen:doubleupeightsides.

AsfarasEuclidgoes,that’sit.Three,four,five,fifteen,andallmultiplesofthose numbers by powers of two. Seventeen? Crazy. Even more so, becauseGauss’s method makes it pretty clear that seven, nine, eleven, thirteen, andfourteen sides are impossiblewith ruler-and-compass constructions.But, crazyornot,it’strue.There’sevenasimplereason(althoughwhyit’sthereasonisbyno means simple). Seventeen is a prime number, and subtracting one givessixteen,apoweroftwo.

This combination, Gauss realises, holds the key to ruler-and-compassconstructions of regular polygons. In a small notebook, he writes: ‘Principiaquibus innititur sectio circuli, ac divisibilitus eiusdem geometrica inseptemdecimpartesetc.’Toparaphrase:thecirclecanbedividedintoseventeen[equal]parts.It’sthefirstentryinhisnotebook.Later,145otherdiscoveriesareadded,eachonerecordedasabrief,oftencryptic,note.

Languages?Ormathematics?Nocontest.

Gausswasbornintoapoorfamily.HisfatherGerhardtookajobinBrunswick

(Braunschweig) as a gardener and later worked as a canal tender and abricklayer.Gauss’smotherDorothea(néeBenze)wassoilliteratethatshedidn’tevenrecordherson’sbirthdate.However,shewasbynomeansunintelligent,andrememberedthathersonhadenteredtheworldonaWednesday,eightdaysbeforethefeastoftheascension.Characteristically,Gausslaterusedthislimitedinformationtofigureouttheexactday.

Gauss’s intellectualbrilliancequicklybecameapparent.Whenhewas threeyearsoldhisfatherwashandingoutwagestosomelabourers.SuddenlyyoungCarl piped up to the effect: ‘No, father, that’s wrong, it should be –’ Arecalculation proved the boy right. Recognising their son’s potential, Gauss’sparentswent to considerable lengths to help himdevelop it.WhenGausswaseight, his schoolteacherBüttner set the class an arithmetic problem. It’s oftenstated that this was to add the numbers from 1 to 100, but that’s probably asimplification. The actual problemwas probablymore complicated, but alongsimilarlines:addalotofequallyspacednumbers.Theadvantageofsuchasumtotheteacherisstraightforward:there’sacunningshortcut.Avoidrevealingittoyour unsuspecting pupils, and you can tie them up for hours in a giganticcalculation,whichtheywillalmostcertainlygetwrong.Theeight-year-oldsatathisdeskforamoment,scribbledasinglenumberonhisslate,marcheduptotheteacher’s desk and slapped it face down. ‘Ligget se,’ he said, in his countrydialect: ‘There it lies.’Thiswas thecommonway topresentyouranswer,andimpliednodisrespect.Astheotherpupilslabouredandtheirslatesslowlypiledup,BüttnerwatchedGauss,whowaitedcalmlyathisdesk.Whentheslateswereinspected,onlyGauss’sanswerwascorrect.

Suppose the problem really was 1 + 2 + 3 +… + 99 + 100.What’s theshortcut? Well, first you need the imagination to appreciate that there is ashortcut. Then you have to find it. The same trick also works for morecomplicatedsumsofthiskind.It’swidelybelievedthatGaussmentallygroupedthenumbersinpairs:onefromthebeginning,onefromtheend.Now

1+100=101

2+99=101

3+98=101

and the pattern continues (because the first number increases by one, but theseconddecreasestocompensate)untileventually

50+51=101

50+51=101

Thereare50suchpairs,eachaddsto101,sothegrandtotalis50×101=5050.Liggetse.

Büttner realised he had a genius on his hands, and gave Gauss the finestarithmetic text he could buy. The boy read it like a novel, andmastered it asquickly.‘Heisbeyondme.Icanteachhimnothingmore,’saidBüttner.Buthecould still help his prodigy protégé. In 1788 Gauss entered the Gymnasium,aidedbyBüttnerandhisassistantMartinBartels.Therehedevelopedatasteforlinguistics,learningHighGermanandLatin.

Bartels knew someof the great and the goodofBrunswick, and told themabout Gauss’s talents. The news reached the ears of Duke Karl WilhelmFerdinandofBrunswick-Wolfenbüttel,andin1791,aged14,Gausswasgrantedanaudience.Hewasshyandmodest–andincrediblybright.Theduke,charmedand impressed in equal measure, promised to provide money for the boy’seducation. In 1792, sponsored by the duke, Gauss entered the CollegiumCarolinum.Therehedevelopedhisinterestinlanguages,especiallytheclassics.Gerharddeclaredsuchimpracticalstudiesawasteoftime,andDorotheaputherfoot down. Their son was to receive the finest education possible, and thatincludedGreekandLatin.Endof.

Gauss had been pursuing dual interests –mathematics and languages – forsome time. He had independently rediscovered (without proofs) five or siximportantmathematical theorems,amongthemthelawofquadraticreciprocityinnumbertheory,whichI’lldescribelater,andheconjecturedtheprimenumbertheorem,which states that the number of primes less than x is approximatelyx/logx. Itwasprovedin1896byCharlesHadamardandCharlesdelaVallée-Poussin,independently.Theyear1795sawGaussleaveBrunswicktoattendtheUniversityofGöttingen.HisprofessorAbrahamKästnermainlywrotetextbooksand encyclopedias, and did no original research. Gauss was unimpressed andmadehisopinioncrystalclear.Hewasheadingforacareerinlanguageswhenthe gods of mathematics came to his rescue in spectacular fashion with theheptadecagon.

TounderstandhowradicalGauss’sdiscoverywas,weneedtogobacktoancient

Greece, over two thousand years ago. In the Elements, Euclid systematicallycodifiedthetheoremsofthegreatGreekgeometers.Hewasasticklerforlogicanddemandedthateverythingshouldbeproved.Well,almosteverything.Youhadtostartsomewhere,withassumptionsthatwerenotproved.Euclidclassifiedtheseintothreetypes:definitions,commonnotions,andpostulates.Wenowcallthelasttwoaxioms.

On the basis of these assumptions, Euclid developed a large part ofGreekgeometry, step by step. To modern eyes, some assumptions were missing –subtleones,suchas‘ifalinepassesthroughapointinsideacircle,thentheline,if extended far enough,mustmeet the circle’. But nit-picking aside, he did awonderfuljob,deducingfar-reachingconsequencesfromsimpleprinciples.

TheculminationoftheElementswastheproofthatthereexistpreciselyfiveregularsolids; theseareshapeswith regularpolygonsas faces,arranged in thesame manner at every corner. They are the tetrahedron, with four equilateraltriangle faces; the cube, with six square faces; the octahedron, with eightequilateraltrianglefaces;thedodecahedron,withtwelveregularpentagonfaces;and the icosahedron, with twenty equilateral triangle faces. Now, if you’reEuclidandinsistonlogicalproofs,youcan’tdothethree-dimensionalgeometryof the dodecahedron unless you’ve previously done the two-dimensionalgeometry of the regular pentagon. After all, the dodecahedron is built fromtwelveregularpentagons.Sotogettotherealmeat,theregularsolids,youhavetodealwithregularpentagons,andmuchelse.

AmongEuclid’sbasicassumptionsisanimplicitrestrictiononhowyoucanconstructgeometric figures.Everythingproceeds in termsof straight lines andcircles. In effect, you can use a ruler and, as they used to say, a pair ofcompasses.Thisisasingleinstrument,formuchthesamereasonthatwewearapair of trousers and cut our hair with a pair of scissors. Nowadays we oftenabbreviate this to ‘compass’, so Euclid’s procedures are called ruler-and-compass constructions. His geometry is a mathematical idealisation, in whichlinesareinfinitelythinandperfectlystraight,andcirclesareinfinitelythinandperfectly round. So Euclid’s constructions are not just good enough forgovernment work, they’re exact: good enough for an infinitely pedanticsupermindwithaninfinitelypowerfulmicroscope.

Gauss’s approach to regular polygons is based on Descartes’s discovery thatgeometryandalgebraare twosidesof thesamecoin,relatedbycoordinates in

theplane.Astraightlineisrepresentedbyanequation,whichthecoordinatesofeverypointonthelinemustsatisfy.Thesamegoesforcircles,buttheequationismorecomplicated. If two linesorcircles intersect, thepointsof intersectionsatisfybothequations.Whenyoutrytofindthesepointsbysolvingthepairofequations, everything is fairly simple for two lines. If a linemeets a circle, ortwocirclescross,youhavetosolveaquadraticequation.There’saformulaforthis,anditskeyfeatureistakingasquareroot.Therestissimplearithmetic:add,subtract,multiply,divide.

Viewedthroughthisalgebraist’stelescope,aruler-and-compassconstructionboilsdown to forminga seriesof square roots.Witha few tricksof the trade,that’sthesameassolvinganequationwhose‘degree’–thehighestpoweroftheunknown – is 2, 4, 8, 16, that is, some power of 2. Not every such equationreducestoaseriesofquadratics,butthatpowerof2isaclue.Whichpowereventellsyouhowmanyquadraticsyouneedtostringtogether.

Regular polygons turn into very simple equations if you use complexnumbers,inwhich−1hasasquareroot.Theequationforthecornersofaregularpentagon,forinstance,is

x5–1=0

whichisverysimpleandelegant.Removingtheobviousrealsolutionx=1,theotherssatisfy

x4+x3+x2+x+1=0

This is still quite pretty, and, crucially, of degree 4, which is a power of 2.Somethingsimilarhappensfortheheptadecagon,butnowtheequationsaddsupallpowersofxuptothesixteenth–and16isagainapowerof2.

Ontheotherhand,aregularheptagon(sevensides)hasasimilarequationofdegree 6, which is not a power of 2. So you definitely can’t get a regularheptagon using a ruler-and-compass construction.6 SinceEuclid constructs thepentagon, its equation must reduce to a series of quadratics. With a bit ofalgebra, it’s not hard to discover how.Thus armed,Gauss discovered that theequationfor the17-gonalso reduces toaseriesofquadratics.First,16=24,apowerof2,whichisnecessaryforaseriesofsquarerootstodothetrick,thoughnotalwayssufficient.Second,17isprime,whichenabledGausstofindsuchaseries.

Any competent mathematician could follow Gauss’s reasoning, once he

showedtheway,butnooneelseevensuspectedthatEuclidhadnotcataloguedallthepossibleconstructibleregularpolygons.

Notbadfora19-year-old.

Undertheduke’ssponsorship,Gausscontinuedtomakegreatstrides,especiallyin number theory. From childhood he had been a lightning calculator, able toperformcomplicatedarithmeticrapidlyinhishead.Inanerabeforecomputers,this ability was very useful. It helped him make rapid advances in numbertheory,andhisearlyreputationwasgreatlyenhancedwhenhewroteoneofthemostfamousresearchtextsinmathematicalhistory,DisquisitionesArithmeticae(ArithmeticalInvestigations).ThisbookdidfornumbertheorywhatEuclidhaddone forgeometry twomillenniaearlier. Itwaspublished in1801, thanks toasubsidy provided by the faithful duke, who was rewarded with a fulsomededication.

One of the basic techniques in the book is a typical example of Gauss’sabilitytosynthesisesimpleconceptsfromdisorganisedandcomplicatedresults.Todaywecallitmodulararithmetic.Manykeyresultsonnumbertheoryrestontheanswerstotwosimplequestions:

Whendoesagivennumberdivideanother?Ifnot,howarethetwonumbersrelated?Fermat’sdistinctionbetween4k+1and4k+3isofthistype.It’saboutwhat

happens when you divide a number by 4. Sometimes it goes exactly. Thenumbers

048121620…

areexactmultiplesof4.Theotherevennumbers

26101418…

arenot.Infact,theyeachleaveremainder2whendividedby4;thatis,theyareamultipleof4plus‘2leftover’.Inasimilarway,theoddnumberseitherleaveremainder1:

159131721…

orremainder3:

3711151923…

3711151923…

BeforeGaussgot his handson things, theusual formofwordswas that theselistscomprisethenumbersoftheforms4k,4k+1,4k+2,and4k+3,puttingtheremaindersintheirusualorder.Gausssaiditdifferently:theyarethelistsofallnumbers thatarecongruent to0,1,2,3 to themodulus4.Ormorebriefly,thankstoLatingrammar,modulo4.

Sofar that’s just terminology,butwhatmatters isstructure. Ifyouadd twonumbers,ormultiplythem,andaskwhichof0,1,2,3theresultiscongruentto,it turns out that the answer depends only on what the original numbers arecongruentto.Forinstance:

Ifyouaddnumberscongruentto2and3,theresultisalwayscongruentto1.Ifyoumultiplynumberscongruentto2and3,theresultisalwayscongruent

to2.Let’s try it out on an example.Thenumber14 is congruent to2 and23 is

congruentto3.Theirsumis37,sothatshouldbecongruentto1.Andsoitis:37=4×9+1.Theproductis322=4×80+2.

Thismay sound a bit simple-minded, but it lets us answer questions aboutdivisibilityby4usingjustthesefour‘congruenceclasses’.Let’sapplytheideato primes that are sums of two squares. Every whole number is congruent(modulo4)to0,1,2,or3.Thereforethesquaresarecongruenttothesquaresofthesefournumbers,thatis,to0,1,4,or9.Theseinturnarecongruentto0,1,0,1,respectively.Thisisaveryquickandeasywaytoprovethateverysquareiseitherof theform4kor4k+1, inold terminology.But there’smore.Sumsoftwosquaresarethereforecongruenttoeither0+0,0+1,or1+1;thatis,0,1,or 2. Conspicuous by its absence is 3. Nowwe’ve proved that a sum of twosquaresisnevercongruentto3modulo4.Sosomethingthatlooksquitetrickybecomesatrivialityinmodulararithmetic.

If themethodwere limited to congruencemodulo4 itwouldn’t be terriblyimportant, but you can replace 4 by any other number. If you choose 7, forexample,theneverynumberiscongruenttopreciselyoneof0,1,2,3,4,5,or6.Againyoucanpredict thecongruenceclassofasumorproduct fromthoseofthe numbers concerned. So you can do arithmetic (hence also algebra) usingcongruenceclassesinplaceofnumbers.

In Gauss’s hands, this idea became the foundation stone of far-reachingtheoremsaboutnumbers.Inparticular,itledhimtooneofhismostimpressivediscoveries,madeattheageof18.Fermat,Euler,andLagrangehadnoticedthepattern previously, but none of them had given a proof. Gauss found one,

publishing it in 1796 when he was 19; altogether he found six. He privatelycalledittheTheoremaAureum,thegoldentheorem.Itsofficialhandle,farmorecumbersomeandlessmedia-friendly,istheLawofQuadraticReciprocity.It’satool to answer a basic question: what do perfect squares look like to a givenmodulus? For instance,we saw that every square (modulo 4) is either 0 or 1.Thesearecalledquadraticresidues(modulo4).Theothertwoclasses,2and3,are quadratic non-residues. If instead we work (modulo 7) then the quadraticresiduesturnouttobe

0124

(thesquaresof0,1,3,2inthatorder)andthenon-residuesare

356

Ingeneral, if themodulus isanoddprimep, slightlymore thanhalfof thecongruence classes are residues, and slightly less than half are non-residues.However,there’snoobviouspatternregardingwhichnumberiswhich.

Supposepandqareoddprimes.Wecanasktwoquestions:

Ispaquadraticresiduemoduloq?Isqaquadraticresiduemodulop?

It’snotclearthesequestionsshouldbearanyrelationtoeachother,butGauss’sgolden theoremstates thatbothquestionshave thesameanswer,unlessbothpandqareoftheform4k+3,inwhichcasetheyhaveoppositeanswers:oneyes,oneno.Thetheoremdoesn’tsaywhattheanswersare,justhowthey’rerelated.Evenso,withsomeextraeffort,thegoldentheoremleadstoanefficientmethodtodecidewhetheragivennumberisorisnotaquadraticresiduemoduloanothergivennumber.However,ifitisaquadraticresidue,themethoddoesn’ttellyouwhich square to use. Even a basic question like this one still holds deepmysteries.

ThecoreofDisquisitionesisarefinedtheoryofthearithmeticpropertiesofquadratic forms– fancyvariationson ‘sumof two squares’–whichhas sincedeveloped into vast and complex theories, with links to many other areas ofmathematics. In case this all seems terribly esoteric, quadratic residues areimportant in the design of good acoustics in concert halls. They tell us whatshape tomake the sound reflectors and absorbers on thewalls.And quadraticformslieattheheartoftoday’smathematics,bothpureandapplied.

Gauss’s writings are concise, elegant, and polished. ‘When one hasconstructed a fine building, the scaffolding should no longer be visible,’ hewrote. Fair enough if you want people to admire the building, but if you’retraining architects and builders, close scrutiny of the scaffolding is vital. Thesamegoesifyou’retrainingthenextgenerationofmathematicians.CarlJacobicomplainedthatGausswas‘likethefox,whoeraseshistracksinthesandwithhistail’.Gausswasn’taloneinthispractice.WesawthatArchimedesneededtoknow the area and volume of a sphere in order tomake the proofs inOn theSphereandCylinderwork,buthekept themuphissleeve in thatbook.Tobefair, he did disclose the underlying intuition in The Method. Newton usedcalculus to discover many of the results in his Principia, and then presentedthemingeometricguise.Pressureon journalspace,and thehabitsof tradition,stillmakemuchpublishedmathematicsmoreobscure than it needbe. I’mnotconvinced this attitude does the profession any favours, but it’s very hard tochange, and there are some arguments in its favour. In particular, it’s hard tofollow a trail that keeps heading off the wrong way, only to retrace its stepswhenitgetsstuck.

Gauss’sacademicreputationwassky-high,andhehadnoreasontosupposethedukewouldstopsponsoringhimatsomefuturedate,butapermanent,salariedpostwouldoffergreatersecurity.Toobtainone,itwouldbeagoodideatohaveapublic reputation aswell.His chance came in1801.On the first dayof thatyear,theastronomerGiuseppePiazzicausedasensationbydiscoveringa‘newplanet’.Wenowconsiderittobeadwarfplanet,butformuchoftheinterveningtime it was an asteroid.Whatever its status, its name is Ceres. Asteroids arerelativelysmallbodiesorbiting(mainly)betweenMarsandJupiter.Aplanethadbeenpredictedatsuchadistanceonthebasisofanempiricalpatterninthesizesofplanetaryorbits,theTitius–BodeLaw.Thisfittedtheknownplanetswiththeexception of a gap betweenMars and Jupiter, just the place for an unknownplanettolurk.

ByJune,aHungarianacquaintanceofGauss’s, theastronomerBaronFranzXaver von Zach, had published observations of Ceres. However, Piazzi hadmanaged to observe the new world only for a short distance along its orbit.WhenitdisappearedbehindtheglareoftheSun,astronomerswereworriedtheywouldn’t be able to find it again. Gauss devised a new method for derivingaccurate orbits from a small number of observations, and Zach published

Gauss’sprediction,alongwithseveralothersthatalldisagreedwitheachother.InDecember,ZachrediscoveredCeres,almostexactlywhereGausshadsaiditwouldbe.ThefeatsealedGauss’sreputationasamathematicalmaestro,andhisrewardwastobemadedirectorofGöttingenObservatoryin1807.

BythenhewasmarriedtoJohannaOstoff,butin1809shediedaftergivingbirth to their second son, and then the son died too.Gausswas devastated bythesefamilytragedies,buthekeptworkingonhismathematics.Maybeithelpedhim cope by distracting him. He extended the study of Ceres into a generaltheoryofcelestialmechanics: themotionofstars,planets,andmoons.In1809hepublishedTheoryof theMotionofCelestialBodiesabout theSun inConicSections. Less than a year after Johanna’s death, he married her close friendFriedericaWaldeck,usuallyknownasMinna.

By now Gauss was enshrined as the top gun of German, indeed world,mathematics;hisopinionswerevaluedandrespected,andafewwordsofpraiseor condemnation from his lips could have far-reaching effects on people’scareers. On the whole, he didn’t abuse his influence, and he did a lot toencourageyoungermathematicians,buthisoutlookwasveryconservative.Heconsciously avoided anything likely to be controversial, working it out to hisown satisfaction, but shying away from publication. This combinationoccasionally led to injustice.Themostglaringexampleoccurred inconnectionwithnon-Euclideangeometry,astoryI’llpostponetothenextchapter.

Gauss’s works were wide-ranging. He gave the first rigorous proof of theFundamentalTheoremofAlgebra,thateverypolynomialequationhassolutionsin complex numbers.He defined complex numbers rigorously as pairs of realnumbers subject to specificoperations.Heprovedabasic theorem incomplexanalysis, later known as Cauchy’s Theorem because Augustin-Louis Cauchyobtained it independently,and published it. In real analysis, a function canbeintegrated over an interval to give the area under the corresponding curve. Incomplex analysis, a function can be integrated along a curved path in thecomplex plane. Gauss and Cauchy proved that if two paths have the sameendpoints, then the value of the integral depends only on those endpoints,provided the function does not become infinite at any point inside the closedcurve obtained by joining the two paths together. This simple result hasprofound consequences for the relation between a complex function and itssingularities–thepointsatwhichitbecomesinfinite.

He made early steps toward topology, introducing the linking number, atopologicalpropertythatcanoftenbeusedtoprovethattwolinkedcurvescan’tbeunlinkedbyacontinuousdeformation.ThisconceptwasgeneralisedtohigherdimensionsbyPoincaré(Chapter18).Itwasalsothefirststeptowardsatheoryofthetopologyofknots,atopicthatGaussalsothoughtabout,whichtodayhasapplicationstoquantumfieldtheoryandtheDNAmolecule.

AsdirectoroftheGöttingenObservatory,Gausshadtodevotealotofhistimetotheconstructionofanewobservatory,finishedin1816.Hekeptbusywithhismathematicstoo,publishingoninfiniteseriesandthehypergeometricfunction,an article on numerical analysis, some statistical ideas, and Theory of theAttraction of aHomogeneous Ellipsoid, about the gravitational attraction of asolidellipsoid–abetterapproximationtotheshapeofaplanetthanasphere.HewasputinchargeofageodeticsurveyofHanoverin1818,improvingsurveyingtechniques.Bythe1820s,Gausswasbecomingveryinterestedinmeasuringtheshape of the Earth. Earlier, he’d proved a result that he called his TheoremaEgregium (remarkable theorem). This characterises the shape of a surface,independentlyofanysurroundingspace.Thistheorem,plushisgeodeticsurvey,wonhimthe1822CopenhagenPrize.

Henowenteredadifficultperiodofhisfamilylife.Hismotherwasill,andin1817hemovedhertohisownhome.ApositionatBerlinbeckoned,andhiswifewantedhimtoaccept,buthewasreluctanttoleaveGöttingen.Then,in1831,hiswifedied.ThearrivalofthephysicistWilhelmWeberhelpedhimovercomehisgrief.GausshadknownWeberforafewyears,andtogethertheyworkedonthemagnetic field of the Earth. Gauss wrote three major papers on the topic,developingbasicresults inthephysicsofmagnetism,andappliedhistheorytodeducethelocationoftheSouthmagneticpole.WithWeber,hediscoveredwhatwenowcallKirchhoff’s laws forelectricalcircuits.Theyalsobuiltoneof thefirstpracticalelectrictelegraphs,abletosendmessagesmorethanakilometre.

WhenWeberleftGöttingen,Gauss’smathematicalproductivityfinallybegantowane.Hemovedintothefinancialsector,organisingtheGöttingenUniversitywidow’s fund. He put the experience this gave him to good use, andmade afortune investing in company bonds. But he still supervised two doctoralstudents,MoritzCantorandRichardDedekind.Thelatterwroteaboutthecalm,clear manner in which Gauss held research discussions, elaborating basicprinciples and then developing them in his elegant handwriting on a small

blackboard.Gaussdiedpeacefullyinhissleepin1855.

11BendingtheRules

NikolaiIvanovichLobachevsky

NikolaiIvanovichLobachevsky

Born:Nizhny-Novgorod,Russia,1December1792

Died:Kazan,Russia,24February1856

FOR MORE THAN two thousand years, Euclid’s Elements was considered theepitomeof logicaldevelopment.Starting froma fewsimpleassumptions,eachstatedexplicitly,Eucliddeducedtheentiremachineryofgeometry,onestepatatime. He began with the geometry of the plane, and then proceeded to solid

geometry.SocompellingwasEuclid’slogicthathisgeometrywasseennotjustasaconvenient idealisedmathematical representationof theapparent structureofphysicalspace,butasatruedescriptionofit.Withtheexceptionofsphericalgeometry–thegeometryofthesurfaceofasphere,widelyusedinnavigationasa good approximation to the form of the Earth – the default view amongmathematicians and other scholars was that Euclid’s geometry is the onlypossiblegeometry, so itnecessarilydetermines thestructureofphysical space.Spherical geometry isn’t a different kind of geometry; it is just the samegeometry, restricted to a sphere embedded in Euclidean space. Just as planegeometryisthatofaplaneinEuclideanspace.

AllgeometryisEuclidean.One of the first to suspect that this was nonsense was Gauss, but he was

reluctanttopublish,believingthattodosowouldbetoopenupacanofworms.Themost likelyresponseswouldbeblankstaresandaccusationsrangingfromignorance to insanity. The prudent trailblazer chooses regions of the junglewherenoonewillscreamabusefromthetreetops.

NikolaiIvanovichLobachevskywasmorecourageous,ormorefoolhardy,ormorenaivethanGauss.Probablyallthree.WhenhediscoveredanalternativetoEuclid’s geometry, just as logical as its illustrious predecessor, with its ownremarkable inner beauty, he understood its importance, and put his thoughtstogether inGeometriya, finished in1823. In1826heasked theDepartmentofPhysico-MathematicalSciencesatKazanUniversitytoallowhimtoreadapaperonthetopic,anditfinallysawprintinanobscurejournal,theKazanMessenger.Healsosubmittedit totheprestigiousStPetersburgAcademyofSciences,butMikhail Ostrogradskii, an expert applied mathematician, rejected it. In 1855Lobachevsky,thenblind,dictatedanewtextonnon-Euclideangeometry,titledPangeometry.Geometriya was finally published in its original form in 1909,longafterhisdeath.

His remarkable discoveries, along with those of an even more unfairlyneglected mathematician, János Bolyai, are now recognised as the start of agiganticrevolutioninhumanthoughtaboutgeometryandthenatureofphysicalspace.Butitiseverthefateofpioneerstobemisrepresentedandmisunderstood.Ideasthatshouldhavebeenhailedfortheiroriginalityareroutinelydenouncedas nonsense, and their originators receive little recognition. Hostility is morelikely; thinkofevolutionandclimatechange. I sometimes feel that thehumanracedoesn’tdeserve itsgreat thinkers.When theyshowus thestars,prejudiceandlackofimaginationdragusallbackintothemud.

Inthiscase,humanitywasunitedinthebeliefthatgeometrymustbeEuclidean.PhilosopherssuchasImmanuelKantwenttoeruditelengthstoexplainwhythatwas inevitable. This beliefwas based on longstanding tradition, reinforced bythe effort required to master the arcane arguments of Euclid, which wereinflicted upon generations of schoolboys as a kind of gigantic memory test.People naturally value knowledge that comes at great effort: if Euclid’sgeometrywerenotthatofrealspace,allthathardworkwouldhavebeenwasted.Anotherreasonwastheseductivelineofthinkingthathassincebeendubbedtheargumentfrompersonalincredulity.OfcoursetheonlygeometrywasEuclid’s.Whatelsecoulditbe?

Rhetorical questions sometimes get rhetorical answers, and this particularquestion,takenseriously,ledmathematiciansintoverydeepintellectualwaters.TheinitialmotivationwasafeatureofEuclid’sElementsthatlookedlikeaflaw.Not a mistake, just something that seemed inelegant and superfluous. Euclidarranged his development of geometry logically, starting with simpleassumptionsthatwerestatedexplicitlyandnotproved.Everythingelsewasthendeduced from these assumptions, step by step.Most of the assumptions weresimpleandreasonable:‘allrightanglesareequal’,forinstance.Butonewassocomplicatedthatitstoodoutlikeasorethumb:

Ifalinesegmentintersectstwostraightlinesformingtwointerioranglesonthesamesidethatsumto less than two right angles, then the two lines, if extended indefinitely,meet on that side onwhichtheanglessumtolessthantworightangles.

It’sknownastheparallelaxiom(orpostulate)becauseit’sreallyaboutparallellines.Whenthetwostraightlinesareparallel,theynevermeet.Inthiscase,theparallel axiom tells us that the sum of the interior angles concerned must beexactly two right angles – 1808. Conversely, if that’s the case, the lines areparallel.

Parallellinesarebasicandobvious:justlookatruledpaper.Itseemsevidentthatsuchlinesexist,andofcoursetheynevermeetbecausethedistancebetweenthemisthesameeverywhere,soitcan’tbecomezero.SurelyEuclidwasmakingamealofsomethingthatshouldbeobvious?AgeneralfeelingarosethatitoughttobepossibletoprovetheparallelaxiomfromtherestofEuclid’sassumptions.Indeed, several people were convinced they’d done just that, but whenindependentmathematicians lookedat theirallegedproofs, therewasalwaysa

mistakeoranunnoticedassumption.IntheeleventhcenturyOmarKhayyammadeoneoftheearliestattemptsto

resolve the issue. I’vementioned hiswork on cubic equations (here), but thiswasbynomeanstheonlystringtohismathematicalbow.HisSharhmaashkalaminmusadaratkitabUqlidis(ExplanationoftheDifficultiesinthePostulatesinEuclid’s Elements) builds on an earlier attempt by Hasan ibn al-Haytham(Alhazen)toprovetheparallelaxiom.Khayyamrejectedthisandother‘proofs’onlogicalgrounds,andreplacedthembyanargumentthatreducedtheparallelaxiomtoamoreintuitivestatement.

AC=BDandanglesatAandBarerightangles.Does

DCcompletetherectangle?

Oneofhiskeydiagramscutstotheheartoftheproblem.Itcanbeseenasanattempt to construct a rectangle – which we might expect to be entirelystraightforward. Draw a straight line, and two lines of equal length at rightangles to it.Finally, join theendsof those lines to form the fourth sideof therectangle.Done!

Or is it?Howdoweknowtheresult isa rectangle? Ina rectangle,all fourcornersarerightangles,andoppositesidesareequal.InKhayyam’spicture,weknow two angles are right angles and one pair of sides is equal.What of theothers?

Agreed, it looks as thoughwe’ve drawn a rectangle, but that’s becauseweuseEuclid’sgeometryasamentaldefault.Andindeed,inEuclid’sgeometrywecan prove thatCD=ABand anglesC,D are also right angles.However, thedeductionrequires…theparallelaxiom.That’sscarcelyasurprise,becauseweexpectCD tobeparallel toAB. If youwant toprove theparallel axiom fromEuclid’sotheraxioms,youhave toprove thatKhayyamhasdrawna rectanglewithoutappealingtotheparallelaxiom.Infact,asKhayyamrealised,ifyoucanfindsuchaproof,jobdone.Theparallelaxiomitselfquicklyfollows.Avoidingthe trap of trying to prove the parallel axiom, he explicitly replaced it by asimplerassumption:‘Twoconvergentstraightlinesintersectanditisimpossiblefor two convergent straight lines to diverge in the direction in which theyconverge.’Andhewasfullyawarethatthiswasanassumption.

GiovanniSaccheritookKhayyam’sdiagramsfurther,perhapsindependently,buttookastepbackwardsbytryingtousethemtoprovetheparallelaxiom.HisEuclid Freed of Every Flaw appeared in 1733. He split his proof into threepossibilities,dependingonwhetherangleCinthefigureisarightangle,acute(lessthanarightangle),orobtuse(morethanarightangle).SaccheriprovedthatwhateverthestatusofangleCisinonesuchdiagram,thesamethinghappensinallotherdiagramsofthesamekind.Theanglesinvolvedareeitherallright,allacute, or all obtuse. So there are three cases altogether, not three for eachrectangle.That’sabigstepforward.

Saccheri’sproofstrategywastoconsiderthealternativesofacuteandobtuseangles,aimingtorefutethembydeducingacontradiction.First,heassumedtheangleisobtuse.ThisledtoresultsthatheconsideredincompatiblewiththeotheraxiomsofEuclid– casedismissed. It tookhimmuch longer todisposeof thecase of the acute angle, but eventually he derived theorems that he believedcontradicted the other axioms. Actually, they don’t: what they contradict isEuclideangeometry,parallelaxiomandall.SoSaccherithoughthe’dprovedtheparallelaxiom,hencehissweepingtitle,whereaswenowseehisworkasabigsteptowardslogicallyconsistentnon-Euclideangeometries.

Nikolai’s father Ivan was a clerk involved in land surveys. His motherPraskovia,likehisfather,wasaPolishemigrant.Nikolai’sfatherdiedwhenhewasseven,andhismothermovedthefamilytoKazaninwesternSiberia.Afterfinishing school there, he went to Kazan University in 1807. He started outstudyingmedicinebutsoonchangedtomathematicsandphysics.HisprofessorsincludedGauss’sfriendandformerschoolteacherBartels.

In1811Lobachevskygainedamaster’sdegreeinmathematicsandphysics,which led to him becoming a lecturer, then extraordinary professor, then fullprofessorby1822.Theuniversity’sadministratorswerebackward-lookingandwary of anything innovative, especially in science and philosophy. Theyconsidered such things to be some sort of dangerous spin-off from theFrenchrevolution, and a danger to the religious orthodoxy of the time. As a result,academiclifewaspolluted,thebeststaff(amongthemBartels)left,othersweresacked,andstandardsdeclined.Itwasn’tthebestplacetobeifyouwereabouttooverthrowmillenniaofunimaginativetraditioningeometry,andLobachevskydidn’t make life any easier for himself by being outspoken and independent.Nevertheless,hekeptupwithhismathematical research,andhiscourseswere

modelsofclarity.Hisadministrativecareer,whichhadbegunwhenhe joined theuniversity’s

buildings committee, flourished. He bought new apparatus for the physicslaboratoryandnewbooksforthelibrary.Hedirectedtheobservatory,wasdeanofmathematicsandphysicsfrom1820to1825,andheadlibrarianfrom1825to1835.HisdisputeswithhigherauthorityimprovedwhenNicholasI,whotookamorerelaxedattitudetopoliticsandgovernment,becameTsar.HeremovedtheCurator (head) of the university, Mikhail Magnitskii, from office. HisreplacementMikhailMusin-PushkinbecameastaunchallyofLobachevsky,andmadehimrector in1827.Theappointment,whichlastednineteenyears,wasagreatsuccess,withnewbuildingsfor thelibrary,astronomy,medicine,andthesciences. He encouraged research in art and science, and student numbersincreased.Hisquickanddecisiveresponsesensured thatacholeraepidemic in1830andafirein1842didminimaldamage,andtheTsarsenthimamessageofthanks.Throughout,hedeliveredlecturesoncalculusandphysics,togetherwithgenerallecturesforthepublic.

In 1832, aged 40, he married a much younger and wealthy woman, LadyVarvaraMoisieva.Duringthisperiodhepublishedtwoworksonnon-Euclideangeometry: a paper on ‘imaginary geometry’ in 1837, and aGerman summarythat appeared in 1840which greatly impressedGauss. TheLobachevskys hadeighteen children, of whom seven survived to adulthood. They owned a poshhouseandhadanextensivesocial life.All these leftNikolaiwith littlemoneyfor his eventual retirement, and the marriage went poorly. His healthdeteriorated, and the university dismissed him in 1846, an event described as‘retirement’. His eldest son died soon after, and he started losing his sight,eventuallygoingblindandbeingunable towalk.Hedied in1856, inpoverty,unaware that anyone would ever take any notice of his discovery of non-Euclideangeometry.

A secondmathematicianwas equally involved in the big breakthrough: JánosBolyai.Hisideassawprintin1832as‘Appendixexhibitingtheabsolutescienceofspace:independentofthetruthorfalsityofEuclid’saxiomXI(bynomeanspreviouslydecided)’inhisfatherWolfgang’sEssayforStudiousYouthsontheElementsofMathematics.BolyaiandLobachevskygenerallygetthelion’sshareof the credit for turning non-Euclidean geometry into a significant area ofmathematics, but the prehistory of the subject includes four otherswho either

failedtopublishtheirideasorwereignoredwhentheydid.FerdinandSchweikart investigated ‘astral geometry’, developing Saccheri’s

caseof theacuteangle.HesentGaussamanuscriptbutneverpublishedit.Heencouraged his nephew Franz Taurinus to continue the work, and in 1825TaurinuspublishedTheoryofParallelLines.HisFirstElementsofGeometryof1826 states that the case of the obtuse angle also leads to a sensible non-Euclidean ‘logarithmic-spherical’geometry. It failed toattractattentionandheburnt his spare copies in disgust.One ofGauss’s students, FriedrichWachter,alsowroteabouttheparallelaxiom,buthetoowasignored.

To complicate the story further, Gauss had anticipated everyone,understandingasearlyas1800 that theproblemof theparallel axiom isabouttheinternallogicofEuclideangeometry,notaboutrealspace.Ruledlinesonapiece of paper can’t decide the answer. Perhaps they would meet a millionkilometresawayifyouusedabigenoughsheet.Andperhaps,ifyoudrawalotof points equidistant from a straight line, the resulting line is not straight.Pursuing this possibility,Gaussmay have started out like Saccheri, hoping toobtainacontradiction.Instead,hejustobtainedanincreasingnumberofelegant,credible, mutually consistent theorems, and by 1817 he was convinced thatlogically consistent geometries different from Euclid’s are possible. But hepublishednothingonthetopic,commentinginaletterof1829that‘itmaytakeverylongbeforeImakepublicmyinvestigationsonthisissue:infact,thismaynothappeninmylifetimeforIfearthe‘‘clamouroftheBoeotians’’.’7

WolfgangBolyaiwasanoldfriendofGauss’s,andhewrotetothegreatmanasking him to comment (favourably, he hoped) on his son’s epic research.Gauss’sreplydashedhishopes:

Topraise [János’swork]wouldbe topraisemyself. Indeed thewholecontentsof thework, thepath taken by your son, the results to which he is led, coincide almost entirely with mymeditations, which have occupied my mind partly for the last thirty or thirty-five years. So Iremainedquite stupefied.So farasmyownwork isconcerned,ofwhichup tillnowIhaveputlittleonpaper,my intentionwasnot to let itbepublishedduringmy lifetime…It is thereforeapleasantsurpriseformethatIamsparedthistrouble,andIamverygladthatitisthesonofmyoldfriend,whotakestheprecedenceofmeinsucharemarkablemanner.

All very well, but distinctly unfair, since Gauss had published nothing. Ofcourse, praising János’s radical ideas would also risk the clamour of theBoeotians.Privatepraisewasabitofacop-out,andWolfgangandGaussbothknewit.

LobachevskywasnotawarethatbothGaussandBolyaihadalsotackledtheproblem.Theparallelaxiomimpliestheexistenceofauniqueparalleltoagivenlinethroughagivenpoint,andhestartedbyconsideringthepossibilitythatthismight be false. He replaced it by the existence of many such lines, where‘parallel’ means ‘not meeting however far extended’. He developed theconsequencesofthisassumptioninconsiderabledetail.Hedidn’tprovethathisgeometric system was logically consistent, but he failed to derive anycontradictions and became convinced that none could occur.We now call hisset-uphyperbolicgeometry.ItcorrespondstoSaccheri’scaseoftheacuteangle.Theobtuseangleleadstoellipticgeometry,verysimilartosphericalgeometry.Bolyai studied both cases, whereas Lobachevsky limited his investigations tohyperbolicgeometry.

Ittookawhileforthevalidityofnon-Euclideangeometrytosinkin,andforitsimportancetoberecognised.JulesHoüel’sFrenchtranslationofLobachesvky’sworkstartedtheprocessin1866,tenyearsafterhedied.Forawhile,onevitalfeaturewasconspicuousbyitsabsence:aproofthatdenialoftheparallelaxiomneverleadstoacontradiction.Thiscamesomewhatlater:thereareactuallythreeconsistent geometries satisfying all of the other axioms of Euclid. These areEuclideangeometryitself;ellipticgeometry,inwhichparallellinesdonotexist;andhyperbolicgeometry,wheretheyexistbutarenotunique.

Theconsistencyproofturnedouttobesimplerthanmightbeexpected.Non-Euclidean geometry can be realised as the natural geometry of a surface ofconstant curvature: positive for elliptic geometry, negative for hyperbolicgeometry. Euclidean geometry is the transitional case of zero curvature. Here‘line’isinterpretedas‘geodesic’,theshortestpathbetweentwopoints.Withthisinterpretation, all of Euclid’s axioms except the parallel axiom can be provedusingEuclideangeometry.Iftherewerealogicalinconsistencyineitherellipticor hyperbolic geometry, it could be translated directly into a correspondinglogicalinconsistencyintheEuclideangeometryofsurfaces.ProvidedEuclideangeometryisconsistent,soareellipticandhyperbolicgeometry.

In1868,EugenioBeltramigaveaconcretemodel forhyperbolicgeometry:geodesics on a surface called the pseudosphere, which has constant negativecurvature.Heinterpretedthisresultasademonstrationthathyperbolicgeometrywasn’treallynew,justEuclid’sgeometryspecialisedtoasuitablesurface.Insodoing,hemissedthedeeperlogicalpoint:themodelproveshyperbolicgeometry

is consistent, so that the parallel axiomcannot be derived fromEuclid’s otheraxioms. Hoüel realised that in 1870when he translated Beltrami’s paper intoFrench.

Amodelforellipticgeometrywaseasiertofind.Itisthegeometryofgreatcircles on a sphere, with one twist. Great circles meet at two diametricallyoppositepoints,notatonepoint,sotheydon’tobeyEuclid’sotheraxioms.Tofixthisup,redefine‘point’tomean‘pairofdiametricallyoppositepoints’,andthinkofagreatcircleasapairofdiametricallyoppositesemicircles.Thisspace,technically a spherewith diametrically opposite points identified, has constantpositivecurvature,inheritedfromthesphere.

InthePoincarédiscmodel

ofhyperbolicgeometry,a

line(black)canhave

infinitelymanyparallels

(threeshowningrey)

passingthroughagiven

point.Theboundaryofthe

circleisnotconsideredto

bepartofthespace.

Meanwhile,non-Euclideangeometrywasstarting toshowup inotherareasofmathematics,notablycomplexanalysis,whereithasconnectionswithMöbiustransformations, whichmap circles (and straight lines) to circles (and straightlines).Weierstrasslecturedonthistopicin1870.Klein,whowentalong,gotthemessage,anddiscussedtheideawithSophusLie.In1872hewroteaninfluentialdocument,theErlangenProgramme,inwhichhedefinedgeometryasthestudyof invariants of transformation groups. This unified nearly all of the disparategeometries that were floating around by then, the main exception beingRiemannian geometry for surfaces of non-constant curvature, where suitable

transformation groups fail to exist. Poincarémade further advances, includinghisownmodelforhyperbolicgeometry.Thespaceistheinteriorofacircle,and‘straight’linesarearcsofcirclesmeetingtheboundaryatrightangles.

Later, hyperbolic geometry was one inspiration for Riemann’s theory ofcurvedspacesofanydimension(manifolds),whichunderpinsEinstein’stheoryofgravity(Chapter15).Itsapplicationsinmodernmathematicsincludecomplexanalysis, Special Relativity, combinatorial group theory, and Thurston’sGeometrisationConjecture(nowtheorem)inthetopologyofthree-dimensionalmanifolds(Chapter25).

12RadicalsandRevolutionaries

ÉvaristeGalois

ÉvaristeGalois

Born:Bourg-la-Reine,France,25October1811Died:Paris,France,31May1832

ON 4 JUNE 1832 the French newspaper Le Precursor reported a sensational,thoughbynomeansuncommon,event:

Paris,1June.Adeplorableduelyesterdayhasdeprivedtheexactsciencesofayoungmanwhogave the highest expectations, but whose celebrated precocity was lately overshadowed by hispoliticalactivities.TheyoungÉvaristeGalois…was fightingwithoneofhisold friends…whowas known to have figured equally in a political trial. It is said that lovewas the cause of thecombat.Thepistolwasthechosenweaponoftheadversaries,butbecauseoftheiroldfriendshiptheycouldnotbeartolookatoneanotherandleftthedecisiontoblindfate.Atpoint-blankrangetheywere each armedwith a pistol and fired.Onlyone pistolwas charged.Galoiswas piercedthroughand throughbyaball fromhisopponent;hewas taken to thehospitalCochinwherehediedinabouttwohours.Hisagewas22.L.D.,hisadversary,isabityounger.

Galois spent the night before the duelwriting a summary of hismathematicalresearches,whichcentredupontheuseofspecialsetsofpermutations,whichhecalled‘groups’, todeterminewhetheranalgebraicequationcanbesolvedbyaformula.Healsodescribedconnectionsbetweenthisideaandspecialfunctionsknown as elliptic integrals. His results easily imply that there is no algebraicformula to solve the general quintic equation, a question that had puzzledmathematicians for centuries before Gabriel Ruffini published an almostcomplete but interminably lengthy proof, and Niels Henrik Abel devised asimplerone.

Several myths about Galois persist to this day, despite the efforts ofhistorians to sort out the actual course of events. The record is patchy andsometimescontradictory.Forexample,whowashisopponent?Thenewspaperarticle is unreliable – it gets his age wrong, for a start – and much remainsobscure.Theimportanceofhismathematics,however,isclear.Theconceptofagroup of permutations was one of the first significant steps towards grouptheory.Thisturnedouttobethekeytothedeepmathematicsofsymmetry,andis amajor research area even today.Groups arenowcentral tomanyareasofmathematics and indispensable in mathematical physics. They have importantapplications to pattern formation in many areas of physical and biologicalscience.

Évariste’sfatherNicolas-Gabriel,aRepublican,wasmayorofBourg-la-Reinein1814 after Louis XVIII once more became king. His mother Adelaide-Marie

(néeDemante)wasthewell-educateddaughterofalegalconsultant.Shestudiedreligionandtheclassics,andsheeducatedÉvaristeathomeuntilhewas12.In1823hewassent totheCollegedeLouis-le-Grand.HegotfirstprizeinLatin,became bored, and took solace in mathematics. He read advanced works:Legendre’sElementsofGeometryandoriginalpapersbyAbelandLagrangeonthe solution of polynomial equations by radicals.This term refers to algebraicformulasexpressingthesolutionsintermsofthecoefficients,involvingthebasicoperationsofarithmeticandtheextractionofsquare,cube,orhigherroots.TheBabylonianshadsolvedquadraticsbyradicals,andtheRenaissancealgebraistshaddonethesameforcubicsandquartics.Nowitwasbecomingapparentthatsuchmethodshadrunoutofsteam.Abelprovedin1824thatthegeneralquintic–equationofthefifthdegree–can’tbesolvedbyradicals,andhepublishedanexpandedproofin1826.

Ignoring his mathematics teacher’s advice, Galois took the entranceexamination for the prestigious École Polytechnique a year early, withoutbotheringtoprepareforit.Unsurprisingly,hefailed.In1829hesentapaperonthe theory of equations to the Paris Academy, but it went astray. Galoisinterpreted this as deliberate suppression of his genius, but it could have beenjustsloppiness.Itwasabadyearallround.Galois’sfatherkilledhimselfduringa political conflict with the village priest, who forged Nicolas’s signature onmaliciousdocuments.Shortlyafter,Galoismadea secondand finalattempt togetintothePolytechnique,failingagain.Instead,hewenttothelessprestigiousÉcolePreparatoire,laterrenamedtheÉcoleNormale.Hedidwellinphysicsandmathematics,thoughnotinliterature,andgraduatedinbothscienceandlettersattheendof1829.A fewmonths laterheenteredanewversionofhisworkonequations for the academy’s grand prize. Fourier, the secretary, took themanuscripthome,butdiedbeforemakinghisreport.Againthemanuscriptwaslost,andagainGaloissawthisasadeliberateploytodenyhimtherewardshisbrilliancedeserved.Thisnarrative fittedneatlywithhisRepublicanviews,andreinforcedhisdeterminationtohelpfomentrevolution.

Whenopportunityknocked,Galoismissedout.In1824CharlesXsucceededLouis XVIII, but by 1830 the king was facing abdication. To avoid this, heintroducedpresscensorship,butthepeoplerevoltedinprotest.Afterthreedaysof chaos, a compromise candidate was agreed, and Louis-Philippe, Duke ofOrléans,becameking.ButthedirectoroftheÉcoleNormalelockedhisstudentsin. This didn’t go down well with the would-be revolutionary, who wrote ablistering personal attack on the director in a letter to theGazette desÉcoles.

Galoishadsignedhisname,buttheeditordidn’tprintit.ThedirectorusedthisasanexcusetoexpelGaloisforwritingananonymousletter.SoGaloisjoinedtheArtilleryoftheNationalGuard,amilitiapackedwithRepublicans.Notlongafter,thekingabolisheditasasecuritythreat.

InJanuary1831Galoissenttheacademyathirdmanuscriptonhistheoryofequations.After twomonthswithout any response, hewrote to the academy’spresident toaskwhatwascausingthedelay,butreceivednoreply.Hismentalstatebecameincreasinglyagitated,almostparanoid.SophieGermain,abrilliantfemalemathematician,wroteaboutGaloistoGuillaumeLibri:‘Theysayhewillgocompletelymad,andIfear this is true.’InAprilof thesameyear,nineteenmembers of the disbanded Artillery of the National Guard were tried forattempting to overthrow the government, but the jury acquitted them. At araucous banquet of about two hundred Republicans, held to celebrate theacquittal,Galoisheldupaglassandadagger.Hewasarrestedthefollowingdayforthreateningtheking.Headmittedhisactions,butinformedthecourtthathehadproposedthetoastwiththewords:‘ToLouis-Philippe,ifheturnstraitor.’Asympatheticjuryacquittedhim.

In July, theacademypronouncedonhis submission: ‘Wehavemadeeveryeffort to understand Galois’s proof. His reasoning is not sufficiently clear,sufficientlydeveloped,forustojudgeitscorrectness.’Therefereesalsoraisedamathematicalcriticism,whichwasentirelyreasonable.Theywereexpecting tobeinformedofsomeconditiononthecoefficientsoftheequation,whichwoulddetermine whether it is soluble by radicals. Galois had proved an elegantcondition,butitinvolvedthesolutions.Namely,eachsolutioncanbeexpressedasarationalfunctionoftwoothers.It’snowclearthatnosimplecriterionbasedonthecoefficientsexists,butnooneknewthatthen.

Galoiswentballistic.OnBastilleDay,hewas at the frontof aRepublicandemonstrationwithhisfriendErnestDuchâtelet,heavilyarmedandwearinghisArtillery uniform. Both were illegal. The two revolutionary comrades werearrestedand incarcerated in the jailatSainte-Pélagie,awaiting trial.After fourmonths, Galois was convicted and sentenced to prison with a six-monthsentence.Hepassedthetimebydoingmathematics,andwhencholerastruckin1832hewassenttoahospitalandthenreleasedonparole.

Having secured his freedom, he became obsessed with a young woman,identifiedonlyas ‘StéphanieD’with therestofhernamescribbledout. ‘HowcanIconsolemyselfwheninonemonthIhaveexhaustedthegreatestsourceofhappinessamancanhave?’hecomplainedtoanotherfriend,AugusteChevalier.

He copied fragments of the lady’s letters into a notebook.One reads: ‘Sir, beassuredthereneverwouldhavebeenmore.You’reassumingwronglyandyourregretshavenofoundation.’HistoryhassometimesportrayedStéphanieassomesortoffemmefatale,withsuggestionsthatitwasatrumped-up‘affairofhonour’that gaveGalois’s enemies an excuse to challengehim to a duel.But in 1968Carlos Infantozzi inspected the original manuscript and reported that she wasStéphanie-FeliciePoterin duMotel, daughter of the doctor atGalois’s lodginghouse.Thisreadingisabitcontroversial,butplausible.

The police report on the duel states that itwas a private dispute about theyounglady,betweenGaloisandanotherrevolutionary.Ontheeveof theduel,Galoiswrote:

Ibegpatriotsandmyfriendsnottoreproachmefordyingotherwisethanformycountry.Idiethevictimofaninfamouscoquette.Itisinamiserablebrawlthatmylifeisextinguished.Oh!whydieforsotrivialathing,forsomethingsodespicable!…Pardonforthosewhohavekilledme,theyareofgoodfaith.

His views of the lady would naturally be biased, but if his enemies hadengineeredthewholething,hewouldhardlyaskforthemtobepardoned.

Whowastheopponent?Therecordissparseandconfusing.InhisMémoires,Alexandre Dumas says he was a fellow Republican, Pescheux d’Herbinville.Which brings us back to the article in Le Precursor and the enigmatic killer‘L.D.’ The ‘D’ might refer to d’Herbinville, but if so, the ‘L’ is yet anothermistake in a rather inaccurate article. TonyRothman8makes a good case that‘D’standsforDuchâtelet,thoughthe‘L’isquestionable.Manyafriendshiphasfallenapartoverawoman.Theduelwaswithpistols–at25paces,accordingtothepost-mortemreport,butmorelikeRussianroulette ifLePrecursor is tobebelieved.Circumstantialevidencesupportsthelatter,becauseGaloiswashit inthestomach;badluckat25pacesbutguaranteedatpoint-blankrange.Refusingthe offer of a priest, he died a day later of peritonitis and was buried in thecommonditchatthecemeteryofMontparnasse.

The day before the duel, Galois summarised his discoveries in a letter toChevalier. It sketched how groups can tell us when a polynomial equation issoluble by radicals, and touched on other discoveries – elliptic functions,integrationofalgebraicfunctions,andcryptichintswhosemeaningwecanonly

guess.Theletterended:

AskJacobiorGausspubliclytogivetheiropinion,notastothetruth,butastotheimportanceofthese theorems. Later therewill be, I hope, some peoplewhowill find it to their advantage todecipherallthismess.

Fortunately for mathematics, there were. The first person to appreciate whatGaloishadachievedwasJoseph-LouisLiouville.In1843,LiouvillespoketotheverybodythathadmislaidorrejectedGalois’sthreememoirs.‘Ihopetointerestthe Academy,’ he began, ‘in announcing that among the papers of ÉvaristeGalois I have found a solution, as precise as it is profound, of this beautifulproblem: whether there exists a solution [of an equation] by radicals.’ Soon,JacobihadreadGalois’spapers,and–asGaloishadhoped–understood theirimportance. By 1856,Galois theorywas being taught at postgraduate level inboth France and Germany. And in 1909 Jules Tannery, director of the ÉcoleNormale, unveiled amemorial toGalois in his home townofBourg-la-Reine,thankingtheMayorfor‘allowingmetomakeanapologytothegeniusofGaloisin the name of this school to which he entered reluctantly, where he wasmisunderstood,whichexpelledhim,but forwhichhewas,afterall,oneof thebrightestglories.’

What,then,didGaloisdoformathematics?The short answer is that he explained the basicmathematics of symmetry,

whichistheconceptofagroup.Symmetryhasbecomeoneofthecentralthemesof mathematics and mathematical physics, underpinning our understanding ofeverythingfromanimalmarkingstovibratingmolecules,fromtheshapeofsnailshellstothequantummechanicsoffundamentalparticles.

Thelongerversionismorenuanced.His ideas were not entirely without precedent. Very few advances in

mathematicsare.Mathematiciansmostlybuildonclues,hints,andsuggestionsfrom their predecessors. A convenient entry point is Cardano’s Ars Magna,whichprovidedsolutionsforalgebraicequationsofthethirdandfourthdegree.Todaywewritetheseasformulasforthesolutionsintermsofthecoefficients.The key feature of these formulas is that they build the solution using thestandardoperationsofalgebra–addition,subtraction,multiplication,division–togetherwithsquarerootsandcuberoots.Anaturalguessisthatthesolutionofaquintic(fifth-degree)equationcanalsobegivenbysuchaformula,mostlikelyrequiring fifth roots aswell. (The fourth root is the square root of the squareroot, so it’s superfluous.)Manymathematicians – and amateurs – sought this

elusive formula. The bigger the degree, the more complicated the formulasbecome, soa formula for thequinticwasexpected tobeprettymessy.Butnoonecouldfindit.Graduallyitbegantodawnthattheremightbeareasonforthisfailure:thequestwasseekingamare’snest/redherring–chooseyourfavouritecliché–forsomethingthatdoesn’texist.

This doesn’tmean that solutions don’t exist. Every quintic equation has atleastonerealsolution,anditalwayshasfiveifweallowcomplexnumberstoo,andcount‘multiple’solutionscorrectly.Butthesolutionscan’tbeencapsulatedinanalgebraicformulathatusesnothingmoreesotericthanradicals.

The first serious evidence that this might be the case had emerged in the1770s,whenLagrangewroteahuge treatiseonalgebraicequations. Insteadofmerelyobservingthat thetraditionalsolutionswerecorrect,heaskedwhytheyexisted at all. What features of an equation make it soluble by radicals? Heunified theclassicalmethods fordegrees two, three,and four, relating them tospecialexpressions in thesolutions thatbehave in interestingwayswhenthosesolutions are permuted. As a trivial example, the sum of the solutions is thesame, in whichever order we write them. So is the product. The classicalalgebraists proved that any completely symmetric expression like these canalwaysbeexpressedintermsofthecoefficientsoftheequation,withoutanyuseofradicals.

Amoreinterestingexample,foracubicequationwithsolutionsa1,a2,a3,istheexpression

(a1–a2)(a2–a3)(a3–a1)

Ifwepermutethesolutionscyclically,sothata1→a2,a2→a3,anda3→a1,thisexpressionhasthesamevalue.However,ifweswaptwoofthem,sothata1→ a2, a2 → a1, and a3 → a3, the expression changes sign. That is, it ismultiplied by −1 but is otherwise unchanged. Therefore its square is fullysymmetric, and must be some expression in the coefficients. The expressionitself is thus thesquarerootofsomeexpression in thecoefficients.Thishelpsexplain why square roots enter into Cardano’s formula for solving cubics. Adifferentpartiallysymmetricexpressionexplainsthecuberoots.

Pursuing this idea,Lagrange found a unifiedmethod for solving quadratic,cubic, and quartic equations by exploiting the permutational properties ofparticular expressions in the solutions. He also showed that this method failswhenyoutryitonthequintic.Insteadofleadingtoasimplerequation,itleads

toamorecomplicatedone,makingtheproblemworse.Thatdoesn’timplythatnoothermethodcansucceed,butit’sadefinitehintofpotentialtrouble.

In 1799 Paolo Ruffini took the hint and published a two-volume book,General Theory of Equations. ‘The algebraic solution of general equations ofdegree greater than four,’ he wrote, ‘is always impossible. Behold a veryimportant theorem which I believe I am able to assert (if I do not err).’ Hecredited Lagrange’s research as inspiration. Unfortunately for Ruffini, theprospect ofwading through a 500-page tome, filledwith complicated algebra,justtoobtainanegativeresult,didn’tappealtoanyoneelse,andhewaslargelyignored.Leadingalgebraistswerestartingtoacceptthatnosolutionwaslikely,which probably didn’t help. Rumours that the book had errors circulated,dampingdownenthusiasmevenmore.Hetriedagainwithrevisedproofs,whichhehopedwouldbeeasiertounderstand.Cauchydidwritetohimin1821,sayingthat his book ‘has always seemed to me worthy of the attention ofmathematicians and which, in my judgement, proves completely theimpossibility of solving algebraically equations of higher than the fourthdegree.’

Cauchy’spraisemighthaveimprovedRuffini’sreputation,buthediedwithina year.After his death, a general consensus emerged that the quintic can’t besolved by radicals, but the status ofRuffini’s proof remained unclear. In fact,manyyearslater,asubtleflawwasfound.Thegapcouldbepatchedup,makingRuffini’sbookevenlonger,butbythenAbelhadpublishedamuchshorterandsimpler proof. Indeed, one of his results turned out to bewhatwas needed tocompleteRuffini’sproof.Abeldiedyoung,ofwhatwasprobably tuberculosis.Thequinticseemstohavebeensomethingofapoisonedchalice.

BothRuffiniandAbelpickedupLagrange’skeyidea:whatmattersiswhichexpressionsareinvariantundercertainpermutationsoftheroots.Galois’sgreatcontributionwastodevelopageneraltheory,basedonpermutations,thatappliestoallpolynomialequations.Hedidn’tjustprovespecificequationsareinsolublebyradicals;heaskedexactlywhichonesaresoluble.Hisanswerwasthatthesetofpermutationsthatpreserveallalgebraicrelationsamongtheroots–hecalledthis the group of the equation – must have a particular, rather technical butpreciselydefined, structure.Thedetailsof this structure explain exactlywhichradicalswillappear,whenasolutionby radicalsexists.Theabsenceof suchastructuremeansthere’snosolutioninradicals.

The structure involved is distinctly complicated, though natural from thegroup viewpoint. An equation is soluble by radicals if and only if its Galois

group has a series of special subgroups (called ‘normal’) such that the finalsubgroupcontainsjustonepermutation,andthenumberofpermutationsineachsuccessivesubgroupisthatofthepreviousone,dividedbyaprimenumber.Theideaof theproof is that onlyprime radicals areneeded– for instance, a sixthrootisthesquarerootofthecuberoot,andboth2and3areprime–andeachsuchradicalreducesthesizeofthecorrespondinggroupbydividingbythesameprime.

The Galois group of the general quartic, for instance, contains all 24permutations of the solutions. This group has a descending chain of normalsubgroups,withsizes

2412421

and

24/12=2isprime

12/4=3isprime

4/2=2isprime

2/1=2isprime

Therefore we can solve the quartic, and we expect to encounter square roots(fromthe2s)andcuberoots(fromthe3s)butnothingelse.

The groups for quadratic and cubic equations are smaller, and again havedescendingchainsofnormalsubgroupswhosesizesarerelatedbydivisionbyaprime.What of the quintic? This has five solutions, giving 120 permutations.Theonlychainofnormalsubgroupshassizes

120601

Since60/1=60isnotprime,therecanbenosolutioninradicals.Galois didn’t actuallywrite down a proof that the quintic can’t be solved.

Abelhadalreadydonethat,andGaloisknewit.Instead,hedevelopedageneraltheorem characterising all equations of prime degree that can be solved byradicals. To show that the general quintic is not among these equations is atriviality–sotrivialforGaloisthathedoesn’tevenmentionit.

WhatmakesGaloissignificantisnotsomuchhistheorems,ashismethod.Hisgroup of permutations – now called the Galois group – consists of allpermutations of the roots that preserve the algebraic relations between them.More generally, given some mathematical object, we can think of alltransformations – perhaps permutations, perhaps something more geometric,suchas rigidmotions– thatpreserve its structure.This iscalled thesymmetrygroupof thatobject. ‘Group’here focusesononeparticularaspectofGalois’sgroupsofpermutations,whichheemphasised,butdidnotdevelopintoamoregeneral concept. Itmeans that a symmetry transformation followedbyanothersymmetrytransformationalwaysyieldsasymmetrytransformation.

Asasimplegeometricexample,thinkofasquareintheplane,andtransformitusingrigidmotions.Youcanslideit,spinit,evenflipitover.Whichmotionsleavethesquareapparentlyunchanged?Youcan’tslideit;thatmovesitscentretoanewlocation.Youcanrotateit,butonlythroughoneormorerightangles.Anyotherangleproducesatiltthatwasn’ttherebefore.Finally,youcanflipitoveraboutanyoffouraxes:thetwodiagonals,andlinesthroughthemiddleofopposite sides.Not forgetting the trivial transformation ‘leave it alone’wegetexactly8symmetries.

Dothesameforaregularpentagon,andyouget10symmetries;foraregularhexagon12,andsoon.Acirclehasinfinitelymanysymmetries:rotationthroughanyangle,andflippingaboutanydiameter.Differentshapescanhavedifferentnumbers of symmetries. Indeed, subtler properties than the mere number ofsymmetries also come into play – not just howmany there are, but how theycombine.

Symmetrypervadesevery fieldofmathematics, fromalgebra toprobabilitytheory, and it has become absolutely central to mathematics and theoreticalphysics.Givenanymathematicalobject,thequestion‘Whatareitssymmetries?’immediately springs to mind, and the answer is often very informative. Inphysics, Einstein’s special theory of relativity is largely about how physicalquantitiesbehaveunderaparticulargroupofsymmetriesofphysicallaws,calledthe Lorentz group, based on the philosophical point that the laws of natureshould not depend on where or when you observe it. Today, all of thefundamental particles of quantum mechanics – electrons, neutrinos, bosons,gluons, quarks – are classified and explained in terms of a single symmetrygroup.

Galoistookavitalstepalongthetrailthatformalisedsymmetryasinvarianceunderagroupof transformations. It led to theabstractdefinitionofagroup,a

keyfeatureofthemodernapproachtoalgebra.HenriPoincaréoncewentasfarassayingthatgroupsconstitute‘thewholeofmathematics’strippeddowntoitsessentials.Itwasanexaggeration,butanexcusableone.

13EnchantressofNumber

AugustaAdaKing

AugustaAdaKing-Noel,

CountessofLovelace(née

Byron)

Born:Piccadilly(nowLondon),England,10

December1815

Died:Marylebone,London,27November1852

ITWASNOTAHAPPYFAMILY.The poet LordGeorgeGordonByronwas convinced that hewas about to

become the proud father of a ‘glorious boy’, but hewas bitterly disappointedwhen his wife Anne Isabella (née Milbanke, and known as ‘Annabella’)presented himwith a daughter. Shewas namedAugustaAda –Augusta afterByron’shalf-sister,AugustaLeigh.ByronalwayscalledherAda.

Amonthlaterthecoupleseparated,andfourmonthsafterthatByronleftthe

shoresofEngland,nevertoreturn.LadyByrongainedcustodyofherdaughterand disdained further contact with Lord Byron, but Ada developed a morenuancedviewandtookaninterestinhisactivitiesandwhereabouts.HetravelledaroundEurope,spendingsevenyearsinItaly,anddiedwhenAdawaseight,ofadiseasecaughtwhilefightingagainst theOttomanEmpireintheGreekWarofIndependence.Muchlatersheaskedtobeburiednexttohimuponherdeath,arequestthatwasdulyhonoured.

AnnabellaconsideredByrontobe insane,aviewthatwasreasonablegivenhisoutrageousbehaviour. Indirectly, this led toAda’s interest inmathematics.Annabellawasmathematically talentedand tookakeen interest in thesubject.Byron’sabilitiesmostdefinitelylayelsewhere.Inan1812letter tohiswifehewrote:

I agree with you quite upon Mathematics too – and must be content to admire them at anincomprehensibledistance–alwaysaddingthemtothecatalogueofmyregrets–Iknowthattwoandtwomakefour–andshouldbegladtoproveittooifIcould–thoughImustsayifbyanysortofprocessIcouldconverttwoandtwointofiveitwouldgivememuchgreaterpleasure.

The studyofmathematicswas therefore, inAnnabella’s eyes, an idealway todistance the child from her father. Moreover, she believed, the subjectencouraged a trained and disciplined mind. To this she added music, whichendowed young ladies with desirable social skills. Apparently Annabelladevotedmoreeffort toorganisingherdaughter’seducation than shedid to thedaughter herself; mainly Ada encountered her grandmother and her nurse. In1816Byronwrotesuggestingthat itwas timeforAdato‘recogniseanotherofherrelations’,namelyherownmother.

Ada enjoyed the advantages and disadvantages of an upper-class Englishupbringing, and was educated by a series of private tutors. A certain MissLamontinterestedheringeography,whichshemuchpreferredtoarithmetic,soAnabellapromptlyinsistedthatoneofthegeographylessonsshouldbereplacedby extra arithmetic. It wasn’t long before Miss Lamont was whisked away.Familymembersbecameconcernedthattoomuchpressurewasbeingplacedonthe girl, with too many punishments and too few rewards. Annabella’s ownmathematicstutor,WilliamFrend,wasropedintoteachAda,buthewasgettingoldandhadn’tkeptup todatewith thesubject. In1829DrWilliamKingwasbroughtin,buthismathematicalabilitieswereslight.Realmathematiciansknowthattheirsubjectisn’taspectatorsport–youhavetodoittoappreciateit.Kingpreferred to read about it. Arabella Lawrence was engaged to tame Ada’s

‘argumentative disposition’. Meanwhile Ada suffered a series of healthproblems,includingasevereboutofmeasleswhichsetherbackforalongtime.

In1833Adawaspresentedatcourt,atraditionalcoming-of-ageformembersof her class.Butwithin a fewmonths, a farmore significant event in her lifeoccurred. She went to a party and met the original but unorthodoxmathematician Charles Babbage. With this chance event, her mathematicalcareertookahugestepforward.

Theencounterwas,perhaps,lessfortuitousthanI’veindicated,becauseEnglishhighsocietymovedinthesamecirclesasprominentindividualsinscience,thearts,andcommerce.Theleadinglightsintheseareasallkneweachother,dinedtogether in small groups, andmaintained an interest in each other’s activities.Adaquicklybecameacquaintedwith the luminariesofherera– thephysicistsCharlesWheatstone,DavidBrewster,andMichaelFaraday,aswellasthewriterCharlesDickens.

TwoweeksaftermeetingBabbage,Ada–withhermother,bothaschaperoneand as an interested party – visited him in his studio. The main object ofattentionwasafantastic,complexmachine:theDifferenceEngine.ThecoreofBabbage’s life’sworkwas thedesign and,hehoped, constructionofpowerfulmachinesforperformingmathematicalcalculations.Babbagefirstconceivedofsuchamachine in1812,whenhewasmusingon thedeficienciesof logarithmtables. Thoughwidely used throughout the sciences and crucial to navigation,thepublished tableswere litteredwithmistakes causedbyhumanerror, eitherwhendoinghandcalculationsorsettingtheresultsintype.TheFrenchhadtriedto improve accuracy by breaking the calculations down into simple stepsinvolving only addition and subtraction, assigning each step to human‘computers’ trained to perform those tasks quickly and accurately, andrepeatedlycheckingtheresults.Babbagerealisedthatthisapproachwasidealforimplementationbyamachine,which,with the rightdesign,wouldbecheaper,morereliable,andfaster.

AsmallpartofBabbage’sDifferenceEngine.

His first attempt in that direction, theDifference Engine, is best seen as amechanical forerunner of the calculator; it could carry out basic operations ofarithmetic. Itsmain rolewas tocomputepolynomial functionssuchas squaresandcubes,ormorecomplicatedrelatives,bymethodsbasedonthecalculusoffinitedifferences.

The underlying idea is simple. Patterns in these functions appear if weconsider thedifferencesbetweensuccessivevalues.For instance,startwith thecubes:

0182764125216

Thedifferencesbetweensuccessivenumbersgo:

1719376191

Takedifferencesagain:

612182430

andyetagain:

6666

whenasimplepatternbecomesobvious. (It’s fairlyobviousat thepreviousstage;lesssoattheonebeforethat.)Whatmakesthiscuriouspatternimportantis the possibility of running the process backwards. Totalling series of 6srecreates the sequence immediatelybefore that; totalling the resultingnumbersgives the sequence before that one; finally, totalling that sequence yields thecubes.Asimilarmethodworksforanypolynomialfunction.Youjusthavetobeabletoadd.Multiplication,whichlooksmorecomplicated,issuperfluous.

Mechanicalaidstocomputationwerehardlyanewidea.Alongtraditionofsuchaidsrunsthroughoutmathematicalhistory,fromcountingonfingerstotheelectronic computer. But Babbage’s planwas unusually ambitious.He’d gonepublicwith the ideainapaperpresentedto theRoyalAstronomicalSociety in1822,andheextracted£1700fromtheBritishgovernmentayearlaterforapilotproject.By1842thegovernment’sinvestmenthadrisento£17,000–aboutthreequartersof amillionpounds (onemilliondollars) in today’smoney–withnoworkingmachineinsight.Adaandhermotherhadviewedaprototype,asmallpart of the overall plan. To make matters worse (in the government’s view),Babbagethenproposedafarmoreambitiousmachine,theAnalyticalEngine–agenuineprogrammablecomputer,builtofexquisitelyengineeredcogsandleversandpawlsand ratchets, inspiration for theentire ‘steampunk’genreof sciencefiction, with its mechanical versions of everything from computers to mobilephones and the internet. Unfortunately both the Difference and AnalyticalEngines remainedprettymuch that: science fiction.However, inmodern timestheDifferenceEnginewasactuallybuilt,inaprojectheadedbyDoronSwadeofLondon’sScienceMuseum.Based onBabbage’s second design, itworks, andcan be inspected in the museum today. Another, constructed according toBabbage’s first design, is in theComputerHistoryMuseum inCalifornia.NoonehasyettriedtobuildanAnalyticalEngine.

In1834Adametoneofthegreatfemalescientists,MarySomerville,whowasaclosefriendofBabbage.Thetwospentmanyhoursdiscussingmathematics,andMarylentAdatextbooksandsetherproblemstosolve.TheyalsotalkedaboutBabbage and the Difference Engine. The two became friends and went toscientificdemonstrations,andothereventssuchasconcerts,together.

In 1835 Ada married William King-Noel, who became the First Earl ofLovelace three years later. The couple had three children, after which shereturned to her first love, mathematics, under the tutelage of the noted

mathematician, logician, and eccentric Augustus De Morgan, founder of theLondonMathematicalSocietyandscourgeofmathematicalcrackpots. In1843shebeganaclosecollaborationwithBabbage,whicharose froma reportonalecture about the Analytical Engine that he’d given in Turin in 1840. LuigiMenabreahad takennotes andwritten themup forpublication.Ada translatedthemfromtheItalian,andBabbagesuggestedsheshouldaddsomecommentaryofherown.Sheagreedenthusiastically, andher commentary soonoutstrippedtheoriginallecture.

TheresultwastobepublishedintheTaylor’sScientificMemoirsseries.Atalatestageofproofreading,Babbagehadsecond thoughts:hercommentarywassogood,hefelt,thatitwouldbebetterifshepublisheditseparatelyasabook.LadyKingblewher aristocratic top.Mostof theworkalreadydonewouldbewasted,theprinterwouldbeannoyedatthebreachofcontract–no,theideawasridiculous. Babbage immediately backed down, as she had surely known hewould. To soften the blow, Ada offered to continue writing about his work,provided no similar change of heart happened again. She also hinted that shemight be able to help secure funding for the construction of the AnalyticalEngine, providedBabbage engaged a groupof practical friends to oversee theproject.Ada’smotherwasalwayscomplainingof ill-health;possiblywhat shehad in mind was her likely inheritance. If so, she was disappointed, for hermotheroutlivedherbyeightyears.

Ada’scommentaryisthemaindocumentuponwhichherscientificreputationrests. As well as explaining the operation of the device, it made two majorcontributionstowhatwenowseeasthedevelopmentofthecomputer.

The first was to illustrate the machine’s versatility. Where the DifferenceEnginewasacalculator,theAnalyticalEnginewasatruecomputer,capableofrunning programs9 that could in principle calculate anything, indeed, run anyspecified algorithm. The idea originated with Babbage, but Ada provided aseries of illustrative examples, showing how the machine could be set up toperformspecificcalculations.Themostambitiousofthesewastoworkoutso-calledBernoullinumbers.ThesearenamedafterJacobBernoulli,whodiscussedtheminhisArtofConjecturingof1713,oneofthefirstbooksoncombinatoricsand probability. The Japanesemathematician SekiKowa had discovered themearlier,buthisresultsweren’tpublisheduntilafterhisdeath.Theyarisefromtheseriesdevelopmentofthetrigonometrictangentfunction,andoccurinavarietyof othermathematical contexts. They are all rational numbers (fractions), andevery secondBernoulli number from the third onwards is zero; these features

aside,theyhavenoobviouspattern.Thefirstfeware:

11/21/60−1/3001/420−1/30

05/660−691/2730

Despitethelackofasimplepattern,Bernoullinumberscanbecalculatedinturnusingasimpleformula.Thisformulawasimplementedintheprogram.I’llcomebacktothethornyissueofAda’spreciserolehereinamoment.

Her secondcontributionwas less specific thanwritingprograms,butmuchmore far-reaching. Ada realised that a programmable machine can do muchmore than mere calculation. Her inspiration was the Jacquard loom, anextraordinarily versatile machine for weaving cloth in rich, complex patterns.The trickwas tousea longchainofcardswithholespunched in them,whichcontrolled mechanical devices that activated threads of different colours, orotherwiseaffectedthepatternoftheweave.Shewrote:

ThedistinctivecharacteristicoftheAnalyticalEngine,andthatwhichhasrendereditpossibletoendowmechanismwithsuchextensivefacultiesasbidfairtomakethisenginetheexecutiveright-hand of abstract algebra, is the introduction into it of the principlewhich Jacquard devised forregulating, by means of punched cards, the most complicated patterns in the fabrication ofbrocadedstuffs.Itisinthisthatthedistinctionbetweenthetwoengineslies.Nothingofthesortexists in the Difference Engine. We may say most aptly that the Analytical Engine weavesalgebraicalpatternsjustastheJacquardloomweavesflowersandleaves.

Thisanalogythentookflight.TheAnalyticalEngine,shewrote,

might act upon other things besides number, were objects found whose mutual fundamentalrelationscouldbeexpressedby thoseof theabstractscienceofoperations,andwhichshouldbealso susceptible of adaptations to the action of the operating notation and mechanism of theengine…Supposing,forinstance,thatthefundamentalrelationsofpitchedsoundsinthescienceofharmonyandofmusicalcompositionweresusceptibleofsuchexpressionandadaptations,theenginemight compose elaborate and scientific pieces ofmusic of any degree of complexity orextent.

Here Ada’s imagination transcends that of her peers. The whole thrust ofVictorian inventionwas a gadget for everything.One gadget to peel potatoes,another to slice boiled eggs, another to practise your riding skills without ahorse…butnow,shesawthatasingleversatilemachinecouldperformvirtuallyanytask.Allthatwasneededwastherightseriesofinstructions–theprogram.

Forthisreason,Adaisoftenseenasthefirstcomputerprogrammer.Shewasarguably the first person to publish sample programs, although it’s alwayspossibletosuggestprecursors,amongthemJacquard.Morecontroversialistheextenttowhichtheprogramsinhercommentaryarehers,ratherthanBabbage’s.Writing in thebiographyCharlesBabbage,Pioneerof theComputer,AnthonyHymanpointsoutthatthreeorfourotherpeoplemusthavedonesimilarthingsbefore:Babbage,afewassistants,andperhapshissonHerschel.Moreover,themost impressive example, the Bernoulli number program, was written byBabbage‘tosaveAdathework’.Hymanconcludesthat‘thereisnotascrapofevidencethatAdaeverattemptedoriginalmathematicalwork’.Nevertheless,hewrites that ‘Ada’s importance was as Babbage’s interpretress. As such herachievementwasremarkable.’

AgainstallthiswemustperhapssetBabbage’sownwords:

Wediscussedtogether thevarious illustrations thatmightbe introduced:Isuggestedseveral,butthe selection was entirely her own. So also was the algebraic working out of the differentproblems,except, indeed, that relating to thenumbersofBernoulli,whichIhadoffered todo tosaveLadyLovelace the trouble.This she sentback tome for anamendment,havingdetectedagravemistakewhichIhadmadeintheprocess.The notes of theCountess ofLovelace extend to about three times the length of the original

memoir. Their author has entered fully into almost all the very difficult and abstract questionsconnectedwiththesubject.These two memoirs taken together furnish, to those who are capable of understanding the

reasoning, a complete demonstration – That the whole of the developments and operations ofanalysisarenowcapableofbeingexecutedbymachinery.

From this scientific pinnacle, Ada’s subsequent trajectory was largelydownwards.Somethingofawildchild,shewasstrong-willedandimpulsive.Aseriesofaffairswithgentlemanfriendswashushedup,andherhusbandhadahundredormorecompromisinglettersofhersdestroyed.Alikingforwinewentoutofcontrolandshealsoindulgedinopium.Shebecameaninveterategamblerand left debts of £2,000onher death.Thegamblingmay evenhave stemmedfromamisguidedattempttoraisemoneyfortheAnalyticalEngine.

Herhealth,nevergood,declined,andshediedofcancerattheageof37.Tothe end, her mind remained active and her intelligence acute. She intuitivelygrasped the big picture, yet she had completemastery of the details. In 1843

Babbagesummedherup:‘Forget thisworldandall its troublesandifpossibleits multitudinous Charlatans – every thing in short but the Enchantress ofNumber.’Nothingevermadehimchangethatopinion.

14TheLawsofThought

GeorgeBoole

GeorgeBoole

Born:Lincoln,England,2November1815

Died:Cork,Ireland,8December1864

WHENGEORGEBOOLEWAS16hedecidedtobecomeanAnglicanclergyman,buthis father’s shoemakingbusiness failed, flinginghimheadlong into the roleoffamily provider. A career in the Church was no longer sensible, because theEnglish clergy were poorly paid. He was also becoming increasingly unsureabout the doctrine of the Holy Trinity, veering strongly towards the moreliterally monotheistic views of the Unitarians, a sect whose stance has beencharacterisedas‘beliefinatmostoneGod’.Thismadeitimpossibleforhimtosign up to the Thirty-Nine Articles of the Church of England without goingagainsthisconscience.

The most – perhaps only – suitable position, given his background and

talents, was in teaching, and in 1831 he took up the post of usher (assistantteacher)atMrHeigham’sSchoolinDoncaster,somefortymilesfromhishometown of Lincoln. In the middle of the nineteenth century this was quite adistance, and he was homesick; one letter says wistfully that nobody inDoncaster couldmakegooseberry pies as good as hismother’s. Itmight havebeennomore thananattempt topayheracompliment,butBoolecomplainedabouthislotformuchofhiscareer.HisUnitariantendencies,combinedwithahabitofsolvingmathematicalproblemsinchapelonSundays,outragedsomeofhis students’ parents, who were staunch Methodists. They complained to theheadmasterandtheirsonsprayedforBoole’ssoulatprayermeetings.Heigham,though happy with Boole’s performance as a teacher, reluctantly fired him,replacinghimwithaWesleyan.

Gooseberry pies and sectarian squabbles notwithstanding, Boole starteddelvingevendeeperintomathematics,pursuinghisstudieswithouttheadviceofatutor.Atfirsthereliedonapublicservice, thecirculatinglibrary,whichhadnumerous textbooks at surprisingly advanced levels, but the library wasdisbanded,soBoolewasforcedtobuyhisowntexts.Mathematicaltextbooks,asithappened,providedthemaximumstimulationfortheminimumoutlay,andhepurchased Sylvestre Lacroix’s Differential and Integral Calculus. A fellowteacher wrote that during an hour set aside for the teaching of writing, fromwhichBoolewasexcused:‘MrBooleisprofoundlyhappy;foranhouratleasthecanstudyoldLacroixwithoutinterruption.’

Later, Boole became convinced that he’dmade amistake buying a text asoutmodedasLacroix’s,butstudyingitgavehimconfidenceinhisownabilities.Oneconsequencewasanideathatstruckhim,brieflybutforcibly,inearly1833,whileoutwalkingacrossafarmer’sfield.Namely,thepossibilityofexpressinglogic in symbolic form. He didn’t develop the idea until many years later,publishing his first book on the topic in 1847: TheMathematical Analysis ofLogic,BeinganEssayTowardsaCalculusofDeductiveReasoning.AugustusdeMorgan,withwhomBoolehadfrequentcorrespondence,encouragedhimtoprepare a more extensive, better thought out book. His interests and Boole’soverlapped substantially. Boole took the advice, and in 1854 his masterworkduly appeared: An Investigation into the Laws of Thought, on Which areFoundedtheMathematicalTheoriesofLogicandProbabilities.Inthisworkhecreatedmathematicallogic,settingupwhateventuallybecameatheoreticalbasisforcomputerscience.

Boole’sfatherJohncamefromalong-establishedLincolnshirefamilyoffarmersandtraders,‘thebestthatchersandthemostreadingmen’inthetinyvillageofBroxholme.Hebecamea shoemaker, and left forLondon,hoping tomakehisfortune.Working alone in a dark cellar, he staved off depression by studyingFrench, science,andmathematics, especially thedesignofoptical instruments.He met and married Mary Joyce, a ladies’ maid, and after six months theymoved toLincoln,where Johnopened a cobbler’s shop.Theywanted a child,butitwastenyearsbeforeonearrived;theynamedhimGeorge.Agirlandtwoboysfollowedinshortorder.

John was much more interested in making telescopes than shoes, so thebusiness stumbled along, but the Booles made a living by renting rooms tolodgers.Georgegrewupinascientificatmosphereandhadanenquiringmind.HisfathertaughthimEnglishandmathematics.Georgelovedmathematicsandfinishedasix-volumegeometrytextbythetimehewaseleven(hisfatherwrotethis in pencil inside the book). He read widely and had an almost eideticmemory,abletorecallanyrequiredfactinstantly.

At the age of 16,Boole became a teacher atHeigham’sSchool.After twomore teachingpositionshe set uphis own school inLincoln at the ageof19;thenhetookoverHall’sAcademyinWaddington.Hisfamilyjoinedhimtohelprun the school.Boolenever lost sightofhighermathematics, andwas readingLaplace and Lagrange. He opened a boarding school in Lincoln, and beganpublishingresearchinthenewlyfoundedCambridgeMathematicalJournal.

In1842hebeganalifelongcorrespondencewithfellowspiritDe’Morgan.In1844 he won the Royal Society’s Royal Medal, and in 1849, buoyed by hisgrowing reputation, hewas appointed as the first professor ofmathematics atQueen’s College, Cork, Ireland. There he met his future wife Mary Everest(nieceofGeorgeEverest,whocompletedthefirstmajorsurveyofIndia,leadingtotheworld’shighestmountainbeingnamedafterhim)in1850.Theymarriedin 1855, and had five daughters, all remarkable: Mary, who married themathematicianandauthorCharlesHowardHinton,abrilliant rascal;Margaret,whomarried the artist Edward IngramTaylor;Alicia,whowas influenced byHintonanddidsignificantresearchonfour-dimensionalregularsolids;Lucy,thefirstfemalechemistryprofessor inEngland;andEthel,whomarriedthePolishscientistandrevolutionaryWilfridVoynichandwrotethenovelTheGadfly.

AmongBoole’searlyworkisasimplediscoverythatledtoinvarianttheory,anarea of algebra that became a very hot topic indeed. In the study of algebraicequations,aformulacansometimesbesimplifiedifitsvariablesarereplacedbysuitableexpressionsinanewsetofvariables.Solvethissimplerequationtofindthevaluesofthenewvariables,thenworkbackwardstodeducethevaluesoftheoriginal ones. This is how the Babylonian and Renaissance solutions ofequationsworked.

Anespecially importantclassofchangesofvariablesoccurswhen thenewvariablesarelinearcombinations–expressionslike2x−3y,involvingnohigherpowersorproductsoftheoldvariablesxandy.Ageneralquadraticform

ax2+bxy+cy2

intwovariablescanbesimplifiedinthismanner.Animportantquantityinthetheoryofsuchformsisthe‘discriminant’b2−4ac.Boolediscoveredthatafteralinearchangeofvariables,thediscriminantofthenewquadraticformisthatoftheoriginal,multipliedbyafactorthatdependsonlyonthechangeofvariables.

This apparent coincidence has a geometric explanation. It really is acoincidence,inthesensethattwofeaturesthatusuallyareseparatecoincide.Ifweequatethequadraticformtozero,itssolutionsdefinetwo(possiblycomplex)lines…unlessthediscriminantiszero,inwhichcasewegetthesamelinetwice.Thequadraticisthenthesquare(px+qy)2ofalinearform.Acoordinatechangeis a geometric distortion, and it carries the original lines to the correspondingones for thenewvariables. If the two linescoincide for theoriginalvariables,theythereforecoincideforthenewones.Sothediscriminantsmustberelatedinsuchamanner that ifonevanishes, sodoes theother. Invariance is the formalexpressionofthisrelationship.

Boole’s observation about the discriminant seemed little more than acuriosity,untila fewmathematicians, themostprominentbeingArthurCayleyandJamesJosephSylvester,generalisedhisinsighttoformsofhigherdegreeintwo ormore unknowns.These expressions also possess invariants,which alsodeterminesignificantgeometricfeaturesoftheassociatedhypersurface,definedby equating the form to zero. An entire industry emerged, in whichmathematicians won their spurs by calculating invariants of ever morecomplicated expressions. Eventually Hilbert (Chapter 19) proved twofundamentaltheoremsthatprettymuchkilledthetopicoffuntilitwasrevivedinamoregeneralform.Itremainsofinteresttoday,withimportantapplicationstophysics,andhadbeengivenanewleaseoflifebythedevelopmentofcomputer

algebra.

The research that made Boole a household name among mathematicians andcomputerscientists–andinanyhouseholdwhosesearchesonGoogleenterintothe heady realm of Boolean searches – had increasingly been occupying histhoughts.Boolealwayssoughttheinnersimplicitiesthatunderpinmathematicalconcepts.Helikedtoformulategeneralprinciples,casttheminsymbolicform,and let the symbols do the thinking. The Laws of Thought carried out thisprogramme for the rules of logic. Its big idea was to interpret these rules asalgebraic operations on symbols representing statements. Because logic is notthesameasarithmetic,someoftheusualalgebraiclawsmightnotapply;ontheother hand, there might be new laws, which don’t apply to arithmetic. Theupshot,knownasBooleanalgebra,makesitpossibletoprovelogicalstatementsbyperformingalgebraiccalculations.

Thebookopenswitharatherdeferentialpreface,andlocatesthediscussioninthecontextofexistingphilosophy.ThenBoolemovesontotherealmeat,themathematics,withadiscussionoftheuseofsymbols.Hespecialisestosymbols(hecallsthem‘signs’)thatrepresentlogicalstatements,focusinginparticularonthe general laws they obey. He tells us that he will represent a class, orcollection,ofindividuals,towhichaparticularnameapplies,byasingleletter,suchasx.Ifthenameis‘sheep’,thenxistheclassofallsheep.Aclassmaybedescribedbyanadjective,suchas‘white’,inwhichcasewemighthaveaclassyof allwhite things.The productxy then denotes the class of all things havingbothproperties,thatis,allwhitesheep.Sincethisclassdoesnotdependontheorder inwhich thepropertiesarestated,xy=yx.Similarly, ifz isa thirdclass(Boole’sexampleisx=rivers,y=estuaries,z=navigable)then(xy)z=x(yz).Thesearethecommutativeandassociativelawsofstandardalgebra,interpretedinthisnewcontext.

Henotesone law,vital to thewhole enterprise, that is not true inordinaryalgebra.Theclassxx is theclassofall thingsthathavethepropertydefiningxandthepropertydefiningx,soitmustbethesameasx.Thereforexx=x.Forexample,theclassofthingsthataresheepandaresheepissimplytheclassofallsheep.Thislawcanalsobewrittenasx2=x,anditrepresentsthefirstpointatwhichthelawsofthoughtdepartfromthoseofordinaryalgebra.

Next,Boolemoves on to signs ‘wherebywe collect parts into awhole, orseparateawhole into itsparts’.Forexample, suppose thatx is theclassof all

menandyistheclassofallwomen.Thentheclassofalladults–eithermenorwomen–isdenotedx+y.Againthere’sacommutativelaw,whichBoolemakesexplicit,andanassociativelaw,whichcomesundertheumbrellastatementthatthe ‘laws are identical’with those of algebra. Since, for example, the class ofEuropean men or women is the same as that of European men or Europeanwomen,thedistributivelawz(x+y)=zx+zyalsoholds,withzbeingtheclassofallEuropeans.

Subtractioncanbeusedtoremovemembersfromaclass.IfxrepresentsmenandyAsiatics,thenx-yrepresentsallmenwhoarenotAsiatics,andz(x–y)=zx–zy.

Perhaps themost striking featureof this formulation is that it’s not overtlyabout logic. It’s about set theory. Instead of manipulating logical statements,Booleworkswiththecorrespondingclasses,comprisingthosethingsforwhichthe statement is true.Mathematicians have long recognised a duality betweenthese concepts: each class corresponds to the statement ‘belongs to the class’;each statement corresponds to ‘the class of things for which the statement istrue’.Thiscorrespondencetranslatespropertiesofclassesintopropertiesoftheassociatedstatements,andconversely.

Boole introduces this idea by way of a third class of symbols ‘by whichrelation is expressed, and by which we form propositions’. For example,representstarsbyx,sunsbyy,andplanetsbyz.Thenthestatement‘thestarsarethe suns and the planets’ can be stated as x = y + z. So propositions areequalitiesbetweenexpressionsinvolvingclasses.It’saneasydeductionthat‘thestars,excepttheplanets,aresuns’;thatis,x–z=y.‘This,’Booletellsus,‘isinaccordancewiththealgebraicruleoftransposition.’Al-Khwarizmiwouldhaverecognisedthisruleasal-muqabala(here).

Theupshotof all this is that thealgebraof classesobeys the same lawsasordinary algebrawithnumbers, plus the strangenew lawx2 =x.At this pointBoolehasaverycleveridea.Theonlynumbersobeyingthatlaware0=02and1=12.Hewrites:

Letusconceive, then,ofanAlgebra inwhich thesymbolsx,y,z,etc.admit indifferentlyof thevalues 0 and 1, and of these values alone.The laws, the axioms, and the processes, of such anAlgebrawillbeidenticalintheirwholeextentwiththelaws,theaxioms,andtheprocessesofanAlgebraofLogic.Differenceofinterpretationwillalonedividethem.

Thisenigmaticstatementcanbeinterpretedasreferringtofunctions f(x,y,z,…),definedonsomelistofsymbols,thattakeonlythevalues0(false)or1(true)We

nowcall theseBooleanfunctions.Onedelightful theoremdeservesmention. Iff(x)isafunctionofonelogicalsymbol,Booleprovesthat

f(x)=f(1)x+f(0)(1–x)

Amoregeneralequationofthesamekindisvalidforanynumberofsymbols,leadingtosystematicmethodsformanipulatinglogicalpropositions.

Armed with this principle and other general results, Boole developsnumerousexamples,andshowshowhisreasoningapplies to topics thatwouldinterestthereadersofhistime.TheseincludeSamuelClarke’sDemonstrationoftheBeingandAttributesofGod,whichconsistsofaseriesoftheorems,provedusingobservationalfactsandvarious‘hypotheticalprinciples,theauthorityanduniversalityofwhicharesupposedtoberecognisedapriori’andtheEthicsofBenedictSpinoza.Boole’saimhereistoexplainexactlywhichassumptionsareinvolved in the deductionsmade by these authors.His quasi-Unitarian beliefsmayalsobemakingacameoappearance.

Previous analysis of logic had been verbal, with a few symbolic mnemonics.Aristotlediscussedsyllogisms–argumentsalongthelines

AllmenaremortalSocratesisamanThereforeSocratesismortal

with variations on the use of ‘all’ and ‘some’. Medieval scholars classifiedsyllogismsinto24types,givingthemmnemonicnames.Forexample,Bocardoreferstosyllogismsoftheform

SomepigshavecurlytailsAllpigsaremammalsThereforesomemammalshavecurlytails

Herethevowelsin‘bOcArdO’indicatetheformat,whereO=‘some’andA=‘all’.Thesameconventionwasused tonameother typesof syllogism.ButnosystematicsymbolicnotationforlogicwasintroducedbeforeBoole.Noticethatifwereplace‘some’by‘all’,obtaining

AllpigshavecurlytailsAllpigsaremammalsThereforeallmammalshavecurlytails

thenewsyllogismbecomesillogical.Ontheotherhand,

AllpigsaremammalsAllmammalshavecurlytailsThereforeallpigshavecurlytails

isalogicallycorrectdeduction–eventhoughinrealitythesecondstatementisfalse.As it happens, the concluding statement is, give or take the odd specialbreedofpig,true.

To explain how his symbolism relates to classical logic,Boole reinterpretsAristotle, showing how the validity (or not) of each type of syllogism can beprovedsymbolically.Forinstance,let

p=theclassofallpigsm=theclassofallmammalsc=theclassofallcreatureswithcurlytails

ThenthefinalsyllogismabovetranslatesintoBoole’ssymbolismasp=pmandm=mc,thereforep=pm=p(mc)=(pm)c=pc.

The rest of the book develops analogous methods for calculatingprobabilities,andendswithageneraldiscussionof‘thenatureofscienceandtheconstitutionoftheintellect’.

Boole wasn’t particularly happy at Cork. In 1850, after returning from adelightfulholidayinYorkshire,heaskedDeMorgan:‘IfyoushouldhearofanysituationinEnglandthatwouldbelikelytosuitme,toletmeknowofit,’andremarked that ‘I no longer feel as if I couldmake this placemy home.’Onesource of discontent was the authoritarian and religiously orthodox universityadministration, which cracked down on anyone who disagreed with it. Theprofessorofmodernlanguages,RaymonddeVericour,hadjustbeensuspendedbecause of anti-Catholic remarks in a book he had written. The university’sCouncil,underPresidentRobertKane,actedinsuchhastethatitcontravenedtheuniversity’sown statutes, anddeVericour’s appeal securedhispositionagain.

Boole sidedwith deVericour, but kept his head down. In 1856, further high-handedactionbyKane,aimedatBoole’swife’suncleJohnRyall,ledBooletowrite a stinging letter to theCorkDailyReporter.Kane’s replywas long andrather defensive, and Boole responded with yet another letter. Finally thegovernment opened an official enquiry, denounced Kane for not spendingenough time at the college, and censured both him andBoole for airing theirdisputeinpublic.KanemovedhisfamilytoCorkandeverythingsettleddown,thoughfromthenonthetwomaintainedacoldpolitenesstowardseachother.

In 1854 Boole’s mind turned to positions that had become available inMelbourne, Australia, but late in 1855 he dropped the idea completely whenMary Everest accepted his proposal of marriage. The Booles rented a largehouse overlooking the sea, close to the newly opened railway line,making iteasyforGeorgetocommute–thoughhedidatonepointaskthecollegetosetitsclocksbackby15minutestomakeiteasierforhimandthestudentstotravelby a later train. The college rejected the proposal. His eccentricity showed inotherways:heoncearrivedatalecturethinkingaboutaproblem,pacedupanddownforanhourcontemplatingit,whiletheranksofstudentsontheirbenchesfelt unable to interrupt, and then left, complaining to his wife that a ‘mostextraordinarythinghappenedtoday.Noneofmystudentscametomylecture.’

Latein1864hewalkedfromhishousetothecollege,adistanceofabout4–5kilometres, inaheavydownpour.Hefell illwithaseverecold,whichaffectedhislungs.MaryBoole,adevoteeofhomeopathy,engagedahomeopathtotreathim.Itdidn’twork,andhediedofpleuro-pneumonia.EthelVoynitch,hisfifthdaughter,wrote:

InAuntMary’s[Boole’ssister]viewatleast,thecauseofFather’searlydeathwas…theMissus’s[MaryBoole’s]beliefinacertaincrankdoctorwhoadvocatedcoldwatercuresforeverything…TheEverestsdoseemtohavebeenafamilyofcranksandfollowersofcranks.

Ironically, Boole himself considered homeopathy ineffective. In 1860 DeMorgan toldhim thathebelieved thathomeopathyhadcuredhispleurisy,butBoolewassceptical:

I have witnessed pleurisy and its former mode of treatment…One would say beforehand thathomeopathycouldhavenoeffectonsuchadisease…Themoralis–ifyouareeverattackedwithaninflammationandhomeopathydoesnot[work]…donotsacrificeyourlifetoanopinion…butcallinsomeaccredited[doctor].

TheareaofmathematicallogicopenedupbyBooleanalgebraisnowknownasthepropositionalcalculus. Itgoesback to the fifthcenturyBC,whenEuclidofMegara (not to be confusedwith the geometer Euclid ofAlexandria) initiatedwhatlaterbecametheStoicschooloflogic.AkeyfeatureofStoiclogicis theuseofconditional reasoning,of the form‘ifA thenB’.DiodorusandPhiloofMegara disagreed on a fundamental issue, which continues to flummoxmathematicsstudentstoday.Namely,giventhetruthorfalsityofAandB,whenistheimplication‘ifAthenB’true?Notethatwhat’sunderdiscussionisnotthetruthofeitherAorB,butthatofthedeductionofAfromB.Philo’sanswerwasthattheimplicationisfalseifAistrueandBisfalse,butotherwiseit’strue.Inparticular, it is true whenever A is false. Diodorus’s answer was different:wheneverAcannotleadtoafalseconclusion.Thatboilsdownto‘bothAandBaretrue’.

Today’smathematicallogicianssidewithPhilo.ThecounterintuitivecaseiswhenAisfalse.IfBisalsofalse,itseemsreasonabletoaccepttheinference‘ifA thenB’ asvalid. Inparticular, ‘IfA, thenA’ seemsa reasonable inference,whatever the truth value of A may be. If B is true, or its current status isunknown, however, it may seem unreasonable to accept its deduction from afalsehood.Forexample,thestatement

If2+2=5thenFermat’sLastTheoremistrue

is considered true – whether Fermat’s Last Theorem is true or false. (Thatdoesn’tleadtoaneasyproofofFermat’sLastTheorem,becausetodeducethatyou must first prove that 2 + 2 = 5, which is impossible if mathematics isconsistent. This is why Philo’s convention does no harm.) To illustrate thereasoningbehindthisconvention,considerthefollowingtwodeductions:

If1=−1then2=0[add1toeachside]

If1=−1then1=1[squareeachside]

Bothdeductionsarelogicallysound,bythereasoninginbrackets.Thefirsttakestheform

If(falsestatement)then(falsestatement)

andthesecondtakestheform

If(falsestatement)then(truestatement)

Sovalidreasoning,startingfromafalsestatement,canleadeithertoafalseone

oratrueone.Another approach that gives the same result is to ask what’s needed to

disprove an implication ‘ifA thenB’. That is, prove it false. For example, todisprove

Ifpigshadwings,they’dfly

wemustexhibitawingedpigthatcan’tfly.So‘ifAthenB’isfalseifAistrueandBisfalse,butinallothercases,theimplicationistrue,sincewecan’tproveitfalse.

This argument isn’t a proof. It’s motivation for the convention used inpredicate logic. In modal logic, conditionals are handled differently. Forexample, thestatementaboutwingedpigswouldbeconsidered true,subject tothewingsbeingfunctionalforflight.Butthesimilarstatement

Ifpigshadwings,they’dplaypoker

wouldbeconsideredfalse,since–evenhypothetically–possessingwingsdoesnot enhance one’s poker-playing abilities. In contrast, the latter statement isconsidered tobe true inpredicate logic,becausepigsdon’thavewings.Pokerdoesn’tcomeintoit.ThisillustratessomeofthedifficultiesthatBooleandotherearlylogiciansweregrapplingwith,anditwarnsusnottoassumethattoday’sconventionsarenecessarilythelastword.

The use ofBoolean algebra, or propositional calculus, in computing stemsfrom representing numerical and other data using the binary system, whichrequires only the digits 0 and 1. In their simplest manifestations, thesecorrespondto‘noelectricalvoltage’and‘someelectricalvoltage’(ataspecifiedlevel, for example 5 volts). In today’s computers, all data, programs included,are encoded in binary. The data are manipulated by electronic circuits that,among other things, implement the operations of the propositional calculus –essentiallyBooleanalgebra.Eachsuchoperationcorrespondstoa‘gate’,andasanelectrical signalor signalspass through thegate, theoutputdependson theinput(s)accordingtothelogicaloperationconcerned.

Thisideawaspioneeredbytheguruofinformationtheory,ClaudeShannon.Operations on digital data performed by computers can be implemented bysuitable electronic circuits, made from logic gates. So Boolean algebra is thenatural mathematical language for this aspect of computer design. Earlyelectronic engineers implemented these operations in relay circuits, and thenvalve (vacuum tube) circuits.With the inventionof the transistor, valveswere

replacedbysolid-statecircuitry;todayweusecomplexarraysofincrediblytinycircuitsdepositedonsiliconchips.

Boole’s formalisation of logic in symbolic terms opened up a new world,pavingthewayforthedigitalerawhosefruitswenowenjoy.And,frequently,curse, forwe’venotyet fullymasteredournewtechnology,despitehanding itevergreatercontrolofeverythinginourlives.

15MusicianofthePrimes

BernhardRiemann

GeorgFriedrichBernhard

Riemann

Born:Breselenz,KingdomofHanover,17September

1826

Died:Selasca,Italy,20July1866

BERNHARD RIEMANN HAD SHOWN immense mathematical talent, technicalmastery,andoriginalityfromtheageof20.MoritzStern,oneofhistutors,latersaid that ‘he already sang like a canary’. His other tutor, Gauss, seemed lessimpressed, but the courses Gauss was teaching were elementary, unlikely toshowcasethestudent’strueabilities.Soon,evenGaussunderstoodthatRiemannwas unusually able, and supervised his doctoral thesis. The topicwas dear toGauss’s heart: complex analysis. Gauss commented on the ‘gloriously fertileoriginality’ of the work and arranged an entry-level position for Riemann atGöttingenUniversity.

InGermany, thenextstepafter thePhDwasHabilitation,amoreadvanced

degree requiring deeper research that opened up a proper academic career byentitling its holder to become a Privatdozent, able to give lectures and chargefees.RiemannhadspenttwoandahalfyearsmakingbigstridesinthetheoryofFourier series (Chapter 9). The research had gone well, but now he wasbeginningtothinkhe’dbittenoffmorethanhecouldchew.

Theproblemwasn’t theworkonFourierseries.Thatwasdoneanddusted,andRiemannwasconfidentofitsqualityandaccuracy.No,theproblemwasthefinal step in qualifying forHabilitation. The candidate had to deliver a publiclecture. He had proposed three topics: two on the mathematical physics ofelectricity, a subject he had also studied under Wilhelm Weber, and, moredaringly,oneonthefoundationsofgeometry,wherehehadsomeinterestingbutratherhalf-bakedideas.ThechoicefromthesethreetopicswasuptoGauss,whoatthattimewasworkingwithWeberanddeeplyinterestedinelectricity.WhatRiemanndidn’t take intoaccountwas thatGausswasalsodeeply interested ingeometry,andwantedtohearwhatRiemannhadtosayaboutit.

SonowRiemannwasworkinghissocksofftryingtodevelophisvagueideasabout geometry into something that would make a real impression on thegreatestmathematician of the age, in an area that luminary had been thinkingabout formuchofhis life.His startingpointwasa resultofwhichGausswasespeciallyproud,hisTheoremaEgregium (here).This specifies the shapeof asurface without reference to any surrounding space, and it inaugurated thesubjectofdifferentialgeometry.ItledGausstostudygeodesics–shortestpathsbetweenpoints–andcurvature,whichquantifieshowmuch the surfacebendscomparedtotheordinaryEuclideanplane.

Riemannplanned to generaliseGauss’s entire theory in a radical direction:spacesofanydimension.Mathematiciansandphysicistswerejustbeginningtoappreciate the power and clarity of geometric thinking in ‘spaces’ with morethantheusualtwoorthreedimensions.Underlyingthiscounterfactualviewpointwas something entirely sensible, the mathematics of equations in manyvariables.Thevariablesplaytheroleofcoordinates,sothemorevariablesthereare,thegreateristhedimensionofthisconceptualspace.

Riemann’s efforts to develop this notion led him to the brink of a nervousbreakdown. Tomakemattersworse, hewas simultaneously helpingWeber tounderstandelectricity.Fortunately,theinterplaybetweenelectricalandmagneticforces ledRiemann to anewconceptof ‘force’, basedongeometry: the sameinsight that led Einstein to General Relativity, decades later. Forces can bereplaced by the curvature of space. Now Riemann had the new viewpoint he

neededtodevelophislecture.Inasomewhatdesperate flurryofactivity,hesortedout the foundationsof

moderndifferentialgeometry,beginningwiththeconceptofamultidimensionalmanifoldandanotionofdistancedefinedbyametric.Thisisaformulaforthedistancebetweenany twopoints thatareveryclose together.Hedefinedmoreelaborate quantities now called tensors, gave a general formula for curvatureexpressedasaspecialkindoftensor,andwrotedowndifferentialequationsthatdetermine geodesics. But he also went further, probably drawing inspirationfrom his work with Weber, and speculated about possible relations betweendifferentialgeometryandthephysicalworld:

Theempiricalnotionsonwhichthemetricaldeterminationsofspacearefounded,thenotionofasolidbodyandofarayoflight,ceasetobevalidfortheinfinitelysmall.Wearethereforequiteatliberty tosuppose that themetric relationsofspace in the infinitelysmalldonotconformto thehypotheses of geometry; andweought in fact to suppose it, ifwe can therebyobtain a simplerexplanationofphenomena.

Thelecturewasatriumph,eventhoughtheonlypersonpresentwhowaslikelytounderstand it fullywasGauss.Riemann’soriginalitymadeabig impressiononGauss, who toldWeber how surprised he was at its depth. The impulsivegamblehadpaidoff.

Riemann’s insights were further developed by Eugenio Beltrami, ElwinBrunoChristoffel,andtheItalianschoolunderGregorioRicciandTullioLevi-Civita.Later,theirworkturnedouttobejustwhatEinsteinneededforGeneralRelativity. Einstein was interested in very large regions of space, whereasRiemann’svisionforphysicslayintheverysmall.Evenso,itallgoesbacktoRiemann’slecture.

Riemann’s father Friedrich was a Lutheran pastor and a veteran of theNapoleonicwars.The familywaspoor.HismotherCharlotte (néeEbell) diedwhenRiemannwas quite young.Hehad a brother and four sisters.His fathereducatedhimuntiltheageoften.In1840hestartedgoingtothelocalschoolinHanover, entering directly into the third class. He was very shy, but hismathematical gifts were immediately apparent. The school’s director allowedRiemanntoreadmathematicsbooksfromhisowncollection.Whenhelenttheboy Legendre’s 900-page text on number theory, Riemann devoured it in a

week.In 1846 he went to Göttingen University, initially to study theology, but

Gauss recognisedhismathematical talents andadvisedhim to switch subjects,which(withhisparents’approval)hedid.Göttingeneventuallybecameoneofthebestplacesintheworldtostudymathematics,butinthosedays,despitethepresenceofGauss,itsmathematicalinstructionwasfairlyordinary.SoRiemanndecamped to Berlin, where he worked under the geometer Jakob Steiner, thealgebraist andnumber theoristDirichlet, and thenumber theorist and complexanalyst Gotthold Eisenstein. There he learned about complex analysis andellipticfunctions.

Cauchyextendedcalculusfromrealnumberstocomplexnumbers.Complexanalysis emerged when Berkeley’s objections to Newton’s fluxions wereeventuallycounteredbyKarlWeierstrass,whoformulatedarigorousdefinitionof‘passingtoalimit’.Oneofthehottopicsinmid-1800scomplexanalysiswasthestudyofellipticfunctions,whichamongotherthingsspecifythelengthofanarc of an ellipse. They are a deep generalisation of trigonometric functions.Fourier exploited one basic property of these – they’re periodic, repeating thesame value if 2π is added to the variable. Elliptic functions have twoindependent complex periods, and repeat the same values on a grid ofparallelograms in the complex plane. They exhibit a beautiful connectionbetween complex analysis and symmetry groups (translations of the grid).Wiles’s proof of Fermat’sLastTheoremuses this idea.Elliptic functions alsocomeupinmechanics,forexampleingivinganexactformulafortheperiodofapendulum.The simpler formuladerived in schoolphysics is an approximationforaswingthroughaverysmallangle.

Dirichlet’s approach tomathematicsappealed toRiemann,beingmuch likehis own. Instead of a systematic logical development, they both preferred tobeginbyacquiringanintuitivegraspoftheproblem,thensortingoutthecentralconceptsandrelationships,andfinallyfillinginthelogicalgapswhileavoidingextensive computations asmuch as possible.Manyof today’smost successfulandoriginalmathematiciansdo the same.Proofs arevital tomathematics, andtheirlogicmustbeimpeccable–butproofsoftencomeafterunderstanding.Toomuch rigour, too early, can stifle agood idea.Riemannadopted this approachthroughouthiscareer.Ithadonebigadvantage:peoplecouldfollowthegeneralline of thinking without spending weeks checking complicated sums. Itsdisadvantage,forsomeat least,wastheneedtothinkconceptually,ratherthanjustploughingthroughcalculations.

ForhisPhD,Riemannrewrotethebookoncomplexanalysisbyintroducingtopologicalmethods.Hewas led to this reformulation by a feature that everystudent has to grapple with: the tendency of complex functions to bemany-valued.Therearehintsofthisphenomenoninrealanalysis.Forexample,everynonzero positive real number has two square roots: one positive, the othernegative. This possibility has to be borne in mind when solving algebraicequations,butitcanbehandledfairlyeasilybysplittingthesquarerootfunctionintotwoseparateparts:thepositivesquarerootandthenegativesquareroot.

Thesameambiguityafflictsthesquarerootofacomplexnumber,butit’snolonger entirely satisfactory to pull it apart into two distinct functions. Thenotions‘positive’and‘negative’havenousefulmeaningforcomplexnumbers,sothere’snonaturalwaytosplitthetwovaluesapart.Butthere’sadeeperissue.Intherealcase,ifwechangeapositivenumbercontinuously,itspositivesquarerootalsochangescontinuously,andsodoesitsnegativesquareroot.Moreover,the two remain distinct. But in the complex case, continuous changes to theoriginal number can turn one of its square roots into the other, while alwaysmovingthemcontinuously.

Thetraditionalwaytosortthisoutwastoallowdiscontinuousfunctions,butthen you have to keep checking whether you’re approaching a discontinuity.Riemannhadabetteridea:modifytheusualcomplexplanetomakethesquarerootfunctionsingle-valued.Thisisdonebytakingtwocopiesoftheplane,oneabove theother; slittingeachalong thepositive real axis; and then joining theslits so the top plane gets joined to the bottom one as you cross the slit. Thesquare root becomes single-valued when interpreted using this ‘Riemannsurface’.Thisisaradicalapproach.Theideaistostopworryingaboutwhichofthe many possible values you’re dealing with, and let the geometry of theRiemannsurface takecareofeverything.And itwasn’t theonly innovation inthethesis.Anotherwastouseanideafrommathematicalphysics,theDirichletprinciple, to prove existence of certain functions. This principle states that afunction that minimises energy is a solution to a partial differential equation,Poisson’s equation,which governs gravitational and electric fields.Gauss andCauchy had already discovered that the same equation arises naturally incomplexanalysisinconnectionwithdifferentialcalculus.

Riemann settled into academic life. His natural shyness made lecturingsomething of a trial, but he slowly adapted and began to understand how to

relatetohisaudience.In1857hewasappointedfullprofessor.Inthesameyearhe published another major work on the theory of Abelian integrals, a broadgeneralisation of elliptic functions that provided fertile ground for histopologicalmethods.WeierstrasshadsubmittedanarticletotheBerlinAcademyon the same topic, but when Riemann’s paper appeared, Weierstrass was sooverwhelmed by its novelty and insight that he withdrew his own work andnever again published in the area. That didn’t stop him pointing out a subtleerror in Riemann’s use of the Dirichlet principle, mind you. Riemann madeheavyuseofafunctionthatmadesomerelatedquantityassmallaspossible.Itledtoimportantresults,buthehadn’tgivenarigorousproofthatsuchafunctionactually exists. (Hebelievedonphysicalgrounds that itmust, but thiskindofreasoninglacksrigourandcangowrong.)Atthisstagethemathematicianssplitinto those who wanted logical rigour, and therefore considered the gap to beserious,andthosewhowereconvincedbythephysicalanalogiesandweremoreinterestedinpushingtheresultsfurther.Riemann,inthesecondcamp,saidthateventhoughtheremightbeaflawinthelogic, theDirichletprinciplewasjustthemostconvenientwaytoseewhatwasgoingon,andhisresultswerecorrect.

It was, in a way, a rather standard disagreement between puremathematicians and mathematical physicists, and the same game plays outregularlytoday,beittheDiracdeltafunctionorFeynmandiagrams.Bothsideswere right,by theirownstandards. Itmakes littlesense toholdupprogress inphysicsjustbecausesomeplausibleandeffectivetechniquecan’tbejustifiedincompletelogicalrigour.Equally,theabsenceofsuchjustificationisasmokinggun for mathematicians, hinting that something vital is missing from ourunderstanding. Weierstrass’s student Hermann Schwartz satisfied themathematicians by finding a different proof of Riemann’s results, but thephysicistsstillpreferredsomethingmoreintuitive.EventuallyHilbertsortedoutthe existence problem by proving a version of the Dirichlet principle that isrigorous and suited toRiemann’smethods. In the interim, thephysicistsmadeprogress that wouldn’t have happened if they’d heeded the objections of themathematicians, and themathematicians’ efforts to justifyRiemann’s intuitionledtoahostofmajorresultsandconceptsthatwouldn’thavebeendiscoveredifthey’dsidedwiththephysicists.Everybodywon.

Manifolds and curvature had made Gauss aware of Riemann’s potential andprowess,buttherestofthemathematicalcommunitygotthemessageonlyafter

he published his research onAbelian integrals. Kummer, Karl Borchardt, andWeierstrass mentioned it when they proposed him for election to the BerlinAcademyin1859.Oneofthetasksfacingnewmemberswastopresentareporton their current work, and Riemann didn’t disappoint. He’d changed tack yetagain, and the report was titled ‘On the number of primes less than a givenmagnitude’. In this work, he posed the Riemann Hypothesis, a conjecture incomplexanalysisrelatedtothestatisticaldistributionofprimes.It’scurrentlythemostfamousunsolvedprobleminthewholeofmathematics.

Prime numbers are central to mathematics, but in many respects they’reinfuriating. They have hugely important properties, but display a remarkableabsence of pattern. Looking along a list of prime numbers, in sequence, it’sdifficult to predict the next one (aside from everything after 2 being odd andavoidingmultiples of small primes like 3, 5, 7). The primes are uniquely andunambiguously defined, yet in some respects they appear random. Statisticalpatterns do exist, however. Around 1793 Gauss noticed empirically that thenumber of primes less than any given number x is approximately x/logx. Hecouldn’t find a proof, but the conjecture became known as the prime numbertheorem because in those days ‘theorem’ was a standard term for unprovedstatements.CompareFermat’sLast.Whenaprooffinallyappeared,itcamefroma totally unexpected direction. Primes are discrete objects, arising in numbertheory.At theoppositeendof themathematical spectrum iscomplexanalysis,about continuous objects, and employing totally different (geometric, analytic,topological)methods.Ithardlyseemedlikelythattherecouldbeaconnection–buttherewas,andmathematicshasneverbeenthesamesinceitsdiscovery.

The link goes back toEuler,who in 1737, in FormulaManmode, noticedthatforanynumbers,theinfiniteseries

1+2–s+3–s+4–s+…

isequaltotheproduct,overallprimesp,oftheseries

1+p–s+p–2s+p–3s+…=1/(1–p–s)

The proof is simple, little more than a direct translation into power serieslanguage of the uniqueness of prime factorisation. Euler was thinking of thisseriesforrealnumberss,indeedmainlyforintegers.Butitmakessenseifsisacomplexnumber,subjecttosometechnicalissuesaboutconvergenceandatricktoextendtherangeofnumbersforwhichit’sdefined.Inthiscontextit’scalled

the zeta function,written as ζ(z). As the power of complex analysis began tomanifest itself, it was only natural to study this kind of series using the newtools, in the hope that a proof of the prime number theorem might emerge.Riemann,anexpertcomplexanalyst,wasboundtogetinvolved.

Thepromiseof this approach first becameapparent in1848,whenPafnutyChebyshevmadeprogress towardsaproofof theprimenumber theoremusingthezetafunction(thoughthisnamecamelater).Riemannmadetheroleofthisfunction clear in his concise but penetrating 1859 paper. He showed that thestatistical properties of primes are closely related to the zeros of the zetafunction, that is, the solutions z of the equation ζ(z) = 0.A high point of thepaperwasaformulagivingtheexactnumberofprimeslessthanagivenvaluexasan infiniteseries,summedover thezerosof thezetafunction.Almostasanaside, Riemann conjectured that all zeros, other than some obvious ones atnegativeevenintegers,lieonthecriticallinez=½+it.

This,iftrue,wouldhavesignificantimplications;inparticular,itimpliesthatvarious approximate formulas involving primes are more accurate than cancurrently be proved. In fact, the ramifications of a proof of the RiemannHypothesis are huge. However, no proof or disproof is known. There’s some‘experimental’evidence:in1914GodfreyHaroldHardyprovedthataninfinitenumber of zeros lie on the critical line. Between 2001 and 2005 SebastianWedeniwski’s programZetaGridverified that the first 100billion zeros lie onthe critical line. But in this area of number theory, that kind of result isn’tentirely convincing, becausemany plausible but false conjectures first fail forabsolutelygigantic numbers.TheRiemannHypothesis is part ofProblem8 inHilbert’sfamouslistof23greatunsolvedmathematicalproblems(Chapter19),andisoneoftheMillenniumPrizeproblemsselectedbytheClayMathematicsInstitutein2000,forwhichthereisaprizeofonemilliondollarsforthecorrectsolution.It’sastrongcontenderforthebiggestunsolvedprobleminthewholeofmathematics.

Riemann proved his exact formula for primes using, among other things,Fourier analysis. The formula can be viewed as telling us that the Fouriertransformofthezerosofthezetafunctionisthesetofprimepowers,plussomeelementary factors. That is, the zeros of the zeta function control theirregularitiesoftheprimes.InTheMusicofthePrimes,MarcusduSautoy’stitleisinspiredbyastrikinganalogy.Fourieranalysisdecomposesacomplexsoundwave into its basic sinusoidal components. In the same way, the glorioussymphonyoftheprimenumbersdecomposesintotheindividual‘notes’played

byeachzeroof thezeta function.The loudnessof eachnote isdeterminedbyhowbigtherealpartof thecorrespondingzerois.SotheRiemannHypothesistellsusthatallofthezerosareequallyloud.

Riemann’s insights into the zeta function entitle him to be considered themusicianoftheprimes.

16CardinaloftheContinuum

GeorgCantor

GeorgFerdinandLudwig

PhilippCantor

Born:StPetersburg,Russia,3March[OS19

February]1845

Died:Halle,Germany,6January1918

THE CONCEPT OF INFINITY, of things that go on for ever without stopping, hasintriguedhumanbeingsformillennia.Philosophershavehadafielddaywithit.Over the last few centuries,mathematicians in particular havemade extensiveuse of the infinite; more precisely, of a variety of different interpretations ofinfinity inmanydifferent contexts. Infinity isn’t just a very largenumber. It’snot really a number at all, because it’s bigger than any specific number. If itwereanumber,itwouldhavetobebiggerthanitself.Aristotlesawinfinityasaprocessofindefinitecontinuation:whichevernumberyou’vecurrentlyreached,

youcanalwaysfindabiggerone.Philosopherscallthispotentialinfinity.Several Indian religions have a fascinationwith very big numbers.Among

themisJainism.AccordingtotheJainmathematicaltextSuryaPrajnapti,somevisionary Indian mathematician stated, around 400 BC, that there are manydifferent sizes of infinity. It sounds like mystical nonsense. If infinity is thebiggest thing that can exist, how can one infinity be bigger than another?Buttowards the end of the nineteenth century, the German mathematician GeorgCantordevelopedMengenlehre–set theory–andusedit toarguethat infinitycan be actual, not just an Aristotelian process of potentiality, and that inconsequencesomeinfinitiesarebiggerthanothers.

At the time, many mathematicians considered this idea to be mysticalnonsensetoo.Cantorhadtofightongoingbattleswithhiscritics,manyofwhomused language that in today’s world would probably result in a lawsuit. Hesufferedfromdepression,possiblyexacerbatedbythederisionheapeduponhim.But most mathematicians now accept that Cantor was right. Indeed, thedistinction between the smallest infinity and any larger one is basic to manyareasofappliedmathematics,inparticularprobabilitytheory.Andsettheoryhasbecomethelogicalfoundationforthewholeofmathematics.Hilbert,oneofthebiggestnamestorealiseearlyonthatCantor’sideasweresound,said:‘NoonewillexpelusfromtheparadisethatCantorhascreated.’

Cantor’s mother, Maria Anna (née Böhm), was a talented musician, and hisgrandfather Franz Böhm had been a solo player in the Russian ImperialOrchestra. Young Georg grew up in a musical family and became anaccomplishedviolinist.Hisfather,alsonamedGeorg,wasawholesaleagentinSt Petersburg, who later joined the city’s Stock Exchange. His mother wasCatholic,buthisfatherwasProtestant,andGeorgwasbroughtupinthatfaith.Initiallyhehadaprivatetutor,transferringtoaprimaryschoolinthecity,butStPetersburg’scoldwinterswerebadforhisfather’shealth,sothefamilymovedtoWiesbadeninGermanyin1856,andlatertoFrankfurt.AlthoughCantorspenttherestofhis lifeinGermany,helaterwrotethathe‘neverfeltatease’ there,andwasnostalgicfortheRussiaofhisyouth.

InFrankfurt,Cantorwasaboarderat theRealschule inDarmstadt. In1860he graduated, being described as an unusually able student, with particularmention of his high skill in mathematics, especially trigonometry. His fatherwanted Cantor to become an engineer, and sent him to the Höhere

Gewerbeschule in Darmstadt. But Cantor wanted to study mathematics, andpesteredhis fatheruntilhegave in. In1862hebeganstudyingmathematicsatthe Zürich Polytechnic. Cantor moved to the University of Berlin when hisfather died in 1863 and left him a substantial inheritance. There he attendedlecturesbyKronecker,Kummer,andWeierstrass.AfterasummeratGöttingenin1866hepresentedhisdissertation‘Onindeterminateequationsoftheseconddegree’,atopicinnumbertheory,in1867.

He then took a position as a teacher at a girls’ school, but worked on hisHabilitation.After being appointed to theUniversity ofHalle, he submitted athesis in number theory, and Habilitation was granted. Eduard Heine, aprominentmathematician atHalle, suggested thatCantor should change fieldsand tackle a famous unsolved problem about Fourier series: prove that therepresentationofa function in this formisunique.Dirichlet,RudolfLipschitz,Riemann,andHeinehimselfhadalltriedtoprovethisresult,butfailed.Cantorsolveditwithinayear.Foratimehecontinuedworkingontrigonometricseries,and his researches led him into areas that we now recognise as prototype settheory. The reason is that many properties of Fourier series rest on delicatefeatures of the function being represented, such as the structure of the set ofpointsatwhichitisdiscontinuous.Cantorcouldn’tmakeprogressintheseareaswithoutcoming face to facewithcomplicated issuesabout infinite setsof realnumbers.

Research into the foundations of mathematics were on the rise, and aftercenturies of informal treatment of the ‘real’ numbers as infinite decimals,mathematicianswerestartingtowonderwhatitallmeant.Forexample,there’snowaytowritedowntheinfinitedecimalexpansionofπ.Allwecandoisgiverules for how to find it. In 1872 one of Cantor’s papers about trigonometricseries introduced a novelmethod for defining a real number as the limit of aconvergentsequenceofrationalnumbers.InthesameyearDedekindpublishedafamouspaper,inwhichhedefinedarealnumberintermsofa‘section’dividingthe rational numbers into two disjoint subsets, such that the members of onesubsetareall lessthananymemberoftheother.Init,hecitedCantor’spaper.Thesetwoapproaches–convergentsequencesofrationalsorDedekindsections– are both standard in courses on the foundations of mathematics and theconstructionofthesetofrealnumbersfromtherationals.

By 1873 Cantor had embarked on the research that qualifies him as asignificant figure of the highest order: set theory and transfinite (his term forinfinite) numbers. Set theory has since become an essential part of any

mathematics course, because it gives a convenient and versatile language inwhichtodescribethesubject.Informally,asetisanycollectionofobjects;theycouldbenumbers,triangles,Riemannsurfaces,permutations,whatever.Setscanbecombinedinvariousways.Forexample,theunionoftwosetsiswhatyougetby combining them into one set, and the intersection is what they have incommon. Using sets, we can define basic concepts such as functions andrelations.Wecanconstruct systemsofnumbers suchas the integers, rationals,reals,andcomplexnumbersfromsimplerconstituents,makingheavyuseoftheemptyset,whichhasnomembers.

Transfinitenumbersareawaytoextendthenotionof‘howmanymembers?’toinfinitesets.Cantorstumbledacrossthisideain1873whenheprovedthattherational numbers are countable; that is, they can be placed in one-to-onecorrespondencewith the natural numbers 1, 2, 3,… (I’ll explain the ideas andterminology shortly.) If there’s only one size of infinity, this result would beobvious,buthe soon foundaproof that the realnumbers arenot countable. Itwaspublishedin1874,ayearofgreatpersonal importanceforCantorbecausehemarriedVallyGuttmann–amarriagethatwouldleadtosixchildren.

Seekingastilllargerinfinitythanthatofthereals,Cantorthoughtaboutthesetofallpoints in theunit square.Surely thesquare,with its twodimensions,hasmorepoints than the real line?Writing toDedekind,Cantor expressedhisopinion:

Canasurface(sayasquarethatincludestheboundary)beuniquelyreferredtoaline(sayastraightline segment that includes the end points) so that for every point on the surface there is acorrespondingpointofthelineand,conversely,foreverypointofthelinethereisacorrespondingpointofthesurface?Ithinkthatansweringthisquestionwouldbenoeasyjob,despitethefactthattheanswerseemssoclearlytobe‘no’thatproofappearsalmostunnecessary.

Soon,however,hefoundthattheanswerwasn’tasobviousasitseemed.(‘Proofappearsunnecessary’toamathematicianislikearedragtoabull,andheshouldhave seen it coming.) In 1877 he proved that such a correspondence does, infact,exist.‘Iseeit,butIdon’tbelieveit!’Cantorwrote.Butwhenhesubmittedhispaper to theprestigiousJournal fürdie reineundangewandteMathematik(JournalforPureandAppliedMathematics),LeopoldKronecker–abrilliantbutultra-conservative mathematician and a leading light of that period – wasunconvinced,andonlytheinterventionofDedekindledtothework’sacceptanceandpublication.Cantor,withsomejustification,neversubmittedanotherpaperto that journal. Instead, between 1879 and 1884, he sent the bulk of his

development of set theory and transfinite numbers to the MathematischeAnnalen(MathematicalAnnals),probablyfacilitatedbyFelixKlein.

Before continuing Cantor’s story, we need to understand the revolutionarynature of his ideas, and what they were about. It would be too confusing topresent them in the terminology of the period, so I’ll apply some modernhindsighttoextractafewbasicideas.

Inhis1638DiscoursesRelatingtoTwoNewSciences,Galileoraisedabasicissue – somewhat paradoxical – about infinity. The book is presented as adiscussion between Salviati, Simplicio, and Sagredo. Salviati always wins,Simpliciodoesn’tstandachance,whileSagredo’sjobistokeepthediscussionmoving. Salviati observes that it’s possible to match counting numbers tosquares,sothateachnumbercorrespondstoauniquesquare,andeachsquaretoauniquenumber.Justmatcheachnumberwithitssquare:

Withfinitenumbers,iftwosetsofobjectscanbematchedinthismanner,theymust contain the same number ofmembers. If everyone seated at a table hastheirownknifeandfork,andjustoneofeach,thenthenumberofknivesequalsthatofforks,andbothequalthenumberofpeople.So,eventhoughsquaresforma rather ‘thin’ subset of all numbers, it seems that there are exactly as manysquaresasnumbers.Salviaticoncludes:‘Wecaninferonlythatthetotalityofallnumbers is infinite, and the attributes “equal”, “greater”, and “less”, are notapplicabletoinfinite,butonlytofinite,quantities.’

Cantorrealisedthat thesituationisn’tquite thatbleak.Heusedthiskindofmatching(whichhecalledaone-to-onecorrespondence)todefine‘samenumberofmembers’ for sets,be they finiteor infinite.Thiscanbedone, interestinglyenough,withoutknowingwhatthenumberactuallyis.Indeed,we’vejustdoneso for the knives and forks. So logically, ‘samenumber’ is prior to ‘number’.There’snothingstrangeaboutthis:wecanseewhentwopeopleareequallytallwithoutknowingtheirexactheights,forexample.

Thewaytointroduceactualnumbersistospecifyastandardset,andsaythatanythingthatmatchesithasthatsetasitscardinal–afancywordfor‘numberofelements’.Theobviouschoiceforaninfinitesetisthesetofallnaturalnumbers,

whichdefinesatransfinitecardinalthatCantordubbed‘aleph-Null’.Herealephis the first letter of the Hebrew alphabet, and Null is German for ‘zero’. Insymbols, it looks like this: 0. By definition, any set thatmatches the naturalnumbershascardinal 0.Salviatiprovedthatthesetofsquaresalsohascardinal0.This seems paradoxical because there are clearly numbers that are not

squares–indeed,‘most’numbersaren’tsquares.Wecanresolvetheparadoxbyaccepting that removing some elements froman infinite set neednotmake itscardinalsmaller.Thewholeneednotbegreaterthanthepart,asfarascardinalsareconcerned.However,wedon’thavetofollowSalviatiandreject thewholeideaofcomparison:wegetsensibleresultsifweassumethatthewholeisgreaterthanorequaltothepart.Afterall,thewholepointaboutinfinityasaconceptisthatitdoesn’talwaysbehavelikefinitenumbers.Thebigquestionishowfarwecanget,andwhatwecansalvage.

Cantor’s next big discoverywas that the rational numbers (let’sworkwiththepositiveonesforsimplicity)alsohavecardinal 0.Theycanbematchedtothenaturalnumberslikethis:

To get the top row,we order rationals differently from their numerical order.Definethecomplexityofarationalnumbertobethesumofthenumeratorandthedenominator.Consideronlyrationalswherethesehavenocommonfactor,toavoidincludingthesamenumbertwice.Forexample,2/3and4/6arethesamerational; we choose only the first form. First split the rationals into classes,orderedbycomplexity.Eachsuchclassisfinite.Then,withineachclass,orderthe fractions according to their numerators. So the classwith complexity 5 isorderedlikethis:

1/42/33/24/1

It’s easy to prove that every positive rational occurs once and once only.Thenaturalnumberthatmatchesitisitspositionintheresultingorderedlist.

Up to thispoint, itmightbe that 0 is justa fancysymbol for infinity,andallinfinitiesareequal.Thenextdiscoveryexplodesthatpossibility.Thesetofreal

numberscan’tbematchedtothenaturalnumbers.Cantor’s firstproofof1874wasaimedat aproblem innumber theory, the

existenceof transcendental numbers.Analgebraicnumber is one that satisfiessomepolynomialequationwithintegercoefficients,suchas ,whichsatisfiesx2–2=0.Ifanumberisn’talgebraic,it’scalledtranscendental.Nosuchequationwasknownforeorπ,andthesewerethoughttobetranscendental,aconjecturethatturnedouttobecorrect.Liouvilleprovedtheexistenceofatranscendentalnumberin1844,buthisexamplewasveryartificial.Cantorprovedthat‘most’real numbers are transcendental, by showing that the set of algebraic numbershas cardinal 0, but the set of reals has a larger cardinal. His proof involvesassumingtherealsarecountable,andconstructingasequenceofnestedintervalsthatomit every realnumber in turn.The intersectionof those intervals (whichcanbeprovednottobeempty)mustcontainarealnumber,butwhicheveronethatis,it’salreadybeenexcluded.

In1891he foundamore elementaryproof, the famousdiagonal argument.Assume (for a contradiction) that the real numbers (say between 0 and 1 forsimplicity) are countable.Then the countingnumbers canbematched to thesereals.Indecimalnotation,anymatchingofthiskindtakestheform

10·a1a2a3a4…20·b1b2b3b4…30·c1c2c3c4…40·d1d2d3d4………

By assumption, every real number occurs somewhere in the list. Now weconstruct one that doesn’t.Define successivedecimal placesx1,x2,x3, of a realnumberxasfollows:

Ifa1=0letx1=1,otherwiseletx1=0.

Ifb2=0letx2=1,otherwiseletx2=0.

Ifc3=0letx3=1,otherwiseletx3=0.

Ifd4=0letx4=1,otherwiseletx4=0.

Continuethisprocessindefinitely,makingxneither0or1,sothatitdiffersfromthenthdecimaldigitoftherealnumbercorrespondington.

Byconstruction,xdiffersfromeverynumberonthe list. Itdiffersfromthe

first number in its first digit, from the second number in its second digit; ingeneral,itdiffersfromthenthnumberinitsnthdigit,soit’sdifferentfromthenth number, no matter what value n has. However, we assumed that the listexists,andeveryrealnumberappearsonit.Thisisacontradiction,andwhatitcontradictsistheassumptionthatsuchalistexists.Thereforenosuchlistexists,andthesetofrealnumbersisuncountable.

AsimilarideaunderpinsCantor’sdiscovery,whichhefoundhardtobelieve,that the plane has the same cardinal as the real line.A point in the plane hascoordinates(x,y)wherexandyarerealnumbers.Forsimplicity,restricttotheunitsquare;thenxandyhavedecimalexpansions

x=0·x1x2x3x4…

y=0·y1y2y3y4…

Match this pair to a point on the line whose coordinate is those of x and yinterleaved,likethis:

0·x1y1x2y2x3y3…

Sincewecanrecoverxandybyselectingonlysuccessivedigitsinodd-oreven-numbered locations, thisdefinesaone-to-onecorrespondencebetween theunitsquare and the unit interval on the line. It’s easy to beef this up to the entireplaneand theentire line. (Afewtechnicalitiesneed tobe takencareof,whichI’vesuppressed,todowiththelackofuniquenessofthedecimalrepresentationofanumber.)

TherewasonequestionthatCantorwasunabletodecide,eitherway.Istherea transfinite cardinal strictlybetween 0 and the cardinalof the realnumbers?Cantorthoughtnot,becausehecouldn’tfindone,despitetryingalotofplausiblecandidates.ThisconjecturebecameknownastheContinuumHypothesis.We’llseehowitfaredinChapter22.

Foradecadefrom1874,Cantorthrewhiseffortsintosettheory,discoveredtheimportance of one-to-one correspondences in the foundations of the numbersystem,anddevisedhisextensionofcountingprinciplestotransfinitenumbers.Hisworkwassooriginalthatmanyofhiscontemporarieswereunabletoacceptitorbelieve ithadvalue.HismathematicalcareerwasblightedbyKronecker,

who found his revolutionary ideas philosophically distasteful. ‘God made theintegers,allelseistheworkofMan,’Kroneckersaid.

Cantor rather set himself up as a philosophical target by statingunequivocallythatsettheorywasaboutactualinfinity,notAristotelianpotentialinfinity.Thiswasaslightoverstatementbecause it’s ‘actual’ infinityonly inaconceptualsense.Inmathematics,it’susuallypossibletopassfromadescriptionthat appears to involve actual infinity to one that looks purely potential.However,thistranslationprocessoftenlookscontrived:Cantorwascorrectthatthenaturalwaytothinkabouthisworkistoviewinfinityasacompletedwhole,notasaprocessthat,whilefiniteatanystage,canbecontinuedindefinitely.ThephilosopherLudwigWittgensteinwasavocalcritic.Hewasespeciallyscathingabout the diagonal argument, and even when Cantor had died he was stillcomplainingabout‘theperniciousidiomsofsettheory’.Butthemainreasonhekeptcomplainingwas thatmathematicians increasingly sidedwithCantor, andnone of them paid much attention to Wittgenstein. This must have beenespecially galling since he was particularly interested in the philosophy ofmathematics, but then mathematicians don’t take kindly to philosophers whoinsistthey’redoingitallwrong.Settheoryworked,andmostmathematiciansarepragmatic,evenaboutfoundationalissues.

Cantor was religious, and struggled to reconcile his mathematics with hisbeliefs. The nature of the infinite was still heavily bound up with religion,becausetheChristianGodwasconsideredinfinite,andwasheldtobetheuniqueactualinfinity.Kronecker’sremarkabouttheintegerswasn’tametaphor.ThenalongcomesCantor,claimingactual infinities inmathematics…Well,youcansee what would happen. Cantor struck back, though, saying: ‘The transfinitespeciesarejustasmuchatthedisposaloftheintentionsoftheCreator…asarethefinitenumbers.’Thiswasacleverargument,becausedenyingitwouldbetoclaimGodhadlimitations,whichwasheretical.CantorevenwrotetoPopeLeoXIIIaboutitall,andsenthimsomemathematicalarticles.GodknowswhatthePopethoughtaboutthem.

OthersunderstoodwhatCantorwasdoing.HilbertrecognisedtheimportanceofCantor’swork,andpraised it.Butashegrewolder,Cantor felt thatset theoryhadnotmadetheimpacthe’dhopedfor.In1899hehadanattackofdepression.Hesoonrecovered,buthelostconfidence,tellingGöstaMittag-Leffler‘Idon’tknow when I shall return to the continuation of my scientific work. At the

momentIcandoabsolutelynothingwithit.’Tocombathisdepressionhewentfor a holiday in the Harz mountains, and attempted a reconciliation with hisacademic enemy, Kronecker. Kronecker responded positively, but theatmospherebetweenthemremainedtense.

Cantor’s mathematics was a worry, too: he was unhappy that he couldn’tprovehisContinuumHypothesis;thoughthe’dproveditwasfalse,butquicklyfoundamistake;thenthoughthe’dproveditwastrue,butagainfoundamistake.At this point Mittag-Leffler asked Cantor to withdraw a paper from ActaMathematicaeventhoughithadreachedproofstage–notbecauseitwaswrong,butbecauseitwas‘onehundredyearstoosoon’.Cantorjokedaboutthis,buthewasveryhurt.Hestoppedwriting toMittag-Leffler, tooknofurther interest inthejournal,andprettymuchgaveuponsettheory.

His depression tended to express itself in twoways.Onewas an increasedinterest in the philosophical implications of set theory. The other was aconviction that the works of Shakespeare were actually written by FrancisBacon.Thisbee inhisbonnet ledhim tomakea serious studyofElizabethanliterature,andby1896hewaspublishingpamphletsaboutthispettheory.Then,in quick succession, his mother, younger brother, and youngest son died. Heshowedincreasingsignsofmentalinstability,andin1911,whentheUniversityofStAndrewsinScotlandinvitedhimasadistinguishedguesttocelebrationsofthe university’s 500th anniversary, he spent much of the time talking aboutBacon and Shakespeare. Depression became a constant companion. He spentsometimeinhospitalforthecondition,andin1918hediedinasanatoriumfromaheartattack.

TheironyisthatMittag-LefflerwasessentiallyrightwhenhetoldCantorhewasa century ahead of his time, though not perhaps in the sense he intended.AlthoughCantor’sideasslowlygainedground,themostsignificantimpactofsettheory onmathematics had towait until the 1950s and 60s,when the abstractapproachtomathematicspromotedbythegroupcallingitselfNicolasBourbakicame into full flower. Bourbaki’s influence on mathematical education has(thankfully) waned, but its insistence that mathematical concepts should bedefinedprecisely, in asmuchgenerality as possible, still holds sway.And thebasisforprecisionandgeneralityistheviewpointaffordedbyCantor’sbelovedsets.Today,everyareaofmathematics,pureandapplied,isfirmlybasedintheformalism of set theory. Not just philosophically, but practically.Without the

language of sets,mathematicians now find it impossible even to specifywhattheyaretalkingabout.

The verdict of posterity is that, yes, there are philosophical issueswith settheory and transfinite numbers, but these are no worse than the very similarphilosophicalissueswithKronecker’sbelovedintegers.Those,too,aretheworkofman,andtheworkofmanisusuallyflawed.Ironically,wenowdefinethemusing… set theory. And we see Cantor as one of the true originals inmathematics. Ifhehadn’t inventedset theory, someonewouldeventuallyhavedone so, but it couldwell have taken decades before anyone else came alongwithhisuniquecombinationofpower,depth,andinsight.

17TheFirstGreatLady

SofiaKovalevskaia

SofiaVasilyevnaKovalevskaia(néeKorvin-Krukovskaya)orSophie/SonyaKowalevski

Born:Moscow,Russia,15January1850Died:Stockholm,Sweden,10February1891

FROMEARLIESTCHILDHOODyoungSofa,as thefamilyaffectionatelycalledher,had a burning desire to understand whatever took her fancy. Her interest inmathematicswas kindled at the age of eleven; remarkably, the causewas the

nurserywallpaper.Her fatherVasilyKorvin-Krukovskywas lieutenant-generalof artillery in the Imperial Russian Army, and her mother Yelizaveta (néeShubert) was from a family of high standing in the Russian nobility. Thewallpaper comes into the story because the family owned a country estate atPalabino,nearStPetersburg.Onmoving toPalabino thefamilyhad thewholehouse redecorated, but failed to buy enough wallpaper for the nursery. As asubstitute, they used pages from an old textbook, which just happened to beOstrogradskii’s lecture course on differential and integral calculus. In herautobiographyMemoriesofChildhoodSofiarememberedspendinghoursstaringatthewalls,tryingtofigureoutthemeaningofthearcanesymbolsthatcoveredthem.Shequicklymemorisedtheformulas,butlaterrecalledthat‘atthetimeIwasstudyingitIcouldnotunderstanditatall’.

Shealreadyhadforminthiskindofself-education.Thefashionat thetimewasnottoteachreadingtoyoungchildren,butSofiahadbeendesperatetoread.At theageofsixshehad taughtherselfbymemorising theshapesof letters innewspapersandthenpesteringanadulttotellherwhattheymeant.Sheshowedoffhernewability toher father,who, though incredulousat first–he thoughtshe’d just memorised a few sentences – was soon convinced, and wasimmenselyproudofherinitiativeandintelligence.

WhenSofia’sbedroomwallpapertriggeredasimilarlyself-propelledinterestin mathematics, her family, remarkably forward-thinking for the period, didnothingtodiscourageit,eventhoughmanyamongtheirsocialcirclewouldnothave deemedmathematics to be a fit subject for a young lady.Circumstancesconspiredtoallowhertopursueherpassion.Mathematicshadbeenoneofherfather’s favourite subjects, andSofiawashis favouritedaughter.Hermother’sfatherFedorFedorovichShuberthadbeenamilitarytopographer,andhisfatherFedor IvanovichShubert had been a leading astronomer and amember of theAcademy of Sciences. So mathematical blood (to use that period’s image ofheredity)flowedinSofia’sveins.Moreover,herfamilyhadlongbeenimmersedin the mathematical subculture, which may well have been a more importantinfluence.

Beginningwiththebasics,thegeneralmadesurethatSofia’stutorsinstructedherinarithmetic.Butwhenheeagerlyaskedhisdaughterhowshelikedit,herinitial responsewasdistinctly lukewarm: itwasn’tcalculus.Herviewchangedwhen she finally realised thatwithout the basics, shewould never progress tothosefascinatingequationsonthewallpaper.Notonlydidshegoontomastercalculus; she progressed to the frontiers of mathematical research, making

discoveries that amazed the period’s leading mathematicians. She worked onpartialdifferentialequations,mechanics,andthediffractionoflightbycrystals.Her mathematical publications number only ten, and one is a translation intoSwedish of one of the others, but their quality is outstanding. She waspenetrating, original, and technically proficient. The prominent AmericanmathematicianMarkKacdescribedherasthe‘firstgreatladyofmathematics’.She was arguably the greatest female scientist of her time, eclipsed only byMarieCurieafewdecadeslater.

SofiawasborninMoscowin1850.Shehadaneldersister,Anna,knowntothefamily as Aniuta, whom she adored; later she was blessed with a youngerbrother,Fedor.HerunclePyotrVasilievichKrukovskyhadastrong interest inmathematics, and often talked to her about it, long before she could possiblyunderstandwhathewassaying.

In 1853,when Sofiawas three years old, Russia became embroiled in theCrimean War. The conflict was ostensibly about the rights of Christianminorities in the Holy Land, but France and the United Kingdom weredetermined to stop Russia from taking over areas of the declining OttomanEmpire.By1856anallianceofFrance, theUnitedKingdom,Sardinia,andtheOttomans had defeated Russia after the siege of Sevastopol. This humiliationbroughtaboutmassivepublicdiscontentinRussia.Peasantsandliberalsrevoltedagainst an oppressive system, which they increasingly viewed as corrupt andincompetent.Thegovernmentfoughtbackwithcensorshipandrepressionbythetsarist secretpolice.Manynoblesownedvast countryestates,but they seldomspentmuchtimethere,preferringStPetersburg’spoliticalimportanceandsocialdelights. Prudence now dictated that even those with liberal leanings shouldspendmoretimeinthecountryandpaymoreattentiontothegrievancesoftheirworkforce. So in 1858 General Korvin-Krukovsky told his wife that it hadbecometheirdutytorelocatetotheirestate.

At first Sofia and her elder sister Aniuta were left to their own devices,exploringthecountrysideandgenerallygettingintoscrapes.Butaftertheytriedtoeatsomeunsuitableberries,andwereillfordays, theirfatherhiredaPolishtutor IosifMalevich and a strict English governessMargarita Smith, who thegirls disliked intensely.Malevich taught Sofia the basics of a youngwoman’seducation,includingarithmetic,butherunclePetrinductedherintosomeofthemysteries ofmore advancedmathematics – topics such as squaring the circle

(constructinga squareofareaequal to thatofagivencircle,which isactuallyimpossiblewith the traditional geometric instruments of ruler and compasses)andasymptotes(linesthatacurveapproachesindefinitelyclosetowithouteverreachingthem).Theseconceptsfiredherimaginationandleftherwantingmore.

EventuallyMissSmithresigned,andpeacereignedintheKorvin-Krukovskyhousehold. In 1864 Aniuta sent two stories that she’d written to Fedor andMikhail Dostoievski, which were published in their journal Epokha. AniutabeganasecretcorrespondencewithFedor,andafterherfatherobjectedandthenrelented,FedorDostoievskibecamepartofthefamily’scircle.Sofiajoinedthesocialwhirl,meetingotherprominentfiguresaswell.ForatimeshedevelopedaschoolgirlcrushonDostoievski.WhenFedorproposedmarriagetoAniuta,Sofiawasoutraged,evenmoresowhenhersisterturnedhimdown.

Ataboutthistimeshebecameabsorbedinthemathematicalmysteriesofherbedroom wallpaper, and one strand of her future life was set. A neighbour,NikolaiTyrtov,wasaphysicsprofessoratthePetersburgNavalAcademy,andhebroughtheracopyofhisintroductorytextbookonphysics.Notknowinganytrigonometry, she struggled until she found a more intuitive geometricapproximation – essentially the clasical use of a chord of a circle. Tyrtov,excitedbythisdemonstrationofherabilities,urgedthegeneraltoletherstudyhighermathematics.

At that time, Russian women were not allowed to go to university, but theycould study abroad with written permission from father or husband. So Sofiacontracted a ‘fictitious marriage’ with Vladimir Kovalevskii, a youngpalaeontology student. This ploy, amarriage of convenience with no genuinerelationship, was fairly common among educated young Russian women as awaytogainsomefreedom.ToSofia’schagrin,herfathersuggestedadelay.Intypically headstrong manner she bided her time until the house was full ofdistinguisheddinnerguests;thenshesneakedout,leavinganotesayingshehadgonetoVladimir’slodgingsunchaperoned,andwouldstaythereuntiltheywereallowed to marry. To avoid social meltdown, the general duly presented hisdaughterandherfiancétotheguests.Sofia’splanwastogetmarried,thendumpVladimirandgoherownway,butVladimirbecame infatuatedwithhis futurewifeandhersocialcircle,andhadnowishforthemtoseparate.Theymarriedin1868,whenSofiawas18,andshebecameSofiaKovalevskaia.

LikemanyyoungRussiansatthattime,Kovalevskaia’spoliticalviewswere

nihilist.Thatis,sherejectedanyconventionthatlackedrationalsupport,suchasgovernment and the law. Vladimir Lenin, quoting the radical writer DmitriPisarev,capturedtheattitude,anextremeformofsocialDarwinismthrownbackinthefacesoftherichandpowerfulwhooftenjustifiedtheirprivilegesinmuchthe sameway: ‘Break,beatup everything,beat anddestroy!Everything that’sbeingbrokenisrubbishandhasnorighttolife!Whatsurvivesisgood.’WhenthenewlywedsarrivedinStPetersburgtheirapartmentsoonbecamethesocialhubforlike-mindednihilists.

In1869theyleftRussia,initiallyforVienna.Vladimir’spublishingbusinesshad collapsed and hewas fleeing creditors; both of themwere also seeking amore intellectual atmosphere. Vladimir set his sights on geology andpalaeontology.Kovalevskaia– toher surprise–waspermitted to takephysicslectures at the university, but in the absence of any equally accommodatingmathematicians the couple moved to Heidelberg. At first the universityauthorities gave her the usual runaround, apparently under the impression shewas awidow and bemusedwhen told shewasmarried, but eventually it wasagreed that she was free to attend lectures provided the professor had noobjection. Soon she was spending twenty hours a week in lectures, bymathematicians such as Leo Königsberger and Paul DuBois-Reymond, thechemicalphysicistGustavKirchhoff,andphysiologistHermannHelmholtz.

Shealsopestered themisogynistchemistWilhelmBunsentoallowherandherfriendIuliaLermontovatoworkinhislaboratory,wherehehadpreviouslyswornnowoman–especiallyaRussian–wouldeversetfoot.‘Nowthatwomanhas made me eat my words,’ he complained to Weierstrass, and spreadscandalous rumours in revenge. His colleagues, in contrast, were enthusiasticabout their talented female student, and the newspapers carried occasionalarticlesabouther.Kovalevskaiarefusedtolettheattentiongotoherhead,andconcentratedonherstudies.

TheKovalevskiistravelledtoEngland,France,Germany,andItaly.VladimirmetCharlesDarwinandThomasHuxley,withwhomhewasalreadyacquainted.Throughsuchcontacts,KovalevskaiawasabletomeetthenovelistGeorgeEliotsocially. In her journal for 5 October 1869 Eliot wrote: ‘On Sunday, aninterestingRussianpaircametoseeus–M.andMme.Kovalevskii:she,aprettycreature, with charming modest voice and speech, who is studyingmathematics…atHeidelberg;he,amiableandintelligent,studyingtheconcretesciencesapparently–especiallygeology.’ThephilosopherandsocialDarwinistHerbert Spencer was also present, and boorishly proclaimed the intellectual

inferiorityofwomen.Kovalevskaiaarguedwithhimfor threehours,andEliotwrotethatshehad‘defendedourcommoncausewellandbravely’.

In 1870 Kovalevskaia moved to Berlin, hoping to study under Weierstrass.Hearingrumoursthathedisapprovedofeducationforwomen,sheworeabonnetmoresuited toanolderwoman,whichhidher face.Weierstrasswassurprisedwhen she asked to study with him, but replied politely, giving her someproblemstosolveandbringback.Aweeklatershereturned,havingsolvedthemall,oftenbyoriginalmethods.Weierstrasslatersaidshehad‘thegiftofintuitivegenius’. The university senate refused her permission to study officially, soWeierstrass offered her private lessons. They began a correspondence thatcontinueduntilherdeath.

Aniutawasbynowliving inPariswithVictorJaclard,ayoungMarxist. In1871 the National Guard declared the Paris Commune, a radical socialistgovernmentthatbrieflyruledthecity.Leninsaiditwas‘thefirstattemptbytheproletarianrevolutiontosmashthebourgeoisstatemachine’.Thestatemachinehadnowishtobesmashed.WhenSofiaheardthatJaclardmightbearrestedforhis political activities, the Kovalevskiis headed for Paris. As the Versaillesgovernment began to bombard the Commune, Sofia and Aniuta nursed thewounded.TheKovalevskiis returned toBerlin,butwhenParis fellandJaclardwasarrested,theywentbacktohelpAniuta,gettinghersafelytoLondon.There,KarlMarxprovidedmorehelp.GeneralKorvinKrukovskyandhiswifewenttoParis intending togetJaclardset free.Theycouldn’tsecureanofficial release,but itwascausallymentioned that Jaclardwasbeingmoved toanotherprison.Astheprisonerswerebeingtakenthroughthecrowds,awomangrabbedhimbythearmandhauledhimaway.Somebelieve itwasAniuta (except shewas inLondon at the time), someKovalevskaia, others Jaclard’s sister; some think itwasVladimirindisguise.Jaclardescaped,Vladimirgavehimhispassport,andhefledtoSwitzerland.Fromthenon,evenwhenimmersedinhermathematics,Kovalevskaiainvolvedherselfinpoliticalandsocialmovements.

BackinBerlin,shedivedintoherstudieswithenthusiasm.Herresearchwasgoing well, but her marriage was not. The couple quarrelled incessantly, andVladimir was muttering darkly about a divorce. By 1874 Kovalevskaia hadwritten three research articles, all of doctoral quality.Themost importantwasthe first; Charles Hermite called it ‘the first significant result in the generaltheory of partial differential equations’. The second was on the dynamics of

Saturn’s rings, and the thirdwas a technical paper about the simplification ofintegrals.

A partial differential equation relates the rates of change of some quantitywith respect to several distinct variables. For instance, Fourier’s heat equationrelateschangesintemperaturewithrespecttospace–alongtherod–tohowitsvalueateachspecificlocationchangeswithrespecttotime.Histrickforsolvingtheequationusingtrigonometricseriesreliesonaspecialfeature:theequationislinear, so solutions can be added to each other to give further solutions.Kovalevskaia’s paper of 1875 proves the existence of solutions for nonlinearpartialdifferentialequations,providedtheysatisfysometechnicalconditions.ItextendedCauchy’sresultsfrom1842,andacombinedversionisnowcalledtheCauchy–Kovalevskaiatheorem.

HerpaperonSaturn’sringswaswrittenwhilesheworkedwithWeierstrass,butthetopicdidn’tinteresthimandshedidtheresearchalone.Shestudiedthedynamicsofrevolvingringsof liquid,whichLaplacehadproposedasamodelfor the rings of Saturn. She analysed the stability of the rings in this model,showingthattheycouldn’tbeellipsesasLaplacehadthought,butmustbeegg-shaped, fat at oneendand thinner at theother.Thepaper is interesting for itsmethods, andwouldhavebeenevenmore so if ithadcontained thenecessaryproofs, but it soon became known that the rings were made of innumerablediscrete particles, so the underlying fluid model was questionable. AsKovalevskaiawrote:‘DuetoMaxwell’sresearchithasbecomedoubtfulwhetherLaplace’sviewofthestructureoftheringsofSaturnisacceptable.’

Nowcametheperennialproblemofacademicpolitics.Thepapershadtobepresented to a university for a doctorate, and it had to be one of the rareinstitutions thatwaswilling toawardone toawoman.WeierstrassapproachedGöttingen,which sometimes awarded doctorates to foreignerswithoutmakingthemundergotheusualformaloralexamination,whichwouldbecarriedoutinGerman. Kovalevskaia obtained the degree of doctor of philosophy inmathematics summa cum laude (with distinction), becoming the first womanafterMariaAgnesiinRenaissanceItalytogainaPhDinmathematics,andoneofpreciousfewtogetascientificdoctorate.

Kovalevskaiawasnowafullyfledgedmathematician.

In 1874 the Kovalevskiis went back to Russia, first to the family home inPalabino,andfromtheretoStPetersburginsearchofacademicpositions.They

failed to secure any job offers. Kovalevskaia’s German degree counted fornothing: she would need a Russian one. However, as a woman, she wasn’tpermitted to sit the exam. Frustrated, the Kovalevskiis went into business tomake somemoney, a decision that quickly proved disastrous. Kovalevskaia’sfatherdied in1875, leavinghera legacyof30,000roubles,whichwouldhaveallowedthemafrugalliving,ifinvestedwisely.Instead,theyputthemoneyintoa property scheme. Initially this seemed to be a success, and theKovalevskiismovedtoanewhouse,withagarden,orchard,andcow.(Havingyourowncowwas de rigueur among wealthy middle-class Russians.) The couple had adaughter,alsonamedSofia.Vladimirputmoremoneyintoaradicalnewspaper,eventually losing 20,000 roubles when it folded. Months later the propertyschemecollapsed.Vladimirhadusedspeculativefutureprofitstobuyland,andwhen his creditors called in his debts, the property empire turned out to befantasy.

In 1878 Kovalevskaia renewed contact with Weierstrass and followed hisadvice to tackle therefractionof lightbyacrystal. In1879she lectured to theSixthCongressofNaturalScientistsonherearlierresearchonAbelianintegrals.In 1881 she and her daughter arrived in Berlin, whereWeierstrass had foundthem an apartment. Vladimir’s finances went from bad to worse, and thecouple’s possessionswere sold to pay off part of his debt. In 1883, sufferingfrom sudden mood swings and likely to face prosecution for his role in afinancial swindle, he committed suicide by drinking a bottle of chloroform.Aguilt-riddenKovalevskaia stopped eating for fivedays, then fainted.Force-fedby her doctor, she regained consciousness and threw herself into her work,completinghertheoryofrefractioninacrystal.ShereturnedtoMoscowtosetVladimir’saffairsinorder,andpresentedherrefractionresearchtotheSeventhCongressofNaturalScientists.

Herhusband’sdeathremovedamajorobstaclebetweenKovalevskaiaandanacademic post, because awidowwasmore acceptable than an independent ormarried woman. Previously, she had met the leading Swedish mathematicianGösta Mittag-Leffler through his sister Anna Carlotta Edgren-Leffler, arevolutionary, actress, novelist, and playwright. Their friendship lasted untilKovalevskaia’s death. Mittag-Leffler, impressed by her research on Abelianintegrals, secured a position for her at StockholmUniversity – temporary andprovisional,butagenuineacademicpostnonetheless.Kovalevskaiabecametheonly woman to hold such a position in the whole of Europe. She arrived inStockholmlatein1883.Sheknewthejobwouldbechallenging,andshe’dhave

to battle against prejudice, but one progressive newspaper described her as ‘aprincess of science’, which was encouraging. Though she did remark that asalarywouldbeevenbetter.

Kovalevskaia’s literary ambitions blossomed, and she and Edgren-Lefflercoauthoredtwoplays:TheStruggleforHappinessandHowItMightHaveBeen.Shealsoattackedamajorclassicalprobleminmechanics:therotationofarigidbodyaboutafixedpoint.Hereshemadeatotallyunexpecteddiscovery–anewtype of solution now called the Kovalevskaia top. Some academic-politicalhorse-tradingconvertedherunpaidpositionintoanextraordinaryprofessorship,whichmightbemadepermanentafterfiveyears.Nowshehadenoughtoliveon–barely–andbeganpayingoffsomeofherlatehusband’sdebts.Shebecamesomethingofalocalcelebrity,whichpersuadedtheUniversityofBerlintoallowher to attend lectures in anyPrussianuniversity.She travelledback toRussia,thentoBerlin,andbacktoSweden.ShejoinedtheeditorialboardofthejournalActaMathematica,anotherfemalefirst.

Wheels were moving; Hermite had persuaded the Paris Academy’s PrixBordintosetaproblemtailoredtoherinterests,andtherewaslittledoubtamongtheinnercirclethatKovalevskaiawouldwin.In1888shedulywontheprizeforherworkontherotationofasolidbody.Asherreputationasamajorresearchmathematiciangrew, theoldbarrierswerestarting tocomedown. In1889shewasappointedprofessorordinariusatStockholmUniversity,a tenured lifetimepost. Shewas the firstwoman to hold such a position at a northernEuropeanuniversity.Aftermuch lobbyingonherbehalf, shewasgrantedaChair in theRussianAcademyofSciences.Firstthecommitteevotedtochangetherulestoallowwomentobeadmitted;threedayslatertheyelectedher.

Kovalevskaia wrote several non-mathematical works, including A RussianChildhood, her playswithAnnaCarlotta, and a partly autobiographical novel,NihilistGirl(1890).Shediedofinfluenzain1891.

Kovalevskaia’s unexpected discovery of a new solution to the problem of arotatingrigidbodywasamajorcontributiontomechanics,whichisabouthowparticlesandbodiesmoveunder theactionofforces.Typicalexamplesare theswingofapendulum, thespinningofa top,and theorbitalmotionofaplanetroundtheSun.AswesawinChapter7,mechanicsreallytookoffin1687whenNewton published his laws ofmotion.The second law is especially importantbecauseittellsushowabodymovesundertheinfluenceofknownforces:mass

timesaccelerationequalsforce.Thislawspecifiesthebody’spositionindirectlyin terms of the rate of change of the rate of change of position, making it a‘secondorder’differentialequation.

If we’re lucky, we can solve the equation, obtaining a formula for thepositionof thebodyat anygiven time. If so, theequation is integrable.Muchearlyworkinmechanicsboilsdowntofindingsystemsmodelledbyintegrableequations.But even for very simple systems, this canbehard.Apendulum isone of the simplest mechanical systems there is, and it does turn out to beintegrable;evenso,anexactformulainvolvesellipticfunctions.

Tobeginwith,integrablecaseswerediscoveredbyintelligenttrialanderror.Asmathematicians gained experience, they started to pin down some generalprinciples.Themostimportantoftheseareknownasconservationlaws,becausethey specifyquantities that areconserved–don’t change–during themotion.The most familiar is energy. In the absence of friction, the total energy of amechanical system always remains the same. Others are linear and angularmomentum.Ifthereareenoughconservedquantities,theycanbeusedtodeducethesolution,and thesystemis integrable.Forhistorical reasons, the integrablecasesforthemotionofarigidbodyarereferredtoas‘tops’.

BeforeKovalevskaia,twointegrabletopswereknown.OneistheEulertop,a rigid body not subject to external twisting forces (torques). The other is theLagrangetop,whichspinsaboutitsaxisonaflathorizontalsurfacewithgravityactingvertically.Lagrangediscoveredthatthissystemisintegrableifthetophasrotationalsymmetry.Thekeyinbothcasesistoconsiderthetop’smomentsofinertia,whichtellushowmuchtorque(twistingforce)isneededtoaccelerateitsangularmotionaboutagivenaxisbyagivenamount.Everyrigidbodyhasthreespecialmomentsofinertia,saidtobeprincipal.Everymathematicianversedinmechanics knew about the Euler and Lagrange tops. They also knew – orthoughttheydid–thattheseweretheonlyintegrablecases.SoKovalevskaia’sdiscoveryofathirdtypewas,tosaytheleast,ashock.Moreover,itdoesn’trelyon symmetry, which mathematicians were starting to get accustomed to andrealised helped to solve equations; instead, her new solution exploitedmysteriousfeaturesofatopwithoneprincipalmomentofinertiahalfthesizeoftheothertwo.Wenowknowtherearenootherintegrablecasestobefound.

Systems that are not integrable can be studied by other means, such asnumerical approximations. Often, they exhibit deterministic chaos: irregularbehaviour resulting from non-random laws. But even today, physicists,engineers, and mathematicians take a serious interest in integrable systems:

they’reeasier tounderstand,providingrare islandsofregularity inanoceanofchaos,andtheirexceptionalnaturemakesthemspecial,henceworthyofdetailedstudy.TheKovalevskaiatophasbecomeaclassicofmathematicalphysics.

18IdeasRoseinCrowds

HenriPoincaré

JulesHenriPoincaré

Born:Nancy,Lorraine,France,29April1854

Died:Paris,France,17July1912

ARCHIMEDESGOTIDEASinthebath.HenriPoincarégotthemsteppingonabus.Poincaréwasoneof themost inventive andoriginalmathematicians of his

time.HealsowroteseveralbestsellingpopularsciencebooksbasedonlecturesgiventotheParisianSociétédePsychologie.Hetookaninterestinthethoughtprocesses ofmathematicians, placing particular emphasis on the subconsciousmind.InScienceandMethodherelatesanexamplefromhisownexperience:

ForfifteendaysIstrovetoprovethattherecouldnotbeanyfunctionslikethoseIhavesincecalledFuchsianfunctions.Iwasthenveryignorant;everydayIseatedmyselfatmytable,stayedanhourortwo,triedagreatnumberofcombinationsandreachednoresults.Oneevening,contrarytomycustom, Idrankblackcoffeeandcouldnot sleep. Ideas rose incrowds; I felt themcollideuntil

pairsinterlocked,sotospeak,makingastablecombination.BythenextmorningIhadestablishedtheexistenceofaclassofFuchsianfunctions,thosewhichcomefromthehypergeometricseries;Ihadonlytowriteouttheresults,whichtookbutafewhours.

Hethengoesintosomedetailabouthisownexperiences,firstpointingoutthat,to paraphrase, you don’t need to know what the technical terms in the storymean.Justconsiderthemplaceholdersforsomeadvancedmathematicaltopic.

Iwantedtorepresentthesefunctionsbyaquotientoftwoseries;thisideawasperfectlyconsciousanddeliberate,theanalogywithellipticfunctionsguidedme.Iaskedmyselfwhatpropertiestheseseriesmusthave if theyexisted, and I succeededwithoutdifficulty in forming the series I havecalledtheta-Fuchsian.JustatthistimeIleftCaen,whereIwasthenliving,togoonageologicalexcursionunder the auspicesof the schoolofmines.The changesof travelmademe forgetmymathematicalwork.HavingreachedCoutances,weenteredanomnibustogosomeplaceorother.AtthemomentwhenIputmyfootonthesteptheideacametome,withoutanythinginmyformerthoughtsseeming tohavepaved thewayfor it, that the transformations Ihadused todefine theFuchsianfunctionswereidenticalwiththoseofnon-Euclideangeometry.Ididnotverifytheidea;Ishouldnothavehadtime,as,upontakingmyseatintheomnibus,Iwentonwithaconversationalreadycommenced,but I felt aperfect certainty.Onmy return toCaen, for conscience’ sake Iverifiedtheresultatmyleisure.

Thestorycontinueswithtwofurthermomentsofsuddenillumination.Introspecting on this and other discoveries, Poincaré distinguished three

phases of mathematical discovery: preparation, incubation, and illumination.Thatis:doenoughconsciousworktoimmerseyourselfintheproblemandgetstuck;waitwhilethesubconsciousmullsitover;thenthelittlelightbulbinyourheadgoesoff,thecelebrated‘aha!’moment.

It’s still one of the best insights we have into the workings of a greatmathematicalmind.

Henri Poincaréwas born inNancy, France. His father Léonwas professor ofmedicineattheUniversityofNancy,andhismotherwasEugénie(néeLaunois).HiscousinRaymondPoincarébecameprimeminister,andwaspresidentoftheFrenchRepublicduringthefirstworldwar.Henrisufferedfromdiphtheriawhenquite young, and his mother gave him special tuition at home until he hadrecovered.HewenttotheLycéeinNancy,spendingelevenyearsthere.Hecametopineverysubjectandwasabsolutelyformidableinmathematics.Histeacher

calledhima ‘monster ofmathematics’ andhewonnational prizes.Hehad anexcellentmemoryandcouldvisualisecomplicated shapes in threedimensions,which helped compensate for eyesight so poor that he could hardly see theblackboard,let’alonewhatwaswrittenonit.

In1870 theFranco-Prussianwarwas in full swing, andPoincaré served inthe ambulance corpswith his father. Thewar ended in 1871, and in 1873 heattendedtheÉcolePolytechniqueinParis,graduatingin1875.Thenheswitchedto the École des Mines, studying mining engineering as well as moremathematics.Hegotadegreeinminingengineeringin1879.Itwasabusyyear.Hebecameamine inspector in theCorpsdesMines in theVesoul region,andcarriedoutanofficialinvestigationintoanaccidentatMagnyinwhicheighteenminerswere killed.He also pursued his doctorate underHermite,working ondifferenceequations,analoguesofdifferentialequationsinwhichtimechangesin discrete steps rather than continuously. He realised the potential of suchequations as models of many bodies moving under gravity, such as the solarsystem, anticipating future developments along those lines, which grew inimportance when computers became powerful enough to carry out the hugenumbersofcalculationsrequired.

Afterobtaininghisdoctorate,hegotajobasjuniorlecturerinmathematicsattheUniversity ofCaen andmet his futurewifeLouise Poulin d’Andesi. Theymarriedin1881,andhadfourchildren–threegirlsandaboy.By1881Poincaréhad secured a far more prestigious job at the University of Paris, where hematured into one of the leading mathematicians of his age. He was highlyintuitive,andhisbestideasoftenarrivedwhenhewasthinkingaboutsomethingelse – as his story about the bus exemplifies. He wrote several bestsellingpopular science books: Science and Hypothesis (1901), The Value of Science(1905), and Science and Method (1908). And he ranged over most of themathematics of his day, including complex function theory, differentialequations,non-Euclideangeometry,topology–whichhevirtuallyfounded–andapplicationsofmathematics to areas asdiverseas electricity, elasticity,optics,thermodynamics, relativity, quantum theory, celestial mechanics, andcosmology.

Topology, you’ll recall, is ‘rubber-sheet geometry’. Euclid’s geometry is builtaroundpropertiespreservedbyrigidmotions,suchaslengths,angles,andareas.Topology throws all this away, seeking properties preserved by continuous

transformations,whichcanbend,stretch,compress,andtwist.Amongthemareconnectedness(onepieceortwo?),knottedness,andhavingoneormoreholes.The subjectmay sound nebulous, but continuity is fundamental, perhaps evenmore so than symmetry. In the twentieth century topology became one of thethreepillarsofpuremathematics,theothersbeingalgebraandanalysis.

ThatitdidsoowesalargedebttoPoincaré,whowentbeyondrubbersheetsto rubber spaces, so to speak. The metaphor of a sheet is a two-dimensionalconcept.Ignoringanysurroundingspace–Gauss’sviewpoint–ittakesonlytwonumberstospecifyapointonasheet,or,moreformally,asurface.Theclassicaltopologists,amongthemGauss’sstudentJohannListing,managedtounderstandthe topology of surfaces in considerable detail. In particular, they classifiedthem;thatis,theylistedallofthepossibleshapes.Todothis,theyexploitedaningeniousmethodtoconstructasurfacefromaflatpolygon(anditsinterior).

Ifoppositeedgesofasquarearegluedtogether,the

resultisatorus.Buttheresultcanbeimaginedand

studiedusingjustthesquareandthegluingrules,without

actuallybendingthesquare.

Asimpleandimportantexampleofasurfaceisthetorus.Whenembeddedinthree-dimensionalspace,thisisshapedlikeanAmericandoughnut,withaholethrough the middle. Amathematical torus is defined to be the surface of thedoughnut–nodough,justtheboundarybetweendoughandthesurroundingair.Conceptually, this shapecanbedefinedwithoutanydoughorair.Startwithasquare, and add rules saying that corresponding points on opposite edges areidentical. If you were to bend the square round to glue corresponding edgestogether,you’dgetatorusshape.Butyoucanstudyeverythingonaflatsquare,providedyouremembertherules.Manycomputergames‘bend’therectangularscreen by implementing the gluing rules graphically, so an alienmonster thatdisappearsoff theleft-handedgereappearsat theright.Noonewithanysensewould tryphysically tobendthescreen toachieve thiseffect.Thisobjectgoesbytheoxymoronicnameofthe‘flattorus’.It’sflatbecauseitslocalgeometryisthesameasthatofaflatsquare.It’satorusbecauseitsglobaltopologyisthatof…well,atorus.

Listing and others showed that any closed surface of finite extent can beobtainedbyconceptuallygluing theedgesofasuitablepolygon.Itusuallyhasmorethanfoursides,andthegluingrulescanbecomplicated.Fromthisitcanbeproved thateveryorientablesurface–having twoseparatesides,unlike thefamousMöbiusband–isak-torus.Thatis,itisasurfacelikeatorusbutwithkholesinit,fork=0,1,2,3,…Ifk=0wegetasphere,ifk=1wegettheusualtorus, and ifk ≥ 2weget somethingmore complicated.There’s an analogousclassificationofnon-orientablesurfacestoo,butlet’snotgointothat.

The2-torusand3-torus.

Poincaréwanted togeneralise topology tospacesofdimensiongreater thantwo, and the obvious first step was to go to three dimensions. Here Gauss’sintrinsicviewofgeometryisvital,becauseitmakeslittlesensetotrytoembedacomplicated topological space in ordinary three-dimensional Euclidean space.It’s like trying to embed a torus in the plane, without the trick of identifyingedges.Itwon’tfit.

To see that interesting three-dimensional topological spaces – three-dimensional manifolds, or 3-manifolds – are possible, we generalise the trickthat Listing used. For example, the flat three-dimensional torus is made bytakingasolidcube(togetsomethingthree-dimensionalweneedtheinterior,notjustthesixsquarefaces)andconceptuallygluingoppositefacestogether.Nowasolidaliencouldvanishthroughonefaceandreappearattheoppositeone,asifthosetwofacesareoppositesidesofaStargate-styledoorwayandthealienhasjustpassedthroughit.

Moregenerally,wecantakeapolyhedronandgluefacestogetheraccordingto some list of rules. This recipe leads to lots of topologically different 3-manifolds,butitnolongergivesthemall.(That’snotobvious,butit’strue.)Infact, classifying the topological types of manifolds with three or moredimensions is essentially impossible; there are toomany topologically distinctshapes.Butwith enough effort, some general patterns can be dug out. In thisconnectionanabsolutelybasicquestiongoesbacktoPoincaré;it’sknownasthePoincaréConjecture.Really itwouldbebetternamedthePoincaréMistake,aswe’ll shortly see, but let’s be charitable. In 1904 Poincaré discovered thatsomething he’d tacitly been assuming was obvious wasn’t even true, and heasked whether it could be fixed up by starting from stronger hypotheses. He

couldn’tsort itout,remarkingthat‘thisquestionwouldleadustoofarastray’,andheleftitasateaserforfuturegenerations.

To understand the conjecture, we beginwith an analogous question in thesimpler context of surfaces. How can you distinguish the sphere from all theotherk-tori?Poincarénoticedthatasimpletopologicalfeaturedoesthetrick.Ifyou draw a loop – a curvewhose ends join together – on a sphere, it can bedeformedcontinuously,alwaysstayingonthesphere,untilit’sallscrunchedupatasinglepoint.Withnoholestogetintheway,youcanjustkeepshrinkingtheloopuntilitallpilesupatthatpoint.Onak-toruswithoneormoreholes(k>0),however,aloopthatwindsthroughaholecan’tbeshrunklikethat.Ithastostaythreadedthroughthehole.

Thejargonfor‘everyloopdeformstoapoint’is‘homotopysphere’.We’vejustsketchedaproofthat,forsurfaces,everyhomotopysphereistopologicallythesameasagenuinesphere.Thischaracterisesaspherebyasimpletopologicalproperty.A hypothetical ant, living on a surface, could in principle figure outwhether it’s on a spherebydragging loopsof string around and trying topilethem up at a single point. Poincaré assumed that the same kind of thingcharacterisesa3-sphere,whichisa3-manifoldanalogoustoasphericalsurface.It’snot justasolidball.Aballhasaboundary, the3-spheredoesn’t.Youcanthinkofitasasolidballwhosesurfaceisscrunchedupintoasinglepoint–justas a disc turns into a sphere, topologically,when you gather all the boundarypointstogether.Thinkofabagwithastringaroundthetop.Whenyoupullthestringtight,theboundaryscrunchesup,andthebaghasthesametopologyasasphere.

Nowdothiswhenyouhaveoneextradimensiontoplaywith.The conjecture came about because Poincaré was thinking about another

topologicalproperty,knownashomology.Thisislessintuitivethandeformingloops,but it’scloselyrelated.There’sasense inwhichloops threadedthroughdistinctholesinak-torusconstituteindependentwaysnottobedeformabletoapoint. Homology captures this idea without reference to holes, which are aninterpretationoftheoutcomeintermsthatappealtoourvisualsense.Thenotionofaholeisabitmisleading,becauseaholeisn’tpartofthesurface:it’saplacewhere the surface is absent. In two dimensions, thanks to the classificationtheorem,aspherecanbecharacterisedbyitshomologyproperties(noholes).

Inoneofhisearlypapers,Poincaréassumedthatthesamestatementistrueforthreedimensions.Itseemedsoobviousthathedidn’tbothertoproveit.Butthenhediscoveredaspace thathas thesamehomologyas the3-sphere,but is

topologically distinct from the 3-sphere. Tomake it, glue opposite faces of asolid dodecahedron together, much likemaking a flat three-dimensional torusfromasolidcube.Toprove that this ‘dodecahedral space’ isnot topologicallyequivalenttoa3-sphere,Poincaréinventedhomotopy–whathappenstoaloopwhenyoudeformit.Unlikethe3-sphere,hisdodecahedralspacecontainsloopsthatdon’tdeformcontinuously toapoint.So thenheaskedwhether thisextraproperty does characterise the 3-sphere. It was a question, not really aconjecture,becausehedidn’texpressanexplicitopinion.However,it’sclearheexpectedtheanswertobeyes,socallingitaconjectureisn’ttoounfair.

The Poincaré Conjecture turned out to be hard. Very hard. If you’re atopologist,usedtotheterminologyandwaysofthinking,thequestionissimpleandnatural. Itought tohaveanaturalanswerwithasimpleproof.Apparentlynot.But the ideas that ledPoincaré to it sparkedanexplosionof research intotopologicalspacesandpropertieslikehomologyandhomotopy,which,ifyou’relucky,candistinguishthem.ThePoincaréConjecturewasfinallyprovedin2002byGrigoriPerelman,usingnewmethodsinspiredinpartbyGeneralRelativity.

ForPoincaré,topologywasn’tjustanintellectualgame.Heappliedittophysics.The traditionalmethod for analysing a dynamical system is towrite down itsdifferentialequationandthensolveit.Unfortunately,thismethodseldomgivesan exact answer, so for centuries mathematicians used approximate methods.Untilcomputersbecamewidelyavailable, theapproximations tooktheformofinfinite series, of which only the first few terms would actually be used;computersmadenumericalapproximationspracticalaswell. In1881,Poincarédeveloped an entirely new way of thinking about differential equations in‘Memoironcurvesdefinedbyadifferentialequation’.Thispaper founded thequalitativetheoryofdifferentialequations,whichseekstodeducepropertiesofthesolutionsofadifferentialequationwithoutwritingdownformulasorseriesforthem,orcalculatingthemnumerically.Instead,itexploitsgeneraltopologicalfeatures of the phase portrait – the collections of all solutions, viewed as aunifiedgeometricobject.

Asolutionofadifferentialequationdescribeshowthevariableschangewiththepassageof time.Asolutioncanbevisualisedbyplottingthesevariablesascoordinates. As time passes, the coordinates change, so the point that theyrepresent moves along a curve, the solution trajectory. The possiblecombinations of variables determine a multidimensional space, with one

dimensionpervariable,calledphasespaceorstatespace. If solutionsexist forall initial conditions,which is commonly the case, every point in phase spacelies on some trajectory.Sophase spacebreaksup into a familyof curves, thephaseportrait.Thecurves fit together like smoothlycombed furmadeof longhairs, except near a steady state of the equation, where the solution remainsconstant for all time and the hair reduces to a point. Steady states are easy tofind,andprovidethebeginningsofa‘skeleton’ofthephaseportrait:adiagramofitsmaindistinctivefeatures.

As described so far, we have to know the solutions, or numericalapproximations to them, to draw the phase portrait. Poincaré discovered thatsomepropertiesof thesolutionscanbedetected topologically.Forexample, ifthesystemhasaperiodicsolution–onethatrepeatsthesamesequenceofstatesoverandoveragain–thetrajectoryisaloop,andthesolutionjustkeepsgoingroundandroundlikeapethamster inawheel.Topologically,anyloopcanbedeformed into a circle, so the problem simplifies to topological properties ofcircles. The presence of a loop can sometimes be detected by considering aPoincarésection.Thisisasurfacethatcutsacrossabundleoftrajectories.Givenanypointonthesection,wefollowitstrajectoryuntil(ifever)ithitsthesectionagain. This determines a map from the surface to itself, the Poincarémap or‘firstreturn’map.Ifthesectioncrossesaperiodictrajectory,thecorrespondingpointreturnstothesamelocation.Thatis,it’safixedpointofthePoincarémap.

Inparticular,supposethatthesectionisadisc,aball,orahigher-dimensionalanalogue,andthatwecanshowthattheimageofthesectionunderthePoincarémap lies inside the same section. Then we can invoke a general topologicalresultknownastheBrouwerfixed-pointtheoremtoconcludethatafixedpointmust exist; that is, the differential equation has a periodic solution passingthrough that section. Poincaré introduced a variety of techniques along theselines, and stated a general conjecture about the long-term behaviour oftrajectoriesfordifferentialequationsintwovariables.Namely,thetrajectorycanconverge to apoint, a closed loop, or aheteroclinic cycle– a loop formedbytrajectories that link a finite number of fixed points together. Ivar Bendixsonproved this conjecture in 1901, and the result is known as the Poincaré–Bendixsontheorem.

Poincaré’s realisation that topological methods potentially offer deep insightsinto the solutions of differential equations, even when there’s no formula for

those solutions, lies behind today’s approach to nonlinear dynamics, withapplications across the scientific board. It led him to another epic discovery:chaos, nowoneof thebig triumphsof topological dynamics.The contextwasthemotionofseveralbodiesunderNewtoniangravity–themany-bodyproblem.

JohannesKeplerdeducedfromobservationsofMarsthattheorbitofasingleplanetroundtheSunisanellipse.Newtonexplainedthisgeometricfactintermsofhislawofgravitation:anytwobodiesintheuniverseattracteachotherwithaforceproportionaltotheirmassesandinverselyproportionaltothesquareofthedistance between them. In principle,Newton’s lawpredicts themotion of anynumberofmutuallygravitatingbodies,suchas theplanetsof thesolarsystem.Unfortunately the law of gravity doesn’t prescribe the movement directly: itprovidesadifferentialequationwhosesolutiongivesthepositionsofthebodiesatanyinstantof time.Newtonfoundthat, for twobodies, thisequationcanbesolved,and theresult isKepler’sellipse.But for threeormorebodies,no tidysolutionof this kind seemed feasible, andmathematiciansworking in celestialmechanicsresortedtospecialtricksandapproximations.

The year 1889 saw the sixtieth birthday of Oscar II, King of Sweden andNorway,which at the time formed a single kingdom. In celebration, the kingoffered a prize for a solution of themany-body problem, a topic proposed byMittag-Leffler. The answer was to be given not as a simple formula, whichalmostcertainlydidn’texist,butasaconvergentinfiniteseries.Theproblemcanthenbesolvedtoanydesireddegreeofaccuracybycalculatingenoughtermsoftheseries.

Poincarédecided tohaveago,andwon theprize,even thoughhismemoirdidn’tsolvethefullproblem.Heconsideredonlythreebodies,andassumedtwowere of equal mass orbiting each other at diametrically opposite points of acircle,andthethirdwassolight that ithadnoeffectonthetwomoremassivebodies.His resultspresentedevidence that, insomecircumstances,nosolutionof the specified kind exists. The system can sometimes behave in a highlyirregularmanner,sothatitsgeometrylooksasthoughsomeonehasaccidentallydropped a loosely wound ball of string on the ground. He described his keygeometricinsight,abouthowtwoimportantcurvesdefiningthedynamicscrosseachother:

Whenonetriestodepictthefigureformedbythesetwocurvesandtheirinfinityofintersections,eachofwhichcorrespondstoadoublyasymptoticsolution,theseintersectionsformakindofnet,webor infinitely tightmesh…One isstruckby thecomplexityof this figure that Iamnoteven

attemptingtodraw.

WenowunderstandthatPoincaréhaddiscoveredthefirstimportantexampleofdynamical chaos: the existence of solutions to deterministic equations, soirregular that some aspects of them seem to be random. But at the time, thisresult–thoughintriguing–seemedmorelikeadeadend.

Untilrecently,whatI’vejustwrittenwastheofficialstory.Butinthe1990sthemathematical historian June Barrow-Greenwas visiting theMittag-LefflerInstitute in Sweden. She came across a printed copy of a different version ofPoincaré’s memoir – and it didn’t mention the possibility of highly irregularorbits. It turnedout that thiswas theversionPoincaré submitted, but after thewinner was announced, he noticed an error. Almost the entire print run wasdestroyed, and a corrected versionwas quickly printed at Poincaré’s expense.Onecopyoftheoriginal,however,survivedintheinstitute’sarchives.10

Poincaré might have given the appearance of the stereotypic impracticalacademic,butheretainedhisminingconnectionsthroughouthis life,andfrom1881to1885hedirectedthedevelopmentofthenorthernrailwayasanengineerat theMinistryofPublicServices. In1893hewasmadechief engineerof theCorps de Mines, and in 1910 he was promoted to inspector general. At theUniversity of Paris he occupied chairs in many subjects: mechanics,mathematicalphysics,probability,andastronomy.HiselectiontotheAcademyofSciencescamewhenhewasonly32,in1887,twoyearsbeforehewonKingOscar’s prize, and he eventually became its president in 1906. In 1893 heworkedfortheBureaudesLongitudes,tryingtosetupaunifiedsystemoftimethroughouttheworld,suggestingthattheworldshouldbedividedupintotimezones.

He came very close to beating Einstein to Special Relativity, showing in1905 thatMaxwell’s equations for electromagnetism are invariant underwhatwenowcalltheLorentzgroupoftransformations,whichimpliesthatthespeedoflightmustbeconstantinamovingframe.Perhapsthemainpointhemissed,butEinstein spotted,was that physics really is like that.He also proposed thenotion of a gravitational wave, propagating at the speed of light, in the flatspacetimeofSpecialRelativity.TheLIGOexperimentdetected suchwaves in2016,butby then the contexthad shifted to the curved spacetimesofGeneralRelativity.

Poincarédiedfromanembolismaftercancersurgeryin1912,andwasburiedin his family’s vault at the cemetery of Montparnasse. His mathematicalreputation continued to grow, as others developed the ideas that he firstproposed.Todayheisconsideredtobeoneofthegreatoriginalsofthesubject,andoneofthelasttorangeoveralmosttheentiremathematicallandscapeofhisday.Hismathematicallegacyremainsaliveandkicking.

19WeMustKnow,WeShallKnow

DavidHilbert

DavidHilbert

Born:Wehlau,nearKönigsberg,Prussia(now

Kaliningrad,Russia),23January1862

Died:Göttingen,Germany,14February1943

AGERMANPROFESSORwhoreachedtheageofsixty-eightwasobligedtoretire.WhenDavidHilbertpassedthismilestonein1930,manypubliceventsmarkedtheofficialendofanoutstandingacademiccareer.He lecturedonhis firstbig

result, theexistenceofafinitebasisfor invariants.MotoristsfoundthemselvesdrivingalongthenewlynamedHilbertstrasse.Whenhiswiferemarked‘Whataniceidea!’Hilbertreplied‘Theidea,no–buttheexecutionisnice.’

MostpleasingofallwasbeingmadeanhonorarycitizenofKönigsberg,thecitynearwhichhehadbeenborn.ThehonourwastobeconferredatameetingoftheSocietyofGermanScientistsandPhysicians,andHilberthadtodeliveranacceptancespeech.Hedecideditmustbewidelyaccessible,andsinceImmanuelKanthadbeenborninKönigsberg,somethingwithaphilosophicalaspectwouldbe appropriate. It must also sum up his life’s work. He settled on ‘Naturalknowledge and logic’. Hilbert had form in such activities, often deliveringlectures in a Saturdaymorning series intended for everyone in the university.Relativity,infinity,theprinciplesofmathematics…hedidhisbesttomakethemaccessibletoanyoneinterested.Nowhefocusedallhiseffortsonalecturethatwouldtrumpthemall.

‘Understandingnatureandlifeisournoblesttask,’hebegan.Hewentontocompare and contrast two ways of understanding the world: thought andobservation. The two are linked by the laws of nature, to be deduced fromobservations and developed by pure logic. It was a view that would haveappealed toKant,whichwas ironic becauseHilbertwasn’t a big fan ofKant.This wasn’t the occasion to say so, and on this particular issue there was nodisagreement, but Hilbert couldn’t resist one dig, a suggestion that Kant hadoverestimated the importance of a priori knowledge, not obtained throughexperience.Geometrywasagoodexample:therewasnoreasontoassumethatspace was necessarily Euclidean, as Kant had argued. Remove theanthropomorphicdross,however,andtrueaprioriconceptsremain,namely,thegeneralities of mathematics. ‘Our entire present culture, insofar as it isconcernedwith the intellectual understanding and conquest of nature, rests onmathematics!’ he declaimed. And he ended by defending pure mathematics,oftencriticisedforlackofpracticalrelevance:‘Purenumbertheoryisthatpartofmathematics for which up to now [my italics] no application has ever beenfound…Thegloryofthehumanspiritisthesoleaimofallscience!’

SosuccessfulwasthetalkthatHilbertwaspersuadedtorepeatitforthelocalradio station, and the recording survives. He emphasised that problemspreviously thought impossible–suchasfindingthechemicalcompositionofastar – had yielded to new ways of thinking. ‘There is no such thing as aninsolubleproblem,’hesaid.The finalwordsof the talkwere ‘Wemustknow.Weshallknow.’Then,justasthetechnicianstoppedthetape,Hilbertlaughed.

Atthetime,Hilbertwasdeepintoamassiveprogrammetosetthewholeofmathematics on logical foundations, and his words were a statement ofconfidencethathisprogrammewouldsucceed.Already,muchprogresshadbeenmade.Afewstubborncasesstillhadtobesortedout.Whentheywerepolishedoff,Hilbertwouldn’tjusthavealogicalbasisforallmathematics:he’dbeabletoprovethathisaxiomsarelogicallyconsistent.

Itdidn’tworkoutthewayhe’dhoped.

Hilbertcamefroma familyof lawyers.Hisgrandfatherwasa judgeandprivycouncillor,hisfatherOttoacountyjudge.HismotherMaria(néeErdtmann)wasaKönigsbergmerchant’s daughter. Her passionswere philosophy, astronomy,andprimenumbers, and it looks as thoughher enthusiasms rubbedoff onherson.AsisterElsiearrivedwhenDavidwassix.Hismothertaughthimathomeuntil he entered school at the late age of eight. The school specialised in theclassics, offering little in theway ofmathematics and no sciencewhatsoever.Rote learning was the order of the day, and Hilbert did poorly at anythingrequiringmemorisingstructurelesslistsoffacts.Hedescribeshimselfashavingbeen ‘dull and silly’.One subjectwas a glorious exception.His school reportreads: ‘For mathematics he always showed a very lively interest and apenetratingunderstanding:hemasteredallthematerialtaughtintheschoolinaverypleasingmannerandwasabletoapplyitwithsurenessandingenuity.’

In1880HilbertbeganstudyingforadegreeattheUniversityofKönigsberg,specialising in mathematics. He took courses at Heidelberg under LazarusFuchs; back in Königsberg he studied under Heinrich Weber, Ferdinand vonLindemann,andAdolfHurwitz.HebecameclosefriendswithHurwitzandwithHermannMinkowski,afellowstudent.ThroughouthislifehecorrespondedwithMinkowski.Lindemann,whowasshortlytobecomefamousforprovingthatπdoes not satisfy any algebraic equation with integer coefficients, becameHilbert’s thesis advisor. He suggested that Hilbert should work in invarianttheory, following the trailblazedbyBooleandextendedbyCayley,Sylvester,andPaulGordan.Theirmethodswerecomputational,andHilbert’sproficiencyin these horrible calculations impressed his friend Minkowski, who wrote ‘Irejoicedoveralltheprocesseswhichthepoorinvariantshadtopassthrough’.In1885Hilbertwasawardedhisdoctorate,aftergivingapubliclectureonphysicsandphilosophy.

AtthattimetheleadingauthorityoninvarianttheorywasGordan,andthebig

unsolvedquestionwas toprove theexistence, foranynumberofvariablesandanydegreeofequation,ofafinitebasis.That is,a finitenumberof invariants,such that all other invariants are combinations of them. List the basis, and ineffectyou’vegotthelot.Fortwo-variablequadratics,thebasisconsistssolelyofthe discriminant. Finiteness had been proved in many cases, always bycalculating all of the invariants and then extracting a basis. By this method,Gordanhadprovedthemostgeneralknowntheoremofthiskind.

Theentire areawas turnedupsidedown in1888,whenHilbertpublishedashort paper proving that a finite basis always exists,without calculating anyinvariantswhatsoever.Infact,heprovedthatanysuitablecollectionofalgebraicexpressionsalwayshasafinitebasis–beitcomposedofinvariantsornot.Thiswasn’t the kind of answer thatGordan had been expecting, andwhenHilbertsubmitted thework toMathematischeAnnalen,Gordanrejected it. ‘This isnotmathematics,’hesaid.‘Thisistheology.’HilbertcomplainedtotheeditorKlein,refusing to change the paper unless some ‘definite and irrefutable objectionagainstmyreasoningisraised’.Kleinagreedtopublishthepaperinitsoriginalform.IsuspectheunderstoodtheproofbetterthanGordan,whowasoutofhisdepthwhencomputationalabilitywasreplacedbyconceptualthinking.

A fewyears laterHilbert extendedhis results and submittedanotherpaper.Kleinacceptedit,describingitas‘themost importantworkongeneralalgebrathat theAnnalen haseverpublished’.As far asHilbertwasconcerned,hehadnow achieved everything he had set out to do in that area. ‘I shall definitelyleavethefieldofinvariants,’hewrotetoMinkowski.Andhedid.

Havingpolishedoffinvarianttheory–thesubjectprettymuchdiedonceHilberthad demolished it, only to be revivedmanyyears later in a stillmore generalcontext,andwithrenewedinterestincomputationsaswellasconcepts–Hilbertfound a new area to work in. In 1893 he embarked on a new project, theZahlbericht(NumberReport).TheGermanMathematicalSocietyhadaskedhimto survey a major area within number theory, to do with algebraic numbers.These are complex numbers that satisfy a polynomial equation with rational(equivalently,integer)coefficients.Anexampleis ,whichsatisfiesx2–2=0;another is the imaginarynumber i,which satisfiesx2+1=0.As remarked inChapter 16, a complex number that is not algebraic is called transcendental(here);examplesincludeπande,thoughthispropertyishardtoproveandforalong time was an open problem. Charles Hermite proved e transcendental in

1873,andLindemanndealtwithπin1882.Themainroleplayedbyalgebraicnumberswasinnumbertheory.Eulerhad

tacitlyused someof theirproperties, for examplewhenprovingFermat’sLastTheoremforcubes,butitwasGausswhobeganasystematicstudy.Whentryingtogeneralisehis lawofquadraticreciprocity tohigherpowers than thesquare,he discovered a beautiful extension to fourth powers, based on algebraicnumbersoftheforma+ibwhereaandbareintegers.Thissystemof‘Gaussianintegers’hasmanyspecialfeatures,andinparticularithasitsownanalogueofprimenumbers,completewithauniquefactorisationtheorem.Gaussalsomadeuse of algebraic numbers related to roots of unity in his construction of theregularheptadecagon.

In Chapter 6, in connection with Fermat’s Last Theorem, we discussedKummer’s use of algebraic numbers and his notion of ideal numbers (here).Dedekind simplified this idea by reformulating it in terms of special sets ofalgebraic numbers, which he called ideals. After Kummer, algebraic numbertheory took off, aided and abetted by Galois’s theory of equations and thegrowing development of abstract algebra (Chapter 20). The phrase ‘algebraicnumbertheory’hastwointerpretations:analgebraicapproachtonumbertheory,orthetheoryofalgebraicnumbers.Bothmeaningswerenowconvergingontothe same thing, and this is what the German Mathematical Society wantedHilbert to sort out. Characteristically, he went much further. He asked aperennial question among mathematicians when faced with a large body ofimpressivebutdisorganised results: ‘Yes,butwhat’s itreally about?’This ledhimtoformulateandprovemanynewtheorems.

Throughout the preparation of the Zahlbericht, Hilbert got extensivefeedback fromMinkowski – sometimes too extensive, so that at timesHilbertbegantodespairofeverfinishingittohisfriend’ssatisfaction,buteventuallythereportwaspublished. It formulated andprovedgeneral analoguesofquadraticreciprocity,providing thebasisofwhat isnowcalledclass field theory, still aflourishing, though highly technical, framework for algebraic number theory.TheprefaceoftheZahlberichtstates:

Thusweseehowfararithmetic,theQueenofmathematics,hasconqueredbroadareasofalgebraandfunctiontheorytobecometheirleader…Theconclusion,ifIamnotmistaken,isthataboveallthemoderndevelopmentofpuremathematicstakesplaceunderthebannerofnumber.

Wemightnotgoquitesofartoday,butatthetimetheclaimwasjustifiable.

Hilbertwouldspendfive to tenyears inonearea,polishoff thebigproblems,and then depart for pastures new – sometimes forgetting completely that he’deverstudiedthattopic.Heonceremarkedthathedidmathematicsbecauseyoucan always work something out again if you forget it. A mathematician’smathematician to the core, he had now ‘done’ algebraic number theory. Hemoved on. His students, who had been bombardedwith lectures on algebraicnumbersforyears,werestartledtofindthatnextyear’stopicwastheelementsofgeometry.HilbertwasgoingbacktoEuclid.

Asalways,Hilberthadhisreasons,andagainthekeyquestionwas:‘Yes,butwhat’sitreallyabout?’Euclid’sanswerwouldhavebeen‘space’,whichiswhyhe illustratedhis theoremswithgeometricdrawings.Hilbert,however,was farmoreinterestedinthelogicalstructureoftheaxiomsforgeometryandhowtheyledtotheoremsthatwereoftenfarfromobvious.HewasalsodissatisfiedwithEuclid’s list of axioms, because the use of pictures had led Euclid to makeassumptionsthathehadnotstatedexplicitly.

Asimpleexampleis‘astraightlinepassingthroughapointthatliesinsideacircle must meet the circle’. This looks obvious in a picture – but it’s not alogical consequence of Euclid’s axioms. Hilbert realised that Euclid’s axiomswereincomplete,andsetouttoremedythedeficiency.Eucliddefinedapointas‘thatwhichhasnopart’,andastraightlinetobe‘alinewhichliesevenlywiththe points on itself’. Hilbert considered these statements to be meaningless.Whatmatters,heargued,ishowtheseconceptsbehave,notsomementalimageof what they are. ‘One must be able to say at all times – instead of points,straight lines, and planes – tables, chairs, and beer mugs,’ Hilbert told hiscolleagues.Inparticular,pictureswereout.

This project was of course related to the deeper question, by then wellunderstood, of non-Euclidean geometries and the parallel axiom (Chapter 11).Hilbertwas trying to establish the basic principles for axiomatic treatments ofmathematical topics. These included consistency (not leading to a logicalcontradiction) and independence (no axiom is a consequence of the others).Other desirable qualities were completeness (nothing vital is missing) andsimplicity(whenpossible).Euclideangeometrywasatestcase.Consistencywaseasy: you can model Euclid’s geometry using algebra, applied to (x, y)coordinates in the plane. That is, you can start with ordinary numbers andconstruct a mathematical system obeying all of Euclid’s axioms. Then theaxioms can’t be self-contradictory, because proof by contradictionwould then

showthattheconstructedmodeldoesnotexist.There’sonepotentialflawinthatargument, however, andHilbert was aware of it early on. It assumes that thestandard system of numbers is itself non-contradictory; that arithmetic isconsistent,whichiswhatmathematiciansmeanby‘exists’.Obviousthoughthismay seem, noonehad actually proved it.Later,Hilbert tried to eliminate thisgap,butitcamebacktohaunthim.

The outcome was a small, concise, and elegant book, Foundations ofGeometry,publishedin1899.ItdevelopedEuclideangeometryfrom21explicitaxioms.ThreeyearslaterEliakimMooreandRobertMoore(norelation)provedthatoneofthemcanbededucedfromtheothers,soonly20areactuallyneeded.Hilbert started with six primitive notions: ‘point’, ‘line’, ‘plane’, and therelations ‘between’, ‘lies on’, and ‘congruent’. Eight axioms govern incidencerelationsamongpointsandlines,suchas‘anytwodistinctpointslieonaline’.Four(whichEuclid’spicturesledhimtoassumewithoutmakingthemexplicit)govern the order of points along a line.Sixmore sort out congruence (of linesegments and triangles; ‘congruent’ basically means ‘same shape and size’).Next came Euclid’s parallel axiom, which by then every competentmathematicianknewhadtobe included.Finally, therewere twosubtleaxiomsofcontinuity,ensuringthatthepointsofalinearemodelledontherealnumbers(andnot,say,therationals,wherelinesthatseemtomeetinadiagrammayfailtodosoatarationalpoint).

ThemainvalueofHilbert’sbookwasn’tinteaching–Euclidwasn’texactlyin vogue any more – but in unleashing a flurry of activity on the logicalfoundationsofmathematics.Americanmathematiciansinparticularwereattheforefront of this wave, from which emerged a logico-mathematical hybrid,metamathematics. This is, in a sense, mathematics applied to itself; moreproperly, to itsownlogicalstructure.Amathematicalproofcanbeviewednotjustasaprocessthatleadstonewmathematics,butasamathematicalobjectinitsownright.Indeed,it’sthisdeepself-referentialaspectthatsowedtheseedsofthe destruction of Hilbert’s dream. In November of the same year came thebombshell, a paper by a young logician named Kurt Gödel (Chapter 22). Itcontainedproofsoftwodevastatingtheorems.First,ifmathematicsisconsistent,this can never be proved. Second, there exist statements in mathematics forwhich neither a proof nor a disproof exists. Mathematics is inherentlyincomplete,itslogicalconsistencycannotbedetermined,andsomeproblemsaretrulyimpossibletosolve.

Hilbert is reported as being ‘very angry’ when he first learned of Gödel’s

work.

No account of Hilbert’s influence is complete withoutmentioning theHilbertProblems,alistof23majoropenquestionsandareasinmathematics,whichhepresented in a talk at the Second InternationalCongress ofMathematicians inParis in 1923.They set the scene for a substantial proportion ofmathematicalresearch in the twentiethcentury.They include findingaconsistencyproof formathematics, a rather vague request for an axiomatic treatment of physics,questions about transcendental numbers, the Riemann Hypothesis, the mostgeneral reciprocity law inanynumber field,analgorithmtodeterminewhenaDiophantine equationhas a solution, andvarious technical issues ingeometry,algebra,andanalysis.Tenhavebeencompletelysolved,threeremainunsolved,afewaretoovaguetorecognisewhatasolutionwouldlooklike,andtwodon’thavesolutionsatall,inaverystrongsense.

MathematicsafterHilbertdidn’tjustconsistofpeopletryingtosolvehis23problems,but they exerted a considerable, and largelybeneficial, influenceonthe development of mathematics over the next half-century. If you wanted tomakeyourmark amongyourmathematical fellows, solving aHilbert problemwasagoodwaytogo.

Hilbert’s interest in mathematical physics became stronger as he aged, acommon phenomenon among mathematicians who begin their careers in the‘pure’campandgraduallydrift towardsapplications.By1909hewasworkingonintegralequations,leadingtothenotionofaHilbertspace,nowfundamentaltoquantummechanics.Healsocameclose todiscoveringEinstein’sequationsfor General Relativity in a 1915 paper, published five days before Einstein’sannouncement,statingavariationalprinciplethatimpliestheEinsteinequation.However,hefailedtowritedowntheequationitself.

Hilbertwasusuallyagenialsoul,lavishwithpraiseforagoodperformance,buthecouldbebrutalifanyoneutteredmeaninglessplatitudesorliedtohim.Inseminars, if a student was labouring some point that Hilbert felt wasstraightforward, he would say ‘But that is completely simple!’ and a wisestudentwouldpromptlymoveon.Inthe1920sHilbertranaMathematicsClub,withweeklymeetings,opentoanyone.Manywell-knownmathematiciansgavetalks, instructedtopresent‘onlytheraisinsoutof thecake’.If thecalculationsbecamedifficult,Hilbertwouldinterruptwith‘wearenotheretocheckthatthesignisright’.

Astimepassed,hebecamelesstolerant.AlexanderOstrowksiremarkedthatonce,whenavisitorgaveanexcellent talkona really importantandbeautifulpiece of research, Hilbert’s only question was a sour: ‘What’s it good for?’WhenNorbertWiener,abrilliantAmericanwhocoinedtheterm‘cybernetics’,talked to the club, everyone went out for supper, as was the custom. Hilbertstarted talking about past lectures at the Club, saying that the quality hadgenerallydeclinedovertheyears.Inhisday,hesaid,peoplereallythoughtaboutcontent and presentation, but nowadays youngpeople usually gave poor talks.‘Recently it has been especially bad,’ he said. ‘But now, this afternoon, therewasanexception–’

Wienerpreparedforthecompliment.‘Thisafternoon’stalkwastheworstthereeverhasbeen!’In1933theNaziswererootingouttheJewsamongtheGöttingenacademics

and dismissing them.OnewasHermannWeyl, one of the greatmathematicalphysicists, who had been appointed Hilbert’s successor on his retirement in1930.Otherswere EmmyNoether (Chapter 20), the number theorist EdmundLandau, and Paul Bernays, Hilbert’s collaborator on mathematical logic. By1943virtuallytheentiredepartmentofmathematicshadbeenreplacedbypeoplemoreacceptabletotheNaziadministration,andwasapaleshadowofitsformergloriousself.Thatyear,Hilbertdied.

He’dseen itallcoming.AfewyearsearlierBernhardRust, theministerofeducation, had asked Hilbert whether Göttingen’s Mathematics Institute hadsufferedbecauseofthedepartureoftheJews.Itwasastupidquestion,becausemostofthefacultyhadeitherbeenJewsortheirspouses.Hilbert’sanswerwasdirectandblunt:

‘Suffered?Itdoesn’texistanylonger,doesit?’

20OverthrowingAcademicOrder

EmmyNoether

AmalieEmmyNoether

Born:Erlangen,Germany,23March1882Died:BrynMawrPA,USA,14April1935

IN1913EMMYNOETHER,afemalemathematicianofgreatrenown,wasinViennagivingacourseoflectures,andshevistedFranzMertens,amathematicianwhoworkedinmanyfieldsbutisbestknownforhiscontributionstonumbertheory.OneofMertens’sgrandsonslaterwrotedownhisrecollectionsofthevisit:

Althoughawoman,[she]seemedtomelikeaCatholicchaplainfromaruralparish–dressedinablack,almostankle-lengthandrathernondescript,coat,aman’shatonhershorthair…andwithashoulderbagcarriedcrosswiselikethoseoftherailwayconductorsoftheimperialperiod,shewasratheranoddfigure.

Two years later this unassuming person was responsible for one of the greatdiscoveries of mathematical physics: a fundamental link between symmetriesand conservation laws. From that point on, symmetries of the laws of naturehaveplayedacentralroleinphysics.Todaytheyunderpinthe‘standardmodel’of subatomic particles in quantum theory, which is virtually impossible todescribewithoutappealingtosymmetry.

Noetherwasaleadingfigureinthedevelopmentofabstractalgebra,inwhichcalculationswithmanydifferenttypesofnumbersorformulaswereorganisedintermsofthealgebraiclawsthatthesesystemsobey.Perhapsmorethananyothermathematician,the‘oddfigure’seenbyMertens’sgrandsonwasresponsibleforthe change that marks the borderline between the neoclassical period of thenineteenth and early twentieth century,with its emphasis on special structuresandformulas,andthemodernperiodfromabout1920onwards,withemphasison generality, abstraction, and conceptual thought. She was the inspirationbehind the subsequent Bourbakiste movement, which originated in the jointeffortsofagroupofyoung,mainlyFrench,mathematicians,whoaimedtomakemathematicspreciseandgeneral.Perhapstoogeneral,insomeeyesatleast,butthereyougo.

EmmyNoetherwasbornintoaJewishfamilyintheBavariantownofErlangen.HerfatherMaxwasanotablemathematicianwhoworkedinalgebraicgeometryand algebraic function theory. He was highly talented but, compared to thegreatsofhisera,abitspecialised.Hisfamilywaswelloff,owningaflourishingwholesalehardwarecompany.ThisbackgroundundoubtedlyinfluencedEmmy’sattitudes to lifeandtomathematics. Initially,sheplannedtobecomea teacher,and obtained the necessary qualifications to teach French and English. But,perhapsnotsosurprisingly,shewasbittenbythemathematicsbugandstudiedattheUniversityofErlangen,whereherfatherworked.

Twoyearsearliertheuniversitysenatehaddeclaredthatmixed-sexeducationwould‘overthrowallacademicorder’,andtherewereonlytwowomenstudentsoutof986.Shewasallowedtoauditclassesbutnottoparticipatefully,andhad

togetpermissionfromindividualprofessorstoattendtheirlectures.Butin1904the rules changed, allowing women tomatriculate on the same basis asmen.Noether duly did so in 1904, andmoved toGauss’s old stamping ground, theUniversity of Göttingen, to do a PhD in invariant theory, supervised by theeminent Gordan. Her thesis calculations were extraordinarily complex,culminating in a list of 331 ‘covariants’ for fourth-degree forms in threevariables. The normally indefatigable Gordan had given up on this giganticcalculation forty years earlier. Noether’s methods were fairly old-fashioned,paying little or no attention toHilbert’s innovations. In 1907 she received thePhDdegreesummacumlaude.

IfNoetherhadbeenaman,shewouldnaturallyhaveprogressedtothenextstage in securingapermanent academicpost.Butwomenwerenot allowed toproceedwithHabilitation, so sheworked unpaid at Erlangen for seven years.She helped her father, who by then was disabled, and continued her ownresearch. A formative experience, which diverted her towards more abstractmethods,wasaseriesofdiscussionswithErnstFischer,whodrewherattentionto Hilbert’s new methods and advised her to use them. This she did,spectacularly,anditseffectsarevisiblethroughouthersubsequentcareer.

Mathematicswasstartingtoopenuptowomen,andNoetherwasadmittedtoseveral major mathematical societies. This led to her visit to Vienna andMertens’s grandson’s recollections. In Erlangen she supervised two PhDstudents,althoughofficiallytheywereregisteredunderherfather.ThenHilbertandKleininvitedhertoGöttingen,whichhadbecomeaworld-renownedcentrefor mathematical research. This was 1915, and Hilbert was moving intomathematical physics, inspired by Einstein’s theories of relativity. Relativityrestsonthemathematicsofinvariants,thoughinamoreanalyticcontextthanthealgebraic invariants that Gordan, Hilbert, and Noether had been studying.Namely differential invariants, which includewhat by then had become basicphysicalconcepts,suchasthecurvatureofspace.

Hilbertwantedanexpertoninvariants,andNoetherfittedthebillperfectly.Within a short time she solved twokey problems.The firstwas amethod forfindingalldifferential covariants forvector and tensor fieldsonaRiemannianmanifold – in effect, to discoverwhat other quantities behave likeRiemann’scurvature tensor. This was vital because Einstein’s approach to physics wasbased on the ‘relativity’ principle that the laws should be the same for anyobserver,whenexpressed inanycoordinateframemovingatuniformvelocity.So the laws ought to be invariant under the transformation group defined by

moving frames. The second was an offshoot of this problem. The naturalsymmetry group for Special Relativity is the Lorentz group, defined bytransformations that mix up space and time but preserve the speed of light,giving relativity its unique flavour. Noether proved that every ‘infinitesimaltransformation’oftheLorentzgroupgivesrisetoacorrespondingconservationtheorem.

We can appreciateNoether’s ideas in themore familiar context ofNewtonianmechanics, where they also apply and provide significant insights. Classicalmechanics boasts several conservation laws, the most familiar beingconservationofenergy.Amechanicalsystemisanysetofbodiesthatmoves,astimepasses,according toNewton’s lawsofmotion. Insuchsystems there isaconceptofenergy,whichtakesseveralforms:kineticenergy,relatedtomotion;potential energy, resulting from interaction with a gravitational field; elasticenergy, such as that contained in a compressed spring; and so on.The lawofconservationofenergystatesthatintheabsenceoffriction,howeverthesystemmoves, consistent with Newton’s laws of motion, its total energy remainsconstant – is conserved. If there’s friction, kinetic energy is converted intoanother kind of energy, heat, and again the total energy is conserved.Heat is‘really’thekineticenergyofvibratingmoleculesofmatter,butinmathematicalphysicsit’smodelledinadifferentmannerfromtheenergyofrigidbodies,rods,andsprings,soitsinterpretationdiffersfromthatoftheothertypesmentioned.Other conservation laws of classical mechanics include conservation ofmomentum (mass times velocity) and angular momentum (a measure of spinwhoserathertechnicaldefinitionisirrelevanthere).

Thanks toGalois (Chapter12)and thosewhofollowedhim, theconceptofsymmetryhadbeenidentifiedwithinvarianceundergroupsoftransformations:collectionsofoperationsthatcanbeperformedonsomemathematicalstructure,whose effect is to leave that structure apparently unchanged.An equation hassymmetrywhensomesuchtransformation,appliedtoasolutionoftheequation,always yields another solution. The laws of physics, when expressed asmathematical equations,havemanysymmetries.Newton’s lawsofmotion, forexample,havethesymmetriesoftheEuclideangroup,whichconsistsofallrigidmotions of space. They’re also symmetric under time-translation –measuringtime from a different starting point – and in some circumstances under time-reflection:reversingthedirectioninwhichtimeflows.

Noether’s insight was the existence of a link between some types ofsymmetryandconservationlaws.Sheprovedthateverycontinuoussymmetry–belongingtoafamilyofsymmetriescorrespondingtocontinuouslyvaryingrealnumbers–givesrisetoaconservedquantity.

Letme unpack that, because as stated it’s rather enigmatic. Some types ofsymmetry naturally live in continuous families. The rotations of a plane, forexample, correspond to the angle of rotation, which can be any real number.Theserotationsformagroup,whoseelementscorrespondtotherealnumbers.Atechnicalissueisworthnoting:realnumbersthatdifferbyafullcircle(360°or2πradians)definethesamerotation.Allofthese‘one-parametergroups’eitherlookliketherealnumbers,orangles.Translationsofspaceinagivendirection,which can be obtained by sliding space rigidly through any distance in thedirectionconcerned,arealsocontinuoussymmetries.Othersymmetriesmaybeisolated,notpartofsuchafamily.Reflectionisanexample.Youcan’tperformhalf a reflection, or one tenth of a reflection, so it’s not part of any one-parametergroupofrigidmotions.TheinfinitesimaltransformationsthatNoetherstudied in her PhD are another way to think of one-parameter groups. TheunderlyingconceptisthatofaLiegroupanditsassociatedLiealgebra,namedaftertheNorwegianmathematicianSophusLie.

InNewtonianmechanics, the conserved quantity corresponding to the one-parameter group of time translations turns out to be energy. This reveals aremarkable link between energy and time, which in turns shows up in theuncertainty principle of quantummechanics.This allows a quantum system toborrow energy (which is temporarily not conserved) provided it pays it backagain before nature notices the discrepancy (wait a split second and it isconserved).Theconservedquantitycorrespondingtoaone-parametergroupofspatialtranslationsturnsouttobemomentuminthecorrespondingdirection,andthat for rotations is angular momentum. In short: the fundamental conservedquantities of Newtonian mechanics all come from continuous symmetries ofNewton’s laws ofmotion – one-parameter subgroups of the Euclidean group.Andthesameistrueforrelativityand,tosomeextent,forquantummechanics.

Not bad for amathematician considered incapable of lecturing in her ownright,whohadonlyrecentlystartedtoworkontheproblem.

On the strength of this and other successes, Hilbert and Klein battled toconvince the university to change its mind about female faculty members.Academicpoliticsaswellasinbuiltmisogynycameintoplay,andtheprofessorsinthePhilosophyDepartmentwerevehementlyopposed.Ifawomancouldgain

Habilitation and charge fees for lectures, what would stop her becoming aprofessorandamemberoftheuniversitysenate?Heavenforfend!WorldWarIwasinfullswing,andthisgavethemanewargument:‘Whatwilloursoldiersthinkwhentheyreturntotheuniversityandfindtheyareexpectedtolearnatthefeetofawoman?’

Hilbert’s reply was scathing. ‘Gentlemen: I do not see that the sex of thecandidate is an argument against her admission as Privatdozent. After all, theSenateisnotabathhouse.’Buteventhatfailedtomovethephilosophersfromtheir entrenched position. Hilbert, inventive and iconoclastic as ever, found asolution.AnoticefortheWintersemesterof1916–17reads:

MathematicalPhysicsSeminarProfessorHilbert,withtheassistanceofDrE.NoetherMondaysfrom4to6,notuition.

Noether spent four years lecturing under Hilbert’s name, until the universityfinallycavedin.HerHabilitationwasapprovedin1919,allowinghertoobtaintherankofPrivatdozent.Sheremainedaleadingmemberofthedepartmentuntil1933.

We can gaugeNoether’s lecturing abilities from a trick that her despairingstudentsonceplayed.Usually therewereonly five to tenstudents in theclass,butonemorningshe turnedup tofindahundredof them.‘Youmusthave thewrongclass,’shetoldthem,butno,theyinsisted,theywerethereintentionally.Soshedeliveredherlecturetothisunusuallylargegathering.

When she finished, one of her regular students passed her a note. ‘Thevisitorshaveunderstoodthelecturejustaswellasanyoftheregularstudents.’

The problem with her lectures was straightforward. Unlike mostmathematicians, she was a formal thinker. To her, the symbols were theconcepts.Tofollowherlectures,youhadtothinkthesameway.Andthatwashard.

Despite this, it was Noether, and her emphasis on formal structures, thatwouldopenupmuchof today’smathematics.Sometimesyou justhave tobitethebullet.

WithHabilitation safely behind her,Noether promptly changed fields, pickingupwhereDedekind left offwhenhe replacedKummer’sobscurenotionof an

ideal number by a conceptually simpler but more abstract notion, that of anideal. The context for this approach was itself abstract: the theory of rings –algebraicsystemsinwhichaddition,subtraction,andmultiplicationaredefined,andsatisfytheusualrules,withthepossibleexceptionofthecommutativelawxy =yx formultiplication.The integers, the real numbers, and polynomials inoneormorevariablesallformrings.

We can get a brief flavour of the set-up using ordinary integers. Thetraditionalway to think about prime numbers and divisibility is toworkwithspecific integers, suchas2,or3,or6.Weobserve that6=2×3, so6 isnotprime; on the other hand, no such decomposition into smaller numbers ispossiblefor2or3,sotheyareprime.But,asDedekindrealised,there’sanotherwaytoseethis.Considerthesetsformedbyallmultiplesof6,2,and3,whichI’lldenotelikethis:

[6]={…,–12,–6,0,6,12,18,24,…

[2]={…,–4,–2,0,2,4,6,8,10,12,14,16,18,20,22,24,…

[3]={…,–6,–3,0,3,6,9,12,15,18,21,24,…

Herethecurlybracketsindicatesets,andweallownegativemultiples.Observethateverymemberof[6]isamemberof[2].Thisisobvious:anymultipleof6isautomatically a multiple of 2 because 6 is a multiple of 2. Similarly, everymember of [6] is amember of [3]. In otherwords, you can spot divisors of agivennumber(here6)byseeingwhichsetsofthiskindcontainallmultiplesof6.

Ontheotherhand,somenumbers in[3]aren’t in [2],andconversely.So2doesn’tdivide3and3doesn’tdivide2.

Withabitoffiddlingaround,theentiretheoryofprimesanddivisibilityforintegers can be reformulated in terms of these sets of multiples of a givennumber. The sets are examples of ideals, which are defined by two mainproperties:thesumanddifferenceofnumbersintheidealarealsointheideal,andtheproductofanumberintheidealbyanynumberintheringisintheideal.

Noether restatedHilbert’s theoremsabout invariants in termsof ideals, andthen generalised his results in a totally new direction. Hilbert’s finite basistheoremforinvariantsboilsdowntoprovingthatanassociatedidealisfinitelygenerated; that is, it consists of all combinations of a finite number ofpolynomials(thebasis).Noetherreinterpretedtheargumentasthestatementthat

anychainofeverlargeridealsmuststopafterfinitelymanysteps.Thatis,everyidealintheringofpolynomialsisfinitelygenerated.Shepublishedthisideain1921 in a far-reaching paper, ‘Theory of ideals in ring domains’. This paperkickedoffgeneralcommutativeringtheory.Noetherbecameadeptatsqueezingimportant theorems out of chain conditions, and a ring that satisfies this‘ascendingchaincondition’ issaid tobeNoetherian.Thisconceptualapproachto invariantswasahugecontrast to the turgidcalculations inher thesis,whichshenowdismissedas‘Formelgestrüpp’–aformulajungle.

Today every mathematics undergraduate is taught the abstract axiomaticapproach to algebra.Here themost important concept is that of a group, nowstripped of all associations with permutations or the solution of algebraicequations. Indeed, an abstract group need not even be composed oftransformations.Itisdefinedtobeanysystemofelementsthatcanbecombinedto yield another element of the system, subject to a short list of simpleconditions: the associative law, the existence of an ‘identity element’ whichcombines with any other element to yield that element, and the existence foreach element of an ‘inverse’ element, which composes with it to yield theidentity. That is, there is an element that has no effect, to each elementcorresponds another that undoes whatever the element itself does, and if youcombinethreeelementsinarowitdoesn’tmatterwhichpairyoucombinefirst.

Slightly more elaborate structures bring into play the full panoply ofarithmetical operations. I’ve alreadymentioned a ring. There’s also a field, inwhichdivisionispossibletoo.Theprecisedevelopmentofthisabstractviewiscomplicated, and many figures contributed to it. Who first did what is oftenunclear. By the time the precise definitions had been sorted out, mostmathematiciansalreadyhadaprettyclearfeelingforwhatwasgoingon.Butatrootweowe thewholeviewpoint toNoether,whoemphasised theneedforanaxiomaticapproachtoallmathematicalstructures.

In1924,DutchmathematicianBartelvanderWaerdenjoinedhercircleandbecametheleadingexpositorofherapproach,encapsulatedinhis1931ModernAlgebra. By 1932, when she delivered a plenary address at the InternationalCongress ofMathematicians, her algebraic ability was recognised worldwide.She was quiet, modest, and generous. Van der Waerden summed up hercontributionwhenhewroteherobituary:

Themaxim bywhich EmmyNoetherwas guided throughout herworkmight be formulated asfollows: Any relationships between numbers, functions, and operations become transparent,

generallyapplicable,andfullyproductiveonlyaftertheyhavebeenisolatedfromtheirparticularobjectsandbeenformulatedasuniversallyvalidconcepts.

Noetherhadmorethanalgebrainhersights.Sheimportedthesameinsightsintotopology.To the early topologists, a topological invariantwas a combinatorialobject, such as the number of independent cycles – closed loops with certainproperties. Poincaré had begun the process of adding extra structure,with theconceptofhomotopy.WhenNoetherfoundoutwhatthetopologistsweredoing,she immediately spotted something they’d all missed: an underlying abstractalgebraic structure. Cycles weren’t just things you could count: with a bit ofcare, you could turn them into a group. Combinatorial topology becamealgebraic topology. Her viewpoint won instant converts, in particular HeinzHopf and PavelAlexandrov. Similar ideas occurred independently to LeopoldVietorisandWaltherMayerinAustriabetween1926and1928,leadingthemtodefineahomologygroup–abasicinvariantofatopologicalspace.Algebrahadtakenover fromcombinatorics, revealing a far richer structure that topologistscouldexploit.

In1929NoethervisitedMoscowStateUniversity,toworkwithAlexandrovand to teach abstract algebra and algebraic geometry.Although not politicallyactive,shequietlyexpressedsupportfortheRussianRevolutionbecauseoftheopportunities it opened up in science and mathematics. This didn’t go downterribly well with the authorities, who evicted her from her lodgings whenstudentscomplainedaboutthepresenceofaMarxist-sympathisingJewess.

In1933,when theNazisdismissedJewsfromuniversitypositions,NoetherfirsttriedtoobtainapositioninMoscow,buteventuallymovedtoBrynMawrUniversityintheUnitedStates,withtheaidoftheRockefellerFoundation.Shealso gave lectures at the Institute for Advanced Study in Princeton, butcomplainedthateveninAmericashefeltuncomfortableina‘men’suniversity,wherenothingfemaleisadmitted’.

Despitethat,sheenjoyedAmerica,butnotforlong.Shediedin1935,fromcomplications after a cancer operation.AlbertEinsteinwrote in a letter to theNewYorkTimes:

In the judgment of the most competent living mathematicians, Fräulein Noether was the mostsignificant creativemathematicalgenius thus farproduced since thehigher educationofwomenbegan. In the realm of algebra, in which the most gifted mathematicians have been busy for

centuries,shediscoveredmethodswhichhaveprovedofenormousimportanceinthedevelopmentofthepresent-dayyoungergenerationofmathematicians.

Notonlythat:shetookthemenonattheirowngame,andbeatthem.

21TheFormulaMan

SrinivasaRamanujan

SrinivasaRamanujan

Born:Erode,TamilNadu,India,22December1887

Died:Kumbakonam,TamilNadu,India,26April1920

ITWASJANUARY1913.TurkeywasatwarintheBalkansandEuropewasbeing

draggeddeeperanddeeperintotheconflict.GodfreyHaroldHardy,professorofmathematicsatCambridgeUniversity,despisedwar;andhetookgreatpridethatthefieldofhislife’swork,puremathematics,hadnomilitaryuses.

Outside, snow drizzled damply down, while begowned undergraduatesscurried through the slush of Trinity Great Court. But in Hardy’s rooms acheerfulfirekeptthecoldatbay.Onthetablelaythemorningpost,readytobeopened.Heglancedattheenvelopes.Onecaughthiseyebecauseofitsunusualpostage stamps. India. Postmarked Madras, 16 January 1913. Hardy slit themanillaenvelope,morethanalittlebatteredbyitslongjourney,anddrewoutasheafofpapers.Anaccompanyingletter,inanunfamiliarhand,began:

DearSir,IbegtointroducemyselftoyouasaclerkintheAccountsDepartmentofthePortTrustOffice

atMadras at a salary of only £20 per annum. I am now about 23 years of age. I have had noUniversityeducation…AfterleavingschoolIhavebeenemployingthesparetimeatmydisposaltoworkatMathematics…Iamstrikingoutanewpathformyself.

OhLord,anothercrank.Probablythinkshe’ssquaredthecircle.Hardynearlythrewtheletterinthewastebasket,butashepickeditupasheetofmathematicalsymbolscaughthiseye.Curiousformulas.Afew,herecognised.Otherswere…unusual.Iftheauthoroftheletterisacrank,hemightatleastproveanentertaining

crank.Hardyreadon:

Very recently I cameacrossa tractbyyoustyledOrdersof Infinity inhereofwhich I findastatementthatnodefiniteexpressionhasasyetbeenfoundforthenumberofprimenumberslessthan any given number. I have found an expressionwhich very nearly approximates to the realresult,theerrorbeingnegligible.

Myword.He’srediscoveredthePrimeNumberTheorem.

Iwouldrequestyoutogothroughtheenclosedpapers.Beingpoor,ifyouareconvincedthereisanythingofvalueIwouldliketohavemytheoremspublished…BeinginexperiencedIwouldveryhighlyvalueanyadviceyougiveme.RequestingtobeexcusedforthetroubleIgiveyou.Iremain,DearSir,YoursTruly,S.Ramanujan

Not a typical crank,Hardymused.A typical crankwould bemore aggressiveandmore conceited. Putting the letter aside, he picked up the enclosed sheets

andbegantoread.Halfanhourlaterhewassittingbackinhischairwithanoddexpressiononhis face.Howstrange.Hardywas intrigued.But itwas timeforhis undergraduate lecture on analysis, so he shrugged into his chalk-spatteredgown,walkedoutoftheroom,andshutthedoorbehindhim.

Thatevening,overhightable,hetalkedaboutthestrangelettertoanyoftheFellows of the college who cared to listen, including his colleague and closecollaborator JohnLittlewood. Littlewoodwaswilling towaste an hour on themattertohelpputhisfriend’smindatease,andthechessroomwasfree.Astheywalkedin,Hardyhelduptheslimsheafofpaper.‘Thisman,’heannouncedtothegatheringatlarge,‘iseitheracrankoragenius.’

Anhourlater,HardyandLittlewoodemergedwiththeverdict.Genius.

I hope you’ll forgivemy dramatisation of these events. I’ve put thoughts intoHardy’shead,but survivingdocumentationmakes it clear that somethingverysimilarmusthavegonethroughhismind,andthegeneralturnofeventsrespectsrecordedhistory.

The author of the letter, Srinivasa Ramanujan, was born into a Brahminfamily in1887.HisfatherK.SrinivasaIyengarwasaclerk inasarishop,andhismotherKomalatammalwasabailiff’sdaughter.Thebirth tookplace inhisgrandmother’shouseinErode,atowninthesouthernprovinceofTamilNadu,India. He grew up in Kumbakonam, where his father worked. But it wascommonforayoungwifetospendtimewithherparentsaswellasherhusband,sohismotherfrequentlytookhimtolivewithherfathernearMadras,some400kilometresaway.Thefamilywaspoor,thehousetiny.Itwasbasicallyahappychildhood,althoughRamanujanwasveryobstinate.For the first threeyearsofhislife,hescarcelysaidaword,andhismotherfearedhewasdumb.Agedfive,hedidn’tlikehisteacheranddidn’twanttogotoschool.Hepreferredtothinkaboutthingsforhimself,askingannoyingquestionssuchas‘Howfarapartareclouds?’

Ramanujan’smathematical talents surfaced early, and by the age of 11 hehadoutstrippedtwocollegestudentswholodgedathishome.Helearnedhowtosolve cubic equations andcould recite thedigitsofπ ande at some length.Ayear later he borrowed an advanced textbook and mastered it completely,without apparent effort. When he was 13 he devoured Sidney Loney’sTrigonometry, which included the infinite series expansions for the sine and

cosine, and was already producing his own new results. His ability inmathematics won him many prizes at school, and in 1904 the headmasterdescribedhimasdeservingmoremarksthanthemaximumpossible.

At theageof15aneventoccurredwhichwas tochangehis life,butat thetime it seemed mundane. He borrowed a copy of George Carr’s Synopsis ofElementaryResultsinPureMathematicsfromtheGovernmentCollegeLibrary.TheSynopsisis,tosaytheleast,idiosyncratic.Itsthousand-pluspageslistsomefive thousand theorems–allwithoutproofs.Carrbased thebookonproblemsthat he posed when coaching his students. Ramanujan likewise set himself aproblem: to establish all the formulas in the book. He had no help, no otherbooks.Effectively,he’dsethimselfaresearchprojectoffivethousandseparatetopics.Toopoortoaffordpaper,hedidhiscalculationsonaslateandjottedtheresultsinaseriesofnotebooks,whichhekeptthroughouthislife.

In 1908Ramanujan’smotherKomalatammal decided to find her son, thenagedtwenty,awife.ShesettledonJanaki,thedaughterofoneofherrelatives,who livedabout100kilometres fromKumbakonam. Janakiwasnine.Theagedifferencewasn’tagreatobstacle ina societyofarrangedmarriagesandchildbrides.Ramanujanwas–itseemed–averyordinaryyoungman;alazyfailurewithnojob,nomoney,andnoprospects.ButJanakiwasoneoffivedaughtersinafamilythathadlostmostofwhatitowned,andherparentswouldbehappymerely to find a husband who would be kind to her. That was enough forKomalatammal,whichordinarilymeantitwasadonedeal.Butonthisoccasionherhusbandblewhis top.Hissoncoulddobetter!Hehadnearlymarried twoyears before, but by mischance a death in the bride’s family put paid to it.Mostly, thefatherwasupset thathiswifehadn’taskedhisadvicefirst.Atanyrate,hesnubbedthebride’sfamilybyrefusingtocometothewedding.

Theweddingdaydawned,andtherewasnosignofthegroomorhisfamily.Thebride’sfatherRangaswamyannouncedtoallandsundrythatifRamanujandidn’t turn up soon, he’d marry Janaki off on the spot to someone else.Eventually the train fromKumbakonam turnedup, hours late, and itwaswellafter midnight when Ramanjuan and his mother (minus father) arrived at thevillage on a bullock cart. Komalatammal quickly made short work ofRangaswamy’s threats, pointing out very publicly that a poor fatherwith fivedaughterswouldrejectagenuineofferathisperil.

After the customary five to six days of celebrations, Janaki found herselfmarried to Ramanujan. She wouldn’t join him until she reached puberty, butboth their lives had changed. Ramanujan started looking for a job. He tried

tutoring students in mathematics, but found no takers. When he became ill,possiblyasa resultofapreviousoperation,he turnedup inahorsecartat thehomeofafriend,R.RadhakrishnaIyer,whotookhimtoseeadoctorandthenputhimonatrainbacktoKumbakonam.Justashewasleaving,Ramanujansaid‘If I die, please hand these over to Professor SingaraveluMudaliar or to theBritishprofessorEdwardRoss.’Andhepressedintohisstartledfriend’shandstwofatnotebooks,stuffedfullofmathematics.

Herewas not justRamanujan’s legacy, but his job ticket: evidence that hewasmorethananindolentwastrel.Hestartedcallingoninfluentialpeoplewithhismathematicalportfoliounderhisarm.InTheManWhoKnewInfinity,RobertKanigelsays:‘Ramanujanhadbecome,intheyearandahalfsincehismarriage,adoor-to-doorsalesman.Hisproductwashimself.’Itwasahardsell.InIndiaatthat time, the best route to employment was the right connections, butRamanujanhadnone.Allhehadwashisnotebooks…andoneotherimportantthing.Hewasfriendly.Everyonelikedhim.Hewaslively,andtoldjokes.

Eventually, his persistence and uncomplicated charm paid off. In 1912 amathematicsprofessor,P.V.SeshuAiyar,senthimtoseeR.’RamachandraRao,a civil servant who was a district collector at Nellore. Rao recollected theinterview:

I condescended to permit Ramanujan to enter my presence. A short uncouth figure, stout,unshaved,notoverclean,withoneconspicuousfeature–shiningeyes…Isawquiteatonce thattherewas somethingoutof theway;butmyknowledgedidnotpermitme to judgewhetherhetalkedsenseornonsense…Heshewedmesomeofhissimplerresults.ThesetranscendedexistingbooksandIhadnodoubtthathewasaremarkableman.Then,stepbystep,heledmetoellipticintegralsandhypergeometricseriesandatlasthistheoryofdivergentseriesnotyetannouncedtotheworldconvertedme.

RaosecuredRamanujananappointment in theMadrasPortTrustOfficeat30rupees per month, a job that left him enough spare time to continue hisresearches.Anotherbonuswasthathecouldtakeawayusedwrappingpapertowritehismathematicson.

It was then that, at the urgings of the same people, Ramanujan wrote hisdiffident letter to Hardy. Hardy immediately sent an encouraging reply.Ramanujan asked him to send a ‘sympathetic letter’ to help him get ascholarship.Hardywasaheadofhim,andmoreambitious.He’dalreadywrittentotheSecretaryforIndianStudentsinLondon,seekingawaytogetRamanujana Cambridge education. But then it transpired that Ramanujan didn’t want to

leave India. The Cambridge network rolled into action. Another Trinitymathematician,GilbertWalker,wasvisitingMadras,andhewrotealettertotheUniversityofMadras,whichgrantedRamanujanaspecialscholarship.Atlasthewasfreetodevoteallofhistimetomathematics.

Hardy continued trying to persuade Ramanujan to come to England.Ramanujanbegantowaver,andthemainobstaclebecamehismother.Then,onemorning, to general family astonishment, she announced that the goddessNamagirihadappearedtoherinadream,commandinghertolethersonfulfilhislife’scalling.Ramanujanwasgivenagranttocoversubsistenceandtravel,setsailforEngland,andbyApril1914wasinTrinityCollege.Hemusthavefeltveryoutofplace,buthestucktoit,publishingmanyresearchpapers,includingsomeinfluentialjointworkwithHardy.

RamanujanwasBrahmin, aHindu caste forbidden to cause harm to livingcreatures.AlthoughhisEnglishfriendsgottheimpressionthathismainreligiousmotivationwasnotbelief,but social custom,heobserved theproper rituals asfar as was possible in wartime England. As a vegetarian, he didn’t trust theCollege cooks to eliminate all meat products, so he taught himself to cook,Indianstyleofcourse.Accordingtofriends,hebecameanexcellentcook.

Around 1916 his friend Gyanesh Chandra Chatterji, in Cambridge as aGovernment of India state scholar, was about to get married, so Ramanujaninvitedhimandhisbride-to-be fordinner.Asagreed,Chatterji, fiancée, andachaperone turned up at Ramanujan’s rooms, and he served them soup.Whentheypolisheditoff,heofferedsomemore,andallthreetookasecondhelping.Sohesuggestedathirdhelping.Chatterjiaccepted,buttheladiesdeclined.

Shortlyafterwards,Ramanujanwasnowheretobeseen.They waited for him to return. After an hour had passed, Chatterji went

downstairs to find a porter.Yes, he had seenMrRamanujan.Hehad called ataxi,andgoneoffinit.Chatterjireturnedtotheroomandthethreeguestswaiteduntilteno’clockatnight,whencollegerulesrequiredthemtoleave.Nosignoftheir host. No sign of him for the next four days… What had happened?Chatterjiwasworried.

Ondayfive,atelegramarrivedfromOxford:couldChatterjiwireRamanujanfivepounds?(Thatwasalotinthosedays,afewhundredpoundstoday.)Moneysent,Chatterjiwaited,andRamanujandulyappeared.Askedwhathadhappened,he explained. ‘I felt hurt and insulted when the ladies didn’t take the food Iserved.’

Itwas an outward sign of inner turmoil. Ramanujanwas at the end of his

tether.HehadnevertrulyadaptedtolifeinEngland.Hishealth,nevergood,wasgettingworse,andheendedupinhospital.Hardyvisitedhimthere,andthevisitled to another story about Ramanujan that also features a taxi. It has becomesomethingofacliché,butitbearsrepetitionallthesame.

Hardy once wrote that every positive integer was one of Ramanujan’spersonalfriends,andillustratedthiswithananecdoteaboutvisitingRamanujaninhospital.‘Ihadriddenintaxicabnumber1729andremarkedthatthenumberseemed tome rather a dull one, and that I hoped it was not an unfavourableomen. ‘‘No,’’ he replied, ‘‘it is a very interesting number; it is the smallestnumberexpressibleasthesumoftwocubesintwodifferentways.”’

Tobeprecise,

1729=13+123=93+103

anditisthesmallestpositivenumberwithsuchaproperty.Thestorymakesitspointwell,butIcan’thelpwonderingwhetheritwasa

bitofaset-up,withHardytryingtobuckuphissickfriendbygettinghimtoriseto the bait.Most peoplewouldn’t spot this feature of the number 1729, to besure,butRamanujanwouldundoubtedlyrecogniseitimmediately.Indeed,manymathematicians, especially those with an interest in number theory – Hardyamongthem–wouldbeawareofit.It’salmostimpossibleforamathematiciantolookat1729andnotthinkof1728,whichisthecubeof12.Andit’salsohardnottonoticethat1000is10cubedand729is9cubed.

Be that as it may, Hardy’s story led to a minor but intriguing concept innumbertheory:thatofataxicabnumber.Thenthtaxicabnumberisthesmallestnumberthatcanbeexpressedasasumoftwopositivecubesinndistinctways.Thenexttwotaxicabnumbersare

87,539,319

6,963,472,309,248

Thereareinfinitelymanytaxicabnumbers,butonlythefirstsixareknown.By1917Ramanujanwas back in his rooms, obsessedwithmathematics to

theexclusionofallelse.Hewouldworkdayandnight,thencollapseexhaustedandsleepfor20hours.Thisdidhishealthnogood,andthewarcausedshortagesof the fruit and vegetables onwhich he relied.By spring, hewas afflicted bysomeundiagnosedbutprobably incurabledisease.Hewasadmitted to a smallprivatehospitalforpatientsfromTrinityCollege.Overthefollowingtwoyears

he saw eight or more doctors and was admitted to at least five hospitals andsanatoriums. The doctors suspected a gastric ulcer, then cancer, then bloodpoisoning;buttheydecidedthatthemostlikelycausewastuberculosis,andthisiswhattheymainlytreatedhimfor.

Finally,muchtoolate,academichonourswerecominghisway.HebecamethefirstIndiantobeelectedaFellowoftheRoyalSociety,andTrinityelectedhimtoaFellowship.Reinvigorated,hepickeduphismathematicsagain.Buthisheath remained poor, the English climate was suspect, and in April 1919 hereturnedtoIndia.Thelongvoyagedidn’tsuithim,andbythetimehearrivedinMadras his health had once more deteriorated. In 1920 he died in Madras,leavingawidowbutnochildren.

TherearefourmainsourcesforRamanujan’smathematics:hispublishedpapers,histhreeboundnotebooks,hisquarterlyreportstotheUniversityofMadras,andhisunpublishedmanuscripts.Afourth‘lost’notebook–abundleofloosesheets–wasfoundagainin1976byGeorgeAndrews,butsomeofhismanuscriptsarestill missing. Bruce Berndt has edited a three-volume work Ramanujan’sNotebooks,includingproofsofallofhisformulas.

Ramanujanhadanunusualbackground,andnoformaltraining.Itwashardlya suprise that his mathematics was a little idiosyncratic. His greatest strengthwas in an unfashionable area – the production of ingenious and intricateformulas.Ramanujanwas the FormulaManpar excellence, unrivalled by anysaveafewOldMasterssuchasEulerandJacobi.‘ThereisalwaysmoreinoneofRamanujan’sformulaethanmeetstheeye,’Hardywrote.Mostofhisresultsare about infinite series, integrals, and continued fractions. An example of acontinuedfractionistheexpression:

which was on the last page of his letter, featuring in a distinctly weird, butcorrect, formula.He applied some of his formulas to number theory, taking aspecialinterestinanalyticnumbertheory,whichseekssimpleapproximationstosuch quantities as the number of primes below a given limit –Gauss’s primenumber theorem (Chapter 10) – or the average number of divisors of a given

number.His publications while at Cambridge were influenced by his contact with

Hardy, and written in a conventional style with rigorous proofs. The resultsrecorded in his notebooks have a very different quality. Because hewas self-educated,hisconceptofproofwaslessthanrigorous.Ifamixtureofnumericalevidence and formal argument led to a plausible conclusion, and his intuitiontoldhimhehad the right answer, thatwas enough forRamanujan.His resultswereusuallycorrect,buthisproofsoftenhadgaps.Sometimesanycompetenttechnician could fill the gaps, and sometimes quite different arguments wereneeded. On rare occasions, his results were wrong. Berndt argues that ifRamanujan ‘had thought likeawell-trainedmathematician,hewouldnothaverecorded many of the formulas which he thought he had proved’, andmathematicswouldhavebeenpoorerasaresult.

A good example is a result thatRamanujan called his ‘Master Formula’.11His proof involves series expansions, interchanges of the order of summationand integration, and other similar manoeuvres. Because he uses infiniteprocesses,eachstepisfraughtwithdanger.Thegreatestanalystsspentmostofthenineteenthcenturyworkingout justwhensuchproceduresarepermissible.Theconditionsthat,accordingtoRamanujan,makehisformulatrue,aregrosslyinsufficient. Nevertheless, almost all of the results that he derives from hisMasterFormulaarecorrect.

SomeofRamanujan’smoststrikingworkisinthetheoryofpartitions,abranchofnumbertheory.Givenawholenumber,weaskinhowmanywaysitcanbepartitioned,thatis,writtenasasumofsmallerwholenumbers.Forexample,thenumber5cansplitupinsevenways:

54+13+23+1+12+2+1

2+1+1+11+1+1+1+1

Therefore p(5) = 7. The numbers p(n) grow rapidly with n. For instance,p(50) = 204,226 and p(200) is a staggering 3,972,999,029,388. No simpleformulaforp(n)exists.However,wecanaskforanapproximateformula,givingthe general order ofmagnitude of p(n). This is a problem in analytic numbertheory, and an especially intractable one. In 1918 Hardy and Ramanujanovercamethetechnicaldifficultiesandderivedanapproximateformula,arather

complicatedseriesinvolvingcomplex24throotsofunity.Theythenfoundthatwhenn=200thefirsttermaloneagreeswiththefirstsixsignificantfiguresofthe exact value. By adding just seven more terms they obtained3,972,999,029,388·004,whoseintegerpartistheexactvalue.Theyobservedthatthisresult‘suggestsveryforciblythatitispossibletoobtainaformulaforp(n)whichnotonlyexhibitsitsorderofmagnitudeandstructure,butmaybeusedtocalculateitsexactvalueforanyn’,andtheywentontoprovepreciselythat.Itmust be one of the very few occasions when the search for an approximateformulaledtoanexactone.

Ramanujan also found some remarkable patterns in partitions. In 1919 heprovedthatp(5k+4)isalwaysdivisibleby5andp(7k+5)isalwaysdivisibleby7. In1920hestatedsomesimilar results: forexamplep(11k+6) isalwaysdivisibleby11;p(25k+24)isdivisibleby25;allofp(49k+19),p(49k+33),p(49k+40),andp(49k+47)aredivisibleby49;andp(121k+116)isdivisibleby121.Noticethat25=52,49=72,and121=112.Ramanujansaidthat,asfarashecould tell, such formulasexistonly fordivisorsof the form5a7b11c,butthiswaswrong.ArthurAtkinfoundthatp(17303+237)isdivisibleby13,andin2000KenOnoprovedthatcongruencesofthiskindexistforallprimemoduli.AyearlaterheandScottAhlgrenprovedtheyexistforallmodulinotdivisibleby6.

Some of Ramanujan’s results remain unproved even to this day. One thatsuccumbedaboutfortyyearsagoisparticularlysignificant.Inapaperof1916hestudiedafunctionτ(n)definedtobethecoefficientofxn–1intheexpansionof

[(1–x)(1–x2)(1–x3)…]24

Thusτ(1)=1,τ(2)=–24,τ(3)=252,andsoon.Theformulacomesfromdeepand beautifulwork in the nineteenth century on elliptic functions. Ramanujanneededτ(n)tosolveaproblemaboutpowersofdivisorsofn,andheneededtoknowhowbigitwas.Heprovedthatitssizeisnolargerthann7,butconjecturedthatthiscanbeimprovedton11/2.Heconjecturedtwoformulas:

τ(mn)=τ(m)τ(n)ifmandnhavenocommonfactor

τ(pn+1)=τ(p)τ(pn)-p11τ(pn-1)forallprimep

Thesemake it easy to compute τ(n) for anyn. LouisMordell proved them in

1919,butRamanujan’sconjectureontheorderofmagnitudeofτ(n)resistedallhisefforts.

In1947AndréWeilwaslookingoveroldresultsofGauss,andherealisedhecouldapplythemtointegersolutionsofvariousequations.Followinghisnose,andacuriousanalogywith topology,heformulatedaseriesofrather technicalresults, the Weil conjectures. These acquired a central position in algebraicgeometry.In1974PierreDeligneprovedthem,andayearlaterheandYasutakaIhara deduced Ramanujan’s conjecture from them. That his innocent-lookingconjecturerequiredsuchamassiveandcentralbreakthroughbeforeitcouldbeansweredisasignofhowgoodRamanujan’sintuitionwas.

Amonghismoreenigmaticinventionswere‘mockthetafunctions’,whichhedescribedinhisfinallettertoHardyin1920;detailswerelaterfoundinhislostnotebook.Jacobiintroducedthetafunctionsasanalternativeapproachtoellipticfunctions. They’re infinite series that transform in a very simple way whensuitable constants are added to the variable, and elliptic functions can beconstructedbydividingonethetafunctionbyanother.Ramanujandefinedsomeanalogousseries,andstatedalargenumberofformulasinvolvingthem.Atthetime,thewholeideaseemedtobejustanexerciseinmanipulatingcomplicatedseries,with no connection to anything else inmathematics. Today,we realisethisisnotthecase.Theyhaveimportantconnectionswiththetheoryofmodularforms,whichariseinnumbertheoryandarealsorelatedtoellipticfunctions.

A similar but distinct concept, the Ramanujan theta function, has recentlyturnedouttobeusefulinstringtheory,themostpopularattemptbyphysiciststounifyrelativityandquantummechanics.

Because Ramanujan functioned in such an extraordinary manner, obtainingcorrect results by non-rigorousmethods, it has sometimes been suggested thathis thought patternswere special or unusual. Ramanujan himself is quoted assayingthatthegoddessNamagiritoldthemtohimindreams.However,hemayhavesaidthisjusttoavoidembarrassingdiscussions.AccordingtohiswidowS.JanakiAmmalRamanujan, he ‘neverhad time togo to the templebecausehewas constantly obsessedwithmathematics’.Hardywrote that he believed ‘allmathematicians think, at bottom, in the same way, and Ramanujan was noexception’,butheadded:‘Hecombinedapowerofgeneralisation,afeelingforform, and a capacity for rapidmodification of his hypotheses, thatwere oftenreallystartling.’

Ramanujan wasn’t the greatest mathematician of his period, nor the mostprolific;buthisreputationdoesnotjustrestonhisremarkablebackgroundandtouching ‘poor boy makes good’ story. His ideas were influential during hislifetime,andtheygrowmoreinfluentialastheyearspass.BruceBerndtbelievesthat, far from being old-fashioned, Ramanujan was ahead of his time. It’ssometimeseasiertoproveoneofRamanujan’sremarkableformulasthanitistoworkouthowhecouldpossiblyhavethoughtofit.AndmanyofRamanujan’sdeepest ideas are only now becoming appreciated. I leave the final word toHardy:

Onegift[thathismathematics]haswhichnoonecandeny:profoundandinvincibleoriginality.Hewouldprobablyhavebeenagreatermathematicianifhehadbeencaughtandtamedalittleinhisyouth;hewouldhavediscoveredmorethatwasnew,andthat,nodoubt,ofgreaterimportance.OntheotherhandhewouldhavebeenlessofaRamanujan,andmoreofaEuropeanprofessor,andthelossmighthavebeengreaterthanthegain.

22IncompleteandUndecidable

KurtGödel

KurtFriedrichGödel

Born:Brünn,Austria-Hungary,28April1906

Died:PrincetonNJ,USA,14January1978

THESTEREOTYPIC IMAGE ofmathematicians, aside fromallof thembeingmaleand elderly, is that they’re a bit strange. Other-worldly, certainly. Eccentric,commonly.Downrightcrazy,sometimes.

We’veseenthatthisimagedoesn’tfitmostmathematicians,asidefromtheirbeingmale,andeven thathaschangeddramaticallyover the last fewdecades.Agreed, mathematicians tend to end their careers by being elderly, but whodoesn’t?Theonlywaytoavoidthisistodieyoung,likeGalois.Reputationsandresponsibilities tend to grow with age, so the elderly are likely to be over-representedamongtheleadersofthesubject.

Whentheirmindsarefocusedonresearch,mathematicianscaneasilyappearother-worldly,butasabiologistcolleagueofminealways insisted, they’renotabsent-minded: they’represent-mindedsomewhereelse. Ifyouwant tosolveadifficultmathematicalproblem,youneedtoconcentrate.Somemathematicians

(bynomeanstheonlyprofessiontodoso)takethislackofworldlyfocustothepoint of eccentricity. Perhaps the most obvious example is Paul Erdős, whonever held an academic position and never owned a house.He travelled fromone colleague to another, spending a night on the sofa ormonths in the spareroom.Yethewroteanextraordinary1500researchpapersandcollaboratedwithastaggering500differentmathematicians.

Asforbeingcrazy:someare,atsomestageoftheirlife,mentallyill.Cantorsuffered from serious bouts of depression. JohnNash, the subject of the bookandmovieA BeautifulMind, won the 1994Nobel Prize in Economics (moreprecisely, theNobelMemorial Prize,which is treated like any of the originalNobel Prizes for most purposes). Yet he suffered for many years from acondition diagnosed as paranoid schizophrenia, and underwent electroshocktherapy.By an effort ofwill, recognising psychotic interludes and refusing togiveintothem,hemanagedtocurehimself.

KurtGödelwasdefinitelyeccentric, andat timeswentbeyond.His chosenareaofmathematicallogicwasnot,atthattime,mainstreammathematics,andinthisrespecthewasifanythingmoreother-worldlythanmostofhiscolleagues.Incompensation,hisdiscoveries in that area revolutionisedour thinkingaboutthe foundations of logic and mathematics, and how these interact. He wasbrilliantlyoriginal,andremarkablydeep.

His interest in logic began in 1933 when Adolf Hitler rose to power inGermany, and was stimulated by seminars given by Moritz Schlick, aphilosopherwhofoundedlogicalpositivismandtheViennaCircle.In1936oneofSchlick’sformerstudents,JohannNelböck,murderedhim.ManymembersoftheViennaCircle had already fledGermany, fearing anti-Semitic persecution,butSchlick,whowasinAustria,stayedonattheUniversityofVienna.Hewaswalking up the steps to give a lecture when Nelböck shot him with a pistol.Nelböckconfessedto themurder,butusedthecourtproceedingsasaplatformfrom which to proclaim his political beliefs. He claimed his lack of moralrestraint had been a reaction to Schlick’s philosophical stance, which wasantagonistic to metaphysics. Others suspected the true cause to have beenNelböck’s infatuation with another student, Sylvia Borowicka. His unrequitedpassion led him to a paranoid belief that Schlick was a competitor for heraffections.Hewassentencedtotenyearsinprison,butthecasecontributedtoagrowinganti-Semitichysteria inVienna,even thoughSchlickwasnot, in fact,Jewish. Post-truth politics is nothing new. Worse, when Germany annexedAustria,Nelböckwasreleased,ameretwoyearsintohissentence.

ThemurderofhismentorhadaterribleeffectonGödel.He,too,developedsigns of paranoia – though it rather fitted the old joke ‘Just because I’mparanoid,itdoesn’tmeanthey’renotouttogetme.’Gödelwasn’tJewisheither,but he hadmany friendswhowere. Living underNazi rule, paranoiawas theultimate in sanity.However,hedevelopedaphobiaaboutbeingpoisoned,andspent severalmonths being treated formental illness. This fear came back tohaunthiminthelastfewyearsofhislife,whenheagaindevelopedsymptomsofmental illness and paranoia. He refused to eat any food that his wife hadn’tcooked.In1977shehadtwostrokes,andwasadmittedtohospitalforalengthyperiod, so she was no longer able to cook for him. He stopped eating, andstarvedhimselftodeath.Itwasagruesomeandfutileendforoneofthegreatestthinkersofthetwentiethcentury.

Gödel’sfatherRudolfmanagedatextilefactoryinBrünn,Austria-Hungary,nowBrno in theCzechRepublic.Fromearlychildhoodandwell intoadulthoodhewasveryclosetohismotherMarianne(néeHandschuh).RudolfwasProtestant,MarianneCatholic;KurtwasbroughtupintheProtestantchurch.HeconsideredhimselfacommittedChristian,believinginapersonalGod,butnotinorganisedreligion.Hewrotethat‘religionsare,forthemostpart,bad–butreligionisnot’.He read the Bible regularly but didn’t attend church. An attempt at amathematical proof of the existence of God, derived using modal logic, wasfoundamonghisunpublishedpapers.Hisnicknameinthefamily,asachild,wasHerrWarum(MrWhy),forreasonsyoucanguess.Whensixorsevenyearsoldhe suffered a bout of rheumatic fever, and although he made a completerecovery, he never lost the belief that the illness had damaged his heart. Hishealthwasoftenfragile,astateofaffairsthatcontinueduntilhisdeath.

From 1916 Gödel was a student at the Deutsches Staats-Realgymnasium,gaininghighmarksinallofhissubjects,especiallymathematics,languages,andreligion.HewasautomaticallymadeaCzechoslovakcitizenwhen theAustro-Hungarian Empire broke up at the end of World War I. He attended theUniversityofViennain1923,unsureinitiallywhethertostudymathematicsorphysics, but Bertrand Russell’s Introduction to Mathematical Philosophy ledhimtosettleonmathematics,withthemainfocusbeingonmathematicallogic.Akeyturningpointinhiscareerhappenedin1928,whenhewenttoalectureinBolognabyDavidHilbertatthefirstInternationalCongressofMathematiciansto be held after the end of World War I. Hilbert explained his views about

axiomaticsystems,especiallywhetherthey’reconsistentandcomplete.In1928Gödel read Principles of Mathematical Logic by Hilbert and WilhelmAckermann,whichprovidedthetechnicalbackboneforHilbert’sprogrammetosettlethesequestions.In1929hechosethattopicforhisdoctoralthesis,workingunderHansHahn.HeprovedwhatwenowcallGödel’sCompletenessTheorem:thatpredicatecalculus(Chapter14)iscomplete.Thatis,everytruetheoremcanbe proved, every false one can be disproved, and there’s no other option left.However,predicatecalculusisverylimited,andinadequateasafoundationformathematics. Hilbert’s programme was formulated within a much richeraxiomaticsystem.

Gödel became an Austrian citizen the same year. (His citizenshipautomaticallychangedtoGermanin1938whenGermanyannexedAustria.)Hewasawardedadoctorate in1930. In1931hedemolishedHilbert’sprogrammebypublishing‘OnformallyundecidablepropositionsofPrincipiaMathematicaand similar systems’, which proved that no axiom system rich enough toformalise mathematics can be logically complete, and that it’s impossible toproveanysuchsystemisconsistent.(I’lltellyouaboutPrincipiaMathematicainamoment.)HeattainedHabilitationin1932,becomingaPrivatdozentattheUniversityofViennain1933.Theharrowingeventsrecountedearlierhappenedduring thisperiodofhis life.Toget abreak fromNaziAustria,hevisited theUnitedStates.Therehemet,andbecamefriendswith,Einstein.

In1938hemarriedAdeleNimbursky(néePorkert),whomhehadmetattheDer Nachtfalter night club in Vienna eleven years earlier. She was six yearsolder thanhewas,hadbeenmarriedpreviously,andbothhisparentsobjected,butheignoredtheirwishes.WhenWorldWarIIstartedin1939,Gödelbecameconcerned that he might be drafted into the German army. His poor healthshouldhaveruledthatout,buthe’dalreadybeenmistakenforbeingJewish,sohe might also be mistaken for being healthy. He managed to wangle anAmericanvisa,andheadedfortheUSAbywayofRussiaandJapan,alongwithhis wife. They arrived there in 1940. In that year he proved that Cantor’sContinuum Hypothesis is consistent with the usual set-theoretic axioms formathematics. He took up a position at the Institute for Advanced Study inPrinceton,firstasanordinarymember,thenapermanentone,then,from1953,aprofessor.Althoughhestoppedpublishingin1946,hecontinuedtodoresearch.

Gödel became aUS citizen in 1948. Apparently he believed he’d found alogicalflawintheUSConstitution,andattemptedtoexplainittothejudge,whosensiblyfailedtotakethebait.HisclosefriendshipwithEinsteinledhimtodo

some work on relativity. In particular, he found a spacetime that possesses aclosed timelike curve – a mathematical formulation of a time machine. Ifsomethingfollowssuchacurvethroughspaceandtime,itsfuturemergesintoitspast. It’s likebeing inLondon in1900, travelling twentyyears into the future,and findingyou’reback inLondonand theyear isagain1900.More recently,closedtimelikecurveshavebecomeahottopic,notsomuchbecausetheymightleadtoapracticaltimemachine,butbecausetheyshedlightonthelimitationsofGeneralRelativity,andsuggestapossibleneedfornewlawsofphysics.

In his final years, Gödel’s health, never good, became worse. His brotherRudolfreportedthathe

hadaveryindividualandfixedopinionabouteverything…Unfortunatelyhebelievedallhislifethathewasalwaysrightnotonlyinmathematicsbutalsoinmedicine,sohewasaverydifficultpatientfordoctors.Afterseverebleedingfromaduodenalulcer…hekept toanextremelystrict(overstrict?)diet,whichcausedhimslowlytoloseweight.

Whathappenedafterthat,youalreadyknow.Onhisdeathcertificatethecauseof death is stated as ‘malnutrition and inanition caused by personalitydisturbance’. Inanition is exhaustion caused by lack of food.Heweighed justthirtykilograms.

Sinceancienttimes,mathematicswasheldupasashiningexampleofsomethingthatwassimplytrue–absolutetruth,noifsorbuts.Twoplustwoisfour:takewhatyougetandnowhining.Itssolecompetitorforabsolutetruthwasreligion(denominationandsectofthebeliever’schoice,ofcourse),butmathematicshada sneaky advantage even then.Religions, as Terry Pratchett has said, are true‘foragivenvalueoftrue’.Mathematicscouldproveitwastrue.

Asphilosophers, logicians,andmathematicians inclined in thosedirections,startedtothinkmoredeeplyaboutwhatthistypeofabsolutetruthinvolves,theyrealised that it’s to some extent illusory. Two plus two equals four forwholenumbers, butwhat, exactly, is a number?For thatmatter,what are ‘plus’ and‘equals’? Mathematicians answered these questions by formulating thecontinuumof real numbers, butKronecker considered this ‘thework ofman’,believingthatonlytheintegersareGod-given.It’shardtoseehowanarbitrarycreation of the human mind can constitute absolute truth. It has to be aconvention,atbest.

Thenotion thatmathematics consistsofnecessary truthswas abandoned infavour of thembeing deductions fromexplicit assumptions according to somespecified system of logic. Honesty then demands following Euclid’s lead andstatingthoseassumptionsandlogicalrulesasasystemofexplicitaxioms.Thisis metamathematics – applyingmathematical principles to the internal logicalstructure ofmathematics itself. Bertrand Russell and Alfred NorthWhiteheadpaved the way in their 1910–13 Principia Mathematica – the title was aconscioushomagetoNewton–andafterseveralhundredpagestheymanagedtodefine the number ‘one’. After that, the pace hotted up, and more advancedmathematical concepts appeared with ever-increasing rapidity until it wasobvious that the rest could be done in the same way and they gave up. Onetechnicalfeature,theirtheoryof‘types’,introducedtoavoidcertainparadoxes,waslaterabandonedinfavourofotheraxiomschemesforset theory, themostpopularbeingthatofErnstZermeloandAbrahamFraenkel.

It’sagainstthisbackgroundthatHilbertsoughttocompletethelogicalcircleby proving that some such axiomatic system is logically consistent (no proofleadstoacontradiction)andcomplete(everymeaningfulstatementeitherhasaproof or has a disproof). The first step is essential because, in an inconsistentsystem, ‘two plus two equals five’ has a proof. Indeed, any statement can beproved.Thesecondidentifies‘true’with‘hasaproof’and‘false’with‘hasnoproof’.HilbertfocusedonanaxiomaticsystemforarithmeticbecausePrincipiaMathematicadeducedeverythingelseinmathematicsfromthat.TopickuponKronecker, once God has given us the integers, Man can sort out the rest.Hilbert’s programmeoutlined a series of steps that he believedwould achievethisgoal,basedon the logical complexityof the statementsconcerned, andhemanagedtosortoutsomeofthesimplercases.Italllookedpromising.

Gödel, I suspect, spotted something philosophically fishy about the wholeenterprise. In effect, an axiomatic system for mathematical logic was beingasked to demonstrate its own consistency. ‘Are you consistent?’ ‘Of course Iam!’Pause.‘Yeah,yeah…WhyshouldIbelieveyou?’Bethatasitmay,somesource of scepticism led him to prove two devastating results: hisincompletenesstheoremandhisconsistencytheorem.

The second rests on the first. Bearing in mind that an inconsistent logicalsystemcanproveanything,itcanpresumablyprovethestatement‘thissystemisconsistent’.(Itcanofcoursealsoprove‘thissystemisinconsistent’,butignore

that.)Sowhatkindofguaranteeof truthcansuchaproofoffer?None.That’swhatthe‘yeah,yeah’responseintuitivelygrasps.There’sonepossiblewaythatHilbert’s programme can escape from this trap: perhaps the statement ‘thissystem is consistent’ makes no sense within the formal axiomatic system.Certainlythestatementdoesn’tlookmuchlikearithmetic.

Gödel’sanswerwastoturnitintoarithmetic.Aformalmathematicalsystemisbuiltfromsymbols,andaproof(orallegedproof)ofsomestatementismerelya string of symbols.The symbols can be given code numbers, and a string ofthemcanalsobegivenauniquenumericalcode.Gödelnumberingachievesthisby turningastringofcodenumbersabcdef…intoasinglenumberdefinedbymultiplyingpowersofprimes:

2a3b5c7d11e13f…

Todecodeitbackintoastring,appealtouniquenessofprimefactorisation.There are other ways to encode symbol strings as numbers: this one is

mathematically elegant and totally impractical. But all Gödel needed was itsexistence.

Not only do statements encode as numbers: so do proofs, which are justsequencesofstatements.Thelogicalrulesfordeducingeachstatementfromthepreviousonesprovideconstraintsonwhichofthesenumberscancorrespondtoalogically valid proof. So the statement ‘P is a valid proof of statement S’ canitselfbethoughtofasastatementinarithmetic:‘ifyoudecodePintoasequenceofnumbers, thefinaloneis thenumbercorrespondingtoS’.Gödelnumberingletsuspassfromametamathematicalstatementabouttheexistenceofaprooftoanarithmeticalstatementaboutthecorrespondingnumbers.

Gödelwanted toplay thisgamewith thestatement‘thisstatement is false’.Hecouldn’tdoitdirectly,becausethatstatementisn’tarithmetical.Butitcanbemade arithmetical usingGödel numbers, and then it effectively becomes ‘thistheoremhasnoproof’.Therearesometechnicaltrickstomakeallthissensible,but that’s the gist of it. Suppose, now, thatHilbert is right, and the axiomaticsystemforarithmeticiscomplete.Then‘thistheoremhasnoproof’eitherhasaproof, or it doesn’t. Either way, we’re in trouble. If it has a proof, that’s acontradiction. If it has no proof, it’s false (we’re assuming Hilbert is right,remember?),soitdoeshaveaproof–anothercontradiction.Sothestatementisself-contradictory… and there’s a theorem in arithmetic that can neither beprovednordisproved.

Gödel quickly parlayed this result into his consistency theorem: if an

axiomatic formulation of arithmetic is consistent, then no proof of itsconsistencycanexist.Thisisthe‘yeah,yeah’pointinitsformalglory:ifanyoneeverfoundaproofthatarithmeticisconsistent,wecanimmediatelydeducethatit’snot.

For a while Hilbert and his followers hoped that Gödel’s theorems justindicated a technical deficiency of the particular axiomatic system set up inPrincipiaMathematica. Perhaps some alternative could avoid the trap. But itsoon became apparent that the same argumentworks inany axiomatic systemrichenoughtoformalisearithmetic.Arithmeticisinherentlyincomplete.Andifit’slogicallyconsistent,whichmostmathematiciansbelieveandallofusassumeasaworkinghypothesis,youcanneverproveit is. Inastroke,Gödelchangedhumanity’sentirephilosophicalviewofmathematics.Itstruthscan’tbeabsolute– because there are statements whose truth or falsity lies outside the logicalsystemaltogether.

We generally assume that an unsolved conjecture, like the RiemannHypothesis,iseithertrueorfalse,soeitherthere’saprooforthere’sadisproof.Post-Gödel,wemustadda thirdpossibility.Maybeno logicalpath leads fromtheaxiomsofset theory to theRiemannHypothesis,andno logicalpath leadsfromtheaxiomsofsettheorytothenegationoftheRiemannHypothesis.Ifso,there’snoproofthatit’strue,andnoproofthatit’sfalse.MostmathematicianswouldbetthattheRiemannHypothesisisdecidable.Infact,mostthinkit’strue,andthatonedayaproofwillbefound.Butifnot,surelyacounterexamplewillbefoundinstead,azerooffthecriticalline.Thepointis,wedon’tknowthat.Weassume that ‘sensible’ theorems either have proofs or disproofs, whileundecidableoneslookabitcontrivedandartificial.However,inthenextchapterwe’llseethatasensiblenaturalquestionintheoreticalcomputerscienceturnedouttobeundecidable.

Classical logic,with its sharpdistinctionbetween truthand falsity,withnomiddleground,istwo-valued.Gödel’sdiscoverysuggeststhatformathematics,athree-valuedlogicwouldbemoreappropriate:true,false,orundecidable.

23TheMachineStops

AlanTuring

AlanMathisonTuring

Born:London,23June1912Died:Wilmslow,Cheshire,7June1954

ACCORDING TO HIS COLLEAGUE Jack Good at Bletchley Park, Alan Turingsufferedfromhayfever.Hecycledintotheoffice,andeveryJuneheworeagasmask to protect him from the pollen. There was something wrong with hisbicycle,too,andeverysooftenthechaincameoff.SoTuringcarriedacanofoil

andaragtocleanupafterhereplacedit.Eventually,becomingtiredofputtingthechainbackon,hedecidedtotackle

theproblemrationally.Hestartedcountinghowmanytimesthepedalsrevolvedbetween one loss of the chain and the next. This number was remarkablyconstant. Comparing it to the number of links in the bicycle chain and thenumberofspokesinthebackwheel,hededucedthatthechainfelloffwheneverboth chain and wheel were in some particular configuration. He then kept arunningcount towarnhimwhen thechainwasabout tocomeoff,andcarriedout a manoeuvre that kept it on. He no longer needed to carry oil and rag.Eventuallyhediscoveredthataslightlybentspokewascomingintocontactwithadamagedlink.

Itwasatriumphofrationality,butanyoneelsewouldhavetakenthebiketoacycleshop,wherethefaultwouldquicklyhavebeenfound.Ontheotherhand,bynotdoingthat,Turingsavedthecostofarepair–andmadesurethatnooneelse could ride his bike.As in somany other things, he had his reasons; theywerejustdifferentfromeveryoneelse’s.

Alan Turing’s father Julius was a member of the Indian Civil Service. HismotherEthel(néeStoney)wasthedaughterofthechiefengineeroftheMadrasRailways.ThecouplewantedtheirchildrentobebroughtupinEngland,sotheymovedtoLondon.Alanwasthesecondoftwosons.Whenhewassix,hewentto school in the coastal townofStLeonard’s,where the headmistress quicklyrealisedhewasunusuallybright.

When he was 13, he attended Sherborne school, an independent ‘public’school, the quaint English term for a private fee-paying school frequentedmainlybychildrenoftherich.Likemostsuch,theschoolemphasisedclassics.Turing had bad handwriting, was poor at English, and even in his favouritesubject ofmathematics he preferred his own answers to those required by theteachers.Eitherdespiteorbecauseofthis,hewonallthemathematicsprizes.Healso enjoyed chemistry, but again preferred to plough his own furrow. Hisheadmasterwrote:‘Ifheis tobesolelyascientificspecialist,heiswastinghistimeatapublicschool.’

Tootrue.The school was unaware that in his spare time Turing was reading about

relativityfromEinstein’spapers,andquantumtheoryfromArthurEddington’sThe Nature of the Physical World. In 1928 he became close friends with

ChristopherMorcom,a studentoneyearhigher, and they sharedan interest inscience.Butwithintwoyears,Morcomwasdead.Turingwasdevastated,buthesoldieredondoggedly,winningaplacetostudymathematicsatKing’sCollege,Cambridge.Hecontinuedtoreadtextbooksthatwerewellaheadof,oroutside,theundergraduatecurriculum.Hegraduatedin1934.

Turing was incorrigibly scruffy. Even when he wore a suit it was seldompressed.He is said to have tied up his trousers using a necktie, or sometimesstring.Hislaughwasaloudbray.Hehadaspeechimpediment,notsomuchastammer as a sudden pause when he would say ah-ah-ah-ah-ah… whensearchinghismindforanappropriateword.Hewasn’tfastidiousaboutshavingand suffered from ‘five o’clock shadow’. He is often portrayed as a nervous,socially inept nerd, but hewas actually quite popular, and a goodmixer. Hisapparenteccentricitieslargelystemmedfromtheoriginalitynotjustofwhathethoughtabout,buthowhethought.Whenworkingonaproblem,Turingfoundanglesthatnooneelseknewexisted.

A year later he was taking a postgraduate course on the foundations ofmathematicsfromMaxNewman,wherehelearnedabouttheHilbertprogrammeanditsrefutationbyGödel.TuringrealisedthatGödel’sundecidabilitytheoremwasreallyaboutalgorithms.Aquestionisdecidableifthereexistsanalgorithmto answer it. You can prove that, for a given problem, by finding one.Undecidability is deeper and more difficult: you must prove that no suchalgorithm exists. It’s hopeless to attempt that unless you have a precisedefinitionofwhatanalgorithm is.Gödelhad ineffectdealtwith this issuebythinkingofanalgorithmasaproofwithinanaxiomaticsystem.Turingstartedthinkingabouthowtoformalisealgorithmsingeneral.

In1935hebecameaFellowofKing’sCollege,forhisindependentdiscoveryofthe central limit theorem inprobability,whichprovides some rationale for thewidespreaduseofthe‘bellcurve’,ornormaldistribution,instatisticalinference.But in 1936 his thoughts about Gödel’s theorems came to the fore, with thepublicationofhisseminalpaper‘Oncomputablenumbers,withanapplicationtothe Entscheidungsproblem’ (decision problem). In it, he proved anundecidabilitytheoremforaformalmodelofcomputation,nowcalledaTuringmachine. He proved that no algorithm can decide in advance whether acomputation will stop with an answer. His proof is simpler than Gödel’s,althoughbothrequirepreliminarymanoeuvrestosetupthecontext.

Although we speak of a Turing machine, the name refers to an abstractmathematical model representing an idealisedmachine. Turing called it an a-machine–‘a’for‘automatic’.Itcanbethoughtofasastripoftapedividedintoadjacent cells, which are either empty or contain a symbol. The tape is themachine’smemory,andits lengthisunlimitedbutfinite.Ifyougettotheend,addsomemorecells.Ahead,positionedoveraninitialcell,readsthesymbolinthatcell.Itthenconsultsatableofinstructions(programsuppliedbytheuser),writesasymbolinthecell(overwritinganythingalreadythere),andmovesthetapeonecellsideways.Then,dependingon the tableandsymbol, themachineeitherstops,orobeystheinstructionsinthetableforthesymbolinthecellithasmovedto.

There are many variants, but all are equivalent in the sense that they cancompute the same things. In fact, this rudimentary machine can in principlecompute anything that a digital computer, however fast and advanced, cancompute.Forexample,aTuringmachineusingthesymbols0–9andperhapsafew others can be programmed to calculate the digits of π to any specifiednumber of decimal places, writing them in successive cells of the tape, andfinallystopping.Thislevelofgeneralitymayseemsurprisingforsuchasimpledevice, but the intricacy of the computation is inherent in the table ofinstructions,whichcanbeverycomplicated,justastheactionsofacomputerareinherent in the software it’s running. However, the simplicity of a Turingmachine alsomakes it very slow, in the sense that even a simple computationinvolvesagiganticnumberofsteps.It’snotpractical,butitssimplicitymakesitwellsuitedtotheoreticalquestionsaboutlimitstocomputation.

Turing’s first important theoremproves theexistenceofauniversalTuringmachine, which can simulate any specific one. The program of the specificmachine is encoded on the universal machine’s tape, before the computationstarts.Thetableofinstructionstellstheuniversalmachinehowtodecodethesesymbols into instructions, and carry them out. The universal machine’sarchitectureisanimportantstepclosertoarealcomputer,withaprogramstoredinmemory.Wedon’tbuildanewcomputerforeachproblem,withahard-wiredprogram–exceptforsomeveryspecialapplications.

HissecondimportanttheoremdoesaGödel,provingthatthehaltingproblemforTuringmachinesisundecidable.Thisproblemasksforanalgorithmthatcandecide, given the program for a Turing machine, whether the machine will(eventually) stopwith an answer, or continue for ever. Turing’s proof that nosuchalgorithmexists–thatthehaltingproblemisundecidable–assumesthatit

does,andthenappliestheresultingmachinetoitsownprogram.However,thisiscunningly transformedso that thesimulationhalts ifandonly if theoriginalmachine doesn’t. This leads to a contradiction: if the simulation stops, then itdoesn’t;ifitdoesn’t,thenitdoes.WesawthatGödel’sproofultimatelyencodesa statementof the form ‘this statement is false’.Turing’s,which is simpler, ismorelikeacardbearingonitstwosidesthemessages:

Thestatementontheothersideofthiscardistrue.Thestatementontheothersideofthiscardisfalse.

Eachstatement,intwosteps,impliesitsownnegation.Turingsubmittedhispaper to theProceedingsof theLondonMathematical

Society, not realising that, a few weeks earlier, the American mathematicallogicianAlonzoChurch had published ‘An unsolvable problem in elementarynumber theory’ in the American Journal of Mathematics. This provided yetanother alternative to Gödel’s proof that arithmetic is undecidable. Church’sproof was very complicated, but he published first. Newman persuaded thejournal to publish Turing’s paper anyway, because it wasmuch simpler, bothconceptually and structurally. Turing revised it to cite Church’s paper, and itappeared in 1937. The tale had a happy ending, though, because Turing thenwenttoPrincetontotakeaPhDunderChurch.Histhesiswaspublishedin1939asSystemsofLogicBasedonOrdinals.

Notanauspiciousyear,1939marked thestartofWorldWar II.Realising thatwarwaslikely,andknowingthatmodernwarfarereliedheavilyoncryptography– secret codes – the head of the Secret Intelligence Service (SIS orMI6) hadboughtapropertysuitableforuseasacipherschool.BletchleyParkconsistedofa mansion built in a strange mixture of architectural styles, standing in 235hectares of grounds. The house had been scheduled for demolition, to build ahousingestate.Itstillexists,alongwithitsoutbuildings, includingsomeofthewartimehuts, andBletchleyPark isnowa tourist attraction themedaround itswartimecodebreakers.

CommanderAlastairDennison,headofoperationsoftheGovernmentCodeandCypherSchool(GC&CS),movedhistopcryptanalysts–codebreakers–toBletchley Park. They included chess players, crossword solvers, and linguists;one was an expert on Egyptian papyri. Seeking to expand their number, he

soughtout‘menoftheprofessortype’.TheAxisforceswereincreasinglyusingmachinestoencryptmessages,basedoncomplexsystemsofrotatingwheelsanddaily settings created by plug-in wires. Advanced technical knowledge wasthereforerequiredtoo,andthatmeantmathematicians.Severaljoinedtheteam,NewmanandTuringamong them.Theyworked in strictest secrecy, supportedbyclericalstaffandadministrators.Atitspeakinearly1945,BletchleyParkhadastaffof10,000.

Themainmachines used by the Axis forces were the Enigma and Lorenzciphers. Both cipher systems were thought to be unbreakable, but themathematicalstructureoftheencryptionalgorithmhadsubtleweaknesses.Thesewereexacerbatedwhenusersbroke the rulesand tookshortcuts, suchasusingthe same settings on consecutive days, sending the same message twice, orstartingmessageswithstandardwordsandphrases.Turingwasakeyfigure inthe team that was trying to break Enigma, working under Dilly Knox ofGC&CS. In 1939 thePolesmanaged to get their hands on aworkingEnigmamachine,andtoldtheBritishhowitworked–howtherotorswerewiredup.ThePolish cryptanalysts also developed methods for breaking the Enigma code,basedontheGermanhabitofprecedingcodemessageswithashortpieceoftextallowingtheoperatortotestthemachine.Forexample,amessagethatcontinueda previous one would often start with FORT (Fortsetzung, ‘continuation’),followedbythetimethefirstmessagewassent,repeatedtwiceandbracketedbythe letterY.ThePolishcryptanalysts inventedamachine, thebomba, tospeedthingsup.

TuringandKnox, realising that theGermanswouldprobablyeliminate thisflaw,soughtmorerobustmethodsofdecryption,anddecidedtheyalsoneededamachine,whichtheynamedthe‘bombe’.Turingdrewupthespecificationsforthe bombe,whichwould implement the same general technique of crib-baseddecryption.Thisisamethodthatcanbetriedwhentheplaintextversionofsomepartofamessagecanbeguessed–suchastheFORTsegment.Typicaltextsofthiskindwere theGermanversionsof ‘nothing to report’and‘weathersurvey[time]’.Amazingly,FieldMarshallErwinRommel’squartermasterstartedeverymessagetohimwiththeidenticalformalopeningphrases.

Turing’s design for the bombe was turned into hardware by an engineernamedHaroldKeen,whoworkedfortheBritishTabulatingMachineCompany(asortofBritishIBM).Thebombe’staskwastousehigh-speedtrialanderrortoidentifysomeofthebasicsettingsoftheEnigmamachine,whichwere(usually)changedeveryday.Itexaminedeachpossibilityinturn,seekingacontradiction.

If it found one, it went on to the next possibility, running through all 17,576combinations until it hit somethingplausible.At that point it stopped, and thesettings could be read off. Turing improved the process with some statisticalanalysis. He also tackled the more difficult version of Enigma used by theGermanNavy. In 1942 hewas seconded to the British Joint StaffMission inWashington DC to advise the Americans on bombes and their uses. Histechniques cut the number ofmachines required from 336 to 96, speeding uptheirimplementation.

TheabilitytodecryptAxiscommunicationscausedastrategicproblem:iftheenemyrealisedtheAlliescoulddothat,theywouldtightenuptheirprocedures.SoevenwhentheAlliesknewtheenemy’sintentions,anyactiontodefeatthemhad tobe indirect and infrequent.Usedwith cunningandmuchdeception, theAllies’ ability to decrypt enemy codemessages helped themwinmanymajorengagements, notably theBattle of theAtlantic.The efforts ofTuring and hiscolleaguesprobablyshortenedthewarbyfouryears.

After the end of thewar, it turned out that theGerman cryptanalystswereaware that the Enigma code could in principle be broken. They just hadn’tbelievedanyonewouldgototheimmenseamountofeffortrequiredtodoit.

Thecryptographicworkwasintenseandsustained,butlifeatBletchleyParkhadits lighter moments. Turing relaxed by playing sports and chess, and bysocialisingwithcolleaguesduring the limited timeallotted for thatpurpose. In1941 he became increasingly friendly with Joan Clarke, a brilliant femalemathematicianwhohadabandonedherstudiesforPartIIIof theMathematicalTriposatCambridgetojointheteamatBletchleyPark.Theywenttothecinematogether and generally enjoyed each other’s company. The relationship grewever closer, and eventually Turing proposed marriage. Joan immediatelyaccepted.

He had made her aware of his homosexual tendencies, but this failed todiscourage her, possibly because they had enough in common – chess,mathematics, cryptography… Few men in those days would have wanted amathematicalprodigyasawife,butthiswasn’tanissueforTuring.Neitherwashis homosexuality, at least, not to beginwith.At that time, respectabilitywasmoreimportant tomanypeople thansexualorientation,andawife’smainrolewasseentobethatofhousekeeper.Turingdid,however,allowJoantobelievethathishomosexualitywasonlyatendency,notactualsexualactivity.Theymet

each other’s parents, without any problems, and Turing bought her anengagementring.Joandidn’twearit towork,andamongtheircolleaguesonlyShaun Wylie officially knew they were engaged, but the others suspectedsomethingofthekind.

As the yearwent by, Turing began to have second thoughts. They spent aweek’s leave walking and cycling in North Wales, but the holiday ran intoproblemswithahotelbooking,andTuringhadforgottentoarrangeatemporaryration card to buy food. Shortly after their return, he decided that marriagewouldnotbeintheinterestsofeitherofthem,andtheengagementwasbrokenoff. He managed to do this without making Joan feel rejected, and theycontinuedtoworktogether,thoughlessfrequentlythanbefore.

Turingwasahigh-classathlete,specialisinginlong-distancerunning,wherehis lack of speed was more than compensated for by unusual stamina. As afellow at King’s College he frequently ran the 50-kilometre round trip fromCambridgetoElyandback,andduringthewarhewouldrunbetweenLondonandBletchleyPark formeetings. In1946 themagazineAthletics listedhimasthewinnerofWaltonAthleticClub’sthree-miletitle,in15minutes37·8seconds– a respectable time but nothing out of the ordinary. He did cross-countryrunning, and the followingyear came third in theKent20-mile road race in atimeof2hours,6minutes, and18 seconds– fourminutesbehind thewinner;thenfifthinanAAAmarathoninatimeof2hours,46minutes,and3seconds.The Club’s secretary wrote: ‘We heard him rather than saw him. Hemade aterriblegruntingnoisewhenhewasrunning,butbeforewecouldsayanythingtohim,hewaspastuslikeashotoutofagun.’In1948,whenBritainhostedtheOlympicGames,Turingcame fifth in the trials for theBritishmarathon team.Thegoldmedallist’stimewasonly11minuteslessthanTuring’spersonalbest.

AfterthewarTuringmovedtoLondon,andworkedonthedesignofoneofthefirst computers,ACE (AutomaticComputingEngine) at theNational PhysicalLaboratory. Early in 1946 he gave a presentation on the design of a stored-program computer – farmore detailed than theAmericanmathematician JohnvonNeumann’sslightlyearlierdesignforEDVAC(ElectronicDiscreteVariableAutomaticComputer). TheACEprojectwas slowed down by official secrecyaboutBletchleyPark,soTuringwentbacktoCambridgeforayear,writinganunpublishedarticleaboutmachineintelligence,hisnextgreattheme.In1948hebecameDeputyDirectoroftheComputingMachineLaboratoryattheUniversity

ofManchester, along with the post of reader (roughly equivalent to associateprofessor in the USA). In 1950 he wrote ‘Computing machinery andintelligence’, proposing the now famous Turing test for intelligence in amachine;basically,youcanhavea longconversationwith itonany topicyouwishandyouwon’tbeabletotellyou’renottalkingtoahuman(aslongasyoucan’t see it). Although controversial, this was the first serious proposal alongsuchlines.Healsostartedworkonachess-playingprogramforahypotheticalmachine.HetriedtorunitonaFerrantiMark1,butthememorywastoosmall,sohesimulatedtheprogrambyhand.Themachinelost.Butonly46yearslater,IBM’sDeepBlue beat chess grandmasterGaryKasparov, and a year later anupdatedprogramwonaseriesagainsthim3½–2½.Turingwasjustaheadofhistime.

From 1952 to 1954 he turned to mathematical biology, especiallymorphogenesis – the creation of form and patterns in plants and animals. Heworked on phyllotaxis, the remarkable tendency of plant structures to involveFibonaccinumbers2,3,5,8,13,andsoon,eachbeingthesumoftheprevioustwo. His biggest contribution was to write down differential equations thatmodel pattern formation. The underlying idea was that chemicals calledmorphogens lay down a cryptic ‘pre-pattern’ in the embryo, which acts as atemplateforthepatternsofcolouredpigmentthatappearasthecreaturegrows.Thepre-patterniscreatedbyacombinationofchemicalreactionsanddiffusion,inwhichmolecules spread fromcell tocell.Themathematicsof suchsystemsshows that they can form patterns by a mechanism known as symmetry-breaking,whichoccursiftheuniformstate(allchemicalconcentrationsthesameeverywhere)becomesunstable.Turingexplainedthiseffect:‘Ifarodishangingfromapointalittleaboveitscentreofgravityitwillbeinstableequilibrium.If,however, a mouse climbs up the rod, the equilibrium eventually becomesunstableandtherodstartstoswing.’Aswingingrodisinalesssymmetricstatethanonehangingvertically.

However, biologists came to prefer a different approach to the growth andformoftheembryo,knownaspositionalinformation.Hereananimal’sbodyisthoughtofasakindofmap,anditsDNAactsasaninstructionbook.Thecellsofthedevelopingorganismlookatthemaptofindoutwheretheyare,andthenatthebooktofindoutwhattheyshoulddoatthatlocation.Coordinatesonthemap are supplied by chemical gradients: for example, a chemical might behighlyconcentratednearthebackoftheanimalandgraduallyfadeawaytowardsthe front. By ‘measuring’ the concentration, a cell can work out where it is.

Evidencesupporting the theoryofpositional informationcame from transplantexperiments, inwhich tissue inagrowingembryo ismoved toanewlocation.For example, amouse embryo starts to develop a kind of striped pattern thateventuallybecomesthedigitsthatmakeupitspaws.Transplantingsomeofthetissue provides insight into the chemical signals it receives from surroundingcells. The experimental results were consistent with the theory of positionalinformation,andwerewidelyinterpretedasconfirmingit.

However, inDecember2012a teamof researchers ledbyRushikeshShethcarried out more complex experiments. They showed that a particular set ofgenesaffectsthenumberofdigitsthatthemousedevelops.Astheeffectofthesegenesdecreases,themousegrowsmoredigitsthanusual–likeahumanwithsixorsevenfingersinsteadoffive.Theirresultsareincompatiblewiththetheoryofpositionalinformationandchemicalgradients,butmakecompletesenseintermsofTuring’sreaction–diffusionapproach.InthesameyearagroupunderJeremyGreen showed that ridge patterns inside amouse’smouth are controlled by aTuring process.12 Themorphogens involved are FibroblastGrowthFactor andSonicHedgehog,socalledbecauselaboratoryfruitflieslackingtheflyversionhaveextrabristlesontheirbodies.

Turingwasgay,andin1952,whenhebeganarelationshipwithanunemployed19-year-oldnamedArnoldMurray,activehomosexualitywasillegal.AburglaryatTuring’shome,bysomeonewhoknewMurray,ledtoaninvestigationbythepolicewhichuncovered thehomosexual relationship.TuringandMurraywerecharged with gross indecency. On solicitor’s advice, Turing pleaded guilty,whileMurray received a conditional discharge. Turingwas given a choice ofimprisonment,orprobationaccompaniedbyhormonetreatmentwithasyntheticoestrogen.InProf:AlanTuringDecodedhisnephewDermotTuring,alawyer,argues that the sentencing was ‘procedurally flawed, partly illegal, andineffective’.Inparticular,othersprosecutedatthesametimeweretreatedmoreleniently,andthepersonwithwhomhecommittedtheoffenceeffectivelywentunpunished. Turing chose probation and hormone treatment, predicting: ‘Nodoubt I shall emerge from it all adifferentman,butquitewho I’venot foundout.’Hewasright.Hebecameimpotentanddevelopedbreasts.

The conviction seems to have been driven by official panic. The recentdiscoverythatGuyBurgessandDonaldMacleanwereKGBdoubleagentshadexacerbated fears about Soviet agents recruiting homosexuals as spies by

threatening to expose them. The Government Communications Headquarters(GCHQ), which had developed from GC&CS, promptly removed Turing’ssecurityclearance,and theUnitedStates refusedhimentry.SoAlanTuring,aman whose mathematical genius had shortened World War II by years (forwhich hewas awarded anOBE– he deserved a knighthood) becamepersonanongrataonbothsidesoftheAtlantic.

In June 1954 his housekeeper found his dead body. The post-mortemreportedcyanidepoisoningasthecause.Therewasapartiallyeatenapplebesidehim,andthiswasassumedtobethesourceofthecyanide,although–bizarrely–itwasn’t tested for the substance.TheCoroner’s verdictwas suicide.Anotherpossibility seems to have been ignored. Turing might have inhaled cyanidefumes from an electroplating experiment in his spare room.He usually ate anapplebeforegoingtobed,andoftenleftithalfeaten.Hehadshownnosignsofbeingdepressedbyhishormonetreatment,andhadjustmadealistoftasksheneededtoperformwhenhegotbacktotheofficeafterapublicholiday.Sohisdeathcouldhavebeenaccidental.

In2009,afteracampaignontheinternet,thePrimeMinisterGordonBrowngave a public apology for Turing’s ‘appalling’ treatment. Continuation of thecampaign led toaposthumouspardon in2013byQueenElizabeth II. In2016theBritish government announced that all gay and bisexualmen convicted ofnow-abolished sexual offences would be pardoned, in an amendment to thePolicing and Crimes Bill informally known as ‘Turing Law’. However, somecampaignerscontinuetoinsistonanapology,notapardon,onthegroundsthatapardonimpliescommissionofanoffence.

24FatherofFractals

BenoitMandelbrot

BenoitB.Mandelbrot

Born:Warsaw,Poland,20November1924

Died:CambridgeMA,USA,14October2010

DISRUPTIONCAUSEDBYWORLDWAR II delayed the 1944 entrance examinationsforParis’stwogreateducationalinstitutions,theÉcoleNormaleSupérieureandtheÉcolePolytechnique,by sixmonths.Theexaminations lastedamonthandwereextremelydifficult,butyoungBenoitMandelbrotcompletedboth.Oneofhisteachersdiscoveredthatoutofallthecandidates,justonehadansweredoneparticularly difficult mathematics question. He guessed it must have beenMandelbrot,andonasking,discoveredhewas right.The teacherconfided thathe’d found the problem impossible himself, because of a ‘truly horrible tripleintegral’whichlayattheheartofthecalculation.

Mandelbrot laughed. ‘It’s very simple.’Thenhe explained that the integralwasactuallythevolumeofasphere,indisguise.Ifyouusedtherightcoordinatesystem, it was obvious. And everyone knew the formula for the volume of asphere. That’s all there was to it. Once you saw the trick…Mandelbrot wasobviously right. Shocked, the teacher wandered off, muttering ‘Of course, ofcourse.’Whyhadn’thespottedthathimself?

Becausehe’dbeenthinkingsymbolically,notgeometrically.Mandelbrot was a natural geometer, with a strong visual intuition. After a

difficult childhood, as a Jew in occupied France in constant danger of beingarrestedbytheNazis,andmostlikelyendingupinadeathcamp,hecarvedoutforhimselfanunorthodoxbuthighlycreativemathematicalcareer, thecoreofwhich was spent as a Fellow at IBM’s Thomas J. Watson Laboratories inYorktownHeights,NewYork state.There,heproduceda seriesof articlesontopicsrangingfromthefrequenciesofwordsinlanguagestothefloodlevelsofrivers.Then, in aburstof inspiration,he synthesised thebulkof thesediverseandcuriousresearchesintoasinglegeometricconcept:thatofafractal.

The traditionalshapesofmathematics, suchasspheres,cones,orcylinders,have a very simple form. The closer you look, the smoother and flatter theyappear to be. The overall detail disappears, andwhat’s left looksmuch like afeaturelessplane.Fractals aredifferent.A fractalhasdetailed structureonanyscale of magnification. It’s infinitely wiggly. ‘Clouds are not spheres,’Mandelbrotwrote,‘mountainsarenotcones,coastlinesarenotcirclesandbarkis not smooth, nor does lightning travel in a straight line.’ Fractals captureaspects of nature that the traditional structures ofmathematical physics don’t.They’veledtofundamentalchangesinhowscientistsmodeltherealworld,withapplicationstophysics,astronomy,biology,geology,linguistics,globalfinance,and many other areas. They also have deep pure mathematical features, andstronglinkstochaoticdynamics.

Fractalsareoneofseveralareasofmathematicsthat,whilenotentirelynew,took off during the second half of the twentieth century, and changed therelationship between mathematics and its applications by providing newmethodsandviewpoints.Therootsoffractalgeometrycanbetracedbacktothesearch for logical rigour inanalysis, leading to the inventionaround1900ofavarietyof‘pathologicalcurves’whosemainrolewastoshowthatnaiveintuitivearguments can go wrong. For instance, Hilbert defined a curve that passesthrougheverypointinsideasquare–notjustcomesclose,buthitseverypointexactly.It’scalledaspace-fillingcurve,forobviousreasons,anditcautionsusto take care when thinking about the concept of dimension. A continuoustransformationcan increase thedimensionof a space,here from1 to2.OtherexamplesareHelgevonKoch’ssnowflakecurve,whichhasinfinitelengthbutenclosesfinitearea,andWacławSierpiński’sgasket,acurvethatcrossesitselfateverypoint.

However, those early works had little significance outside specialist areas,and were mainly seen as isolated curiosities. In order for a subject area to‘arrive’, someone has to pull the pieces together, understand their underlyingunity, formulate the requiredconcepts in sufficientgenerality, and thengooutand sell the ideas to the world. Mandelbrot, though by no means amathematicianin theorthodoxsense,hadthevisionandthetenacity todojustthat.

Benoit was born into an academic family of Lithuanian Jews, in Warsaw,between thewars.HismotherBella (néeLurie)was a dentist.His fatherKarlMandelbrojt,whohadhadnoformaleducation,madeandsoldclothes,buthissideof the family largelyconsistedof scholars,goingback forgenerations, soBenoitwasraisedinanacademictradition.KarlhadayoungerbrotherSzolem,who later became a distinguished mathematician. Because his mother hadalready lost one child through an epidemic, she keptBenoit out of school forseveral years, to avoid the possibility of infection. Another uncle, Loterman,taught him at home, but hewasn’t a very effective teacher. Benoit learned toplay chess, and listened to classical myths and stories, but did little else. Hedidn’t even learn the alphabet or his multiplication tables. He did, however,developanaptitudeforvisualthinking.Hischessmovesweredictatedmorebythe shapeof the game– the pattern of pieces on the board.He lovedmaps, apredilectionthatheprobablygotfromhisfather,whowasanavidmapcollector.

Theywerehungonall thewalls.Healsoreadanythinghecouldlayhishandson.

The family left Poland in 1936 as economic and political refugees. Hismother had been unable to continue as amedic and his father’s business hadcollapsed.TheymovedtoParis,wherehisfatherhadasister.LaterMandelbrotcreditedherwithsavingtheirlivesandhelpingthemwardoffdepression.

SzolemMandebrojt was moving up in the mathematical world, and whenBenoit was five his uncle became a professor at the University of Clermont-Ferrand. Eight years later he advanced to the position of professor ofmathematics at the Collège de France, Paris. Mandelbrot, impressed, beganthinkingaboutacareerinmathematicshimself,althoughhisfatherdisapprovedofsuchanimpracticaloccupation.

When Mandelbrot was in his teens, uncle Szolem took charge of hiseducation.HewenttotheLycéeRolininParis.ButoccupiedFrancewasabadplaceand time tobeaJew,andhischildhoodwasmarkedbypovertyand theconstantthreatofviolenceordeath.In1940,thefamilyfledagain,thistimetothetinytownofTulleinsouthernFrancewherehisunclehadacountryhouse.ThentheNazisoccupiedsouthernFranceaswell,andMandelbrotspentthenexteighteenmonths evading capture.Hedescribed this period of his life in bleakterms:13

For somemonths Iwas in Périgueux as apprentice toolmaker on the railroads. For later use inpeacetime,theexperiencewasbetterthananotherwartimestintashorsegroom,butIdidnotlookortalklikeanapprenticeorgroomand,atonepoint,narrowlyescapedexecutionordeportation.SomegoodfriendseventuallyarrangedforadmissiontotheLycéeduParc,inLyons.Whilemuchof theworldwas in turmoil, itwas almost business as usual in a class preparing for the fearedexaminations of the French elite universities called ‘Grandes Écoles’. The few months thatfollowedinLyonswereamongthemostimportantofmylife.StarkpovertyanddeepfearoftheGermanbossofthecity(welaterdiscoveredhisnametobeKlausBarbie)tiedmetomydeskformostofthetime.

Barbie was a Hauptsturmführer in the dreaded Schutzstaffel (SS, literally‘protection squad’) and a member of the Gestapo (secret police). He becameknownastheButcherofLyonfortorturingFrenchprisonersinperson.AfterthewarhefledtoBolivia,butwasextraditedtoFrancein1983andimprisonedforcrimesagainsthumanity.

AtLyonin1944,studyingmathematics,Mandelbrotdiscoveredthathehadahighdegreeofvisual intuition.Whenhis teacherposedsomedifficultproblem

in symbolic form, such as an equation, he instantly transformed it into ageometricequivalent,whichwasusuallymucheasiertosolve.Hewasadmittedto theÉcoleNormaleSupérieure inParis, to studymathematics.However, themathematicalstylethatwaspractisedtherewasverymuchthatoftheBourbakischool–abstract,general,focusedonpuremathematics.Hisunclehadasimilarmathematicalphilosophy,andhadbeenanearlymemberofBourbakibeforethegroup began its systematic revision ofmathematics on rigorous abstract lines.This formal style of mathematical thinking, without pictures or concreteapplications, did not appeal to Mandelbrot. After a few days at the ÉcoleNormale,hedecidedhewasinthewrongplace,andresigned.Instead,hetookup a place in themore practically oriented École Polytechnique (he’d alreadypassed the entrance examination for that, along with the exam for the ÉcoleNormale).Herehehadmuchmorefreedomtostudydifferentdisciplines.

His uncle continued to push him towards more abstract mathematics, andsuggestedthatMandelbrotshouldchooseaPhDtopicrelatedtoworkofGastonJuliaoncomplexfunctions,whichhadbeenpublishedin1917.Thissuggestiondidnotappeal.WhenacceptingtheWolfPrize,Mandelbrotlaterwrote:14

Myuncle’sbelovedTaylorandFourierserieshadstartedcenturiesagointhecontextofphysics,but in the 20th century developed into a field self-described as ‘fine’ or ‘hard’ mathematicalanalysis. Inmyuncle’s theorems, theassumptionscould takepages.Thedistinctionsheenjoyedwere so elusive that no condition was both necessary and sufficient. The long pedigree of theissues,forhimasourceofpride,wasfortheyoungermeasourceofaversion.

Oneday,stillseekinga topic,MandelbrotaskedSzolemforsomethingtoreadon the metro. His uncle remembered having thrown an article into thewastebasket, and fished it out, saying that it was ‘crazy, but you like crazythings’.ItwasareviewofabookbythelinguistGeorgeZipf,aboutastatisticalpropertythatwascommontoalllanguages.Nooneseemedtounderstandwhatit was about, but Mandelbrot decided on the spot that he would explain thisproperty,nowcalledZipf’slaw.Hemadesomeprogress,aswe’llsoonsee.

From1945to1947MandelbrotstudiedunderPaulLévyandGastonJuliaatthe École Polytechnique, and then went to the California Institute ofTechnology, obtaining amaster’s degree in aeronautics.He thenwent back toFrance, getting a PhD degree in 1952. He was also employed at the CentreNational de la Recherche Scientifique. He spent a year at the Institute forAdvancedStudy inPrinceton,New Jersey, under the sponsorshipof JohnvonNeumann. In 1955 he married Aliette Kagan and moved to Geneva. After

severalvisitstotheUSAtheMandelbrotsmovedtherepermanentlyin1958,andBenoitworkedasanIBMresearcherinYorktownHeights.HeremainedatIBMfor35years,becominganIBMFellowandthenaFellowEmeritus.Hereceivednumerous awards, including the Légion d’Honneur (1989), the Wolf Prize(1993),andtheJapanPrize(2003).HisbooksincludeFractals:Form,Chance,andDimension(1977)andTheFractalGeometryofNature(1982).Hediedofcancerin2010.

TheworkonZipf’slawsetthepatternofMandelbrot’sfuturecareer,whichforalongtimeseemedtobeaseriesofapparentlyunrelatedinvestigationsofstrangestatistical patterns, hopping like a butterfly from oneweird flower to another.OnlywhenhewasatIBMdiditallstarttocometogether.

Zipf’slawintroducedhimtoasimplebutuseful(andunderestimated)ideainstatistics, that of a power-law relationship. In one standard compilation ofAmericanEnglish,thethreemostcommonwordsare:

the,occurring7percentofthetime,of,occurring3.5percentofthetime,and,occurring2.8percentofthetime.

Zipf’s law states that the nth word (ranked by the frequency with which itoccurs)isthefrequencyofthefirstword,dividedbyn.Here7/2=3·5and7/3=2·3. The latter figure is lower than observed, but the law isn’t perfect, it justquantifiesageneraltendency.Herethefrequencyofthenthwordintherankingisproportionalto1/n,whichwecanwriteasn−1.Otherexamplesshowsimilarpatterns, but with a power that differs from −1. For example, in 1913 FelixAuerbach noticed that the size distribution of cities follows a similar law butwiththepowern−1·07.Ingeneral, if therank-n itemhasfrequencyproportionaltonc,forsomeconstantc,wespeakofacthpowerlaw.

Classical statistics pays little attention to power-law distributions, focusinginsteadonthenormaldistribution(orbellcurve),foravarietyofreasons,somegood.Butnatureoftenseemstousepower-lawdistributionsinstead.LawslikeZipf’s apply to the populations of cities, the numbers of people watching aselectionofTVshows,andhowmuchmoneypeopleearn.Thereasonsforthisbehaviour are still not fully understood, but Mandelbrot made a start in histhesis,15 and Wentian Li has offered a statistical explanation: in a language

where each letter in the alphabet (plus a space character to separate words)appears with the same frequency, the distribution of words by rank obeys anapproximation toZipf’s law.VitoldBelevitchproved that the samegoes for avariety of statistical distributions. Zipf’s own explanation was that languagesevolve over time to provide optimal understanding for the least effort (inspeakingorlistening),andthepower−1emergesfromthisprinciple.

Subsequently,Mandelbrotpublishedpapersonthedistributionofwealth,thestockmarket,thermodynamics,psycholinguistics,thelengthsofcoastlines,fluidturbulence,populationdemographics, thestructureoftheuniverse, theareasofislands,thestatisticsofrivernetworks,percolation,polymers,Brownianmotion,geophysics, random noise, and other disparate topics. It all seemed a bitdisjointed.Butin1975everythingcametogetherinaflashofinsight:therewasacommonunderlyingthemetoalmostallofhiswork.Anditwasgeometric.

Thegeometryofnaturalprocessesseldomfollowsthestandardmathematicalmodelsof spheres, cones, cylinders, andother smooth surfaces.Mountainsarejagged and irregular. Clouds are fluffy with bulges and wisps. Trees branchrepeatedly,fromtrunktoboughtotwig.Fernshavefrondsthatlooklikealotofsmaller frondsstrung together inoppositepairs.Underamicroscope, soot isalot of tinyparticles clumped together,withgaps andvoids.They’re all a longway from the smooth rotundity of a sphere.Nature abhors a straight line, andshe’snottookeenonmuchelsefromEuclidandthecalculustexts.Mandelbrotcoined a name for this type of structure: fractal. And he energetically andenthusiastically promoted the use of fractals in science, to model many ofnature’sirregularstructures.

‘Model’ is a key word here. The Earth may appear roughly spherical –ellipsoidalifyouwantmoreaccuracy–andthoseshapeshavehelpedphysicistsandastronomersunderstandsuchthingsastidesandthetiltoftheplanet’saxis,but themathematical objects aremodels, not reality itself. They capture somefeatures of the natural world in an idealised form, simple enough for humanbrainstoanalyse.ButthesurfaceoftheEarthisroughandirregular:themapisnottheterritory.Norshoulditbe.AmapofAustraliacanbefoldedandputinyour pocket, ready for usewhen needed, but you can’t do thatwithAustraliaitself.Amapshouldbesimplerthantheterritory,butprovideusefulinformationabout it.Amathematical sphere is perfectly smooth nomatter howmuch youmagnify it, but reality turns intoquantumparticles at the atomic level.This isirrelevant to a planet’s gravitational field, however, so it can and should beignored in that context. Water can profitably be modelled as an infinitely

divisiblecontinuum,even thoughrealwaterbecomesdiscretewhenyouget tothelevelofmolecules.

It’sthesamewithfractals.Amathematicalfractalisn’tjustarandomshape.Ithasdetailedstructureonallscalesofmagnification.Often,ithasvirtuallythesamestructureonallscales.Suchashapeissaidtobeself-similar.Inafractalmodelofafern,eachfrondismadeofsmallerfronds,whichinturnaremadeofevensmallerones,andthisprocessneverstops.Inarealfern,itstopsafterfouror five stages, atmost.Nevertheless, the fractal is a bettermodel than, say, atriangle.JustasanellipsoidcanbeabettermodeloftheEarththanasphere.

Mandelbrotwasveryawareof theprominentroleofPolishmathematiciansin the prehistory of fractals, a highly abstract approach to analysis, geometry,and topologydevelopedby a small coterie ofmathematicians,manyofwhommet regularly in theScottishCafé inLvov (Lwów, nowLviv).They includedStefanBanach,whofoundedfunctionalanalysis,andStanisławUlam,whowasheavilyinvolvedintheManhattanProjecttobuildanatomicbomb,andcameupwith the main idea for the hydrogen bomb. Wacław Sierpiński, at WarsawUniversity,wasofalikemind,andheinventedashapethatwas‘simultaneouslyCantorianandJordanian,ofwhicheverypoint isapointoframification’.Thatis,acontinuouscurvethatcrossesitselfateverypoint.

Thefirstfewstagesin

constructingaSierpiński

gasket

LaterMandelbrotjokinglycalledthisshapetheSierpińskigasketbecauseofitsresemblancetothemany-holedsealthatjoinsthecylinderheadofacartotheengine.RecallthattheSierpińskigasketisoneofasmallzooofexamplesthatcame into being in the early twentieth century, collectively known aspathological curves – although they’re not pathological to nature, or even tomathematics: they just seemed that way to themathematicians of that period.Patterns like the gasket appear on seashells. Anyway, the gasket can be

constructedbyaniterativeprocedure,appliedtoanequilateraltriangle.Divideitinto four congruent equilateral triangles, each half the size.Delete the centraltriangle,which isupsidedown.Apply thesameprocess to the threeremainingtriangles, and repeat indefinitely. The gasket is what remains when all theupside-downtriangles,butnottheiredges,aredeleted.

Mandelbrot took inspiration from such curves, now seen as early fractals.Laterhefoundthisamusing:16

MyuncleleftforFranceagedabouttwenty,arefugeedrivenbyanideologythatwasnotpoliticaloreconomicbutpurelyintellectual.Hewasrepelledbythe‘Polishmathematics’,thenbeingbuiltup as amilitantly abstract field byWacław Sierpiński (1882–1969). By profound irony,whoseworkwastobecomeafertilehuntinggroundwhen,muchlater,Ilookedfortoolstobuildfractalgeometry?Sierpiński!Fleeing[Sierpiński’s]ideology,myunclejoinedtheheirsofPoincaréwhoruledParisinthe1920s.Myparentswerenotideologicalbuteconomicandpoliticalrefugees;theirjoining my uncle in France later saved our lives. I never met Sierpiński but his (unwitting)influenceonmyfamilyhadnoequal.

A few pure mathematicians, following up such notions, discovered that thedegreeof roughnessofa fractalcanbecharacterisedbyanumber,which theycalled its ‘dimension’ because it agreeswith the usual dimension for standardshapeslikealine,asquarewithitsinterior,orasolidcube,whichrespectivelyhavedimensions1,2,and3.However,thedimensionofafractalneednotbeawhole number, so the interpretation ‘how many independent directions’ nolonger applies. Instead, what matters is how the shape behaves undermagnification.

If you make a line twice the size, its length multiplies by 2. Doubling asquaremultipliesitsareaby4,anddoublingacubemultipliesitsvolumeby8.Thesenumbersare21,22,and23:thatis,2raisedtothepowerofthedimension.Ifagasketisdoubledinsize,itcanbesplitintothreecopiesoftheoriginal.So2raised to the power of the dimension ought to equal 3. The dimension istherefore log 3 /log 2, which is about 1·585. A more general definition, notconfined toself-similar fractals, iscalled theHausdorff–Besicovichdimension,and a more practical version is called the box dimension. The dimension isuseful inapplications,and isoneway to test fractalmodelsexperimentally. Inthismanner, for example, it hasbeen shown that clouds arewellmodelledbyfractals,with a dimension of a photographic image (projection into the plane,easiertoworkwithandmeasure)beingroughly1·35.

A final irony illustrates the danger of passing snap value judgements inmathematics.In1980,seekingnewapplicationsoffractalgeometry,MandelbrottookanotherlookatJulia’s1917paper,theonehisunclehadrecommendedandhehad rejected as too abstract. Julia, and anothermathematicianPierreFatou,hadanalysedthestrangebehaviourofcomplexfunctionsunderiteration.Thatis,startwithsomenumber,applythefunctiontothattogetasecondnumber,thenapply the function to that to get a third number, and so on, indefinitely.Theyfocused on the simplest nontrivial case: quadratic functions of the form f(z) =z2+c for a complex constant c. The behaviour of thismap depends on c in acomplicated manner.17 Julia and Fatou had proved several deep and difficulttheorems about this particular iteration process, but it was all symbolic.Mandelbrotwonderedwhatthepicturelookedlike.

Left:TheMandelbrotset.Right:Magnificationofpartofit.

Thecalculationswerefartoolengthytocarryoutbyhand,whichisprobablywhyJuliaandFatouhadn’t investigated thegeometryof theprocess.Butnowcomputers were starting to gain real power, and Mandelbrot was working atIBM.Soheprogrammedacomputertodothesumsanddrawthepicture.Itwasmessy(theprinterwasrunningoutofink)andcrude,butitrevealedasurprise.ThecomplicateddynamicsofJuliaandFatouisorganisedbyasinglegeometricobject–andit,ormorepreciselyitsboundary,isafractal.Thedimensionoftheboundary is 2, so it’s ‘almost space-filling’. We now call this fractal theMandelbrot set, a name coined by Adrien Douady. As always, there wereprediscoveriesandcloselyrelatedwork; inparticular,RobertBrooksandPeterMatelski drew the same set in 1978. The Mandelbrot set is the source ofcomplexandbeautifulcomputergraphicimages.It’salsothesubjectofintensemathematicalstudy,leadingtoatleasttwoFieldsmedals.

So the abstract pure-mathematical paper that Mandelbrot had initiallyrejected turned out to contain an idea that became central to the theory of

fractals,whichhehaddevelopedpreciselybecauseofitslackofabstractionandits links with nature.Mathematics is a highly connected whole, in which theabstractandtheconcretearelinkedbysubtlechainsoflogic.Neitherphilosophyissuperior.Thebigbreakthroughsoftencomebyusingboth.

25OutsideIn

WilliamThurston

WilliamPaulThurston

Born:WashingtonDC,30October1946

Died:RochesterNY,21August2012

MATHEMATICIANS LIKE NOTHING BETTER than to talk to othermathematicians –abouttheirwork,inthehopeoflearningofanewideatohelpwiththeirlatestproblem;aboutthenewThairestaurantthat’sjustopenedupontheedgeofthecampus;aboutfamilyandmutualfriends.Theygenerallydothissittingaround

insmallgroupsdrinkingcoffee.AsAlfrédRényioncesaid,‘Amathematicianisamachine for turning coffee into theorems.’ It’s a pun inGerman,where thewordSatzmeansboth‘theorem’and‘(coffee)grounds’.

Theseinformaldiscussionsoftenoccurinamoreformalcontext–aseminar(a technical lecture for specialists), a colloquium (a less technical lectureostensibly for professionals or graduate students who may be working in adifferent area, although sometimes it’shard to tell thedifference), aworkshop(smallish specialised conference), a sandpit (smaller and less formal), or aconference (biggerandpossiblybroader). InDecember1971 theUniversityofCaliforniaatBerkeleyhostedaseminarondynamicalsystems.Thishadrecentlybecome a hot topic because Stephen Smale, Vladimir Arnold, and theircolleagues and students in Berkeley and Moscow were continuing wherePoincaréleftoffwhenhediscoveredchaos,devisingnewtopologicalmethodstotackleapparentlyintractableoldproblems.Adynamicalsystemisanythingthatdevelops over time according to specific non-random rules. The rules for acontinuous dynamical system are differential equations, which determine thestate of the system a tiny instant into the future in terms of its current state.There’s an analogous concept of a discrete dynamical system, in which timeticks in isolated instants, 1, 2, 3,… The seminar speaker presented abreakthrough solution of a problem that boiled down to looking at a finitenumberofpointsintheplane.Thespeakerexplainedakeytrick:howtomoveany given number of points to new positions, not too far away, so that theydidn’tstraytoofaratanystageofthemotion.(Afewotherconditionshadtobeobeyed too.) This theorem was easy to prove for spaces of three or moredimensions, but now a long-sought proof for two dimensions had, it wasclaimed,beenfound.Lotsofinterestingresultsindynamicsfollowed.

At the back of the room sat a shy young graduate student, looking like ahippiewithathickbeardandlonghair.Hestoodup,andratherdiffidentlysaidhedidn’tthinktheproofwascorrect.Advancingtotheblackboard,hedrewtwopictures,eachshowingsevenpointsintheplane,andbegantousethemethodsexplainedinthetalktomovethepointsofthefirstconfigurationtothepositionsinthesecondone.Hedrewthepathsalongwhichthepointsweresupposedtomove,andthesestartedtogetinthewayofeachother,requiringthenextpathtomake a longer excursion to avoid the obstacle,which in turn created an evenlonger obstacle. As curves sprouted like the heads of the mythical hydra, itbecame clear that the student was right. Dennis Sullivan, who was present,wrote:‘Ihadneverseensuchcomprehensionandsuchcreativeconstructionofa

counterexample done so quickly. This combined with my awe at the sheercomplexityofthegeometrythatemerged.’

ThestudentwasWilliamThurston–‘Bill’ tofriendsandcolleagues.Therearedozensofsimilarstoriesabouthim.Hehadanaturalintuitionforgeometry,especiallywhenitgotreallycomplicated,andthenewlydevelopinggeometryofmanydimensions–four,five,six,younameit–providedamplescopefortheexerciseofhisastonishingabilitytoconverttechnicalproblemsintovisualform,and then solve them. He had a knack for seeing through the complexity anduncovering simple underlying principles. He became one of the leadingtopologists of his age, solvedmany important problems, and introduced a fewkey conjectures of his own that defeated even his prodigious talents. BillThurston, a truly significant figure of modern pure mathematics, is a worthyrepresentativeofthisesotericspecies.

Ironically,Thurstonhadpooreyesight.Hesufferedfromcongenitalstrabismus–hewas‘cross-eyed’,andcouldnotfocusbotheyesonthesamenearbyobject.Thisaffectedhisdepthperception,so thathefound itdifficult toworkout theshapeofathree-dimensionalobjectfromatwo-dimensionalimage.HismotherMargaret (neé Martt) was a proficient seamstress, able to sew patterns socomplicated that neither Thurston nor his father Paul could understand them.Paul was an engineer-physicist at Bell Labs with a penchant for hands-onconstructionofgadgets.And,ononeoccasion,handsin:heshowedyoungBillhow to boil water with your bare hands. (Use a vacuum pump to lower theboilingpointtojustaboveroomtemperature;thenstickyourhandsintowarmitup.)TohelpcombatBill’sstrabismus,Margaretspenthourswithhimwhenhewastwoyearsold, lookingatbooksfullofcolouredpatterns.His later loveofpatterns, and his facility as a handyman, probably stem from these earlyactivities.

Hisearlyschoolingwasunusual.NewCollegeFloridatookasmallnumberof students, selected for outstanding ability, and imposed very few limits onwhat they studied,or evenwhere they lived.Onoccasion,Thurston lived in atent in thewoods; at other timeshe slept in the school buildings, dodging thejanitor. The school nearly collapsed after eighteen months when half of itsteachersresigned.HisuniversitydaysatBerkeleyweremoreorganised,butthattoo was a turbulent time, with much student opposition to the Vietnam war.Thurstonjoinedacommitteethatwastryingtopersuademathematiciansnotto

acceptmilitaryfunding.HehadbythenmarriedRachelFindley,andtheirfirstchild was born. The birth, Rachel said, was in part intended to make sureThurston wasn’t drafted into the army. Labour coincided with his PhDqualifyingexamination,andThurston’sperformancewaserratic,but,asalways,original.HisPhDthesiswasonsomespecialproblemsinthethen-hottopicoffoliations,inwhichamultidimensionalspace(ormanifold)isdecomposedintoclosely fitting ‘leaves’, like a book decomposes into pages but with lessregularityintheirarrangement.Thistopicisrelatedtothetopologicalapproachto dynamical systems. The thesis contains several important results, but hasneverbeenpublished.FoliationswereThurston’sfirstmainareaofresearch,andhecontinuedworkingonthemattheInstituteforAdvancedStudyinPrincetonin 1972–73, and atMIT in 1973–74. Indeed, he solved somany of the basicproblemsoftheareathatasfarasothermathematicianswereconcerned,intheendheprettymuchkilleditoffcompletely.

In1974ThurstonbecameaprofessoratPrincetonUniversity(nottobeconfusedwiththeInstituteforAdvancedStudy,whichhasnostudents).Afewyearslaterhis research moved into one of the most difficult areas of topology, three-dimensional manifolds. These spaces are analogous to surfaces, but have oneextradimension.TheirstudygoesbackmorethanacenturytoPoincaré(Chapter18),butuntilThurstongotinvolvedtheyhadalwaysseemedratherbaffling.Thetopologyofhigher-dimensionalmanifoldsiscurious.Theeasiestdimensionsareone(trivial)andtwo(surfaces,solvedclassically).Thenexteasiestturnedouttobedimensionfiveorhigher,mainlybecausehigh-dimensionalspaceshavealotof room for performing complicatedmanoeuvres.Even then, theproblems arehard.Harderstillarefour-dimensionalmanifolds,andthehardestofallarethethree-dimensionalones–enoughroomforhugecomplexity;notenoughroomtosimplifyitinanystraightforwardway.

Astandardwaytoconstructn-dimensionalmanifoldsistotakelittlepatchesof n-dimensional space and prescribe rules for gluing them together.Conceptually,notinreality.InChapter18wesawhowthisapproachworksforsurfacesand3-manifolds(here).Wealsoencounteredafundamentalquestionin3-manifold topology, the Poincaré conjecture. This characterises the three-dimensionalsphereintermsofasimpletopologicalproperty:allloopsshrinktoapoint.Astandardwaytosneakuponsuchquestionsis togeneralisethemtohigher-dimensional analogues. Sometimes a more general question is easier;

then you read off the special case you started from. Initial progress wasencouraging. In 1961 Stephen Smale proved the Poincaré conjecture in alldimensions greater than or equal to 7. Then John Stallings polished offdimension 6 andChristopherZeeman did dimension 5.Theirmethods fail fordimensions3and4,andtopologistsbegantowonderwhetherthosedimensionsbehave differently. Then, in 1982, Michael Freedman found an extremelycomplicated proof of the four-dimensional Poincaré conjecture using radicallydifferenttechniques.AtthisstagetopologistshadprovedthePoincaréconjecturein every dimension except the one Poincaré had originally asked about. Theirmethodsshednolightwhatsoeveronthatfinal,resistantcase.

EnterThurston,turningthewholesubjectonitshead.Topologyisrubber-sheetgeometry,andPoincaré’squestionwastopological.

Naturally, everyonehadbeen attacking it using topologicalmethods.Thurstonthrew away the rubber sheet and wondered whether the problem was reallygeometric.He didn’t solve it, but his ideas inspired a youngRussian,GrigoriPerelman,todojustthat,someyearslater.

Recall (Chapter 11) that there are three kinds of geometry: Euclidean,elliptic, and hyperbolic. These are respectively the natural geometry of spaceswithzerocurvature,constantpositivecurvature,andconstantnegativecurvature.Thurstonbeganwithacuriousfact,whichlooksalmostaccidental.Herevisitedthe classification of surfaces – sphere, torus, 2-torus, 3-torus, and so on, as inChapter18–askingwhatkindsofgeometryshowup.Thespherehasconstantpositivecurvature,soitsnaturalgeometryiselliptic.Onerealisationofthetorusistheflattorus,asquarewithoppositeedgesidentified.Asquareisaflatobjectintheplane,soitsnaturalgeometryisEuclidean,andthegluingrulesgivetheflattorusthesametypeofgeometryasthesquare.Finally,althoughthisislessobvious,everytoruswithtwoormoreholeshasanaturalhyperbolicgeometry.Somehow,theflexibletopologyofsurfacesreducestorigidgeometry–andallthreepossibilitiesoccur.

Of course, surfaces are very special, but Thurston wondered whethersomething similar works for 3-manifolds. With his amazing intuition forgeometry he quickly realised that it couldn’t be as simple. Some three-dimensionalmanifolds,suchastheflat torus,areEuclidean.Some,suchasthe3-sphere, are elliptic.Somearehyperbolic.Butmost arenoneof these things.Undaunted, he investigatedwhy, and found two reasons. First, there are eightsensible geometries for 3-manifolds. One, for instance, is analogous to acylinder: flat in some directions, positively curved in the others. The second

obstacleismoreserious:many3-manifoldsarestillnotaccountedfor.However,whatseemedtoworkwasakindofjigsaweffect.Every3-manifoldseemedtobebuiltfrompieces,suchthateachpiecehadanaturalgeometryselectedfromthoseeightpossibilities.Moreover,thepiecesweren’tanyoldpiece:theycouldbechosen to fit together ina fairlystringentway.These ideas ledThurston tostate,in1982,hisGeometrisationConjecture:everythree-dimensionalspacecanbe cut up, in an essentially uniquemanner, into pieces, each having a naturalgeometric structure given by one of his eight geometries. The PoincaréConjecturefor3-manifoldsisasimpleconsequence.Buttherethematterstuck.The Clay Mathematics Institute made the Poincaré conjecture one of itsMillenniumPrizeproblems,withonemilliondollarsasarewardforaproof.

In 2002,Perelmanput a preprint on awebsite called the arXiv (‘archive’),aboutsomethingcalledtheRicciflow.ThisideaisrelatedtoGeneralRelativity,in which gravity is an effect of the curvature of space-time. Earlier RichardHamiltonhadwonderedwhethertheRicciflowmightgiveasimpleproofofthePoincaréconjecture.The ideawas tostartwithahypothetical3-manifoldsuchthat every closed curve shrinks to a point. Think of it as a curved three-dimensional space in Einstein’s sense – an idea that originated in Riemann’sHabilitationthesis(Chapter15).

Nowcomes thecleverbit: try to redistribute thecurvature tomake itmoreeven.

Imaginetryingtoironashirt.Ifyou’renotcarefulputtingitontheironing-board, it has lots of bumps and ridges. These are regions of high curvature.Elsewheretheshirtisflat–zerocurvature.Youcantrytoflattenoutthebumpsbyironingthem,butclothdoesn’tstretchorcompressterriblywell,soeitherthebumps move somewhere else, or you create wrinkles. A simpler and moreeffectivemethod,which stops the bumpsmoving around or reappearing, is tograb theedgesof theshirtandstretch itout.Then thenaturaldynamicsof theclothflattensoutthebumps.TheRicciflowdoesthesamesortofthingfora3-manifold. It redistributes the curvature from regions of high curvature intoregionsoflowercurvature,asifthespaceistryingtoevenitscurvatureout.Ifeverything works nicely, the curvature keeps flowing until it’s the sameeverywhere.Maybe the result is flat, maybe not, but either way its curvatureoughttobethesameateverypoint.

Hamiltonshowedthatthisideaworksintwodimensions:abumpysurfaceonwhicheveryclosedcurveshrinkstoapointcanbeironedoutbyitsRicciflowuntilitendsupwithconstantpositivecurvature,whichimpliesthatit’sasphere.

But in three dimensions there are obstacles, and the flow can get stuckwherebitsofthemanifoldbunchtogetherandcreatewrinkles.Perelmanfoundawaytogetroundthis–basically,bycuttingoffthatbitofshirt,ironingitseparately,and sewing it back in. His preprint and a follow-up claimed that thismethodprovesboththePoincaréConjectureandThurston’sGeometrisationConjecture.

Usually, claims to have solved some huge conjecture initially meet withscepticism.Mostmathematicianshavefoundpromisingproofsoftheirown,forsomedifficultproblemthatintereststhem,onlytodiscoverasubtlemistake.Butevenearlyon,therewasageneralfeelingthatPerelmanmightwellhavecrackedit. His method for proving the Poincaré Conjecture looked plausible; theGeometrisation Conjecture was perhaps more troublesome. However, aconsensual feeling isn’t enough: the proof must be checked. And the ArXivversion–allthatexisted–leftalotofgapsforreaderstofillin,onthegroundsthat these stepswere obvious.Actually, filling in gaps and checking the logictookseveralyears.

Perelman was extraordinarily talented, and what seemed obvious to himwasn’tanythinglikeasobvioustothosemathematicianswhotriedtocheckhisproof.Tobefair,theyhadn’tbeenthinkingabouttheproblemthewayhehad,orforanything likeas longashehad,whichput thematadisadvantage.Hewasalso a bit reclusive, and as time went by and no one had yet pronounceddefinitivelyonwhateventuallyturnedout tobeepic,groundbreakingwork,hebecame annoyed and disillusioned. By the time his proof was accepted, he’dabandonedmathematics altogether. He refused themillion-dollar prize, whichwas offered to him even though technically he hadn’t complied with theconditionsbecausehisproofhadn’tbeenpublishedinarecognisedjournal.Herejected the award of a Fields medal, generally considered the mathematicalequivalentofaNobelPrize, thoughcarryingamuchsmallermonetaryreward.EventuallytheClayInstituteusedtheprizemoneytosetupashort-termpositionforoutstandingyoungmathematiciansattheInstitutHenriPoincaréinParis.

Todaymanymathematiciansusecomputers,notjustforemailandtheweb,noteven just for big numerical calculations, but as a tool to help them exploreproblems in an almost experimental fashion. Indeed, computer-assisted proofsturnupfromtimetotime,ofteninconnectionwithimportantproblemsthathaveresistedthemoretraditionalmethodsofpen,paper,andhumanbrainpower.Thisrelaxedattitudetocomputersisrelativelyrecent;notbecausemathematiciansare

stick-in-the-muds who resist new technology, but because computers werepreviously too limited, both in speed and memory. A serious mathematicalproblem can stretch even the fastest supercomputer; one recent investigationwouldhavehadanoutputthesizeofManhattanifithadbeenprintedout.

Thurston’s revival of three-dimensional hyperbolic geometry led him topioneertheuseofcomputersatthefrontiersofgeometry.Inthelate1980stheNationalScienceFoundationfundedanewGeometryCenterattheUniversityofMinnesota,whichhostedresearchmeetingsandpublicoutreachactivities.Italsoadvanced the use of computer graphics, and two of its videos achievedconsiderable fame.They’re still available on theweb, even though theCenteritselfisnowdefunct.Thefirst,NotKnot,fliestheviewerthroughvariousthree-dimensional hyperbolic manifolds that Thurston discovered. Its complex andintriguinggraphicsaresopsychedelicthatextractshavebeenusedinconcertsbythe Grateful Dead.Outside In is an animation of a remarkable theorem thatSmalediscoveredasagraduatestudentin1957.Namely,youcanturnasphereinsideout.18

Imagineaspherewhoseoutsideispaintedgoldandinsidepurple.Ofcourseyoucanturnitinsideoutbymakingaholeandpushingitthroughthehole,butthat’snotatopologicaltransformation.Thetrickisclearlyimpossiblewitharealsphere,suchasaballoon(thoughit’snottotallyobvioushowtoprovethis),butmathematicallywe can allow transformationswhere the spherepasses throughitself, which you can’t do with a balloon. Now you could try pushing fromopposite sides so that two purple bulges poke through the golden surface, butthatleavesanever-tighteningtubularringofgold.Whentheringcontractsdowntoacircle, thesurfaceceasestobesmooth.Smale’stheoremsaysthatthiscanbeavoided:there’satransformationsuchthatatallstagesthesphereissmoothlyembedded inspace, thoughperhapscutting through itself.Fora long time thiswas a pure existence proof: no one actually knew how to do it. Then varioustopologistsworkedoutdifferentmethods;one,BernardMorin,hadbeenblindfrom the age of six. Themost elegant and symmetricalmethod is Thurston’s,andthat’sthestarofInsideOut.

Thurstonhadan impacton thepublic appreciationofmathematics inotherways,too.Hewroteaboutwhatit’sreallyliketobeamathematician,andhowhethoughtaboutresearchproblems,givingoutsidersaninsideview.WhenDaiFujiwara,afashiondesigner,heardoftheeightgeometries,hegotintouchwithThurston,andtheirinteractionledtoanextensivearrayofwomen’sfashions.

Thurston’scontributiontomanyareasofgeometry,rangingfromtopologyto

dynamics, is extensive. His work is characterised by a remarkable ability tovisualise complex mathematical concepts. When asked for a proof he wouldusually draw a picture. Thurston’s pictures often revealed hidden connectionsthatnooneelsehadnoticed.Anothercharacteristicwashisattitudetoproofs:heoftenleftoutdetailsbecausetohimtheyseemedobvious.Whenanyoneaskedhim to explain a proof that they didn’t understand, he would often invent adifferentoneonthespotandsay:‘Perhapsyou’llpreferthisone.’ToThurston,allmathematicswasasingleconnectedwhole,andheknewhiswayrounditlikeotherpeopleknowtheirownbackgarden.

Thurstondiedin2012,aftersurgeryforamelanomainwhichhelosthisrighteye. While undergoing treatment, he continued to do research, provingfundamental new results in the discrete dynamics of rational maps of thecomplex plane. He went to mathematical conferences, and worked to inspireyoungpeopleabouthisbelovedsubject.Whatevertheobstacles,henevergaveup.

MathematicalPeople

WHAT, THEN,HAVEWELEARNED fromour significant figures,whosepioneeringdiscoveriesopenedupnewmathematicalvistas?

Themostobviousmessageisdiversity.Mathematicaltrailblazerscomefromallperiodsofhistory,allcultures,andallranksoflife.ThestoriesI’veselectedhere span a period of 2500 years. Their protagonists lived in Greece, Egypt,China, Persia, India, Italy, France, Switzerland, Germany, Russia, England,Ireland, andAmerica. Somewere born intowealthy families – Fermat, King,Kovalevskaia. Many were middle class. Some were born poor – Gauss,Ramanujan.Somecamefromacademicfamilies–Cardano,Mandelbrot.Somedidn’t –Gauss andRamanujan again,Newton,Boole. Some lived in troubledtimes – Euler, Fourier, Galois, Kovalevskaia, Gödel, Turing. Some werefortunatetohavelivedinamorestablesociety,oratleastamorestablepartofit–Madhava,Fermat,Newton,Thurston.Somewerepoliticallyactive–Fourier,Galois, Kovalevskaia. The first two were imprisoned as a result. Others kepttheirpoliticstothemselves–Euler,Gauss.

Therearesomepartialpatterns.Manygrewupinintellectualfamilies.Someweremusical.Someweregoodwith theirhands,otherscouldn’t fixabicycle.Many were precocious, showing unusual talent at an early age. Minorcoincidences,suchasthechoiceofbedroomwallpaper,achanceconversation,aborrowedbook,triggeredalife-changinginterest inmathematics.Manystartedouttryingtopursueadifferentcareer,especiallyinlawortheclergy.Somewereencouragedbytheirproudparents,somewereforbiddentostudymathematics,someweregrudginglypermittedtofollowtheircalling.

Some were eccentric. One was a rogue. A few were crazy. Most werenormal,inasmuchasanyofuscountsasnormal.Mostmarriedandhadfamilies,butsome–Newton,Noether–didn’t.

Most were male – a cultural bias. Until recently, women were often

consideredunsuited,bybiologyandtemperament,tomathematics,indeedtoanyscience.Theireducation, itwassaid,shouldbe indomesticskills:crochet,notcalculus.Theirsocietyreinforcedthisview,andwomenwereoftenasvocalasmen about the unsuitability of mathematics as a womanly pursuit. Even ifwomenwanted to study the subject, theywere forbidden to attend lectures, totake examinations, to graduate, and to join the ranks of academe.Our femaletrailblazersblazed two trails:one through the jungleofmathematics, theotherthrough the jungle ofmale-dominated society. The second trailmade the firstevenharder.Mathematicsisdifficultenoughwhenyoucangettraining,books,andtimetothink.It’salmostimpossiblewhenyouhavetobattletoobtainanyofthosethings.Despitetheseobstacles,afewgreatwomenmathematiciansbrokedown thebarriers, blazing a trail forothers to follow.Even today,womenaregenerally under-represented in mathematics and science, but it’s no longersocially acceptable to attribute this to differences in ability or mentality, asseveralprominentmenhavediscoveredtotheirdismay.Noristhereashredofevidencetosupportthoseviews.

It’s tempting to lookforneurologicalexplanationsofunusualmathematicaltalent. In the early days of phrenology, Franz Gall proposed that importantabilitiesareassociatedwithspecificregionsofthebrain,andcanbeassessedbymeasuringtheshapeoftheskull.Ifyou’regoodatmathematics,yourheadwillhaveamathematicsbump.Phrenologyisnowseenaspseudoscience,althoughspecificregionsofthebraindosometimesplayspecificroles.Today’sobsessionwithgeneticsandDNAmakesitnatural toaskwhether there’sa‘mathematicsgene’.It’shardtoseehow,becausemathematicsgoesbackonlyafewthousandyears,soevolutionhasn’thadtimetoselectformathematicalability,anymorethanitcanhaveselectedfortheabilitytopilotafighterjet.Mathematicaltalentpresumablyexploitsotherattributes,moreconducivetosurvival–keenvision,aretentive memory, skill at swinging through trees. Sometimes mathematicsseemstoruninfamilies–theBernoullis–butmostlyitdoesn’t.Evenwhenitdoes,theinfluencesareoftennurture,notnature:amathematicaluncle,calculusonthewallpaper.Evengeneticistsareslowlycomingtorealise thatDNAisn’teverything.

The pioneering mathematicians do have some generalities in common.They’re original, imaginative, and unorthodox. They seek patterns and relishsolvinghardproblems.Theypaycloseattentiontologicalfinepoints,buttheyalso indulge in creative leaps of logic, becoming convinced that some line ofattackisworthpursuingevenwhenthere’slittletojustifythatview.Theyhave

strong powers of concentration, yet, as Poincaré urged, they shouldn’t be soobsessive that theykeepbanging theirheadsagainstbrickwalls.Theyneed togive their subconsciousminds time tomull everything over. They often haveexcellentmemories,butsome–Hilbert,forinstance–don’t.

Theycanbelightningcalculators,likeGauss.Euleroncesettledanargumentbetweentwoothermathematicians,aboutthefiftiethdecimalplaceinthesumofacomplicatedseries,bydoingthesumsinhishead.Ontheotherhand,theycanbe terrible at arithmetic without any obvious disadvantage. (Most lightingcalculators are hopeless at anythingmore advanced than arithmetic;Gauss, asever, was an exception.) They have the ability to absorb huge quantities ofprevious research, distilling its essence andmaking it their own, but they canalsoignoreconventionalpathsaltogether.ChristopherZeemanusedtosaythatitwas a mistake to read the research literature before starting on a problem,becausedoingsowouldslotyourmindintothesamegrooveseveryoneelsewastrapped in. Early in his career, the topologist Stephen Smale solved whateveryone thoughtwasa trulyhorribleproblembecausenoonehad toldhim itwashard.

Nearly all mathematicians have a strong intuition, either formal or visual.Here I’m referring to the visual areas of the brain, not to eyesight: Euler’sproductivity increasedwhenhewentblind. InThePsychologyof Invention inthe Mathematical Field, Jacques Hadamard asked a number of leadingmathematicianswhether theythoughtaboutresearchproblemssymbolically,orusing somekindofmental image.Withvery fewexceptions, theyusedvisualimagery, even when the problem and its solution were mainly symbolic. Forexample,Hadamard’smental image forEuclid’s proof that there are infinitelymanyprimesinvolvednotalgebraicformulas,butaconfusedmasstorepresentthe known primes, and a point far from that mass to represent a new prime.Vaguemetaphoricalimageswerecommon,formaldiagramslikeEuclid’srare.

Thetendencyto invokevisual(andtactile) images isevidentasfarbackasAl-Khwarizmi’sAlgebra,whosetitlerefersto‘balancing’.Theimageinvokedisonethatteachersoftenusetoday.Thetwosidesofanequationarethoughtofascollectionsofobjectsplacedinthecorrespondingpansofapairofscales,whichmustbalance.Algebraicoperationsare thenperformed in the samemanneronboth sides, to ensure that it remains balanced. Eventuallywe end upwith theunknown quantity in one pan and a number in the other: the answer.Mathematicians solving equations often imagine the symbols moving around.(That’s why they still like blackboard and chalk: a bit of rubbing out and

rewritingcanachievemuchthesameeffect.)Moreobviouslygeometricthinkingalso occurs in Al-Khwarizmi’s Algebra, with its diagram of the process ofcompleting the square to solve aquadratic equation.According to legend,onemathematiciangaveaverytechnicallectureonalgebraicgeometry,drawingjusta single dot on the blackboard to represent a ‘generic point’.He referred to itfrequently, and the lecture made far more sense as a result. Blackboards andwhiteboards across the planet, not to mention napkins and sometimestablecloths, are crammed with a jumble of esoteric symbols and weird littledoodles.Thedoodlescanrepresentanythingfromaten-dimensionalmanifoldtoanalgebraicnumberfield.

Hadamardestimatedthatabout90percentofmathematiciansthinkvisually,and10percentthinkformally.Iknowofatleastoneprominenttopologistwhohas trouble visualising three-dimensional shapes. There’s no universal‘mathematicalmind’–onesizedoesn’t fitall.Mostmathematicalmindsdon’tproceedone logical stepat a time;only thepolishedproofsof their resultsdothat. Usually, the first step is to get the right idea, often by thinking vaguelyabout structural issues, leading to somekind of strategic vision; the next is tocomeupwith tactics to implement it; the final step is to rewrite everything informal terms to present a clean, logical story (Gauss’s removal of thescaffolding).Inpracticemostmathematiciansalternatebetweenthesetwowaysof thinking, resorting to imagerywhen it’s not clear how to proceed, orwhentryingtogetasimpleoverview,butresortingtosymboliccalculationwhentheyknowwhattodobutareunsurewhereitleads.Some,however,seemtoploughaheadregardless,usingonlysymbols.

Extrememathematicalabilitydoesn’tcorrelatestronglywithanythingelse.Itseemstostrikeatrandom.Some,suchasGauss,‘getit’whenthey’rethreeyearsold.Some,amongthemNewton,fritter theirchildhoodawaybutblossomlaterinlife.Youngchildrengenerallyenjoynumbers,shapes,andpatterns,butmanyloseinterestastheygrowolder.Mostofuscanbetrainedinmathematicsuptohigh-school level, but fewgobeyond.Somenever reallyget togripswith thesubjectatall.Manyprofessionalmathematicianshaveastrong impression thatwhen it comes tomathematicalability,we’renotallbornequal.Whenyou’vegone through life finding most school mathematics easy and obvious, whileothers struggle with the basics, it sure looks that way. When some of yourstudentsfindeasyconceptsbaffling,whileothersgrasphardonesimmediately,thisfeelingisreinforced.

Perhapsthisanecdotalevidenceiswrong.Alotofeducationalpsychologists

thinkso.There’sbeenavogue inpsychology for the ‘blankslate’viewof thechild’s mind. Anyone can do anything: all they need is training and lots ofpractice. If you want it badly enough, you’ll get it. (If you don’t get it, thatprovesyoudidn’twant it…aneatpieceofcircularreasoninggreatlyfavouredby sports commentators.) It would bewonderful if this were true, but StevenPinkertookthiskindofpoliticallycorrecthopeapartinTheBlankSlate.Also,many educators detect a disability, discalculia, which impairs the learning ofmathematics like dyslexia affects reading and writing. I’m not sure bothpositionscanbemaintainedsimultaneously.

Physically,we’renotallbornthesame.Butforsomereason,alotofpeopleseemtoimagine–orwanttoimagine–thatmentally,weare.Thismakeslittlesense. Brain structure affects mental abilities just as bodily structure affectsphysicalones.Somepeoplehaveeideticmemoriesthatremembereverythingingreat detail. It seems implausible that anyone canbe taught tohave an eideticmemory if only they train and practise. The blank slate hypothesis is oftenjustifiedbypointingoutthatalmosteveryonewhoishighlysuccessfulinsomearea of human activity practises a lot. That’s true – but it doesn’t imply thateveryone who does a lot of practice in some area of human activity will behighly successful.AsAristotle andBoolewell knew, ‘A impliesB’ is not thesameas‘BimpliesA’.

Before you get too annoyed, I’m not arguing against trying to teachmathematics,or anythingelse, to anyone.Almost allofus improvewithgoodteachingandplentyofpractice,whatever theactivity.That’swhyeducation isworth theeffort.GeorgePólya revealedsomeuseful tricks inHowtoSolve It.It’s abit like those ‘how tohavea super-powermemory’books, teachingyoutechniques that help you remember things, but directed towards solvingmathematical problems. However, people with eidetic memories don’t usemnemonic tricks. What they want to recall is there as soon as they need it.Similarly,evenifyoumasterPólya’sbagoftricks,you’reunlikelytobecomeanewGauss, howevermuchwork you put in. TheGausses of thisworld don’tneedtobetaughtspecialtricks.Theyinventedthemforthemselvesinthecradle.

On the whole, people don’t make themselves successful by working theirsocksoff at something inwhich theyhaveno real interest.Theypractisehardbecauseevennaturaltalentneedsplentyofexercisetokeepithealthy;becauseyouhavetokeepinpracticetostaytalented;butmainlybecausethat’swhattheywanttodo.Evenwhenit’sdifficultorboring,insomecuriouswaytheyenjoyit.Youcanonlystopbornmathematiciansdoingmathematicsbylockingthemup,

andeventhenthey’llscratchequationsonthewalls.Andthat,ultimately,isthecommonthread that runs throughallofmysignificant figures.They love theirmathematics.They’reobsessedwithit.Theycandonoother.Theygiveupmoreprofitable professions, they go against their families’ advice, they plough onregardlessevenwhenmanyoftheirowncolleaguesconsiderthemmad,they’rewillingtodieunrecognisedandunrewarded.Theylectureforyearsfornosalary,just to get a foot in the door. The significant figures are significant becausethey’redriven.

Whatmakesthemthatway?It’samystery.

PhotographbyAvrilStewart

IanStewartisaprofessorofmathematicsattheUniversityofWarwickandtheauthorofnumerousbooksonmathematics.HehaswrittenforNewScientistandScientificAmerican,amongotherpublications.StewartlivesinCoventry,UnitedKingdom.

BythesameauthorConceptsofModernMathematicsDoesGodPlayDice?FearfulSymmetry(withMartinGolubitsky)Game,Set,andMathAnotherFineMathYou’veGotMeIntoNature’sNumbersFromHeretoInfinityTheMagicalMazeLife’sOtherSecretFlatterlandWhatShapeIsaSnowflake?(revisededition:TheBeautyofNumbersinNature)TheAnnotatedFlatland

MathHysteriaTheMayorofUglyville’sDilemmaHowtoCutaCakeLetterstoaYoungMathematicianTamingtheInfinite (alternative title:TheStoryofMathematics)WhyBeautyIsTruth

CowsintheMazeProfessorStewart’sCabinetofMathematicalCuriositiesMathematicsofLifeProfessorStewart’sHoardofMathematicalTreasuresSeventeenEquationsthatChangedtheWorld(alternativetitle:InPursuitoftheUnknown) The Great Mathematical Problems (alternative title: Visions ofInfinity)Symmetry:AVeryShortIntroduction

Jack of All Trades (science fiction eBook) Professor Stewart’s Casebook ofMathematicalMysteries

ProfessorStewart’sIncredibleNumbersCalculatingtheCosmosInfinity:AVeryShortIntroduction

WithJackCohenTheCollapseofChaosEvolvingtheAlien(alternativetitle:WhatDoesaMartianLookLike?)Figments

ofRealityWheelers(sciencefiction)Heaven(sciencefiction)

ScienceofDiscworldseries(withTerryPratchettandJackCohen)TheScienceofDiscworldTheScienceofDiscworldII:TheGlobeTheScienceofDiscworldIII:Darwin’sWatchTheScienceofDiscworldIV:JudgementDay

WithTimPostonTheLivingLabyrinth(sciencefiction)RockStar(sciencefiction)

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”InSignificantFigures, IanStewart bringsmathematics to lifewith intriguingaccountsoftwenty-fiveextraordinarycontributorstothefield.Hisbiographicalsketchesblendequalpartspassion—loveaffairs and rivalries—with insights—groundbreaking discoveries—to offer vivid, complete portraits of his subjects.By showing how even mathematical geniuses face all-too-human challenges,Stewartoffersarivetingchronicleofoneofhumankind’sloftiestendeavors.”

—PAULHALPERN,authorofTheQuantumLabyrinth

FurtherReading

GeneralReadingEric Temple Bell. Men of Mathematics, Simon and Schuster 1986. (Firstpublished 1937.) Carl Benjamin Boyer. A History of Mathematics, Wiley1991.

Morris Kline.Mathematical Thought from Ancient to Modern Times, OxfordUniversityPress1972.

MacTutor History of Mathematics archive: http://www-groups.dcs.st-and.ac.uk/~history/

Wikipedia:https://en.wikipedia.org/wiki/Main_Page

Chapter1ArchimedesEduardJanDijksterhuis.Archimedes,PrincetonUniversityPress1987.Mary Gow. Archimedes: Mathematical Genius of the Ancient World, Enslow2005.

ThomasL.Heath.TheWorksofArchimedes(reprint),Dover1897.RevielNetzandWilliamNoel.TheArchimedesCodex,Orion2007.

Chapter2LiuHuiGeorgeGhevergheseJoseph.TheCrestofthePeacock,I.B.Tauris1991.

Chapter3Muhammadal-KhwarizmiAliAbdullahal-Daffa.TheMuslimContributiontoMathematics,CroomHelm1977.

GeorgeGhevergheseJoseph.TheCrestofthePeacock,I.B.Tauris1991.RoshdiRashed.Al-Khwarizmi:TheBeginningsofAlgebra,SaqiBooks2009.

Chapter4MadhavaofSangamagrama

GeorgeGhevergheseJoseph.TheCrestofthePeacock,I.B.Tauris1991.

Chapter5GirolamoCardanoGirolamoCardano.TheBookofMyLife,NYRBClassics2002.(Firstpublished1576.)GirolamoCardano.TheRuleofAlgebra(ArsMagna)(reprint),Dover2007.(Firstpublished1545.)Chapter6PierredeFermat

Michael SeanMahone.TheMathematicalCareer of Pierre deFermat, 1601–1665(secondedition),PrincetonUniversityPress1994.

SimonSingh.Fermat’sLastTheorem–TheStoryofaRiddlethatConfoundedthe World’s Greatest Minds for 358 Years (second edition), Fourth Estate2002.

Chapter7IsaacNewtonRichard S. Westfall. The Life of Isaac Newton, Cambridge University Press1994.

RichardS.Westfall.NeveratRest,CambridgeUniversityPress1980.MichaelWhite.IsaacNewton:TheLastSorcerer,FourthEstate1997.

Chapter8LeonhardEulerRonald S. Calinger. Leonhard Euler – Mathematical Genius in theEnlightenment,PrincetonUniversityPress2015.

WilliamDunham.Euler –TheMaster ofUsAll,MathematicalAssociationofAmerica1999.

Chapter9JosephFourierIvorGrattan-Guinness.JosephFourier1768–1830,MITPress1972.JohnHervel. Joseph Fourier – theMan and the Physicist, Oxford UniversityPress1975.

Chapter10CarlFriedrichGaussWalterK.Bühler.Gauss–ABiographicalStudy,Springer1981.G.Waldo Dunnington, Jeremy Gray, and Fritz-Egbert Dohse.Carl FriedrichGauss:TitanofScience,MathematicalAssociationofAmerica2004.

M.B.W.Tent.ThePrinceofMathematics–CarlFriedrichGauss,A.K.Peters/CRCPress2008.

Chapter11NikolaiIvanovichLobachevskyAthanase Papadopoulos (editor). Nikolai I. Lobachevsky, Pangeometry,EuropeanMathematicalSociety2010.

Chapter12ÉvaristeGaloisLauraTotiRigatelli.ÉvaristeGalois 1811–1832 (VitaMathematica), Springer2013.

Chapter13AugustaAdaKingMalcolm Elwin. Lord Byron’s Family: Annabella, Ada and Augusta, 1816–1824,JohnMurray1975.

JamesEssinger.Ada’sAlgorithm–HowLordByron’sDaughterAdaLovelaceLaunchedtheDigitalAge,GibsonSquareBooks2013.

Anthony Hyman. Charles Babbage – Pioneer of the Computer, OxfordUniversityPress1984.

Sydney Padua. The Thrilling Adventures of Lovelace and Babbage – The(Mostly)TrueStoryoftheFirstComputer,Penguin2016.

Chapter14GeorgeBooleDesmondMacHale.TheLifeandWorkofGeorgeBoole(secondedition),CorkUniversityPress2014.

GerryKennedy.TheBoolesandtheHintons:TwoDynastiesThatHelpedShapetheModernWorld,Atrium2016.

PaulJ.Nahin.TheLogicianandtheEngineer:HowGeorgeBooleandClaudeShannonCreatedtheInformationAge,PrincetonUniversityPress2012.

Chapter15BernhardRiemannJohn Derbyshire. Prime Obsession: Bernhard Riemann and the GreatestUnsolvedProbleminMathematics,PlumeBooks2004.

Marcus Du Sautoy. The Music of the Primes: Why an Unsolved Problem inMathematicsMatters(secondedition),HarperPerennial2004.

Chapter16GeorgCantorAmirD.Aczel.TheMysteryoftheAleph:Mathematics,theKabbalah,andtheSearchforInfinity,FourWallsEightWindows2000.

JosephWarrenDauben.GeorgCantor:HisMathematicsandPhilosophyofthe

Infinite(secondedition),PrincetonUniversityPress1990.

Chapter17SofiaKovalevskaiaAnnHibner Koblitz.AConvergence of Lives – Sofia Kovalevskaia: Scientist,Writer,Revolutionary,Birkhäuser1983.

Chapter18HenriPoincaréJean-MarcGinouxandChristianGerini.HenriPoincaré:ABiographyThroughtheDailyPapers,WSPC2013.

Jeremy Gray. Henri Poincaré, A Scientific Biography, Princeton UniversityPress2012.

Jacques Hadamard. The Psychology of Invention in the Mathematical Field,Princeton University Press 1945. (Reprinted Dover 1954.) FerdinandVerhulst.HenriPoincaré,Springer2012.

Chapter19DavidHilbertConstanceReid.Hilbert,Springer1970.BenYandell.TheHonorsClass:Hilbert’sProblemsandTheirSolvers(secondedition),A.K.Peters/CRCPress2003.

Chapter20EmmyNoetherAugusteDick.EmmyNoether:1882–1935,Birkhäuser1981.M.B.W. Tent.Emmy Noether: TheMother of Modern Algebra, A.K.’Peters /CRCPress2008.

Chapter21SrinivasaRamanujanBruceC.BerndtandRobertA.Rankin.Ramanujan:LettersandCommentary,AmericanMathematicalSociety1995.

RobertKanigel.TheManWhoKnewInfinity–ALifeoftheGeniusRamanujan,Scribner’s1991.

S.R.Ranganathan.Ramanujan;TheManand theMathematician (reprint),EssEssPublications2009.

Chapter22KurtGödelGabriella Crocco and Eva-Maria Engelen. Kurt Gödel, Philosopher-Scientist,Publicationsdel’UniversitédeProvence2016.

JohnDawson.LogicalDilemmas:TheLifeandWorkofKurtGödel,A.K.Peters/CRCPress1996.

Chapter23AlanTuringAndrewHodges.AlanTuring:TheEnigma,BurnettBooks1983.MichaelSmith.TheSecretsofStationX:HowtheBletchleyParkCodebreakersHelpedWintheWar,BitebackPublishing2011.

DermotTuring.Prof:AlanTuringDecoded,TheHistoryPress2016.

Chapter24BenoitMandelbrotMichael Frame andNathan Cohen (eds.).BenoitMandelbrot: a Life inManyDimensions,WorldScientific,Singapore2015.

BenoitMandelbrot. The Fractalist: Memoir of a Scientific Maverick, Vintage2014.

Chapter25WilliamThurstonDavid Gabai and Steve Kerckhoff (eds.). William P. Thurston, 1946–2012.Notices of the American Mathematical Society 62 (2015) 1318–1332; 63(2016)31–41.

Notes

1.WhenreferringtoabookorpaperoriginallyinLatinoraEuropeanlanguage,IusuallyusetheEnglishtranslationofthetitle,exceptwhenhistorianscommonlyuseanabbreviatedformoftheoriginalone.ThefirsttimesuchaworkappearsIgivetheoriginaltitle,withatranslation,unlessit’sobvious.ThetitlesofancientChinese,ArabicandIndiantextsaretransliteratedandoftenabbreviated,andatranslationisusuallyprovided.

2.GeorgeGhevergheseJoseph.TheCrestofthePeacock,I.B.Tauris1991.

3.AlexandreKoyré.AnunpublishedletterofRobertHooketoIsaacNewton,Isis43(1952)312–337.

4.TheRoyalSocietyplannedtocommemorateIsaacNewton’stercentenaryin1942,butWorldWarIIintervenedsothecelebrationswerepostponedto1946.Keyneshadwrittenalecture‘Newton,theman’,buthediedjustbeforetheeventtookplace.HisbrotherGeoffreyreadthelectureonhisbehalf.

5.RichardAldington.FrederickIIofPrussia,LettersofVoltaireandFredericktheGreat,LetterH7434,25January1778,Brentano’s1927.

6.Moreprecisely,thepolynomialmustalsobeirreducible–notaproductoftwopolynomialsofsmallerdegreewithintegercoefficients.Ifnisprimethenxn–1+xn–2+…+x+1isalwaysirreducible.

7.BoeotiaisaregionincentralGreece.InclassicaltimestheAtheniansdescribedtheBoeotiansasdullandunintelligent.Thenamebecameaproverbialreferencefordumbstupidity.

8.TonyRothman.Geniusandbiographers:thefictionalizationofÉvaristeGalois,AmericanMathematicalMonthly89(1982)84–106.

9.InUKEnglishtheAmericanspellingisoftenusedtodistinguishacomputer

programfromanyotherkindofprogramme,andisstandardintheindustry.

10.JuneBarrow-Green.PoincaréandtheThreeBodyProblem,AmericanMathematicalSociety,Providence1997.

11.Ramanujan’sMasterFormulastatesthatif

isacomplex-valuedfunctionthen

whereГ(s)isEuler’sgamma-function.

12.AndrewEconomou,AtsushiOhazama,ThantriraPorntaveetus,PaulSharpe,ShigeruKondo,AlbertBasson,AmelGritli-Linde,MartynCobourne,andJeremyGreen.PeriodicstripeformationbyaTuringmechanismoperatingatgrowthzonesinthemammalianpalate,NatureGenetics(2012);DOI:10.1038/ng.1090.

13.BenoitMandelbrot.AMaverick’sApprenticeship,TheWolfPrizesforPhysics,ImperialCollegePress2002.

14.Seenote13.

15.BenoitMandelbrot.Informationtheoryandpsycholinguistics,inR.C.OldfieldandJ.C.Marchall(eds.),Language,PenguinBooks1968.

16.Seenote13.

17.Letc=x+iybeacomplexnumber.Startatz0=0anditeratethefunctionz2

+c,getting

z1=(z02+c)

z2=(z12+c)

z3=(z22+c)

andsoon.ThencliesintheMandelbrotsetifandonlyifallthepointsznlieinsomefiniteregionofthecomplexplane.Thatis,thesetofiteratesis

bounded.

18.https://www.youtube.com/watch?v=wO61D9x6lNY.

Index

Abel,NielsHenrik,122,123,129,130Abelianintegrals,160,161,184ACE(AutomaticComputingEngine),251Ackermann,Wilhelm,PrinciplesofMathematicalLogic,237ActaMathematica(journal),174,185Agnesi,Maria,183Ahlgren,Scott,231Aiyar,P.V.Seshu,225Alexandrov,Pavel,219algebra:Booleanalgebra,146–9,152–4;andCardano’sArsMagna,48–51;

‘invented’byal-Khwarizmi,30–4algebraicequations,127–30algebraicnumbers,204–5algorithms,29–30,34–5al-Haytham,Hasanibn(Alhazen),113al-Kashi,Jamshid,Miftahal-Hisab(TheKeytoArithmetic),38al-Khwarizmi,MuhammadibnMusa:ongeographyandastronomy,35–6;

‘invention’ofalgebra,5,29–34,148;Algebra,29,30–4,280;OnCalculationwithHinduNumerals,34–5

al-Ma’mun,Caliph,29,30al-Rashid,Harun,29AnalyticalEngine(ofCharlesBabbage),137–41Andrews,George,229Anne,Queen,64,67AnthemiusofTralles,BurningGlasses,12Antikytheramechanism,13anti-semitism,209,219,235–6,256,258–9

Apollonius,55Arago,François,88Archimedeanscrew,5,13Archimedes:approximatesπ,5;claw(crane-likedevice),5,13;death,20;‘heat

ray,’11–12;lawofthelever,5,13;life,13–14;CattleProblem,14,18–20;OnConoidsandSpheroids,17;OnFloatingBodies,17–18;MeasurementofaCircle,18;TheMethodofMechanicalTheorems,14,16–17,69;OnPlaneEquilibria,15;QuadratureoftheParabola,14;TheSandReckoner,13,18;OntheSphereandCylinder,15–16,20;OnSpirals,17

Archimedespalimpsest,16Archimedes’sPrinciple,1,13,17–18Aristarchus,18Aristotle,1,149,164Arnold,Vladimir,268Aryabhata(Indianmathematician-astronomer),39,41,44Atkin,Arthur,231Auerbach,Felix,261Ayscough,James,65Ayscough,Margery,65

Babbage,Charles,6,135–41Babylonians,solvequadraticequations,1,48BachetdeMéziriac,Claude,59Bacon,Francis,67,174–5Banach,Stefan,263Barbie,Klaus,259Barrow,Isaac,67Barrow-Green,June,198Bartels,Martin,99,115Baselproblem,82–3Bate,John,TheMysteriesofNatureandArt,66Battuta,Muhammadibn,Rihla,27Baytal-Hikma(HouseofWisdom),Baghdad,29,32–3Beaumont-de-Lomagne,France,55Belevitch,Vitold,261Beltrami,Eugenio,118–19,157Bendixson,Ivar,197

Berkeley,BishopGeorge,70,158BerlinAcademy,78,81,160,161Bernays,Paul,209Berndt,Bruce,229,233Bernoulli,Daniel,81,83Bernoulli,Jacob,80,83,139Bernoulli,Johann,80–1,83,87Bernoulli,Nicolaus,81Bernoullinumbers,139Bézout,Étienne,89Bhaskaracharya(Indianmathematician),41Bhattathir,MelpathurNarayana,39Biot,Jean-Baptiste,88BletchleyPark(GovernmentCodeandCypherSchool),243,248–50Bolyai,János,6,111,116–18Bolyai,Wolfgang,116,117Bombelli,Rafael,51Boole,George:backgroundandeducation,142–5;andBooleanalgebra/logic,

146–50,152–4;andinvarianttheory,145–6;ProfessorofMathematicsinCork,Ireland,145,150–2;AnInvestigationintotheLawsofThought,6,144,146;TheMathematicalAnalysisofLogic,143

Boole,Lucy,145Boole,MaryEverest,145Booleanalgebra/logic,146–50,152–4Borchardt,Karl,161Bouguer,Pierre,81Bourbaki,Nicolas,175Bourbakistmovement,175,211,259Boyle,Robert,66Brahmagupta(Indianmathematician),41,44,57Brewster,David,135BritishTabulatingMachineCompany,249Brooks,Robert,266Brouncker,Lord,57Brown,Gordon,254Brunswick-Wolfenbüttel,DukeKarlWilhelmFerdinandof,99,102–3BrynMawrUniversity,USA,219

Bullialdus,Ismaël,74Bunsen,Wilhelm,181Burgess,Guy,254Byron,LadyAnnabella,133–4Byron,LordGeorgeGordon,133–4

CairoInstitute,Egypt,90calculus:andIsaacNewton,65,67,68–72;andtheKeralaschoolofastronomy

andmathematics,43–4CambridgeMathematicalJournal,144CambridgeUniversity:AlanTuringat,245,251–2;IsaacNewtonat,64,66,67;

SrinivasaRamanujanat,221,226CambridgeUniversityLibrary,16Cantor,Georg,7,93,164–75,235Cantor,Moritz,109Cardano,Fazio,46–7Cardano,Girolamo,5,45–52;ArsMagna,46,49–51,127,128;BookonGames

ofChance,47Carr,George,SynopsisofElementaryResultsinPureMathematics,224Carvaci,Pierrede,54Cauchy,Augustin-Louis,108,129,158,160Cauchy–Kovalevskaiatheorem,183Cauchy’sTheorem,108Cayley,Arthur,146,203Ceres(asteroid/planet),106–7Champollion,Jean-François,90ChangHeng,23,25ChaoChunChin,23chaos,dynamical,197–8CharlesX,King,123Chatterji,GyaneshChandra,227Chebyshev,Pafnuty,162Chevalier,Auguste,125,126China,mathematicsin,21–4Christoffel,ElwinBruno,157Church,Alonzo,247Cicero,20

Clairaut,Alexis,72Clarke,Joan,250–1Clarke,Samuel,DemonstrationoftheBeingandAttributesofGod,149Clarke,William,65ClayMathematicsInstitute,163,272,274combinatorics,84–5,139complexanalysis,61,82,84,108,119,120;andBernhardRiemann,155,158–

60,161computergames,192ComputerHistoryMuseum,California,138computers:ACE(AutomaticComputingEngine),251;Babbage’sDifference

andAnalyticalEngines,135–41;andBletchleyPark,248–50;andBooleanalgebra,153–4;EDVAC(ElectronicDiscreteVariableAutomaticComputer),251;IBM’sDeepBlue,252;andtheMandelbrotset,265–6;theTuringmachine,245–7;useingeometry,274–5

Condorcet,Nicolasde,80CononofSamos,14ContinuumHypothesis,172,174,238cosinesseesinesandcosinesCrimeanWar,178cubicequations,48–51,68,128–30Curie,Marie,178

daGama,Vasco,43daVinci,Leonardo,46d’Alembert,JeanleRond,81–2,87Darwin,Charles,181DeMorgan,Augustus,138,143,144,150Dedekind,Richard,109,167–8,216delFerro,Scipione,49delNave,Annibale,49Delambre,Jean,88Deligne,Pierre,232Dennison,Alastair,248Desargues,Girard,56Descartes,René,12,54,56,60,66,73;LaGéometrie,67d’Herbinville,Pescheux,125

Dickens,Charles,135DifferenceEngine(ofCharlesBabbage),135–6,139differentialcalculus,56,69,71–2,160differentialequations,73,86,87–8,178,182–3,186,190,195–7,268differentialgeometry,156–7Diodorus,152Diophantineequations,19,58–9,61,208DiophantusofAlexandria,19,33,48,58–9Dirichlet,Peter,60,158,166Dirichletprinciple,160–1discretemathematics,84–5DNAmolecule,108Dostoievski,Fedor,179Douady,Adrien,266DuBois-Reymond,Paul,181Duchâtelet,Ernest,124,126Dumas,Alexandre,125dynamicalsystems,268

Eddington,Arthur,TheNatureofthePhysicalWorld,244Edgren-Leffler,AnnaCarlotta,184,185EDVAC(ElectronicDiscreteVariableAutomaticComputer),251Einstein,Albert:onEmmyNoether,220;friendshipwithKurtGödel,237–8;

GeneralRelativity,7,131,157,208,216;SpecialRelativity,199,212–13;theoryofgravity,120

Eisenstein,Gotthold,158Ekholm,Nils,94Eliot,George,181Elizabeth,EmpressofRussia,77ellipticcurves,61ellipticfunctions,158,160,186,232–3ellipticgeometry,118–19,272Enigmacypher,248–50EratosthenesofCyrene,14,19Erdős,Paul,235ErlangenProgramme,120Euclid,Elements,1,32–3,58,97,100–1,110–11,112–14

EuclidofMegara,152Euclideangeometry,67,111–12,117–19,205–6Eudoxus,16Euler,Christoph,78Euler,Johann,78Euler,Leonhard,5,60,92,104;appliedmathematics,86;blindness,279;

calculatingabilities,279;lifeandcareer,77–82;andprimenumbers,162;provesFermat’sLastTheorem,57,204;puremathematics,82–5;solvestheBaselproblem,82–3;IntroductiontoAnalysisoftheInfinite,82;Mechanics(1736),86;MethodforFindingCurvedLines,86;TheoryoftheMotionofSolidBodies(1765),86

Euler,Paul,80Eulertop,186–7‘Eureka!’,17exhaustion(method),16,17,18,25

Faraday,Michael,135Fatou,Pierre,265–6Fermat,Pierrede,5,53–62,69,72,103,104;‘LittleTheorem’,84Fermat’sLastTheorem,4,54,58–62,204Ferrari,Lodovico,50–1Fibonaccisequence,252Findley,Rachel,270Fior,Antonio,49Fischer,Ernst,212fluxions,44,70–1,158foliations,270Fontana,Niccolò(‘Tartaglia’),49–51Fourier,Joseph,5–6,123,158;andglobalwarming,93–4;heatequation,91–3;

lifeandcareer,89–90;AnalyticTheoryofHeat,88Fourieranalysis,163Fourierseries,91,94–5,156,166fractals,9,256–7,262–5Fraenkel,Abraham,239FredericktheGreat,King,78–9,81Freedman,Michael,271Frend,William,134

Frey,Gerhard,61Fuchs,Lazarus,202Fujiwara,Dai,275Fuss,Nikolai,78,80

GalileoGalilei,1,66,72,168Gall,Franz,278Galois,Évariste:education,careerandearlydeath,121–6;mathematicsof

symmetry,6,122,127–32,214Gassendi,Pierre,66Gauss,CarlFriedrich,6,57–8,160;calculatingabilities,279;constructs

heptadecagon,96–7,101–5;DirectorofGöttingenObservatory,107–9;anddiscoveryofCeres,107;earlylife,98–100;andFermat’sLastTheorem,54;LawofQuadraticReciprocity,84,105;andnon-Euclideangeometry,111,117,193;andprimenumbers,162;tutortoBernhardRiemann,155–7,158;DisquisitionesArithmeticae,103–6;TheoremaEgregium,109,156;TheoryoftheAttractionofaHomogeneousEllipsoid,108;TheoryoftheMotionofCelestialBodiesabouttheSuninConicSections,107

‘Gaussianintegers’,204GeneralRelativity,7,8,157,195,199,208,238,273geodesics,118,156,157geometryseedifferentialgeometry;Euclideangeometry;hyperbolicgeometry;

non-EuclideangeometryGermain,Sophie,124German,R.A.,19GermanMathematicalSociety,204–5Girard,Albert,57globalwarming,93–4Gödel,Kurt:incompletenessandconsistencytheorems,8,207,239–42,245–7;

lifeandcareer,234–8Goldbach,Christian,60,83Gordan,Paul,203,212GöttingenObservatory,107,108GöttingenUniversity,100,109,158,183,209,212GovernmentCodeandCypherSchool(GC&CS),BletchleyPark,248,254GovernmentCommunicationsHeadquarters(GCHQ),254graphtheory,85gravity,lawof,64,67,73–5,197

Greekgeometry,14–17Green,Jeremy,253‘greenhouseeffect’,94Gregory,James,42GregoryXIII,Pope,52Gregory’sseries,38,42Gsell,Katarina,81

Hadamard,Charles,100Hadamard,Jacques,ThePsychologyofInventionintheMathematicalField,

279–80Hahn,Hans,237HaidaoSuanjing(SeaIslandMathematicalManual),24Halifax,CharlesMontagu,Earlof,63Halley,Edmond,74Hamilton,Richard,273Hardy,GodfreyHarold,163,221–3,226–7,229,231,233Heath,Thomas,16Heiberg,Johan,16HeidelbergUniversity,180,202Heine,Eduard,166Hellegouarch,Yves,61Helmholtz,Hermann,181heptadecagon,97,100,102,204Hermite,Charles,182,185,190,204Hersh,Reuben,WhatisMathematics,Really?3–4Hilbert,David,7–8,161,174;oncurves,257;andEmmyNoether,212–13,215;

theHilbertProblems,163,208–9;Hilbertprogramme,202,239–40,245;invarianttheory,146,203,217;lifeandcareer,200–2,208–9;FoundationsofGeometry(1899),206–7;PrinciplesofMathematicalLogic,237;andtheZahlbericht(NumberReport),204–5

Hindunumerals,34Hinton,CharlesHoward,145Hinton,Mary(néeBoole),145Hipparchus,39,40Hitler,Adolf,235Hobbes,Thomas,66

Hooke,Robert,2,73,74–5,87Hopf,Heinz,219Hoüel,Jules,118–19Hurwitz,Adolf,202Huxley,Thomas,181hydrostatics,17,73,79Hyman,Anthony,CharlesBabbage,PioneeroftheComputer,140–1hyperbolicgeometry,6,118–20,272,275

Ihara,Yasutaka,232Infantozzi,Carlos,125‘infinitedescent’,54infinitenumbers,164–75infiniteseries,38,42–3,83,84,88,162–3,195InternationalCongressofMathematicians:(1923),208;(1928),237;(1932),218invarianttheory,145,146,203,212,217IsidorusofMiletus,16IslamicEmpire,28–9

Jaclard,Anna(néeKorvin-Krukovskaya),178–9,181–2Jaclard,Victor,181–2Jacobi,Carl,106,126,232Jacquardloom,139–40Jainism,165Jewishmathematiciansseeanti-semitismJiuzhangSuanshu(NineChaptersontheMathematicalArt),22,23–5Joseph,GeorgeGheverghese,41,43;TheCrestofthePeacock,22,26Julia,Gaston,259,260,265–6Jyesthadeva(Indianmathematician),39;Yuktibhasa,41,44

Kac,Mark,178Kane,Robert,150–1Kanigel,Robert,TheManWhoKnewInfinity,225Kant,Immanuel,112,201Kasparov,Gary,252Kästner,Abraham,100KazanUniversity,111,115Keen,Harold,249

Keill,John,71Kepler,Johannes,66,72,73,74,197Keralaschoolofastronomyandmathematics,India,39–44Keynes,JohnMaynard,75Khayyam,Omar,48,113–14King,AugustaAdaseeLovelace,AugustaAdaKing,CountessofKing,DrWilliam,135King’sCollege,Cambridge,251Kirchhoff,Gustav,181Kirchhoff’slawsforelectricalcircuits,109Klein,Felix,119–20,168,203,212,215Knox,Dilly,248Koch,Helgevon,257Königsberg,Prussia,84–5,201,202Königsberger,Leo,180Korvin-Krukovskaya,Yelizaveta,176–7Korvin-Krukovsky,Vasily,176,178Kovalevskaia,Sofia:impedimentstogaininguniversityeducation,179–81;life

andcareer,7,176–9,183–5;NihilistGirl,185;onpartialdifferentialequations,182–3;ARussianChildhood,185;onSaturn’srings,183

Kovalevskaiatop,185,186–7Kovalevskii,Vladimir,180–4Kowa,Seki,139Krafft,Wolfgang,78Kronecker,Leopold,166,168,173,174,239Krukovsky,PyotrVasilievich,178,179Kummer,Ernst,60,161,166,204,216

Lacroix,Sylvestre,DifferentialandIntegralCalculus,143Lagrange,Joseph-Louis,58,90,104,123,127–8Lagrangetop,186–7Lamé,Gabriel,60Landau,Edmund,209Laplace,Pierre-Simonde,86,183Lawrence,Arabella,135Lebesgue,Henri,93Legendre,Adrien-Marie,60,90,123,158

Leibniz,Gottfried,69–72,83Lenin,Vladimir,180,182LeotheGeometer,16LeoXIII,Pope,174Lermontova,Iulia,181Lessing,Gotthold,18lever,lawof,5,13,15,17Levi-Civita,Tullio,157Lévy,Paul,260Lexell,Anders,78,80Li,Wentian,261Libri,Guillaume,124Lie,Sophus,120LIGOexperiment(2016),199Lindemann,Ferdinandvon,202–3,204Liouville,Joseph,60,126,171Lipschitz,Rudolf,166Listing,Johann,191–2Littlewood,John,223LiuHsing,24–5LiuHui,5,22–6Lobachevsky,NikolaiIvanovich:lifeandcareer,115–16;andnon-Euclidean

geometry,6,110–11,117–18;Pangeometry,111logic,Boolean,149–50,152–4LondonMathematicalSociety,138Loney,Sidney,Trigonometry,224Lorentzgroupoflaws,131–2,199,213Lorenzcypher,248Louis-PhilippeI,King,124Lovelace,AugustaAdaKing,Countessof,6,133;backgroundandeducation,

133–5;commentaryonBabbage’sAnalyticalEngine,138–41;meetsCharlesBabbage,135

Lovelace,WilliamKing-Noel,1stEarlof,138Lucian,12

Maclean,Donald,254MadhavaofSangamagrama,5,37–44

Malevich,Iosif,179Malus,Étienne-Louis,90Mandelbrot,Benoit:andfractals,9,256–7,262–6;lifeandcareer,255–60;and

Zipf’slaw,260–1Mandelbrot,Szolem,257,258,260Mandelbrotset,265–6Marcellus,MarcusClaudius,20Marx,Karl,182Matelski,Peter,266mathematicalbiology,252–3Maupertuis,PierreLouisMoreaude,81Maxwell,JamesClerk,199Mayer,Tobias,86Mayer,Walther,219Menabrea,Luigi,138Menelaus,39Mengenlehre(settheory)seesettheory(Mengenlehre)Mersenne,Marin,54,57Mertens,Franz,210MillenniumPrizeproblems,163,272Minkowski,Hermann,202,203,205Mittag-Leffler,Gösta,174,175,184,198Möbiustransformations,119Moivre,Abrahamde,66,83Monge,Gaspard,90Moore,Eliakim,207Moore,Robert,207Morcom,Christopher,245Mordell,Louis,232Morin,Bernard,275Motel,Stéphanie-FeliciePoterindu,125Mudaliar,Singaravelu,225Muhammad,Prophet,27,28Murray,Arnold,253Musin-Pushkin,Mikhail,115Muybridge,Eadweard,1Mythbusters(TVshow),12

NapoleonBonaparte,90Nash,John,235NationalScienceFoundation(US),275negativenumbers,32,49,51Nelböck,Johann,235Neumann,Johnvon,251,260Newman,Max,245,248Newton,Isaac:andalchemy,75–6;andcalculus,68–72;fluxions,44,70–1,158;

infiniteseriesforthesineandcosine,38,43;inventionofthecatflap,68;andlawofgravity,67,74–5,197;lawsofmotion,87,186,213–15;lifeandcareer,65–8;onthe‘shouldersofgiants,’2;asWardenoftheRoyalMint,63–5;MethodofFluxionsandInfiniteSeries(1671),71;‘Principia’(PhilosophiaeNaturalisPrincipiaMathematica,1687),5,64,68–9,71,72–4

NicholasI,Tsar,115–16NilakanthaSomayaji,Kelallur,Tantrasamgraha(1501),39,44Nimbursky,Adele(néePorkert),237Noether,Emmy,8;abstractalgebra,216–19;backgroundandeducation,211–

12;symmetryandconservationlaws,210,213–15;topology,219–20Noether,Max,211NotKnot(videodemonstratinghyperbolicspace),275numbertheory,33,57–62,83,204–5;andCarlGauss,102–3;andPierrede

Fermat,54

Oldenburg,Henry,43Ono,Ken,231OscarII,KingofSwedenandNorway,197Ostrogradskii,Mikhail,111,177Ostrowski,Alexander,209Oughtred,William,TheKeyofMathematics,67OutsideIn(videodemonstratingturningasphereinsideout),275

π:andArchimedes,18;andtheChinese,23,24–6;andMadhavaofSangamagrama,38,42–3;transcendentalnumber,204

Papadopoulos-Kerameus,Athanasios,16Parameshvara(Hinduastronomer),39ParisAcademyofSciences,60,81,88,123–4,185,199

ParisCommune,182partitions,theoryof,230–1Pascal,Blaise,54Pellequation,19–20,57Perelman,Grigori,9,195,271,272–4PhiloofMegara,152Piazzi,Giuseppe,106–7Pinker,Steven,TheBlankSlate,281Pisarev,Dmitri,180Plutarch,13,14,15,20Poincaré,Henri:andchaostheory,197–9;differentialequations,195–7;disc

modelofhyperbolicgeometry,119,120;ongroupsofpermutations,132;lifeandcareer,190–1;andthemathematicalthoughtprocess,188–9;topology,7,191–5;ScienceandHypothesis(1901),191;ScienceandMethod(1908),188,191;TheValueofScience(1905),191

Poincaré,Raymond,190PoincaréConjecture,193–5,271Poincaré–Bendixsontheorem,197Poisson’sequation,160Pólya,George,HowtoSolveIt,282Pratchett,Terry,239primenumbers,35,57,161–2,217,230Ptolemy,36,39,40Pythagoras’sTheorem,23,26,41,58

quadraticequations,1,48quantummechanics,8,86,132,208,215,233quantumtheory,191,211quarticequations,50–1quinticequations,127–30

Rajagopal,Cadambur,44Ramanujan,S.JanakiAmmal,233Ramanujan,Srinivasa:backgroundandself-education,223–6;theoryof

partitions,230–1;thetafunctions,232–3;atTrinityCollege,Cambridge,226–9;writestoGodfreyHaroldHardy,221–3,226

Rao,R.Ramachandra,225–6

relativity,theoryofseeGeneralRelativity;SpecialRelativityRényi,Alfréd,267Ricci,Gregorio,157Ricciflow,273Riemann,Bernhard,6–7,93,120,166,213;backgroundandeducation,158–9;

anddifferentialgeometry,155–7RiemannHypothesis,7,161–3,208,242RobertofChester,LiberAlgebraeetAlmucabola,30Robespierre,Maximilien,89RosettaStone,90Ross,Edward,225Rothman,Tony,125Ruffini,Gabriel,122Ruffini,Paolo,GeneralTheoryofEquations,128–9Russell,Bertrand:IntroductiontoMathematicalPhilosophy,237;Principia

Mathematica(1910–13),239–40RussianAcademyofSciences,185Rust,Bernhard,209Ryall,John,151

Saccheri,Giovanni,114,118Sakkas,Ioannis,11–12Saltykov,GeneralIvan,77SanssouciPalace,Prussia,78–9Saturn’srings,182–3Saussure,Horace-Bénédictde,93–4Sautoy,Marcusdu,TheMusicofthePrimes,163Schlick,Moritz,235Schooten,Fransvan,GeometryofRenéDescartes,67Schwartz,Hermann,161Schweikart,Ferdinand,116ScienceMuseum,London,137Serre,Jean-Pierre,61settheory(Mengenlehre),165,167–8Shakespeare,William,174–5Shannon,Claude,154Sheth,Rushikesh,253

Shimura–Taniyama–Weilconjecture,61Shubert,FedorIvanovich,177Sierpiński,Wacław,257,263–4Sierpińskigasket,263sieveofEratosthenes,35SimaQian,RecordsoftheGrandHistorian,23sinesandcosines,41,43,91–2,99Smale,Stephen,268,271–3,275,279Smith,Margarita,179Somayaji,Putumana,KaranaPaddhati,44Somerville,Mary,138SpecialRelativity,120,131,199,213Spencer,Herbert,181Spinoza,Benedict,Ethics,149Stallings,John,271statics,15Steiner,Jakob,158Stern,Moritz,155Stirling,James,68,83StockholmUniversity,184–5Stoicschooloflogic,152Stott,Alicia(néeBoole),145stringtheory,233Stukeley,William,66sudoku,85Sullivan,Dennis,269sundials,ofIsaacNewton,66SuperweaponsoftheAncientWorld(TVdocumentary),13SuryaPrajnapti(Jainmathematicaltext),165SuryaSiddhanta(Hinduastronomytexts),41Swade,Doron,137Sylvester,JamesJoseph,146,203symmetry,mathematicsof,127,131–2,214–15Syracuse,13,20;siegeof(c.214–212BC),12

Taniyama,Yutaka,61Tannery,Jules,126

Tarry,Gaston,85Taurinus,Franz,116Taylor,EdwardIngram,145Taylor,Margaret(néeBoole),145Taylor,Richard,61thetafunctions,232–3ThomasJ.WatsonLaboratories,NewYorkstate,256thoughtprocesses,ofmathematicians,188–90,230,278–82Thurston,William(‘Bill’):backgroundandeducation,269–70;pioneersuseof

computers,274–5;thoughtprocessesofmathematicians,275–6;andthree-dimensionalmanifolds,270–4

Thurston’sGeometrisationConjecture,120,272–4Tischendorf,Constantinvon,16Titius–BodeLaw,107topology,191–5,219–20,270–4torus,192–3,272transfinitenumbers,167trigonometry,39–41TsuCh’ungChih,25Turing,Alan:backgroundandeducation,243–5;atBletchleyPark,248–51;

homosexuality,250–1,253–4;andmathematicalbiology,252–3;post-warwork,251–3;andtheTuringmachine,245–7

Turing,Dermot,253TuringLaw,254Tyrtov,Nikolai,179

Ulam,Stanisław,263

Vallée-Poussin,Charlesdela,100Varahamihira(Indianmathematician),41Varman,Sankara,Sadratnamala,44Vericour,Raymondde,150–1Viète,François,67Vietoris,Leopold,219Virgil,Aeneid,78Voynich,Ethel(néeBoole),145,151Voynich,Wilfrid,145

Wachter,Friedrich,116–17Waerden,Bartelvander,ModernAlgebra,218Walker,Gilbert,226Wallis,John,60,67,69,72waveequation,87–8Weber,Heinrich,202Weber,Wilhelm,109,156–7Wedeniwski,Sebastian,163Weierstrass,Karl,119,158,160,161,166;andSofiaKovalevskaia,181,183,

184Weil,André,232Weyl,Hermann,209Wheatstone,Charles,135Whish,Charles,44Whitehead,AlfredNorth,PrincipiaMathematica(1910–13),239–40Wiener,Norbert,209Wiles,Andrew,4,61,158Williams,Hugh,19Wittgenstein,Ludwig,173women,inmathematics,212,215–16,277–8Wren,Christopher,74Wright,John,68

Zach,BaronFranzvonXaver,107Zahlbericht(NumberReport),204–5Zarnke,Charles,19Zeeman,Christopher,271,279Zermelo,Ernst,239ZhouBiSuanJing(ArithemeticalClassicoftheGnomonandtheCircularPaths

ofHeaven),21–2,23Zipf,George,260–1